computational methods in multiphase flow iv

COMPUTATIONAL METHODS IN

MULTIPHASE FLOW IV

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FOURTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL

METHODS IN MULTIPHASE FLOW

MULTIPHASE FLOW IV

CONFERENCE CHAIRMEN

A.A. MammoliUniversity of New Mexico, USA

C.A. BrebbiaWessex Institute of Technology, UK

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andUniversity of New Mexico, USA

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J. Adilson de CastroS. AliabadiM. GorokhovskiC. Koenig

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W-Q. LuJ. MlsA. NieckeleK. Sefiane

WIT Transactions on Engineering Sciences

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COMPUTATIONAL METHODS IN

MULTIPHASE FLOW IV

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Preface

Multiphase flow remains one of the unsolved problems in fluid mechanics. Thereare many factors which make it difficult to deal with such flows. First, they are verydiverse in nature, and as a consequence the laws governing them are similarlyvaried. In addition, there are generally several length scales at play, and they are inmany cases fully coupled. Although constitutive equations and simulation methods for treating simple‘model’ flows have been developed with partial success, it is still extremely difficultto develop equations which describe realistic multiphase flows at the macroscopicscale, and even when such models are developed, it is difficult to calibrate them byexperiment or simulation. However, as large-scale computation becomes moreprevalent, it is becoming possible to dissect various features of a flow which wouldbe difficult to examine experimentally, increasing our understanding of the importantfeatures that must be treated in a model. For example, it is possible to characterizespatial distributions of components in great detail by simulation, while by experimentone can only obtain overall features. It has become apparent that flow-inducedstructure must be treated by a realistic constitutive model. Close-range interactionsbetween dispersed phase particles are also important in determining large-scaleflow behavior. It remains to be seen whether large-scale simulation will continue to enableconstitutive modelling, as has traditionally been the case, or whether simulation willbecome the only step in the treatment of multiphase flow problems, as computationalpower continues to increase and computational techniques become more efficient. Complementing the interesting mathematical and numerical problems thatmultiphase flows pose is an equally interesting array of real-life problems which arecharacterized by multiphase flow: groundwater transport, river and sea-bedsedimentation, boiling and condensation, sprays and aerosols, combustionprocesses in power generation equipment, and many others. This conference is all-inclusive, representing a very broad spectrum of the manyfeatures of multiphase flows. Papers in the proceedings cover several of themathematical and numerical aspects of multiphase flows, as well as many practicalapplications. Because of the diversity of papers, we are confident that many fruitfulinteractions between researchers will occur as a result of this meeting. As always, we would like to thank the contributors for their excellent work andthe scientific advisory committee for their help with the review and selection process.

The EditorsBologna, Italy2007

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Contents

Section 1: Multiphase flow simulation Aerosol modelling and pressure drop simulation in a sieving electrostatic precipitator M. Telenta, H. Pasic & K. Alam ...........................................................................3 A CFD Lagrangian particle model to analyze the dust dispersion problem in quarries blasts J. T. Alvarez, I. D. Alvarez, S. T. Lougedo & B. G. Hevia....................................9 Modeling of dispersion and ignition processes of finely dispersed particles of aluminum using a solid propellant gas generator A. Rychkov, H. Miloshevich, Yu. Shokin, N. Eisenreich & V. Weiser.................19 A methodology for momentum flux measurements in two-phase blast flows R. G. Ames & M. J. Murphy................................................................................29 Two-phase flow transient simulation of severe slugging in pipeline-risers systems G. Hernández, M. Asuaje, F. Kenyery, A. Tremante, O. Aguillón & A. Vidal ...........................................................................................................39 CFD simulation of gas–solid bubbling fluidized bed: an extensive assessment of drag models N. Mahinpey, F. Vejahati & N. Ellis ...................................................................51 An advanced gas–solid flow engineering model for a fluidized bed reactor system D. Mao & M. Tirtowidjojo ..................................................................................61 CFD simulation of a stratified flow at the inlet of a compact plate heat exchanger M. Ahmad, J. F. Fourmigue, P. Mercier & G. Berthoud....................................75

Numerical computation of a confined sediment–water mixture in uniform flow L. Sarno, R. Martino & M. N. Papa....................................................................87 Experimental validation of multiphase flow models and testing of multiphase flow meters: a critical review of flow loops worldwide O. O. Bello, G. Falcone & C. Teodoriu ..............................................................97 Section 2: Flow in porous media Modelling groundwater flow and pollutant transport in hard-rock fractures J. Mls .................................................................................................................115 Transient groundwater flow in a single fracture M. Polák & J. Mls .............................................................................................125 Petroleum reservoir simulation using EbFVM: the negative transmissibility issue C. R. Maliska, J. Cordazzo & A. F. C. Silva .....................................................133 An integral treatment for heat and mass transfer along a vertical wall by natural convection in a porous media B. B. Singh.........................................................................................................143 Application of integrated finite differences to compute symmetrical upscaled equivalent conductivity tensor C. Vassena & M. Giudici ..................................................................................153 A parallelizable procedure for contaminant diffusion in waste disposal A. S. Francisco & J. A. de Castro .....................................................................163 Permeability, porosity and surface characteristics of filter cakes from water–bentonite suspensions V. C. Kelessidis, C. Tsamantaki, N. Pasadakis, E. Repouskou & E. Hamilaki ...................................................................................................173 Section 3: Interfaces Investigation of slug flow characteristics in inclined pipelines J. N. E. Carneiro & A. O. Nieckele ...................................................................185

Behaviour of an annular flow in the convergent section of a Venturi meter G. Salque, P. Gajan, A. Strzelecki & J. P. Couput ...........................................195 Micro-scale distillation – I: simulation M. Fanelli, R. Arora, A. Glass, R. Litt, D. Qiu, L. Silva, A. L. Tonkovich & D. Weidert...........................................................................205 Viscoelastic drop deformation in simple shear flow investigated by the front tracking method C. Chung, M. A. Hulsen, K. H. Ahn & S. J. Lee................................................215 Section 4: Bubble and drop dynamics Numerical modelling of bubble coalescence and droplet separation Y. Y. Yan & Y. Q. Zu..........................................................................................227 Simulation of radial oscillations of a free and a contrast agent bubble in an ultrasound field A. V. Teterev, N. I. Misychenko, L. V. Rudak & A. A. Doinikov .......................239 Visualization method for volume void fraction measurements in gas–liquid two-phase flows of a water turbine outlet channel R. Klasinc, M. Hočevar, T. Baicar & B. Širok..................................................249 Nonlinear dynamics of lipid-shelled ultrasound microbubble contrast agents A. A. Doinikov & P. A. Dayton .........................................................................261 Lagrangian Monte Carlo simulation of spray-flow interaction T. Belmrabet, R. Russo, M. Mulas & S. Hanchi................................................271 Dynamic hydraulic jumps in oscillating containers P. J. Disimile, J. M. Pyles & N. Toy .................................................................281 Section 5: Suspensions Experimental investigations of sedimentation of flocs in suspensions of biological water treatment plants B. Zajdela, A. Hribernik & M. Hribersek .........................................................293 Modelling molecular gas suspension diffusion and saturation processes in liquid media R. Groll..............................................................................................................303

Analysis of two- and three-particle motion in a Couette cell M. Popova, P. Vorobieff & M. Ingber...............................................................315 Micropolar fluid flow modelling using the boundary element method M. Zadravec, M. Hriberšek & L. Škerget..........................................................325 Numerical modelling of colloidal fluid in a viscous micropump H. El-Sadi & N. Esmail .....................................................................................333 Section 6: Turbulent flow Computational and experimental analyses of a liquid film flowing down a vertical surface S. Sinkunas, J. Gylys & A. Kiela .......................................................................339 The transition of an in-line vortex to slug flow: correlating pressure and reaction force measurements with high-speed video B. J. de Witt & R. J. Hugo.................................................................................349 A DNS approach to stability study about a supersonic mixing layer flow F. Guan, Q. Wang, N. Zhu, Z. Li & Q. Shen .....................................................359 Hydrodynamic transmission operating with two-phase flow M. Bărglăzan, C. Velescu, T. Miloş, A. Manea, E. Dobândă & C. Stroiţă .......................................................................................................369 A note on crossing-trajectory effects in gas-particle turbulent flows B. Oesterlé.........................................................................................................379 Large eddy simulation and the filtered equation of a contaminant F. Gallerano, L. Melilla & G. Cannata ............................................................389 Author Index ...................................................................................................399

Section 1 Multiphase flow simulation

Aerosol modelling and pressure drop simulation in a sieving electrostatic precipitator

M. Telenta, H. Pasic & K. Alam Ohio University, Mechanical Engineering Department, Athens, OH, USA

Abstract

This paper first describes so-called sieving electrostatic precipitator suitable for efficient and cost-effective cleaning of polluted gases of both large and ultra-fine particulates in a very broad temperature range. In SEP the particulate-laden gas is passed through a set of closely packed and charged fine-wire screens. In the last three years, a large number of fly ash collection-efficiency experiments have been conducted—first, on a bench-size unit both at room and elevated temperatures and, in a laboratory pilot-scale setting. Most recently, a consortium led by American Electric Power (AEP), Ohio University, Ohio Coal Development Office and PECO have built and started tests on a pilot slip-stream unit in AEP’s plant in Conesville, Ohio. However, deeper understanding of SEP calls for numerical treatment of particulates charging, their agglomeration, and various particulate-capturing mechanisms (field and diffusion charging, interception by screen wires etc.) simultaneously taking place in laminar flow conditions. The paper describes our attempt to model this process. Keywords: sieving electrostatic precipitator, modelling, particle charging, coagulation, particulate capture.

1 Introduction

Sieving electrostatic precipitator (SEP), developed at Ohio University, is the next generation of electrostatic precipitators. It could offer better particle collection efficiency than conventional precipitators. Also, the step forward is its small size, lower operational and overall cost, and enhanced ability to collect submicron-sized particles. The main difference between SEP and conventional electrostatic precipitators is in the collecting units: conventional precipitators

© 2007 WIT PressWIT Transactions on Engineering Sciences, Vol 56, www.witpress.com, ISSN 1743-3533 (on-line)

Computational Methods in Multiphase Flow IV 3

doi:10.2495/MPF070011

have plates parallel to the air flow, while SEP utilizes screens that are set perpendicular to the gas flow and therefore fly ash is being sieved—hence the term “sieving”. This difference results in new particle-capturing mechanisms which differ from those in conventional precipitators. In SEPs screens are under high DC voltage of about 40-60 kV. The SEP typically operates at gas velocities about 1 m/s, particulate concentration 3-10 g/m3, DC current of 40-60 kV. Screen openings are 500 microns or less and the screen spacing is about 5 mm. For more details see Pasic et al. [1]. This paper attempts to recognize the complexity of the particle behaviour in the collecting equipment—in particular the SEP. Furthermore, it suggests necessary steps to resolve some of these problems utilizing numerical methods and existing software packages or combining those specialized packages into a single one capable of handling this multidisciplinary modelling.

2 Modelling/results

The SEP is a new technology. It is not completely tested and therefore fully optimized. Many parameters have yet to be tested and validated. This could be done by elaborate and expensive laboratory parametric testing. Hopefully, some or eventually a large number of these research steps could be replaced or at least supplemented with numerical treatments. This could greatly reduce research time and the overall cost. Computational fluid dynamic software FLUENT is one such example, offering opportunity to make SEP research more rapid. In addition, with various software plug-ins it could possibly depict most of the processes which particles undergo in the SEP. In SEP, particles are captured with almost all possible mechanisms. The dominant ones are due to field and diffusion charging, coagulation (of small into larger particles that are easier to capture), and capture-by-obstacles, such as by impaction and interception. Nowadays, most of these mechanisms are quite well described in the literature and are (or could be) easily software-implemented. There exist numerous numerical approaches and the corresponding software for numerical simulations of some of those specific aspects, such as particle interaction with other particles or interacting with obstacles to which they could possibly attach. However, most of these simulations are restricted to applications in a limited space domain or to small particle numbers, etc.—issues primarily related to a limited computer capacity. Indeed, as computer technology advances, new opportunities emerge for better implementation of those already developed numerical methods. In what follows, we will illustrate just some aspects of that modelling through simulations of the gas pressure drop and screen clogging. For other modelling results, such as collision frequencies of charged particles and their agglomeration, and more detailed simulation see Telenta [2].

2.1 Pressure drop

Pressure drop is one of the most important design parameters related to efficacy and efficiency of any particle collection device. In SEP, particulate-laden gas is


4 Computational Methods in Multiphase Flow IV

forced through tens of screens. Determining the pressure drop requires the gas flow simulation as the first step. Many CFD software packages are available and could be used to accomplish this step. One such package—FLUENT was used in this work (for more details see Telenta [2]). Also, in Telenta [2] the user defined functions (UDFs) have been used, as well, since FLUENT, as it is, is limited in certain aspects of pressure drop simulation. UDFs are additional features that are easily implemented in FLUENT. In the case of SEP, UDF is utilized in conjunction with so-called porous media to properly represent the pressure drop created by sets of screens, since velocity and, therefore, pressure profiles in front of the screens are not uniformed (Figs 1 and 2, Telenta [2]).

Figure 1: Velocity profile with streamlines in front of the first screen [2].

Figure 2: Static pressure profile [2].

This gas flow analysis takes care of the fluid flow part and gives a solid basis for the future upgrades concerning the particle collection. However, FLUENT offers only limited options regarding the particle simulation, Triesch et al. [6], and needs to be supplemented with additional UDFs in order to be applied to SEP simulations. Without UDFs, it cannot be used for modelling particulates charging, coagulation, and obstacle collection.



2.2 Particle charging

Particle charging is an important issue in SEP. Particles are charged in a DC high-voltage electric field which creates a strong corona field near tips of discharge electrodes. Thus, after acquiring the charges, particles stick to each other, due to agglomeration, or to screen wires. These phenomena, such as Coulomb equations for interactive forces between particles, for example, are well known and documented in the literature and adequate UDFs can be developed and implemented in FLUENT. One such example is DEM Solutions’ [3] software jointly developed with NASA.

2.3 Coagulation

Coagulation process could be implemented in and modelled by FLUENT. Namely, once particles’ position are tracked by FLUENT, and when two or more particles get close enough, their coagulation can be modelled by a UDF which is based on well established theory; for more details and results see Telenta [2].

2.4 Collection by obstacles and screens clogging

These processes can be dealt with in a manner similar to that used in coagulation studies. Particle position, which is calculated by FLUENT, can be compared using UDF in reference with the screen wire position, and if the obstacle is in the particle way, the particle is captured. After a certain amount of particles are captured and piled, clogging of the screen can occur. Some work has already been done in software different than FLUENT, Figs. 3-5, Tafreshi at al. [4]. This is done on a micro-level analyzing a small number of particles and obstacles.

Figure 3: Flow path lines between fibers [4].



Figure 4: Particle deposition on a cylindrical obstacle/fiber [4].

Figure 5: Progression of particle deposition on a cylindrical obstacle/fiber [4].

Also, FLUENT can be combined with EDEM software to do this kind of simulation, Fig. 6 [3].

Figure 6: Filter screen designed to catch large particles; stream view [3].

3 Conclusions

This paper attempts to recognize the complexity of the particulate behaviour and its capture in sieving electrostatic precipitator. A deeper understanding of this process calls for numerical treatment of particulates charging, agglomeration,



and various particulate-capturing mechanisms, such as field and diffusion charging, interception by screen wires, etc., all simultaneously taking place in laminar flow conditions. The paper describes our attempt to model this process. Furthermore, we have made an attempt to resolve some of these issues by utilizing numerical methods and existing software packages or combining those specialized packages into a single one capable of handling this multidisciplinary modelling, Telenta [2].

References

[1] Pasic, H. et al. “Current Status of Development of Sieving Electrostatic Precipitator”, 23-rd Annual International Coal Conference, Pittsburgh, PA, USA, 2006.

[2] Telenta, M. “Aerosol Calculation and Pressure Drop Simulation for Sieving Electrostatic Precipitators”, MS Thesis, Ohio University, 2007.

[3] www.dem-solutions.com, 2007. [4] Tafreshi, H. V., Maze, B., Pourdeyhimi, B., “Filtration by Micro and

Nano-fiber Filters: Simulation and Experiments”, North Carolina State University, Raleigh, NC, 2004.

[5] Fluent Manual, 2007. [6] Triesch, O., Bohnet, M., “Measurements and CFD Prediction of Velocity

and Concentration Profiles in a Decelerated Gas-Solids Flow,” Powder Technology 115, 101-113, 2001.



A CFD Lagrangian particle model to analyze the dust dispersion problem in quarries blasts

J. T. Alvarez, I. D. Alvarez, S. T. Lougedo & B. G. Hevia GIMOC, Mining Engineering and Civil Works Research Group, Oviedo School of Mines, University of Oviedo, Spain

Abstract

In the framework of the Research Project CTM2005-00187/TECNO, “Prediction models and prevention systems in the particle atmospheric contamination in an industrial environment” of the Spanish National R+D Plan of the Ministry of Education and Science, 2004-2007 period, a CFD model to simulate the dispersion of the dust generated in blasts located in limestone quarries has been developed. This is a complex phenomenon that is being studied through the use of several digital video recordings of blasts and the dust concentration field data measured by light scattering instruments, as well as the subsequent simulation of the dispersion of the dust clouds using Multiphase Computational Fluid Dynamics. After several tests with multiphase methods, both Eulerian and Lagrangian, finally the latter was used due to its ease in implementing calculations of discrete phases composed by multisized particles with affordable memory requirements. Keywords: Bench Blasting, dust dispersion modelling, CFD, Discrete Lagrangian methods.

1 Introduction

This paper explains the simulations done to model the dust dispersion generated in a production blast located in a medium sized limestone quarry, ranging around 1Mt/year, exploited through drill and blast method. Limestone is a key natural resource, base of multiple materials used in Civil Works and main raw material of the cement. In order to achieve this production level there are needed at least two blasts per week, blasts that generate several possible environmental risks as the aerial



doi:10.2495/MPF070021

wave, the ground vibrations and the dust thrown to the atmosphere and dispersed by the wind. Our research group is developing a project named CTM2005-00187/TECNO, “Prediction models and prevention systems in the particle atmospheric contamination in an industrial environment” granted by funds of the Spanish National R+D Plan of the Ministry of Education and Science, 2004-2007 period. Within the research objectives appears one relating to the determination of the amount of dust produced in a blast and its immersion in the atmosphere surrounding the quarry area by means of two main tools: measurement campaigns of the dust concentration using “Light scattering” dust sensors and computerized simulations through commercial CFD software. These are done through the combined use of Solidworks to generate the 3D models, ICEM CFD to adequately mesh the domain and Ansys CFX 10.0 in case of the calculation and analysis of the results. There are several numerical methods that can be used to study the particulated material dispersion. One good summary of them was done by Reed [1] and the authors have already used classical dispersion models with acceptable results [2], but much more sophisticated tools are needed in order to study in more detail the dispersion effects and the possible mitigation methods. Among the possible methods it was decided to use the “Particle Tracking” one, a Lagrangian method implemented in CFX 10.0 with undoubted advantages in case of simulations of multisized dispersed phases.

2 Blast characteristics

The start conditions basic or determinate a dust dispersion problem are two: the emission conditions, which in this case depend on the parameters that define the blast, and the atmospheric conditions.

Figure 1: Quarry bench and blast design.

Table 1 shown below summarizes the main blast parameters. This is a classical bench blast, see figure 1, with only one row of shots that are loaded with two explosive types: one high power explosive placed at the bottom based on nitro-glycerine and another one of medium power based on ammonium nitrate that creates the so called column load. The upper part of the shot is filled with a compacting material, usually the product resulting of the drill, which is called clay stopper.



Regarding the atmospheric conditions during the blast there were registered 24ºC of ambient temperature, 1015 milibars of ambient pressure, sun and null rainfall. Wind velocity was measured using a Met-One wind sensors and was appreciably constant at 2.5 m/s. The wind bearing was at right angle to the bench face, which was completely dry. Relative humidity was measured at 55%. These meteorological parameters will define the wind profile used in the simulation.

Table 1: Blast characteristics.

Bench height 18 m Bottom charge depth 5 m

Burden 4.5 m Column charge depth 9 m

Spacing 5 m Depth detonator EZDET

Drill diameter 120 mm Head detonator MS-16 (450 ms)

Shot slope 15º Initiation type Electric

Stemming 2,83 m Shot number 12

Drill depth 18 m Total explosives load 1586 kg

Bottom charge 50 kg Detonating cord 19m/shot

Column charge 82 kg

Another important parameter that has to be defined is the particle size distribution of the material that forms the dust cloud. As is shown by Almeida [3] et al and Jones et al [4] the particulated material thrown to the air by a blast has two main sources. First, rock pulverized by the several phenomena that take place in the blast (shock wave, high pressure gases or dynamic breaking mechanisms, etc.) and second the dusty products of the explosive chemical reaction. Both Almeida and Jones estimate the size distribution of the dust clouds in ranges from submicron sizes up to 50 microns. Particles over this size are also produced but have not been considered in the simulation as are quickly settled into ground by its own weight. The several blasts studied were registered in digital video. There can be clearly noticed in those recordings the dust cloud generation and the subsequent movement and dispersion by the wind. Figure 2 shows a couple of still images taken from the tapes. In addition to the digital video there was also used continuous measurement instrument to record the dust concentration values. The technology “light scattering” was employed, two equipment named E-Sampler manufactured by Met One Instruments Inc., both equipped with electronics capable of transferring the concentration meteorological data to a computer. These dust sensors were located at several distances form the bench one at 120 m from the blast and the other one at 200 m, distances that were confirmed as enough in order to Project the equipment form the flyrocks. There were obtained concentration peaks between 500 and 900 µg/m3 in case of the



120 m sensor and much lower values, between 150 and 400 µg/m3 in case of the equipment installed at 200 m. Therefore it can be inferred that the dust cloud is starting its dispersion as it moves away from its source.

Figure 2: Blast and dust cloud.

3 CFD simulation

3.1 Approach

The numerical modelling of the dispersion of contaminant species in the atmosphere is with no doubt reaching its climax and several different techniques are available in the software market. The particle tracking method is a multiphase modelling tool implemented in Ansys CFX 10.0, where a dispersed phase, the particles in this case, follow in a Lagrangian way the pressure and velocity field calculated in the continuum phase, which is the air. The particle trajectories are not evaluated for each and every one of them, but only for a limited number that will be representing the millions that compose the dust cloud. Each representative particle path will be assigned a certain amount of mass, and the movement of this mass through the continuum domain will be studied in a time-dependent way, as the emission will be considered not constant. Ansys CFX can represent dispersed phases as solid particles, liquid particles or even bubbles. In this case it will be used a dispersed phase made from 7 families of particles sizes, with mass distributions obtained from Almeida and Jones. The method is used following the classical procedure used in the CFD simulations: geometry definition, meshing of the domain, definition of the problem physics, solution and finally post processing of the results.

3.2 Geometry and meshing

Starting from the information taken form the field and the surveying of the area it was developed a 3D model of the bench and its surrounding air using Solidworks. This parametric software woks through the use of Boolean operations among geometries in a similar way as a sculptor creates its works.



Dimensions are approximately 400x500x250 m. In order to simulate the blasted material it will be created a semicone in the bench area with a 56º slope, as it can be seen below in figure 3. In the same figure is shown the final 3D model created, with different colours for each geometry feature. Please notice the semicircle shape present over the bench, which will be defined, as the other surrounding volumes, as air but will be used only as a meshing tool to define different density in the mesh structure. Once the geometry is created it is exported to the meshing tool, ICEM CFD. This software allows the creation of structured or unstructured grids composed by tetrahedrons, prisms or hexahedrons or a mixture of any of these elements. It will be defined a much more fine grid in the areas where the dust is present, not only in the blast area but also in the areas occupied by the dust cloud, as there is the place where it can occur more steep gradients in both the continuum and disperse phase defining variables. Another parameter that has to be taken into account is the quality of the mesh, quality understood as the proximity of the tetrahedral elements that compound the mesh to the perfect tetrahedron. The use of perfect tetrahedral allows an homogeneous placing of the calculation nodes over the domains, which benefits the easiness of convergence of the CFD calculation and frequently can be a key issue that can define even the existence of a solution [6]. Figure 4 shows the final mesh used, after two previous attempts that were rejected due to the calculation differences shown. The final mesh includes a global value of 706.324 elements with a final RMS quality of 0.38. A much finer grid will make almost unfeasible the calculation in the computer used, Pentium 4 2.4 GHz and 1GB RAM, making essential the use of calculation clusters in a multiprocessing scheme. Calculation duration ranged from 6 to 8 hours depending on the simulation time.

Figure 3: Bench after the blast and 3D model.

3.3 Problem physics and resolution

Once the domain is created and is adequately meshed the next step is to define the boundary conditions. These conditions are necessary to define how the different geometrical areas that limit the simulation domain are being affected by the variables affecting both the continuum and dispersed phase. Each surface has to be defined as an “Inlet”, “Opening”, “Outlet” or “Wall”.



Figure 4: Model meshing.

There will be considered two Inlet conditions, the surfaces closer to the spectator in the figure 4, where the air enters in a subsonic regime. The blowing air will be defined as a velocity profile following a classical logarithmic equation, as can be seen in documents from the U.S. Environmental Pollution Agency [7] as well as adequately oriented following the wind bearing measured in the field. The remaining vertical surfaces and the ceiling will be defined as “Openings”, areas where there will be free flow both incoming or out coming. The remaining surfaces will be defined as “Walls” with a roughness factor of 0.5 cm. The software automatically applies wall functions to define adequately the air flow near these surfaces. As has been pointed out the simulation will be done using a dispersed phase, the dust cloud, moving through a continuum, the air. A key parameter that has to be defined is the turbulence model used to calculate the continuum variables field. Medium complexity turbulence models were selected in order to obtain affordable resolution times in single processor machines (Temmerman et al [8] Silvester et al [9]). Similar studies done by the authors in similar applications (see [10, 11]) guided to the final selection of a roughness k-epsilon model using a logarithmic wind profile. Simulations were methodically repeated using several dust mass flow values until the concentration curves calculated were almost equal to the measured ones. These curves were compared simultaneously at both measurement points located at 120 and 200 m from the blast. The dust injection is achieved through the combination of several surface sources both planar and spherical in the blast surrounding area. The dust is injected in a pulsed shape with a duration ranging 25 to 35 s.



3.4 Results and interpretation

The analysis of the results of a CFD simulation is generally speaking easy and intuitive when using the modern post processing tools. The valuable time spent on the 3D-modelling, meshing and the establishing of the physics of the simulation is worthwhile once it is realized the wide possibilities of visualization and interpretation of calculation results. Among the possible studies that can be done there can be highlighted the velocity vectors fields in planes parallel to the benches, the contours of the concentration variables in each time that will show the position of the dust cloud, contours of the maximum concentration values independently of time, etc. Due to space limitations, in this paper there will be shown just the concentration isosurfaces in several cases of simulation times and concentrations, isosurfaces that are considered representative of both the simulation results and the potentiality of the methodology used. The CFX-Solver stores in the output files the values of the variables that define the behaviour of the continuum phase in each and every one of the calculated points defined by the volume finite technique. It also assigns to each control node of the mesh a value of the volume fraction of the dispersed phase in the continuum. This value will serve as base to develop that calculation of the dust concentration expressed, as is usual in these environmental studies in ranges of µg/m3. One of the most useful features of the postprocessor is the isosurface generation of a certain concentration value. This is, there can be shown the surface that fit all the points in the simulation domain where the concentration value is equal to a selected value. This will make appear in the display the shape and position of the dust cloud. Figure 5 shows several isosurfaces of concentrations ranging 100 µg/m3 to 750 µg/m3. Two columns of figures appear, the left one representing simulation time at 55 s and the right one at 75 s with concentration values increasing towards the bottom of the figure. Comparing the two figures appearing in the row signed as 0.1 mg/m3 we can observe how the isosurface of 100 µg/m3 has been displaced towards the right side of the figure as the dust cloud is moved by the air. The second row of figures, clouds with concentration levels at 200 µg/m3 show a similar behaviour, although the size of the isosurface is clearly smaller, as it was expected, as it has to be included within the surface relative to 100 µg/m3. The third row of figures, 500 µg/m3, shows a much more shrink cloud and the dust concentrated in the low part of the bench. It is also useful to point out how the dust clouds are smaller as the time gets higher, which is showing the effects of the dispersion of the dust. The fourth and last row shows high concentration values, 750 µg/m3, that are again only appearing in the figures of the left side, not being almost represented in the right ones. This is, as time evolves, higher concentrations disappear from the domain.



0.1 mg/m3

0.2 mg/m3

0.5 mg/m3

0.75 mg/m3

Figure 5: Dust cloud evolution at Time=55 s (left column) and 75 s (right one).



4 Conclusions

The dispersion of the particles generated in blasts located in quarries can be simulated using CFD tools as Ansys CFX 10.0. The use of digital video, detailed topography and continuous measurement of dust concentration allow the tuning and verification of the developed models. The use of transitory particle tracking models allows the detail study of the dust cloud movement and will allow future studies conducted to mitigate the possible environmental impact generated.

Acknowledgements

We want to acknowledge the help and advices from the Ansys CFX Technical Support Team in the development of these studies.

References

[1] Reed, W.R. Significant Dust Dispersion Models for Mining Operations. Information Circular 9478. National Institute for Occupational Safety and Health (NIOSH). September 2005

[2] J. Toraño, R. Rodriguez, I. Diego and A. Pelegry, “Contamination by particulated material in blasts: analysis, application and adaptation of the existent calculation formulas and software”. Environmental Health Risk III, pp. 209-219, (2004).

[3] Almeida, S.M. Eston, S.M. and De Assunçao, J.V. “Characterization of Suspended Particulate Material in Mining Areas in Sao Paulo, Brazil.” I.T. International Journal of surface Mining, Reclamation and Environment 2002, Vol. 16, no. 3, pp. 171-179

[4] Jones, T., Morgan, A. and Richards, R. “Primary blasting in a limestone quarry: physicochemical characterization of the dust clouds”. Mineralogical Magazine, April 2003, Vol 67(2), pp. 153-162

[5] ANSYS CFX-Solver, Release 10.0: Theory; Particle Transport Theory: Lagrangian Tracking Implementation; page 173.

[6] ANSYS CFX-Solver, Release 10.0: ANSYS CFX-Solver, Release 10.0: Modelling 7.

[7] Environmental Pollution Agency. AP-42, 13.2.5.1, Miscellaneous Sources. Pp2. 1998.

[8] Temmerman L., Wang C. and Leschziner M.A. (2004) A Comparative Study Of Separation From A Three-Dimensional Hill Using Large Eddy Simulation And Second-Moment-Closure Rans Modelling. European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS

[9] S.A. Silvester, I.S. Lowndes and S.W. Kingman, “The ventilation of an underground crushing plant”, Mining Technology (Trans. Inst. Min. Metall. A), Vol. 113, pp. 201-214 (2004)



[10] Surface velocity contour analysis in the airborne dust generation due to open storage piles. Toraño J., Rodríguez R. and Diego I. European Conference on Computational Fluid Dynamics. ECCOMAS CFD 2006, Delft The Netherland, 2006

[11] Toraño J. et al., Influence of the pile shape on wind erosion CFD Emission Simulation, Appl. Math. Modell. (2006), doi:10.1016/j.apm.2006.10.012



Modeling of dispersion and ignition processes of finely dispersed particles of aluminum using a solid propellant gas generator

A. Rychkov1, H. Miloshevich2, Yu. Shokin2, N. Eisenreich3 & V. Weiser3 1Institute of Computational Technologies SB RAS, Russia 2Prishtin University, Serbia 3Fraunhofer Institute of Chemical Technologies, Germany

Abstract

Using numerical modeling, we studied the formation and propagation of a cloud of finely dispersed aluminum particles generated by a special unit under the action of high-temperature combustion products from solid propellant gas generator, as well as the ignition conditions of these particles. We used the Favre-averaged system of Navier-Stokes equations closed by the q – ω turbulence model to simulate the formation and motion of the cloud of finely dispersed particles. The motion of the polydisperse second phase was described within a stochastic approach that takes into account the effect of the turbulent character of the flow field of the carrier gas on the motion of particles. The finite volume method using the second-order upwind LU difference scheme with TVD-properties is applied for numerically solving this system of equations. The results obtained are in a qualitative agreement with experiments carried out at the Fraunhofer Institute for Chemical Technology (Pfinztal, Germany). Keywords: mathematical modeling, two-phase flow, processes of ignition and burning, solid propellant gas generators.

1 Introduction

The scheme of a unit for the fast dispersion of finely dispersed particles is shown in fig. la. It is a cylinder whose central part (a gas generator) is filled with solid monopropellant spherical-form granules with a diameter of a few



doi:10.2495/MPF070031

millimeters. The remaining peripheral part of the cylinder contains bulk pulverized finely dispersed aluminum powder that was simulated by a polydisperse medium consisting of spherical particles. There is a metallic net between the gas generator and the bulk material, which does not produce appreciable hydrodynamic resistance to the motion of the gas generator combustion products. The upper and the bottom lids of the cylinder were assumed to be impermeable and fixed so that the particles in the cylinder after the ignition of propellant granules under the effect of gaseous high-temperature high-pressure combustion products moved in a radial direction only. The compaction of particles by the nonstationary action of the combustion products pressure is a rather complex process and an independent problem [1]. Therefore, at the given stage of simulation, the bulk medium between the cylinder heads was supposed to move in the regime of ‘plug’ pneumatic transport with the maximum permissible level of porosity for spherical particles equal to 0.42 [1]. After the left boundary of the ‘plug’ left the cylinder, the process of its destruction and the formation of a cloud of particles under the effect of outflowing high-temperature combustion products started, as well as their ignition and combustion.

Figure 1: The scheme of the unit.

2 Description of model and basic equations

The pressure of the combustion products of propellant granules at the moment the bulk material plug leaves the cylinder is 10—15 MPa, the efflux into the ambient medium is supersonic and the turbulent character of the flow is to be taken into account. Therefore, in this paper, we used the Favre-averaged system of Navier-Stokes equations closed by the ω−q turbulence model to simulate the formation and motion of the cloud of finely dispersed particles [2]. The motion of the polydisperse second phase was described within a stochastic approach [3] that takes into account the effect of the turbulent character of the flow field of the carrying gas on the motion of particles. In describing the mathematical model of the processes studied, we adopted the following assumptions.

• The flow is turbulent, two-phase, axisymmetric, and nonstationary. The outflow occurs into the static atmosphere of the standard composition.



The gas generator combustion products represent a nonreacting mixture of an inert component and an oxidizer whose oxidation potential was simulated by some weight fraction of oxygen β .

• The second phase consists of polydisperse aluminium spherical particles and comprises N fractions, inside each of them the size of the particles is identical. The particle collisions at the given stage of simulation are disregarded.

• The temperature distribution over a particle volume is assumed to be homogeneous.

• The gas output from solid propellant granules was simulated by the source terms in the mass and energy conservation equations.

Figure 2: The particle combustion model.

In order to describe the combustion of aluminum particles, we use the model of a contractile metallic core of diameter kid , and an oxide shell of diameter sid , . The scheme of this model is shown in fig. 2. The combustion of the i-th particle is supposed to be described by the one-stage overall reaction 322 2/14/3 OAlOAl =+ , its linear velocity is determined by the formula in [4].

8.06

2.01.052

3,

)10(00735.0

)10(10)(

⋅⋅

⋅⋅−=

−−

io

Oki

d

TpYdtdd

(1)

where 0id is the initial particle diameter; 2OY is the mass fraction of the oxidizer; Tp, are the pressure and the temperature. The decrease in the mass of a particle nucleus in combustion is described by the dependence

dtdd

ddt

dm kikiAl

Ali )(

2,2

,

)( πρ= (2)

The oxide shell mass, on the one hand, increases due to the formation of the oxide in the combustion process and, on the other hand, may decrease due to the escape of 32OAl from the shell surface in the form of submicron particles. Therefore, the change in the mass of the oxide shell was written as



)431( 2

)()32(

α−+−=Al

OAl

iOAl

i

MM

dtdm

dtdm (3)

where α is an empirical coefficient taking into account the process of the oxide escape from the shell surface. The submicron particles are assumed to be in a local equilibrium with the carrying gas. Therefore, their mass is added to the inert component of the gas generator combustion products and their energy to the total energy of the carrying gas. Further, an aluminum particle is supposed to start burning when the particle temperature reaches a certain value, i.e., the ignition temperature igT .

The system of equations describing this flow has the form:

∑=

><−+=∂∂

+∂∂

+∂∂ N

ii

Ali

Al

Ogen n

dtdm

MM

Mvrrr

uxt 1

2 )43()(1)( αρρρ (4)

2

2 2 2 2

2

, ,

1

1( ) ( )

34

OO x O O r O

AlNO i

i geniAl

YuY q r vY q

t x r rM dm n MM dt

ρρ ρ

β=

∂ ∂ ∂+ + + + =

∂ ∂ ∂

< > + ⋅∑ (5)

2

2 2 2 2, ,1( ) ( ) 0N

N x N N r N

YuY q r vY q

t x r rρ

ρ ρ∂ ∂ ∂

+ + + + =∂ ∂ ∂

(6)

, ,

1

1( ) ( )

(1 )

MM x M M r M

AlNi

i geni

Y uY q r vY qt x r r

dm n Mdt

ρ ρ ρ

α β=

∂ ∂ ∂+ + + + =

∂ ∂ ∂

− < > + −∑ (7)

2

,1

1( ) ( )N

ixx xr i p i

i

u duu p r uv n mt x r r dt

ρ ρ τ ρ τ=

∂ ∂ ∂+ + + + + = − < >

∂ ∂ ∂ ∑ (8)

2

,1

1( ) ( )N

irx rr i p i

i

v dvvu r v p p n mt x r r dt

ρ ρ τ ρ τ=

∂ ∂ ∂+ + + + + = − < >

∂ ∂ ∂ ∑ (9)

( )

, ,1

,

1( ( ) ) ( ( )

) [

( )

]

xx xr x xr

Ni i

rr r i p i p ii

Ali

p i i i i

Ali

p gen gen

E u E p u v r v E p ut x r r

du dvv n u m v mdt dtdmd Nu T T C T T

dtdm Q Q M

dt

ρ ρ τ τ ϕ ρ τ

τ ϕ

π λ α

=

∂ ∂ ∂+ + + + + + + + +

∂ ∂ ∂

+ = − < > + < > +

< ⋅ − > − < − > −

< > +

∑ (10)



, .1( ) ( )x q r q q

q uq r uq St x r r

ρ ρ τ ρ τ∂ ∂ ∂+ + + + =

∂ ∂ ∂ (11)

, .1( ) ( )x ru r u S

t x r rω ω ωρω ρ ω τ ρ ω τ∂ ∂ ∂

+ + + + =∂ ∂ ∂

(12)

)(2

2

2

20

M

M

N

N

O

O

MY

MY

MY

TRp ++= ρ (13)

The equations of the motion of the i-th particle along its path are written as

16

2.01.052

32, )10(

)10(102 −

−−

⋅⋅

⋅⋅⋅−= n

io

OikiAl

Ali

dA

TPYNd

dtdm πρ (14)

dtdm

MM

dtdm Al

i

Al

OAlOAl

i )2

( 3232

α−⋅

−= (15)

( )isipi

ipDii uuud

Cdt

du−′+= 2

,,

.Re43

ρ

µ (16)

( )isipi

ipDii vvvd

Cdtdv

−′+= 2,,

.Re43

ρ

µ (17)

( )isiipi

i TTdNu

CdtdT

−⋅

= 2,,

6ρλ (18)

ii u

dtdx

= (19)

ii v

dtdy

= (20)

where kkq εω == , are ‘turbulent’ variables related to the turbulent kinetic energy k and its dissipation rate ε ; 2)( 22 vuTCE v ++= is the total specific energy; ρ/pEH += is the specific enthalpy; µ and tµ are the molecular and turbulent viscosity, respectively; p, T are the static pressure and temperature; R is the gas constant; 2222 ,,,, NOMNO MMYYY and MM are the mass fractions and molecular weights of oxygen, nitrogen, and the inert component, respectively. The values genM and genQ are the mass and energy sources simulating the inflow of high-temperature combustion products from the gas generator.

qDSCCS qq ρωωωµ )1)3/(2/( 2

1 −−= , 2

232

1 ])//([ ρωωω ωωµωω CDCSCCS −−=



The constants in the description of the turbulence model have the following values: 55.01 =ωC , 833.02 =ωC , 666.03 =ωC , 09.0=µC , 5.01 =qC ,

ωρµ µ /2qCt = ,

∂∂

+∂∂

==xv

rru

rexrrx1µττ , ,

xT

ex ∂∂

−= λφ

,rT

er ∂∂

−= λφ ,te µµµ += te λλλ += , tptt C Prµλ = ,rrv

rxuD

∂∂

+∂∂

=1 ,

2,,,, 32)( kkjiijji uuuuS −+= , ,)(, x

YSc

Dq k

t

tkkx ∂

∂+−=

µρ

,)(, rY

ScDq k

t

tkkr ∂

∂+−=

µρ k = MNO ,, 22

and tPr is the turbulent Prandtl number. The terms in the angular brackets indicate averaging over the volume of the cell in a difference grid; in is the concentration of particles of the i-th fraction in the cell at the given instant

,10Re,

10Re,44.0

6Re

1Re

243

3

3/2

≤

>

+

= pi

pi

pi

piDi if

if

Cµ

ρ isipi

VVd −=

,Re

where vu ′′, are the random vector components of the disturbed gas velocity

),( vuv ′′=′ ; ||/)(32 1 VVNerfckv r−=′ , V is the averaged velocity vector;

rN is a random number from the range [-1, +1]; )(1−erfc is the inverse error function; iC is the specific heat of the i-th particle, ix and iy are its

coordinates. The diameter of the aluminium oxide shell sid , of each i-th particle was determined from the equality

])([6 32

3,

3,

3,

32OAlkisiAlki

OAli

Ali dddmm ρρπ

−+⋅=+ (21)

The mean particle density is

3,

32

,)(6

si

OAli

Ali

pi dmm

⋅

+=

πρ (22)

The domain of the solution to the system of eqns (4) - (20) and its size in meters are shown in fig. 1 b, where the OY-axis is horizontally directed, the OX-axis is



vertically directed. The domain was bounded from below either by an underlying surface (pulverization near the Earth surface) or was not bounded at all (pulverization in an infinite space). The boundary conditions for the carrying gas were given as follows. The flow symmetry condition was given along the OX-axis, the impermeability condition on the surfaces of the cylinder lids and on the underlying surface, and the nonreflective boundary condition on the other boundaries. The inelastic reflection condition was given for the particles on the underlying surface

+− −+−−= npinpi VV ,32

, )49.056.176.1993.0( ΘΘΘ , +− = ττ ,, 75

pipi VV

where −+−+

ττ ,,,, ,,, pipinpinpi VVVV are the normal and tangential velocities of the i-th particle before (+) its drop and after ( -) its reflection; Θ is the angle of incidence. The initial condition for the gas and the particles was a stationary state. The mass increase genM of the gas generator combustion products in eqns (5), (7), (10) was determined from the solution of the equation

0ggbggr

gen NrSdt

dM⋅−= ρ (23)

where Sgr ,ρ and bgr are the density of a propellant granule, its current

surface and the combustion rate, respectively; 0ggN is the number of

granules in the unit volume of a solid propellant charge, which remains constant in combustion and is determined from the initial conditions.

3 Numerical method

The finite volume method using the second-order upwind LU difference scheme with TVD-properties [8] is applied for solving numerically the system (4) – (13) for carrier gas. The scheme is close to the scheme from [5]. As well know the stiffness of this system is main difficult by numerical solution of it at low Max numbers. To circumvent this problem we used the preconditioned matrix much as in [5]. We calculated the particle motion by the A-stable difference scheme of the second-order of accuracy [6]. The iterative process is organized to take into account effect of the second phase on movement of carrier gas.

4 Some computational results

The calculations were carried out for the following initial conditions: the mass of finely dispersed aluminum particles is 2 kg; the solid propellant charge of the gas generator is 0.2 kg; the combustion rate of granules 6.0

0 )/(5 pprbg = [mm/s];



the heat output of the propellant genQ = 1780 [KJ/kg]. The number of the fractions of particles was five, the ignition temperature of an aluminium particle

igT = 1300 K. Fig. 3 shows the time variations in the total heat generation of burning aluminium particles Q[J].

Figure 3: The heat generation of burning particles.

Figure 4: Flow field isotherms for particles with

43 0.8 µm.d =


43 1.3 µmd = .


43 28 µmd = .

The digits refer to the mean mass sizes of particles of 43d = 0.8 µm , 1.3 µm , and 28 µm , respectively. The solid lines indicate the computational results in the presence of an oxidizer in the gas generator combustion products ( β = 0.2), the dotted lines give results in its absence ( β = 0). As can be seen, the presence of an oxidizer contributes to an earlier ignition of aluminium particles and increases the completeness of its combustion. For the particles with 43d = 0.8 µm the completeness of combustion amounted to 24.5% and 9.2%, for 43d = 1.3 µm to



2.9% and 2.3%, and for 43d = 28 µm to 0.25% and 0.23%, respectively. The calculations showed that the structure of the flow, with a forming particle cloud streamlined by the carrying gas, is rather complex and to a large degree specified by the interphase reaction intensity and the heat generation of burning particles. Figs. 4–6 depict the flow field isotherms at time t = 5 ms for the above three mean mass sizes of particles. The positions of instant paths of gas particles, which give an idea of the eddy structure of the flow are also shown.

Figure 7: Parameters for particles with m8.0d43 µ= .

Figure 8: Parameters for particles with 43 1.3 µmd = .

Figure 9: Parameters for particles with 43 28 µmd = .

As can be seen, for small-sized particles with rather high completeness of combustion (fig. 4) the temperature of the front part of an outflowing jet is rather high. At the same time in the case of coarse particles, the ‘breakdown’ of the jet occurs faster and its penetration into the environment is less intense. Figures 7–9 give the distributions of temperature T [K] and pressure P [MPa] along the OY-axis for x = 1.1 m for the same versions of the flow, which can be used to judge the wave pattern of the outflow process.



For small-sized particles (see figs. 7 and 8), we observe a strong increase in pressure behind the rarefaction wave due to the strong retardation of the gas by the particles and its subsequent intense oscillations, which induces reverse eddy flows. The maximum pressure peak even at a distance of one meter remains relatively safe for a human organism. In the case of coarse particles, whose force interaction with the carrying gas is weak, the pressure peak is lower. The computational results were compared with physical experiments carried out for a segment (a quarter of the cylinder) of this unit. Quantitative comparisons are impossible in view of the lateral interaction of the outflowing jet with the environment in this experiment. However, as to the dynamics of the formation and propagation of the particle cloud, the qualitative agreement proved sufficiently satisfactory [7].

References

[1] Dullien, F. A. Porous Media Transport and Pore Structure. Academic Press, New York, 1979.

[2] Coakley, T. J. Turbulence modeling for high speed flows. AIAA Paper 92-0436, 1992.

[3] Crow, C. T. Review - Numerical models for dilute gas-particles flows. Trans. ASME J. Fluid Engrg., 104, pp. 297-303, 1982.

[4] Beckstead, M.W. Correlating aluminum burning times. Combustion, Explosion, and ShockWaves, 41(5), pp. 533-546, 2005.

[5] Chen, K. and Shuen Ji. Three-dimensional coupled implicit method for spray combustion flows at all speed. AIAA Paper 94-307, 1994.

[6] Rychkov, A. D. Mathematical Modelling of Gasdynamic Processes in Channels and Nozzels. Novosibirsk, Nauka, 1988 (in Russian).

[7] Rychkov, A. Schneider, H. Shokina, N. and Eisenreich, N. Numerical and experimental investigation of the gas-dynamic aspect in the dispersal process of µ -sized energetic particles. Proc. of the 33rd International Annual conference of ICT ‘Energetic Materials-Synthesis, Production and Application’, Karlsruhe, Germany, pp. 140.1-140.12, 2002.

[8] Yoon, S. and Jameson, A.. An LU-SSOR scheme for the Euler and Navier-Stokes equations. AIAA Paper, 87-600, 1987.



A methodology for momentum flux measurements in two-phase blast flows

R. G. Ames1 & M. J. Murphy2 1Naval Surface Warfare Center, Dahlgren Division, USA 2Lawrence Livermore National Laboratory, USA

Abstract

Modern diagnostics for air blast waves have been developed to the point that they are sufficiently accurate and robust to capture most blast phenomena of interest (most often pressure). Two-phase blast flows also transport momentum and energy in a non-gas phase and, as such, the properties of this second phase must be accounted for in estimating flowfield parameters. Standard air blast diagnostics are not sufficient to capture these effects and the diagnostics that do exist for multiphase flows normally rely on steady conditions or sampling periods of at least a few milliseconds. The extreme transients associated with multiphase blast flows (microsecond-scale) preclude the use of such techniques. For this reason, novel approaches are required in order to capture the total momentum and energy flux in two-phase blast flows. This paper outlines a novel suite of diagnostics and an analysis technique that allows for momentum flux measurements in two-phase blast flows. Keywords: multiphase blast, two-phase blast, diagnostics, momentum flux, impulse.

1 Introduction

The problem of separate momentum flux measurements in two-phase blast flows is difficult because of the difference in length scales over which the two phases interact with measurement devices. When trying to measure a reflected pressure using a standard pressure gauge, the gas-phase loads are relatively constant over the sensor surface area. As such, the average load is very close to the local loads across the sensing surface. For a solid-phase particle blast, the local gradients are very large due to particle impacts against the sensing surface. The average load across the sensing surface is much lower, however, because the particles



doi:10.2495/MPF070041

appear at discrete locations with spacing that is too large to be considered continuum. As such, the average load across the gauge sensing surface might be well within the calibrated range of a standard pressure gauge but the local loads are much higher and are likely to produce damage to the sensing surface. For this reason, novel diagnostic techniques for two-phase blast flows have been developed and documented in a previous publication [1]. While it is relatively straightforward to collect integrated measurements of multiple-phase blast effects (e.g. impulse measurements using the momentum trap technique) it is more difficult to extract the time-history of the loading that produces that impulse. As explained above, standard pressure gauges, which are sufficient to obtain detailed time-histories for gas-phase loads, are not suitable for flowfields that contain a significant amount of solid particulate. In order to facilitate this type of measurement it is often necessary to measure parameters such as the number density flux of solid-phase particles. This type of measurement, combined with measurements of the time-varying speed of the particles, will give a measure of the momentum and energy flux in the solid phase. If the speed of the particles cannot be measured (as is usually the case) then their behavior can be estimated using a suitable model with parameters derived from measurable quantities. In either case, the combination of time-varying and integrated measurements serves to increase the accuracy of the technique. This paper describes one approach to the momentum flux measurement problem using a combination of number density flux measurements and impulse measurements.

2 Measurement techniques

The methodology described here uses two main measurements: one for the number density flux of particles and one for the impulse delivered to a rigid target. The number density flux is measured using a particle streak recorder that uses a rotating drum placed in a multiphase blast field to produce a time history of the particulate passing through a thin aperture. The impulse measurements are performed using the momentum trap technique.

2.1 The momentum trap technique

The momentum trap technique was first described by Hopkinson in his classic paper describing the pressure bar technique for measuring blast and impact loads [2]. More recently, Held [3, 4] applied a similar technique to the problem of measuring the impulse generated by air blast waves against cylindrical targets. The Held technique uses a block of known mass and geometry suspended above a level surface at some distance d from an explosive charge. When the charge is detonated and the blast wave produces loads on the block, it is thrown from its initial location. So long as the time scales associated with the blast loads are much less than the time scales associated with the motion of the block it can be assumed that the load is impulsive. As such, the block of mass mb will be instantly accelerated to a speed v in the direction of the blast wave motion.



Figure 1: Diagram showing the momentum trap measurement technique.

So long as the speed of the block is sufficiently low it will travel ballistically to a landing distance l from its initial location. Given the known initial center of gravity height h it can be shown that the change in momentum of the block (which is equal to the impulse delivered to it) is given by

g/h2

lmI b= (1)

where g is the acceleration due to gravity. It is usually the case that the impulse is normalized to the frontal surface area of the block to give a measure of the impulse per unit area exposed to the blast. This area, A, and impulse per unit area IA are then related by the expression

g/h2

lI Aφ

= (2)

where φ is the frontal area density of the block, mb/A. The measurement problem, then, reduces to the problem of measuring the throw distance l. This is usually achieved by placing the momentum trap within a level sand pit: when the block is thrown from its initial location it impacts on the sand and leaves a mark to indicate the throw distance.

2.2 The particle streak recorder

The particle streak recorder is a device that measures the flux of particles in a two-phase blast flow. It does so by employing a spinning drum within a protective shroud. The shroud includes a thin aperture that is aligned toward the oncoming blast wave. As the two-phase blast wave passes the front of the shroud, the aperture allows a small amount of particulate to pass onto the rotating drum. The rotating drum is designed such that the surface will allow the particulate to leave an impression as the disc is spinning. As a result, the drum produces a time history of the particulate that passes through the aperture via the record of particle impacts across its surface.



Figure 2: The Particle Streak Recorder (above) and typical impact surface images (below).

The data produced by the particle streak recorder include a collection of impact surface images that need to be analyzed in order to produce a history of number density and area density of holes. This analysis technique can be conducted in a number of ways. For large, uniformly sized particulate (e.g. uniform mm-scale spheres), it is possible to visually inspect the surface at regular spatial intervals (which correspond to regular temporal intervals). This technique has been employed in, e.g., [5]. For smaller, non-uniform-size particulate (as in the bottom of Fig. 2), the number and sizes of impact locations preclude the use of a visual inspection technique and an image processing technique must be used to analyze the data. This technique normally consists of three steps: the first is the production of a series of digital micrographs of the surface of the drum. These micrographs must be of appropriate magnification to capture both the smallest feature of interest (normally the smallest particle size) as well as the smallest time span of interest. This image must then be converted into a two-color “binary image” where one color represents undisturbed surface and the other represents an impact hole (or crater for lower-energy particulate). Finally, this binary image is then analyzed for connectivity among pixels to determine the statistics associated with the impact locations. These statistics include parameters such as number of impact points, mean impact point size, and max/min impact point size. These statistics, when collected at various spatial locations across the drum surface, provide the time-history of the number and size of particulate that passed through the aperture.



3 Analysis technique

Analyzing the data collected using the momentum trap technique is straightforward: the throw distance is directly proportional to the total impulse delivered to the block. The mass of the block and geometry of the setup define the constant of proportionality. The analysis technique employed for the particle streak recorder is somewhat more complicated, however, due to the uncertainties associated with the measurement technique. Though the particle number density is known as a function of time, the particle speeds are not. As such, the time history of the particle speeds must be inferred from the particle streak recorder data and a suitable model for their behavior. The methodology described here begins with an appropriate form of the momentum flux rate equation for the solid-phase particles and postulates models for the terms therein.

3.1 Particle mass and momentum flux

Before continuing with the form of the momentum equation used for the solid-phase particles it is instructive to review the momentum flux rate equations for continuum (i.e. single-phase) flows. For this class of flows, the total mass flux rate per unit area is given by

ccc,A V

dtdm

ρ= (3)

where the subscripts c are used here to denote an equivalent-continuum description of the solid-phase particulate flow. Note that the mass mA,c is the mass flux per unit area in the equivalent continuum description. While this description is useful for a continuum problem, it is more instructive to write the mass flux in terms of the number density of particles in a particulate blast flow. In this case, the mass flux is characterized by a number density of particles per unit area, NA, and an average particle mass, mp. Using this description the mass flux rate is given by

dt

dNmdt

dm Ap

A = (4)

where mA is the mass flux of particles per unit area. The momentum flux rate associated with this mass flux rate is simply the product of the mass flux rate and speed of the moving mass. For the equivalent continuum case, it is the well-known parameter ρcVc

2. For the solid-phase particulate the expression is given as

dt

dNVmdt

dp App

A = (5)



where pA is the momentum flux per unit area and Vp is a representative average speed of the particulate. Note that Vp is, in general, a function of time.

3.2 Models for particle number density and speed

Eq. (5) provides the relationship between momentum flux rate, particle number density flux rate, and particle speed. In order to conduct the analysis that follows these parameters must be expressed in analytical form. Insight gained from previous testing has shown that these parameters behave roughly as the structure of the air blast wave and, for this reason, it is reasonable to approximate their behavior using decaying exponentials (as for air blast properties in, e.g., [6]). For the number density flux rate this relationship can be expressed as

)tt(k2N

A 01etAdt

dN −−= for t ≥ t0 (6)

where AN, k1, and t0 are chosen to provide an appropriate representation of the measured particle number density flux rate. The t2 term is included to allow for the observed behavior of an initial rise in the particle number density flux rate; this rise is quickly overcome, however, by the decaying exponential behavior. Note that t0 is the time at which the particulate first arrives at the measurement location. Note also that the time t is taken as time from detonation and that the expression in Eq. (6) is only valid for times greater than or equal to the arrival time t0. The particle speed can be modeled using a similar approach but with a different modification to account for behavior in the early time. Because the solid-phase particulate is generally a high-density material, it is likely that it suffers relatively little momentum change after the detonation process provides its initial speed. As such, the particulate is likely to remain at a constant speed as it moves outward after the detonation. This behavior will begin to change, however, as the particulate moves out of the high induced velocities behind the air shock wave. In this case, the particles will move either ahead of the shock or fall far behind and the drag on the particulate is likely to be substantial and cause a significant reduction in particle speed. The location at which this behavior is observed will be different for different multiphase blast fields; it is a necessary consideration in applying the model given here. In practice, the model described below will be valid (to first approximation) so long as the particulate remains behind the initial shock. The model captures the decay due to particles falling behind the shock but assumes that the initial portion of the particulate blast is contained in the air shock structure. If the constant-initial-speed approximation is appropriate then the initial particulate speed at any standoff d is given simply by d/t0 where t0 is the time after detonation at which the particulate reaches distance d. Allowing for decay from this initial value then gives a particle speed function of the form



)tt(kp

02etdV −−= for t ≥ t0 (7)

where k2 is chosen to provide a good match to the observed data. This model simply states that the initial particle speed at distance d is equal to the initial particle speed after detonation followed by a decay given by the combination of the decaying exponential and 1/t behavior.

3.3 Determining the values of AN, k1 and k2

The methodology described here requires an estimate for three parameters: AN, k1, and k2 in addition to a measure of the particulate arrival time t0. All of these parameters are available from the data collected by the particle streak recorder. The value of t0 can be measured using a variety of techniques but is usually achieved by including some type of timing fiducial on the rotating drum. This fiducial is measured relative to the detonation time (using, e.g., a once-per-revolution signal recorded on a data acquisition system with timing referenced to the detonation time) and is marked on the drum. During the analysis procedure, the data are collected using this fiducial mark as a timing reference, thereby allowing a determination of the time at which the particulate first arrives at the particle streak recorder. The values of An and k1 are determined by fitting the measured data to the form given in Eq. (6). The value of k2 is more difficult to determine but can be approximated if the “minimum marking speed” Vmin of the particulate is known. The minimum marking speed is the lowest speed that will produce evidence of an impact on the rotating drum. This speed is a function of several parameters including particle mass, particle density, particle shape, and drum material. There are a number of impact theories and/or computational methods available to determine the value of Vmin given the combination of particle and drum properties. It is important to note that the methodology described here captures only the properties of the particulate with speed greater than Vmin (because slower particulate does not produce measurable data). As such, the combination of particle properties and rotating drum properties must be chosen carefully to make certain that the range of measured speeds (i.e. Vmin and greater) captures the preponderance of the momentum and energy associated with the solid-phase particulate. If Vmin is too high, a significant portion of the momentum and energy flux will not be measured and the methodology will produce erroneous results. The particle streak recorder data show both the start time t0 and end time t1 of the particle impacts. As described above the end time t1 corresponds to the minimum marking speed and can be used in the particle speed equation, Eq. (7), to give

)tt(k

1min

012etdV −−= (8)

or



−=

dtVln

tt1k 1min

102 (9)

Eq. (9), then, gives the value of k2 for given values of Vmin and t1.

3.4 Computing the impulse per unit area

The impulse per unit area produced by the solid particulate will be the integral of the momentum flux rate equation given in Eq. (5). In other words,

dtdt

dpI0t

AA ∫

∞

= (10)

Using the form of the number density flux relationship given in Eq. (6) and particle speed given in Eq. (7) gives the impulse per unit area as

dttedemAI0

21021

t

t)kk(t)kk(pNA ∫

∞+−+= (11)

which can be integrated to give

[ ]( )221

021NpA kk

t)kk(1dAmI

+

++= (12)

Once the values of AN, k1, and k2 are determined from the data, Eq. (12) can be used to compute the impulse per unit area associated with the particle impacts.

4 Discussion

The accuracy of Eq. (12) is limited by the assumptions about the behavior of the particles. This accuracy can be improved by requiring the analysis to match the data collected by the momentum trap technique. However, the momentum trap technique captures the impulse produced by both the solid-phase and gas-phase loads; as such, the value of IA as computed in Eq. (12) should be less than the impulse per unit area measured using the momentum trap technique. In order to provide a rigorous check against measured data, the momentum trap technique must be able to distinguish between solid-phase and gas-phase loads. This requirement can be addressed in a number of ways, the simplest of which is to provide a gas-phase pressure measurement on the momentum trap. This is most often achieved using a pressure gauge located behind a conical



particle stripper. The stripper serves to remove the particles from the flow in the region of the pressure gauge, thereby preventing damage to the gauge and allowing a gas-phase-only measurement. However, the presence of the conical stripper is a disturbance to the flow and must be accounted for. Shock tube testing has shown that a conical stripper will produce significant changes in peak pressure (as much as 50% for some configurations) but that the impulse values are relatively unaffected (10% or less). Beyond this correction is the correction required for the fact that the gas-phase pressures produce a load in the direction opposite the blast wave direction once the shock passes the momentum trap. The air shock loads on the rear of the momentum trap serve to reduce the overall impulse delivered to the block (i.e. they load the block in the opposite direction of the initial blast). A single pressure gauge on the front of the momentum trap will produce a measurement that is much higher than the actual gas-phase impulse as a result of the back-face loading. It is usually possible, however, to create a correction for this phenomenon using an experimental or numerical approach. The momentum trap technique, alone, can provide separate measurements of solid-phase and gas-phase impulse. However, it provides no temporal resolution and is limited by assumptions about the behavior of the air shock as it interacts with the block and particle stripper. Likewise, the particle streak recorder technique is limited by assumptions about the behavior of the particulate but it is able to provide a time-history of the loads produced by the solid-phase particulate. It cannot, however, provide gas-phase loads. The combination of the techniques, in conjunction with the methodology described here, provides a higher-confidence measurement than can be produced by either technique, alone.

5 Conclusions

This paper has presented a methodology that allows for measurements of momentum flux in two-phase blast flows. It outlined both the experimental apparatus and analysis techniques that are required to conduct such measurements. The paper has also discussed the validity of the approaches outlined here and has provided guidance on the circumstances under which the methodology is appropriate.

Acknowledgements

This work was jointly sponsored by the United States Office of Naval Research and the United States Office of the Undersecretary of Defense/Land Warfare and Munitions. In particular, Dr. Judah Goldwasser provided excellent support and guidance for the work presented here. The experimental apparatus was designed in conjunction with a number of contributors, including Sean Treadway, Mitch Moffett, Greg Larsen, and Perry Fridley.



References

[1] Ames, R.G.; Murphy, M.J.; Groves, S.E.; Cunard, D.; “Diagnostics for Multiple-Phase Blast Flows”, in Proceedings of the 3rd International Conference on Fluid-Structure Interaction, September, 2005, La Coruna, Spain

[2] Hopkinson, B.; Proc. Roy. Soc. A, v. 87, 498; 1905 [3] Held, M; “Impulse Method for the Blast Contour of Cylindrical High

Explosive Charges”; Propellants, Explosives, Pyrotechnics, Vol. 24, 1999 [4] Held, M; “Improved Momentum Method”; Propellants, Explosives,

Pyrotechnics, Vol. 26, 2001 [5] Frost, D; Zhang, F et al.; “Near-Field Impulse Effects from Detonation of

Heterogeneous Explosives”; in Proceedings of the Conference on Shock Compression of Condensed Matter, 2002

[6] Kinney, G.F.; Graham, K.J.; Explosive Shocks in Air; 2nd Ed., Springer-Verlag, New York (1985)



Two-phase flow transient simulation of severe slugging in pipeline-risers systems

G. Hernández, M. Asuaje, F. Kenyery, A. Tremante, O. Aguillón & A. Vidal

Laboratory of Energy Conversion, Simón Bolívar University, Venezuela

Abstract

Pipeline-risers systems are frequently encountered in the petroleum industry, especially in the offshore platforms. Single-phase flow does not involve significant troubles in the operations through these arrangements. However, during multiphase flow, flooding of the separation facilities could be expected due to the generation of severe slugs at the bottom of the riser. The size and frequency of the slugs are functions of the accumulation and displacement of liquid at the base of the riser and can be controlled with an adequate model. An improved transient model is presented to simulate severe slugging phenomenon in pipeline-risers systems. Gas penetration is described thoroughly since the first bubble penetrates into the riser until it reaches the top of it. The model presents improvements in the characteristics method applications including a correction for the gas density deviation caused by the nonfixed space-time resolution during the gas penetration. The results were compared with experimental data and previous models showing better accuracy. The model can be used to design new pipeline riser-systems or to adjust the operation of existing systems to prevent the occurrence of severe slug flow. Keywords: two-phase flow, pipeline-risers systems, transient model, severe slugging.

1 Introduction

Severe Slugging process is well known for disturbing operations in pipeline-risers systems due to the production of long liquid slugs in a short period of time, causing flooding of the separation facilities downstream. It occurs when the liquid and gas superficial velocities are relatively low to maintain stratified flow in the pipeline. Once the liquid intends to climb through the riser, the gravity



doi:10.2495/MPF070051

force makes difficult the continuity of the process and breaks the steady state condition. Since this moment, accumulation liquid process starts at the bottom of the riser until the gas pressure upstream becomes greater enough. Therefore, this pressure not only overcomes but also penetrates the liquid head, causing the displacement and production of the liquid accumulated in form of severe slugs. The pressure upstream decreases due to gas expansion until it is not enough to hold the remaining liquid in the riser that fall in order to start a new liquid process accumulation. Thus, the production operations remain in an unsteady state or transient conditions. The severe slugging process has been previously studied. Schmidt et al. [1] described the process in four-step cycle as follows: slug formation, slug movement (production into the separator), blow out and liquid fall back. Taitel et al. [2] presented a model to describe the physical phenomenona but the results were not accurately when compared with experimental data due to the unsatisfied gas continuity in the riser. Fabre et al. [3] developed a model based on the continuous gas penetration through the riser and did not consider the slug formation blocking the gas passage. The model was not able to simulate certain specifics conditions obtained in their own experimental facilities. Sarica and Shoham [4] presented a simplified transient model to describe the phenomenona physically. The simulation of the slug generation, slug production and liquid fall back showed better accuracy than above mentioned models. It is important to remark that the model did not present a procedure to describe the gas penetration into the riser resulting in a cycle time period shorter than experimental data. This paper presents an improved two-phase transient model to simulate severe slugging phenomenon in pipeline-risers systems where the four-step cycle are described physically. An algorithm is proposed to simulate the gas penetration step, which is considered the most complex. It includes a procedure to correct the gas density deviation caused by the nature of the characteristic method. The model predicts accurately the time period of the cycle along with other variables, which are very important to design separation facilities or to adjust operations in order to ovoid humans and economics risks. The results were compared with Sarica’s model [4] showing better performance to simulate the experimental data measured by Fabre et al. [3].

2 Model description In order to simulate the four-step cycle described previously the continuity and momentum equations are formulated for each step of the cycle. The development of the model is based on one-dimensional analysis where the gravity domains and wall shear stress is neglected. This approach can be improved in near future studies when more experimental data become available to simulate highly viscous liquids.

2.1 Equations during the slug generation step

The continuity equation for liquid and gas are formulated for this step considering the pipeline void fraction αP as constant. The fig. 1 shows the slug generation process.



vLSo

vGSoαp Pp

x ZL

PSEP

θL vLS,pen

vLSo

vGSoαp Pp

x ZL

PSEP

θL vLS,pen

Figure 1: Slug generation process.

vLSo

vGSoαp Pp

x ZL=hriser

PSEP

θL vLS,pen

vLSo

vGSoαp Pp

x ZL=hriser

PSEP

θL vLS,pen

Figure 2: Slug production process.

Taking the pipeline as a volume, control the liquid and gas continuity equations along with the combined momentum equation for this step are respectively:

dtdxvv Po,LSpen,LS α−= (1)

( )[ ]dt

PxLdPv PPoo,GS

−= α (2)

( ) gsinxZPP LLsepP ρθ−+= (3)

2.2 Equations during slug production step

In this step, the liquid level in the riser ZL has reached the separator and remains constant as shown in the fig. 2. Thus, eqn. (3) can now be written as follows:

( ) gsinxhPP LrisersepP ρθ−+= (4) Eqns. (2) and (4) can be solved simultaneously, in order to calculate the liquid length and the pressure in the pipeline x and PP respectively. The slug flow rate production into the separator can be calculated with eqn. (1) since the liquid is incompressible.



T

pen,LSL A

vq = (5)

2.3 Equations during gas penetration step

Once the liquid length in the pipe line x is zero from the set of equations presented above, the gas penetration takes place as shown in the fig. 3a.

vLSo

vGSoαp Pp ZL=hriser

PSEP

θL vLS,pen & vGS,pen

Zj

Zj

Zi

dz

(a) (b)

vLSo

vGSoαp Pp ZL=hriser

PSEP

θL vLS,pen & vGS,pen

Zj

Zj

Zi

dz

(a) (b) Figure 3: Gas penetration process.

The boundary conditions at the base of the riser can be obtained from the continuity equations in the pipeline given by

LSopen,LS vv = (6)

−=

dtdPLPv

P1v P

Poo,GSP

pen,GS α (7)

Another boundary needed to solve the problem is located at the position of the first bubble that penetrates into the riser Zj. In this step, the system variables P, αr, ρG, vLS and vGS in the riser are functions of both time and space, while in the pipeline they continue being only function of time. This is due to the no uniformity of the gas void fraction in the riser αr and the hydrostatic pressure. The continuities for the liquid and gas respectively, and combined momentum equations are formulated for a riser differential control volume shown in the fig. 3b, and these are:

( )0

zv

t1 LSr =

∂∂

+∂−∂ α

(8)

( ) ( )0

zv

tGSGrG =

∂∂

+∂

∂ ραρ (9)

( ) gdzgZZPP L

Z

ZrLjLsepi

j

i

ραρ

+−+= ∫ (10)



Eqns. (8)–(10) have five unknowns, namely, P, αr, ρG, vLS and vGS for each control volume position Zi. To close the model, two additional equations are needed. Assuming ideal gas behavior one equation would be:

G

G

MRT

Pρ

= (11)

Another equation can be obtained from the drift flux formulation for the flow in the riser given by Zuber and Findlay [5] which is used to obtain the gas void fraction in terms of superficial phase velocities:

( ) oLSGSor

GS vvvcv++=

α (12)

where vo = 0.35 gd and represents the bubble-rise velocity in stagnant liquid.

2.4 Equation during gas blowdown

It occurs since the first gas bubble reaches the top of the riser as shown in the fig. 4. The pipeline pressure decreases drastically in this step until it is not enough to push the thin liquid film remaining in the pipe wall into the separator, causing an instantaneous liquid fall to begin the cycle again. The set of eqns. (8)–(12) can be used to solve the five unknown in this step.

vLSo

vGSoαp Pp Zj=hriser

PSEP

θL

Zi

differential control volume

dz

vLS,pen & vGS,pen

vLSo

vGSoαp Pp Zj=hriser

PSEP

θL

Zi

differential control volume

dz

vLS,pen & vGS,pen Figure 4: Gas blowdown process.

3 Characteristic method in the gas penetration step

In order to calculate ρG and αr during the gas penetration step, an initial-value problem with a free boundary is formulated and solved with the characteristic method. This method consists to reduce two partial differentials equations into one ordinary differential equation, where characteristic conditions are satisfied. Thus, eqns. (8) and (9) were reduced to the following ordinary differential equation (see appendix A for details),

( ) 0Dt

DC1

DtD G

orG

rr =−+ρ

αραα

(13)



Solving this ordinary differential equation, the following constant is obtained

( )1Ck

ro

rG

−=

ααρ

(14)

along with the characteristic:

Gvdtdz

= (15)

Eqns. (14) and (15) replace the differentials eqns. (8) and (9) to calculate ρG and αr as it will be shown in the algorithm furthermore.

4 Velocities in the gas penetration step

vLS and vGS are obtained from the same differentials equations (8) and (9), but now through the finite-difference method. Addition of the two mentioned equations gives the following relationship (see appendix A.1 and A.2 for details):

( )

∂∂

+∂

∂−=

∂+∂

zv

tzvv G

GG

G

rLSGS ρρρα

(16)

Substituting eqn. (12) into eqn. (16) the gas superficial velocity is given by

−+

−

∆∆

−

=

∆+∆+

∆+∆+∆+

∆+∆+

∆+∆+∆+∆+

∆+

∆+∆+

∆+∆+

ttzzG

ttzGtt

zzro

ttzzG

tzzGtt

zzrottzr

ttzGS

ttzzr

ttzzGS

C

tzC

v

v

ρρ

α

ρρ

αα

α11

1

(17)

Finally, liquid superficial velocity is obtained from eqn. (12).

5 Correction of the gas deviation

For each time-step, the coordinate frame in the riser is different due to the nature of the characteristic method. Analysing eqn. (15), it is obtained ∆z = vG∆t, where ∆t is established previously as constant and vG increases for each time-step due to the gas expansion. As a result, ∆z is different for each time-step as shown in the fig. 5 for t5 and t6. As a consequence of the nonfixed space-time resolution, a gas density deviation is presented in the eqn. (17) which is formulated under finite-difference method criteria and requires the gas density in the previous time-step, staying in the same position respect to z which does not keep constant. The fig. 6, illustrates better this case. This happens for all the points during the grid resolution. The correction proposed in this paper is based on the density average between the two closest points e.g. (3,5) and (4,5), around the correct point in the previous step-time, during the grid resolution. This is shown in the algorithm furthermore.



z

timet1

k1

z1

z2

z3

z5

z

z1

z2

z3

z4

k2k3

k4

k5

z6

t2 t3 t4 t5 t6

z5

z4

t1 t2 t3 t4 t5 t6

time

k1k2k3

k4

k5

hriser hriser

z

timet1

k1

z1

z2

z3

z5

z

z1

z2

z3

z4

k2k3

k4

k5

z6

t2 t3 t4 t5 t6

z5

z4

t1 t2 t3 t4 t5 t6

time

k1k2k3

k4

k5

hriser hriser

Figure 5: Characteristic method illustrated.

timet1

z

z1

z2

z3

z4

z6

t2 t3 t4 t5 t6

z5

hriser

(1,6)

(3,5)

(4,5)

(2,5)

(1,5)

(5,5)

correct point

taken point

(2,6)

(3,6)

(4,6)

(5,6)

(6,6)

calculation point

timet1

z

z1

z2

z3

z4

z6

t2 t3 t4 t5 t6

z5

hriser

(1,6)

(3,5)

(4,5)

(2,5)

(1,5)

(5,5)

correct point

taken point

(2,6)

(3,6)

(4,6)

(5,6)

(6,6)

calculation point

Figure 6: Error presented taking the gas density in the previous time-step, staying in the same position respect to z.

6 Algorithm proposed to simulate gas penetration step

When the gas is about to penetrate into the riser, the conditions can be determined. Considering i as space counter and j as time counter. Initial conditions [i=1(the riser base), j=1(t=0)] 1. vLSo, vGSo and Po are given and αP is calculated with any stratified flow

model. 2. z(1,1)=0, this means the first gas bubble is in the riser base at t=0. 3. P(1,1) and ρG(1,1) are calculated from eqns. (10) and (11). 4. vLS(1,1) and vGS(1,1) are then calculated from eqns. (6) and (7), dPp/dt = 0. 5. αr(1,1) is calculated from eqn. (12).



( ) ( )( ) ( )1,1

1,1++−

=++jiCqk

qkjiGo

r ρα

( ) ( )

( )( ) ( ) ( ) ( ) ( )

( )

( ) ( )( )

++

+−+++

++

+−

∆+−++

++−++

++=++

1,11,11,11

1,1,111,1,11,1

1,1,

1,11,1

jijijiC

jiji

tjizjizjiC

jijiv

jijiv

G

Gro

G

Gro

r

GS

rGS

ρρα

ρρα

αα

For j = 1 to (tmax/∆t)For i = j to 1

If i=j then, calculate

End if

If i ≠ j then, calculate with this new counter q=j+1-i

End if

If i = 1 then, calculate (boundary conditions)

End ifNext iFor i = 1 to j

Next iNext j

( ) ( ) ( ) ( )[ ] ( ) ( )

+++++

−++−+++++=++2

1,21,111,11,21,21,1 jijigjizjizjiPjiP rrL

ααρ

( ) ( )[ ]1,11,1 ++−+=++ jizzgPjiP LLsep ρ

( )1,1 ++ jiGρ

( )1,1 ++ jirα

( ) ( )RT

jiMPjiG1,11,1 ++

=++ρ

( ) ( ) ( ) ( )[ ] ( ) ( )

++++

−+−+++++=+2

1,11,11,1,11,11, jijigjizjizjiPjiP rrL

ααρ

( ) ( )( ) ( )

∆−+

−+

=+t

jiPjiPPvjiP

jiv poGSoGS,1,

1,11, α

( ) ( )( )[ ] oLSoGSo

GSr vvjivC

jivji+++

+=+

1,1,1,α

( )1, +jiGρ

( )1+jk

( ) ( )( ) t

jijivjiz

r

GS ∆++

=+∆1,1,1,

α( ) ( ) ( )1,1,2,1 +∆++=++ jizjizjizand( ) 02,1 =++ jiz

( )1,1 ++ jivLS

calculated from eqn. (11)

calculated from eqn. (14), taking the k1 calculated previously




obtained from eqn. (11)





[Introduce here the corrrection for the deviation of ρG ( i+1 , j ) fig.8]

( ) ( )( ) ( )1,1

1,1++−

=++jiCqk

qkjiGo

r ρα

( ) ( )

( )( ) ( ) ( ) ( ) ( )

( )

( ) ( )( )

++

+−+++

++

+−∆

+−++++−++

++=++

1,11,11,11

1,1,111,1,11,1

1,1,

1,11,1

jijijiC

jiji

tjizjizjiC

jijiv

jijiv

G

Gro

G

Gro

r

GS

rGS

ρρα

ρρα

αα

For j = 1 to (tmax/∆t)For i = j to 1

If i=j then, calculate

End if

If i ≠ j then, calculate with this new counter q=j+1-i

End if

If i = 1 then, calculate (boundary conditions)

End ifNext iFor i = 1 to j

Next iNext j

( ) ( ) ( ) ( )[ ] ( ) ( )

+++++

−++−+++++=++2

1,21,111,11,21,21,1 jijigjizjizjiPjiP rrL

ααρ

( ) ( )[ ]1,11,1 ++−+=++ jizzgPjiP LLsep ρ

( )1,1 ++ jiGρ

( )1,1 ++ jirα

( ) ( )RT

jiMPjiG1,11,1 ++

=++ρ

( ) ( ) ( ) ( )[ ] ( ) ( )

++++

−+−+++++=+2

1,11,11,1,11,11, jijigjizjizjiPjiP rrL

ααρ

( ) ( )( ) ( )

∆−+

−+

=+t

jiPjiPPvjiP

jiv poGSoGS,1,

1,11, α

( ) ( )( )[ ] oLSoGSo

GSr vvjivC

jivji+++

+=+

1,1,1,α

( )1, +jiGρ

( )1+jk

( ) ( )( ) t

jijivjiz

r

GS ∆++

=+∆1,1,1,

α( ) ( ) ( )1,1,2,1 +∆++=++ jizjizjizand( ) 02,1 =++ jiz

( )1,1 ++ jivLS


calculated from eqn. (14), taking the k1 calculated previously









[Introduce here the corrrection for the deviation of ρG ( i+1 , j ) fig.8]

Figure 7: Algorithm proposed to solve the gas penetration step by the characteristic method.

6. K1 is obtained from eqn. (14) and will remain constant for the first gas bubble along the time.

7. ∆z(1,1) is calculated from eqn. (15) with an arbitrary ∆t established previously.

8. z(2,2) is calculated with z(1,1)+ ∆z(1,1). This is the first gas bubble position in t+ ∆t and represents the second point of the curve described for the first



bubble on which the solution of the ordinary differential equation k1 is satisfied.

9. z(1,2)=0 is established. This means there is another bubble about to penetrate into the riser at t+ ∆t.

For the next step time (t+ ∆t) follow the algorithm shown in the fig.7

7 Comparison of experimental and numerical results

The numerical results were compared with experimental data of Fabre et al. [3] and Sarica and Shoham [4] model. The data was taking in a laboratory-scale flow loop made of 0.053-m-ID transparent polyvinyl pipes. The air/water mixture flowed through a 25-m-long inclined pipeline with an angle of -0.57º, and 13.5-m-long vertical riser. The inlet flow conditions were vLSo = 0.13 m/s and vGSo = 0.20 m/s. For the gas, the velocity was calculated from the mass flow rate using the density at 20ºC and 100 kPa. The Sarica and Shoham [4] model was chosen for the comparation because it showed better approximation than the existences models at the moment. The fig. 9 shows the pressure pipeline vs. time in the four-step cycle of severe slugging. Pressure increase corresponds to the slug generation step. Then, the pressure is maintained due to the slug production and finally the pressure decreases due to the gas penetration and gas blow down steps. The same figure compares the results and it can be seen clearly that the two models present good agreement in the slug generation and slug production according to the pipeline pressure. However, there is remarkable difference in the prediction of the gas penetration step due to the rigorous procedure followed through the present model. Finally, the model with the correction of the gas density shows better agreement than the Sarica and Shoham [4] model.

If then, calculate

End if

( ) ( )jizjiz ,1,1 >++

( ) ( ) gjiZZPjiP LLsep ρ1,1,1 ++−+=+

( ) ( )RT

jiPMji GG

,1,1 +=+ρ

Else

If then( ) ( )1,1, ++< jizjqzgo to *

End if

*

For i = j to 1

( ) ( ) ( ) ( ) ( ) ( )

++

−++−+++=+2

,,111,1,1,1,1 jqjqgjiZjqZjqPjiP rrL

ααρ

Next q

( ) ( )RT

jiPMji GG

,1,1 +=+ρ

Figure 8: Correction proposed for the gas deviation.



120000

140000

160000

180000

200000

220000

240000

0 25 50 75 100 125 150 175 200 225 250Time, s

Pres

sure

, Pa

ExperimentSarica et al. ModelPresent model

Figure 9: Transient simulation of the pipeline pressure during the severe slugging cycle.

8 Conclusions

a. An important contribution has been achieved for transient simulations of two phase flow.

b. A thoroughly algorithm is proposed to simulate transient conditions for the gas penetration step in the severe slugging cycle.

c. Even though the characteristic method is a powerful numerical tool to solve initial-value problems with free-boundary in transient conditions, it requires a correction for the gas deviation generated for the nonfixed space-time resolution.

d. A correction for the characteristic method is proposed in order to simulate the gas penetration step in transient conditions.

e. It has been shown the good agreement of the model when compared with experimental data and Sarica and Shoham [4] model.

f. The four-step severe slugging cycle has been explained physically.

9 Recommendations

1. Future studies can adequate the algorithm and the correction proposed for terrain slugging where the riser is not completely vertical.

2. The wall shear stress can be incorporated to the model proposed in order to simulate highly viscous liquids in either severe or terrain slugging cycle.



Nomenclature

vLS and vGS = liquid and gas superficial velocities, m/s vLSo and vGSo = liquid and gas superficial velocities at the pipeline inlet, m/s ρL and ρG = liquid and gas densities, kg/m3

αp and αr = pipeline and riser gas void fraction L = pipeline length, m t = time, second g = acceleration of gravity, m/s2 R = universal gas constant, 8314.5 Nm/kmol K MG = gas molecular mass, kg/Kmol T = temperature, K d = diameter, m Co = distribution coefficient, 1.2. PP = pipeline pressure, Pa x = coordinate used to measure the liquid length in the pipeline, m ZL = coordinate used to measure the liquid level in the riser, m Zj= coordinate used to measure the first gas bubble into the riser, m Zi = coordinate used to measure the riser differential control volume, m Pi = pressure in a riser point at the coordinate Zi , Pa qL = liquid flow rate into the separator, m3/s AT = pipe transversal area, m2

Appendix

Rearranging eqn. (9) gives

0z

vtz

vt

GG

G

G

rGSr =

∂∂

+∂

∂+

∂∂

+∂

∂ ρρραα

(A1)

Combining eqns. (8) and (A1) yields ( )

∂∂

+∂

∂−=

∂+∂

zv

tzvv G

GG

G

rLSGS ρρρα

(A2)

Substituting eqn. (12) in eqn. (A2), the following expression is obtained

( ) 0z

vt

C1z

vt

GG

Gor

G

rrG

r =

∂∂

+∂

∂−+

∂∂

+∂

∂ ρρα

ρααα

(A3)

According to the following operator ( ) ( ) ( )

( )∂∂

+∂

∂=

dtdz

tDtD

(A4)

If

Gvdtdz

= (A5)

the terms in the square bracket in eqn. (A3) can be rewritten and eqn. (A3) yields



( ) 0Dt

DC1Dt

D Gor

G

rr =−+ρα

ραα

(A6)

This ordinary differential equation is satisfied along the characteristic direction defined by eqn. (A.5)

References

[1] Schmidt, Z., Doty, D.R. & Dutta-Roy, K., Severe slugging in offshore pipeline-riser pipe system. SPE J 12334, pp. 27-38, 1985.

[2] Taitel, Y., Vierkandt, S., Shoham, O. & Brill, J. P., Severe slugging in a riser system, experimental and modelling. Int. J. Multiphase Flow, 16, pp. 57-68, 1990.

[3] Fabre, J., Peresson, L. L., Corteville, J, Odello, R. & Bourgeois, T., Severe slugging in pipeline/riser system. SPE 16846, pp. 299-305, 1990

[4] Sarica, C. & Shoham, O., A simplified transient model for pipeline-riser systems, Chemical Engineering Science. Vol. 46, No. 9, pp. 2167-2179, 1991.

[5] Zuber, N. & Findlay, J. A., Average volumetric concentration in two-phase flow systems. J. Heat Tranfer, Ser. C87, pp. 453-458, 1965.



CFD simulation of gas–solid bubbling fluidized bed: an extensive assessment of drag models

N. Mahinpey1, F. Vejahati1 & N. Ellis2

1Environmental Systems Engineering, Faculty of Engineering, University of Regina, Regina, Saskatchewan, Canada 2Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada

Abstract

In the computational fluid dynamics modeling of gas–solid two phase flow, drag force is one of the dominant mechanisms for interphase momentum transfer. Despite the profusion of drag models, an extensive comparison is missing from the literature. In this work the drag correlations of Syamlal-O’Brien, Gidaspow, Wen-Yu, Arastoopour, Gibilaro, Di Felice, Zhang-Reese and Koch et al. are reviewed using a multifluid model of FLUENT software with the resulting hydrodynamics parameters being compared with experimental data. Also adjustment of drag models based on minimum fluidization was studied. A new method adopted to adjust the drag function of Di Felice showed a quantitative improvement compared to the adjusted drag model of Syamlal-O’Brien. Prediction of bed expansion and pressure drop showed excellent agreement with results of experiments conducted in a Plexiglas fluidized bed. A mesh size sensitivity analysis with varied interval spacing showed that mesh interval spacing with 18 times the particle diameter and using higher order discretization methods produces acceptable results. Keywords: multiphase flow, fluidization, computation, modeling, CFD, drag models, two-dimensional.

1 Introduction

Studies conducted on the dynamics of a single particle in a fluid have proven several mechanisms of momentum transfer between phases: drag force, caused by velocity differences between the phases; buoyancy, caused by the fluid



doi:10.2495/MPF070061

pressure gradient; virtual mass effect, caused by relative acceleration between phases; Saffman lift force, caused by fluid-velocity gradients; Magnus force, caused by particle spin; Basset force, which depends upon the history of the particle’s motion through the fluid; Faxen force, which is a correction applied to the virtual mass effect; Basset force to account for fluid-velocity gradients; and forces caused by temperature and density gradients [1]. Several factors should be considered in extension of single particle model to describe interaction forces in multi-particle systems, including the effect of the proximity of other particles, which implies that the drag force is a function of solid volume fraction, in addition to the particle Reynolds number. Also the single-particle interaction force must be corrected to account for the effect of mass transfer between the phases, and the momentum transfer accompanying such mass transfer must be included in the interaction force. Buoyancy, drag, and momentum transfer due to mass transfer have been considered as controlling mechanisms of gas–solid momentum transfer, since they are the dominant forces as a result of the large density difference between the particles and the fluidizing gas and also due to lack of satisfactory formulations of the other forces. Whilst the inherent instabilities due to inclusion of buoyancy are still not resolved, prediction of a drag model that covers the whole range of Reynolds number and phasic volume fraction has been looked at as the main challenge of numerous of the studies in multiphase flow modeling [2]. These attempts have resulted in the appearance of a substantial number of drag correlations in the literature. The copiousness of drag models available in the literature and the selective attitudes of some researchers have resulted in some inconsistencies regarding the appropriate comparison of available drag models. Almost all the available studies have included efforts to compare two, or at most three, drag correlations, and occasionally the discrepancies between the reported results in modeling fluidization hydrodynamics are easily observed. In this respect, the underlying objective of this study is to accomplish an extensive assessment of frequently used drag correlations in a large selection of published literature and provide a comprehensive comparison between simulation and experimental results using the variety of the drag models. Also, a new approach to adjust the drag model, based on minimum fluidization velocity, is proposed and compared with experimental values. CFD simulation was carried out using the commercial CFD code, FLUENT.

2 Model equations

The drag force depends on the local relative velocity between phases and void fraction and some other factors, such as particle size distribution, particle shape, etc. However, void fraction dependency is very difficult to be determined for any conditions other than a packed bed or infinite dilution (single particle). Also, some factors, like particle size distribution, particle shape, and particle clustering have not been considered in deriving drag correlations. In an ideal case, it could only be determined how the drag for specific material varies with local “slip” velocity and packing, although, totally unrealistic. On the other hand, most



researchers have information on the minimum fluidization velocity of their own material. In this respect, Syamlal and O’Brien [3] introduced a method to adjust drag law using minimum fluidization velocity as a calibration point. This adjustment has been introduced in order to make the drag law more accurate for a specific system under study. However, this method requires measurement of the minimum fluidization velocity and void fraction of the bed at minimum fluidization velocity by means of experimentation. As another alternative based on the same concept used by Syamlal and O’Brien [3], we developed the following method to adjust the Di Felice drag model. Di Felice [4] expressed the drag coefficient model as the product of drag force on an unhindered particle subjected to the same volumetric flux of fluid and a voidage function:

)(4

3sfgs

sdgs

DCgsK αννρα

−= (1)

where )( sf α is defined as

( ) xssf −−= )1( αα (2)

and the empirical coefficient (x) as a function of Res is expressed as

( )

−−−=

2

25.1exp

βQPx (3)

( )sRe10log=β (4)

65.0 & 7.3 == QP (5)

In the absence of gas-wall friction and solid stress transmitted by the particles, the momentum balance at minimum fluidization can be written as follows [5]:

( ) ( ) ( )svgvg

gsKgsgg

DragBuoyancy

−=−⋅−

=

αρρα1

(6)

At the minimum fluidization velocity, considering that 0=sν and mfUg =ν ,

the equation (6) will be reduced to:

( ) ( ) ( )mfUmfg

gsKgsgmfg

,,1

αρρα =−⋅− (7)



Plugging the drag model into Equation (7) and utilizing a nonlinear optimization algorithm the drag model parameters P & Q in Equation (3) can be adjusted for the system under study using experimental data at minimum fluidization velocity. However, when adjusting the drag models it should be kept in mind that the adjustment should not alter the behavior of the drag correlation when voidage approaches 1. Most drag correlations are formulated such that in that limit, the single sphere CD can be recovered.

3 Experimental set-up

The experimental set-up used in this study has been shown in fig. 1. Experiments were carried out in the Department of Chemical and Biological Engineering at the University of British Columbia. The Column is a 2D Plexiglas of 1.2 m height, 0.28 m width, and 0.025 m thickness. Spherical glass beads of 250–300 µm diameter and density 2500 kg/m3 were fluidized with air at ambient conditions. Pressure drops were measured using three differential pressure transducers located at elevation 0.03, 0.3, and 0.6 m above the gas distributor, respectively. The static bed height of 0.4 m with a solid volume fraction of 0.6 was used in all the experiments. Pressure drop and bed expansion were monitored at different superficial gas velocities ranging from 0 to 0.8 (m/s).

Figure 1: Geometry of 2D Plexiglas fluidized bed.

4 Simulation set-up

The two-dimensional (2D) geometry was discretized using 13440 structured rectangular cells. Performing a grid size sensitivity analysis using different mesh sizes, 5 mm mesh interval spacing was chosen for all the simulation runs. The discussion on the effect of the different mesh sizes has been brought up in a later section. A preliminary case study proved that using fixed time step; in order of magnitude 10-3, which has been reported in literature, is not sufficient to avoid the instability in convergence for 2D multiphase simulations. Therefore, an adaptive time-stepping algorithm with 100 iterations per each time step was used



to ensure a stable convergence. The automatic determination of the time step size is based on the estimation of the truncation error associated with the time integration scheme (i.e., first-order implicit or second-order implicit). If the truncation error is smaller than a specified tolerance, the size of the time step is increased; if the truncation error is greater, the time step size is decreased. A minimum value of order 10-5 was used for the lower domain of time step. A convergence tolerance of 10-4 for each scaled residual component was specified for the relative error between two successive iterations. The governing equations were solved using the finite volume method. The Phase-Coupled SIMPLE algorithm (PC-SIMPLE) [6], which is an extension of the SIMPLE algorithm to multiphase flow, was applied for the pressure-velocity coupling. In this algorithm, the velocities are solved, coupled by phases, in a segregated fashion. Subsequently, the block algebraic multigrid scheme used by the couple solver was used to solve the equation formed by the velocity components of phases at the same time. Also, a pressure correction equation is built based on total volume continuity. Pressure and velocities are then corrected so as to satisfy the continuity constraint. Second-order upwind discretization schemes were used for all the simulation runs. Including the adjusted drag model cases, 9 drag correlations in total, were studied in this work (i.e., Arastoopour, Di Felice, Gibilaro, Gidaspow, Syamlal-O’Brien, Wen-Yu, Zhang-Reese, Koch et al.). FLUENT employed an approximate CPU time of 32 hours for 30 s of real-time simulation at a mean time step of 0.0005 s on a double core Sun Microsystems workstation W2100Z with 2 AMD/Opteron 64-bit processors and 4 GB RAM.

5 Results and discussion

Experimental runs were conducted to measure the pressure drop and bed expansion ratio, H/H0, at different superficial gas velocities. The gas-phase volume fraction from pressure drop measurement across the bed was obtained [7]. At experimentally determined minimum fluidization velocity, Umf = 0.065 m/s, the overall pressure drop, bed expansion ratio, and voidage found to be 4.4 KPa, 1.1, and 0.5, respectively. A wide range of gas superficial velocity (0.011-0.75 m/sec) was considered to measure these parameters. The CFD simulations were carried out using the transient Eulerian-granular model in FLUENT v6.3. Several superficial gas velocities, 0.11, 0.21, 0.38, and 0.46 m/s, which correspond to 1.6, 3.2, 5.8, and 7Umf, respectively studied. The drag coefficient values as a function of solid volume fraction for different drag models are plotted in Fig. 2. All the drag functions show a rising trend of drag coefficient value with increasing the solid volume fraction. The values of drag coefficients were calculated at a typical Reynolds number, Res =10. At low volume fraction of solids (<0.18), excluding the Syamlal-O’Brien adjusted model, which overestimated and the Arastoopour function, which slightly underestimated the drag coefficient values, all the drag models represent almost the same value of drag function. Also, it should be noted that at the limit of extremely dilute suspension all the drag models approach the single particle drag value. For the values of solid volume fraction above 0.2, the Di Felice adjusted



model precipitously separates from the other model toward the higher values of drag coefficient. This trend continues until it crosses the Syamlal-O’Brien adjusted drag model at the value of solid void fraction equal to 0.46. From this point on, Di Felice adjusted model gives the highest values of the drag coefficient. With the exception of the Syamlal-O’Brien adjusted drag model which shows an decreasing trend regarding the slope of the curve for values of solid volume fraction greater than 0.14, all other models show an approximately constant slope (i.e. linear growth in drag coefficient value). It is also noted that the differences among the drag models mainly occur when the solid volume fraction is higher than 0.2. The graph also reveals that adjustment of drag models based on minimum fluidization velocity results in the prediction of higher values of drag coefficient through the whole range of solid volume fraction.

Figure 2: Variation of drag coefficient vs. solid volume fraction in different drag laws.

5.1 Bed expansion

Fluctuation and vigorous motion of the bed surface in fluidized beds have made the determination of bed expansion by visual observation a challenging task. The general method employed to determine the bed expansion is normally based on the bed voidage measurement, which in turn is deduced from the mean pressure drop [8]. Also, bed height has been measured experimentally by means of the overhead observation of a probe tip. Such measurement has been believed to be highly biased, and, more importantly, no standard errors or deviations of data have been reported [8]. Fryer and Potter [9] reported that the experimental technique might well underestimate the bed expansion due to the diffusing characteristic of the bed surface. The other frequently employed method is to plot the time-mean gauge pressure (single-point pressure) against the height of the pressure transducer taps, where the intersection of the two slopes corresponds to the height of the expanded bed. However, this method requires an adequate number of pressure transducers at different elevations along the bed and freeboard [10]. In case of a limited number of pressure transducers, Ellis [10], using the time-mean differential pressure drop data across a certain interval



inside the dense bed and across another section extending from the lowest pressure tap to a tap in the freeboard, adopted the following correlation to estimate the expanded bed height:

probebottomZbedZbedPtotalP

H +∆∆

∆= (8)

However, due to the fact that this technique relies on a single pressure drop measurement inside the bed, one should be more cautious about the accuracy of data rather than the method based on the gauge pressure measurement profile. To determine the bed expansion, from modeling perspective, we considered the height of the bed that contains 95% of the bed weight as the bed height. The results of this method, as reported by Syamlal and O’Brien [8], are not sensitive to the percent bed-weight value chosen within a small range, due to the fact that most of the time, experimental values are reported as bed expansion percent rather than as actual bed height. For this series of simulations, a static bed height of H0=0.4 m over a range of superficial velocities 11.7, 21, 38, and 46 cm/s was used. All the simulations show the correct qualitative behavior of bed expansion. Ascending trend of bed expansion with increasing superficial gas velocity can be observed from the graph (fig. 3). Also, all the available drag correlations with the exception of two adjusted drag models (i.e. Di Felice and Syamlal-O’Brien) and the original Di Felice drag model at high superficial gas velocity (0.46 m/s), underestimate the bed expansion.

Figure 3: Comparison of simulated bed expansion ratio with experiment data.

5.2 Pressure drop

In order to eliminate the large temporal fluctuation of pressure drop in the early seconds of the simulation, the time-average of pressure drop for comparing simulation and experimental results were taken after statistical steady state was established. Numerical results for all the models showed that 3 s of simulation is adequate to reach statistical steady state behavior. Time-averaging was carried out over a range of 4-20 s of real time simulation. As indicated in Fig. 4, the pressure drop inside the bed between two specific elevations (i.e. 0.03 m and 0.3 m as demonstrated in Fig. 1) for all the models showed a declining trend with



increase of the superficial gas velocity, which is in good qualitative agreement with the experimental data. Here again, two adjusted drag models (i.e. Di Felice and Syamlal-O’Brien) showed their superiority in predicting the pressure drop inside the bed. At higher gas velocities, 7Umf, deviation from reported experimental data was observed, which may be explained by the underestimation of the effect of particle clustering at high superficial gas velocities and the influence of the gas distributor at higher velocities.

Figure 4: Pressure drop inside the bed ( )mZmZ PPP 3.003.01 == −=∆ .

5.3 Grid size sensitivity analysis

To study the effect of mesh size resolution on numerical results, a grid size sensitivity analysis was carried out using three distinctive mesh intervals spacing of 5 mm, 4 mm, and 2 mm for 20 seconds of real-time simulation. The results indicate that the grid size spacing selected for simulation in this work (i.e. 5 mm) was adequate for satisfactory prediction of the hydrodynamics in computational geometry. On the other hand, the results did not support the previously proposed criteria (i.e. adequacy of mesh size less than or equal to 10 times the particle diameter) in the literature, [8,11], for CFD simulation of fluidized beds. Table 1 compares the time required for 20 s of real-time simulation. Required time for simulating 20 seconds of 2D fluidized bed drastically increases from 32 hr to almost one week for a decrease in grid interval spacing from 5 mm to 2 mm, respectively.

Table 1: Grid size sensitivity results.

Mesh spacing (mm) 1P∆ (kPa) 2P∆ (kPa) Voidage Simulation time

(hr) 2mm 3.36 5.50 0.55 168 4mm 3.38 5.37 0.54 80 5mm 3.39 5.39 0.54 32



6 Conclusion

The influence of most widely used drag functions, including the Wen-Yu, Gidaspow, Di Felice, Syamlal-O’Brien, Zhang-Reese, Arastoopour, Gibilaro, Koch et al. models, on CFD simulation of a 2D fluidized bed using FLUENT software was studied. All the models showed an acceptable qualitative agreement with the experimental data. Also, adjustment of drag models based on minimum fluidization velocity showed a quantitative improvement in prediction of hydrodynamics parameters. In this respect, the new method of adjustment based on minimum fluidization velocity in absence of gas-wall friction and solid stress transmission applied on the Di Felice drag model showed excellent agreement with experimental results regarding the prediction of bed expansion and pressure drop inside the bed. The mesh size sensitivity analysis carried out in this study demonstrated that even grid interval spacing of 18 times of the particle size (i.e. 275 µm) was able to give acceptable results which is contradictory with some other mesh size sensitivity analyses reported in the literature. However, further modeling efforts are required to study the influence of other parameters such as gas distributors, which have not been studied; comparison of 2D and 3D modeling of fluidized bed reactors and also, effect of particle size distribution which has been underestimated using the mean particle diameter. Moreover, new experimental studies should be carried out using recent advancements in instrumentation engineering in order to resolve the available experimental discrepancies reported in the literature such as void fraction measurements, bed expansion ratio etc.

Nomenclature

DC drag coefficient, dimensionless d particle mean diameter, m

dragf drag force per unit volume, N/m3

g gravitational acceleration, m/s2

H expanded bed height, m gsK gas/solid momentum exchange, kg/m3.s

P gas pressure, Pa Z height coordinate measured from distributor, m Greek letters

α gas void fraction ρ density, kg/m3 ν instantaneous velocity vector, m/s



Subscripts g gas mf minimum fluidization s solid

References

[1] Johnson, G., Massoudi, M. & Rajagopal, K.R., A review of interaction mechanisms of fluid-solid flows, U.S. DOE, 1990.

[2] Syamlal, M., Rogers, W. & O’Brien, T. J. MFIX Documentation Theory Guide, 1993.

[3] Syamlal, M. & O’Brien, T. J. Derivation of a drag coefficient from velocity- voidage correlation; U.S. Department of Energy, Office of Fossil Energy, National Energy Technology Laboratory, Morgantown, WV; April, 1987.

[4] Di Felice, R., The voidage functions for fluid-particle interaction system. International Journal of Multiphase Flow, 20 (1), pp. 153–159, 1994.

[5] Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. San Diego: Academic Press, 1994.

[6] Vasquez, S. A., Ivanov, V. A. A phase coupled method for solving multiphase problems on unstructured meshes. In Proceedings of ASME FEDSM’00: ASME 2000 Fluids Engineering Division Summer Meeting; ASME Press: New York, 2000.

[7] Yang W.C., Handbook of fluidization and fluid-particle systems, New York: Marcel Dekker Inc., 2003.

[8] Syamlal M., O’Brien T., Fluid dynamic simulation of O3 decomposition in a bubbling fluidized bed. AIChE J, particle tech. and Fluidization, 49(11), pp. 2793- 2801, 2003.

[9] Fryer, C., and O. E. Potter, Experimental investigation of models for fluidized bed catalytic reactors. AIChE J., 22(1), pp. 38-47, 1976.

[10] Ellis N., Hydrodynamics of gas–solid turbulent fluidized beds, PhD thesis, The University of British Columbia, February 2003.

[11] Zimmermann, S. & Taghipour, F., CFD modeling of the hydrodynamics and reaction kinetics of FCC fluidized bed reactors. Ind. Eng. Chem. Res., 44(26), pp. 9818-9827, 2005.



An advanced gas–solid flow engineering model

D. Mao & M. Tirtowidjojo The Dow Chemical Company, Freeport, TX, USA

Abstract

A literature survey shows that there are two approaches to mathematically describe gas–solid two-phase flow and reaction in the fluidized bed reactors. One approach is a very simplistic engineering reactor model, and the other approach is based on a rigorous Computational Fluid Dynamic (CFD) model. Despite significant progress on numerical methods, CPU capacity and performance, and theories describing gas–solid flow, CFD based reacting flow modeling is far from being a routine tool for designing new reactors. Thus, simpler reactor models are still the main method for providing guidance for fluidized bed reactor design and optimization. However, traditional reactor models have considered too many simplifying assumptions that create inconsistencies in the governing equations of the boundary conditions such that it can result in erroneous prediction. In the current paper, an advanced gas–solid two-phase engineering reactor model has been developed for flow, heat transfer and reaction in the riser reactor of a circulating fluidized bed system. In contrast to CFD, this model can use as input parameters the hydrodynamic information obtained from published literature or experimental measurement such as axial and radial profiles for solid volume fraction, gas velocity and gas dispersion coefficient. While the hydrodynamics can be as accurate as measured, this advanced engineering reactor model is also coupled with either simple power law or the most rigorous kinetic models that involve elementary reactions. In this way, we can extract heterogeneous reaction kinetics from pilot plant or production reactor data directly. Model validation with pilot plant data and tracer data for a riser reactor will be presented and the short falls of traditional treatment of engineering fluidized bed reactor models will also be discussed. Keywords: fluidized bed, gas–solid two-phase flow, reaction, CFD.



doi:10.2495/MPF070071

for a fluidized bed reactor system

1 Introduction

It is well known that circulating fluidized bed (CFB) is an important technology with significant industrial applications for gas-phase catalytic reacting system (Wang et al, [27]; Kunni and Levenspiel, [15]). Despite the wide use of this technology, the opportunity to improve and to reduce the risk of scaling up CFB system using computational fluid dynamics (CFD) is significant. However, the use of reacting flow Navier-Stoke based CFD to simulate CFB process is still very far from being routine. The difficulties arise due to the needs to describe the fluid dynamics associated not only of the gas-phase but also the flowing solid particulates that perform catalytic reaction. Accurate description of the dynamic for the solid flow is needed to predict the solid void fraction such that active site concentration of the catalyst can be accurately determined to calculate reaction rates in the reactor. Because of the complexity of multiphase flow theory, it appears that the CFD based method is still insufficient to provide accurate prediction of solid fraction profile (Mao, [19]; Mao et al, [20, 21]; Das et al, [6, 7]). An alternative to the CFD reacting flow approach is to consider a more empirical engineering reactor model for gas–solid two-phase flow. This approach makes use of solid flow information based on some experimental data or published data (Kruse et al, [13]; Kruse and Werther, [14]; Abba et al, [1]). Review of the many available models, however, show that fundamental equations such as mass balance, momentum and energy balances, and assumptions of boundary condition used are not adequately described (Abba et al, [1]). In addition, numerical methods used in these models may result in numerical instability that render the prediction to be questionable. In this work, an advanced gas–solid two-phase engineering reactor (AGS2D) model has been developed for flow, mass and heat transfer and reaction in the riser reactor of a circulating fluidized bed system. As in the conventional engineering CFB models, this approach will still use solid velocity and density profiles that could be determined experimentally or from published data. The current engineering model is equivalent to the two-fluid dynamic model for multiphase flow (Gidaspow, [11]; Mao et al, [20]). Velocity profile in the two-fluid dynamic model is solved based on momentum equations without simplification, while velocity profile in the current AGS2D model is solved based on mass balance and velocity profiles. These profiles can be obtained from published data (Rhodes et al, [25]; Du et al, [8]) or cold flow experiment. Conceptually, this model can be used to scale up CFB reacting system from a pilot demonstration unit (PDU) and cold flow experimental data at various scale where data for gas and solid velocity profiles (Ug and Us) and solid void fraction and εs profiles are obtained. The kinetics can in principle be determined from experimental data or using a detailed elementary reaction based approach such as used by fundamental kinetic modeling approach (Tirtowidjojo and Pollard, [26]). However, the known kinetics for catalytic decomposition of ethylbenzene (EB) to styrene is considered here for illustration of AGS2D model utility.



2 Model development

2.1 Governing equations for both mass and energy balance

Equations 1a, 1b and 1c are governing equations for gas species concentration, gas temperature and solid temperature. Equations 2a and 2b govern gas pressure and the last gas species concentration based on ideal equation of state, respectively. Equations 1 are solved based on an upwinding numerical scheme. As described above, the velocity profile used in Equations 2a is known. Heat transfer between gas and solids is coupled by heat transfer coefficient, h, and temperature difference Ts-Tg. In the present case, h is considered to be infinite such that gas and solid have the same temperature.

( ) ( )ig

iggrg

iggzg

igggigg sr

CrD

zrz

CD

zz

Cu

t

C,

,,

,,

,, 1+

∂

∂

∂∂

+

∂

∂

∂∂

=∂

∂+

∂

∂αα

αα (1a)

( ) ( ) ( )gsgTg

grgg

gzggggpggggpgg TThs

rT

rzrz

Tzz

TuCt

TC−++

∂

∂

∂∂

+

∂

∂

∂∂

=∂

∂+

∂

∂,,,

,, 1 αλαλαραρ (1b)

( ) ( ) ( )gssTs

srss

szsssspsssspss TThs

rTr

zrzT

zzTuC

tTC

−−+

∂∂

∂∂

+

∂∂

∂∂

=∂

∂+

∂

∂,,,

,, 1 αλαλαραρ (1c)

Equations 1a, 1b and 1c are written in a general conservation form to account for changing gas and solid volume fractions from along reactor height. However, in a fully developed zone at the middle of a riser reactor, for example, the αs is approximately constant, and many times this is typically assumed (Abba et al [1]) in the formulation. In the present work, we consider the most general case where solid flow is different from gas flow and dispersion and heat conduction for gas and solid phases. Mass density of a gas mixture is a function of axial location because both temperature and pressure change axially with development of endothermic or exothermic reactions. Thus, mixture gas mass density should be inside the derivative, rather than outside of the derivative as typically assumed (Abba et al, [1]).

( ) guudzd

dzdp

ssssssg αραρ −−=

(2a)

∑=

−=M

iig

g

gg C

TR

pC

2,1,

(2b)

Radial Peclet number, Per, is defined from radial dispersion coefficient in Equation 2c, while the Axial Peclet number, Pez, is defined from axial dispersion coefficient in Equation 2d. The influence of axial dispersion is ignored here due to the overwhelming convection. The source term, Equation 2e, is a function of gas-phase and catalytic reactions. Influence of radial gas velocity profile (n), radial solid volume fraction (f) and radial gas Peclet number (Per) on EB conversion and Styrene selectivity will be assessed. The parameter q is fixed at 2 or 3 (see Figure 1) due to Rhodes et al [25].

rg

avgr D

RuPe

,

= (2c)



zg

avgz D

RuPe

,

= (2d)

( ) thermaligcatigpig rrs ,,, 1 ααρ +−= (2e)

2.2 A hybrid of engineering gas–solid two-fluid model

In this work, pressure drop for momentum balance due to solid acceleration and gravitational force are considered since pressure has a direct influence on reaction rates. Energy balances for heterogeneous system are also considered because temperature plays a key role in both gas and catalytic reaction rates. However, traditional reactor models (Abba, et al [1]) typically assume that both solid volume fraction and mixture gas mass density are fixed. The AGS2D model can directly make use of available velocity and solid fraction. Despite available velocity and solid volume fraction, literature search indicates that there has not been any attempt of directly coupling this flow information with heterogeneous reaction in a CFB or riser reactor model to evaluate and scale up riser reactor or CFB in general. The gas phase and solid phase mass balances are calculated from radial profiles as shown in Equations 3a, 3b and 3c.

∫==R

gggavggavggavggg drruAum0,,, παραρ (3a)

∫==R

sssavgsavgsss drruAum0,, παραρ (3b)

1,, =+ avgsavgg αα (3c)

Equation 4a describes average variables for gas mixture mass density, ρg,avg, gas void fraction, αg,avg, gas velocity, ug,avg, solid void fraction, αs,avg, solid velocity, us,avg, gas pressure, pg,avg, and temperature, Tavg. Since αs,avg is typically larger in the bottom acceleration region and top deceleration region and lowest in the middle fully developed region, fixed mass flow rate requires that the axial average solid velocity vary accordingly as described in Equation 3b.

20

R

drrR

avg π

πφφ

∫= (4a)

Ideal gas equation of state is utilized in the present work as shown in Equation 4b.

avg

avg

R

gavg RT

pR

rdr== ∫ 2

0

π

πρρ (4b)

Profiles for gas velocity in Equation 5a can be taken from Martin et al [23] and Derouin et al [7]. In a riser, index n may be around 1/7 and less than 1.0 because of turbulence (Bird et al, [4]).



Figure 1: Solid volume fraction radial profile (Rhodes et al, [25]).

−

++

=

+n

n

avgg

avgg R

rnnu

u1

,

1113

α

(5a)

Rhodes et al [25] provided solid volume fraction radial profile as show in Equation 6a. Note that this equation tends to under predict the exp data at the center and close to the wall as shown in Figure 1. The AGS2D model can use either correlation equations or discrete data. The extension of Equation 6a is shown in Equation 6b with four parameters, αs,0, αs,1, a, and b. Rhodes et al [25] analyzed that plot shape, b to be in the range of 1 and 3.

2

,

2

=

Rr

avgs

s

αα

(6a)

b

ss

ss

Rra

=

−

−

0,1,

0,

αααα

(6b)

Average solid volume fraction is defined in Equation 6c. The relationship among average solid volume fraction and parameters is listed in Equation 6d when a is equal to 1.

ss

RRrdr απαπ 2

02 =∫ (6c)

1,0, 22

2 sssbb

b ααα+

++

= (6d)

Solid volume fractions at both wall (αs,1) and core center (αs,0) can be related by “f” in Equation 6g. Thus, two-parameter correlation equation for solid volume fraction is shown in Equations 6h and 6i. It is interesting to note from experimental data in Figure 2 that “b” is in order of 4 and “f” is in order of 10 or even 20.

ss f αα 11, = (6e)

ss bfb

αα 10,

22 −+= (6f)



1

1

0,

1,

22 fbbf

fs

s

−+==

α

α (6g)

( )

−+

++

=b

avgs

s

Rrf

fbb 11

22

,αα

(6h)

( )

−+

++

=−

− q

avgg

g

Rrf

fqq 11

22

11

,αα

(6i)

Note that Equation 6h and 6i automatically average value when integrating gas velocity radial profile or radial solid void fraction.

2.2.1 Axial average velocity profile for both gas phase and solid phase Axial profiles for average solid velocity and gas velocity in Equations 7a and 7b can be calculated based on mass balance and axial average solid void fraction.

Amu

avgss

savgs

,, αρ

= (7a)

Am

uavggavgg

gavgg

,,, αρ

= (7b)

Mass balance for both gas and solid two phases should be satisfied not only for reactor scale (whole cross radial section) based on Equations 7a and 7b, but also for each control volume cell. Axial average solid void fraction can be obtained from pressure drop data or from cold flow experiments. Equation 7c describes the pressure drop on reactor scale and Equation 7d is for pressure on each cell. Equation 7e shows the gas density as a function of temperature and species composition.

( ) guudzd

dzdp

avgssavgsavgsavgssavgg

,,,,, αραρ −−= (7c)

( ) guudzd

dzdp

ssssssg αραρ −−= (7d)

∑=

= M

m m

m

gg

MYRT

p

1

ρ (7e)

2.3 Reaction kinetics Reaction kinetics for EB reaction is taken from Ahari et al [2] as shown in Equations 8a to 8b. Kinetics parameters are shown in Table 1.

( ) ( )( ) ( )( ) ( ) 435623256

42663256

22563256

CHTOLCHHCHEBCHCHHCHCBZHCEBCHCHHC

HSTYCHCHHCEBCHCHHC

+→++→

+↔ (8a)



The species heat capacity, Cp, is calculated from Equation 8c and the corresponding constants are listed in Table 2. Heat of reaction is obtained from NIST database.

( )

( )

23

33

22

11

10194.227.126157.122725

exp

/exp3600

/

2

2

TTFRT

FK

RTEAk

PPkr

Pkr

KPPPkr

EB

iii

HEB

EB

EBHSTYEB

−×−−=∆

∆−

=

−=

=

=

−=

(8b)

Table 1: Kinetic parameters for dehydrogenation catalyst reactions.

Reaction No. A E (kJ/kmol) 1. STY 0.02 8.071E4 2. BZ 19.4 25.0E4 3. TOL 0.3 8.73E4

22

TDCTBTAC p +++= (8c)

Table 2: Coefficients for thermal capacity.

Species name A B C D EB 1.314E+01 9.539E-02 -3.248E-05 -691200 Sty 1.800E+01 7.150E-02 -2.150E-05 -917000 H2 7.130E+00 -7.120E-04 7.570E-07 -2900 BZ 7.280E+00 6.810E-02 -2.390E-05 -543000 Ethylene 3.480E+00 2.790E-02 -8.730E-06 -62400 Toluene 1.060E+01 8.050E-02 -2.740E-05 -568000 Methane 6.910E-01 2.260E-02 -6.320E-06 141000 N2 6.760E+00 6.060E-04 1.300E-07 0

2.4 Boundary and initial conditions

Initially, the riser is filled with nitrogen only. Dirichlet boundary condition or fixed number is used at the bottom inlet section, Neumann boundary condition is used at the top outlet riser, symmetric boundary condition is used at the core center and wall boundary condition is used at the riser wall.

2.5 Numerical scheme

Equations 1a, 1b and 1d can be written in a compact conservation in Equation 9a based on numerical flux. E and F are convection fluxes along z and x directions, respectively. Ev and Fv are diffusion or heat conduction flux along z and x directions, respectively.



Second order term as diffusion or heat conduction are discretized with central-difference scheme. But for convection term, Patankar [24] found central-difference scheme is not stable and developed a power and mixed scheme. Liou [17, 18] developed a AUSM and AUSM+ scheme for all speeds, Edwards [9] developed a low-diffusion flux-splitting scheme (LDFSS). Both AUSM and LDFSS are similar based on flux splitting with low diffusion. Mao et al [20, 21] developed LDFSS and AUSM for gas–solid two-phase flow and reaction. Here, first order upwinding scheme is utilized in Equation 9c for steady state results. Central-difference scheme is shown in Equation 9d.

( ) ( ) 0=∂−∂

+∂−∂

+∂∂

xFF

zEE vv

τφ (9a)

xFF

zEuE

xv

zv

∂∂

Γ==

∂∂

Γ==

φ

φφ

0

(9b)

( )( )

( )

<

≥==

+

++

0,

0,

1

2/12/1

uifu

uifuuE

i

i

ii

φ

φφ (9c)

( ) ( ) ( )2

12/12/1

iiii

uuuE φφφ +== +

++ (9d)

(a) (b)

Figure 2: (a): Radial concentration profile; (b): Radial –C profiles (Bader et al, [3]).

3 Results and discussions

3.1 Validation

Comparison against analytical solution derived by Klinkenberg et al [12] shows good comparison at wide range of Per for flat gas velocity (gas velocity is constant along radial direction) as shown in Figure 2a. Numerical method



residual is found to be as small as 1.0e-9. Finally, a good comparison with experimental data from Bader et al [3] is obtained in Figure 2b. The AGS2D model also compares favourably against the analytical solution for 1D flow and reaction without dispersion (Levenspiel, [16]) for both first and second order reaction kinetics.

3.2 Influence of numerical scheme

Central difference numerical scheme (CD) shows oscillation or unstable phenomena as shown in Figure 3a, and in contrast simulation by upwinding numerical scheme (UW) is very stable also observed by Patankar [24]. Gas velocity is constant along radial direction and is 1.0 m/s. Inlet gas concentration may be unity or (1/a), where a is tracer injection radius.

(a)

(b) (c)

Figure 3: (a): C-contours by different numerical scheme using 21(r) by 101 (z) mesh; (b): Tracer-C close to the wall; (c): Error evolution.

When dispersion coefficient increases ten times larger from 1.0E-3 to 1.0E-2, oscillation still takes place with CD while not observed with UW. The fluctuation of CD predicted concentration particularly close to the wall depicted



in Figure 3b is so large that negative concentration predicted near the inlet of the reactor. To overcome this numerical instability problem the typical practice is to force the negative dimensionless concentration to be zero and “unity” when concentration is larger than the unity. While, this method may work for simple dispersion convection flow, it is difficult to conceive how CD will predict a reasonable result when reaction exists. The numerical diffusion problems associated with UW has been solved by a low-diffusion flux-splitting scheme and advection upstream splitting method developed by Mao et al [20]. In addition, UW provides much faster convergence than CD as shown in Figure 3c.

3.3 Influence of radial and axial solid volume fraction profile

Figure 4a shows different axial solid volume fraction profiles considered. “case-1” uses a uniform distribution along axial direction, “case-2” and “case-3” profiles are taken from literature (Levenspiel, [16]), “case-4” corresponds to a step profile with same average solid volume fraction as “case-1”. “case-5” has just as half of solid volume fraction as “case-1”. Figure 4b shows that solid profile has great influence on temperature profile as expected since the catalyst also provides heat input to the system. Figures 4c and 4d show that the axial solid fraction profile has some influence on EB conversion and Styrene selectivity even though total mass of solid is same (i.e. case 1 and 4) at Peclet number is 250. As can be seen in Figure 4c, EB conversion is lower in case 4 than in case 1 even though the total solid fraction in the reactor is the same. The model shows that the more uniform solid density profile gives higher conversion even though the end temperature point is very close as can be seen in figure 4b. Figure 4c also shows that a significant conversion can be achieved at lower riser height where the axial solid density is larger as in cases 2, 3, and 4 as compared to case 1 By decreasing solid density in the riser by half (compare case 5 and 1), EB conversion was predicted to drop while Styrene selectivity increased (see Figures 4d) as expected since the reaction rates also drop with lower catalyst in the riser. This is also confirmed by comparison of the more realistic solid profiles described by cases 2 and 3. The model prediction clearly shows that both the solid hold up and profile in the reactor are important factors in the design of riser reactor for endothermic reaction where highest temperature is the most significant at the beginning of the riser reactor. Note that the use of piece-wise linear function in cases 2-4 as an input to the AGS2D model demonstrate the robustness of the numerical solver used in this model.

4 Conclusions

An advanced hybrid of CFD and engineering reactor model in 2D model has been developed for gas–solid two-phase flow, heat transfer and reaction for the circulating fluidized bed riser. Mass balance, momentum balance and energy balance are considered for the coupling of two-phase flow and reaction. A robust



numerical scheme, upwinding scheme, is used to solve the partial differential equations. The model and the numerical scheme are validated by comparison with analytical solution, experimental data, mass balance and residual check. The developed model has also been shown to accommodate various input from experimentally determined solid and gas velocity profiles and density. The model can be used to extract kinetics from pilot reactor and scale-up for production reactor. Sensitivity analysis shows that gas velocity, gas dispersion, and solid volume fraction has influence on reaction rate. The model shows also that catalyst profile inside the reactor is a critical parameter for reactor design.

(a)

(b)

(c) (d)

Figure 4: (a): Average axial solid volume fraction; (b): Temperature contours under different solid profiles; (c): Average EB conversion; (d): Average Styrene selectivity.



References

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[7] Derouin, C., Nevicato, D., Forissier, M., Wild, G. and Bernard, J.R. “Hydrodynamics of riser units and their impact on FCC operation”, Ind. Eng. Chem. Res. Vol 36, 4504-4515, 1997

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[10] Elnashaie, S.S.E.H., Abdallah, B.K., Elshishini, S.S., Alkohwaiter, S., Noureldeen, M.B. and Alsoudani, T. “On the link between intrinsic catalytic reactions kineics and the development of catalytic processes: Catalytic dehydrogenatin of ethylbenzene to styrene”, Catalysis Today, Vol 64, 151-162, 2001

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CFD simulation of a stratified flow at the inlet of a compact plate heat exchanger

M. Ahmad, J. F. Fourmigue, P. Mercier & G. Berthoud Commissariat à l’Energie Atomique, Grenoble, France

Abstract

The work described in this paper considers 3D CFD (computational fluid dynamics) simulations of an adiabatic stratified liquid-vapor flow at the inlet of a compact plate heat exchanger using the commercial CFD code “FLUENT” and an in-house code “NEPTUNE 3D” developed by CEA and EDF. An experimental loop is built up that represents a compact plate heat exchanger in which the liquid and vapor flow rates in the different channels are measured and the flow inside the cylindrical distributor of diameter greater than that of the inlet tube can be visualized. The numerical predictions showed the good agreement with the experimental measurements. The interfacial shear stress was calculated in a steady stratified flow and compared with the computed shear stresses by the two codes. Keywords: two-phase flow, maldistribution, CFD simulation, separated phase model, compact heat exchangers.

1 Introduction

Mal-distribution of two phase flow is the main cause of the deterioration of the thermal and hydraulic performances of heat exchangers and it is mainly affected by the flow pattern at the condensers or evaporators inlets (Ahmad et al [1]). The purpose of our study is to find a modeling tool capable of simulating some possible two phase flow patterns in compact heat exchangers. One of the most current flow patterns observed in the inlet tube of a heat exchanger is the stratified flow. Few authors have tested the CFD simulation models to study the distribution in heat exchangers. Jones and Galliera [4] used FLUENT to simulate single-phase flow distribution in the manifold that performed well in calculating the



doi:10.2495/MPF070081

larger scale features of branching and manifold flows. Lalot et al [5] used the computer code STAR-CD to study the flow maldistribution in an electric heater. Zhang and Li [7] used FLUENT to predict the flow distribution in plate-fin heat exchangers. Fei [2] used the mixture model implemented in FLUENT to simulate a two-phase flow distribution in the header. CFD simulation technique can provide the flexibility to construct computational models that can be easily adapted to a wide variety of physical conditions without constructing a large scale prototype or expensive test rigs. In our study, a stratified liquid-vapour flow in a compact heat exchanger is simulated using the commercial CFD code “FLUENT” and an in-house code “NEPTUNE CFD” developed by CEA and EDF. The interfacial shear behavior has a great influence in this kind of configurations. Therefore, the simulations were first carried out using the single-fluid approach in combination with the VOF model implemented in FLUENT and then the two phase approach of NEPTUNE CFD with the adaptation of a dispersed flow shear force model for a separated-phases flow. The numerical results of the two codes are compared with the experimental data.

2 Study of the flow distribution

2.1 Experimental apparatus

A distribution experimental loop representing a compact plate heat exchanger was built up. The test section consists of a horizontal manifold and eight parallel downward branches (Figure 1).

Figure 1: Test section.

The manifold is 127 mm long and its diameter is 50 mm. It is horizontally supplied by a 17.3 mm in diameter and 100 mm long glass pipe to visualize the two phase flow at the header inlet with a 1,500 mm tube made of stainless steel of the same diameter. The end of the manifold is closed by a transparent



polycarbonate plug. Each branch is 2 × 50 mm rectangular. The channels are regularly 10 mm spaced along the manifold. Figure 2 shows an isometric view of the test section.

2.2 CFD models

In the two codes, the conservation equations of mass and momentum are solved using the finite volume method. The used mesh involved 250,000 cells (figure 3) making a compromise between the accuracy in representing the physical phenomena needing refined grid and the cpu-time. An average necessary cpu-time for each treated case varies from ten days with FLUENT to three weeks with NEPTUNE 3D. In FLUENT, an automatic meshing allows the grid refinement at the interface level depending on the gradient of the void fraction and thus can decrease the numerical diffusion in the value of the void fraction in this zone. The turbulence was modelled using the RNG-based ε−k turbulence model in FLUENT [3] and

ε−k EDF model [6] in NEPTUNE.

Figure 2: An isometric cross section of the header.

2.2.1 FLUENT-VOF The VOF model implemented in FLUENT is a one-fluid model. It relies on the fact that the two phases are not interpenetrating. The volume fraction of each phase is calculated in each computational cell. The variables and properties of each cell are either representative of one phase or representative of the mixture of the two phases, depending on the volume fraction value. The tracking of the interface between the two phases is accomplished by the solution of the continuity equation for the volume fraction of one of the phases.



Figure 3: Over view of the mesh.

0. =∇+∂

∂q

q vt

αα

(1)

qα is the volume fraction of phase q and v is the vector velocity. A single momentum conservation equation is solved through out the domain and the resulting velocity field is shared among the two phases. The momentum equation, shown below, is dependent on the volume fraction of all phases through the properties ρ and µ .

( ) ( ) ( )[ ] FgvvPvvvt

T ++∇+∇∇+−∇=∇+∂∂ ρµρρ .. (2)

where P is the pressure, qq ραρ ∑= is the average mass density, qqµαµ ∑= is

the average dynamic viscosity and F is the surface tension force.

2.2.2 NEPTUNE 3D-two fluid model The balance equations of the two fluid model where the Navier–Stokes equation apply for each phase can be written: • Two mass balance equations:

0).()( =∇+∂∂

qqqqq vt

ραρα (3)

• Two momentum balance equations:

( ) ( )[ ] qiqqqqqqqq

qq MgPvvt

v++∇−+∇=

∇+

∂

∂ραατταρα Re.. (4)



qτ is the shear tensor and qiM is the interfacial momentum transfer term that accounts for mass transfer, drag force, added mass, lift,... In our treated cases, the drag force is the only considered interfacial force, thus iqiqi aM τ= , where qiτ is

the interfacial friction per unit area and ia is the volumetric interfacial area. • Drag force The interfacial drag force is calculated with the adaptation of a dispersed flow shear force model to compute a separated-phases flow. The interfacial drag transfer term between phases p and q has the following form: ( ) ( ) pq

rDqpqp VFI αα−=→ (5)

DF , drag coefficient between phases p and q. pqrV , relative velocity between

phases p and q, α is the volumetric fraction. The drag coefficient is written in terms of the particle drag-relaxation time pτ . In the following notation, p represents the dispersed phase and q represents the continuous phase:

pq

pDF

ταρ 1

= , p

pqr

p

D

p

q

p

VdC

><=431

ρρ

τ, [ ]687.0Re15.01

Re24

pp

DC += (6)

The model considers either dispersed gas (vapour V) bubbles in a continuous liquid (L) flow, or dispersed liquid droplets in a continuous gas (vapour) flow with regard to the volumetric fraction.

- Bubbly flow ( )3.0<Vα VL

VbubbleDF

ταρ 1

= , (7)

- Misty flow ( )7.0>Vα LV

LdropletDF

ταρ 1

= (8)

- Mixing ( )7.03.0 << Vα :

( ) ( )7.03.07.03.03.0

3.07.07.0 droplet

DVbubble

DV

D FFF

−−

+

−−

=αα (9)

we can note that for this closure, two characteristic diameters have to be defined. In our case of liquid-vapour separated flow (stratified), the volumetric fraction ( )pα tends to zero (or a residual value) in the two single phase zones, and included between 0.3 and 0.7 (arbitrary) in the interface zone.

2.3 Results and discussion

The validation of the computed results is built up basically on the comparison between the numerical predictions of the two-phase distribution in the different channels and the experimental data of the treated cases. In the presentation of the



results of the two-phase distribution measurements, the non-dimensional liquid (resp. vapour) flow rate in channel i is the ratio of the liquid (resp. vapour) flow rate measured in the channel over the mean liquid (resp. vapour) flow rate:

∑=

=8

1k

k*k

8j

j

ii

M

MM

(10)

where =k (liquid) or υ=k (vapour). Two different cases are treated corresponding to a stratified flow at the header inlet. In the first case, a total inlet flow rate of 86 kg/hr (average superficial mass velocity is 100 kg/m²s) is imposed with a mass quality of 20%. In the second case, we increased the flow rate to 129 kg/hr (average superficial mass velocity is 150 kg/m²s) with the same mass quality and void fraction. A homogeneous mean velocity profile ( )KV is fixed for each phase in the boundary section.

)1()1(

αρ −−

=L

TL

xGV , αρV

TV

xGV = (11)

TG (kg/m².s) is the average inlet superficial mass velocity, Lρ and Vρ are respectively the mass densities of liquid and vapour, x is the mass quality and α is the void fraction calculated using the model of Lockhart-Martinelli for a steady state two-phase flow. Being in an instationary flow, comparisons requested averaged time values for computed and experimental date.

Figure 4: Comparison between numerical predictions and experiment (case1: G=100 kg/m²s, x=20%).

2.3.1 Case 1 Figure 4 shows that both numerical predictions and experimental data have a similar distribution profile for the liquid phase. The liquid flow rates in the channels 2 and 4 are under-predicted by the two solvers. In last four channels, the computed liquid flow rates are higher than the measured values. FLUENT-

Distribution of liquid phase

0

0,5

1

1,5

2

2,5

3

3,5

1 2 3 4 5 6 7 8Tube

Liqu

id fl

ow fr

actio

n Experimental resultsFluentNeptune

Distribution of vapour phase

0

0,5

1

1,5

2

2,5

3

1 2 3 4 5 6 7 8Tube

Vapo

ur fl

ow fr

actio

n Experimental resultsFluentNeptune



VOF solver seems to perform better than the code NEPTUNE in the last four channels, whereas Neptune gives better results in channel 3. The prediction of the vapour distribution in figure 4 shows a maximum discrepancy of 60% of the mean vapour flow with FLUENT in channel 7 and 70% of the mean vapour flow rate with NEPTUNE in channel 4 when compared to experimental measurements.

Figure 5: Contours of computed volume fractions and visualizations (case 1: G=100 kg/m²s, x=20%).

The liquid vapour interface traced by the two codes as illustrated in figure 5 shows almost the same shooting length after the inlet section enlargement. Figure 5 also shows in the right side, the visualization of the instationary flow at two instants 1t and 2t . At 1t , it shows the shooting point at channel 3 level as predicted by the two codes and at 2t , it shows the liquid jump at the header cap. The interfacial shear forces did not seem to have a significant influence in this part of the flow compared to the inertia force. However, NEPTUNE slightly over estimated shear forces (see section 3) which resulted in a slightly higher liquid jump after the impact than that was shown by visualizations (instant 2t ).

2.3.2 Case 2 In the second case, the total inlet flow rate was increased, mass quality was fixed and thus the void fraction was the same but with higher liquid momentum. The two codes also give a similar distribution profile for the liquid phase (figure 6). The uncertainty of FLUENT-VOF code prediction is less than 25% of the mean liquid flow rate except for channels 3 and 4 where it reaches 60%. The prediction of Neptune code reaches 200% of the mean liquid flow rate in the last channel.



The vapour distribution is better predicted by FLUENT code. The error in the prediction is less than or equal to 25% of the mean vapour flow rate except for the last channel where the difference is estimated to 45%.

Figure 6: Comparison between numerical predictions and experiment (case 2: G=150 kg/m²s, x=20%).

Figure 7: Contours of volume fractions and visualizations (case2: G=150 kg/m²s, x=20%).

The computed mass and velocity of the liquid after channel 8 are much higher in NEPTUNE code which is illustrated by a much higher liquid jump after the impact when compared with visualizations (figure 7). This discrepancy might be explained by the different grid resolution used in the two codes and the important shear force calculated in NEPTUNE.

Distribution of liquid phase

0

0,5

1

1,5

2

2,5

3

1 2 3 4 5 6 7 8Tube

Liq

uid

flow

frac

tion

Experimental resultsFluentNeptune

Distribution of vapour phase

0

0,5

1

1,5

2

2,5

3

1 2 3 4 5 6 7 8Tube

Vap

our

flow

frac

tion

Experimental resultsFluentNeptune



3 Calculation of the average interfacial shear in a steady stratified flow

In this section, a model was carried out to estimate the average of the interfacial shear stress in a steady stratified flow between two parallel plates. The velocity profiles were defined for the two phases using the one-seventh law for a turbulent flow. The vapour velocity is defined in two zones, the first zone is between the wall and the point of maximum velocity value (zone of thickness b) and the second zone is between the liquid-vapour interface and that point of maximum velocity value (zone of thickness a) as shown in figure 8.

Figure 8: Liquid-vapour flow in a steady state in a parallel plate channel.

The equilibrium between the pressure gradient motive force and the boundary shear forces can be applied on each phase as well as on both phases. The pressure drop in the liquid and in the vapour phase is the same as it is in the two phases, so we can write:

Del

p LpVp

L

iLp ττττ +=

−=

∆∆ (12)

iτ is the average value of the interfacial shear stress, between liquid and vapour

phases in a steady state flow. Lpτ and Vpτ are respectively the averaged shear stresses between the liquid and the vapour with wall. D is the thickness of the channel and Le is the thickness of the liquid phase, and thus we can write:

)1( α−= DeL (13)

α being the void fraction that can be calculated using the model of Lockhart-Martinelli: Dba α=+ (14)

To describe the velocity profiles of both phases, the one seventh-law of a turbulent flow was used. For the vapour, this law was applied in the two zones that are joined at the point of the same maximum vapour velocity.

71

2

74

71

1 74.8 yUV

VpVV

=

−

ρτ

ν , by << 20 (15)



i

V

iVV uyU +

=

−71

1

74

71

2 74.8ρτν , ay << 10 (16)

At the point of maximum vapour velocity, equality between equations (15) and (16) is attained, at ay =1 and by =2 , we have:

⇒= (max)2(max)1 VV UU

717

4

71

74.8 bV

VpV

−

ρτ

ν = i

V

iV ua +

−717

4

71

74.8ρτν (17)

The vapour flow rate can be expressed as a function of the mean vapour velocity as follows: )( LVV eDVQ −= (18)

By integration and summation of equations (15) and (16), we can deduce the vapour flow rate:

( ) aubaeDVQ iVpiVVLVV +

+=−=

−−74

78

74

78

74

71

6125.7. ττρν (19)

Similarly, using the one-seventh law we can describe the liquid velocity profile:

717

4

71

74.8 yUL

LpLL

=

−

ρτ

ν , Ley <<0 (20)

The interfacial velocity can be then deduced from equation (20):

717

4

71

74.8 LL

LpLi eu

=

−

ρτ

ν (21)

As for vapour phase, by integration of equation (20), we can get the liquid flow rate function of the mean liquid velocity:

787

4

71

6475.7. LL

LpLLLL eeVQ

== −

ρτ

ν (22)

Knowing the hydraulic diameter of the plate, the mass quality and inlet flow rate, the resolution of the system of equations gives the average steady state value of the interfacial shear force. Figure 9 represents the comparison between the calculated value of the shear force in a channel having the same hydraulic diameter (17.3mm) and initial boundary conditions as in case 1 (section 2.3) and the computed shear forces. The shear force value of FLUENT-VOF solver in the interface cell is computed using an average dynamic viscosity value between the two phases. Increasing the bubble characteristic diameter decreases the NEPTUNE computed shear force value and decreases the error of the prediction.



Figure 9: Comparison between model and computed values of shear forces.

4 Conclusion

A stratified liquid-vapour flow at the inlet of a compact heat exchanger was simulated using “FLUENT” and “NEPTUNE” CFD codes and experimental measurements have been carried out. The profile distributions of both phases in all channels were rather well predicted by the two codes. However FLUENT-VOF solver seems to perform better than NEPTUNE in the case of high liquid inlet momentum (case 2). This error in the prediction by NEPTUNE might be due to the insufficient grid resolution and the over-estimation of shear force of dispersed flow model.

References

[1] Ahmad, M., Mercier, P. and Berthoud, G., Experimental study of two-phase distribution in a compact plate heat exchanger, Inter Conf, Heat SET, GRETh/CEA, Chambery 2007.

[2] Fei, P., Adiabatic developing two-phase refrigerant flow in manifolds of heat exchangers, thesis of University if Illinois at Urbana Champaign, 2003.

[3] FLUENT 6, Dec. 2001 User Guide. Fluent Inc. [4] Jones, G. F. and Galliera, J. M., Isothermal Flow Distribution in Coupled

Manifolds: Comparison of Results from CFD and an Integral Model, Proc. ASME Int. Cong. and Exhib., Anaheim, CA, 1998.

[5] Lalot, S., Florent, P. and Lang, S. K., Flow maldistribution in heat exchangers. Appl. Thermal Eng. 26, pp. 847-863, 1999.

[6] Lavieville, J. Bouker, M. and al., NEPTUNE CFD V1.0, Theory Manual, EDF R&D, Chatou, 2006.

[7] Zhang, Z. and Li, Y. Z., CFD simulation on inlet configuration of plate-fin heat exchangers, School of Energy and Power Engineering, Xi'an 710049, China, 2003.

0,1

1

10

0 20 40 60 80 100 120 140 160 180 200

Distance from the inlet (mm)

inte

rfac

ial s

hear

forc

e (N

/m²)

Fluent-VOF ModelNEPTUNE d10mm NEPTUNE d20mm

Entry effect

Steady state



Numerical computation of a confinedsediment–water mixture in uniform flow

L. Sarno1, R. Martino2 & M. N. Papa3

1 Private Consultant, Naples, Italy2 Department of Hydraulic and Environmental Engineering ‘G.Ippolito’,Naples University, Italy3 Department of Civil Engineering, Salerno University, Italy

Abstract

The purpose of this paper is to simulate a laminar mud flow confined in anarrow rectangular open channel. The flow bed is an erodible layer made upof the same material involved in the flow; the equilibrium condition betweenthe moving and non-moving layer is assumed. The mud mixture under study isruled by the Herschel–Bulkley’s (H–B) shear thickening rheological law. It issupposed that the local volumetric concentration is linearly increasing with thedepth and it is constantly equal to its maximum value where the local velocityis smaller than a threshold value. Relations among rheological parameters andconcentration have been obtained through laboratory rheometric tests. Turbulenceeffects and Coulombian stresses have been ignored. The momentum equation hasto be integrated along the flow cross section for the flow velocity to be obtained.Unfortunately, it is very difficult to integrate this equation using H–B rheologicallaw, since there are different stress functions and it is not possible to know a priorithe sub-domains of them (plug, non-plug and bed regions). In the present worka modified rheological law, continuous over the whole domain of integration isemployed and the momentum equation is numerically integrated. This modifiedlaw has been obtained by adding a constant correcting the denominator in the H–Bstress functions. Therefore, there are no longer any dead zones or plug regions.However it is noteworthy that, using a small constant, the model produces agood simulation of plug and dead zones: i.e. the velocity gradient is very smallthere. The mathematical model has two parameters: maximum concentration andthreshold velocity. These parameters have been adjusted by back-analysis withmeasurements from laboratory flume experiments in uniform flow conditions.Keywords: mud flow, Herschel–Bulkley rheological law, equilibrium, plug.



doi:10.2495/MPF070091

1 Introduction

Mud flows are very dangerous for anthropic settlements and so, during the lastyears, they have been widely studied in environmental engineering. Mud flow ischaracterized by motion of a two-phase mixture, consisting of water and high-concentrated fine-granulometry solid matter; therefore, its mechanical behavioris solid-like while the acting shear stresses are smaller than a fixed yield stress,and it is similar to a non-Newtonian fluid when the acting stresses are bigger.Researcher aim at obtaining a resistance law correlating the flow rate and the flowdepth, taking into account both the natural mixtures and irregular-shape flow crosssections.

The present work presents a computational model to study the mud flowunder some simplifying hypotheses. The numerical computation was performedin MATlab environment, by implementing a finite-difference method.

2 Constitutive law and definitions

There are two ways to approach the mechanical problem;

1. considering independently solid and liquid phases2. using an equivalent fluid model with a rheological law, which, taking into

account all the modes of resistance inside the mixture, relates the shearstress τ with the shear rate γ.

In this work the problem is tackled according to the second approach, i.e. arheological law, where parameters depend on the local volumetric concentrationof solid matter, has been used. The mixture ability to support shear stress dependson the relative distance existing between solid particles and, consequently, on thesolid concentration.

The motion, under study, takes place in a rectangular-shaped open channel.Uniform and laminar conditions have been assumed: that means the velocity isa scalar function defined over the flow cross section. The motion develops onerodible layer, consisting of the same solid matter which is in the mixture, andin equilibrium condition: there is a dynamic equilibrium between solid depositand particles at motion inside the flow. Therefore, at equilibrium there is a zone,called dead zone, where solid matter is at rest. To tell the cross section sub-domain,where velocity is non-null, from the whole cross section, the first one will becalled “active cross section” and the other one simply “cross section”. The velocityfunction, in equilibrium condition, is marked out by null gradient at the boundary,between active flow cross section and dead zone.

The following anticlockwise system of axes has been assumed as referenceframe: the x axis is parallel to the motion direction and so it is perpendicular to thecross section, y is perpendicular to the erodible bed and it lies on the cross section,z is parallel to the absolute bed plane and it lies on the cross section.

A scheme of the channel and the reference frame is reported in fig. 1.



Figure 1: View of xy plane, z-axis is perpendicular to the sheet. It is shown thatusually erodible layer slope is different from flume bed slope. The dashedline represents the cross section.

The mixture considered is ruled by Herschel–Bulkley’s law, which presents thefollowing one-dimensional form:

τ = τB + µ

(du

dn

)η

if τ > τB (1)

du

dn= 0 if τ ≤ τB (2)

where u is the velocity, τ the shear stress, τB the yield stress, µ the apparentviscosity. The second form is due to the fact that, when acting stress is smallerthan yield stress, the behavior of mixture is solid-like.

In steady conditions, everywhere the acting shear stress is equal to the resistantone.

The general expression of H–B’s law [1], valid for three-dimensional problems,is the following one:

T− pI =τBD√−DII

+2ηµD√−D1−η

II

if τ > τB (3)

D = 0 if τ ≤ τB (4)

where T is the stress tensor, I the unit tensor, D the strain rate tensor, DII thesecond invariant of the secular equation associated with tensor D.

There are two different rheological forms: the second one postulating that,where acting stress is smaller than yield stress τB , there is no strain, i.e. the wholeshear stress is supported by the solid matter. The presence of a plug region, wherethe velocity vector is constant, set on the top of the flow, close to the free surface,is a direct consequence of the above.

Fig. 2, reproduced from [2], shows an example of a typical velocity distribution,where dead and plug zones can be seen.



Figure 2: Velocity distribution, observed in laboratory flume experiments: deadand plug zones are evident. [2].

Rheological parameters τB and µ strongly depend on the local concentration,but η is only dependent on the chemical-physical nature of the solid suspension.For solid matter used in laboratory flume experiments, coming from the area ofSarno (Italy), the following fitting forms have been obtained, through rheometrictests:

τB = 0.0589 · e12.071·c, µ = 0.0020 · e9.382·c, η = 1.722 (5)

where c is the local volumetric concentration. This dependence is exponential,therefore a good estimation of c is essential.

In this paper three kinds of concentration have been used: volumetriclocal concentration c, mean concentration cm over the cross section and flowconcentration ct. They are defined in the following expressions:

c(x) := limV →0

Vs

Vs + Vw, cm :=

∫Ω

c(x) dA

Ω, ct :=

∫Ω

c(x)u(x) · n dA∫Ω

u(x) · n dA

(6)

where Vs the volume taken up by the solid suspension, Vw the volume of water.Ω is the cross section domain, u(x) the velocity at x point, n the unit vector normalto the cross section.

The local concentration is defined over the whole section, whereas the meanconcentration and the flow concentration are features of the entire motion. Theflow concentration means also the ratio between the solid flow rate and the totalone.

Projecting on x-axis eqn (3), the following expressions have been obtained:

τyx =

τB + µ

[(∂u

∂y

)2

+(

∂u

∂z

)2] η

2

√√√√[(∂u

∂y

)2

+(

∂u

∂z

)2] ∂u

∂y(7)



τzx =

τB + µ

[(∂u

∂y

)2

+(

∂u

∂z

)2] η

2

√√√√[(∂u

∂y

)2

+(

∂u

∂z

)2] ∂u

∂z, (8)

where τzx is the x-axis component of the shear stress vector, acting on the surfacewith normal z, and similarly τyx is the x-axis component of the one, acting on thesurface with normal y.

3 Differential problem

Momentum equation, valid everywhere over the cross section domain, can bewritten as:

ρ (g − u) = ∇ ·T (9)

where u is the Lagrangian acceleration and g the gravity constant.The x-axis component of eqn (9) can be written as follows:

ρg sin θ +∂τzx

∂z+

∂τyx

∂y= 0, (10)

where ρ is the mean density (ρ = c ρsolid +(1 − c) ρwater) and θ is the flow slope,usually different from that of non-erodible layer, which lies below.

If velocity boundary conditions and local concentration distribution wereknown, since functions τzx and τyx depend on u because of eqns (7) and (8), itwould be possible to solve numerically the differential problem, associated to eqn(10), for the only function u(y, z).

The domain of integration was a reference cross section, arbitrarily chosen inthe whole flume. It is assumed that solution does not vary with total flow depth, ifCoulombian stresses can be ignored.

3.1 Corrective term ε2

The main difficulty in integrating eqn (10) derives from its being a free-boundaryproblem: the size of plug sub-domain and the velocity value in the plug cannotbe a priori fixed. Besides, eqn (10) is not defined in the plug. To overcome theproblem, the functions (7) and (8) were replaced by the following ones, which aredefined and continuous over the whole cross section:

τyx =

τB + µ

[(∂u

∂y

)2

+(

∂u

∂z

)2] η

2

√√√√[ε2 +(

∂u

∂y

)2

+(

∂u

∂z

)2] ∂u

∂y(11)



τzx =

τB + µ

[(∂u

∂y

)2

+(

∂u

∂z

)2] η

2

√√√√[ε2 +(

∂u

∂y

)2

+(

∂u

∂z

)2] ∂u

∂z. (12)

The critical effect, due to the ε2, is the absence of yield stress, therefore thereare not dead or plug zones any more [3]. The constant ε2 should be as small aspossible, for a Bingham fluid in [4] a value smaller than 10−16 is recommended.Obviously, the smaller is ε2, the closer come the expressions (11) and (12) to theoriginal H–B’s law and the more the u solution will have a zone, where the velocityis almost null with its gradient, and a zone with almost constant velocity. In thispaper it has been used an ε2 = 10−3.

3.2 Concentration distribution

Unfortunately, it is not yet possible to obtain a reliable estimation of localconcentration c through experimental measures, therefore hypotheses about itshould be formulated. Undoubtedly, c is increasing with the depth and there aresome experimental results that confirm a nearly linear trend of c at solid boundary[5]. A good estimation of c is essential, since rheological parameters depend on it.

In this work following hypotheses have been assumed:• concentration linearly increasing with the depth: c = c0 + k y;• existence of a maximum packing value of concentration cmax, independent

of parameters of motion which vary from case to case (e.g. slope, cm);• existence of a threshold value of velocity uthr, under which concentration

is equal to maximum packing value.Values between [0.66 − 0.69] for cmax and between

[10−4 − 10−3 m/s

]for

uthr were tried.

3.3 Boundary conditions

The following boundary conditions have been assumed:• no-slip condition, that is null velocity everywhere at solid boundaries of

channel;• null shear stress at free surface.

The model could be tested also in slip condition at side solid border, whichseems to be more realistic, but it is very difficult to obtain a reliable experimentalestimation of velocity there.

4 Numerical implementation

A finite-difference discretization of differential problem was performed. First-order derivative of u in eqns (11) and (12) were replaced by their centralapproximations.



The integration domain was discretized in rectangular-shaped cells, of size ∆z×∆y. Continuous functions u, ρ become discrete functions, pertinent to the centreof gravity of cells. Therefore they have been implemented as matrices. Similarlyfunctions τyx and τzx are implemented as matrices, with the following convention:τzx(i, j) is the shear stress acting over the right-hand side of the cell (i, j), andτyx(i, j) is the shear stress acting over the lower side of the cell (i, j). Stresses areregarded as positive when concordant with x-axis.

A scheme of conventions about τ stresses is reported in fig. 3.Therefore, instead of a momentum equation, a forces balance can be written for

the generic cell (i, j):

[gρ(i, j) sin θ] + [τzx(i, j) − τzx(i, j − 1)] /∆z+

[τyx(i, j) − τyx(i − 1, j)] /∆y = 0. (13)

Figure 3: The convention assumed for τ stresses.

There are as many equations as are the cells, and so, as are unknown variablesu(i, j). A 64 × 64 cells discretization has been used: the differential problemhas been changed in a non-linear system of 4096 equations. Boundary conditionshave been implemented, by using ghost null values of u at solid boundary and byimposing τyx = 0 at free surface.

To solve the non-linear system, it was used “Fsolve”, which is a trust-regionalgorithm, included in Optimization Toolbox [6]. Solution tolerance was set to10−6. To improve performances, the symmetry of the problem was exploited (forthe non-linear system to have only 2048 equations) and it was used a patternmatrix, which informs the computer about zeros, in order to obtain a fastercomputation of Jacobian matrix.

4.1 Loops to define c matrix and convergence of algorithm

Every time the algorithm solves the differential problem, a guessing distributionof c is assumed. The computation finishes when the solution is congruent with itsdistribution of c. It is possible to state two different congruence conditions:



• “threshold velocity condition”, which is verified when the velocity solutionis smaller than threshold velocity in cells where c = cmax and only in them;

• “flow concentration condition”, which is verified when cf , calculated sincethe differential problem for u has been solved, is equal to cf to simulate.

The second condition is used to assure that the specific motion observed inlaboratory flume experiments is simulated and not any other.

Outside of the code which solves the differential problem, there are two nesteddo-while loops, responsible to verify the congruence conditions: the outer one ispertinent to the “flow concentration condition”, the inner one to the “thresholdvelocity condition”. The hypothesis that cm is constant in every columns of thecross section is assumed. Therefore, to fix a local concentration distribution, thereare n + 1 freedom degrees, where n is the number of columns (in the case of thiswork 64): a degree is cm and the n others ones are the packing positions in eachcolumn, i.e. the positions where c becomes equal to its maximum value cmax.

The flowchart of algorithm is reported in fig. 4.Having obtained the u solution, the flow rate Q, which is useful for the analysis

of results, has been calculated.

Figure 4: Algorithm flowchart.



Due to loops outside of differential problem, which can be regarded as Turing-computable functions but not analytical, it is very difficult to obtain a strict proof ofconvergence, by using spectral methods. An heuristic way has been tackled: it hasbeen observed that even increasing the discretization level (up to 160 × 160 cells)the solution weakly changes. Furthermore, the solution seems to be independentof first guessing concentration distribution [7].

5 Analysis of results and conclusions

Velocity and local concentration distributions, obtained by the simulation, arereported in fig. 5. There is a discontinuity of the first type in c distribution: itis, of course, a loophole of the model, on which future studies will be focused.

Figure 5: Velocity and local concentration distributions obtained by the simulation:constant velocity zones are highlighted.

Although a modified rheological law with a rather big value of ε2 (10−3) hasbeen used, a velocity distribution with well defined dead and plug zones can beeasily seen (fig. 5): in other words, the integration method seems to be suitable forthis kind of problems.

The expected boundary points of plug and dead zones, for the generic columnj, occur when the total shear stress is equal to the yield stress:√

τ2yx + τ2

zx = τB(c). (14)

It is very interesting, now, to compare boundary points of constant velocityzones, which can be observed in velocity distribution after simulation, with theexpected boundary points, for each column of integration. The remarkable resultof this work is that, everywhere in the integration domain, one can notice anencouraging correspondence between these points. Fig. 6 reports the comparisonat middle column of the flume. That is a further confirmation that this modified-rheology numerical method seem to be working, even with high values ofconstant ε2.



Figure 6: Comparison among total stress, yield stress, velocity distribution andlocal concentration.

The model needs two parameters, cmax and uthr, not yet obtainable by directmeasurements. So they have to be fixed, by a back-analysis of some laboratoryflume experiments.

In future research simulated flow rates with experimental measures will becompared and the condition of threshold velocity will be improved, with thecontribution of further experimental results. Moreover, it will be interesting torun the model in slip condition at solid boundary and, hopefully, to implementCoulombian stresses, which seem to be not totally negligible in the dead zones.

References

[1] Macosko, C.W., Rheology - Principles, Measurements and Applications.Wiley-VCH: New York, 1994.

[2] Dalrı, C., Fraccarollo, L., Larcher, M. & Armanini, A., Analisi sperimentaleper la caratterizzazione del flusso di miscugli iperconcentrati di originesintetica e naturale. Proceedings of XXIX Convegno Nazionale di Idraulicae Costruzioni Idrauliche – IDRA, Trento, 2004.

[3] Whipple, K.X., Open channel flow of bingham fluids: application in debris-flow research. Journal of Geology, 105, pp. 243–263, 1997.

[4] Bercovier, M. & Engleman, M., A finite element method for incompressiblenon-newtonian flows. Journal of Computational Physics, 36, pp. 313–326,1980.

[5] Martino, R. & Papa, M.N., Effetto delle pareti nelle correnti detritiche: primirisultati. Proceedings of XXIX Convegno Nazionale di Idraulica e CostruzioniIdrauliche – IDRA, Trento, 2004.

[6] AA.VV., Optimization toolbox – Trust-region methods for nonlinearminimization. Matlab 70 R14 Help, 2004.

[7] Sarno, L., Simulazione di una colata di fango in regime visco-plastico econdizioni di moto uniforme. Degree thesis – Naples University, Naples, 2006.Supervisor: Martino R.



Experimental validation of multiphase flow models and testing of multiphase flow meters: a critical review of flow loops worldwide

O. O. Bello1, G. Falcone2 & C. Teodoriu2 1Clausthal University of Technology, Institute of Petroleum Engineering, Germany 2Texas A&M University, Department of Petroleum Engineering, USA

Abstract

Around the world, research into multiphase flow is performed by scientists with hugely diverse backgrounds: physicists, mathematicians and engineers from mechanical, nuclear, chemical, civil, petroleum, environmental and aerospace disciplines. Multiphase flow models are required to investigate the co-current or counter-current flow of different fluid phases under a wide range of pressure and temperature conditions and in several different configurations. To compliment this theoretical effort, measurements at controlled experimental conditions are required to verify multiphase flow models and assess their range of applicability, which has given rise to a large number of multiphase flow loops around the world. These flow loops are also used intensively to test and validate multiphase flow meters, which are devices for the in-line measurement of multiphase flow streams without separation of the phases. However, there are numerous multiphase flow varieties due to differences in pressure and temperature, fluids, flow regimes, pipe geometry, inclination and diameter, so a flow loop cannot represent all possible situations. Even when experiments in a given flow loop are believed to be sufficiently exhaustive for a specific study area, the real conditions encountered in the field tend to be very different from those recreated in the research facility. This paper presents a critical review of multiphase flow loops around the world, highlighting the pros and cons of each facility with regard to reproducing and monitoring different multiphase flow situations. The authors suggest a way forward for new developments in this area. Keywords: multiphase flow loop, multiphase flow modelling.



doi:10.2495/MPF070101

1 Introduction

Multiphase flow can occur in pipes as well as in porous media. The focus of this paper is on the former, although it will be shown that more research on the interaction between multiphase flows in porous media and those in pipes is needed, both theoretical and experimental. Multiphase flows consist of the simultaneous passage through a system of a stream composed of two or more phases. They are very common natural phenomena: the flow of blood in our body, the rising gas bubbles in a glass of beer and the steam condensation on windows are all examples of naturally occurring multiphase flows. However, it the large scale multiphase flows, such as those that occur in the petroleum industry, on which this paper will focus. For example, in a typical oil and gas development, multiphase flow is encountered in the wells, in the flow lines and risers transporting the fluids from the wells to the platform and in the multiphase flow lines that carry the produced fluids to the treatment facilities at shore. Multiphase flow systems can be very complex, due to the simultaneous presence of different phases and, usually of different compounds in the same stream. Thus, the development of adequate models presents a formidable challenge. The combination of empirical observations and numerical modelling has proved to enhance the understanding of multiphase flow. Models to represent flows in pipes were traditionally based on empirical correlations for hold-up and pressure gradient, but it is more usual nowadays to use codes based on the multi-fluid model, in which averaged and separate continuity and momentum equations are written for the individual phases. For these models, closure relationships are required for interface and pipe wall friction. To compliment the theoretical effort, experimental measurements under controlled conditions are required to verify multiphase flow models and assess their range of applicability. This is why there exists a large number of multiphase flow loops around the world, each of them with specific capabilities and limitations. This paper attempts to review all the major world-wide facilities that allow a wide range of two- and three-phase flow experiments, but the authors accept that their review may not be exhaustive. Flow loops may be operated by academic organisations, independent research centres or individual companies and there is a special category for oil and gas applications, where real hydrocarbon fluids and field operating conditions are used. The review is based on information available in the public domain and focuses on large scale facilities. This choice reflects the specific need for multiphase flow loops for studies related to hydrology, petroleum and environmental engineering, geothermal energy plants, underground gas storage and CO2 sequestration. For studies on nano-technology, life science and medical systems, different flow loops are necessary to reproduce “reality” in a laboratory.



Finally, there are ad hoc facilities for the investigation of boiling and condensation processes and for nuclear engineering applications. No flow loop can be representative of all possible situations. Even when experiments in a given flow loop are believed to be sufficiently exhaustive for a specific study area, the conditions that will be encountered in a real application can be very different from those recreated in the research facility. The objective of this paper is therefore to review some of the major world-wide flow loop facilities for two- and three-phase flow investigation that are reported in the public domain, to point out unresolved problems in reproducing real processes in a laboratory environment.

2 Multiphase flow

The phases present in a multiphase flow are: solid, which is incompressible and has non-deformable interfaces with the surrounding fluids; liquid, which is relatively incompressible, but has deformable interfaces with the other phases; and gas, which is compressible and deformable. The different phases of a multiphase flow may consist of different chemical substances. It is also possible to have the two phases of a two-phase flow made of the same pure component. Two-phase flows include: (1) Gas-solid flows, where solid particles are suspended in gases. (2) Liquid-liquid flows, as in oil-water emulsions in pipelines. (3) Liquid-solid flows, where solids are suspended in liquids. (4) Gas-liquid flows, which is the flow most widely found in industrial applications. Three-phase flows include: (1) Gas-liquid-solid flows, as in froth flotation for the separation of minerals. (2) Gas-liquid-liquid flows, as in natural gas-oil-water. (3) Solid-liquid-liquid flows, as in sand-oil-water. Four-phase flows are the most difficult case and include: (1) Liquid-liquid-gas-solid, as in oil-water-gas-sand (or asphaltenes or hydrates) mixtures. Flow regimes. The shape and behaviour of the interfaces between phases in a multiphase mixture dictate what is referred to as “flow regime” or “flow pattern”. There are competing forces or mechanisms occurring within the multiphase fluid at the same time. The balance between them determines the flow pattern. Flow pattern classifications were originally based on visual observations of two-phase flow experiments, which were mapped on two-dimensional plots (called “flow pattern maps”) and the boundaries between regimes determined. Different investigators used different coordinates for the maps (e.g. mass flow rates, momentum fluxes or superficial velocities), in search for parameters that were independent of the given experimental set-up. Inevitably, the judgement of the observed regime was very subjective. For three-phase flow, the investigation of oil-water-natural gas flow regimes for the petroleum industry immediately showed the complexity of defining the



liquid-liquid mixing patterns, superimposed on the existing complexities of flow regimes arising from the gas–liquid interactions per se’ [1]. The factors that dictate the flow pattern of a multiphase flow are: the fluid properties; the operating pressure and temperature; the pipe diameter, shape, inclination and roughness; and the presence of any upstream or downstream pipe work (e.g. choke valves, T-junctions). Even more complexity is introduced when trying to define the flow regime under transient flow conditions; when all the key flow parameters are changing in time and space. More recently, visual observations have been combined with a modelling effort.

3 Multiphase flow modelling

Briefly, multiphase flow models can be categorised as follows [2]. • Empirical: data for frictional pressure gradient and void fraction are related

to system variables through empirical equations. A thorough review of the historical development of empirical multiphase flow models for hydrocarbon mixtures is given in [3]. The empirical models are the simplest and fastest to run, but their accuracy may be unacceptable outside the range of applicability for which the models were developed and validated. On the other hand, the more rigorous and complex models, based on the numerical solution of the conservation equations, are costly, time consuming and have intrinsic problems with convergence and the definition of the closure relationships. Such problems become even more important when solutions for transient flow are sought.

• Multifluid: formal governing equations (mass, momentum and energy) are solved with appropriate closure laws (usually based on empirical data). An evaluation of mechanistic two-phase flow models is given in [4]. The multi-fluid models do not cope well with intermittent flows. In slug flows, for instance, the liquid phase flows upwards in the slugs, but downwards in the Taylor bubble regions. This type of intermittency makes averaging difficult and does not fit well in a multi-fluid model framework.

• Phenomenological: observations are made of the flow patterns and models constructed with appropriate closure laws to represent the flow based on the pattern features. An example of flow models of this type is discussed in [2]. The flow regime boundaries are established through transition models (e.g. the model of Hewitt and Jayanti [5] for the slug-churn transition) and each regime is modelled by taking account of the phenomena occurring within it (such as droplet entrainment and deposition in annular flow).

• Interface tracking: calculates the details of the interfacial structure by various techniques. A review of interface tracking methods is given by [6]. The applicability of interface tracking methods is usually limited to simple flow configurations.



4 Review of existing flow loops

In the following text, a selection of major flow loops is presented and reviewed. A thorough investigation and ranking of world-wide flow loops for multiphase flow experiments should include all of the following factors: loop geometry, dimensions, operating pressure and temperature, range of phase flow rate, equipment and instrumentation, piping material, fluid properties, data acquisition and information processing systems. However, the objective of this paper is to illustrate how to approach such an investigation and to identify future needs for niche experimental investigations. Thus, only a selection of the above key parameters that define the potential of a flow loop was considered for this study.

4.1 Distribution of flow loops world-wide

The locations of the flow loops identified for this study are indicated in Table 1. The authors were unable to find details of flow loops in certain regions of the world (e.g. former Soviet Union, Eastern Europe and Asia), although it is quite possible that facilities do exist there. A general observation is that the facilities are located either in areas of large scale oil and gas production (e.g. Norway, UK and USA) or in areas where research on nuclear power is important (e.g. UK, France, USA and Germany).

4.2 Common aspects of flow loop design

No flow loop can represent of all possible multiphase flow situations, instead a loop is built to meet a specific need or to mimic a specific process. However, some recurrent design aspects of multiphase flow loop facilities are evident, such as low-pressure flow loops tend to have pipes made of polyvinyl chloride (PVC) material, with special test sections made of Perspex or transparent PVC material. This feature allows visual investigation of the flow to identify flow patterns. High-pressure facilities, on the other hand, are built with carbon steel or stainless steel pipe work. To protect the carbon steel against corrosion, inhibitors are added to the test fluids. One loop only is reported to be made of Copper [7]. There is also a trend in the type of fluids used for multiphase flow experiments. Water and air prevail, although stabilised oil, kerosene and nitrogen are becoming more commonplace for studies related to oil and gas applications. For investigations focused on flow pattern identification, inert tracers are often used to enhance the contrast between phases. Finally, all flow loops adopt similar strategies regarding the choice of equipment (e.g. valve, compressors and pumps) and instrumentation (e.g. pressure transducers and hold-up measurement systems).

4.3 Flow loop capabilities

Each flow loop, in the database created by the authors, has been classified according to the following criteria: total reported length, maximum working diameter, inclination, operating pressure, length of test section and type of fluid. Table 1 summarises the flow loops selected for this review.



Table 1: Selection of flow loops reviewed in this paper.

Notation in this paper Flow direction

Fluids/Piping Diameter [mm]

Ref.

SINTEF* (SINTEF Petroleum Research), NO

Horizontal & vertical

Hydrocarbons, H2O, N2 Carbon steel / PVC

100 200 304.8

[4, 8]

SwRI (South West Research Institute), US

Vertical Gas, water Carbon steel

25.4 [9]

IFE, NO Vertical N/A 100 [31] NFL (Memorial University of Newfoundland), CA

Vertical 76.2 [30]

TUFFP1 (Tulsa), US Vertical N/A [32] TUFFP2 (Tulsa), US Hilly terrain,

horizontal N/A Carbon Steel

[32]

NEL (National Engineering Laboratory), UK

Vertical N/A Carbon Steel

[23]

IFP (Institut Français du petrole), FR

Vertical Air, water Carbon steel

4 [23]

CRAN (Cranfield University), UK

Horizontal N/A Carbon steel

250 [14]

BHRA (BHR Group Limited), UK

Horizontal Stainless Steel/ PVC

200 400

[28]

SHELL (Rijswijk), NL Horizontal & vertical

Stainless Steel/ Perspex pipe

82 [29]

ITE (Petroleum Engineering Institute of TU Clausthal), DE

Horizontal & vertical

Air/water/sand Plexiglas

40 [21]

TAMU1 (Texas A&M University), US

Vertical Air/water Transparent PVC

127 [10]

CSM (Colorado school of Mines), US

Vertical N/A Transparent PVC

52, 140, 153.2

[27]

MPC (Middle East Technical University), TR

Horizontal Water and air Transparent PVC

57, 114.3 [26]

CEESI 1 (Colorado Engineering Experiment Station, Inc.), US

Special flow (Hydrates)

Water, gas, hydrates N/A

[24]

CEESI 2 (Colorado Engineering Experiment Station, Inc.), US

High pressure gas flow loop

Gas, Oil and water Carbon steel

[16]

ICL (Imperial College London), UK

Vertical Air/Water Copper

31.8 [7]

WASP (Imperial College London), UK

Horizontal Water, Air, Sand and Petroleum Stainless Steel

76.2 [25]

Atalaia (Petrobras), BR Field equipment

N/A N/A [18]

Trecate (ENI), I Field equipment

N/A N/A [17]

K-Lab (Statoil), NO Field equipment

N/A N/A [15]



Total length. The maximum length of a flow loop affects the development of different flow regimes, particularly when transient flow is investigated. When performing experiments at high gas fractions, the longer the test section, the greater the effect of wellbore storage. Typically, wells used in hydrology, petroleum and environmental engineering, geothermal energy plants, underground gas storage and CO2 sequestration are orders of magnitude longer than the tubes used in experiments. However, as experiments on actual wells are difficult to perform, it is usually assumed that the conditions for flow pattern transitions are similar to those occurring in short tubes. Changes in pipe inclination and flow direction also affect the nature of the flow generated within the system. Some loops have a total length of hundreds of metres, but actual tests sections of just a few metres. The maximum total length found during this review is that of the SINTEF large scale facility [4, 8], with approximately 1000 metres of pipe work. Operating pressure. A flow loop’s operating pressure is another key parameter in mimicking real multiphase flow phenomena, especially when compressible fluids are involved. The magnitude of absolute operating pressure, pressure drop in the pipe and pipe length all have an impact on the type of flow regime that can be developed. High pressure facilities are used to extend the validity of empirical multiphase flow models, which were originally developed for lower pressures. During this review, a flow loop was found with a maximum working pressure of 25 MPa [9], while the average pressure for the remaining facilities was found to be equal to or less than 10 MPa. If the test sections are made of Perspex or PVC material, the maximum operating pressure is limited to approximately 1 MPa. About half of the investigated loops have this operational limitation. The total length versus operating pressure for the selected flow loops is shown in Figure 1. Length of test section. For multiphase flow investigations related to wells, a vertical test section is needed. Only two flow loops among those considered for this review have a vertical elevation higher than 40 metres: the SINTEF large scale facility [8] and the Texas A&M PETE Tower Lab (referred to as TAMU1 in the figures) [10]. Besides the difference in pipe diameter, the major difference between these two loops is their maximum working pressures, which are 9 MPa and 0.8 MPa, respectively. The vertical height versus operating pressure, for those loops that reportedly have vertical test sections, is shown in Figure 2. Range of phase flow rates. The range of flow regimes that can be reproduced in a flow loop is related to the flow rates that can be circulated in the system. The maximum reported flow rates of gas, liquid and solids for the flow loops identified for this study are given in Figure 3. It must be noticed that no indication of the individual phase velocities in a two- or three-phase flow situation is provided in Figure 3. A reference was found [11] where the maximum flow rates of seven flow loops are expressed in terms of phase superficial velocity, as showed in Table 2. However, there is no indication of how the phase superficial velocities vary in relation to different phase fractions. Without this information, it is impossible to assess the full potential of a multiphase flow loop or infer which flow patterns it can reproduce.



Figure 1: Total length vs. max. working pressure for selected flow loops.

Figure 2: Vertical height vs. max. working pressure for selected flow loops.

Instrumentation. Key flow parameters are required to accurately model multiphase flows and measurements taken during multiphase flow experiments are used to validate and fine-tune the models. Flow loops are therefore equipped with ad hoc sensors and devices to record phase hold-up, temperature, absolute pressure and differential pressure. Each device or sensor can be characterised by its rangeability, repeatability and accuracy of the measurement. Most devices are unable to provide meaningful outputs under transient flow conditions, due to the high instability of key flow features. All of the loops investigated for this study

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12

Pressure [MPa]

Vert

ical

Loo

p H

eigh

t [m

]

Sintef

TAMU

NEL

ITENFL

Shell

ISF

ICL

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30

Max. Working Pressure [MPa]

Tota

l Rep

orte

d Le

ngth

[m] Sintef

NFL

CALTEC

NEL ShellIFE

SwRI

ITE

TAMU

ICL

MPC

TUFPP2BHRA

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5

Max. Working Pressure [MPa]

Tota

l Rep

orte

d Le

ngth

[m]

NFL

NELIFEITE

TAMU

ICLMPC

TUFPP1

WASP

IFP

CRAN



have solutions implemented to measure the reference phase flow rates circulated through the system. Table 3 summarises the most common techniques for measuring the reference phase flow rates, the phase hold-up and for identifying the flow patterns.

Figure 3: Maximum phase flow rates for selected multiphase flow loops.

Table 2: Properties of different test rigs as reported by [11].

450100 140 133

3600

18501500

42504000

720

0

500

1000

1500

2000

2500

3000

3500

4000

4500

SINTEF NEL CRAN IFP BHRA

Loop Name

Flow

rate

[m3/

hr] Liquid flow rate

Gas flow rate



Table 3: Techniques for measuring reference phase flow rates, phase hold-up and for identifying flow patterns.

Gas flow rate

Liquid flow rate

Phase velocity

Phase volumetric fraction or mixture density

Flow pattern visualization

Thermal wave flow meter

Electro-magnetic meter

X-correlation techniques

Gamma ray densitometry

High speed video system (visible spectrum)

Hot wire flow meter

Ultrasonic flow meter

Laser Doppler Anemometer

Capacitance/conductivity probes

Coriolis flow meter

Coriolis flow meter

Particle Image Velocimetry

X-ray or gamma-ray tomography

Vortex flow meter

4.4 Flow loops for the testing of Multiphase Flow Meters

Multiphase Flow Metering (MFM) is the measurement of the flow rates of each individual phase in a multiphase flow. A review of MFM techniques is presented by [12]. There are a few accepted standards for evaluating the performance of multiphase flow and wet gas meters for oil and gas applications, but, as yet, no International Regulations exists. At present, the following options are available to the industry for the verification of a meter’s performance: • Testing is carried out at the manufacturer’s own test facilities, such as the

Schlumberger flow loop in Cambridge, UK or in a third-party test loop. Independent facilities include the National Engineering Laboratory (NEL) [4, 13] Cranfield University [14] and Imperial College London, all in the UK [7] the K-Lab Wet Gas loop in Norway [15] and the Colorado Engineering Experiment Station (CEESI) in the USA [16] have already carried out assessments of the performance of commercial meters and research prototypes.

• Testing is carried out in the field by the end user and the meter is tested against conventional test separators. There are only a few field flow loops, such as ENI’s Trecate field in Italy [17] Petrobras’ Atalaia field in Brazil [18] and the K-lab at Statoil Kaarstoe gas terminal [15] that allow controlled flow tests with real fluids. In these cases, the reference measurements are sufficiently accurate, but care must be taken when carrying out the tests (flow instabilities in the loop, possible malfunction of the test separator, etc.).

The initial testing of a meter is carried out in specialised laboratories where two- or three-phase flows can be established. In this type of testing, fluids with well known properties are used (e.g. water, air, synthetic oil or stabilised crude oil) and flow rates are controlled (corresponding to fixed gas and water fractions), which greatly reduces and even eliminates many uncertainties. This initial step defines the operational envelope of the meter and its measurement



errors. The laboratory testing is then followed by field trials, which are required to identify potential operational problems, but may introduce more sources of error. These can be due to different upstream conditions (small variations in facilities layout may change the history of the flow), using real fluids instead of laboratory fluids of known properties, and the need for fluid property correlations to reconcile reference measurements with the meter readings taken at field operating conditions. Typically, the results of a field trial indicate the presence of error compensation. Whatever the testing and verification environment is, the issue remains of comparing the flow rates predicted by the meter with those taken as reference measurements taken at the separator (in the case of field testing) or with conventional single-phase metering devices (in the case of laboratory testing). The results of calibrations are only as accurate as the reference measurements provided by the calibration facility [19]. When evaluating the results of a calibration campaign, the uncertainty of the reference measurements must be accounted for. Flow loops used to verify and calibrate MFM’s have either vertical or horizontal (or both) test sections in order to accommodate some or all of the possible metering configurations. Some loops have been specifically designed for testing at high GVF. Each facility has its own specifications in terms of operating pressure, phase flow rates, fluid properties, pipe diameter, length of the test section and available instrumentation and equipment.

4.5 Flow loops for flow assurance studies

The term “flow assurance” is thought to have been coined by Petrobras in the early 1990s as ‘Garantia de Fluxo’ which literally translates as ‘Guarantee the Flow’, or Flow Assurance [20]. It was originally related to the chemistry issues associated with deepwater oil exploitation. In a broader sense, flow assurance deals with anything that may impair the flow of hydrocarbons from reservoir to sales point. If the pressure drop in the overall production system is such that the reservoir fluids cannot be brought to surface, then this is a flow assurance issue. The same applies to corrosion or erosion of the pipe work of a production system and to the deposition of wax, paraffins, asphaltenes and hydrates. Some of the flow loops identified for this study are dedicated to the investigation of flow assurance issues. They are the Cold-Oil-Water Flow Assurance Loop, the Single- and Multi-Phase Paraffin Deposition Flow Loop and the Marathon Hydrate Assurance Loop at Tulsa University, the Multiphase Corrosion Flow Loop at the Ohio University and CEESI Hydrates flow loop.

5 Future needs for niche experimental investigations

This review of existing flow loops worldwide revealed that some specialist areas of research are still lacking dedicated test facilities. These niche areas include the study of sand transport in single- or multi-phase flows and the investigation of the dynamic interactions between flow in porous media and flow in pipes under transient flow conditions.



Sand transport. The relevant mechanisms of sand particle transport are still poorly understood. Sand transport in oil-gas-sand production and transfer systems is governed by pressure, temperature, fluid composition, sand properties (density, grain dimension and shape) and momentum transfer between fluid and solid phases. In order to develop an insight into sand transport in multiphase flow systems, experimental testing is required. Several experimental and modelling techniques have been developed for the measurement and prediction of sand particle transport in oil-gas multiphase flow through pipes [21]. There exist non-intrusive techniques for taking flow measurements in solid-liquid-gas mixtures in pipes, including acoustic ultrasound, x-ray tomography, magnetic resonance imaging, neutron radiography, particle image velocimetry (PIV), laser Doppler anemometry, radioactive particle tracking, holographic interferometry and digital image analysis technique. The visual measurement of sand hold-up and the development of a dedicated mechanistic model require purpose-built flow loops. Previous studies on sand transport were focused on the modelling of low sand fractions, which does not apply to situations where sand loading is more significant. To date, laboratory research on sand transport in multiphase flow systems has been limited to small-scale studies, which do not provide an adequate environment for the simulation of gas-oil-sand and oil-water-sand multiphase flow behaviour through slotted liners, perforated tunnels, wellbores, flowlines and production riser systems [21]. Also, the effects of high pressure and high temperature on such types of multiphase flows have been neglected. The specific goals of an ad hoc sand transport research program should be as follows: • Develop of a large-scale, high-pressure and high-temperature flow loop with

dedicated instrumentation for the real-time monitoring of sand particle velocity, sand hold-up and sand distribution in pipes. This would allow one to evaluate the effects of sand particles on the characteristics of gas-oil and oil-water two-phase flows.

• Generate an experimental database to enhance the validation of mechanistic flow models for gas-oil-sand and oil-water-sand multiphase flow problems.

• Implement the validated flow models into in a user-friendly simulator for the design, performance analysis and optimisation of gas-oil-sand and oil-water-sand systems.

To this aim, work is currently ongoing at Petroleum Engineering Department (ITE) Technical University of Clausthal. Dynamic interactions between flow in porous media and flow in pipes under transient flow conditions. To date, a fully integrated solution that describes the dynamic interactions between multiphase flows in porous media and in connected wellbores under fully-transient conditions, and copes with compressible and incompressible fluids does not exist. Classical models of these interactions employ steady-state inflow performance relationships (IPR’s) where the inflow from the porous medium is related to the pressure at the bottom of the pipe, which is related to the multiphase flow behaviour in the wellbore. The latter is also calculated from steady-state relationships (though these often lack a



fundamental basis). Transitions between flow regimes can occur in the wellbore, often over a relatively small range of flow rates (i.e. over a relatively short time), in which case the use of steady-state IPR’s may be erroneous. The best solution would be to couple together transient models for porous media and pipes. To develop and validate an integrated model of this type, laboratory experiments are required to mimic the behaviour of the near-wellbore region under unsteady-state flow conditions and the dynamic interactions between the porous medium and the well. To date, no flow loop exist for this specific purpose, although preliminary design calculations have been carried out by Falcone [2] and Costantini [22]. A main tank would contain both air and water at the same constant pressure, reproducing a boundary limit of the system: the constant reservoir pressure conditions at the edge of the drainage radius. A cylindrical porous medium, which reproduces the near-wellbore region, would be built using small glass beads accurately located in a Plexiglas cylinder in isotropic and homogeneous conditions. It would be installed at the bottom of a vertical pipe section (i.e. the well). Air and water from the main tank would be fed into the porous cylinder via a distribution manifold. After the flow becomes steady, transient flow periods could be imposed on the system by operating a surface valve or by varying the input flow rates. During the transient flow periods, measurements of pressure, temperature and void fraction could be made at different locations along the rig. Work is currently ongoing at Texas A&M University to design and build a dedicated flow loop capable of simulating the integrated system made of reservoir, near-wellbore region and wellbore.

6 Conclusions

The development and validation of theoretical multiphase flow models requires measurements at controlled experimental conditions. This has given rise to a large number of multiphase flow loops around the world, some of which are also used intensively to test and validate multiphase flow meters and to investigate flow assurance issues. This review shows the main features of a selection of facilities for the investigation of large scale multiphase flows, such as those that occur in the petroleum industry. Each flow loop in this review has been classified according to total reported length, maximum working diameter, inclination, operating pressure, length of test section and type of fluid. However, it appears that some form of standardisation is required in the way flow loop capabilities are reported, particularly with regards to the flow rates that can be circulated in a given system. In most references, there is no indication of how the phase superficial velocities can vary in relation to different phase fractions in a two- or three-phase flow. Without this information, it is impossible to assess the full potential of a multiphase flow loop or infer which flow patterns it can reproduce. Hence, based on the information available in the public domain only, it is very difficult to identify the most appropriate facility for a given study area.



Finally, this review shows that a flow loop cannot represent all possible situations and that some specialist areas of research are still lacking dedicated test facilities. These niche areas include the study of sand transport in single- or multi-phase flows and the investigation of the dynamic interactions between flow in porous media and flow in pipes under transient flow conditions.

References

[1] Hewitt, G.F., Three-phase gas–liquid–liquid flows in the steady and transient states, Nuclear Engineering and Design 235 (2005) 1303–1316.

[2] Falcone, G., “Modelling of flows in vertical pipes and its application to multiphase flow metering at high gas content and to the prediction of well performance”, Ph.D. thesis, Imperial College, 2006.

[3] Brown, K.E., The Technology of Artificial Lift Methods I, PennWell, 1977.

[4] Dhulesi, H., Lopez, D., Critical Evaluation of Mechanistic Two-Phase Flow Pipeline and Well Simulation Models, SPE36611, SPE Annual Technical Conference & Exhibition, Denver, Colorado, 6-9 October, 1996.

[5] Hewitt, G. F. and Jayanti, S., Prediction of the slug-to-churn flow transition in vertical two-phase flow, Int. J. Multiphase Flow, Vol. 18, pp 847-860, 1992.

[6] Hewitt, G.F., Reeks, M.W., Computational modelling of multi-phase flows, Chapter 7 in “Prediction of Turbulent Flows”, Edited by G.F. Hewitt and J.C. Vassilicos, Cambridge University Press, 2005.

[7] Falcone, G., Hewitt, G.F., Lao, L., Richardson, S.M., ANUMET: A Novel Wet Gas Flowmeter, SPE84504, SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 5-8 October 2003.

[8] SINTEF, www.sintef.no [9] SwRI, www.swri.org [10] Scott, S., Introduction to the Goals of the Event & Texas A&M Research,

Multiphase Measurement Roundtable, Houston, 3 May 2006 [11] Valle, A. Multiphase Pipeline Flows in Hydrocarbon Recovery,

Multiphase Science and Technology, Quarterly, Vol. 10, No. 1, 1998. [12] Falcone, G., Hewitt, G.F., Alimonti, C., Harrison, B., Multiphase Flow

Metering: Current Trends and Future Developments, Distinguished Author Series, Journal of Petroleum Technology, April 2002.

[13] Henry, M. Tombs, M., Duta, M., Two-Phase (Gas/Liquid) Floe Metering of Viscous Oil Using a Coriolis Mass Flow Meter: A Case Study, 24th International North Sea Flow Measurements Workshop, 24-27 October 2006.

[14] Multiphase flow facility at Cranfield University, www.cranfield.ac.uk [15] K-lab - advanced test facility for wet gas equipment, www.statoil.com [16] Steven, R., A Discussion on Horizontally Installed Differential Pressure

Meter Wet Gas Flow Performances, 24th International North Sea Flow Measurements Workshop, 24-27 October 2006.



[17] Mazzoni, A., Halvorsen, M, Aspelund, A., Field Qualification FlowSys TopFlow Meter, Agip Test Facility Trecate, Italy, Milano, April 2001.

[18] Marruaz, Keyla S., Goncalvez, Marcelo A. L., Gaspari et al., Horizontal Slug Flow in a Large-Size Pipeline: Experimentation and Modeling. J. Braz. Soc. Mech. Sci. [online]. 2001, vol. 23, no. 4 [cited 2007-02-12], pp. 481-490.

[19] Corneliussen, S., Couput, J., Dahl, E., Dykesteen, E., Frøysa, K., Malde, E., Moestue, H., Moksnes, P.O., Scheers, L., Tunheim, H., Handbook of Multiphase Flow Metering, Revision 2, The Norwegian Society for Oil and Gas Measurement and The Norwegian Society of Chartered Technical and Scientific Professionals, March 2005.

[20] FEESA Limited, Flow Assurance & Optimisation of Oil & Gas Production, What is Flow Assurance? www.feesa.net.

[21] Bello, O. O., Reinicke, K. M. and Teodoriu, C., Experimental Study on Particle Behaviour in Simulated Oil-Gas-Sand Multiphase Production and Transfer Operations, ASME Fluids Engineering Division Summer Meeting & Exhibition, July 17-20 2006, Miami, FL, USA

[22] Costantini, A., Dynamic interaction between the reservoir and the well during well testing, Dip.Ing. thesis, University “La Sapienza” of Rome & Imperial College, October 2005

[23] Vilagines, R., Hall, A.R.W., Comparative Behaviour of Multiphase Flowmeter Test Facility, Oil and Gas Science and Technology, Rev. IFP, Vol. 58 (2003), No. 6, pp. 647-657

[24] Savidge, J., Flow Data for Natural Gas with Water and Hydrates, 24th International North Sea Flow Measurements Workshop, 24-27 October 2006

[25] King, M.J.S., Hale, C.P., Lawrence, C.J., Hewitt, G.F., Characteristics of flowrate transients in slug flow, Int. J. Multiph. Flow, 1998, vol.24, no.5, pp.825-854

[26] Omurlu, C., Ozbayoglu, M.E., Friction Factors for Two-Phase Fluids for Eccentric Annuli in CT Applications, SPE 100145, SPE/ICoTA Coiled Tubing & Well Intervention Conference & Exhibition, The Woodlands, TX, 4-5 April 2006

[27] Sutton, R.P., Skinner, T.K., Christiansen, R.L., Wilson, B.L, Investigations of Gas Carryover with a Downward Liquid Flow, SPE 103151, 2006 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 24-27 September 2006

[28] BHRA, www.bhrgroup.co.uk [29] SHELL, www.shell.com [30] Memorial University Newfoundland, www.mun.ca [31] IFE – Well Flow Loop, www.ife.no [32] University of Tulsa, www.utulsa.edu [33] King, M.J.S., Hale, C.P., Lawrence, C.J., Hewitt, G.F., Characteristics of

flowrate transients in slug flow, Int. J. Multiph. Flow, 1998, vol. 24, no5, pp. 825-854



Section 2 Flow in porous media

Modelling groundwater flow and pollutanttransport in hard-rock fractures

J. MlsFaculty of Science, Charles University, Prague, Czech Republic

Abstract

There are several reasons for the fact that fracture flow has been a subject ofactive research for the last three decades. Rock fractures commonly exist in theEarth’s upper crust and, therefore, significantly control groundwater movement.Fracture-dominated flow has become increasingly important in various problemsof geotechnical interest. A particularly important reason for investigating fractureflow and contaminant transport is the necessity of building repositories of nuclear-fuel waste which are often situated deep in granite massifs. The possibility oftheir damage during the long periods of storage requires the study of problemsconnected with the risk of possible contaminant displacement.

The aim of this article is to model water flow and contaminant transport in hard-rock fractures. Such results are required that make it possible to determine thehydraulic parameters of real fractures by comparison with data measurable underfield conditions.

Making use of the known hydromechanical characteristics of the modelledfracture and the aqueous phase, a problem with a set of three partial differentialequations and related boundary conditions was formulated and numerically solved.The unknown functions of the problem are the liquid-phase pressure, the flowvelocity and the contaminant concentration. The method of solution is describedand the achieved results are visualized and presented.

1 Introduction

The safety of deep repositories of spent nuclear fuel is a matter of great relevanceand importance. In the course of long duration storage, radionuclides can leach outof their containers and spread into the surrounding host-rock materials. Predictionof the rate of contaminants migration will depend on the hydraulic characteristics



doi:10.2495/MPF070111

Figure 1: Dependence of probability of permeable fracture on percolationprobability for different numbers of sites.

of the fracture system and the sorption characteristics of migrants and the host-rockmaterial.

In order to successfully simulate the fracture flow of the aqueous phase, we needto define the fracture geometry and to find a correspondence between the hydrauliccharacteristics of the fracture and data measurable in situ.

There are several approaches to the problem of fracture-geometry description.The variable aperture model is the most general one. The model incorporatesvariable aperture within the fracture space and makes it possible to consider openand closed regions of the fracture. The assignment of the aperture values is usuallydone by means of geostatistical methods (Moreno et al. [9], Nordqvist et al. [11]).The aperture values are then defined by an aperture probability distribution. Boththe normal distribution (Mourzenko et al. [10], Volik et al. [17]) and the lognormalaperture distribution (Nordquist et al. [11], Ewing and Jaynes [3]) have been used.Another approach to fracture geometry is an application of the percolation theory,e.g. Stauffer and Aharony [15], Berkowitz [2], Renshaw [14]. The percolationtheory works with lattices formed of different elements (triangles in the presentcase) and studies their statistics. For all the latices, each site (triangle) or bond(common boundary of neighboring sites) is randomly occupied (open for aqueousphase) with a given probability p or empty (closed) with probability 1 − p.



According to the prescribed value p, the resulting fracture becomes permeableor not permeable for the aqueous phase. After that, the proper value of fracturepermeability is modelled by means of aperture distribution.

Vesely and Mls [16] studied the influence of the fracture grid size on the relationbetween the percolation probability p and the probability of the fracture to bepermeable. It was found that the increasing number of sites narrows the transitionzone of p between the impermeable and the permeable fracture. These results areshown in Figure 1, where the percolation probabilities are depicted horizontallyand the probabilities of permeable fractures vertically.

The governing equation of the liquid-phase flow is the continuity equationtogether with an equation defining the relation between the discharge density andthe gradients of pressure and geodetic head. The well known Hagen-Poiseuilleequation is generally used for this relation. The resulting equation is usuallycalled the Reynolds equation, e.g. Zimmerman et al. [18]. Ge [4] further extendedthe Reynolds equation by incorporating tortuosity and the effect of the angle offracture walls. Konzuk and Kueper [8] pointed out that using locally held Hagen-Poiseuille equation leads to overestimation of the fracture-flow rate by a factorranging from 1.3 to 1.9.

2 Equations of liquid-phase flow

The liquid-phase flow within the fracture space is governed by two basic laws. Thefirst one is the mass balance equation

σ d(x)∂p

∂t(x, t) +

∂qi

∂xi(x, t) = 0 , (1)

where x = (x1, x2) are coordinates in the fracture plane, d is the aperture, σis specific storage of the fracture, p is the liquid-phase pressure, t is time, andq = (q1, q2) is specific discharge of the liquid phase. This equation is accompaniedby the Hagen-Poiseuille equation

qi(x, t) = −d3(x)12 µ

(ρ g

∂z

∂xi+

∂p

∂xi(x, t)

), i = 1, 2, (2)

where z is vertically upward oriented coordinate, g is gravity acceleration, and µis the dynamic viscosity of the liquid phase. Putting these equations together, weobtain second order partial differential equation

σ d(x)∂p

∂t(x, t) − ∂

∂xi

(d3(x)12 µ

(ρ g

∂z

∂xi+

∂p

∂xi(x, t)

))= 0 , (3)

Under the assumption of incompressible liquid phase, Equation (3) simplifies byintroducing hydraulic head u,

u(x, t) = z(x) +p(x, t)

ρ g. (4)



Assuming further steady-state flow or zero value of specific storativity, Equations(1) and (2) become

∂qi

∂xi(x, t) = 0 , (5)

qi(x, t) = −ρ g d3(x)12 µ

∂u

∂xi(x, t) , i = 1, 2. (6)

3 Equations of solute transport in fractures

Let us consider a presence of a contaminant in the water within the fracturespace. Correct prediction of the contaminant movement depends on severalcharacteristics of the hydrogeological environment (e.g. fracture geometry,discharge density) and the contaminant itself (e.g. solubility, sorptive binding).Denoting c the contaminant concentration (mass of contaminant per volume of theaqueous phase), the transport equation is

d(x)∂c

∂t(x, t) =

∂

∂xi

(d(x)D

∂c

∂xi(x, t) − c(x, t) qi(x, t)

) − 2∂a

∂t(x, t) , (7)

where D is coefficient of molecular diffusion and a is the mass of the contaminantsorbed on unit surface of the solid phase. In most cases, isotherms of Langmuirand Freundlich are used to asses the effects of the concentration on sorption, e.g.Park and Hahn [12]. The Freundlich isotherm is

a = K cn (8)

and the Langmuir isotherm is

a =K1 c

K2 c + 1, (9)

where K, K1, K2, n are constants depending on the solid phase and thecontaminant in question.

4 Measured data

Measurements of real fracture apertures and asperities are difficult to obtain inpractice. It is even more complicated to obtain such data for deep laying fractures.Hence, the inverse modelling is probably the most promising approach to thisproblem.

Several laboratory measurements were performed and published aimed atgetting knowledge of real aperture and asperity sizes, e.g. Pyrak-Nolte et al. [13],Hakami et al. [5], Hakami and Larsson [6]. For crystalline rock, it is possible toconclude that the typical average apertures range from 10 to 600 micrometers.The asperity sizes are in millimeters. According to the laboratory conditions of themeasurements, it may be expected that the asperity data are more reliable.



A relatively well described fracture flow experiment was conducted at the Stripamine in Sweden (Abelin et al. [1]). The experiment took place in a granite massifat the depth of 360 m in several excavated test drifts. Using a large-scale plasticsheeting technique, a very detailed monitoring of water inflow was performed.It was found that the water inflow rates ranged from 0.01 l m−1 h−1 to 0.06l m−1 h−1 with one exception of 0.26 l m−1 h−1. The presence of tritium insome locations and its absence in others indicated that there were several isolatedchannels which led water down to the depth of 360 meters in less than 30 yearswhereas most of the water had a longer residence time.

The obtained experimental data are of crucial importance when modelling thefracture geometry. The parameters of the fracture generation must be determinedin such a way that the computed flow is in agreement with the measured values.

Knowledge of the process of contaminant transport through the real fractionenables us to get further data for the fraction geometry calibration. Under theassumption that the transport parameters of the migrant and the rock material areknown, particularly parameters of sorption and molecular diffusion, the fracturegeometry can be calibrated by reaching agreement between the measured andcomputed values of the migrant concentration.

Geologic media may consist of variety of sorbing materials such as micas,iron, silicate or manganese oxides, each with their own sorption preferencesfor individual solutes and each with different sorption capacity. Park and Hahn[12] investigated sorption and desorption of selected radionuclides on granite.Particularly, the sorptive binding of 60Co, 85Sr and 137Cs on the Bulguksa granitewas studied. The obtained results enabled the authors to determine values ofthe coefficients K, n and K1, K2 of Freundlich and Langmuir isotherms (8)and (9). The Langmuir isotherm and the values K1 = 5.0 × 10−2 m3/Ci andK2 = 4.63 × 10−4 m3/m2 of 137Cs were used in numerical solutions presentedbelow.

5 The mixed formulation of the flow equation

The transport equation (7) requires values of specific discharge q. Hence, thesystems of Equations (5) to (7) or (3), (2) and (7) must be solved in order toobtain required value of concentration c(x, t). If the influence of c values uponthe phase density ρ is negligible, Equation (3) can be solved separately and theparameter q can be obtained by means of Equation (2). It is well known that theRothe method and subsequently the Galerkin method are efficient when solvinginitial-boundary value problems in Equation (3). Unfortunately, specific dischargeobtained in this way does not satisfy the requirements of Equation (7) which isvery sensitive namely to this parameter. The flow through individual bonds ofthe applied lattice (balance between neighbouring elements) has to determinedwith high level of accuracy. Consequently, it is necessary to reformulate the flowproblem in the following way, known as the mixed formulation, e.g. (Kaasschieterand Huijben [7]). Starting with a boundary value problem in Equations (5)



and (6) on a domain Ω, the mixed variational formulation is to find functions(u,q) ∈ L2(Ω) × H∗(div, Ω) such that

∫Ω

∂qi

∂xi(x)ϕ(x) dx = 0 (10)

and

−∫

Ω

12 µ

ρ g d3(x)qi(x) vi(x) dx +

∫Ω

u∂vi

∂xi(x) dx =

∫ΓD

uD vi νi dS (11)

∀ϕ ∈ L2(Ω) and ∀v ∈ HN (div, Ω), where

H∗(div, Ω) = (v1, v2); v1, v2 ∈ L2(Ω) ,∂vi

∂xi∈ L2(Ω), vi νi = qN on ΓN ,

HN (div, Ω) = (v1, v2); v1, v2 ∈ L2(Ω) ,∂vi

∂xi∈ L2(Ω), vi νi = 0 on ΓN ,

(ν1, ν2) is the unit outward normal to Ω, ∂Ω = ΓD ∪ ΓN , and u = uD on ΓD andqi νi = qN on ΓN are the imposed Dirichlet and Neumann boundary conditionson ΓD and ΓN , respectively. Using further the Raviart-Thomas finite elementformulation and hybridizing the mixed method a system of linear equationsis formulated which has symmetric positive-definite matrix, (Kaasschieter andHuijben [6]). The values of specific discharge q obtained in this way satisfy therequirements of the transport equation. Having obtained functions q1, q2, initial-boundary value problems in Equation (7) can be solved by means of the Rothemethod.

6 Numerical solution

A fracture was generated using the above method based on the percolation theoryapproach. Making use of the defined fracture parameters, several flow problemswere numerically solved. For the obtained specific discharge values, severaltransport problems were solved separately. S-curves were modelled by imposingproper initial and boundary conditions. The reason for this choice was that s-curvesreflect the most common tracer experiments. Different values of the flow rate wereobtained by changing the slope of the fracture. Fig. 2 shows the response: s-curves1, 2, and 3 were obtained for the same fracture and for slopes of π/3, π/4 and π/6,respectively.

The influence of concentration and aperture values on the sorption process isshown in Figure 3. Two different values of mean aperture (5×10−5 and 5×10−4)and two different values of concentration (0.2 Ci/m3 and 1.5 Ci/m3) were chosenand the computations were carried out for their combinations. The differencesbetween the resulting s-curves demonstrate the sensitivity. Curves 2 and 4 wereobtained with the lower asperity. The specific discharge was kept unchanged.



Figure 2: Dependence of the outflow concentration on time for three differentvalues of specific discharge.

Figure 3: Dependence of the outflow concentration on time for different values ofaperture and inflow concentration.



Acknowledgements

This paper is based upon work supported by the Grant Agency of the CzechRepublic, under grant No. 205/04/0614 and by the Ministry of education of theCzech Republic under grant No. MSM 0021620855.

References

[1] Abelin, H., Birgersson, L., Gidlund, J., Neretnieks, I., 1991, A Large-ScaleFlow and Tracer Experiment in Granite. 1. Experimental Design and FlowDistribution, Water Resour. Res., 27, 12, 3107-3117

[2] Berkowitz, B., 1995, Analysis of Fracture Network Connectivity UsingPercolation Theory, Math. Geol., 27, 4, 467-483

[3] Ewing, P. and Jaynes, B., 1995, Issues in single-fracture transport modeling:Scales, algorithms, and grid types, Water Resour. Res., 31, 303-312

[4] Ge, S., 1997, A Governing Equation for Fluid Flow in Rough Fractures,Water Resour. Res., 33, 1, 53-61

[5] Hakami, E., Einstein, H. H., Gentler, S. and Iwano, M., 1995,Characterisation of Fracture Apertures - Methods and Parameters, in Fujii, T.,editor, Proceedings of the Eighth International Congress on Rock Mechanics,Vol. II, A. A. Balkema, Rotterdam, 751-754

[6] Hakami, E., and Larsson, E., 1996, Aperture Measurements and FlowExperiments on a Single Natural Fracture, Int. J. Rock Mech. Min. Sci. &Geomech. Abstr. 33(4), 395-404

[7] Kaasschieter, E. F. and Huijben, A. J. M., 1992, Mixed-Hybrid FiniteElements and Streamline Computation for the Potential Flow Problem,Numerical Methods for Partial Differential Equations, 8, 221-266

[8] Konzuk, J. S. and Kueper, B. H., 2004, Evaluation of cubic law based modelsdescribing singe-phase flow through a rough-walled fracture, Water Resour.Res., 40, W02402

[9] Moreno, L, Tsang, Y.W., Tsang, C.F., Hall, F.V., Neretnieks, I., 1988, Flowand tracer transport in a single fracture: A stochastic model and its relationto some field observations, Water Resour. Res., 24 (12), 2033-2048

[10] Mourzenko, V.V., Thovert, J.F. and Adler, P.M., 1996, Geometry of simulatedfractures. Physical Review E, 53, 6

[11] Nordqvist, A.W., Tsang, Y.W., Tsang, C.F., Dverstorp, B., Andersson, J.,1992, A variable aperture fracture network model for flow and transport infractured rocks, Water Resour. Res., 28, 6, 1705-1713

[12] Park, C-K. and Hahn, P-S., 1999, Reversibility and Linearity of Sorption forSome Cations onto a Bulguksa Granite, Korean J. Chem. Eng., 16, 758-763

[13] Pyrak-Nolte, L. J., Cook, N. G. W. and Nolte, D. D., 1988, Fluid percolationthrough single fractures, Geophys. Res. Lett., 15, 1247-1250

[14] Renshaw, C. E., 1996, Influence of subcritical fracture growth on theconnectivity of fracture works. Water Resour. Res., 32, 6, 1519-1530



[15] Stauffer, D. and Aharony, A., 1994, Introduction to Percolation Theory,Taylor & Francis, London

[16] Vesely, M., Mls, J., 2004, Water Flow in a Single Fracture with VariableAperture, Journal of Hydrology and Hydromechanics, 2, 77-90

[17] Volik, S., Mourzenko, V.V., Thovert, J.F. and Adler, P.M., 1997, ThermalConductivity of a Single Fracture. Transport in Porous Media, 27, 305-326

[18] Zimmerman, R. W., Chen, D., Cook, N. G. W., 1992, The effect of contactarea on the permeability of fractures, Journal of Hydrology, 139, 79-96



Transient groundwater flow in a single fracture

M. Polák & J. Mls Institute of Hydrogeology, Engineering Geology and Applied Geophysics, Charles University, Czech Republic

Abstract

Understanding the water flow behaviour in a single fracture is essential for investigating groundwater flow and solute transport in fractured crystalline rock formations in the context of water supply, remediation of dissolved contaminant or projecting long-term nuclear waste repositories. Numerical modelling was used as a tool for studying the geometrical and hydraulic properties of a single fracture. Two computer codes were developed for this purpose. The first one simulates the fracture-free space geometry and the other one solves the transient flow equation. The fracture geometry simulation consists of defining the contact zones where the fracture is closed for the water flow, and of generating the aperture values in zones of water flow. The distribution of contact areas within the simulated fracture is governed by principles of percolation theory. The positive values of aperture in water-flow zones were distributed according to the knowledge of real fracture data. Transient flow simulation is based on principles of fluid mass balance in the fracture space and on locally valid cubic law that quantifies flow rate through the fracture profile. The problem is solved numerically by means of the discretization in time and the finite element method. The percolation theory approach makes it possible to get a faithful evaluation of the fracture permeability and the transient form of the flow equation enables us to simulate the flow and pressure field propagation in accordance to laboratory or field test conditions. Keywords: modelling, fracture geometry, transient flow, hydraulic head.

1 Introduction

Research of the groundwater flow through fractured rock has increased in the last three decades. A lot of mathematical modelling work has been concerned with characterization of such natural environment. Basic approaches to the water flow



doi:10.2495/MPF070121

in the fractured media can be divided into two main groups. For the modeling of the medium containing a lot of interconnected fractures, the equivalent continuum approach is suitable. Effective use of this conceptual model supposes extremely dense fracture network that causes homogenization of environment. Otherwise a discrete fracture model will be more appropriate. Analyzing of fluid flow behaviour in a single fracture is essential for understanding of flow mechanism in the fracture networks. There are three modeling approaches to describe single fracture geometry. The parallel plate model is simple but it is not able to describe spatial heterogeneity of inner fracture space [1]. The flow tube model is characterized by incorporating of circular or rectangular tubes within the fracture plane. It was shown that the real flow channels within the fracture are not fixed but they change with the direction of potential gradient [2]. The variable aperture approach includes aperture variation in whole fracture plane. This conceptual model is the best fit of real fracture conditions. Aperture is usually generated by variety of stochastic random functions. Usually the fracture void space geometry is represented by random three-dimensional functions which describe the topology of surfaces that confine the free fracture space and determine its flow and transport properties [3]. In the presented work we have used this model approach. In the single fracture consisting of two parallel surfaces the flow is governed by “cubic law” (Hagen–Poiseuille’s equation). The important implication finding is that fluid flow through the fracture may be fully characterized by aperture, although the velocity varies across that distance [4]. Using the cubic law for solution of the fluid flow in more natural fracture with variable aperture is based on assumption of local validity of this equation in each segment of the fracture. We used this principle to investigate the flow in a single fracture. Some authors [5–7] found that using locally held cubic law overestimates the flow rate through fracture in the range from 1.3 times to 1.9 times.

2 Problem definition

Mathematical modeling tools were used for analyze the pressure changes inside the void fracture space in a laboratory rock sample (figure 1) before and after the steady state flow was established. We used mathematical model of transient flow for the estimation of time that is needful for setting the pressure equilibrium in the fracture up. Tested sample of granite has shape of a cube with the size of edge about 0.6 m. There is one visible fracture with a slope of about 70 that divides the sample almost in the middle into two parts. Fracture was sealed on the surface of the sample except of two gabs that were used for steady state flow field creation. One borehole drilled into the sample was used for water injection during the test.

3 Fracture simulation

A computer code was developed for the single fracture space simulation. The fracture geometry simulation consists of two main parts. The first part defines



the contact zones where the walls of the fracture are in direct contact to each other and make these fracture areas closed for the water flow. The second part generates the fracture aperture in zones accessible for flowing water.

Figure 1: Laboratory sample of the fractured rock.

3.1 Fracture plane generation

Fracture permeability is sensitive to applied normal stress. Especially in the deep parts of crystalline massifs, the normal stress causes the closure of significant parts of fractures. The distribution of contact areas within the simulated fracture in our model is governed by principles of percolation theory and random numbers generation. Some authors, i.e. Mourzenko et al. [3] or Renshaw [8], have suggested that the real fractures have fractal properties. Veselý and Mls [9] studied minimal requirements on the element number in percolation meshes in order to obtain statistically relevant percolation cluster formations. Two-dimensional plane fracture is simulated by rectangular tetragon. This tetragon represents central plane of the fracture. It is discretised into finite number of the linear triangular elements. The program makes use of the percolation theory and defines the inner fracture structure based on percolation probability p. Each triangle of the mesh is open with probability p and closed with probability 1-p. Any group of adjacent open triangles that is large enough for connection of the opposite sites of the system forms a percolation cluster. Closed triangles are labeled as primarily closed. If percolation cluster appears others triangles are closed due to blocking of pathways by primarily closed triangles. These triangles are labeled as secondarily closed. All nodes included in closed triangles are labeled as closed. If two adjacent nodes are closed the bond between them is labeled as closed too and it is impermeable for flowing water. Fracture plane generated according to probability p=0.92 is shown in figure 2.

3.2 Aperture generation and fracture orientation

The positive values of aperture in water flow zones of the simulated fracture were distributed according to the knowledge of real fracture data [5, 10] and



according to the apertures range that was calibrated by a steady state model of hydraulic laboratory test. All nodes labeled as closed got zero apertures. Open nodes have randomly assigned aperture value according to the exponential probability distribution in the range 40–1200 µm (figure 3). Fracture aperture is distributed symmetrically along the central fracture plane.

Figure 2: Central fracture plane discretised into 8743 active triangular elements.

Figure 3: Random aperture distribution within the generated fracture.

The last step in fracture generation process is to set orientation of the fracture. The simulated fracture can be arbitrarily rotated around the horizontal axes. In our case the rotation was 70 around the y axis.



4 Transient flow simulation

4.1 Theory

Transient flow simulation in randomly generated fracture is based on two basic principles. Principle of mass conservation of the fluid in the void fracture space, eqn. (1), and principle of locally valid cubic law that quantifies flow rate perpendicular to the fracture profile eqn. (2). Principle of mass conservation is valid in each point in the fracture and can be written as follows:

( ) ( ) ( ), , 0,pd x x t div q x tt

σ ∂+ =

∂ (1)

where x=(x1,x2) are coordinates in the fracture plane, p is the pressure inside the fluid, d is aperture perpendicular to the central fracture plane, σ is specific storage of the fracture and q=(q1,q2) is specific flow rate across transversal line perpendicular to the central fracture plane. Hagen–Poiseuille’s equation (cubic law) is expressed as follows:

( ) ( ) ( ) ( )3 ,, , , 1,2,

12ii i

d x p x tzq x t x t g ix x

ρµ

∂ ∂= − + = ∂ ∂

(2)

where z is vertically upward oriented coordinate, g is gravitational acceleration, ρ is density and µ is dynamical viscosity of the fluid in the fracture. We assumed constant value of fluid density ρ. Then the eqn. (3) can be written as follows:

( ) ( ) ( )3

, , ,12g d x

q x t div u x tρ

µ= − (3)

where u(x,t) is hydraulic head expressed:

.pu zgρ

= + (4)

As a result of substitution of eqn. (3) into the eqn. (1) the governing equation for fluid flow in the fracture is set up and it can be written as follows if the summative rule is used:

( ) ( ) ( )3

, .12i i

g d xp ud x x tt x x

ρσ

µ ∂ ∂ ∂

= ∂ ∂ ∂ (5)

In the randomly generated fracture the governing equation (5) is solved together with boundary condition of 1st and 2nd type, eqn. (6), by finite element method.

( ),( , ) ., 0.

u x tu x t const

x∂

= =∂

(6)

Rothe’s method has been used to solve initial boundary value problems with this equation.

4.2 Boundary conditions and storativity

There were two parts of the fracture outer boundary where stable boundary condition of constant hydraulic head was established. These two parts (in upper left corner and lower right corner) represent the gaps used for inflow and outflow



of the water. Constant value of hydraulic head was prescribed according to the elevation of these boundaries above the reference plane. The rest of outer boundary was assigned as no-flow. During the first 10 s of the simulation (for time t>0 s), that represent water injection, the constant head of 0.3m was assigned into the nodes of the mesh which represent intersection of the fracture and injection borehole. For simulation time t > 60 s, this boundary condition was removed. An important parameter of any simulation is the storativity. In the general case, the water is assumed to be compressible and the solid matrix is deformable and, hence, the changes in pressure will cause the water content to vary with time. The value of the specific storativity of the fracture space used in our simulation was 1×10-3 Pa-1.

Figure 4: Hydraulic head in specific simulation times.

4.3 Simulation results

From analysis of the hydraulic head distribution in different times of the simulation, it is possible to determine the time that is necessary for pressure field stabilization. During the injection stage of simulation, the time necessary to reach the steady-state flow was 8 seconds. After that, the changes in the distribution of hydraulic head were negligible. The process of the pressure field rebalancing is much longer. Simulations show that the second stage, the rebalancing, takes almost 22 days. Absolute balancing of the hydraulic head to the original state takes more than 100 days. Changes in the hydraulic head field for specific simulation times are documented in figure 4. The estimation of



velocity distribution in the void fracture space is shown in figure 5. In the simulation time t=0 s the velocity in the fracture ranged between 0 m/s and 0.15 m/s (in the triangles with nodes with inflow boundary condition). The highest velocity during simulation was calculated at the very beginning of the injection stage of simulation. Its estimation was over 3 m/s in some triangles with bigger aperture close to the injection borehole. During the simulation time the velocity decreased and the zones of heightened velocity level moved through the main pathways.

Figure 5: Velocity distribution in specific simulation time.

5 Conclusions

Two numerical models were developed and used to perform simulation of a single fracture geometry and transient flow calculation. These codes were used for estimation of time dependent distribution of the hydraulic head before and after the steady-state flow was established in the laboratory tests. Simulation results show that changes in pressure are balanced very quickly after injection stage of test has started. Much longer time is necessary for the new balance after removing the injection. More than 20 days are necessary for fracture pressure rebalancing. Adjusted results show that the described method can be simply and effectively employed for fracture flow investigation.



The presented work was supported by Czech science foundation under grant No. 205/04/0614 and by the Ministry of Industry and Trade under project No. 1H-PK/31 MPO ČR.

References

[1] Abelin, H. & Birgersson, L. & Gidlund, J. & Neretnieks, I., A Large-Scale Flow and Tracer Experiment in Granite. 1. Experimental Design and Flow Distribution. Water Resour. Res., 27(12), pp. 3107-3117, 1991.

[2] Tsang Y.E. & Tsang C.F., Flow channeling in a single fracture as a two dimensional strongly heterogeneous permeable medium, Water Resour. Res., 33, pp. 2076-2080, 1989.

[3] Mourzenko, V.V. & Thovert, J.-F. & Adler, P. M., Geometry of simulated fractures, Phys. Rev. E, 53(6), pp. 5606-5625, 1996.

[4] Ge, S., A governing equatition for fluid flow in rough fractures, Water Resour. Res., 33(1), pp. 53-61, 1997.

[5] Hamaki, E. & Larsson, E., Aperture measurements and flow experiments on a single natural fracture, Int. J. Rock Mech. Min. Sci. Geomech Abstr., 33(4), pp. 395-404, 1996.

[6] Nichol N. J. & Rajaram H. & Glass R.J. & Detwiler R., Saturated flow in a single fracture: Evaluation of the Reynolds equation in measured aperture fields, Water.Resour.Res., 35(11), pp. 3361-3373, 1999.

[7] Konzuk, J. S. & Kueper, B.H., Evaluation of cubic law based models describing singe-phase flow through a rough-walled fracture, Water Resour. Res., 40, W02402, 2004.

[8] Renshaw, C. E., Influence of subcritical fracture growth on the connectivity of fracture works. Water Resour. Res., 32(6), pp. 1519-1530, 1996.

[9] Veselý, M., Mls, J., Water Flow in a Single Fracture with Variable Aperture, Journal of Hydrology and Hydromechanics, 2, pp. 77-90, 2004.

[10] Pyrak-Nolte & L.J. & Myer, L.R. & Cook, N.G.W. & Witherspoon, P.A., Hydraulic and Mechanical Properties of Natural Fractures in Low Permeability Rock, Proc. of the Sixth Int. Congress on Rock Mechanics, eds. G. Herget and S. Vongpaisal, Montreal, Canada, pp. 225-231, 1987.



Acknowledgements

Petroleum reservoir simulation using EbFVM: the negative transmissibility issue

C. R. Maliska, J. Cordazzo & A. F. C. Silva Computational Fluid Dynamics Laboratory, Mechanical Engineering Department, Federal University of Santa Catarina, Florianópolis, SC, Brazil

Abstract

Pioneer methods for simulating petroleum reservoirs were developed in the framework of finite difference methods with Cartesian grids. Therefore, the concept of transmissibility was readily applied for calculating the fluxes at the control volume interfaces. With the advent of new methods using curvilinear non-orthogonal and unstructured grids, the concept of transmissibility was maintained, probably for taking advantage of the simplicity in the programming. However, it is well known that for non-orthogonal grids, unstructured or not, the fluxes can not be exactly calculated using only two grid points, what precludes the use of the transmissibility for the flux calculation in such situations. On the other hand, it is common to find in the literature a recommendation that triangles, as used in unstructured grids should not have internal angles greater than 90o in order to avoid the appearance of a negative transmissibility. It is shown that this is a misinterpretation of the transmissibility concept, since transmissibility is always a positive quantity. Keywords: unstructured grids, transmissibility, petroleum reservoir simulation, element-based finite volume methods.

1 Introduction

The numerical techniques embodied in the pioneering industrial petroleum reservoir simulators employed finite-difference method with Cartesian grids. In this approach the connecting coefficients of a 5-point stencil in 2D can be written using the well know concept of transmissibility, [1,2]. The use of non-orthogonal curvilinear coordinates were also introduced seeking generality and flexibility of



doi:10.2495/MPF070131

the numerical schemes, [3–5]. Defining each grid independently, with its coordinates given by the grid generator, as seen in Fig. 1, unstructured grids were introduced.

Figure 1: Non-orthogonal grid. Difficulties in evaluating the mass flux.

In this case, the line joining the two grid points is not orthogonal to the flux area. Therefore, if one insists in applying the transmissibility concept to this situation, errors will arise in the mass fluxes evaluation, since it is mathematically impossible to exactly calculate the mass flux using only two grid points. This error, of course, is a conceptual error, and does not disappear as the grid is refined. Despite this error, several traditional methods used in petroleum engineering, as reported by Sammon [6], employ two-point flux approximation schemes in order to reduce the computational cost and to simplify the code implementing. This paper addresses the two-grid point approximation issue and presents the basic ideas of a numerical scheme which perform exact fluxes calculation and preserves the conservation principles at control volume level. The method can mix triangular and quadrilateral elements. In this method, the transmissibility concept, according to its definition, is no longer valid. Therefore, the negative transmissibility, which is reported in the literature [7,8] to appear in certain situations with triangular grids, is a misinterpretation of its concept. This is demonstrated by solving a well selected problem where this situation occurs.

2 Transmissibility approach in reservoir simulation

As mentioned, several reservoir simulation models use two-point flux approximation schemes. This is based on the physical idea that a flow of a quantity is directly proportional to a potential difference and inversely to the corresponding resistance to this flow. Therefore, the mass flux of a component between two adjacent grid-blocks i and j in the discrete solution of the transport equations is given, according to Heinemann and Brand [9], by

( ) ( )1

Pij

ij p ijpp ij

AQ k j ih=

= Λ Φ −Φ∑ (1)



where pΛ is the mobility of phase p, P is the number of phases; k is the absolute

permeability, Φ is the phase potential, Aij and hij are, respectively, an flow area and a diffusion-like coefficient for the gradient determination at the interface. In Eq. (1), the terms which do not depend on of pressure and saturation can be grouped, resulting in

( ) ( )ij p

P

pijpijij TQ Φ−Φ∑

=Λ=

1

(2)

where Tij is called transmissibility which is, therefore, defined as

ij

ijijij h

AkT = (3)

Figure 2: Transmissibility using the analogy with electrical conductance.

Eq. (3) requires the evaluation of the permeability (a physical property) at the control volume interfaces. As the domain may be heterogeneous, frequently with large differences in physical properties in adjacent grid-blocks, the definition of average properties at the interface can result in errors in the flux calculation. Therefore, the most appropriate procedure is to define the transmissibility for each grid block. For orthogonal grids with fully coincident interfaces this procedure leads to the exact flux determination, as already mentioned. On the other hand, for non-orthogonal grids, for grids with partial contact between the grid-blocks or with local refinement, the fluxes areas and lengths are not clearly defined. Even following the physical insight for this choice, the resulting flux will be non-exact. Applying electrical resistance concept, we obtain for a full contact area

1 212

1 2

1 2

11 1

TTTT T

T T

= =++

(4)

If there is partial contact between two grid-blocks, as shown in Fig. 2, the total transmissibility can be calculated as



2

2

1

112

TA

TA

AT c

+= (5)

where Ac is the contact surface, A1 and A2 are, respectively, the surface of each block, and T1 and T2 are their inner transmissibility. We can observe that in this case we chose to use the area Ac, the contact area, to calculate the transmissibility of grid-blocks 1 and 2, which is an approximation that will not lead to the exact value, even with grid refinement.

If one decides to use only two grid points, it must be clear that this procedure will always furnish a non-exact flux calculation. The choice of the flux area and the diffusion-like length may alleviate this problem by choosing physically consistent values for these parameters, Hegre et al. [10].

3 The Element-based Finite Volume Method (EbFVM)

3.1 Fundamentals

As reported by Tamin et al. [11], a great amount of research was dedicated in the last decade in evaluating the available tools for numerical reservoir simulation. In contrast, there were little efforts in developing new technologies and new approaches using conservative numerical schemes. In this section it is presented a numerical algorithm to be applied for simulating porous media flow with heterogeneities. It employs the ideas of Raw [12] applied to the Navier–Stokes equations. It belongs to the class of the Element-based Finite Volume Methods (EbFVM) with new features for mobility, relative and absolute permeability evaluation and local refinement near wells and/or faults. In this paper attention is devoted to clarify the alleged appearance of negative transmissibility when triangular grids with internal angles greater than 90o are employed.

Figure 3: Elements and control volumes in triangular meshes.

In a finite volume methodology the domain is covered by non-overlapping control volumes where the balances are done, as shown in Fig. 3, where triangular elements and control volumes are identified. In the cell vertex construction, the control volumes are created joining the centre of the elements



to its medians. The resulting control volume is formed by portions (sub-control volumes) of neighbouring elements. In this case, all fluxes at one specified integration point can be calculated using data from the element where the integration point lies. The saturation equation for a multiphase flow, not considering capillary pressure, is

( ) mmm

m

S qpkt Bφ λ ∂ = +⋅ ∇ ∇∂

(6)

where m identifies the phase, p is the pressure common to the existing phases, λ is the mobility of the phase, φ the porosity and S the saturation of the phase. One is interested in evaluating the mass fluxes at the interfaces. The integration of divergent term in Eq. (6), yields

( ) .m m ipipS

k p Sk p dSλ λ ∇ ⋅∆∇ ∑∫ (7)

corresponding to the evaluation at all integration points located at the surface of the control volume. The number of integration points depends on how many triangular elements contribute for forming the control volume where balances are performed. Integrating the remaining terms and collecting them, the approximate form of Eq. (6) appears as

( ) ( )o

mmmo m P ipm ipm P PP P

SS tk pq SBB Vλ

φ ∆∇ ⋅= + + ∆ ∆

∑ (8)

in which the subscript P refers to the control volume, while ip refers to the integration points located at the surface of the control volume.

The right-hand side term inside the bracket in Eq. (8) needs to be evaluated at the integration points, while pressure and saturation are available at the nodes. The mobility evaluation at the integration points is one of the key points in simulating multiphase flow in porous media, since it is the main responsible for the appearance of the well known grid orientation effects. Mobility is a function of the water saturation and when multiplied by the normal pressure gradient gives rise to the flow rate. This requires the specification of an adequate interpolation function which should take into account the direction of the flow. The traditional schemes perform a coordinate-oriented up-winding only, therefore, introducing the undesirable grid orientation effects. This matter is not in the scope of this work, and for now it suffices to mention that a type of flow direction up-winding interpolation is performed. Details about the method can be found in [13].



The evaluation of ( )ip

p S∇ ⋅∆ is realized using the shape function defined for

triangular elements. This is consistent since pressure is an elliptic term and its approximation using the bilinear shape functions

( )( )( )

,

,

,

1

2

3

N 1

N

N

ξ η ξ η

ξ η ξ

ξ η η

= − −

= =

(9)

as

( )3

1

,j jj

p N pξ η=

=∑ (10)

keeps the consistency between the numerical approximation and physics. Therefore, the numerical approximation reads

( )m mipip ip ip

p pk p kS y xx y

λ λ ∂ ∂∇ ⋅∆ = ∆ − ∆ ∂ ∂ ∑ ∑ (11)

where the partial derivatives for pressure are obtained using the shape functions. Enforcing the global mass conservation and substituting all terms in Eq. (8) one obtains an equation for pressure determination. In this paper the IMPES methodology is used, whereby the pressure is solved implicitly and saturation explicitly.

3.2 Mass flux calculation at the integration point in the EbFVM

Eq. (11) allows the calculation of the mass flux at any integration point which belongs to a specified element. For example, the flow through integration point ip2 in Fig. 3, is given by

( )2m ip

k p Sλ ∇ ⋅∆ ( ) ( )2 13 3 1 12 2 1ip T p p T p pλ = − + − (12)

demonstrating that the mass flux calculation requires one to take into account the pressure of the three grid points belonging to the element. This is obvious, of course, since the area where integration point ip2 lies is not orthogonal to the line joining grid points 1 and 2. Therefore, grid points 1, 2 and 3, must be considered. Inspecting Eq. (12), it becomes clear that it is not possible to define a transmissibility which will permit one to calculate the exact mass flux involving only two grid nodes. In Eq. (12) there are geometrical and physical information amalgamated in the coefficients T13 and T12, but they cannot be viewed as transmissibility according its definition. As already advanced, one is free to



calculate the mass flux using only grid points 1 and 2 and defining T12 as the transmissibility. It must be understood that this procedure, however, gives the wrong mass flux. It is possible to alleviate the errors by a better choice of the areas and lengths involved. Results for different choices and its influence on the error of the mass flux calculation can be found in [14]

4 Negative transmissibility: does it exist?

Fig. 4 depicts the domain for the reservoir simulation where three production wells with prescribed pressure at the left side of the domain, and two injection wells (oil and water) located in the top and bottom right, respectively, of the domain. In this problem the flow will be parallel to the x-axis. Water and oil viscosities are set identical and the flow is incompressible. The grid is constructed with 3 elements and 5 control volumes. Element 2, defined by nodes 1, 2 and 3 is constructed such that it displays one internal angle more than 90o, what would cause the appearance of negative transmissibility. The appearance of a negative transmissibility would not be admissible, since transmissibility is always a positive quantity. This, as reported in the literature [7,8], would cause convergence problems. The recommendation, therefore, would be avoiding such elements when building the grid.

Figure 4: Idealized problem to show the negative transmissibility [14].

Element 2 and its fluxes areas connecting with sub-control volumes of neighbouring control volumes are presented in Fig. 5. Point B is chosen to lie in the same horizontal line of point C. The mass flux through faces AB and BC, entering the sub-control volume 1 is given by

( ) ( )12112131131 ppppQQQ SvcSvcBCAB −ℑ+−ℑ=+= (13)

Since pressures at nodes 1 and 2 are equal, Eq. (13) results in

( )131131 ppQ Svc −ℑ= (14)



The geometrical and physical coefficient in Eq. (14) is written as

13113

13112113 SvcBCSvcABSvc τλτλ +=ℑ (15)

Figure 5: Element 2 with its fluxes areas identified.

The second term in the right hand side of Eq. (15) is zero, since the area normal of the flow of the segment BC is zero. Therefore, Eq. (15), reads

( )13

131121 ppQ

SvcAB −= τλ (16)

Since the pressure difference is positive and the mass flux is negative, one

obtains

( ) 01313

1121 <−= ppQSvcABτλ (17)

which demonstrates that

0131<

SvcABτ (18) This coefficient, as shown, can be negative and no longer can be interpreted

as transmissibility, which is always a positive quantity. The reason for the existence of convergence problems, as reported in the literature, is linked with the fact that the equation for multiphase flows were obtained from the discretized equations for single phase flows, multiplying each term by the respective mobility. This, of course, ends up in a wrong discretized equation for multiphase flows, working properly only for constant viscosity and permeability. Details can be found in [14].

5 Conclusions

In this paper it was discussed important aspects related to the type of grids used in petroleum reservoir simulation. It became clear that only in presence of locally



orthogonal grids the two-point approach yields exact results, precluding the use of transmissibility when non-orthogonal grids are employed, if exact fluxes are to be calculated.

It was also presented a flexible numerical scheme, the EbFVM (Element based Finite Volume Method), which can mix triangular and quadrilateral elements. The method, also referred as CVFEM in the literature, retains the geometric flexibility of the finite-element procedure and derives the governing discrete algebraic equations by using a conservation balance applied to discrete control volumes laid-out throughout the domain. The solution of a 2D problem demonstrates that the negative coefficient which appears when CVFEM is used with triangular grids with internal angle greater than 90o is not transmissibility, as defined when two grid-blocks are used. This coefficient can be negative causing no difficulties for the convergence characteristics of the scheme.

The authors are grateful to ANP-Brazilian Agency for Petroleum and Energy and Petrobras for the partial support of this work.

References

[1] Todd, M. R., O´Dell, P. M. And Hirasaki, G. J., Methods for Increased Accuracy in Numerical Reservoir Simulators, SPE Journal, 515-530, 1972.

[2] Aziz, K. and Settari, A., Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979.

[3] Heinemann, Z. E., Brand, C. W., Margit, M. and Chen, Y. M., Modeling Reservoir Geometry With Irregular Grids, SPE Reservoir Engineering, pp. 225-232, 1991.

[4] Rozon, B. J., A Generalized Finite Volume Discretization Method for Reservoir Simulation, SPE paper 18414 presented at the Reservoir Simulation Symposium, Houston, Texas, February 6-8, 1989.

[5] Maliska, C. R., Silva, A.F.C, Czesnat, A.O., Lucianetti, R.M., Maliska Jr., C.R., “Three-Dimensional Multiphase Flow Simulation in Petroleum Reservoirs using the Mass Fractions as Dependent Variables”, Proceedings Fifth Latin American And Caribbean Petroleum Engineering Conference And Exhibition, Rio de Janeiro-RJ, Brasil, 1997.

[6] Sammon, P. H., Calculation of Convective and Dispersive Flows for Complex Corner Point Grids. Paper SPE 62929, Computer Modelling Group, Ltd., 2000.

[7] Fung, L. S., Hiebert, A. D. and Nghiem, L., Reservoir Simulation With a Control-Volume Finite-Element Method, SPE paper 21224 presented at the 11th SPE Symposium on Reservoir Simulation, Anaheim, California, February 17-20, 1991.

[8] Sonier, F., Lehuen, P. and Nabil, R., Full-Field “Gas Storage Simulation Using a Control-Volume Finite Model”, SPE paper 26655 presented at



Acknowledgements

the 68th Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October, 1993.

[9] Heinemann, Z. E. & Brand, C. W., Gridding Techniques in Reservoir Simulation. 1st/2nd Stanford Univ. & Leoben Mining Univ. Reservoir Simulation Inf. Fórum, 1989.

[10] Hegre, T.M., Dalen, V., and Henriquez, A.,” Generalized Transmissibilities for Distorted Grids in Reservoir Simulation”, SPE 15622.

[11] Tamin, M., Abou-kassem, J. H. and Farouq Ali, S. M., “Recent Developments in Numerical Simulation Techniques of Thermal Recovery”, SPE paper 54096 presented at the SPE International Thermal Operations and Heavy Oil Symposium, Bakersfield, California, 17-19 March, 1999.

[12] Raw, M., “A New Control Volume Based Finite Element Procedure for the Numerical Solution of the Fluid Flow and Scalar Transport Equations, Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, Canada, 1985.

[13] Maliska, C.R., “Computational Heat Transfer and Fluid Mechanics” , Editora Livros Técnicos e Científicos S/A, 464 pp. ,2004 (in portuguese).

[14] Cordazzo J., Simulação de Reservatórios de Petróleo utilizando o Método EbFVM e Solver Mutigrid, Ph.D. Thesis, Federal University of Santa Catarina, Mechanical Engineering Department, 2005



An integral treatment for heat and mass transfer along a vertical wall by natural convection in a porous media

B. B. Singh Department of Mathematics, Dr. Babasaheb Ambedkar Technological University, Lonere-402103, Dist. Raigad (M.S.), India

Abstract

This paper deals with the free convective heat and mass transfer along a vertical wall embedded in a fluid saturated porous medium by using an integral method of the Von-Karman type in the presence of temperature and concentration gradients. Mathematical expressions for the local Nusselt number and local Sherwood number have been derived in terms of boundary layer thickness ratio. The governing parameters for the flow-field are buoyancy ratio (N) and Lewis number (Le). The numerical values of the local Nusselt number and local Sherwood number have been computed for a wide range of values of N and Le. The variations of local Nusselt number and local Sherwood number with N have also been studied with the help of graphs for the different values of Le. Similarly, the variations of local Nusselt number and local Sherwood number with Le have been studied for different values of N with the help of graphs. It has been found that the local Nusselt number increases as N increases for the decreasing value of Le, whereas the local Sherwood number increases as N increases for the increasing values of Le. The local Nusselt number and the local Sherwood number increase as Le increases for increasing values of N. The numerical values of the thermal boundary layer and concentration boundary layer thicknesses have also been computed for the flow-field. It has been found that the results obtained by the integral method are in good agreement with those obtained by Bejan and Khair [Heat and Mass Transfer by Natural Convection in a Porous medium, Int. J. Heat Mass Transfer, 28, pp. 909-918, 1985]. Keywords: natural convection, porous media, heat and mass transfer.



doi:10.2495/MPF070141

1 Introduction

The natural convection flows arising out of the combined buoyancies due to thermal and mass diffusion in a porous medium are of importance because of the fundamental nature of the problem and broad range of their applications pertaining to manufacturing and process industries such as geothermal systems, fibre and granular insulation, storage of nuclear waste materials, usage of porous conical bearings in lubrication technology, chemical catalytic reactors, dispersion of chemical contaminant through water saturated soil , natural gas storage tanks , etc. On account of the afore-mentioned applications, Bejan and Khair [1] used the Darcy’s law to study the features of the free convection boundary layer flow driven by temperature and concentration gradients. Recently, Lai and Kulacki [2] have re-examined the free convection boundary layer along a vertical wall with constant heat and mass flux including the effect of wall injection. The heat and mass transfer by natural convection near a vertical wall in a porous medium under boundary layer approximations has been studied by Nakayama and Hossain [3] and Singh and Queeny [4]. A further review of coupled heat and mass transfer by natural convection in porous medium is given by Nield and Bejan [5]. The objective of the present paper is to apply integral method to analyze free convection problem along a vertical wall in the presence of temperature and concentration gradients. A comparison of the numerical values of the local Nusselt and local Sherwood numbers obtained by the integral method has been done with those obtained by Bejan and Khair [1] for different values of the buoyancy ratio. It has been found that the results obtained by the present method are in good agreement with those obtained by Bejan and Khair.

2 Mathematical analysis

We consider a two-dimensional laminar flow over a vertical flat plate in a porous medium embedded in a Darcian fluid. The co-ordinate system and the physical model are shown in figure 1. In the mathematical formulation of the problem, we note the following conventional assumptions:

i) the physical properties are considered to be constant, except for the density term that is associated with the body force;

ii) flow is sufficiently slow so that the convecting fluid and the porous matrix are in local thermodynamic equilibrium;

iii) Darcy’s law, the Boussinesq and boundary layer approximations have been employed.

With these assumptions, the governing equations are given by



0=∂∂

+∂∂

yv

xu

(1)

( )∞∞ −+−= CCTTv

gKu CT (( ββ (2)

2

2

yT

yTv

xTu

∂∂

=∂∂

+∂∂ α (3)

2

2

yCD

yCv

xCu

∂∂

=∂∂

+∂∂

(4)

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

N

Nu/Ra x1/2

Le

0.1

1

10

100

Figure 1: Physical model and co-ordinate system.

Figure 2: Heat transfer coefficient as a function of buoyancy ratio.

The symbols have got their meanings as mentioned in the Nomenclature. The boundary conditions at the wall are

y = 0 : v = 0 , T = Tw , C = Cw (5)

and at infinity are

y→∞ ; u = 0 , T → T∞ , C→C∞ (6)

3 Integral method

The boundary layer equations (2)–(4) along with boundary conditions (5) and (6) have been solved by using integral method. The partial differential equations get converted into the ordinary differential equations by making use of the following transformations:



( ) 2/1xRa

xy

=η (7)

)()( 2/1 ηαψ fRax= (8)

∞−−

=TT

TT

W

wθ (9)

∞−−

=CCCC

w

wφ (10)

where v

TTKxgRa wT

x αβ )( ∞−

= is the modified local Rayleigh number, ψ is

the stream function. After transformation the resulting equations become

f″(η) - θ′(η) – N φ′(η) = 0 (11)

θ″(η) + ½ f(η) θ′( η) = 0 (12)

φ″(η) + ( Le/2) f(η) φ′( η) = 0 (13)

with boundary conditions

f(0) = 0 , θ (0) = φ(0) = 1 (14)

f'(∞) = θ(∞) = φ(∞) = 0 (15)

where primes denote the differentiation with respect to ‘η’, η∈ [ 0, ∞ ). Here, f'(η) is non-dimensional velocity related to the stream function ψ(x,y). In the above equations (11) – (13), N is the buoyancy ratio defined by

)()(

∞

∞

−−

=TTCC

NwT

wC

ββ

(16)

and Le is the Lewis number defined by

Le = α / D (17)

From (12) and (13), we get

∫∞

=−0

'21)0(' ηφθ df (18)

∫∞

=−0

'2

)0(' ηφθ dfLe (19)



The infinity is boundary layer thickness for temperature and concentration. We now assume the exponential temperature and concentration profiles as follows:

θ(η) = exp (- η/δT) (20)

φ (η) = exp (- ξ η/δT) (21)

Here δT is arbitrary scale for the thermal boundary layer thickness whereas ξ is its ratio to the concentration boundary thickness δC. With the help of above profiles, and using equation (11), the equations (18) and (19) can be obtained in two distinct expressions as

)1(4211

2 +++

=ξ

ξδ

N

T

(22)

+++

= LeN

T )1(4)1(21

22 ξξ

ξξδ

(23)

The above two equations (22) and (23) can be combined together to give the following cubic equation for determining the boundary layer thickness ratio ξ as

ξ3 + ( 1 + 2N) ξ2 – [ (2 + N) Le ] ξ – N Le = 0 (24)

As ξ is determined by using the computer programming like MATLAB from the equation (24), the local Nusselt and Sherwood numbers which are of our main interest in terms of heat and mass transfer respectively, are given as

2/1

2/1 1215.0

)(

+++

=ξ

ξ NRa

Nu

x

(25)

and 2/1

2/1 1215.0

)(

+++

=ξ

ξξ NRa

Sh

x

(26)

The accuracy acquired in the above approximations may be examined by comparing the heat and mass transfer results against those obtained by Bejan and Khair [1]. It is not unusual to have an error of 5 % or more, depending on the assumed profile. However, the situation can be remedied by adjusting the multiplicative constant, namely, replacing 0.5 by 0.444. Thus, the following approximate formulae are proposed:



2/1

2/1 121444.0

)(

+++

=ξ

ξ NRa

Nu

x

(27)

2/1

2/1 1215.0

)(

+++

=ξ

ξξ NRa

Sh

x

(28)

4 Results and discussions

The formulae (27) and (28) give the values of the local Nusselt number (Nu) and Sherwood number (Sh) as 0.444 for N = 0 and Le = 1. These values are the same as obtained by Bejan and Khair [1]. The above assertion is clear from table 1. We have done calculations for a wide range of the parameters N (buoyancy ratio) and Le (Lewis number) in order to understand their influence on the combined heat and mass transfer along a vertical wall due to free convection. These values have been given in table 1. From the table, it is evident that the values of local Nu and Sh obtained by the integral method for different values of Le are in excellent agreement with those obtained by Bejan and Khair who obtained the corresponding values by the similarity solution technique. From the table, it is clear that the thermal boundary layer thickness δT shows an increasing trend for N = 1, 4 for the increasing values of the Lewis number Le. On the contrary, the concentration boundary layer thickness δC shows a decreasing trend for N = 0, 1, 4 for the increasing values of Le. From the table, it is obvious that the Lewis number has more pronounced effect on the concentration field than it has on temperature field. From the table, it is further evident that the magnitudes of the thermal boundary layer and concentration boundary layer thicknesses are equal for N = 0, Le = 1; N = 1, Le = 1 and N = 4, Le =1. The local Nusselt number has been plotted in figure 2 as a function of buoyancy ratio for various values of Lewis number (Le = 0.1, 1, 10, 100). It is found that the rate of heat transfer decreases with increasing Lewis number for N > 0. Similarly the local Sherwood number has been plotted in figure 3 against the buoyancy ratio N for various values of the Lewis number (Le = 1, 10, 50, 100). It is found that the rate of mass transfer increases with increasing Lewis number for all N. The local Nusselt number has been plotted in figure 4 as a function of Lewis number for various values of buoyancy ratio N = 0, 2 and 4. It is found that the local Nusselt number decreases with increasing Lewis number for N > 0. Similarly the local Sherwood number is plotted in figure 5 as a function of Lewis number for various values of buoyancy ratio N = 0, 1 and 4. It is found that the local Sherwood number increases with increasing Lewis number for all N. From figures 4 and 5, also it is evident that the values of local Nusselt and local Sherwood numbers in the present case are in excellent agreement with those obtained by Bejan and Khair.



Table 1: Comparison of Local Nusselt and Sherwood numbers.

Nu/(Rax)1/2 Sh/(Rax)1/2 N Le Numerical Present Numerical Present δT δC 0 1 0.444 0.444 0.444 0.444 2 1 2 0.444 0.444 0.683 0.693 2 1.2807 4 0.444 0.444 1.019 1.053 2 0.8430 6 0.444 0.444 1.275 1.332 2 0.6666 8 0.444 0.444 1.491 1.568 1.9999 0.5663 10 0.444 0.444 1.680 1.776 2 0.5 100 0.444 0.444 5.544 6.061 2 0.1455

1 1 0.628 0.628 0.628 0.628 1.4142 1.4142 2 0.593 0.591 0.930 0.937 1.5015 0.9478 4 0.559 0.557 1.358 1.383 1.5935 0.6418 6 0.541 0.539 1.685 1.728 1.6459 0.5138 8 0.529 0.528 1.960 2.019 1.6806 0.4395 10 0.521 0.520 2.202 2.276 1.7074 0.3901 100 0.470 0.4692 7.139 7.539 1.8733 0.1166

4 1 0.992 0.992 0.992 0.992 0.8944 0.8944 2 0.899 0.896 1.431 1.436 0.9905 0.6180 4 0.793 0.797 2.055 2.017 1.1138 0.4284 6 0.742 0.743 2.533 2.562 1.1951 0.3464 8 0.707 0.707 2.936 2.976 1.2543 0.2979 10 0.681 0.681 3.290 3.341 1.3030 0.2657 100 0.521 0.519 10.521 10.792 1.6630 0.08

0

2

4

6

8

10

12

0 2 4 6

N

Sh/Ra x1/2

Le

100

50

10

1

Figure 3: Mass transfer coefficient as a function of buoyancy ratio.



0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

Le

N=0 (Present)

N=0 (Numerical)

N=2 (Present)

N=2 (Numerical)

N=4 (Present)

N=4(Numerical)

N=4N=2

N=0

Figure 4: Heat transfer results.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 3 6 9 12Le

Sh/Ra x1/2

N=0 (Present)

N=0 (Numerical)

N=1(Present)

N=1 (Numerical)

N=4 (Present)

N=4 (Numerical)

N=4

N=1

N=0

Figure 5: Mass transfer results.

5 Concluding remarks

This paper deals with the free convective heat and mass transfer along a vertical wall embedded in a fluid saturated porous medium. The heat and mass transfer coefficients obtained in the present study by the integral method agree very well



with those obtained by Bejan and Khair. In the present analysis, the results have been presented in such a way that any practicing engineer can easily obtain the physical characteristic of the problem for arbitrary values of the buoyancy ratio and Lewis number. The advantage of this method is that it also provides with great freedom the approximate solutions to non-linear problems. The further advantage of this method is that the results are obtained with more ease as compared to Bejan and Khair.

Nomenclature

N buoyancy ratio T temperature C concentration D mass diffusivity of porous

medium f dimensionless stream function g gravitational acceleration h local heat transfer coefficient k thermal conductivity K permeability Le Lewis number Nu local Nusselt number Sh local Sherwood number Rax modified Rayleigh number u Darcy’s velocity in x- direction v Darcy’s velocity in y- direction x, y cartesian co-ordinate

Greek Symbols α thermal diffusivity of porous

medium η similarity variable βT coefficient of thermal expansion βC coefficient of concentration

expansion δT arbitrary length scale for thermal

boundary layer δC arbitrary length scale for

concentration boundary layer ψ stream function φ dimensionless concentration ξ boundary layer thickness ratio ν kinematic viscosity Subscripts ∞ condition at the infinity w condition at the wall

References

[1] Bejan, A. and Khair, K.R., Heat and Mass Transfer by Natural Convection in a Porous medium, Int. J. Heat Mass Transfer, 28, pp. 909-918, 1985.

[2] Lai, F.C. and Kulacki, Coupled Heat and Mass Transfer by Natural Convection from Vertical Surface in Porous Media, Int. J. Heat Mass Transfer, 34, pp. 1189-1194, 1991.

[3] Nakayama, A. and Hossain, M. A., An Integral Treatment for Combined Heat and Mass Transfer by Natural Convection in a Porous Media, Int. J. Heat Mass Transfer, 38, pp. 761-765, 1995.

[4] Singh, P. and Queeny, Free convection Heat and Mass Transfer along a Vertical Surface in a Porous Media; Acta Mechanica, 123, pp.69-73, 1997.

[5] Nield, D. A. and Bejan, A., Convection in Porous Media, second edition, Springer-Verlag, New York, 1999.



Application of integrated finite differences tocompute symmetrical upscaled equivalentconductivity tensor

C. Vassena & M. GiudiciUniversita degli Studi di Milano, Dip. di Scienze della Terra “A. Desio”,Milano, Italy

Abstract

The equivalent conductivity tensor is computed with a method based on thesolution of the balance equation at the fine scale. In particular the balance equationis solved on a block assigning Dirichlet boundary conditions that vary linearly withthe space coordinates and the equivalent conductivity tensor is the linear tensorrelating average flux and hydraulic gradient. Previous works prove that this methodyields a symmetric equivalent conductivity tensor both for continuous domainsand for discrete models based on integrated finite differences. Here the equivalentconductivity tensor is computed for two lateral faces of a volume of glacio-fluvialsediments and the results are compared with those obtained with a standard finitedifferences method on square grids with different spacings.Keywords: upscaling, equivalent conductivity, symmetry, integrated finitedifferences.

1 Introduction

Discrete models of ground water flow are usually based on the discretisationof the subsurface in grid-blocks for which homogeneous equivalent block-scale hydraulic conductivities must be specified. In real porous media thelocal scale K tensor is heterogeneous within a block and therefore it isnecessary to find an equivalent conductivity tensor, K, for each grid-block.The basic idea for upscaling is that the block-averaged Darcy’s velocity,〈q〉, and hydraulic gradient, 〈J〉, are related by a block-scale Darcy’s law:



doi:10.2495/MPF070151

〈q〉 = −K 〈J〉 . (1)

Reviews and classifications of the different approaches used to transform a detaileddescription of the spatial variability of K to a coarser description can be foundin [1, 8–10, 13, 16].

The block conductivity tensor is in general non diagonal, because thegeometrical regularity of heterogeneity at the fine scale, e.g. layering, yieldsanisotropy at the coarse scale [14]. Moreover, when eqn. (1) is applied, K is notan intrinsic property of the block, but depends on the boundary conditions and thesource terms. However we can decompose 〈q〉 as

〈q〉 = −K 〈J〉 + qnD, (2)

where qnD represents the non-Darcian block-averaged flow rate. The equivalentconductivity tensor, K , defined by eqn. (1) and computed with the techniqueproposed in the second section is a local property of the medium at the coarse scale.Non-local effects are described by qnD, which depends on the whole conductivityfield, on the boundary conditions, and on the source terms.

Theoretical studies show that the computation of K with arbitrary boundaryconditions might yield a non symmetric tensor, which therefore could not admitprincipal directions of anisotropy, whereas with some assumptions the symmetryof the block conductivity tensor is demonstrated for instance by [3, 11] and[17]. Farmer [4] states sufficient conditions that guarantee the symmetry of Kfor a continuous domain: he shows that K is symmetrical for any conductivitydistribution and for any shape of the domain, if it is obtained from eqn. (1) and if〈q〉 is computed by solving forward problems with Dirichlet boundary conditionssuch that the prescribed head is a linear function of the space coordinates.

In [6] we extend this theorem to a physically based conservative discrete modelusing the integrated finite differences method (IFD), proposed in the ground waterliterature by [7] and applied by some authors to model regional flow [5, 12].In [6] the focus is on 2D stationary flow in a confined aquifer, so that the physicalparameter to be considered is the aquifer transmissivity; the proof of the symmetryof the upscaled transmissivity tensor given by [4] is followed and it is rigorouslydemonstrated that K is symmetric even in the discrete case for physically basedconservative numerical models.

The goal of this paper is to test the result obtained in [6] on some numericalexamples: the equivalent K tensor is computed for two faces of a volume of glacio-fluvial sediments belonging to Pleistocene sequences of the Ticino basin (NorthernItaly) described in [15]. We show that IFD can be useful to model media which arethe union of many zones, each of which characterized by a single hydrofacies: infact this method permits the discretisation of the discontinuities between differentzones in a more accurate way than standard finite differences techniques whichcannot handle non-rectangular grid cells.



2 The computation of the symmetric equivalent conductivitytensor

We refer to discrete models for which the porous medium under study issubdivided into coarse blocks, and each coarse block is subdivided into a fine gridof cells, so that at each fine cell the conductivity takes a different value. For thesake of simplicity we refer to 2D flow in the vertical x-z plane, but the basic resultscan be easily extended to 3D flow.

The block conductivity tensor is computed with a local-numerical approach [9],or local-local technique according to the classification of [4]. In the local methodsthe equivalent block conductivity is assumed to depend only on the local K valuesinside the block. Numerical techniques are based on the numerical solution ofbalance equations: the spatially averaged flow through each block is computedfrom the solution of the flow problems at the fine scale and eqn. (1) permits tocompute an upscaled conductivity tensor for the coarse scale blocks.

Virtual experiments are conducted on a block, solving the discrete balanceequation on the fine grid; the boundary conditions are assigned at the border ofeach block as prescribed heads that are linear functions of the space coordinates.

The fine grid is based on a set of nodes, around which Voronoi polygons (fig. 1)are built as the union of the mediators of each segment joining adjacent nodes [2,p. 353]. Each node will be denoted either by an integer index, e.g. j, or by itsposition vector, e.g. xj = ((x1)j , (x2)j); the node index is used to label the cellcorresponding to the domain enclosed by a Voronoi polygon.

Figure 1: Grid built with Voronoi polygons. di,j is the length of the internodesegment connecting nodes i and j; li,j is the length of the side separatingthe cells i and j; the area of the gray region is (di,j li,j) /2 (from [6]).

Let N be the set of nodes which belong to a coarse block, and let N = N0∪Nb,where N0 is the subset of nodes for which the balance equation can be written, Nb

is the subset of the boundary nodes.



The balance equation for any cell j ∈ N0, assuming that the source terms arenull, has the form:

1Aj

∑i∈Sj

Ki,jhi − hj

di,jli,j = 0 (3)

where: Sj is the subset of nodes connected to the j-th node; Aj is the area of thej-th cell; Ki,j is the internode conductivity; hj is the piezometric head at the j-thnode; di,j = ‖xi − xj‖ is the distance between the i-th and j-th nodes; li,j is thelength of the side separating the i-th and j-th cells.

Each of the terms Ki,j (hi − hj) d−1i,j li,j appearing in the summation of eqn. (3)

corresponds to the flux per unit length along y direction entering in the j-th cellthrough the side separating it from the i-th cell.

Notice that the standard finite differences method can be viewed as a specialcase of the IFD method, in particular for square cells li,j = di,j = ∆x, where ∆xis the grid spacing.

Let the boundary conditions on the piezometric head assigned at the border ofthe coarse block be linear:

h(r)i = xi · e(r), i ∈ Nb, (4)

wheree(1) = (1, 0), e(2) = (0, 1)

and the index r refers to the r-th virtual experiment; the piezometric head h(r)

satisfies eqn. (3).In the numerical tests the equivalent tensor K is computed from eqn. (1) and is

given by:

Ksr =1A

∑j∈N

∑i∈Sji≤j

Ki,j

h(r)i − h

(r)j

di,j

li,jdi,j

2(xs)i − (xs)j

di,j, (5)

where

A =∑j∈N

∑i∈Sji≤j

li,jdi,j

2.

In [6] the same track of the proof given by [4] for the continuous case isfollowed; gradients are substituted with finite differences approximations andintegrals are substituted with sums over a block.

The following expression, equivalent to eqn. (5), is obtained for the componentsof K and implies the symmetry of the equivalent conductivity tensor at the coarsescale:

Ksr =1A

∑j∈N

∑i∈Sji≤j

Ki,j

h(r)i − h

(r)j

di,j

h(s)i − h

(s)j

di,j

li,jdi,j

2. (6)

The index s refers to the direction of flow, whereas r refers to the component ofthe hydraulic gradient. Notice that li,jdi,j/2 is the surface of the gray area in fig. 1.



An alternative way to compute K [6] is based on the equivalence betweenthe dissipated energy averaged over a block, −〈q · J〉, and computed from blockaveraged quantities, −〈q〉 · 〈J〉, i.e. the equivalent conductivity tensor is obtainedwith the condition that

−〈q · J〉 = −〈q〉 · 〈J〉 =2∑

m=1

2∑n=1

Km,n 〈Jm〉〈Jn〉 . (7)

In [6] we show that the block conductivity tensors computed from the criteriaof equivalence of flux and of energy dissipation coincide if fixed head boundaryconditions that vary linearly in space are assigned.

3 A case study

The local-numerical approach described in section 2 is applied to Pleistocenesequences of the Ticino basin (northern Italy), where some volumes of glacio-fluvial sediments outcropping at a quarry site are investigated at the meter scale[15]. Here we consider two lateral faces (labelled by A and C in [15]) of one ofthese volumes. In [15] a simplified scheme of “operative facies” is obtained bygrouping facies into five categories, each characterised by a constant conductivityvalue, obtained from laboratory tests or estimated with empirical formulas. Theboundaries between individual facies and depositional units are drawn in fig. 2,where different operative facies are represented with different shadings.

The flow model is applied to each face under the assumption of 2D flow, and Kis computed at the block scale, considering the whole face as a block.

Figure 2: Sedimentological interpretation and operative facies of Faces A and C.Grey: open framework gravel (K = 5 · 10−2 m/s); horizontal lines:sandy gravel well sorted (K = 2 · 10−3 m/s); dots: sandy gravel poorlysorted (K = 6 · 10−4 m/s); vertical lines: coarse to medium sand(K = 5 · 10−4 m/s); white: fine sand (K = 10−4 m/s).

Notice that in [15] cobbles with diameter greater than 2 cm are considered asimpermeable bodies to perform the 3D flow modeling; here we do not considerthose features, because their use in a 2D flow model introduces impermeable



structures with a large and non realistic lateral extension along the directionorthogonal to the face.

The tensor K is computed using the local-numerical approach based on thesolution of the balance equation with the IFD method: for each face two examplesof Voronoi grids are considered and the results are then compared with thoseobtained with a standard finite differences technique on square grids with differentspacings. The Voronoi grids are a coarse one (IC), with nodes placed only alongthe discontinuities among operative facies, and a fine one (IF) obtained by addingsome nodes far from the discontinuities.

Table 1: Equivalent conductivity tensors (10−4 m/s). Rn indicates the regular gridwith spacing of n cm; IC indicates the results obtained with the coarseirregular grid, IF those obtained with the fine grid.

Face A

Grid R8 R4 R2 R1 R0.5 IC IF

Kxx 17 26 29 30 31 32 32

Kxz 3.9 3.4 4.3 4.0 3.9 4.2 4.2

Kzx 4.4 3.6 4.5 4.1 3.9 4.2 4.2

Kzz 5.4 5.2 5.8 5.8 5.8 6.3 6.2

Face C

Grid R8 R4 R2 R1 R0.5 IC IF

Kxx 4.0 4.1 4.0 4.1 4.1 4.1 4.1

Kxz 0.15 0.06 0.11 0.10 0.10 0.087 0.095

Kzx 0.18 0.10 0.12 0.11 0.10 0.094 0.099

Kzz 3.7 3.4 3.1 3.1 3.1 3.0 3.0

In table 1 the components of K computed with different grids are listed.The dominant components are the diagonal ones, especially for face C, for

which the off-diagonal terms are less than the diagonal values by more than oneorder of magnitude: we can conclude that x and z-axis can be considered as theprincipal axes of K.

The off-diagonal terms, Kxz and Kzx, are computed with eqn. (5) and showdifferences which are less than 20% of the computed values for the coarsest grids(R8), but become negligible for fine grids and for irregular grids. The differenceKxz − Kzx is due to the approximations introduced in the solution of the balanceequation and in the computation of K with eqn. (5), which does not explicitlyimply symmetry, and is an estimate of the uncertainties on the components of K .

Table 1 also permits to draw some conclusions about the dependence of theresults on the grid spacings. For these tests the results obtained with the IFD



method do not noticeably depend upon the refinement of the grid: both grids (ICand IF) accurately reproduce the discontinuities between the facies, which is themost important aspect; moreover, the differences between the results obtained withIC and IF are small if compared with the differences between the sizes of the cellsof the grids (table 2). The surface of the cells of the regular grids varies between64 cm2 (R8) and 0.25 cm2 (R0.5) and, as noticed before, the results are influencedby the the size of the cells.

A direct comparison between the IC and IF is given in fig. 3, where a small areaof face C is represented as an example, whereas a more complete analysis of thecharacteristics of the irregular grids can be found in table 2.

Figure 3: Comparison between the irregular grids IC (black) and IF (gray) in thesmall area belonging to face C indicated by a square in fig. 2.

Fig. 3 shows that the coarse grid reproduces in a sufficiently accurate way theboundary of the discontinuities but it is constituted by cells of irregular shapes andnoticeably different size from each other.

Quantitative information about the shape of the cells is given by a shape factor,defined as σj = lj/rj , where lj is the diameter of the j-th cell and rj is the radiusof the inscribed circle. We have σ = 2 if the cell is round, σ = 2

√2 for a square

cell, whereas higher values of σ characterize a cell with a more irregular shape.In table 2 the maximum and the average values of the shape factor are given

for each irregular grid: in particular the values of the maximum and the standarddeviation are useful to assess the enhanced regularity of the fine grids with respectto the coarse ones and the wide range of different shapes and sizes of the Voronoicells in comparison with the regular ones.

4 Conclusions

In [6] the following theorem is proven. Let the discrete balance equation (3)be valid over a Voronoi diagram, with which a block is discretized to applyan IFD model. Let h(r) be the solution to (3) if Dirichlet boundary conditionsare assigned so that the prescribed head is a linear function of xr . The block



Table 2: Parameters of the irregular grids for both faces. In the last column averageand standard deviation of the shape factor σ are listed.

Grid Number of nodes maxj∈N0(Aj) (cm2) maxj∈N0(σj) σ

IC-A 503 68 34.1 5.8 ± 4.1

IF-A 752 35 18.1 3.8 ± 2.1

IC-C 1157 127 40.4 6.0 ± 4.2

IF-C 1685 18 12.6 4.1 ± 1.7

scale equivalent conductivity tensor computed from (1) is symmetric for anyconductivity distribution and for any shape of the block.

A local-numerical approach is adopted: the discrete balance equation is solvedon the fine grid with boundary conditions assigned at the border of each block andthe equivalent conductivity tensor is computed as the coefficient of proportionalitybetween the block averaged Darcy’s velocity and the block averaged hydraulicgradient.

This result is obtained in [6] for the IFD method that is based on a balanceequation and permits to approximate the discontinuities between different zonesin a more accurate way than standard finite differences techniques.

The numerical tests show that, as expected from the theory, the equivalentconductivity tensors that control the Darcian term of the block-averaged flux aresymmetric but for differences between off-diagonal terms due to approximationand rounding errors in the computation of K.

Moreover, the IFD method is useful to model media which are the union of manyhydrofacies and permits the discretisation of the discontinuities between differentzones in a more accurate way than standard finite differences techniques whichcannot handle non-rectangular grid cells.

This work has been supported by the Italian Ministry for University and ScientificResearch (PRIN 2005) and the University of Milan within the project “Field andnumerical studies to model the sedimentary architecture and water flow in aquifersystems of the Po plain at different scales” (principal investigator: M. Giudici).

References

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[2] de Marsily, G., Quantitative Hydrogeology, Academic Press: Orlando,pp. 440, 1986.



Acknowledgements

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[15] Zappa, G., Bersezio, R., Felletti, F., & Giudici, M., Modeling heterogeneityof gravel-sand, braided stream, alluvial aquifers at the facies scale. Jour.Hydrology, (325), pp. 134-153, 2006.

[16] Zijl, W., Scale aspects of groundwater flow and transport systems. Hydrogeol.J., (7), pp. 139-150, 1999.

[17] Zijl, W. & Trykozko, A., Numerical homogeneization of the absolutepermeability using the conformal-nodal and mixed-hybrid finite elementmethod. Transp. Porous Media, (44), pp. 33-62, 2001.



A parallelizable procedure for contaminant diffusion in waste disposal

A. S. Francisco & J. A. de Castro

Departamento de Engenharia Mecânica, Universidade Federal Fluminense, Volta Redonda, Brazil

Abstract

The contaminant transport problem is typically modeled by advection-diffusion equations. In this paper we apply a parallelizable iterative procedure to approximate the solution of the diffusion term of such equations. The spatial discretization is done by using mixed finite elements, and the resulting algebraic problems are handled by a domain decomposition procedure. This procedure permits one to implement the algorithm in distributed machines in order to save on computer memory and execution time. Numerical results are compared with experimental, which indicate that the numerical procedure is accurate and very efficient in a computational point of view. Keywords: contaminant transport, parallelizable procedure, mixed finite elements, porous medium.

1 Introduction

The contaminant transport throughout the soil in waste disposal has been reasonably modeled considering the mechanisms of sorption, diffusion and advection of pollutants. We propose in this paper to investigate such transport taking into account the kinetic parameters of mass transfer that occur simultaneously along with the transport mechanisms. Here, we apply the parallelizable computational technique as an efficient numerical solver for resolution of the governing equation.

Contaminant transport is typically modeled by advection-diffusion equations. Pinto [1] developed a model that realizes simulations of environment impacts caused by deposition of solid waste in the soil. This model has considered the molecular diffusion as the main mechanism of contaminant transport in this



doi:10.2495/MPF070161

porous medium, and the sorption as the interaction process between the contaminant and soil. This model was constructed based on experimental data obtained by Ritter and Gatto [2] in molecular diffusion and equilibrium tests for soil and contaminant solution from a waste repository. Now we propose a parallelizable scheme for the numerical solution of the transport-diffusion equation, in which a mixed finite element method is applied for the spatial discretization. The resultant linear algebraic problems from this discretization are accomplished by an iterative procedure of domain decomposition [3]. This procedure permits to implement the parallel processing of a computational code in several machines, in order to obtain efficiency. It is naturally parallelizable in machines with distributed memory and allocates small memory space. Once it does not require the resolution of large linear equation systems, it is rapid. And it is of simple implementation concerning its code development.

In this paper we consider the process of soil-contaminant interaction described by a mass transfer coefficient of the contaminant, which values can be incorporated into the source/sink term of the mathematical model. Our investigation is conducted by comparing numerical simulations to experimental data of molecular diffusion tests.

2 Mathematical model

2.1 The contaminant solute

Solid wastes accumulated along the years in soil are not inactive. A contaminant solution of several chemical components is composed by physical, chemical and biological mechanisms. This solution moves into the groundwater, where takes place processes as such advection, diffusion, adsorption, degradation etc.

In order to investigate accurately the performance of our numerical solver we consider only one contaminant solute in the groundwater. The solute considered is the ammonium (NH4

+). This contaminant ion suffers from adsorption process in the soil, and it has reliable experimental data available in the literature [4].

2.2 Governing equations

The objective of a contaminant transport model in a porous medium is to estimate the concentration of some solute as a function of the time and space. For this, the initial point is at the statement of the mass balance for the contaminant solute.

Let cw be the contaminant mass fraction in the water phase and let cs be the mass fraction in the soil phase. Then we can write two balance equations involving such variables; in the water phase:

( ) ( ) ( )w w w w w w w wc u c D c Stρ ε ρ ε ρ ε∂

+∇• +∇• − ∇ = −∂

; (1)

and in the soil phase:



( ) ( )(1 ) (1 )s s s s sc D c Stρ ε ρ ε∂

− +∇• − − ∇ =∂

; (2)

where ρw is the water specific mass, ε the porosity, uw is the water velocity, Dw the diffusion coefficient in the water, S the sink term, ρs the solid specific mass, and Ds the diffusion coefficient in the soil.

In some cases, this system of equations is conveniently solved assuming that there is no diffusion in the soil phase and considering that the solute concentration in both water and soil phases are in equilibrium. Thus we could derive a model based only on the sorption isothermal, in which is defined a parameter called distribution coefficient.

Otherwise, we solve the contaminant problem using a model for the sink term as a function of a mass transfer coefficient that represents the dissolution rate of the contaminant solute in the soil and the water. Here, the sink term is associated to the mass transfer kinetic of the contaminant solute as a function of time. This model permits to solve the contaminant problem treating only the eqn. (1). Once we are simulating molecular diffusion tests, in which the advection can be neglected, the contaminant transport is then described by

( ) ( )w w w w wc D c Stρ ε ρ ε∂

+∇• − ∇ = −∂

. (3)

The mass transfer rate, S, of the contaminant ion (NH4

+) between the soil and water phases can be written by the following equation [5, 6]:

( )e sw w w eq wS k A c c cη

β ρ ε= − , (4)

where k is a kinetic constant, βe the mass transfer coefficient, Asw the contact area between the soil and water phases, ceq the equilibrium concentration of the contaminant ion in the leakage within the granular soil, and η an exponent of the equation.

The contact area is calculated by

6(1 )sw

s s

Ad

εϕ−

= , (5)

where ds is the average diameter of soil particle, and φs the form factor of soil particle.

The mass transfer coefficient is calculated by

h we

s s

S Dd

βϕ

= , (6)



where Sh is the Sherwood number. The Sherwood number represents the non-dimensional concentration gradient

on the soil surface [7] and is determined by

130,5851,17 2hS Re Sc= + , (7)

where Re is the Reynolds number, and Sc is the Schmidt number. The Reynolds and Schmidt numbers can be determined by the respective equations:

(1 )w w s

w

u dRe

ρ εµ−

= (8)

and

w

w w

ScDµ

ρ= , (9)

where µw is the water viscosity.

3 Mixed finite element approximation

3.1 Numerical solution

For the contaminant transport eqn. (3), a parabolic problem, we employ an implicit time discretization along with mixed finite elements in the spatial discretization (see references [8–10]). This technique is appropriate to obtain accurate diffusion flux computations. A domain decomposition procedure is applied towards the solution of the resulting algebraic problems.

3.2 Time and space discretization

We write a discretized form for the contaminant transport equation as

( )1

1 1n n

n nw ww

c c d St

ρ ε+

+ +−+∇• = −

∆ , (10)

in which 1 1n n

w w wd D cρ ε+ += − ∇ .

Let 2Ω⊂ℜ be a bonded domain with a Lipschitz boundary∂Ω . Let

, 1,...,j j MΩ = be a partition of Ω with

, ,j j jk kj j kΓ = ∂Ω Γ = Γ∩∂Ω Γ = Γ = ∂Ω ∩∂Ω .



We consider decomposing eqn. (10) over partition jΩ . In addition to

requiring that ,w jc d be a solution of eqn. (10) for jΩ , it is necessary to

impose the consistency conditions

,j kw w jkc c on= Γ (11)

and 0,j j k k jkd v d v on⋅ + ⋅ = Γ , (12)

where jv is the unit outer vector normal to j∂Ω .

In order to define an iterative method for solving the above algebraic problems, it is convenient to replace eqn. (11) and (12) by Robin transmission boundary conditions. Thus, consistency conditions for the contaminant concentration will be given by

,j j jk k k kj jkd v l d v l onχ χ− ⋅ + = ⋅ + Γ , (13)

,k k kj j j jk kjd v l d v l onχ χ− ⋅ + = ⋅ + Γ , (14)

where χ is a positive function on jk∪Γ , and jkl is a Lagrange multiplier

defined on jkΓ . We shall consider lowest index Raviart-Thomas spaces for the spatial

discretization with square elements of size h. Then the discretized form of the system of equations can be written as

1 1 11n n n nw ww wc d S c

t h tββ

ρ ε ρ ε+ + ++ = − +∆ ∆∑ , (15)

( )1 1 12n n nw ww

Dd l chβ β

ρ ε+ + += − − , (16)

where dβ denote the value of the outgoing diffusive flux on the edge β , , , ,L R U Dβ = (see the fig. 1).

All the variables which appear in the above system refer to a single element. In order to define an iterative scheme, we use eqn. (11) and (12) in the above system to express all Lagrange multipliers in terms of the Lagrange multipliers and the fluxes of the adjacent elements. We introduce the superscript “~” to denote variables of adjacent elements, and let β’ denote the edge β of the element under consideration. Then, the eqn. (16) takes the form



Figure 1: Diffusive fluxes on the edge of an element.

( )1 1' '1 1

n n n nwd c d lβ β β

ξ ξ χχξ χξ

+ += − ++ +

, (17)

where 2 w wDh

ρ εξ = . See the Douglas et al [11] for details about the iterative

scheme. Substituting eqn. (17) into eqn. (15), we have

1 1 11

1n n n nw w

w w wc c S ct h tβ

ρ ε ρ εξχξ

+ + ++ = − +∆ + ∆∑

( )' '1

1n nd l

h β ββ

ξ χχξ

+ ++∑ . (18)

If we take a linear scheme for the discretization of the mass transfer rate, then it results that

( )1 1n n ne sw w w eq wS k A c c c

ηβ ρ ε+ += − . (19)

Finally, the numerical scheme to solve the contaminant transport is given by

( )

( )

' '1 1

1

n n nww

nw

nwe sw w w eq

hd l ct

c h k A h c ct

β ββ

η

β

ρ εξ χχξρ εξ β ρ ε

χξ

+

+ ++ ∆

=+ + −

+ ∆

∑

∑ . (20)

dR

dU

dD

dL



4 Numerical simulations

In this section we apply our numerical procedure for simulating the experimental results of a molecular diffusion test [2]. This laboratory experiment were performed with samples of soil and leachate of a solid waste landfill, conform the model presented by [12]. According this model, such samples with distinct concentrations are put into an experimental cell in order to measure the diffusion coefficient. At the top, we have a leachate repository where occurs the diffusion of the ammonium in free solution. And at the bottom, we have the effective diffusion of the ammonium over through the interstices of a soil column due to the tortuosity and porosity effects.

Our numerical experiments are performed in a two-dimensional square domain (0, ) (0, )L Lx yΩ = × , with boundary conditions 0d v⋅ = , on 0,x Lx=

and 0,y Ly= . The domain has 0.1 m 0.1 m× discretized by 50×50

computational grid, and we use a time-step length of 60s . We have as initial condition the ammonium (NH4

+) concentration of 2 39.2 10 kg/m−× in the

leachate repository and the concentration of 31.815 kg/m in the soil column. The following data are held fixed in our experiments: porosity 0.76ε = , diffusion coefficient in the water 10 26.342 10 m /sDw

−= × , water velocity 61.0 10 m/suw−= × , kinetic constant 22.5 10k −= × , equilibrium concentration

1 37.544 10 kg/mceq−= × , exponent number 1.75η = , soil particle average

diameter 65.0 10 mds−= × , and soil particle form factor 1.0sϕ = .

The numerical results are presented in fig. 2 at 72 hours. A concentration profile of the ammonium can be observed as a result of the solute transport that takes place in the experimental cell. The diffusion process tends to smooth the sharp concentration discontinuity at the interface between the leachate repository and the soil column. Our numerical solver is able to capture such physical behavior, even near that interface. In this figure, experimental results are also depicted. We can note that our results are in good agreement with the experimental results.

In order to investigate the evolution in time of the concentration profile, we simulate the experiment at different times: 18, 36 and 72 hours. Fig. 3 shows the respective concentration profiles. Also, we can confirm that our numerical solver duplicates the physical behavior along the time quite well.

5 Discussions

A two-dimensional method for the simulation of the contaminant transport in experimental cell is developed by combining a sorption model to determine the rate of mass transfer of the contaminant with a numerical scheme to solve accurate and efficiently the transport problem.



Figure 2: Concentrations of the ammonium as a function of the height at the experimental cell, at 72 hours.

Figure 3: Profiles of the concentrations of the ammonium at times t = 18, 36 and 72 hours.



We conclude that the model with a sink term representing the sorption process describes adequately the behavior of the contaminant concentration in the experimental cell. Others models of sorption process do not take into account the rate of mass transfer that takes place on the interface between the soil and the contaminant solution. In reference [4] several simulation results are obtained for different models of sorption process, where is evidenced the advantage of the sorption model that we have applied in this paper. The accuracy of the parallelizable iterative procedure is tested against observed values for this transport problem. Such procedure permits to implement the algorithm in distributed machines in order to save on computer memory and execution time. This parallel computation can also be an efficient way to simulate the transport in the experimental cell considering the soil and leachate regions separately, at which we can use distinct diffusion coefficients.

Acknowledgement

The authors wish to acknowledge the financial support by the FAPERJ/Rio de Janeiro through Grant E-26/171.222/2006.

References

[1] Pinto, I.C.R, Modelamento e Simulação Computacional da Migração dos Íons do Chorume em Meio Poroso, Dissertação de Mestrado, Programa de Pós Graduação UFF, Volta Redonda, Brazil, 2004.

[2] Ritter, E., and Gatto, R., Personal communication, 2003, Report PIBIC/ Universidade do Estado do Rio de Janeiro, Brazil.

[3] Douglas, J., Paes Leme, P.J., Roberts, J.E., and Wang, J., A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods. Numerical Mathematics, 65, pp. 95, 1993.

[4] Foster, A.S., Simulação Computacional e Validação da Migração de Íons de Chorume no Solo através de um Modelo Baseado nos Fenômenos de Transferência de Massa, Dissertação de Mestrado, Programa de Pós Graduação UFF, Volta Redonda, Brazil, 2006.

[5] Poirier, D.R., and Geiger, G.H., Transport Phenomena in Materials, Ed. TMS, 509p, 1994.

[6] Kawasaki, N., Kinoshita, H., Oue, T., Nakamura, T., and Tanada, S., Study on Adsorption Kinetic of Aromatic Hydrocarbons onto Activated Carbon in Gaseous Flow. Journal of Colloid and Interface Science, 275, pp. 40-43, 2004.

[7] Incropera, F.P., and Wiit, D.P., Fundamentos de Transferência de Calor e de Massa, Ed. Livros Técnicos e Científicos S.A.: Rio de Janeiro, Brazil, 1990.

[8] Douglas, J., Paes Leme, P.J., Pereira, F., and Yeh, L.M., A massively parallel iterative numerical algorithm for immiscible flow in naturally fractured reservoirs, International Series of Numerical Mathematics, ed. J.



Douglas, Jr., and U. Hornung, eds, Birkhäuser Verlag: Basel, 114, pp. 75-94, 1993.

[9] Douglas, J., Pereira, F., and Yeh, L.M., A parallelizable characteristic scheme for two phase flow I: Single porosity models. Computational and Applied Mathematics, 14, pp. 73-96, 1995.

[10] Douglas, J., Furtado, F. and Pereira, F., Parallel methods for immiscible displacement in porous media, Wuhan University Journal of Natural Sciences, 1, pp. 502-507, 1996.

[11] Douglas, J., Furtado, F., and Pereira, F., On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Computational Geosciences, 1(2), pp. 155-190, 1997.

[12] Barone, F.S., Yankful, E.K., Quigley, R.M., and Rowe, R.K., Effect of multiple contaminant migration on diffusion and adsorption of some domestic waste contaminants in a natural clayey soil. Canadian Geotechnical Journal, 26, pp. 189-198, 1989.



Permeability, porosity and surface characteristics of filter cakes from water–bentonite suspensions

V. C. Kelessidis, C. Tsamantaki, N. Pasadakis, E. Repouskou & E. Hamilaki Department of Mineral Resources Engineering, Technical University of Crete, Chania, Crete, Greece

Abstract

Water – bentonite suspensions behave as non-Newtonian fluids with exceptional rheological and filtration characteristics at low temperatures which deteriorate at temperatures higher than 1200C. Additives restore these characteristics but many of them are thermally unstable at the temperatures encountered, for example in oil-well and geothermal drilling. Greek lignite has been proven to be an excellent additive for water–bentonite suspensions at temperatures up to 1770C. In this work we attempt to assess the reason for such good performance by studying the surface characteristics and the permeabilities of filter cakes of water – bentonite suspensions with and without the additive (various lignite types) after exposing the samples to thermal static aging at 1770C for 16 hours. The filter cakes are produced with an American Petroleum Institute filter press allowing filtration for sufficient time to produce a filter cake with adequate thickness. The surface morphology of the filter cakes has been assessed with a scanning electron microscope. The permeabilities of the filter cakes were determined with an in-house technique which creates a ‘wet core’ of the filter cake of sufficient thickness and the water permeability is measured in a Hassler type meter. The differences between the reference samples (cakes from bentonite suspensions at room temperature) with cake samples from thermally aged water–bentonite suspensions and water–bentonite–lignite suspensions both in surface characteristics and in cake permeability are noted and discussed. Keywords: filter-cake, permeability, lignite, bentonite, high-temperature.



doi:10.2495/MPF070171

1 Introduction

The creation of low permeability filter cakes is one of the desirable properties of water–bentonite suspensions used as drilling fluids in order to minimize fluid loss into permeable formations which could be detrimental to hydrocarbon identification and production. The filtration properties of bentonite-water suspensions are greatly affected by the way bentonite particles associate and the state of the suspension, being flocculated or deflocculated, aggregated or dispersed. The best filtration performance is when a clay suspension is deflocculated and dispersed since the very small clay particles give low porosities and permeabilities of the filter cake that is formed. When bentonite particles are flocculated, they are larger, giving higher porosities and permeabilities. Soluble salts in muds increase cake permeability but thinners usually restore these permeabilities because they disperse clay aggregates into smaller particles. Filter cake permeabilities are of the order of 10-2 mD for flocculated suspensions, of the order of 10-3 mD for untreated fresh water muds and of the order of 10-4 mD for muds treated with thinners [1]. Fluid loss through such filter cakes is generally measured in the laboratory in a single pressure drop, usually 6.9 bar (100 psi), with an American Petroleum Institute standardized procedure [2, 3]. In reality, however, the filter cakes are exposed to different differential pressures and different drilling fluid formulations requiring thus a general understanding of the properties of filter cakes [4] which will help predict not only fluid loss in conditions different from the lab but also the behavior under extreme conditions like high temperatures which are encountered now more often in oil-well exploration. Filter cakes of bentonite-water suspensions are low in permeability, compressible and compactable [5]. To obtain the permeability of the filter cake, k , a permeating fluid of known pressure gradient should be applied to the sample and the permeable flux should be measured. If the applied pressure is large in order to give a measurable flux, the sample may deform. In addition, sealing of the boundaries of a wet and deformable sample may be difficult. Thus, measuring the permeability of the filter cakes in a direct way over a large porosity range is very difficult. It is for this reason that specific values of the permeabilities of filter cakes are not usually reported in the literature. Indirect techniques for measuring cake permeability have been reported by Meeten and Sherwood [5] using an inversion technique, which requires data obtained from filtration measurements at a series of pressure gradient values. The good properties of bentonite-water suspensions deteriorate at temperatures above about 120°C [1, 6]. When drilling stops, the drilling fluid may stay static for a long time while it is exposed to high temperatures and strong gels may develop which cause excessive pressure drop when flowing and do not form good filter cakes. Therefore, water–bentonite suspensions are treated with various materials, to enable them to withstand these high temperatures [7]. The stability of bentonite suspensions at high temperatures may be improved either by modifying the surface charge of bentonite particles or by introducing a



steric barrier against agglomeration using various additives which may be either modified, or non-modified, natural products like lignosulfonate complexes with various metals, tannins, humic acid, lignite and modified lignite, synthetic polymer products, mono- or poly-acrylic acid [8, 9]. Lignite has been used together with many other substances like sodium chromate as alkali solubilized lignite to improve filtration and thermal stability of chrome-lignosulfonate drilling fluids [1, 7, 10]. Results have been reported [11, 12] which show that several Greek lignite types can be used as additives, at optimum concentration of 3% in water–bentonite suspensions giving exceptional filtration control properties even after the bentonite-lignite-water suspension has been exposed to 177 0C for 16 hrs. The performance has been equal or sometimes even better to suspensions with a commercial lignite additive. However, the mechanism of action of the lignite additives for improving the performance of water–bentonite suspensions at high temperatures has not been understood. It is the intention of this ongoing research work to attempt to understand some of the mechanisms involved and processes that take place in such systems. This is accomplished by examining the surface characteristics of filter cakes using scanning electron microscopy and by directly measuring filter cake permeabilities, of filter cakes created with different water–bentonite suspensions, using two different bentonites and various lignite types as additives. In the present work the methodology that was developed will be presented together with preliminary experimental results.

2 Experimental procedure

Two sodium bentonites were used, a Greek bentonite (Zenith, kindly provided by S & B Industrial Minerals S.A.) and a Wyoming bentonite (kindly provided by Baroid – Cebo Holland). Various lignite types from different places in Greece, which were provided by the Greek Institute of Geological and Minerealogical Exploration [11] as well as a commercial lignite (Caustilig) kindly provided by M-I Drilling Fluids were used as additives. The particle size of both bentonites is finer than 70 µm, thus meeting the API 13A specifications [2], while lignite samples were ground, when needed, to less than 70 µm. The suspension preparation procedure followed the specifications of the American Petroleum Institute for drilling fluids [2, 3]. The water–bentonite suspension constitutes of deionized water and bentonite clay in the proportion of 6.42 gr of bentonite in 100 gr of deionized water, while in case of additive addition the proportion is 3 gr of lignite in 100 gr of deionized water. After mixing, the suspensions were either stored in sealed containers for full hydration for about 16 hours at room temperature (hydrated samples) or placed in a high temperature aging cell, pressurized at 100 psig to avoid the evaporation of water and statically aged at 177°C for 16 hours in a portable oven (thermally aged samples) [11]. The static aging procedure simulates the behaviour of the static muds in high temperature wells. The pH of the suspensions was 9.82 for Zenith and 9.00 for Wyoming bentonite. At pH range of 9.0 to 10.0 the rheological and filtration properties of water–bentonite suspensions are insensitive to changes in pH [13].



After aging, the filtration properties of the suspensions were measured in a Low Pressure – Low Temperature (LPLT) filter press (Fann 30201). The Whatman filter paper used in the API filter press has a retention size of 2.7 µm, an area of 4560 cm2 and offers no hydraulic resistance to the flow of water which flowed 100 to 1000 times faster compared to case when there was drilling fluid in the filter press [14]. Moreover, different filter paper retention size has also no effect on the filtration properties. The API filtration procedure allows for 30 min filtration time and the filtrate is measured over this period and reported as fluid loss per thirty minutes. The filter cake that is produced over this period is fairly thin with a thickness of one to three millimetres [1, 11]. Investigation of surface morphology of the cake can be performed on a cake of such thickness as the material required is extremely small. However, for the permeability measurements, a cake of sufficient thickness is required for use on the permeability apparatus. This was created by allowing filtration in the filter press for about 16 hrs and at that time only a very low volume of filtrate was flowing from the press. This procedure gave a fairly thick filter cake of approximately 10-15 mm thickness while at the same time it resulted in an almost uniform cake concentration allowing full compaction of the filter cake. Similar procedures have been followed by Meeten and Sherwood [4] and Sherwood et al. [15] who have confirmed experimentally the uniform cake thickness. Drying of the filter cake for use in SEM was accomplished under room temperature conditions (~25°C), without placing the filter cake in an oven or other drying equipment. It was observed that few days were needed (~3-4 days) for the filter cake to become completely dry. The cake was then prepared for SEM observations. Analysis of morphology of filter cakes created by water–bentonite suspensions have been performed in particular for drilling fluid characterization by Porter [16], Hartmann et al. [17], Plank and Gossen [18] and Chenevert [19]. In this work the microstructure of the fillter cakes was studied by scanning electron microscope (SEM) JEOL JSM 5400, working at 15 kV of electron accelerating voltage. The dried filter cakes were mounted and gold coated with a layer about 10 nm thick by using a vacuum of 10-3 Torr metal-coating process. Microchemical qualitative analyses of clay and lignite particles were carried out using an EDS energy dispersive X-ray analyzer INCA Energy 300. Each sample was studied at several magnifications. The x3500 and x7500 magnifications were taken as optimal for study of the microstructure details and the results presented are at the x7500 magnification. The permeability of the mud cake was measured using an in-house developed experimental setup based on a Hassler type core holder. The mud cake as it was produced from the API filter press was loaded in a ring with 2.54 cm external diameter, 0.1cm thickness and 0.5 cm length. The ring was subsequently placed between two Berea core samples, 2.54 cm diameter and 2 cm long each, of known permeability. The specimen, consisting of the two cores and the ring containing the mud cake was encased in a thermo-shrinkage plastic (Figure 1) and placed in the sample holder of a Hasser type core holder. Water was injected in the sample using an Isco positive displacement pump at a constant pressure of



105 psi. After equilibration, the flow though the core was measured and the permeability of the filter cake was determined assuming flow though porous media in series.

Figure 1: In-house developed specimen used for permeability experiments.

3 Experimental results

SEM and permeability results are presented for filter cakes created using two bentonites and two lignites, a commercial product (Γ) and a Greek lignite (ΤΗ7). In Figure 2 the SEM pictures of the tested filter cakes are shown. All the pictures were taken in a plane perpendicular to the flow plane of the filter cakes. Evident differences in the character of the formed microstructure of the different filter cakes can be observed. The microstructure of the hydrated Wyoming filter cakes (Fig. 2(a)) are characterized by large amounts of leafs placed very closely to each other and thus creating compact orientated layers of smectite particles (EDS analysis). The hydrated Zenith filter cakes (Fig. 2(b)) create very similar microstructure to the Wyoming ones, with comparable particle sizes, densities and compactness of individual grains. The filter cakes from the thermally aged Wyoming and Zenith suspensions (Fig. 2(c), 2(d)) present a more permeable microstructure, which is characterized by a large amount of leafs with open-air voids having small interfacial zones and mutual bonds. The SEM micrographs of the filter cakes treated with commercial lignite Γ (Fig. 2(e) and 2(f)) and with Greek lignite TH7 (Fig. 2(g) and 2(h)) show that the cakes have undergone a reduction in porosity compared to the corresponding thermally aged ones. The smectite particles draw closer and their interaction increases, so they give a less permeable microstructure. In Figure 3 the measured permeabilities of the tested filter cakes are shown. The results reveal first of all that on the two identical tests that have been performed with Wyoming-bentonite filter cakes, the measured permeabilities are within 5% of each other, indicating the measurement capabilities and the repeatability of the system set-up. Secondly, the values of the permeabilities of all samples are very small, of the order between 10-4 and 10-3 mD, close to the permeability values of filter cakes observed from fresh-water muds and muds treated with thinners [1]. Evaluation of the Wyoming-bentonite results shows the significant reduction in permeability values on the samples treated with



Figure 2: Scanning electron micrographs (SEM) of tested filter cakes.

Magnification x7500: (a) H-W, (b) H-Z, (c) TA-W, (d) TA-Z, (e) TΑ-W+Γ, (f) TA-Z+Γ, (g) ΤΑ-W+TH7, (h) ΤΑ-Ζ+TH7 (H: hydrated, TA: thermally aged, W: Wyoming, Z: Zenith).

9 µm 9 µm

(a) (b)

(c) (d)

(e) (f)

(g) (h)



0,0E+00

2,0E-03

4,0E-03

6,0E-03

8,0E-03

1,0E-02

W W W+Γ W+TH7 Z Z+Γ Z+TH7

perm

eabi

lity

(mD

)

Figure 3: Measured permeabilities of filter cakes of the tested bentonite-water suspensions. All suspensions have been thermally aged.

lignite. The effect of the different lignite types is different, with the commercial lignite (W+Γ) giving a permeability ratio to the permeability with only bentonite of 4.1/1.21=3.4, better than the ΤΗ7 lignite, which gives a ratio of 4.1/2.89=1.4. Evaluation of the Zenith bentonite suspension results shows the higher permeability values of the filter cake of the Zenith suspension, compared to the Wyoming counterpart, with a ratio of the two permeabilities of 9.17/4.1=2.2. Lignite addition lowers again the permeability of the filter cakes, with the different lignites having different effect. Better performance is observed with the commercial lignite (Ζ+Γ) with a ratio of permeabilities of 9.17/1.45=6.3, while lignite ΤΗ7 exhibits a very good performance as well, giving a ratio of 9.17/1.69=5.4. Furthermore, the values of the permeability of the filter cakes obtained with the Zenith-suspensions and with the two lignite types are comparable to the Wyoming counterparts.

4 Discussion

Based on the analysis of the SEM micrographs of the filter cakes created with water–bentonite suspensions, it appears that the structure of the filter cake changes according to the additive in the suspension and to the treatment it has undergone. When no additive is present and the suspensions has been only hydrated for 16 hrs, the bentonite platelets, Zenith or Wyoming, are aligned in a direction almost normal to the flow direction, creating a network structure that results in a very low permeability of the filter cake, thus giving low fluid loss value and hence making the suspension excellent for use in drilling applications. When the same suspensions are thermally aged to 177°C, the filter cakes created afterwards offer a more permeable structure, which results both from the association of several clay platelets and the opening up of the platelets thus



leaving more open space. This is observed for both bentonite types. The addition of the two lignites studied, Γ and TH7, presents a different filter cake structure from the previous two. The micrographs show that the clay platelets are smoothly associated, although not in the same way as when the bentonite is only hydrated, thus leaving probably similar open space to the hydration case resulting in similar permeability values among each other and with the hydrated case and much smaller than the case when the suspension is thermally aged. No identifiable differences can be reported from these SEM micrographs from the filter cakes derived with the different lignite types. The permeability measurements are similar to the SEM micrograph observations and show the decrease in permeability values for the filter cakes derived with the suspensions treated with lignite. However, the permeability measurements indicate differences also among the lignite types. However, it should be noticed that the values of the permeability of the filter cakes are extremely small when compared, for example with permeabilities of the permeable formations, which are of the order of 0.1 to 1000 mD. Thus, the developed technique allows for the identification of differences among the lignite types, but one should look at the essential differences which are the ones between bentonite suspensions and bentonite-lignite suspensions.

5 Conclusions

A methodology has been established for the evaluation of the permeability and the surface characteristics of filter cakes created by using different water–bentonite suspensions. The methodology involves the creation of adequate thickness filter cake in an API filter press, the evaluation of surface morphology of the filter cake by scanning electron microscopy and the measurement of the permeability of the filter cake. The permeability measurement presented significant challenges which have been resolved. The thick filter cake is placed between two cores of known permeability and thickness, all held together in a thermo-shrinkage plastic and put in a Hassler type holder for the permeability measurement with water. The technique has been tested and gives repeatable results. It should be stressed that permeability measurements of filter cakes rarely appear in the literature. The technique has been applied to study the characteristics of filter cakes from Wyoming and Zenith bentonite-water suspensions at 6.42% wt. which were hydrated, thermally aged and thermally aged after adding different lignites at 3% wt. SEM micrographs reveal that hydrated bentonite suspension filter cakes form a network of platelets almost normal to the flow direction. This structure opens up when the suspension is thermally aged and multiparticle association is observed increasing the permeability of the filter cake. The addition of the two lignite types studied gives a structure which is closed again and not leaving much open space for the flow of the filtrate. There were not many differences observed among the filter cakes of the two different lignites that were analyzed with SEM.



The permeability measurements give permeability values for the thermally aged suspensions for both bentonites used which are typical of flocculated suspensions, with a variation among the two bentonite types. Additions of lignite at 3% wt. results in drastic reduction of the permeability of filter cakes derived after thermal aging of the suspension. Variations in permeability values have been observed between the different lignites tested.

This work has been funded by the Greek State, under the contract Pythagoras II, Technical University of Crete - Project 8. The provision of Greek lignites by IGME, Greece, of Zenith bentonite by S & B Industrial Minerals S.A., of Wyoming bentonite by Baroid-Cebo Holland and of Caustilig by M-I Drilling Fluids are greatly appreciated.

References

[1] Gray, Η. C. Η. and Darley, G. R., Composition and properties of oil - well drilling fluids, Gulf Publishing Co., 6th Edition, Houston, USA, 1980.

[2] American Petroleum Institute Specifications 13A, Specification for drilling fluid materials, 1993.

[3] American Petroleum Institute Specifications 13I, Recommended practice standard procedure for laboratory testing drilling fluids, 2000.

[4] Sherwood, J. D. & Meeten, G. H., The filtration properties of compressible filter cakes. Journal of Petroleum Science and Engineering, 18, pp. 73-81, 1997.

[5] Meeten, G. H. and Sherwood, J. D., The hydraulic permeability of bentonite suspensions with granular inclusions. Chemical Engineering Science, 49(19), pp. 3249-3256, 1994.

[6] Bleler, R., Selecting a drilling fluid. J. Petr. Techn., 42(7), pp. 832 – 834, 1990.

[7] Clark, R. K., Impact of environmental regulations on drilling fluid technology. J. Pet. Techn., 46(9), pp. 804 – 809, 1994.

[8] Rabaioli, M.R., Miano, F., Lockhart, T. P., and Burrafato, G., Physical/chemical studies on the surface interactions of bentonite with polymeric dispersing agents, SPE 25179. Intern. Symposium on Oilfield Chemistry, New Orleans, LA, U.S.A, 1993.

[9] Burrafato, G., Miano, F., Carminati, S. and Lockhart, T. P., New chemistry for chromium free bentonite drilling fluids stable at high temperatures, SPE 28962. SPE Intern. Symposium on Oilfield Chemistry, San Antonio, TX, USA, 1995.

[10] Nyland, T., Azar, J. J., Becker, T. E. and Lummus, J. L., Additive Effectiveness and Contaminant Influence Control on Fluid Loss control of Water-Based Muds. SPE Drill. Engr., 6, pp. 195–203, 1988.

[11] Mihalakis, A., Makri, P., Kelessidis, V.C., Christidis, G., Foscolos, A. and Papanikolaou, K., Improving Rheological And Filtration Properties of



Acknowledgements

Drilling Muds with Addition of Greek Lignite. Proceedings of the 7th National Congress on Mechanics, edited by A. Kounadis, K. Providakis and G. Exadaktylos, Chania, Greece, pp. 393-398, 2004.

[12] Kelessidis, V. C., Mihalakis, A., Tsamantaki, C., Rheology and rheological parameter determination of bentonite–water and bentonite–lignite–water mixtures at low and high temperatures. Proceedings of the 7th World Congress of Chem. Engr., Glasgow, 2005.

[13] Alderman, N., Ram Babu, D., Hughes, T. & Maitland, G., The rheological properties of water-based drilling fluids - effect of bentonite chemistry, Speciality Chemicals, Production, Marketing and Applications, 9 (5), pp. 314-326, 1989.

[14] Meeten, G. H., Shear and compressive yield in the filtration of a bentonite suspension. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 82, pp. 77-83, 1994.

[15] Sherwood, J. D., Meeten, G. H., Farrow, C. A., Alderman, N. J., The concentration profile within non-uniform mud cakes. J. Chem. Soc. Faraday Trans., 87(4), pp. 611-618, 1991.

[16] Porter, K. E., A basic scanning electron microscope study of drilling fluids, paper SPE 8790. Presented at the 4th Symposium on Formation Damage Control, Bakersfield, CA, 1980.

[17] Hartmann, A., Ozerler, M., Marx, C. and Neuman, H-J., Analysis of mudcake structures formed under simulated borehole conditions. SPE Drill. Engr., pp. 395-402, 1988.

[18] Plank, J. P. and Gossen, F. A., Visualization of fluid-loss polymers in drilling mud filter cakes. SPE Drill. Engr., pp. 203-208, 1991.

[19] Chenevert, M. E., Filter cake structure analysis using the scanning electron microscope, paper SPE 22208, unsolicited. Society of Petroleum Engineers, Richardson, TX, 1991.



Section 3 Interfaces

Investigation of slug flow characteristics in inclined pipelines

J. N. E. Carneiro & A. O. Nieckele Department of Mechanical Engineering, Pontifícia Universidade Católica de Rio de Janeiro - PUC/Rio, RJ, Brazil

Abstract

In the present work a numerical analysis of the slug flow in inclined pipelines is performed with an aim to improve the understanding of slug flow characteristics over hilly-terrain section. The solution is obtained with the two-fluid model on its one-dimensional form. It consists of two sets of conservation equations of mass and linear momentum for the liquid and gas phases. The slug capturing methodology involves the numerical solution of the equations using a finite volume formulation, which is capable of naturally predicting the onset of slugging from a stratified flow regime, as well as the growth and collapse of the slugs. Flows of an oil–gas mixture in slightly inclined pipe configurations are investigated. Three types of pipelines were considered: horizontal, descending and a V-section pipeline. The influence of the gravity effect in average slug parameters, such as frequency, velocity and length is addressed. Qualitative comparisons with experimental observations in the literature show that the methodology seems to be able to correctly predict the effect of pipe inclination on the occurrence (or not) of the slug regime, as well as different overall slugging behaviour in V-section pipes if different inlet gas and liquid superficial velocities are imposed. Keywords: slug flow, incline pipeline.

1 Introduction

Slug flow is a two-phase flow pattern which is characterized by a sequence of packs of liquid separated by long gas (Taylor) bubbles flowing over a liquid film inside the pipe, and is normally associated with high pressure-drops and a considerable degree of intermittency in the system. In offshore production



doi:10.2495/MPF070181

systems, for example, stabilized gas and liquid flow rates are normally sought to ensure a proper operation of the plant, and separation equipments are often designed for such conditions. In this sense, previous knowledge of the flow patterns expected are of extreme importance, and if slug flow is likely to occur, it is not only important to know its mean behaviour but also the statistical details such as the maximum slug length expected, which dictates an proper sizing of receiving equipments. The onset of slugging in horizontal or nearly horizontal pipes is caused by two mechanisms: the natural growth of small random fluctuating disturbances at the gas-liquid interface of stratified flow (namely by a Kelvin–Helmholtz mechanism); and/or the liquid accumulation at valleys of hilly terrain pipelines with sections of different inclinations, also called terrain slugging [1-5]. Wave coalescence was also observed to be an important mechanism acting on slug formation, especially at higher gas flow rates in horizontal pipes [6-7]. Also in the V-section studied by Al-Safran et al. [5], this initiation mechanism was observed at relatively high gas-flow rates and low liquid flow rates, where smaller waves were unable to block the elbow. Slug front and tail do not necessarily travel at the same velocities. A complex flow dynamics exists in which slugs may grow, collapse and merge with each other, different slugs having also different speeds [3, 8]. In this process, the mean slug length normally increases in the flow direction, because small slugs are unlikely to be stable due to bubble wake effects and often degenerate into long waves that are absorbed by faster slugs [3]. As a consequence of the fact that slug length, velocity and frequency are interrelated quantities, it follows that the slug frequency is likely to diminish towards the pipe ending [3, 9]. As pointed out by several authors [9, 10], the spatial evolution of slugging in the pipe may require at least about 200 – 300 diameters from the inlet region to achieve a developed flow. Since slugs evolve from randomly generated waves at the gas-liquid interface, the flow can also be expected to have a stochastic behaviour. In this sense, one speaks of a statistical steady state condition when the time averaged slug parameters (e.g., close to the pipe end section) do not change. Flow pattern studies [2] have shown that the pipe inclination can have a very significant effect on the stratified to slug transition, even at very small angles. It was found out that the stabilizing effect of gravity causes transition from downward stratified flow to occur at higher liquid superficial velocities (for a given gas superficial velocity), while for upward sections the transition is anticipated. When slugs travel through a hilly terrain pipeline with different pipe inclinations, they probably undergo a change in its characteristics when moving from section to section. In addition, slugs can be generated at low elbows (V-sections) or dissipated at top elbows (λ-sections) originating a very complex pattern [5, 11]. Al-Safran et al. [5] conducted an experimental study investigating the slug flow characteristics over a hilly terrain pipe with a V-section, focusing on the mechanisms of slug initiation and characteristics of slugs initiated at the lower dip. An attempt is also made in order to group sets of superficial gas and liquid velocities in flow categories (superimposed on typical



steady state flow pattern maps for the downhill upstream pipe), according to the influence of the V-section on the characteristics of developed slug flow upstream of the downward pipe. As described by Issa and Kempf [8], transient models in the context of pipeline slugging usually solve the Two-Fluid Model equations [12] in its transient one-dimensional version and can be grouped into three categories: empirical slug specification, slug tracking and slug capturing. In contrast to the other ones, a key feature of the slug capturing methodology is the capability of predicting the evolution from stratified to slug flow in a natural manner, i.e., there is no need to incorporate any transition criteria assuming that slugs were generated somehow in the pipe (e.g., by use flow pattern maps). This means that the natural outcome of the solution of the equation system can be either the maintenance of stratified flow in the pipe, or the change in regime if conditions are such that slugs develop in the system. If transition occurs, slugs may grow or decay as they travel downstream in the pipe, and no empirical correlations for slug parameters need to be specified. Thus, the set of equations is maintained even when the regime changes and the slug dynamics is an automatic consequence of the solution of the system. At the present work, the Two Fluid Model is employed to predict the slug formation in horizontal and slightly inclined pipeline. To validate the methodology, comparison is performed with the experimental data of Al-Safran et al. [5].

2 Mathematical modelling

The mathematical model selected is based on the slug capturing technique, in which the slug formation is predicted as a result of a natural and automatic growth of the hydrodynamic instabilities [8, 13]. Both stratified and slug pattern are modeled by the same set of conservation equations based on the Two-Fluid Model. Additionally, closure relations are also included. The liquid is considered as incompressible, while the gas follows the ideal gas law, ρG=P/(RT), where R is the gas constant and T is its temperature, which was considered here as constant. Pressure P was considered constant long the cross section, being the same, for the liquid PL, gas PG and interface (P=PG=PL). Additionally, it was assumed that there is no mass transfer between phases. The governing mass and momentum equations in the conservative form can be written as

0=∂

∂+

∂∂

xu

tGGGGG )()( αραρ ; 0=

∂∂

+∂

∂x

ut

LLLLL )()( αραρ, (1)

,)cos(

)sin()()(

iGwGG

GGGGGGGGG

FFxhg

gxP

x

u

tu

−−∂∂

−

−−∂∂

−=∂

∂+

∂∂

βαρ

βαρααραρ

2

(2)



,)cos(

)sin()()(

iLwLL

LLLLLLLLL

FFxhg

gxP

x

u

tu

+−∂∂

−

−−∂∂

−=∂

∂+

∂∂

βαρ

βαρααραρ 2

(3)

where αG +αL = 1. The subscripts G, L, and i concern the gas, liquid phases and interface, respectively. The axial coordinate is x, ρ and α are the density and volumetric fraction, u is the velocity. The pipeline inclination is β, h is the liquid level inside the pipe, and g is the gravity acceleration. The third term on the right side of eqs. (2) and (3) are related with the hydrostatic pressure at the gas and liquid, respectively. The term F=τ S / A is the friction force per unit volume between each phase and the wall and between the phases (at the interface), where τ is the shear stress, S is the phase perimeter and A is the pipe cross section area. The shear stress is τ = f ρ | ur| ur / 2, where ur is the relative velocity between the liquid and wall, the gas and wall, or gas and liquid. Closure relations are needed to determine the friction factor f. The flow was considered in the laminar regime, when the Reynolds number Re, was smaller the 2100 (ReG ; Rei and ReL for the gas, interface and liquid, respectively). The Hagen–Poiseulle formulas were employed for the gas-wall and interface laminar friction factor and the correlation of Hand [14] for the liquid-wall laminar friction factor, while the correlation of Taitel and Dukler [1] was adopted for the turbulent gas-wall and interface friction factor and the Spedding and Hand [15] correlation for turbulent the liquid-wall friction

laminar: sL

Li

iG

G fffRe

,Re

,Re

241616===

turbulent: ,)Re(

.,Re

.,Re

.... 1390250250

0262004600460sLL

Li

iG

G fffα

=== (4)

where αL is the hold-up (liquid volumetric fraction). The Reynolds numbers are defined as

( ) ,ReGiGGGG

G SSuA

µρ

+=

4 ( ) ,Re

GiG

GLGGi SS

uuAµ

ρ+

−=

4

LL

LLLL S

uAµ

ρ4=Re ,

LLLs

LDUs

µρ

=Re (5)

where µ is the absolute viscosity and D is the pipe diameter. The last Reynolds in eq. (5) is based on the liquid superficial velocity, i.e., the ration of the liquid volume flow rate to the total cross section area of the pipe UsL=QL/A = αL uL.



The geometric parameters such as gas and liquid areas (AG, AL), wetted perimeters (SG , SL), and interface width Si where obtained from the liquid height h [13].

3 Numerical method

The conservation equations were discretized by the Finite Volume Method [16]. A staggered mesh was employed, with both phases’ velocities stores at the control volume faces and all other variables at the central point. The interpolation scheme upwind and the implicit Euler scheme were selected to evaluate the space and time derivates, respectively. The set of resulting equations consists of two momentum equations, one pressure equation (global mass conservation) and one gas volumetric fraction (gas mass conservation). These equations were solved sequentially, through an iterative method [13]. The time step was specified to guarantee a Courant number equal to 0.5 [8], therefore, the time step was obtained from ∆t= 0.5 ∆xi | umax |, where umax is the maximum velocity in the domain. For each time step, due to the non linearities of the problem, the sequence of conservation equations were solved in an iterative process, until convergence was obtained, that is, until the residue of all equations became smaller than 0.0001.

4 Results

At the present work a numerical analysis of the slug flow in inclined pipelines is performed aiming to improve the understanding of slug flow characteristics over hilly-terrain section. Three types of pipelines are investigated: horizontal, descending and a V-section pipeline. A V-section pipeline was defined based on the experimental work of Al-Safran et al. [5]. The pipeline consists of descending and ascending sections with length of 21.34 m, and inclination of β =–1.93º and β = + 1.93º in relation to the horizontal direction as illustrated in Fig. 1a. To guarantee a smooth transition between the downward and upward sections, a small horizontal section of 0.3 m joining the two parts was added. The pipeline diameter is equal to D=0.0508 m. The total length of the pipelines is equal to L= 42.98 m.

β β

L=21.34 m

L=0.3m

L=21.34 m

αG, UsG UsL

patm

(a)

αG, UsG UsL

pa tm

L=42.9 8 m

β

αG, U sG U sL

patm

L=42 .98 m

(b)

(c)

Figure 1: Configurations considered: (a) V-section pipeline; (b) horizontal pipeline (c) slightly inclined pipeline.



To investigate the effect of small inclinations in the slug flow parameters, the same conditions were tested in a horizontal pipeline, Fig. 1b, and in a pipeline with a downward inclination of β =–1.93º, Fig. 1c. All pipelines have the same total length L. The same two-phase fluid mixture (air and oil) employed by [5] was defined. The air was considered as ideal gas with gas constant R=287 N m /(kg K), with molecular viscosity of µG=1.796 × 10-5 Pa s. The oil density was ρL =890.6 kg/m3, and molecular viscosity a µL =1.02 × 10-2 Pa.s. The inlet liquid holdup αL was defined as 0.4 (αG=0.6) and a constant atmospheric pressure patm was kept at outlet. The initial condition was defined as a stratified steady state flow, that is, constant liquid height along the pipeline, with constant liquid and gas velocities, and pressure distribution obtained by solving the momentum conservation equation, considering equilibrium stratified flow. The flow field is determined based on the two-fluid model [8, 13] by the solution of the momentum conservation equations for each phase, continuity of the gaseous phase and total mass conservation. Two situations were considered, classified as Category I and II. The gas and liquid superficial velocities were defined as UsL = 0.6 m/s and UsG = 0.64 m/s for Category I and UsL = 1.22 m/s and UsG = 1.3 m/s for Category II. Figures 2 illustrates successive liquid hold-up profiles along the pipeline in time for the horizontal and descending cases for Category I, while Fig. 3 corresponds to Category II. The liquid hold-up profile in time for the V-section pipeline is shown in Fig. 4 for both Categories.

0 7 14 21 28 35 42x ( m )

20 s

30 s

40 s

50 s

60 s

0 7 14 21 28 35 42

x ( m )

20 s

30 s

40 s

50 s

60 s

(a) (b)

Figure 2: Successive hold-up profiles in time. Category I: UsL= 0.60 m/s, UsG = 0.64 m/s: (a) horizontal, (b) descending.

As it can be seen in Fig. 2, for Category I, slug pattern is observed in the horizontal pipeline, Fig. 2a, but it is not observed in the descending pipeline, Fig. 2b, due to the gravity stabilizing effect which inhibits small perturbations to



grow at the interface, inducing the slug. For the V-section pipeline shown in Fig. 4a, the slug flow is formed by the accumulation of liquid at the dip. This was the same behavior observed experimentally in [5], for the same superficial velocities.

0 7 14 21 28 35 42x ( m )

20 s

30 s

40 s

50 s

60 s

70 s

0 7 14 21 28 35 42

x ( m )

20 s

30 s

40 s

50 s

60 s

70 s

(a) (b)

Figure 3: Successive hold-up profiles in time. Category II: UsL= 1.22 m/s, UsG = 1.30 m/s: (a) horizontal, (b) descending.

0 7 14 21 28 35 42

5

10

15

20

25

30

35

40

0 7 14 21 28 35 42

x ( m )

20 s

30 s

40 s

50 s

60 s

70 s

(a) (b)

Figure 4: Successive hold-up profiles in time. V-section pipeline: (a) Category I: UsG=0.64 m/s, UsL=0.60 m/s, (b) Category II: UsL=1.22 m/s, UsG=1.30 m/s.

The hold-up profiles in time for Category II, shown in Figs. 3 and 4b, show that, as oppose to the Category I case, it can be seen that due to same slug formation mechanism, the slugs are formed approximately at 7 m from the inlet



for all cases. The effect of gravity is to delay just a little the slug formation. It can also be seen, that due to the high frequency, there is not enough time to occur liquid accumulation at the dip of the V-section pipeline, therefore, there are no additional slugs being formed. Once again, these observations agree with the experiments of [5]. For the horizontal and slightly inclined pipeline a pattern map was built based on the studies of Taitel and Dukler [1] and Barnea and Taitel [17]. The predictions obtained with the present work agreed perfectly with the pattern maps. The slug translation velocity Ut, length Ls and frequency νs were determined for the three pipelines configurations. The mean slug translation velocity Ut is inferred by the dimensionless parameter Co, based on the same correlation employed in [8],

dMot UUCU += (6)

>==

<==

53Fr020153Fr540051

.,..,..

Mdo

MdoifUandCifgDUandC (7)

where the mixture velocity UM is equal to the sum of the inlet liquid and gas superficial velocities, UM = UsL + UsG. The Froude number FrM is based on the mixture velocity as

gDU MM /Fr = (8)

The slug parameters corresponding to x = 37 m are shown in Table 1. It can be seen that for the first category the liquid accumulation at the dip leads to a superior frequency for the V-section than the horizontal case, since the length is smaller, once the velocities are similar. This tendency was also experimentally observed by [5]. It should be mentioned here, that it was only possible to perform a qualitative comparison, since the data of [5] were not available due to proprietary restriction.

Table 1: Slugs characteristics.

Co νs (1 / s) Ls/D Category I II I II I II

Horizontal 1.25 1.40 0.33 0.95 42.6 13.9 Descending - 1.39 - 0.94 - 13.5 V-section 1.23 1.37 0.38 0.83 29.6 21.3

Table 1 shows that a slightly higher velocity is found for Category II. The slug length of the horizontal and descending cases differed by 5%, and the frequency was approximately constant. However for the V-section the slug length was 58% larger, leading to a 14% reduction of the frequency in relation to other two cases. The increase in the length is due to the accumulation of liquid at the



dip, which did not induce the formation of new slugs, but increased its length. Further the liquid velocity at the ascending section is smaller, what also contributed to increase the slug length. Figure 5 shows the average slug length along the pipelines, for Category II. It can be seen that mean length is approximately the same for the horizontal and descending case, were the gravity effect is negligible due to the high velocities. Larger slug lengths are observed along the V-section pipeline, especially at the ascending section due to the accumulation of liquid at the dip as described previously. It can be clearly seen that slug length distribution changes across a symmetrical pipeline, since the gravity effect is not symmetrical.

Figure 5: Average slug length along horizontal, descending and V-section

pipelines: UsL = 1.22 m/s and UsG = 1.3 m/s.

5 Final remarks The Two Fluid Model was employed to predict the slug formation along horizontal, slightly inclined and V-section pipelines. The results obtained qualitatively agreed with the experimental data of [5]. The flow can be classified in different Categories, depending in the gravity influence to damp the slug formation. The accumulation of liquid in lower sections of the pipeline can increase not only the size of the slug, but also its velocity.

Acknowledgements

The authors thank the Brazilian Research Council, CNPq for the support awarded to this work.

References

[1] Taitel, Y. & Dukler, A.E., A model for predicting flow regime transitions in horizontal and near horizontal pipes, AIChE Journal, 22, pp. 47−55, 1976.

Ls/ D

0

5

10

15

20

25

30

35

0 100 200 300 400 500 600 700 800 x / D

Horizontal

Descending V - section



[2] Barnea, D., A unified model for predicting flow-pattern transitions for the whole range of pipe inclinations, International Journal of Multiphase Flow, 13, pp. 1−12, 1987.

[3] Taitel, Y. & Barnea, D., Two-phase slug flow, Advances in Heat Transfer, 20, pp. 83−132, 1990.

[4] Fabre, J. & Liné, A., Modeling of two-phase slug flow, Annual Review of Fluid Mechanics, 24, pp. 21−46, 1992.

[5] Al-Safran, E., Sarica, C., Zhang, H.Q. & Brill, J., Investigation of slug flow characteristics in the valley of a hilly terrain pipeline, International Journal of Multiphase Flow, 31, 337−357, 2005.

[6] Lin, Y.P. & Hanratty, T. J., Prediction of the initiation of slugs with the linear stability theory, International Journal of Multiphase Flow, 12, pp. 79−98, 1987.

[7] Woods, B. D., Fan, Z. & Hanratty, T. J., Frequency and development of slugs in a horizontal pipe at large liquid flows, International Journal of Multiphase Flow, 32, pp. 902−925, 2006.

[8] Issa, R. I. & Kempf, M. H. W., Simulation of slug flow in horizontal and nearly horizontal pipes with the two fluid model, International Journal of Multiphase Flow, 29, 69−95, 2003.

[9] Tronconi, E., Prediction of slug frequency in horizontal two-phase slug flow, AIChE Journal, 36, pp. 701−709, 1990.

[10] Barnea, D. & Taitel, Y., A model for slug length distribution in gas-liquid slug flow, International Journal of Multiphase Flow, 19, pp. 829−838, 1993.

[11] Zheng, G., Brill, J. P. & Taitel, Y. Slug flow behavior in a hilly terrain pipeline, International Journal of Multiphase Flow, 20, pp. 63−79, 1994.

[12] Ishii, M., Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[13] Carneiro, J. N. E., Ortega, A. J., Nieckele, A. O., Influence of the Interfacial Pressure Jump Condition on the Simulation of Horizontal Two-Phase Slug Flows Using the Two-Fluid Model, Proceedings of 3rd International Conference on Computational Methods in Multiphase Flow 2005, Portland, Maine, USA, pp. xxx−xxx, 2005.

[14] Hand, N.P. Gas–liquid co-current flow in a horizontal pipe, Ph.D. Thesis, Queen`s University Belfast, 1991.

[15] Spedding, P.L.; Hand, N.P., Prediction in stratified gas–liquid co-current flow in horizontal pipelines, International Journal Heat Mass Transfer, 40, pp. 1923–1935, 1997.

[16] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, 1980.

[17] Barnea, D. & Y. Taitel, Y., Interfacial and structural stability of separated flow, International Journal of Multiphase Flow, 19, pp. 387-414, 1994.



Behaviour of an annular flow in the convergent section of a Venturi meter

G. Salque1, P. Gajan2, A. Strzelecki2 & J. P. Couput1 1TOTAL, Allocation and metering group, Pau, France 2

Abstract

This paper concerns the flow metering of wet gas by a Venturi meter. In many industrial applications dealing with the gas production, annular/dispersed two phase flows are mainly observed. A theoretical approach based on mass and momentum exchange permits prediction of the pressure, droplet velocity and film thickness distributions between the upstream and the downstream pressure taps. To improve these models, an experimental work dealing with the wall film thickness distribution in the convergent part of the Venturi meter is undertaken from flow visualizations and film thickness measurements. Averaged film thickness and wave characteristics in the convergent section are quantified and compared to theoretical results. Keywords: Venturi meter, convergent section, annular flow, liquid film, thickness, interface waves, resistive probes.

1 Introduction

Liquid films flowing on solid surfaces may be observed in different fields (medicine, agriculture, automotive, spatial and aeronautics industry, energy plants, petrol and gas industry etc). Their behaviours were largely studied since few decades. Nevertheless, as far as we know, it seems that no work was published concerning the influence of a longitudinal pressure gradient. This is the case in a Venturi meter used to measure the gas and liquid production of natural reservoirs. When the gas is brought up to surface, condensation



doi:10.2495/MPF070191

Models for Aeronautics and Energetics Department, Toulouse, France Office National d'Etudes et de Recherches Aérospatiales,

phenomena occur and a liquid phase appears, formed from condensates and water. In many cases the gas volume fraction (GVF) is higher than 95% (wet gas), and the flows encountered are of an annular dispersed type. Recently, the modelling of the mass and momentum exchanges between the gas and the liquid phases was done in order to calculate the over reading differential pressure induced by the liquid phase (Lupeau et al [1]). The use of this model shows that the liquid film thickness plays an important role on the Venturi meter behaviour. So, it seems important to verify that the liquid film patterns in the meter are accurately taken into account in the calculations. In this paper, an experimental work developed to analyse the liquid film behaviour in the convergent section of the meter is presented. Two techniques were used; the first based on visualizations gives a qualitative information of the liquid film characteristics, while the second based on resistive measurements permits to quantify the longitudinal distributions.

Film thickness probes



Probe number X (mm)

1 9.6 2 31.7 3 53.8 4 75.9 5 98.1 6 117.9

Figure 1: Sketch of the model used to study the liquid film behaviour in the convergent section.

2 Experimental set up

2.1 Test rig

The tests were carried out at low pressure on the ONERA experimental flow loop. The gas flow (air) is generated by means of high pressure tanks. The gas flow rate is controlled by a sonic nozzle. The mass flow rate of liquid (water) is measured with electromagnetic flow meters. The experiments were performed at atmospheric pressure. The test section is placed in a vertical downwards orientation. It is composed of a flow conditioner, a liquid film injector, and a Venturi meter. The pipe diameter (D = 2.R) is 100 mm. For flow visualizations, a Venturi flow meter with a β of 0.6 was machined in Perpex. The half angles of the upstream convergence and the downstream diffuser were respectively 10.5° and 7.5°. The film thickness measurements were performed on a new model including only the convergent section and half of the throat section of the meter. This new device is equipped with six film thickness probes (figure 1).



Two different techniques are used to analyse the film behaviour. For flow visualizations, an argon laser sheet parallel to the pipe axis is used to illuminate the film and the images are recorded by a CCD camera placed perpendicular to the laser sheet. The film thickness is determined from a film conductance method described by Hewitt [2]. It consists of measuring the electric impedance between two flush-mounted electrodes. The geometry of the probe (pin diameter d and pin spacing s) depends on the thickness range to be explored (Hewitt et al [3]). Here, s/d = 4 with a pin diameter equal to 1 mm. Each pair of electrodes is connected to an electric circuit to measure the conductance and, after a calibration procedure, the instantaneous thickness. After a signal processing, the average film thickness, the wave amplitude the speed of the superficial waves and their wavelength are deduced.

3 Modelling of the gas liquid interaction inside the Venturi meter

To simulate the two phase flow phenomena inside a Venturi meter, it is necessary to take into account the gas/liquid film interaction near the wall, the gas/droplet interaction in the core region and also the mass flux of liquid exchanged between the film and the spray (entrainment and deposition of droplets). Two and three-dimensional calculations of this two phase flow can be considered to take into account the gas liquid interactions of the core region (droplets) [4], but even if recent developments exist to simulate the flow phenomena in the liquid film region (Volume Of Fluid (VOF) [5] or level set techniques [6]), the high grid refinement needed in the three spatial directions does not permit to follow this modelling option. An alternative approach initially developed for a Venturi scrubber has been followed by Azzopardi and co-workers [7]. They use a one-dimensional approach to describe the momentum exchanges between the gas and the liquid phase along the Venturi meter. The same approach is used by Lupeau et al [1]. The flow is divided into two regions: the convergent section and the throat. In each zone, integrated balance equations (mass and momentum conservation) are applied on the gas flow, the liquid film and the dispersed flow. In each pipe section, each flow is defined by its local velocity V and its flowing area S. In these equations, source terms are used to describe the momentum and mass exchanges. This concerns the momentum gas/liquid film interaction at the interface, the momentum exchange between the gas and droplets and the mass exchange between the film and the droplets due to the entrainment. The model supposes that no mass exchange between the liquid and the gas occurs in the meter (evaporation and condensation). The momentum transfer between the film and the gas is described through the interfacial stress τi modelled by the Wallis correlation. An atomization of the liquid film is taken into account at the convergent/throat junction. Different



2.2 Experimental techniques used for analysing the film behaviour

correlations are proposed to estimate the atomised mass flow rate (Fernandez Alonso et al [8]). Lupeau et al [1] used a correlation deduced from a weighting measurement of the liquid film upstream and downstream of the Venturi. The size of the droplet newly atomized at the throat entrance are defined from an empirical correlation (Azzopardi and Govan [9]) and their initial velocity is equal to the average liquid film velocity at the end of the convergent section. This code permits to calculate the distribution of the pressure, the film thickness and the droplets velocity from the inlet of the meter to its throat section.

4 Experimental results

4.1 Visualization of the liquid film behaviour through the Venturi meter

Visualizations of the liquid film through the Venturi meter (Lupeau et al [1]) show that the film behaviour mainly depends on the location and on the liquid volume flow rate Qvl. As a matter of fact, even if an amplification of the disturbance is observed when the gas volume flow rate Qvg is increased, the influence of the gas velocity does not seem to be the main parameter. On the contrary, the film characteristics change greatly when the liquid flow rate is increased. This can be illustrated by figure 2 which presents different snapshots of the liquid film obtained upstream of the Venturi meter for one gas velocity condition and different liquid flow rates. At low liquid flow rate, the liquid interface is formed of regular small amplitude waves associated with relatively large wavelengths.

Figure 2: Snapshots of the liquid film upstream of the convergent section (Qvg = 630 m3/h; ReD = 14 104).

When the liquid flow rate is increased, the amplitude of the waves increases and intermittent large waves, up to 4 times the averaged film thickness, are observed on the movies. On their periphery, liquid ligaments are formed which can induce the entrainment of liquid packets in the gas region and the formation of droplets (Azzopardi et al [10]). In the convergent part of the meter, the gas acceleration induces a diminution of the wave amplitude and a flattening of the film. Nevertheless, the movies reveal instantaneous breaking of waves linked to entrainment. For the higher liquid flow rates tested, an intermittent appearance of large waves is observed.



At the end of the convergent section, the waves seem to be disorganised. At the throat section, the film becomes thicker and the wavelength greatly diminishes (figure 3). Close to the interface, the number of liquid filaments increases. For the higher liquid flow rates, intermittent large waves are often noticed linked to high entrainment processes. When the film flows through the diffuser, the amplitude of the disturbances decreases and the wavelength increases. The appearance of large waves is still observed at high liquid flow rate.

Figure 3: Snapshots of the liquid film at the end of the convergent section (Qvg = 440 m3/h; ReD = 10.7 104).

Figure 4: Influence of the flow parameters on the upstream liquid film thickness.

4.2 Film thickness measurements

4.2.1 Film behaviour upstream of the convergent section The behaviour of the liquid film upstream from the meter was presented by Lupeau et al [1]. From a simple model, it was shown that the averaged film thickness is proportional to a length scale Y taking into account the liquid flow rate, the gas velocity, the gas and liquid viscosity, the gas density and the pipe radius.

( )

YUR

Q

ggg

filmvlm α

µρ

µαδ =⋅=

81

87

83

_ (1)



In Figure 4, δm variation with respect to Y is plotted and compared to the data given by Asali and Hanratty [11]. This graph shows that the Y parameter correlates the results of Asali and Hanratty [11] obtained for different flow conditions and different liquids. In the ONERA case, a scatter of the results is obtained when the air velocity changes. Nevertheless, these results indicate that the film thickness is linearly dependent on the square root of the liquid film flow rate.

Probe 1

050

100150200250

0 0.5 1 1.5 2t(s)

δ(µm

)

Probe 2

050

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δ(µm

)

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050

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)

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)

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)

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050

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0 0.5 1 1.5 2t(s)

δ(µm

)

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050

100150200250

0 0.5 1 1.5 2t(s)

δ(µm

)

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050

100150200250

0 0.5 1 1.5 2t(s)

δ(µm

)

Probe 6

050

100150200250

0 0.5 1 1.5 2t(s)

δ(µm

)

Figure 5: Signals of liquid film thickness inside of the convergent section (Qvg = 650 m3/h; Qvl = 50 l/h).

4.2.2 Modification of the film characteristics in the convergent section Typical film thickness signals are presented in figure 5. Close to the convergent section inlet (probe 1), large peaks are observed. As before, their amplitude augments with the liquid flow rate and the signal becomes noisy. As the film flows into the convergent section, these peaks decrease and fluctuations with low amplitude and high frequencies appear. The average thicknesses computed from these signals are plotted in figure 6. They are compared to distributions predicted by the code. Bold symbols correspond to film thickness measurements performed 1D upstream of the meter. First of all, experimental results indicate that the liquid film thickness increases slightly between the two first probes then decreases inside the convergent section. Comparisons between the upstream location (bold symbol) and the first



probe position indicate that the liquid film flattens at the inlet of the convergent section. This phenomenon can be induced by the transverse pressure gradient linked to the streamline curvature in this region. New experiments are planned to verify this hypothesis. The numerical results only predict a monotone thickness diminution. A great discrepancy appears on the absolute value. Further treatments are done on the signals in order to determine the amplitude, the convection velocity and the corresponding wavelengths of the longitudinal interfacial waves. In a first step, the wave amplitude evolution is analysed. Such information is used to calculate the interfacial stresses (Giroud-Garapon [13]). In a previous experimental work on liquid film behaviour on a hot inclined plate, Giroud-Garapon [13] obtained the following correlation:

⋅−⋅=

048.1111.0 *456.201Re136.1

σδτ

δg

lA (2)

In this expression, A is the amplitude of the wave, Rel is the Reynolds number of the liquid film, τg is the wall stress in dry gas and σ is the surface tension coefficient. The amplitude of the wave is deduced from the film thickness distribution D (δ) calculated from the thickness probability function P(δ).

∫=δ

δδδ0

)()( dPD (3)

%)1(%)99( =−== DD δδA (4) Comparisons between correlation and measurement results are presented in figure 7. If this correlation gives good results upstream of the plate, it seems that additional phenomena appear in the convergent which amplify the wave oscillations especially at high liquid flow rate. In a second step, the convection velocity Uc of the waves is determined from intercorrelations. In upper graph of the figure 8, the acceleration of the waves is observed. It depends on the liquid and gas flow rates. Figure 5 shows that the waves are divided into two classes, long waves with high amplitude and short waves or ripples with small amplitude. Signal processing was developed to quantify these two wave configurations. The long waves Λ which correspond to high thickness variations are measured from FFT. The small wavelengths λ are deduced from time period histogram. The evolution of these wavelengths in the convergent section is plotted on the lower graphs of the figure 8. It is obvious that the long waves scale Λ is equivalent or larger than the convergent length. So it is expected that these waves cannot be seen as a roughness. A stretching effect is clearly observed. The influence of the gas and liquid flow rates is not obvious. For the higher gas flow rate, the wavelength augments with the liquid flow rate while a non-monotone evolution is obtained for the lower gas flow rate. Further analyses are needed to explain this phenomenon. On the opposite, the size of the ripple waves λ is always inferior to the length of the convergent section. For the higher gas flow rate, this scale diminishes continuously as the film flows into the convergent section. For the lower flow rate, it increases first then decreases quickly. As before, this behaviour has to be analysed in more detail in the future.



450 m3/h

0

100

200

300

400

0 50 100 150

x(mm)

δ(m

m)

Exp ; 50 l/hExp ; 100 l/hExp ; 150 l/hExp ; 200 l/hExp ; 250 l/hModelling ; 50 l/hModelling ; 100 l/hModelling ; 150 l/hModelling ; 200 l/hModelling ; 250 l/h

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0

100

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400

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m)

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m)

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m)


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0

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m)

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0

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m)


360 m3/h

0

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m)


450 m3/h

0

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0 50 100 150

x(mm)

δ(m

m)


360 m3/h

0

100

200

300

400

0 50 100 150x(mm)

δ(m

m)


Figure 6: Evolution of the mean film thickness inside of the Venturi meter (bold symbols corresponds to measurements obtained 1D upstream of the convergent section).

360 m3/h

0

500

1000

1500

2000

0 50 100 150

x(mm)

A(µ

m)

Exp ; 50 l/hExp ; 100 l/hExp ; 150 l/hExp ; 200 l/hExp ; 250 l/hCorrelation ; 50 l/hCorrelation ; 100 l/hCorrelation ; 150 l/hCorrelation ; 200 l/hCorrelation ; 250 l/h

450 m3/h

0200400600800

10001200

0 50 100 150

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m)

CorrelationExperiment

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m)


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x(mm)

A(µ

m)


250 l/h200 l/h150 l/h100 l/h50 l/h


250 l/h200 l/h150 l/h100 l/h50 l/h

Figure 7: Wave amplitude distribution in the convergent section (bold symbols corresponds to measurements obtained 1D upstream of the convergent section).



012345

0.0 50.0 100.0 150.0x(mm)

Uc(m

/s)

50 l/h 150 l/h 250 l/h50 l/h 150 l/h 250 l/h

650 m3/h360 m3/h

00.010.020.030.040.050.060.07

0.0 50.0 100.0 150.0x(mm)

λ(m

)

00.10.20.30.40.50.6

0.0 50.0 100.0 150.0x(mm)

Λ(m

)

250 l/h150 l/h50 l/h

650 m3/h

360 m3/h

00.010.020.030.040.050.060.07

0.0 50.0 100.0 150.0x(mm)

λ(m

)

00.10.20.30.40.50.6

0.0 50.0 100.0 150.0x(mm)

Λ(m

)

250 l/h150 l/h50 l/h

650 m3/h

360 m3/h

Figure 8: Convection velocity and wavelengths measured at the interface (left: long waves; right: ripples).

5 Conclusion

The mass and momentum interactions between the liquid phase and the gas phase have to be accurately modelled in order to calculate the impact of the liquid phase on the differential pressure measured on a Venturi meter used in wet gas condition. Among these different interactions, the behaviour of the liquid film on the pipe wall during its flow in the convergent section has to be studied in detail. For this purpose, a dedicated experiment was performed with air and water at atmospheric pressure. The results are compared with 1D flow modelling. The flattening of the film in the convergent section is shown. Nevertheless a great difference appears with the modelling results which indicates that further phenomena not taken into account in the 1D model, appear. From signal analysis, two types of wave are distinguished (high amplitude waves and ripples). The long waves are stretched in the Venturi and their amplitude diminishes. On the contrary, the ripple wavelengths seem to diminish. Further experiments and data analyses are underway to complete these first observations.



References

[1] Lupeau A, Platet B., Gajan P., Strzelecki A., Escande J., Couput J.P. Influence of the presence of an upstream annular liquid film on the wet gas flow measured by a Venturi in a downward vertical configuration. Flow Meas. and Inst. 18(1), (2007), 1–11.

[2] Hewitt G.F., Measurement of two phase flow parameters, Academic Press, (1978)

[3] Hewitt G.F., King R.D., Lovegrove P.C., Techniques for liquid film pressure drop studies in annular two phase flow, Rept. AERE-R3921, UKAEA, Harwell, (1962)

[4] Bissières D, Couput J.P, Estivalezes J.L., Gajan P., Lavergne G., Strzelecki A., Wet gas flow simulation for venturi meters, 1st North America Conference on Multiphase Technology, Banff, (1998)

[5] Hirt C. and Nichols B., Volume of fluid method for the dynamics of free boundaries, Journal of Computational Physics, 39(1), (1981), 201-225

[6] Osher S., Fedkiw. R., Level Set Methods: An Overview and Some Recent Results, Journal of Computational Physics, 169(2), (2001), 463-502

[7] van Werven M., van Maanen H. R. E., Ooms G. and Azzopardi B. J., Modeling wet-gas annular/dispersed flow through a Venturi, AIChE Journal, 49(6), (2003), 1383-1391

[8] Fernandez Alonso D., Azzopardi B.J., Hills J.H., Gas/liquid flow in laboratory-scale venturis, IChem, 77, (1999), 205-211

[9] Azzopardi B.J., Govan, A.H., The modeling of Venturi scrubbers, Filtration and Separation, 21, (1984), 196-200

[10] Azzopardi, B. J., Taylor, S. and Gibbons, D. B. Annular two-phase flow in large diameter pipes. Int. Conf. on Physical Modelling of Multi-Phase Flow, April 19-21, Coventry, (1983), 267-282.

[11] Asali J.C., Hanratty T.J., Ripples generated on a liquid film at high gas velocities, Int. J. of Multiphase flows, 19(2), (1993) 229-243

[12] Webb D.R., Hewitt G.F., Downwards co-current annular flow, Int. J. of Multiphase flows, 2, (1975) 35-49.

[13] Giroud-Garapon S., Etude du comportement d'un film liquide dans les chambres de combustion de statoréacteurs et/ou turboréacteurs, PhD Thesis, Ecole Nationale Supérieure de l'Aéronautique et de l'Espace, 2003



Micro-scale distillation: simulation

M. Fanelli, R. Arora, A. Glass, R. Litt, D. Qiu, L. Silva, A. L. Tonkovich & D. Weidert Velocys, Inc., USA

Abstract

Microchannel technology as applied to chemical processing has resulted in impressive improvements in performance thresholds. Studies published for more than a decade show that enhanced performance in chemical reactors can be largely attributed to the reduction of transport distances. Chemical distillation is now emerging as a new area for the application of microchannel technology.

A simplified method for simulating a microchannel distillation process has been developed and validated with experimental data. Both simulation and experiments show that the height of a theoretical transfer unit for the separation of hexane and cyclohexane in a microchannel distillation unit is reduced to centimetres. Vapour-side resistance was found to control mass transfer for the cases considered. The current simulation can serve as a tool for optimizing and refining the design of multiphase microchannel processes. Keywords: distillation, simulation, separation, vapour-liquid, microchannel.

1 Introduction / background

Microchannel technology as applied to chemical processing has led to impressive improvements in performance thresholds. Several studies presented throughout the last decade show that reduction in transport distances significantly enhances the performance of chemical reactors. Reviews by Boger et al. [1], Hessel et al. [2], and Kreutzer et al. [3] provide excellent summaries of some of the key research and development efforts.

The Battelle Memorial Institute has conducted research in microchannel distillation since the early 2000s and holds U.S. and international patents in the field (Battelle [4–6]). A 2004 study by Wootton and deMello (Wootton [7])



doi:10.2495/MPF070201

demonstrated continuous laminar microchannel evaporation of acetonitrile and dimethylformamide, adopting very small flow rates and co-current flows. The pioneering efforts reported in studies to date involved relatively small production quantities that are not directly applicable to industrial application. Our work is directed towards expanding microchannel distillation from the laboratory scale to the industrial scale.

Recognizing the potential for obtaining significant reduction in HETP (Height Equivalent to a Theoretical Plate) by microchannel distillation, a series of experiments and simulations was initiated to quantify the achievable enhancements. A representative laboratory distillation unit was built and distillation tests were run. The collected data were compared with the simulated results obtained with the FluentTM Computational Fluid Dynamics (CFD) package. Comparative evaluations, adopting a binary hexane-cyclohexane system, have shown good order of magnitude correspondence between the model and experimental distillation trials. This information is now being used to refine and design alternate approaches to microchannel distillation for applications of industrial relevance.

2 Simulation methodology

The current simulation method allows relatively fast and direct estimation of distillation performance in a microchannel device. This method is not intended to be comprehensive. Its scope is to capture and model the primary phenomena that impact the distillation process with sufficient confidence for scoping analyses and future process refinement.

The present data manipulation involved the solution of species mass and momentum balances, but did not consider an energy balance or surface tension effects. Flow instability and heat transfer were not considered. The simulation was conducted under the following assumptions:

• a stationary interface, with no shear; • interfacial concentrations based on ChemCAD provided distribution

coefficients for each species, AK , (linearly interpolated between the column extremes), such that

Ai

AiA x

yK = , (1)

where Aiy and Aix refer to the interfacial vapour and liquid compositions of species A, respectively;

• uni-directional equimolar counter-diffusion at the interface (Bird [8]), such that, within each phase, j,

j

AjAjjA dydxDcN −= , (2)



where jAN is the molar flux of species A in the direction perpendicular to

the interface, jc is the molar density, jAD is the diffusivity of species A,

and j

A

dydx

is the mole fraction gradient of species A in the direction

perpendicular to the interface; • equal molecular weight for the two species (an average of the actual

molecular weights, given their similarity); • constant fluid properties within each phase.

HETP estimates for each phase were calculated using the number of transfer units for that phase, jn , and the length of the mass transfer channel, totZ , such that

∫ −=

jAout

jAin

x

x jAjAi

jAj xx

dxn and

j

totj n

ZHETP ~ (3)

where jAx is the cross-sectional area averaged concentration and jAix is the

interfacial concentration along the channel, within each phase. Overall HETP’s were estimated by combining the HETP for each phase; i.e.,

liquidvaportotal HETPL

mGHETPHETP += , (4)

where m is the slope of the equilibrium line (with the liquid mole fraction in the abscissa, the vapour mole fraction in the ordinate), and G/L is the ratio of the molar gas and liquid flow rates through the column (McCabe et al. [9]; Taylor and Krishna [10]).

The work was performed using the FluentTM CFD package, currently used for some of our more intensive simulation work. Results were analyzed in terms of change in concentration profile along the channel axial length.

3 Experimental configuration and parameters

Experiments were performed using a stainless steel distillation unit involving counter-currently flowing vapour and liquid phase mixtures of hexane and cyclohexane. Liquid flowed vertically downward along a 178 µm deep stainless steel mesh. Vapour was fed to the device from a lower port, opposite the liquid outlet port. Liquid inlet and vapour outlet ports and vapour inlet and liquid outlet ports were slightly offset relative to each other, with the liquid ports lower than the vapour counterparts. Figure 1 shows representative schematics of the experimental setup, viewed from the top and side. The step in the cross section of the vapour channel, visible in the top view of the device, was a result of fabrication requirements.



top view

813 µm 483 µm

178 µm thick mesh

2.22 cm2.86 cm

side view

liquidin

out

in liquidout

12.7 cm

Figure 1: Representative schematics of the experimental distillation device.

Flow rates, feed concentrations and temperatures for the runs are summarized in Table 1. Testing was conducted at atmospheric pressure. The number of theoretical plates corresponding to the achieved separation were calculated using a rigorous ChemCAD distillation model with the liquid and vapour feeds input at the upper and lower plates, respectively.

Table 1: Relevant experimental run parameters.

Case 1 Case 2 Experimental Run Parameter Liquid Vapour Liquid Vapour

feed flow rate (liquid ml/min at ambient) 0.5 0.5 1.0 1.0

feed T (°C) 69 83 68 84 inlet hexane mole fraction 0.839 0.085 0.839 0.085

4 Simulated configuration and parameters

Two potential channel configurations were simulated to evaluate the sensitivity to different flow geometries.

1. The “mesh flow” configuration assumed the liquid flowed along the width of the channel wall as a continuous, uniform film with a depth equivalent to the 178 µm supporting mesh thickness and the uniform feed velocity that would result with the imposed volumetric feed rate.

2. The “falling film” configuration assumed the liquid flowed as a freely falling film along the width of the vertical wall, with the thickness and velocity dictated by the imposed volumetric feed rate. Calculations were based on the analysis of Bird et al. [8], according to which the liquid



vapour

vapour

Reynolds number corresponded to fully laminar flow, with no ripples. The shearing effect of the flowing vapour phase was assumed to be insignificant.

For each configuration, the section of the channel not filled with liquid was assumed to be filled with the counter-currently flowing vapour.

Physical properties were obtained from a ChemCAD distillation simulation of hexane-cyclohexane at the same temperature/pressure operating range as the experimental trials. Interfacial compositions were calculated from linear interpolation of the distribution coefficients between the column extremes. Material properties were assumed constant (averaged from the corresponding phase in the ChemCAD simulation); they are listed in Table 2. The feed compositions and representative flow rates for the simulations matched experimental values.

Table 2: Material properties and interfacial parameters used in the simulation. Gas and liquid diffusivities were estimated adopting standard methodologies (Poling et al. [11]).

Material Property Liquid Vapour density (kg/m3) 660 3.2 viscosity (kg/m.s) 3.0E-04 8.0E-06 diffusivity (m2/s) 5.0E-09 4.5E-06

Hexane cyclohexane molecular weight (g/mol) 85 (86 actual) 85 (84 actual) distribution coefficients at the column top 0.79607 1.06210 distribution coefficients at the column bottom 0.94510 1.50565

The channel was assumed to be long and rectangular, and although the

simulations were 3-dimensional, they were effectively run as 2-dimensional problems by defining the sidewalls as symmetric boundary conditions. The key parameters for the cases considered (named to correspond with the experimental runs) are listed in Table 3.

Table 3: Key run parameters for the simulated cases.

Case ID

Assumed Flow Type

Liquid FilmGap (µm)

Full Channel Flow Rate

(liquid ml/min)

Liquid Velocity

(m/s)

Vapour Velocity

(m/s)

1a falling film 36 0.5 0.0086 0.045 1b mesh flow 178 0.5 0.0017 0.051 2a falling film 43 1.0 0.0136 0.091 2b mesh flow 178 1.0 0.0033 0.102




Run results for the experimental and simulated cases are summarized in Table 4 and taken as key indicators of performance. As confirmed by the close correspondence of the experimental and simulated HETP’s, the current simulation methodology can be deemed acceptable for predicting microchannel distillation performance.

Comparison of the simulated cases shows there is little impact of assumed liquid film gap on HETP, but the mesh flow assumption leads to:

• slightly better performance • closer adhesion to experimental results.

Closer scrutiny of the concentration profiles for representative cases can shed light on these observations.

Table 4: Run results for the experimental and simulated cases.

Flow Rate Case HETP (cm)

(liquid ml/min) ID Description Liquid Vapour Overall

0.5 1 experiment -- -- 1.3 1a simulation - falling film 1.6 0.5 2.1 1b simulation - mesh flow 1.2 0.5 1.7

1.0 2 experiment -- -- 2.1 2a simulation - falling film 1.7 1.0 2.7 2b simulation - mesh flow 1.5 0.9 2.4

a b caxial position (m) axial position (m) axial position (m)

Figure 2: Representative hexane concentration profiles (mole fraction) as a function of axial position with respect to the channel top for simulated Case 1a (falling film). Dotted lines represent liquid-side, solid lines represent vapour-side profiles. Plot a presents interfacial concentrations, Plot b presents concentrations in the cells adjacent to the interface, Plot c shows the concentration difference driving the mass transfer at the interface (values in Plot b less values in Plot a).



Figure 2 shows three plots representing hexane concentration profiles for the low flow, falling film configuration:

a. at the interface, b. in the cells adjacent to the interface, c. as the difference between these values, representing the driving force for

mass transfer along the channel length. Figure 2c shows that for this case, as for all cases considered, the vapour layer near the interface equilibrates very quickly to the interfacial composition. Given the ease of this equilibration, the ability of the vapour species to diffuse through the vapour layer becomes the limiting phenomenon.

0.0

0.2

0.4

0.6

0.8

.000 .020 .040 .060 .080 .100 .120

axial channel position w.r.t. top (m)

hexane

mol frac

vapour bulkvapour interfaceliquid bulkliquid interface

0.0

0.2

0.4

0.6

0.8

.000 .020 .040 .060 .080 .100 .120

axial channel position w.r.t. top (m)

hexane

mol frac

vapour bulkvapour interfaceliquid bulkliquid interface

b

a

Figure 3: Representative surface area averaged concentration profiles and corresponding interfacial compositions for Cases 1a and 2a. Area averaged (bulk) concentration profiles were used to determine HETP values for each of the simulated runs.

The profiles in Figure 3 show the relative area averaged (bulk) concentrations along the length of the distillation unit for Cases 1a and 2a (these concentration differences were used to calculate HETP values for the simulated runs). Although comparison of the two plots shows no difference in liquid-side



concentration profiles, vapour-side concentration profiles show some dependence on flow velocity. At the higher flow rate, deviation between bulk and interfacial concentrations increased. Overall, since the vapour side controls mass transfer, the vapour-side HETP is more directly impacted by the change in flow rate.

6 Conclusion

A simulation approach has been successfully developed for simple and direct modelling of microchannel distillation processing using the FluentTM CFD package. The methodology, which allows easy probing of different distillation geometries, was validated using a microchannel distillation device.

Experimental and simulated HETP’s for the distillation of hexane from a hexane-cyclohexane mixture were found to be less than 3 cm. The mass transfer for the cases considered was found to be vapour-phase controlled. Additional comparative evaluations are ongoing to allow methodology refinement. The current approach can serve:

1. as a predictive tool, 2. as a means of investigating fundamental phenomena and their effect on

multi-phase mass transfer performance.

Acknowledgements

The authors would like to acknowledge the Department of Energy for sponsoring this effort and the assistance of Fluent in troubleshooting the implementation of the simulation.

References

[1] Boger, T., Heibel, A.K., Sorensen, C.M., Monolithic catalysts for the chemical industry. Industrial and Engineering Chemistry Research, 43, pp. 4602-4611, 2004.

[2] Hessel, V., Angeli, P., Gavriilidis, A., Lowe, H, Gas-liquid and gas-liquid-solid microstructured contacting principles and applications. Industrial and Engineering Chemistry Research, 44, pp. 9750-9769, 2005.

[3] Kreutzer, M.T., Kapteijn, F., Moulijn, J.A., Heiszwolf, J.J., Multiphase monolith reactors: chemical reaction engineering of segmented flow in microchannels. Chemical Engineering Science, 60, pp. 5895-5916, 2005.

[4] Battelle Memorial Institute, Improved Conditions for fluid separations in microchannels, capillary-driven fluid separations, and laminated devices capable of separating fluids. International Patent No. WO 03/049835 A1, 2003.

[5] Battelle Memorial Institute, Conditions for fluid separations in microchannels, capillary-driven fluid separations, and laminated devices capable of separating fluids. U.S. Patent No. 6,875,247 B2, 2005.



[6] Battelle Memorial Institute, Methods of contacting substances and microsystem contactors. U.S. Patent No. US 6,869,462 B2, 2005.

[7] Wootton, R.C.R., deMello, A.J., Continuous laminar evaporation: micron-scale distillation. Chemical Communications, pp. 266-267, 2004.

[8] Bird, R.B., Stewart, W.E., Lightfoot, E.N., Transport Phenomena, John Wiley & Sons: NY, 1960.

[9] McCabe, W.L., Smith, J.C., Harriott, P., Unit Operations of Chemical Engineering, 4th edition, McGraw-Hill Book Company: New York, 1985.

[10] Taylor, R., Krishna, R., Multicomponent Mass Transfer, John Wiley & Sons: New York, 1993.

[11] Poling, B.E., Prausnitz, J.M., O'Connell, J.P., The Properties of Gases and Liquids, 5th ed., McGraw-Hill Book Company: New York, 2001.



Viscoelastic drop deformation in simple shear

C. Chung1, M. A. Hulsen2, K. H. Ahn1 & S. J. Lee1

1School of Chemical and Biological Engineering, Seoul National University, Seoul, Korea 2Mechanical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

Abstract

The two-dimensional deformation of immiscible drop in simple shear flow was investigated using the front tracking method. Interface particles were traced by Runge–Kutta 2nd order method and the boundary immersed method was used for calculation of surface tension force at the global mesh. Isothermal, incompressible and creeping flow was assumed. The main purpose of this research is to analyze the effect of viscosity ratio and elasticity on the drop deformation. Oldroyd-B model was used as a constitutive equation with stabilizing schemes such as DEVSS-G/SUPG and matrix logarithm. As for the Newtonian drop deformation in the Newtonian matrix, there was no breakup until Ca=1 when the viscosity ratio was one. And the damped oscillation was observed when the viscosity ratio was not unity. The effect of elasticity on the drop deformation was also investigated. As De increased, the drop was more deformed and orientation angle declined to the shear direction. Keywords: drop deformation, viscosity ratio, elasticity, front tracking method, immersed boundary method, DEVSS-G/SUPG, matrix logarithm.

1 Introduction

The study of drop deformation in simple shear flow provides an insight for understanding the physics of complex flows. Numerical analysis is required for analyzing applications including polymer processing and multi-phase flow in micro-channels. There have been several studies on the deformation of a drop in shear flow since the pioneering works of Taylor [1,2]. Also numerical approaches [3,4] for the same problem showed good agreements with experiments for



doi:10.2495/MPF070211

flow investigated by the front tracking method

Newtonian fluids. As for viscoelastic fluids, however, there seems to be a gap to reach a general consent among numerical solutions [5–7]. Here, the first attempt of the front tracking method was made to describe the deformation of the viscoelastic drop in the viscoelastic medium under shear flow.

2 Mathematical formulation To understand the deformation of the interface in the viscoelastic multi-phase fluid, the viscoelastic instability and the capillary instability should be resolved simultaneously. Here, the numerical schemes such as DEVSS-G [8], SUPG [9] and matrix logarithm [10] were applied to stabilize governing equations. And Lagrangian particles (markers) on the interface were traced with Runge–Kutta 2nd order method.

2.1 Front tracking method Front tracking method is the method which tracks the position of the Lagrangian mesh (front mesh) presented as interface in the Eulerian mesh (global mesh) where real calculation is conducted using the values from the nodes of the Eulerian mesh [11]. Established method for the calculation of normal vector and curvature was used to fit a polynomial for the interface. This method requires the information on the position of the markers nearby, where the quadratic interpolation for the position of the interface was good enough for accurate results [12]. Besides this method, another approach for the calculation of surface tension using the tangential vector of the front element was developed [13]. This approach is not only simple but also very accurate since total force on any closed surface is calculated as zero. Here, the surface tension at the interface was calculated according to this method. Using Frenet–Serret theorem [14], the surface tension at the interface, ef is as below:

( )B

e B As A

df ds dsds

σκ δ σ σ∆

= = = −∫ ∫tn t t (1)

where σ is the surface tension coefficient, δ is the Dirac delta function and s∆ is the average length of the segment of front element. n , t are the outward normal vector and tangential vector at the node of front mesh, κ is the twice mean curvature of the interface. Though the surface tension is calculated at the front mesh, it is necessary to transfer the values of the front mesh to the global mesh. The interchange of the information between two meshes is possible using immersed boundary method [15,16]. In this method, the information between front mesh and global mesh can be transferred using Dirac delta function as follows:

( ) ( ) ( )ij l l ll

F f sδ= − ∆∑x x x x , (2)

( ) ( ) ( )l x y ij lij

f h h F δ= −∑x x x x , (3)

where h is the size of the global element, ( )ijF x and ( )lf x are the values at the node of global mesh and front mesh, respectively. For two dimensional case, delta function is defined as below:



( ) ( ) ( )d x d yδ =x . (4) According to Peskin [15]

1 cos2 2( )

40 2

rh r hd r

hr h

π + ≤=

>

. (5)

In this research, the characteristic length h was set to the minimum of the diagonal length of the global element. Markers representing the interface are positioned with equal distance at the initial state. However, the size of the front element changes as the calculation proceeds since the deformation of the interface changes with time and position. Because large variation of the sizes of front elements would exert a bad influence on the solution, a remeshing algorithm for maintaining the size of the front element in a certain range is necessary. In other words, addition or deletion of the front element should be conducted in case of need. In this research, the length of the front element, ls∆ was kept from 20% to 50% with respect to h .

2.2 Governing equations

Governing equations for the Newtonian fluid consist of momentum equation and continuity equation. When viscoelasticity is considered, the constitutive equation for the polymer stress should be added. Here, Oldroyd-B model was used as a constitutive equation. Isothermal, incompressible and creeping flow was assumed. To stabilize the calculation, DEVSS-G scheme [8] and SUPG scheme [9] were adopted to the governing equations. Also, matrix logarithm [10] was applied to the constitutive equation.

( ) ( )

11

1 1

1 1 1 1 1

1

1

( ) 0

( ) ( 1) ( )

( ) 0

0

n nn n n

n n T

n n n T n T ns

n

sn

t

p

ds

η η β

σκ δ

++

+ +

+ + + + +

+

∆

+

−+ ⋅∇ =

∆− ∇ =

−∇ +∇ ⋅ ∇ + ∇ + − ∇⋅ +

+∇ ⋅ + =

∇ ⋅ =

∫

s s u s S

G u

u u G G

τ s n

u

(6)

Here, u is the velocity, p the pressure, η the solution viscosity and λ the relaxation time of polymer.

( )2 2 2

1 1 12

1 (

.

)

i jiii i i j ij i ji i j

i i ji i ji j

s sfL c L c L

c c c

λ

= = =≠

− = + + + −

= − −

∑ ∑∑s n n n n

f c I

, (7)

where S and f are the transformed matrix of s and f into the global frame. More details for the matrix logarithm were referred to the literature [10]. I is the



identity tensor and the relation between the conformation tensor, c and the extra stress tensor, τ is as follows:

( )pηβλ

= −τ c I . (8)

The polymer contribution to the viscosity, pβ is defined as:

p pp

p s

η ηβ

η η η= =

+. (9)

pη and sη mean the polymer viscosity and the solution viscosity, respectively. In this research, pβ was set to 8/9. The problem was defined with a computational domain of width equal to 4 and height equal to 12. The initial drop was positioned at the center of the domain, coordinates (2,6) with a radius, a as 0.5. Center region was refined as a structured type for the accurate calculation of the interface (figure 1). For the simulation of large deformation or viscoelastic fluids, the mesh was more refined with structured elements as in UC type. More details on the meshes were shown in table 1. Side walls moved with a constant velocity to make simple shear flow with 1γ = . And the steady condition or periodic condition was imposed at the inlet.

(a) UM1 (b) UM2 (c) UM3 (d) UC1 (e) UC2

Figure 1: Mesh configuration.

Table 1: Mesh information.

Name Elements Nodes DOF Δxmin=Δymin UM1 2,300 9,333 47,070 0.1 UM2 7,935 31,909 160,058 0.05 UM3 26,868 107,689 539,102 0.025 UC1 4,032 16,361 82,510 0.1 UC2 14,494 58,353 292,902 0.05




Dimensionless numbers in this problem are capillary number and Deborah number;

m aCa

η γσ

= , (10)

De λγ= . (11) As Ca means the ratio between the surface tension force and viscous force, mη is the viscosity of the outside matrix and σ is the surface tension coefficient. λ is the relaxation time of the fluid. To characterize the shape of deformed drop, the deformation parameter, D is defined as ( ) /( )D L B L B= − + , where L and B are two principal directions of deformed shape, respectively. And the orientation angle is defined as longitudinal direction with respect to the shear direction, as illustrated in figure 2.

Figure 2: Characterization of the deformed drop.

In order to verify our algorithm, the steady solutions for the Newtonian drop in the Newtonian matrix were compared with previous results (figure 3). As Ca increases, D increases and orientation angle decreases. As the mesh is more refined, D value converges to the prediction of boundary integral method [17]. And other results of VOF [5] and of diffuse interface method [6] were in line with our results for the small deformation. Though the critical Capillary number,

cCa beyond which the drop is not capable of sustaining a stationary shape is known to be around 0.875 [17], here, the steady drop was obtained until 1Ca = when the viscosity ratio, α was one. The major difference between the reference [17] and ours is the size of the computational domain with respect to the drop size and inlet condition. Though there was no remarkable difference whether periodic condition or steady condition was imposed since the flow domain is large enough comparing with drop size, the periodic effect could not be negligible if small computational domain was used as in reference [17]. With imposing periodic boundary condition at small domain, Zhou and Pozrikidis [17] solved the problem for periodically packed drops closely. While, the problem for the single drop was solved with steady inlet condition in this research. Considering hindering effect of the periodic condition on the drop deformation [18], the difference for cCa could be reasonable.



Ca

0.0 0.2 0.4 0.6 0.8 1.0

D

0.0

0.2

0.4

0.6

0.8

1.0UM1 UM2 UM3 Chinyoka et al. [5]Yue et al. [6]Zhou & Pozrikidis [17]Taylor [2]

(a) Deformation parameter

Ca

0.0 0.2 0.4 0.6 0.8 1.0

orie

ntat

ion

angl

e (d

eg.)

0

10

20

30

40

50

UM1 UM2 UM3

(b) Orientation angle

Figure 3: Steady drop deformation with increasing Ca.

Shear flow is a mixture of stretching and rotation [19]. Both drop deformation and the rotation of interface are progressed simultaneously. The orientation angle decreases while the initial drop changes to the elongated shape as time passes. When the viscosity ratio was lower than unity, the elongated drop showed somewhat higher orientation angle at early stage because of higher mobility of the drop as shown in figure 4 (a). Then, the drop was more elongated due to high shear force while orientation angle was getting decreased to minimize viscous dissipation [20]. If the orientation angle was close to zero like figure 4 (c), the capillary force became higher than shear force at the head of elongated drop. Hence the drop shows a tendency of coming back to the initial shape. And the shrunk drop was deformed again by shear force. Repeating these behaviours, a damped oscillation in D was observed. Basically, oscillatory behaviour comes from different viscosity ratio. At the experiments of drop with zero interfacial tension when the viscosity ratio was 21, the rotation of the drop was reported



[21]. The same phenomenon was reproduced using perturbation theory [22] even though it showed some discrepancy with experimental results [21]. The simulation of three dimensional drop at similar condition reproduced periodic behaviour [23]. At the zero interfacial condition, namely Ca = ∞ and 10α = , the periodic oscillation in D was reproduced with our algorithm. Therefore, oscillation is attributed to different viscosity ratio. The damping behaviour may be originated from the surface tension, in other words Ca.

(a) t=10 (b) t=20 (c) t=30 (d) t=40

Figure 4: Oscillatory motion of the drop (UM2, Ca=1, α =0.01).

Damped oscillation behaviour in D was obvious in figure 5 when α was 10. At the same viscosity ratio, damped oscillation in D was reported using boundary integral method independent of Ca [17]. When 1α < , the smaller the viscosity ratio, the shorter the frequency of the oscillation, while vice versa for 1α > as shown in figure 5. The larger drop viscosities and matrix lead to larger damping rate, which means that it would take more time to reach steady state.

time

0 20 40 60 80 100

D

0.0

0.2

0.4

0.6

0.8

1.0

α= 0.01α= 0.1α= 1α= 10α= 100

Figure 5: Deformation parameter according to viscosity ratio (UM2, Ca=1).

When compared with the Newtonian drop, an interesting phenomenon was observed in the viscoelastic case. It was reported that the viscoelastic drop was less deformed than the Newtonian case [5]. As De increases, D increases and orientation angle decreases as shown in figure 6. In this case, both drop and matrix are viscoelastic. Though relaxation times should be defined for both fluids, the same relaxation times were used for both fluids in this study as a preliminary step.



time

0 5 10 15 20 25 30

D

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

De= 0.5 / 0.5De= 1 / 1De= 2 / 2De= 3 / 3

time

0 5 10 15 20 25 30

orie

ntat

ion

angl

e

0

10

20

30

40

50

De= 0.5 / 0.5De= 1 / 1De= 2 / 2De= 3 / 3

(a) Deformation parameter (b) Orientation angle

Figure 6: Transient viscoelastic drop deformation in the viscoelastic flow (UC2).

5.54.53.52.51.50.5

201482

2-1-4-7

(a) xxc (b) yyc (c) xyc

Figure 7: Distribution of conformation tensor (UC2, De=2/2, Ca=0.1).

1.310.70.40.1

21.40.80.2

0-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8

(a) xxc (b) yyc (c) xyc

Figure 8: Distribution of conformation tensor (UC2, De=0.5/0.5, Ca=0.1).

At higher De, the drop was more elongated due to higher yyc (yy-component of the conformation tensor) at the waist of the drop than xxc at the head of the drop, and the orientation angle was more tilted to the shear direction than the Newtonian case because of xyc distribution near the drop (figure 7). At lower De, however, the drop was less deformed than the Newtonian case due to the additional yyc distribution at the head of the drop. And the orientation



angle was less tilted to the shear direction due to the opposite sign of xyc near the drop in comparison with higher De case.

4 Conclusions

Deformation of two dimensional drop in simple shear flow was investigated. Front tracking method showed stable results for interface position. In the Newtonian case, mesh convergence was accomplished. Damped oscillation behaviour in the prediction of drop deformation was reproduced when the viscosities of drop and matrix were different. With stabilizing schemes such as DEVSS-G/SUPG and matrix logarithm, numerical simulation for viscoelastic fluids was also successful. One of the main contributions is that this is the first application of front tracking method to the viscoelastic problem. Comparing with other results, we would get the reliability of our solutions. In this research, the viscoelastic drop showed an interesting phenomenon due to the distribution of conformation tensor near the drop. As De increases, the drop more deforms and inclines to the shear direction. Major contribution on the deformation of drop seems to come from the elasticity of fluid outside than from inside.

Acknowledgements

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-213-D00033). The authors wish to acknowledge the National Research Laboratory Fund (M10300000159) of the Ministry of Science and Technology in Korea.

References

[1] Taylor, G.I, The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. London, Ser. A, 138, 41-48, 1932.

[2] Taylor, G.I., The formation of emulsions in definable fields of flow. Proc. R. Soc. London, Ser. A, 146, 501-523, 1934.

[3] Li, J., Renardy, Y.Y. & Renardy M., Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids, 12(2), 269-282, 2000.

[4] Cristini, V., Blawzdziewicz, J. & Loewenberg M., An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence. J. Comput. Phys., 168, 445-463, 2001.

[5] Chinyoka, T., Renardy, Y.Y., Renardy, M. & Khismatullin, D.B., Two-dimensional study of drop deformation under simple shear for Oldroyd-B liquids. J. Non-Newtonian Fluid Mech., 130, 45-56, 2005.

[6] Yue, P., Feng, J.J., Liu, C. & Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech., 515, 293-317, 2004.



[7] Pillapakkam, S.B. & Singh, P., A level-set method for computing solutions to viscoelastic two-phase flow. J. Comput. Phys., 174, 552-578, 2001.

[8] Liu, A.W., Bornside, D.E., Armstrong, R.C. & Brown, R.A., Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields. J. Non-Newtonian Fluid Mech., 77, 153-190, 1998.

[9] Brooks, N. & Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng., 32, 199-259, 1982.

[10] Hulsen, M.A., Fattal, R. & Kupferman, R., Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech., 127, 27-39, 2005.

[11] Unverdi, S. & Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys., 100, 25-37, 1992.

[12] Zhao, P., Heinrich, J.C. & Poirier, D.R., Fixed mesh front-tracking methodology for finite element simulations. Int. J. Numer. Meth. Engng., 61, 928-948, 2004.

[13] Shin, S. & Juric, D., Modelling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity. J. Comput. Phys., 180, 427-470, 2002.

[14] Millman R.S. & Parker G.D., Elements of differential geometry (Chapter 2). Local curve theory, Prentice-Hall, pp. 13-48, 1977.

[15] Peskin, C.S., Numerical analysis of blood flow in the heart. J. Comput. Phys., 25, 220-252, 1977.

[16] Mittal, R. & Iaccarino, G., Immersed boundary methods. Annu. Rev. Fluid Mech., 37, 239-261, 2005.

[17] Zhou, H. & Pozrikidis, C., The flow of suspensions in channels: single files of drops. Phys. Fluids A, 5(2), 311-324, 1993.

[18] Renardy, Y.Y. & Cristini, V., Effect of inertia on drop breakup under shear. Phys. Fluids, 13(1), 7-13, 2001.

[19] Macosko, C.W., Viscous liquid (Chapter 2). Rheology: principles, measurement, and applications, VCH Publishers. Inc., pp. 65-108, 1994.

[20] Kennedy, M.R., Pozrikidis, C. & Skalak, R., Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Computers Fluids, 23, 251-278, 1994.

[21] Torza, S., Cox, R.G. & Mason, S.G., Particle motions in sheared suspensions XXVII. Transient and steady deformation and burst of liquid drops. J. Colloid Interface Sci., 38, 395-411, 1972.

[22] Rallison, J.M., Note on the time-dependent deformation of a viscous drop which is almost spherical. J. Fluid Mech., 98, 625-633, 1980.

[23] Wetzel, E.D. & Tucker III, C.L. Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J. Fluid Mech., 426, 199-228, 2001.



Section 4 Bubble and drop dynamics

Y. Y. Yan & Y. Q. Zu School of the Built Environment, University of Nottingham, UK

Abstract

Both bubble coalescence and droplet separation are important physical phenomena in the natural world and a variety of process industries. This paper presents results of numerical simulation of behaviours of bubble coalescence and droplet separation. The velocity distribution functions of two particles are used in lattice Boltzmann equations. Based on the lattice Boltzmann method (LBM), both the phenomena of two rising bubbles coalescing in liquid and a liquid droplet break-up on wetting boundaries are simulated. Typically, such two-phase problems of large ratio of liquid–gas densities up to 1000 are studied. Keywords: bubble coalescence, droplet separation, numerical modelling, lattice Boltzmann method.

1 Introduction

Both bubble coalescence and droplet separation are important physical phenomena in the natural world and a variety of process industries. It is a common occurrence in two phase flow and flow boiling that the evolution of bubbly flow to slug and annular flows accompanies processes of bubble coalescences. The coalescence or separation of droplets are also popular in droplet and film cooling condensations under difference surface conditions. Numerical modelling of bubble coalescence or droplet separation has been attempted by researchers for many years. Conventional CFD methods based on solving Navier-stokes equations can simulate free surface flow and bubble shape evolution with time [1-3] but can not effectively simulate problems of bubbles or droplets coalescences. Although the phenomena of bubble coalescences have been simulated by VOF method in [4] but only a two dimensional problem was discussed and the ratio of gas-liquid densities was also limited.



doi:10.2495/MPF070221

and droplet separation Numerical modelling of bubble coalescence

In recent years, the lattice Boltzmann method (LBM) has become an established numerical scheme for simulating multiphase fluid flows. Several researchers have applied LBM to study multiphase flow including bubbles or droplets coalescence [5-7]. The key idea behind the LBM is to recover the correct macroscopic motion of fluid by incorporating the complicated physics of the problem into simplified microscopic models or mesoscopic kinetic equations. In this method, kinetic equations for particle velocity distribution function are solved; and macroscopic quantities are then obtained by evaluating hydrodynamic moments of the distribution function. LBM has many computational advantages, such as parallel of algorithm and simplicity of programming [8]. In LBM modelling of multiphase fluid flows, Gunstensen et al. [9] developed a multi-component model based on the two-component lattice gas model; Shan and Chen [10] presented a model with mean-field interactions for multi-phase/component fluid flows; Swift et al. [11, 12] proposed a LBM model for multi-phase flows using the idea of free energy; He et al. [13] also developed a model using the index function to track the interface of multi-phase flow. To overcome the shortcoming that the above LBM schemes can only simulate two-phase fluids with small density ratios (less than 20) due to numerical instability, Inamuro et al. [6] proposed a LBM for incompressible two-phase flows with large density differences by using the projection method. Briant et al. [14-15] developed an approach based on the free energy model introduced in [11, 12] to simulate partial wetting and contact line motion in two-phase fluids. However, as the method naturally inherits the disadvantage of original free energy model of Swift et al. [11, 12] and can only be used to simulate problems with a small density ratio which was around 2 [14]. In the present paper, based on a new LBM scheme developed in [16], both bubble coalescence in liquid with unconfined boundary and a liquid droplet separation on a wetting boundary are studied.

2 The Lattice Boltzmann Model

Based on the three-dimensional nine-velocity (D3Q15) LBM model, as shown in Fig. 1, the particle velocity in the thα direction, αe , is given by [6]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14[ , , , , , , , , , , , , , , ]0 1 0 0 1 0 0 1 1 1 1 1 1 1 10 0 1 0 0 1 0 1 1 1 1 1 1 1 10 0 0 1 0 0 1 1 1 1 1 1 1 1 1

− − − − − = − − − − − − − − − −

e e e e e e e e e e e e e e e

. (1)

To simulate a two-phase flow problem, two velocity distribution functions of particles, αf and αg , are introduced. Function αf is used to calculate the order parameter, φ , which distinguishes two phases, and function αg is used to calculate the predicted velocity, *u .



The evolution of particle distribution functions )t,x(fα and )t,x(gα with particle velocity αe at point x and time t is calculated by following equations:

)t,x(f)t,ex(f )eq(tt ααα δδ =++ (2)

)t,x(g)t,ex(g )eq(tt ααα δδ =++ (3)

where u , ρ and µ are the macroscopic velocity, density and dynamic viscosity respectively; 1=tδ is the time step during which the particles travel the lattice spacing; )eq(fα and )eq(gα are the corresponding equilibrium states of αf and αg , which have been given in details in [16].

11

8

x

zy

1 5

3

4

2

6

9 14

7

12

13 10

0

Figure 1: Discrete velocity set of three-dimensional fifteen-velocity model.

The macroscopic quantities, *u , φ , ρ , µ in the LBM can be evaluated as

∑=α

αφ f , ∑=α

αα ge*u (4)

≥

≤≤

<

+

+

∆−∆

=

*L

*L

*G

*G

L

G*

*

G

,

,sin

,

φφ

φφφ

φφ

ρ

ρπφφφρ

ρ

ρ

12

(5)

GGLGL

G )( µµµρρρρ

µ +−−−

= (6)

where *

Lφ and *Gφ are the cut-off values of the order parameter, Lρ and Gρ are

the density of liquid and gas phases respectively. Lµ and Gµ are the dynamic viscosity of liquid and gas phases respectively. GL ρρρ −=∆ , *

G*L

* φφφ −=∆ and

2/)( *G

*L

* φφφ += . To obtain the velocity field which satisfies the continuity equation ( 0u =⋅∇ ), the predicted velocity *u is corrected by using the following equations,



ρδp*uu

t

∇−=

− (7)

t

*upδρ⋅∇

=

∇⋅∇ (8)

where p is the pressure of the two-phase/component fluid, which can be obtained by solving Eq. (8) in the following LBM framework for velocity distribution function:

*u)]n,x(p)n,x(h[)n,x(h)n,ex(h ⋅∇−−−=++ρ

ωω

τα

ααααα

13

11 (9)

where n is the number of iterations, αω is wetting coefficient and ρτ /. 150 += is a relaxation time. The pressure at step 1+n is given by

∑ +=+α

α )n,x(h)n,x(p 11 (10)

The convergent pressure p is determined when

ε<+−+∈∀ |)n,x(p)n,x(p|,Vx 11 (11) where V denotes the whole computational domain. Substituting the newly obtained pressure p into and solving Eq. (7) gives the corrected velocity field u . As stated and demonstrated in [6, 16], this method can be used to simulate two-phase flow with density ratio up to 1000.

3 Wetting boundary condition

The liquid–gas surface tension force LGσ is given in [17] as

βφφ

σ fGL

LG k)(

26

3−= ; (12)

where Lφ and Gφ are the order parameters of liquid and gas, respectively; fk is a constant parameter to decide the width of interface and the strength of surface tension; β is the constant relating to interfacial thickness. According to Young’s law [18], when a liquid–gas interface meets a partial wetting solid wall, the contact angle, wθ , measured in the liquid, can be calculated from a balance of surface tension forces at the contact line as



LG

SLSGw σ

σσθ

−=cos ; (13)

where SGσ and SLσ are the solid-gas and solid–liquid surface tensions, respectively, which can be represented as [16]: If 0>λ ,

232 1

22222 /LGLGGL

fSG )(dkG

Ω−−++

−=+−= ∫σσφφ

λφψλφσφ

φ, (14)

234 1

22224 /LGLGGL

fSL )(dkL

Ω+−++

−=+−= ∫σσφφ

λφψλφσφ

φ; (15)

and if 0<λ ,

231 1

2222

1

/LGLGGLfSG )(dkG

Ω−−++

−=+−= ∫σσφφ

λφψλφσφ

φ, (16)

233 1

2222

3

/LGLGGLfSL )(dkL

Ω+−++

−=+−= ∫σσφφ

λφψλφσφ

φ; (17)

where 4 3, ,,i 21=φ are the solutions of order parameter; ψ is the free energy density, and Ω the wetting potential given by

βφφ

λ

fGL k)( 24

2−=Ω (18)

where ( )sfk φψλ 2±= , sφ is the order parameter at solid wall [16]. The wetting angle can be determined by substituting Eq. (12) and Eqs. (14)-(17) into Eq. (13) and written as

2

11 2323 //

w)()(cos Ω−−Ω+

=θ (19)

For a given wetting angle at πθ << w0 , Ω can be obtained from Eq. (19) as,

21

31

322

/

w coscossgn

−

−=Ω

γγθπ (20)

where )arccos(sin wθγ 2= and )sgn(ξ gives the sign of ξ . It is noted from Eq. (20) that the required wetting potential Ω can be obtained by choosing a desired contact angle wθ and then calculating λ by Eq. (18) with a newly obtained Ω . In order to introduce the partial wetting boundary condition to the LBM simulation through imposing it through equilibrium distribution functions )eq(fα and )eq(gα , the following boundary conditions should be imposed:



fz kzλφ

−=∂∂

=0

, (21)

∂∂

−∂∂

+∂∂

−≈∂∂

==== 21002

2

4321

zzzzzzzzφφφφ ; (22)

where z is the perpendicular direction to the wall. In this scheme, Eq. (21) is used to determine the first term on the right hand side of Eq. (22). While the second term is calculated using a standard centred finite-difference formula. Finally, Briant et al. [14] found empirically that the best choice for the third term is a left-handed finite-difference formula taken back into the wall, i.e.

( )012

2

4321

====

+−≈∂∂

zzzzz

φφφφ (23)

Figure 2: Computational domain.


4.1 Bubble coalescence in unbounded liquid

The method is firstly applied to bubble coalescence; the coalescence of two rising bubbles is simulated and two cases are calculated. The computational domain is shown in Fig. 2. In an initial study, two bubbles with the same diameter D are placed 5D/4 apart in a liquid inside a rectangular domain and is released at time 0=t . Calculations are carried out for the cases of liquid and gas phases with different density ratios, GL / ρρ , and viscosity ratios, GL / µµ . Dimensionless parameters, Morton number: )/()(gM LGLL

324 σρρρµ −= and Eötvös number: 32 σρρ /D)(gE GL −= are applied for the simulated phenomena. Periodic boundary conditions are imposed on all sides of the computational domain, which is divided into 64 × 64 × 128 cubic lattice. The diameter of each initial bubble occupies 24 lattice spaces, i.e. xD δ24= . The



behaviour of the two bubbles evolutes with time, typically how the lower bubble catches up and finally coalesces with the upper bubble is studied. Velocity vectors of both inside and around the bubbles during the evolution are also studied. Fig. 3 shows time evolution of two bubble coalescence and velocity vectors at section of 2/Ly y= . The two gas bubbles rising in an unbounded liquid with

50=GL / ρρ , 50=GL / µµ , dimensionless Morton number 5101 −×=M , and Eötvös number 10=E are simulated; where *t refers to dimensionless time ( D/tU*t = ), here U is the terminal averaged velocity of gas phase. Fig. 4 shows two bubble coalescence when gas bubbles rise in an unbounded liquid at 1000=GL / ρρ , 50=GL / µµ , Morton number 1=M , and Eötvös number 15=E . The upper figure shows time evolution of bubble shapes and the lower figure shows the velocity vectors at section of 2/Ly y= ;

0=*t 1.3647=*t 3.3696=*t

4.0435=*t 4.4816=*t 4.8859=*t

Figure 3: Coalescence of two rising bubbles in liquid with 50=GL / ρρ , 50=GL / µµ , 5101 −×=M , 10=E .

4.2 Droplets separation on a wetting boundary

The method is applied to calculate a water droplet spreading on a uniform wetting surface. Initially, as shown in Fig. 5, the shape of droplet is spherical, the distance between the centre of the sphere and the wall is m101 3−×=r , where r is the radius of the initial droplet. The computational domain is filled with air except the location occupied by the water and is divided into 40100120 ×× uniform cubic lattices. The motion of water droplets at normal temperature surrounded by air on partial wetting walls is considered. Naturally, the densities of two fluids are set at

3kg/m 3101×=L~ρ , 3kg/m 291.~

G =ρ , and meanwhile the dynamic viscosities of them are at s kg/m 3101 −×=L

~µ , s kg/m 5109351 −×= .~Gµ , respectively. The initial

surface tension between water and air is 2kg/s 3101 −×=LG~σ and the gravitational



acceleration is set at 2m/s 89.g~ = . To relate the physical parameters with simulation parameters, a length scale of m 101 4

0−×=L , a time scale of

s 60 101 −×=T and a mass scale of kg 101 12

0−×=M are chosen; these lead to the

dimensionless parameters: 3101×=Lρ ; 29.1=Gρ ; 1.0=Lµ ; 3109351 −×= .Gµ ; 40.L =φ ; 10.G =φ ; 05.0=k ; 8108.9 −×=g . Unless otherwise specified, the

flowing simulations are within a cuboid computational domain with a no-slip boundary at the lower surface, i.e. the flat partial wetting wall, and the free outflow/inflow boundaries at the other five surfaces. ε in Eq. (11) is set as

6101 −×=ε .

0.2044=*t 0.6132=*t 1.0221=*t 1.4309=*t 1.8397=*t

2.2485=*t 2.6573=*t 3.0662=*t 3.4750=*t 3.8838=*t

Figure 4: Time evolution of bubble shapes and Velocity vectors at section of 2/Ly y= of coalescence of two rising bubbles in liquid with

1000=GL / ρρ , 50=GL / µµ , 1=M , 15=E .

Fig. 6 shows how a small hemispherical water droplet evolves with time on a heterogeneous surface. A narrow hydrophobic strip with width of ml 4106 −×= is located at the centreline of the surface where 65 /w πθ = , and the other area is



occupied by the hydrophilic surface with 6/w πθ = . The initial droplet which has a radius m31051 −×= .r is set at the centre of the wetting surface. As shown in Fig. 6, the droplet stretches over the area occupied by the hydrophilic surface in the early stages of flow evolution due to the adhesive force of the surface. At the same time, the droplet rapidly contracts inward along the hydrophobic strip. With the development of time, the droplet spreads further on the hydrophilic area, and meanwhile contracts inward along the hydrophobic strip and finally breaks up into two smaller droplets. The newly formed droplets continue spreading until an equilibrium state is reached. For a uniform hydrophilic surface separated by a hydrophobic strip, the spreading dynamics of the droplet is affected by three parameters, namely, the width of the hydrophobic strip, the gravity and the wetting property of the hydrophilic surface [19]. A further examination and analysis of the effects of these three parameters on the spreading and break-up of the droplet will be done in the near future.

Figure 5: Computational domain.

s.t 00=

s.t 0150=

s.t 140=

s.t 150=

s.t 1520=

s.t 1540=

Figure 6: Snapshots of droplet spreading and its break-up on heterogeneous surface.



5 Conclusions

A lattice Boltzmann method which can simulate two-phase fluids with large density ratio, and meanwhile deal with interactions between a fluid-fluid interface and a partial wetting wall is developed. Based on this method, the dynamics of two rising bubbles in a liquid with liquid–gas density up to 1000 is simulated. In addition, it is also simulated that a liquid drop breaks up on uniform and heterogeneous walls with liquid–gas density ratio of 1000:1.29. The results of simulations can generally confirm that the current LBM is suitable to study such two-phase flow problems with high ratios of liquid–gas densities and with such partially wetting boundaries. A further experimental validation of the numerical method will be carried out in the near future.

Acknowledgement

The project is supported by British EPSRC under grant EP/D500125/1.

References

[1] Ryskin, G. and Leal, L.G., 1984, Numerical solution of free-boundary problems in fluid mechanism, Part 1, Finite-Difference Technique, J. Fluid Mech., 148, 1-17.

[2] Li, W.Z., Yan, Y.Y. and Smith, J.M., 2003, A numerical study of the interfacial transport characteristics outside spheroidal bubbles and solids. Int. J. of Multiphase Flow, 29(3), 435-460.

[3] Yan, Y.Y. and Li, W.Z., 2006. Numerical modelling of a vapour bubble growth in uniformly superheated liquid, Int. J. of Numerical Methods for Heat & Fluid Flow, 16(7), 764-778.

[4] Krishna, R. and Baten, J.M. van, 1999, Simulating the motion of gas bubbles in a liquid, Nature, Vol. 398, 208.

[5] Yang, Z.L., Do, Q.M., Palm, B. and Sehgal, B.R., 2000, Numerical simulation of bubble dynamics: lattice Boltzmann approach, Proc. of 5th Int. Symp. on Heat Transfer, 12-16 Aug., Beijing, 598-603.

[6] Inamuro, T., Ogata, T., Tajima, S. and Konishi, N., 2004, A lattice Boltzman method for incompressible two-phase flows with large density differences. J. Comput. Phys., 198, 628-644.

[7] Zheng, H.W., Shu, C. and Chew, Y.T., 2006, A lattice Boltzmann model for multiphase flows with large density ratio, J. of Compt. Phys., 218, 353-371.

[8] Chen, S. and Doolen, G.D., 1998, Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 30, 329-364.

[9] Gunstensen, A.K., Rothman, D.H., Zaleski, S. and Zanetti, G., 1991, Lattice Boltzmann Model of Immiscible Fluids. Phys. Rev. A 43, 4320-4327.



[10] Shan, X.W. and Chen, H.D., 1993, Lattice Boltzmann Model for Simulating Flows with Multiple Phases and Components. Phys. Rev. E 47,1815-1819.

[11] Swift, M.R., Osborn, W.R. and Yeomans, J.M., 1995, Lattice Boltzmann Simulation of Nonideal Fluids. Phys. Rev. Lett. 75, 830-833.

[12] Swift, M.R., Orlandini, E., Osborn, W.R. and Yeomans, J.M., 1996, Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Phys. Rev. E 54, 5041-5052.

[13] He, X.Y., Chen, S.Y. and Zhang, R.Y., 1999, A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability, J. of Compt. Phys. 152, 642-663.

[14] Briant, A.J., Papatzacos, P. and Yeomans, J.M., 2002, Lattice Boltzmann simulations of contact line motion in a liquid-gas system. Phil. Trans. of the Royal Society Series a-Mathematical Phys. and Eng. Sci. 360, 485-495.

[15] Dupuis, A. and Yeomans, J.M., 2005, Modeling droplets on superhydrophobic surfaces: Equilibrium states and transitions. Langmuir 21, 2624-2629.

[16] Yan, Y.Y. and Zu, Y.Q., 2006, A lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio, submitted to J. Compt. Phys.

[17] Rowlinson, J.S. and Widom, B., 1989, Molecular Theory of Capillarity, Clarendon, Oxford.

[18] Young, T., 1805, An essay on the cohesion of fluids. Phi. Trans. R. Soc. Lond. 95, 65-87.



Simulation of radial oscillations of a free and a contrast agent bubble in an ultrasound field

A. V. Teterev, N. I. Misychenko, L. V. Rudak & A. A. Doinikov Belarus State University, Belarus

Abstract

A one-dimensional numerical model has been developed for the interaction of an ultrasound field with a free and an encapsulated gas bubble immersed in a liquid. The model includes several approaches to simulations of this type. First, the calculation of the radius of the bubble can be performed by Rayleigh–Plesset-type equations, while the distribution of the gas-dynamic parameters within the bubble is calculated by solving the equations of gas dynamics which are represented in terms of Lagrangian coordinates. Second, a through calculation can be carried out by solving the equations of fluid dynamics for both the interior of the bubble and the surrounding liquid. These numerical approaches can be applied to both free and encapsulated (contrast agent) bubbles. In the case of an encapsulated bubble, the equations describing the radial oscillation of a bubble enclosed in a fluid or solid shell are used. Simulations for a wide range of driving frequencies and bubble radii have been conducted. The obtained results demonstrate wide capabilities of the developed model. Keywords: contrast agents, encapsulation, ultrasound, fluid dynamics, radial oscillation, rheological behaviour, numerical simulation.

1 Introduction

The wide application of ultrasound contrast agents in medicine and the variety of materials used for the encapsulation of contrast agent microbubbles have given rise to numerous theoretical, numerical and experimental investigations in this field [1]. The non-Newtonian behaviour of blood and intricate rheological properties of encapsulating shells hamper the theoretical description of the dynamics of contrast agents in an ultrasound field. Depending on their material,



doi:10.2495/MPF070231

encapsulating shells can exhibit the properties of a viscoelastic solid (Kelvin–Voigt solid), a viscoelastic fluid with stress relaxation (Maxwell fluid), as well as properties whose rheological law is still not understood. Buckling of lipid monolayer coatings is an example [2]. In [3], a model for large-amplitude oscillations of thin-shelled microbubbles has been developed. Theoretical analysis of small-amplitude oscillations of encapsulated bubbles with shell thickness exceeding 15 nm was carried out in [4]. The behaviour of the gas within a free or an encapsulated bubble can also be different from the assumptions underlying Rayleigh–Plesset-type equations. First, heat transfer between the gas and the surrounding liquid (or the shell) can impact the flow pattern around the bubble. This dissipative process can change significantly the dynamics of the surrounding liquid. Second, the distribution of the gas-dynamic parameters within the bubble can be nonstationary rather than homogeneous, as commonly supposed.

The purpose of the present paper is to develop a theoretical and a numerical models that make it possible to examine different scenarios for the behavior of free and encapsulated bubbles in both small- and finite-amplitude ultrasound fields, taking adequate account of existent physical processes.

2 Theoretical model

The proposed model describes oscillations of a free or encapsulated bubble in two different formulations. In the first case, the velocity of the bubble surface is found by solving a Rayleigh–Plesset-type equation. This velocity is then used as the boundary condition at the gas-liquid interface in the gas-dynamic problem which describes the gas motion within the bubble for a given time layer. The gas pressure obtained by solving this problem is then used to determine the velocity of the bubble surface for the next time step. In the second case, the motion of both the gas within the bubble and the surrounding liquid (as well as of the shell if it is present) is simulated by solving a respective gas-dynamic problem.

2.1 Limitations and assumptions

It is supposed that the gas inside the bubble is heat-conducting and its calorific and thermal state equations are known and can be specified either analytically or in tabulated form. If a Rayleigh–Plesset-type equation is used, the gas motion is described by a polytropic equation, normally in the adiabatic approximation. The surrounding liquid is assumed to be compressible with a barotropic equation of state, such as the empiric Tait equation. It is also supposed that the surrounding liquid is at constant temperature and behaves as a Newtonian or a more complex fluid. The shell of the bubble is also at constant temperature, can be either compressible or incompressible, and its rheological behaviour can follow either a fluid or a solid. Mass transfer between the gas and the surrounding liquid (or the encapsulating layer) is assumed to be absent.



2.2 Rayleigh–Plesset-type model for bubble oscillations

The present model allows the calculation of oscillations of a free or encapsulated bubble by a Rayleigh–Plesset-type equation. This approximation is applied both as an independent approach and in combination with gas-dynamic equations for the determination of the boundary conditions at the interface with the gas.

2.2.1 Model for a free bubble The radial velocity of the surface of a bubble can be obtained from a Rayleigh–Plesset-type equation the general form of which can be represented as

( )γρησ ,,,,,,,,, 00 cPPPvRFv acg= , Rv = , (1)

where ν is the radial velocity of the bubble surface, the overdot denotes the time derivative, R is the time-varying radius of the bubble, P0 is the initial pressure in the surrounding liquid, Pg is the gas pressure, Pac is the imposed acoustic pressure, σ is the surface tension, η is the liquid viscosity, ρ0 is the equilibrium liquid density, c is the speed of sound in the liquid, and γ is the ratio of specific heats. A particular form of eqn. (1) is chosen depending on the problem parameters. For example, if the radial motion of the bubble is described by the Keller–Miksis model [5], eqn. (1) takes the form

−−+

−

+

−=

−

cvvP

cR

gPP

cv

cv

Rv R

R

31

23111 2

00

01

ρ, (2)

where RP is given by

( ) ( )2 2 4R g ac

v vP P t P tR R R

σ η= − − − − , (3)

The time-varying gas pressure ( )tPg is calculated either from the adiabatic law,

( )3

00

0

2g

RP t P

R R

γσ = +

, (4)

in the case that the calculation is carried out by using the Keller–Miksis model alone, or from the gas-dynamic problem, as the value of the gas pressure at the inner bubble surface.

The imposed acoustic pressure ( )tPac is specified by

( ) ( )ftPtP Aac π2sin= , (5)

where PA is the pressure amplitude and f is the driving frequency. Let us denote eqns. (2)-(5), or similar equations of the family of

Rayleigh–Plesset-type equations such as the original Rayleigh–Plesset equation [6] or the Herring–Flynn equation [7], by S0B.



2.2.2 Model for a contrast agent bubble In the case of an encapsulated bubble, equations for the velocities of the inner and outer surfaces of the shell can also be written in the form similar to eqn. (1). The arguments of the function F on the right-hand side of eqn. (1) will now be geometrical, physical, thermodynamic and rheological parameters of the bubble, the shell and the surrounding liquid.

In [8], an equation for the radial motion of an encapsulated bubble was derived assuming that the shell is incompressible, the surrounding liquid is weakly compressible, and both the shell and the liquid are viscoelastic fluids following the linear 3-constant Oldroyd constitutive equation [9]. Using the results of [8], one obtains

( ) ( )( )

( )( )( ) ( ) ( ) ,121212

432

11

13

11

2233311

1

2

321

11

−

+−+

−++

+

−+−+=

βδνρρ

βδβδνρρ

δβν

βν

cGR

c

GR

(6)

where the function G is given by

( ) ( )

−−

−−

= ∑

=

tPPR

tDRR

RPG aci i

iii

i

ig 0

2

13

3

1

100

1

21 αησ

ρ

γ

and the following nondimensional quantities are introduced

( ) ( )3 3 31 20 10 2 2 1 2 2 1 1, 1, ,R R R R Rα α δ β ρ ρ δ ρ= − = = = − .

From this point on, the subscript 1 corresponds to the parameters of the shell or its inner surface, while the subscript 2, to the surrounding liquid or the outer surface of the shell. The subscript i = 1 or 2 with the same meaning and the subscript 0 denotes the initial values. Di, R2 and Pg0 are calculated from

( )2111

2121

211 2 vRvRvRDD iiii ++=+ λλ , (7)

( ) 3/131

310

3202 RRRR +−= , (8)

( )0 0 1 10 2 202 2gP t P R Rσ σ= + + , (9) where λi1 is the relaxation time and λi2 is the retardation time. These times are the parameters of the Oldroyd constitutive equation [9].

Thus, the velocity of the surface of an encapsulated bubble can be obtained by solving the differential equations (6)-(9). Let us denote this system as S0S.

2.3 Model for the gas dynamics inside a bubble

As the problem under consideration is one-dimensional, it is reasonable to solve it using the Lagrangian method. The continuity equation, written in Lagrangian



mass coordinates, for the case of spherical symmetry, and with respect to density per unit spatial angle, takes the form [10]

( )m

urt ∂

∂=

∂∂ 21

ρ, (10)

where ρ and u are the density and the velocity of the gas, respectively, and the relation between the Euler and the mass coordinates is given by dm = ρr2dr .The equation of motion in the Lagrangian coordinates is written as

mpr

tu

∂∂

−=∂∂ 2 , (11)

where p is the gas pressure. The small size of the bubble and the high value of the speed of sound lead to

very small time steps. As a result, the solution of the problem is reached over a very large number of time steps. In addition, in some cases it is necessary to trace the medium parameters for a large number of oscillations. These circumstances make high demands to the accuracy of energy computation. Therefore it is reasonable to take the equation of energy in the divergent form taking into account heat conductivity,

( )

∂∂

∂∂

+∂∂

−=

+

∂∂

rTr

mpur

mu

t T22

2

2λε , (12)

where ε is the internal energy per unit mass, λT is the gas heat conductivity coefficient and T is the temperature.

The surface tension pressure on the bubble surface is given by

2PRσσ

= . (13)

To close the set of eqns. (10)-(13), it should be supplemented with a state equation, p = p(ρ,ε). If the temperature distribution within the gas is required, an equation for temperature, T = T(ρ,ε), should be added as well. For a perfect gas, these equations can be written as p = (γ-1)ρε and T = ε/cv, where cv is the specific heat at constant volume.

The boundary condition Ru v= at the spherical surface of the bubble is sort of a piston, whose velocity v is calculated from the Rayleigh–Plesset-type equation (1). Let us denote the system of eqns. (10)-(13), which describes the dynamics of heat-conducting gas inside the bubble, as S1.

2.4 Model for the encapsulating shell and the surrounding liquid

If both the surrounding liquid and the shell are compressible, the continuity equations for them are given by eqn. (10) with the respective density for each medium. The equations of motion for both media take the form



ρr

SmSr

mpr

tu rrrr 322 +

∂∂

+∂∂

−=∂∂ , (14)

where Srr is the radial component of the stress deviator. For example, for a viscoelastic plastic solid, Srк and the viscous stress tensor qrr are given by [11]

∂∂

+∂∂

−=∂∂

tru

tS

irr ρ

ρµ

312 , (15)

∂∂

+∂∂

=tr

uq irrρ

ρη

312 , (16)

where µi is the shear modulus and ηi is the shear viscosity for the liquid or the shell. For a viscoelastic fluid, Srr is specified by the Oldroyd equation [9],

∂∂

+=+∂∂

tuuS

tS rr

irrirrrr

i 21 2 ληλ , (17)

where urr = ∂u/∂r denotes the radial component of the rate-of-strain tensor. The energy equation in this case is given by

( ) ( )uSrm

purm

ut rr

222

2 ∂∂

+∂∂

−=

+

∂∂ ε . (18)

To close the set of the equations, a state equation for each media is also required. For the surrounding liquid, as well as if the shell is treated as a fluid, the Tait equation can be used,

( )0

n

p A Bρρρ

= −

, (19)

where A, B and n are constants. If the shell is treated as a solid, the Tillotson equation can be applied. Let us denote eqns. (14)-(19), with eqns. (15) and (16), or eqn. (17), or a different rheological law, by S2.

3 Numerical model

Differential equations with partial derivatives are solved by finite-difference schemes [11, 12] and ordinary differential equations are solved using methods described in [13].

3.1 Variants of the model

The numerical model has three regimes for the computation of the radial oscillation of a free or encapsulated bubble. If the systems S0B or S0S, based on



Rayleigh–Plesset-type equations, are solved, this approximation is called zero-order approximation and denoted by M0I, where I = B for a free bubble and I = S for an encapsulated bubble.

The next approximation is that the dynamics of the gas within the bubble is modeled by the gas-dynamic equations S1. In this case, the boundary condition on the surface of a bubble with radius R or R1 is obtained from the systems S0B and S0S for a free and an encapsulated bubble, respectively. The gas pressure at surface of the bubble that is used in the systems S0 is calculated from the system S1. This variant of the numerical model is called M1I.

Finally, a through calculation can be carried out where the motion of the liquid around a free or encapsulated bubble is simulated by the system S2 and the motion of the gas within the bubble by the system S1. This approximation is called second-order approximation and denoted by M2I. In this case, a simultaneous calculation by the model M0 is also possible, which allows comparing these two solutions in real time directly in the course of computer simulations.

3.2 Software implementation of the model

The software suite for solving the above-mentioned differential equations is based on the program package OLYMPUS. It has an easy-to-use interface that allows the variation of the control parameters of the problem in real time, if necessary. The visual control of the simulation and on-line comparison of its results is provided by the graphics display system.

4 Numerical examples

As an example of the capabilities of the developed numerical model, let us consider oscillations of an air bubble of radius R0 = 1 µm in water at the atmospheric pressure P0 = 105 Pa, which undergoes an ultrasonic wave field with a frequency of 1 MHz and a pressure amplitude of PA = 2P0.

Figure 1 shows the time-dependent radius of a free bubble that was calculated for the three approximations M0B, M1B and M2B. The solid line represents the radius of a bubble with a fluid shell the thickness of which is approximately 3.5% of the initial radius of the bubble and the density is larger than the density of water by a factor of 1.1. The data are given for six acoustic cycles. One can see that for first two cycles, all the curves are close to one another. After the third cycle, however, significant differences are observed in both the amplitude and the frequency of the oscillations. The maximum amplitudes of the oscillations correspond to the calculations in the fully hydrodynamic approximations M2 for both free and encapsulated bubbles. It should be noted that for the calculation by the model M1, the amplitude of the oscillation gradually increases as well.

Let us consider the behaviour of the radial velocity of a free bubble shown in fig. 2. The results were obtained by using the models M0 and M1. The scale of the plot is limited to ± 100 m/s in order to have a possibility to trace the behaviour of the velocity near zero. It should be noted that the absolute values of velocities during the expansion and compression of the bubble reached 700 m/s.



Figure 1: The time-dependent radius of a free bubble for three variants of calculation by the models M0B, M1B and M2B. The solid line shows the results obtained by the model M2S for an encapsulated bubble with a fluid shell.

Figure 2: The velocity of the bubble surface calculated by the model M0B with eqn. (2) and by the model M1B taking account of the gas motion within the bubble.

This fact shows that the compressibility of the surrounding liquid should be taken into account. A visible difference between the two velocity curves appears at the end of the third cycle, whereupon one can see that the period of the main oscillation of the bubble obtained by the model M1 becomes different from the period calculated from the model M0. This effect can be accounted for by the irreversibility of the oscillatory process when the dynamics of the gas within the bubble is modeled in the regime of sonoluminescence, see [14]. The violation of



reversibility arises from the fact that the pressure at the surface of the bubble is different for the stages of expansion and compression at the same value of the radius of the bubble. As a consequence, a gradual pumping of energy into the bubble occurs, which in turn affects the frequency characteristic of the oscillation of the bubble.

5 Conclusions

The developed model is a logical extension and improvement of the model proposed in [14]. It makes possible both widening the circle of numerical simulations on oscillations of free gas bubbles in liquids in response to an imposed strong ultrasound field and solving similar problems for encapsulated bubbles with different rheological models for the encapsulating shell. The model also allows one to apply different rheological laws to the surrounding liquid, which makes possible the simulation of more complicated media than Newtonian fluids, such as blood.

Comparing the results of numerical simulations carried out in different formulations, one can determine the validity of one or another of the models describing the oscillation of a free or contrast agent bubble. For example, in the simulation performed in the present paper, the models M0B, M1B and M2B provide quite comparable results. Whereas results that are obtained when the compressibility of the surrounding liquid is neglected (not presented here) show that this neglect is inadmissible.

The implementation of the described model is a handy and flexible tool for simulating various aspects of the oscillatory dynamics of free and contrast agent bubbles in both small- and large-amplitude ultrasound fields. Further improvement of the model will lie in the description of mass transfer at the surface of a gas bubble and taking account of the vapour component of this process.

Acknowledgement

This work was supported by the US member of the International Science and Technology Center (ISTC) under Contract B-1213.

References

[1] Doinikov, A.A., (ed). Bubble and Particle Dynamics in Acoustic Fields: Modern Trends and Applications, Research Signpost: Kerala, India, 2005.

[2] Borden, M., Pu, G., Runner, G. & Longo, M., Surface phase behavior and microstructure of lipid/PEG-emulsifier monolayer-coated microbubbles. Colloids and Surfaces B, 35, pp. 209–223, 2004.

[3] Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N., Hilgenfeldt, S. & Lohse, D., A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture. Journal of the Acoustical Society of America, 118(6), pp. 3499–3505, 2005.



[4] Khismatullin, D.B. & Nadim, A., Radial oscillations of encapsulated microbubbles in viscoelastic liquids. Physics of Fluids, 14(10), pp. 3534–3557, 2002.

[5] Keller, J.B. & Miksis, M., Bubble oscillations of large amplitude. Journal of the Acoustical Society of America, 68, pp. 628–633, 1980.

[6] Plesset, M.S. & Prosperetti, A., Bubble dynamics and cavitation. Annual Reviews of Fluid Mechanics, 9, pp. 145–185, 1977.

[7] Herring, C., Theory of the pulsations of the gas bubble produced by an underwater explosion, OSRD Report 236, 1941.

[8] Doinikov, A.A. & Teterev, A.V., Dynamics of ultrasound contrast agents with lipid coating, CD-ROM Proc. of the 13th Int. Cong. on Sound and Vibration (ICSV13), Vienna, Austria, July 2-6, 2006.

[9] Bird, R.B., Armstrong, R.C. & Hassager, O., Dynamics of Polymeric Liquids, Wiley: New York, 1987.

[10] Zeldovitch, Y.B. & Raiser, Yu., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic: New York, 1967.

[11] Mader, C.L., Appendix A & B, Numerical Modeling of Detonations, University of California Press, 1985.

[12] Samarsky, A.A. & Popov, J.P., Difference Methods for Solving Gas Dynamic Equations, Nauka: Moscow, 1980.

[13] Haiser, E., Norsett, S.P. & Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag: Berlin and New York, 1987.

[14] Teterev, A.V., Misychenko, N.I., Rudak, L.V. & Doinikov, A.A., Numerical model for the interaction of a gas bubble with a strong acoustic field, CD-ROM Proc. of the 13th Int. Congress on Sound and Vibration (ICSV13), Vienna, Austria, July 2-6, 2006.



Visualization method for volume void fraction measurements in gas–liquid two-phase flows of a water turbine outlet channel

R. Klasinc1, M. Hočevar2, T. Baicar2 & B. Širok2 1Graz University of Technology, Department for Hydraulic Structures and Water Resources Management, Austria 2University of Ljubljana, Faculty for Mechanical Engineering, Slovenia

Abstract

The Pelton turbines are the part of the pump-storage scheme that is working under permanent backpressure conditions. The knowledge of air-absorbing and degassing in the vessel which follows the turbine chamber is important for the safe and economical working of the turbine. In the case of so called hydraulic short circuit the influence of air bubbles for the safe operation of turbines and pumps is very important. To estimate the volume void fraction in two-phase flow, the measurement method was developed, which is based on the computer-aided visualization. Acquisition of flow images by a fast video camera followed by the computer processing of the acquired images enables the determination of the void fraction via the average value of grey level intensity of the acquired flow images. A relationship between the void fracture and the average grey level intensity was obtained through calibration of the measurement system. Simultaneous measurements with the other measurement method revealed suitability and applicability of the visualization method in a real industrial environment such as a tailwater channel of a Pelton turbine. Apart from that, a study revealed that by further processing the images a number of other important information on the flow, such as bubble size and shape, average bubble direction and velocity, can be obtained by the described method. Keywords: air-water two-phase flow, computer-aided visualization, tailwater channel.



doi:10.2495/MPF070241

1 Introduction

The void fraction, which denotes the phase distribution in multiphase flows, is one of the most important parameters of two-phase flows, especially in two-phase gas–liquid systems. Measurement of the void fraction is essential in many industrial applications in order to define or monitor the parameters such as pressure drop, heat transfer rate or any other transfer mechanism or process between the flow and the confining walls or between phases themselves within the two-phase flow. There are many intrusive as well as non-intrusive methods, which are commonly used to measure void fraction in multiphase flows. Some of the most popular are the method of gamma rays absorption (Stahl and von Rohr [9]), the method of magnetic resonance (Daidzic et al., [1]), tomography and impedance methods (Dong et al., [2]; Jaworek et al., [6]; Huang et al., [5]), double sensor probes (Hogsett and Ishii, [4]; Hibiki and Ishii, [3]) or isokinetic (suction) methods (Mayr and Arch, [7]). It is sometimes hard to identify the most appropriate method for measuring the void fraction. As far as industrial measurements are concerned, there is often a need to conduct fast yet accurate enough measurements in harsh industrial conditions, e.g. in purification facilities, waste water processing, powerplants etc. The above-mentioned methods are often not particularily suitable for such measurements, for they are either meant to be used in laboratory conditions and are not robust enough or they demand extensive (and expensive) preparations or they are simply too complex or clumsy for quick and easy handling. The visualization method, which is described below, tends to fill such a gap, for it is developed mainly to be used in the industrial environment.

2 Description of the system

The measurement system for the volume void fraction measurements in open channels is based on the acquisition of the images of the water flow using the high speed camera. The measurement system consists of the the following main parts, which are pointed out in Fig. 1 as well:

- black & white industrial high speed camera SONY XC HR50 with power supply 12 V,

- lens cosmicar / pentax H1212B 12 mm F 1.2 with close up lenses 6+5 D,

- illumination LED CCS LDL-TP-51x51 12V, 4.4W, - cables for connection of the camera to the framegrabber and power

supply of length 30 m, - water resistant measurement probe with chassis for the camera, lens and

illumination, - framegrabber National Instruments PCI NI 1409, - base software package National Instruments Labview together with

Vision module, - personal computer, monitor, keyboard and mouse.



Figure 1: Schematics of the measurement system.

The spatial resolution of the camera is 640×480 pixels. Each pixel corresponds to 0.04185 mm, calculated in the center of gravity of the volume part between the camera and the illumination. The acquisition frequency of the camera is set to 60 s-1. The black & white resolution is 8 bits or 256 grey levels. Void fraction measurement by the described system is based on the average grey level intensity of the images of the flow in a measurement volume between the camera lens and the illumination source. As a type of illumination we have selected background illumination, therefore images of the illuminated water have a high value of grey level intensity, which depends on the water turbidity and is close to white (grey level intensity 1). On the other hand, the illuminated air bubble has a very low value of the grey level intensity along its circumference, which is close to black (grey level intensity 0) – Fig. 2.

Figure 2: Image of an air bubble in liquid (water).

Liquid phase (high value of grey level intensity)

Gaseous phase (low value of grey levelintensity along the bubblecircumference)



Average grey level intensity (A) can be obtained from N images by eqn. 1 (Trdič et al., [10]):

∑ ∑= =

=

N

i

n

jjE

nNA

1 1

11 , (1)

where E denotes the grey level intensity of the j-th pixel of the image. n denotes the number of all pixels in a single image. The variable E as well as the variable A have values between 0 (black) and 1 (white). A calibration procedure has to be carried out in order to obtain the relationship between the average grey level intensity A and the void fraction α.

3 Calibration

The calibration of the developed measurement system for volume air void fraction measurements in water was performed using newly manufactured calibration device (Fig. 3). The calibration procedure is based on the measurement of the increase of the height of the water upper surface in the water column when bubbles are injected in the flow. The calibration device consisted of vertical vessel of square cross section, openings for inclusion of measurement system, air supply in the bottom part of the vessel through porous inserted block, and measurement slit for measurement of water level in the vessel.

Figure 3: Configuration of the calibration device.



The calibration device enables immediate calibration of the measurement system in the form of functional relationship between the water height level and void fraction of the gaseous phase in the liquid phase, expressed with the variable α as defined by the expressions:

vz

z

VVV+

=α (2)

( ) vz ρααρρ −+= 1 (3)

( )zv

ov H

H

ρρ

ρα

−

−

=1

, (4)

where ρz and ρv denote air and water density, respectively, Ho denotes the water level before the inclusion of gaseous phase, and H denotes the water level during the inclusion of gaseous phase. The calibration procedure was performed using the assumption that the two phase bubble flow is homogeneous and that there is the same distribution of bubbles size in the calibration device and in the field experiment. The effect of compressibility was considered as well. Calibration results are given in Figs. 4 and 5 using presumed functional relationships. These enable calculation of the air void fraction in the test section in dependence on the measured grey level intensity. Figs. 4 and 5 show monotonous functional relationship between the both variables. The non-linear functional relationship can be represented by exponential dependence in the form:

α21

kekA −= , (5) where A denotes measured grey level intensity, α is the volume air void fraction in the control volume, whereas k1 and k2 denote experimentally derived constants. The inverse functional relationship can be determined and expressed in the form:

43 )ln( kAk +=α , (6) where k3 and k4 are experimentally derived constants. The value of the correlation coefficient (r) between measured values and the trend line function (y) for α (Fig. 4) and A (Fig. 5) is r2 = 0.99 in both cases. It should be emphasized that for proper determination of functional relationship between the air volume void fraction and grey level intensity a correction due to



turbidity of the water should be performed where necessary on site of the measurement. In some cases a correction due to different bubble sizes as well as the effect of bubbles coalescence should also be performed. For the present work the calibration was performed using clear potable water.

Figure 4: Functional dependence of volume air void fraction on measured grey level intensity.

Figure 5: Functional dependence of grey level intensity on volume air void fraction.



4 Measurement setup

Void fraction measurements were performed in the outlet channel of the Koralpe hydroelectric plant (Drau river) according to the order placed by the Technische Universität Graz (TUG). Schematics of the measurement points selection is shown in Fig. 6. Eight transverse measurement planes were selected with distance from the turbine axis shown in table 1. In every measurement plane six measurement points were selected with vertical positions shown in table 2. The positioning was provided by servo positioning system from TUG (Mayr and Arch, [7]). Water depth in the channel was 1.63 m. For the transverse measurement planes e to h in the middle position, the height of the measuring point 6 at the vertical position 6 is 1.50m (vertical distance from the channel bottom).

Figure 6: Schematics of measurement points selection. Side view.

Table 1: Distance of measurement planes from the turbine axis. Schematics are shown in Fig. 6 and 7.

Transverse measurement

plane a b c d e f g h

Distance from the

turbine axis (m)

5.915 7.915 9.915 11.915 13.915 15.915 17.915 19.915

Table 2: Vertical positions of measurement points. Schematics are shown in Figs. 6 and 7. For the measurement planes e to h, the height of the measuring point 6 is 1.50 m, all distances are measured vertically from the channel bottom.

Vertical measurement position 1 2 3 4 5 6 Distance from the bottom (m) 0.25 0.5 0.75 0.1 1.25 1.4



There were three parallel longitudinal (x-y) measurement planes – middle, left and right measurement plane (Fig. 7). Longitudinal axis of the channel (x) lay in the central x-y measurement plane; left and right x-y measurement planes were positioned 0.95 m to the left and right, respectively, from the central x-y plane according to the waterflow direction (from turbine to the Drau river). The central x-y measurement plane was shifted 0.66 m to the left due to mounting of the probe as compared to the measurement probe from TUG (Mayr and Arch, [7]). Each longitudinal measurement plane contained 8 × 6 measurement points, which were positioned as depicted in Figs. 6 and 7.

Figure 7: Schematics of measurement points selection. Shown are three

longitudinal measurement planes – middle, left and right, top view.

In every measurement point on a particular measurement plane 240 images of the flow were processed. The exposure time was set to 1/1000 s.

5 Results

Fig. 8 show typical sample images of the two phase flow during measurements taken at each horizontal position x and at three different vertical positions y = 0.25 m, y = 0.75 m and y = 1.25 m. It is clearly seen from Fig. 8 that the volume void fraction decreases with decreasing height from the bottom (y) and we can expect the same with increasing distance from the turbine vertical axis (x).

Figure 8: Sample images taken at a transverse measurement plane d, vertical position y = 1.25 m (left), y = 0.75 m (middle) and y = 0.25 m (right).



From series of consecutive images of two-phase bubble flow structures, the basic variable is presented through time-averaged grey level intensities and appropriate standard deviation of the latter. Equivalent diagrams of void fraction distribution (according to the calibration protocol) were calculated according to the eqn. 6. By applying the functional relationship between the measured grey level intensity and the volume void fraction, depicted in Fig. 4, it is possible to obtain similar diagrams for void fraction (Fig. 9).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Distance to turbine axis [m]

00.61.21.8

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.61.21.8

Hei

ght [

m]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.61.21.8

Left

Middle

Right

1%

3%

5%

7%

9%

11%

13%

15%

17%

19%

21%

Figure 9: Average volume void fraction.

0,25

0,5

0,75

1

1,25

1,5

0 5 10 15 20 25

Volume void fraction [%]

Hei

ght [

m]

average

standard deviation

Figure 10: Average volume void fraction and its standard deviation in the transverse measurement plane c (middle longitudinal measurement plane).



It can be seen from Fig. 9 that the flow in the outlet channel of the turbine is two-phased up to the distance of approx. 16-17 m from the turbine vertical axis; from this point on the flow is mainly single-phased (liquid). The two-phase flow exists in the whole transverse section of the channel only up to a distance of approx. 8-9 m from the turbine veritical axis. Afterwards, the flow stays two-phased only in the upper part of the channel due to buoyancy effects. Standard deviation of the average grey level intensity as well as of the volume void fraction (Fig. 10) is the highest in the region of transition from two-phase to single-phase flow.

6 Conclusions

The paper describes a visualization method for volume void fraction measurement in the gaseous-liquid flow. The volume void fraction is measured by means of image acquisition via a fast industrial camera and by determination of the average grey level intensity of the acquired images. The relationship between the average grey level intensity of the images and the void fraction was determined by a calibration procedure at known values of void fraction. It turned out that such a relationship could be described by a simple law, which is based on the exponential function. Apart from the void fraction measurement it is possible to use the described method for the determination of shape and size of the gas bubbles. By means of statistical processing of the succesive images the assessment of the average bubble direction and their velocity could be achieved at every measuring point (future works). Measurements conducted in the tailwater channel of a Pelton turbine proved the suitability of the described method not only by the comparable results with the other measurement method, but also with respect to robustness, relative simplicity, ease of handling and applicability for operation in industrial environments (hydro powerplants, purification facilities, etc). One of the most important features of the described visualization method is the fact that the volume void fraction can be easily monitored during the operation of the station/facility without the need to stop or hamper the operation of such a facility.

References

[1] Daidzic, N.E., Schmidt, E., Hasan, M.M., Altobelli, S., Gas–liquid phase distribution and void fraction measurements using MRI, Nuclear Engineering and Design 235, pp. 1163-1178, 2005.

[2] Dong, F., Jiang, Z.X., Qiao, X.T., Xu, L.A., Application of electrical resistance tomography to two-phase pipe flow parameters measurement, Flow Measurement and Instrumentation 14, pp.183–192, 2003.

[3] Hibiki, T., Ishii, M., Experimental study on interfacial area transport in bubbly two-phase flows, International Journal of Heat and Mass Transfer 42, pp. 3019-3035, 1999.



[4] Hogsett, S., Ishii, M., Local two-phase flow measurements using sensor techniques, Nuclear Engineering and Design 175, pp. 15–24, 1997.

[5] Huang, Z., Xie, D., Zhang, H., Li, H., Gas–oil two-phase flow measurement using an electrical capacitance tomography system and a Venturi meter, Flow Measurement and Instrumentation 16, pp. 177–182, 2005.

[6] Jaworek, A., Krupa, A., Trela, M., Capacitance sensor for void fraction measurement in water/steam flows, Flow Measurement and Instrumentation 15, pp.317-324, 2004.

[7] Mayr, D., Arch, A., Entgasung von Wasser-Luft-Gemischen in Unterwasserkanälen von Pelton-Wasserkraftanlagen, Österreichische Wasser- und Abfallwirtsch. 11-12, pp. 167-175, 2005.

[8] Sach, L., Angewandte Statistik: Anwendung statistischer Methoden, Springer-Verlag: Berlin,1997.

[9] Stahl, P., von Rohr, P.R., On the accuracy of void fraction measurements by single-beam gamma-densitometry for gas–liquid two-phase flows in pipes, Experimental Thermal and Fluid Science 28, pp. 533-544, 2004.

[10] Trdič, F., Širok, B., Bullen, P.R., Philpott, D.R., Monitoring mineral wool production using real-time machine vision, Real-time imaging 5, pp. 125-140, 1999.



Nonlinear dynamics of lipid-shelled ultrasound microbubble contrast agents

A. A. Doinikov1 & P. A. Dayton2 1Belarus State University, Belarus 2University of California, Davis, USA

Abstract

Encapsulated gas microbubbles, known as contrast agents, are widely used in ultrasound medical applications. The present study is devoted to modelling of the spatio-temporal dynamics of lipid-shelled contrast agents. A theoretical model is proposed that describes the radial and translational motion of a lipid-shelled microbubble in an ultrasound field. The model approximates the behaviour of the lipid shell by the linear 3-constant Oldroyd constitutive equation, incorporates the translational motion of the bubble, and accounts for acoustic radiation losses due to the compressibility of the surrounding liquid. The values of the shell parameters appearing in the model are evaluated by fitting simulated radius-time curves to experimental ones. The results are then used for the simulation of the translational motion of contrast agent bubbles of various radii and the evaluation of the relationship between equilibrium radii of lipid-shelled agents and their resonance frequencies in the regime of nonlinear oscillation. Keywords: contrast agents, encapsulated bubbles, lipid shell, ultrasound, radial oscillation, translational motion, resonance frequencies.

1 Introduction

Ultrasound contrast agents are micron-sized encapsulated gas bubbles which are produced by pharmaceutical companies for medical ultrasound applications [1]. They are normally injected into the bloodstream of the patient in order to increase blood-tissue contrast during an ultrasonic examination and thereby to improve the quality of ultrasonic images. Contrast agents are also used in targeted imaging and ultrasound-assisted localized drug delivery [2, 3]. Targeted agents are taken up by specific tissues or adhere to specific sites in the body.



doi:10.2495/MPF070251

By enhancing the acoustic differences between normal and abnormal parts of organs, these tissue-specific agents improve the detectability of abnormalities, such as lesions, inflammatory processes and thrombi. In addition, targeted agents can carry drugs or genes to be delivered to a specific site or tissue, which provides great possibilities for a highly selective therapeutic action. The shell is necessary to prevent bubbles from fast dissolution in blood and coalescence. Proper theoretical description of the shell is of primary importance as it is the shell that determines many of the functional properties of contrast agents. The shell of currently available contrast agents is made of albumin, polymer, or lipid. The present study is devoted to modelling of the spatio-temporal dynamics and investigation of resonant properties of lipid-shelled contrast agents.

2 Theoretical model

Consider a spherical encapsulated gas bubble immersed in an unbounded liquid and undergoing radial oscillations in response to an imposed acoustic field. The gas in the bubble is separated from the encapsulating layer by interface 1 while the encapsulating layer is separated from the surrounding liquid by interface 2. The radial oscillation of the bubble will be described by the generalized Rayleigh–Plesset equation [4, 5]:

33 31 2 1 1 10 1 22

1 01 1 32 2 2 1 1 2

3 4 1 2 212 2 g

S

R R R R RR PR RR R R R R R

γσ σβ β

ρ

− + + + = − −

2

1 2

( ) ( ) 0

( , ) ( , )( , ) 3 3 S LR rr rr

acR R

r t r tx t dr drP P r rτ τ∞

− − + +

∫ ∫ , (1)

where 1( )tR and 2( )tR are the inner and the outer radii of the bubble shell, respectively, the overdot denotes the time derivative, ( )L S Sβ ρ ρ ρ= − , Sρ and Lρ are the equilibrium densities of the shell and the surrounding liquid, respectively, 0gP is the equilibrium gas pressure inside the bubble, γ is the ratio of specific heats of the gas, 10R is the inner radius of the bubble shell at rest, 1σ and 2σ are the surface tension coefficients for the corresponding interfaces, 0P is the hydrostatic pressure in the surrounding liquid, ( , )ac x tP is the driving acoustic pressure at the location of the bubble, ( )x t is the spatial position of the centre of the bubble in an inertial frame, ( )S

rrτ and ( )Lrrτ are the stress deviators of

the shell and the liquid, respectively, and r is the distance from the centre of the bubble. Note that eqn. (1) assumes that the surrounding liquid and the encapsulating layer are incompressible so that both the liquid velocity and the velocity inside the bubble shell are given by

2 21 1( , ) ( ) ( )v r t t tR rR= , (2)



where ( , )v r t denotes the radial component of either the liquid velocity, if 2r R> , or the velocity inside the encapsulating layer, if 1 2rR R≤ ≤ . The

assumption of incompressible shell also gives 3 3 3 32 1 20 10R R R R− = − , (3)

where 20R is the outer radii of the bubble shell at rest. These equations are used in further calculations.

The rheological behaviour of the bubble shell will be approximated by the linear 3-constant Oldroyd model which can be expressed as [6]

( )( )

1 22S

rr rrSrr S rr SS

vvt tτ ητ λ λ∂ ∂ + = + ∂ ∂

, (4)

where rr v rv = ∂ ∂ is the radial component of the rate-of-strain tensor, 1Sλ is the relaxation time of the shell, Sη is the shear viscosity of the shell and 2Sλ is the retardation time of the shell. Substitution of eqn. (2) into eqn. (4) yields

( )( )

2( ) 2 21 1 11 21 1 13

42

Srr SS

rr S SR R RR R Rt rηττ λ λ

∂ + = − + + ∂. (5)

Equation (5) suggests that ( )( , )Srr r tτ can be written as

( ) 3( , ) 4 ( )SSrr Sr t tD rητ = − . (6)

Substituting eqn. (6) into eqn. (5) shows that the function ( )S tD obeys the equation

( )22 21 1 11 21 1 12S S SSD R R RD R R Rλ λ+ = + + . (7)

The motion of the surrounding liquid will also be described by a linear 3-constant Oldroyd equation

( )( )

1 22L

rr rrLrr L rr LL

vvt tτ ητ λ λ∂ ∂ + = + ∂ ∂

, (8)

where Lη denotes the shear viscosity of the liquid, 1Lλ is the relaxation time and 2Lλ is the retardation time. Note that for 1 2 0L Lλ λ= = , eqn. (8) reduces to the equation that describes a viscous Newtonian liquid so that equations of bubble motion derived below can be used in both cases, when the surrounding liquid is water, which is usually the case in laboratory experiments, and when the surrounding liquid is blood as in clinical applications, since the Oldroyd constitutive equation is an acceptable model for blood [7]. Substituting eqn. (2) into eqn. (8) and representing ( )( , )L

rr r tτ as

( ) 3( , ) 4 ( )LLrr Lr t tD rητ = − , (9)



one finds an equation for the function ( )L tD to be

( )22 21 1 11 21 1 12L L LLD R R RD R R Rλ λ+ = + + . (10)

Equations (6) and (9) make possible the calculation of the integral terms in eqn. (1). Equation (1) can also be modified to take account of the translation motion of the bubble and acoustic radiation losses due to the compressibility of the surrounding liquid. The modification can be performed by directly adopting necessary corrections from a recent paper by Doinikov and Dayton [8]. The final equation takes the form

323 31 2 1 1 102

1 01 1 32 2 2 1

3 4 1 112 2 4

L Lg

S S S

xR R R R RHR PR R cR R R R

γρ ρβ βρ ρ ρ

− + + + − = +

( )3 320 101 2

03 3 31 2 2 1 2

( )2 2 ( )4 4 ( , )SLacL S

tD R RtD x tP PR R R R Rσ σ η η

−− − − − − −

, (11)

where the function H is defined as

( )1 4 3 341 2 11 1 3

1 1 1 1 14 72 2 2

21 2 1 2 1

R R RdGR RH R R R R RdtR R Rβ β β

− − = + + + + +

, (12)

c is the speed of sound in the liquid and G denotes the right-hand side of eqn. (11). The compressibility correction is given by the last term on the left-hand side of eqn. (11), while the first term on the right-hand side of eqn. (11) provides the coupling with the translational equation. This latter is given by

( )3 32 2

2 4 ( , )3 3 ac db L

dx x x tm R R P Fdt xπ πρ

∂+ = − +

∂, (13)

where bm is the mass of the bubble and dF is the viscous drag force which is taken in the form of Oseen’s law

( )2 2( 4) 24 9d L L Lx xF R Rπ η ρ η= − + . (14)

Finally, for (0) (0) 0S LD D= = , from eqn. (11) it follows that 0gP is given by

0 0 10 201 22 2gP P R Rσ σ= + + . (15)

Thus, we have the set of four ordinary differential equations: radial equation (11), translational equation (13) and eqns. (7) and (10) for the functions ( )SD t and ( )LD t . The set is supplemented with eqns. (3), (12), (14) and (15). The initial conditions can be specified as 1 10(0) RR = , 2 20(0) RR = ,



1 2(0) (0) 0R R= = , 0(0)x x= , (0) 0x = and (0) (0) 0S LD D= = . Solving these four equations simultaneously, we can obtain a radius-time curve and translational displacement for an encapsulated bubble of given size.

3 Validation of theoretical model

In order to estimate the values of the shell parameters appearing in the theoretical model, experimental data obtained for the contrast agent MP1950 by Dayton et al [9] were used. MP1950 is a phospholipid-shelled microbubble with a decafluorobutane core. The surrounding liquid used in [9] was water, which allows one to set 1 2 0L Lλ λ= = . Bubbles were insonified with a single 20-cycle acoustic pulse with a pressure amplitude of 180 kPa and a centre frequency of 2.25 MHz. To evaluate the shell parameters, simulated radius-time curves were fitted by the least squares method to experimental curves for 18 bubbles with equilibrium radii from about 0.79 to 2.81 µm. The fit has revealed that the retardation time for lipid coatings is in fact zero. Therefore all simulations described below were performed at 2 0Sλ = . Also, following considerations made by Marmottant et al [10], the surface tension at the gas-shell interface, 1σ , was set equal to zero. Thus, the fitting was in fact carried out by varying the three shell parameters: 1Sλ , Sη and 2σ . The values of the other model parameters were 0 101.3P = kPa, 1000Lρ = kg/m3, 0.001Lη = Pa·s, 1500c = m/s, 1.07γ = , 1100Sρ = kg/m3 and 20 10 2SR R R= − = nm.

It has been found that the surface tension 2σ varies in random manner, i.e., regardless of bubble size, between 0 and 0.038 N/m with a mean of 0.0133 N/m. Whereas the relaxation time 1Sλ and the shell viscosity Sη demonstrate a clear increase with increasing equilibrium radii. The best-fit values of 1Sλ and Sη for the 18 experimental radius-time curves are shown by circles in figs. 1(a) and 1(b), respectively, as a function of equilibrium bubble radius. In both figures, the solid line represents a polynomial regression for the best-fit values of the shell parameters. It is interesting to note that for both parameters, a relatively good agreement between the best-fit values and the regression curve is achieved assuming that 1Sλ and Sη are linearly dependent on the equilibrium volume of the bubble, 3

0 20(4 / 3)bV Rπ= . The equations of the regression curves shown in figs. 1(a) and 1(b) are given by

31 20 00.0125 0.0024 0.0125 0.00057S bR Vλ = + = + , (16)

320 01.25 0.14 1.25 0.033S bR Vη = + = + , (17)

where the units of 20R , 1Sλ and Sη are microns, microseconds and Pa s, respectively.



Figure 1: Best-fit values of (a) the relaxation time and (b) the shell viscosity as a function of equilibrium bubble radius. Each circle represents the best fit for one experimental radius-time curve. The solid lines show the polynomial regression of the fit values.

4 Numerical simulations

4.1 Translational motion

Equations (16) and (17), with 2 0.0133σ = N/m, were used to model translational displacement of lipid-shelled bubbles. The results obtained are displayed in fig. 2. The experimental translational displacement, adopted from [9], is shown by circles. The solid line indicates the simulated displacement calculated by the shell model proposed here, which treats the lipid shell as a viscoelastic fluid following the Oldroyd constitutive equation. For comparison, the dashed line shows the displacement that is predicted by the elastic shell model which treats the encapsulation as a viscoelastic solid following the Kelvin–Voigt constitutive equation, and the dotted line represents the results given by the viscous shell model which assumes that the encapsulating layer behaves as a viscous Newtonian fluid. The data for plotting these two curves were adopted from [11, 12]. One can see that the Oldroyd shell model provides better agreement with the experimental measurements than the two other models.

4.2 Resonance frequencies

The linear resonance frequency of a free bubble is known to be given by the Minnaert formula [13]. For the regime of nonlinear oscillations, there is no analytical formula and resonance frequencies can be evaluated solely by numerical calculations. In a classical paper by Lauterborn [14], resonance frequencies of free gas bubbles in the nonlinear regime are evaluated as follows. The quantity ( )max 0 0R R R− , called the normalized amplitude, is calculated numerically as a function of the frequency of the applied sound field, f , for different values of equilibrium bubble radius 0R . maxR denotes the maximum



Figure 2: Experimental and simulated translational displacement as a function of equilibrium bubble radius. Circles indicate experimental results. The solid line corresponds to the Oldroyd shell model, the dashed line to the elastic (Kelvin–Voigt) shell model and the dotted line to the viscous shell model.

radius of the bubble during its steady-state oscillation. Normalized amplitude-frequency plots are called frequency response curves. Note also that the applied sound field is assumed to be a continuous sinusoidal wave. The frequency of the main resonance of a bubble with equilibrium radius 0R is then determined as that corresponding to the main peak of the frequency response curve obtained for this bubble. This approach is, however, not quite adequate in the case of medical ultrasonic applications where insonation is in the form of pulses which can consist of only a few acoustic cycles. It is also not convenient if resonance frequencies are evaluated from experimentally measured radius-time curves, since, due to random fluctuations and measurement errors, the amplitude of the measured oscillation is not constant even at the steady state. In addition, when experimental data are processed, we normally deal with a fixed driving frequency and a set of radius-time curves measured at this frequency for bubbles of different size. In other words, in experiments, the equilibrium bubble radius is a variable quantity rather than the driving frequency. For such cases, a different approach is proposed to be used. The following quantity is calculated:

2

000

1 ( )( , ) 1T R tW f R dt

T R

= − ∫ , (18)

where ( )R t is the radius-time curve for a bubble with equilibrium radius 0R which occurs at the driving frequency f , and T is the duration of the bubble oscillation. The quantity 0( , )W f R will be referred to as the oscillation power hereinafter. The oscillation power is plotted as a function of 0R at a fixed



frequency f . The resonance radius for this frequency is then determined as that corresponding to the main peak of this plot. Clearly this method can be applied to both theoretical and experimental radius-time curves. We have applied it to evaluate resonance frequencies of lipid-shelled bubbles using eqns. (16) and (17) for the shell parameters. The bubbles were assumed to be surrounded by water and insonified with a 20-cycle, 2.25 MHz, 180 kPa acoustic pulse. The results obtained are displayed in fig. 3. The dependence between the equilibrium radius and the frequency of resonance response for lipid-shelled bubbles is shown by the solid line. For comparison, the dashed line represents the dependence obtained for free bubbles at the same acoustic parameters. Figure 3 reveals that, in contrast to albumin-shelled bubbles whose resonance frequencies are always higher than those of free bubbles, the resonance frequencies of lipid-shelled bubbles, depending on bubble size, can be both lower and higher than those of free bubbles of equivalent size. There are two regions in the behaviour of the resonance frequencies of lipid-shelled bubbles; namely, the resonance frequencies of larger lipid-shelled bubbles are lower than those of free bubbles of equivalent size, while the resonance frequencies of smaller lipid-shelled bubbles exceed those of free bubbles. It should be noted, however, that the sharpness of resonance response decreases with decreasing bubble size so that the resonance response of smaller bubbles is less pronounced, much more flattened, than that of larger bubbles. This effect is illustrated by fig. 4 which shows the oscillation power as a function of equilibrium radius for lipid-shelled bubbles at two values of the driving frequency, all other parameters being the same as in fig. 3. One can see that the higher the driving frequency, the more flattened is the resonance peak. For sufficiently high frequencies, the resonance response vanishes totally.

Figure 3: Dependence between the equilibrium radius and the frequency of resonance response for lipid-shelled bubbles in the regime of nonlinear oscillations. The excitation is a 20-cycle, 2.25 MHz, 180 kPa acoustic pulse. The dashed line represents the dependence for free bubbles at the same acoustic parameters.



Figure 4: Oscillation power as a function of equilibrium radius for lipid-shelled bubbles at two values of the driving frequency.

This effect is explained by a strong damping impact of the shell viscosity on the oscillation of small encapsulated bubbles. A similar effect was pointed out earlier for polymeric- and albumin-shelled bubbles by Khismatullin [15] and Doinikov and Dayton [8].

5 Summary

A new theoretical model for a lipid-shelled contrast agent microbubble has been proposed. The model treats the lipid coating as a viscoelastic fluid following the linear Oldroyd constitutive equation and incorporates the translational motion of the bubble. The translational displacement predicted by the new model was compared to the experimentally measured displacement and the predictions of two existing models which treat the encapsulation as a viscoelastic solid or a simple viscous fluid. It has been shown that the new model provides better agreement with the experimental measurements than the two other models.

An approach has been proposed for evaluating resonance frequencies of lipid-shelled bubbles in the regime of nonlinear oscillation. The approach is based on calculating the time-averaged power of bubble oscillation as a function of equilibrium bubble radius at a given driving frequency. The resonance radius is then determined as that corresponding to the main peak of the oscillation power-radius function. The proposed method was applied to estimate resonance frequencies of lipid-shelled bubbles insonified with a 20-cycle, 2.25 MHz, 180 kPa pulse. It has been shown that the lipid shell can both increase and decrease the resonance frequencies of encapsulated bubbles with respect to those of free bubbles of equivalent size.

Acknowledgement

This work was supported by the International Science and Technology Center (ISTC) under Contract B-1213.



References

[1] Becher, H. & Burns, P.N., Handbook of Contrast Echocardiography, Springer Verlag: Frankfurt and New York, 2000.

[2] Bloch, S.H., Dayton, P.A. & Ferrara, K.W., Targeted imaging using ultrasound contrast agents. IEEE Engineering in Medicine & Biology Magazine, 23, pp. 18–29, 2004.

[3] Klibanov, A.L., Microbubble contrast agents: Targeted ultrasound imaging and ultrasound-assisted drug-delivery applications. Investigative Radiology, 41(3), pp. 354–362, 2006.

[4] Roy, R.A., Church, C.C. & Calabrese, A., Cavitation produced by short pulses of ultrasound. Frontiers of Nonlinear Acoustics, eds. M.F. Hamilton & D.A. Blackstock, Proc. of the 12th ISNA, Elsevier: London, pp. 476–481, 1990.

[5] Church, C.C., The effect of an elastic solid surface layer on the radial pulsations of gas bubbles. Journal of the Acoustical Society of America, 97(3), pp. 1510–1521, 1995.

[6] Bird, R.B., Armstrong, R.C. & Hassager, O., Dynamics of Polymeric Liquids, Wiley: New York, 1987.

[7] Khismatullin, D.B. & Nadim, A., Radial oscillations of encapsulated microbubbles in viscoelastic liquids. Physics of Fluids, 14(10), pp. 3534–3557, 2002.

[8] Doinikov, A.A. & Dayton, P.A., Spatio-temporal dynamics of an encapsulated gas bubble in an ultrasound field. Journal of the Acoustical Society of America, 120(2), pp. 661–669, 2006.

[9] Dayton, P.A., Allen, J.S. & Ferrara, K.W., The magnitude of radiation force on ultrasound contrast agents. Journal of the Acoustical Society of America, 112(5), pp. 2183–2192, 2002.

[10] Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N., Hilgenfeldt, S. & Lohse, D., A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture. Journal of the Acoustical Society of America, 118(6), pp. 3499–3505, 2005.

[11] Doinikov, A.A. & Dayton, P.A., Modeling of the oscillation and translation dynamics of lipid-shelled ultrasound microbubble contrast agents: Theory. Journal of the Acoustical Society of America, submitted.

[12] Doinikov, A.A. & Dayton, P.A., Modeling of the oscillation and translation dynamics of lipid-shelled ultrasound microbubble contrast agents: Comparison to experiments. Journal of the Acoustical Society of America, submitted.

[13] Leighton, T.G., The Acoustic Bubble, Academic Press: San Diego, 1994. [14] Lauterborn, W., Numerical investigation of nonlinear oscillations of gas

bubbles in liquids. Journal of the Acoustical Society of America, 59(2), pp. 283–293, 1976.

[15] Khismatullin, D.B., Resonance frequency of microbubbles: Effect of viscosity. Journal of the Acoustical Society of America, 116(3), pp. 1463–1473, 2004.



Lagrangian Monte Carlo simulation of spray-flow interaction

T. Belmrabet1, R. Russo2, M. Mulas2 & S. Hanchi1

1LMF, EMP, BP 17 Bordj el Bahri, 16111 Alger, Algeria 2Department of Computational Methods for Engineering, CRS4, CFD Area, Uta (Ca), Italy

Abstract

The aim of this work is to determine the interaction between a droplet’s stationary spray and a fluid flow, accounting for droplet vaporization, breakup and turbulences effects. In order to achieve this, a Lagrangian Monte Carlo code (McSpray) has been developed, besides a volume-finite Navier–Stokes solver (Karalis). The two codes work sequentially on the same computational grid, each influencing the other, so that a complete two-way coupling might be modelled. To validate the McSpray code in the case of two-way coupling, two cases have been performed: the first is a surface injection parallel to the inflow continuum velocity; the second is a conic point injection having an injection angle equal to 15 degrees. To verify these results, they are compared with the ones provided by Fluent commercial code. Keywords: finite volume, spray, Monte Carlo.

1 Introduction

The subject of multiphase flow modelling processes is a quite vast research field of utmost practical interest. When two or more phases move relatively to each other, they may exhibit a large number of possible flow regimes. There are several classifying ways of these multiphase flows. In dispersed flows all the phases except one exist as dispersed (discontinuous) particles flowing through the continuous fluid. Using the Eulerian–Lagrangian approach, trajectories of dispersed phase particles are simulated by solving an equation of motion for each particle. Motion of the continuous phase is modelled using a conventional Eulerian



doi:10.2495/MPF070261

framework. Depending on the degree of coupling, solutions of both phases interact with each other. For two-way or four-way coupling, an iterative solution procedure needs to be adopted. In simple, one-way coupling, a continuous phase flow field can be obtained independently of the motion of the dispersed phase.

2 Mean equation in the gas phase

The main features of the gaseous flow are deduced from the resolution of the Reynolds Averaged Navier–Stokes (RANS) equations written for a non-reactive single fluid. The Spalart-Allmaras turbulence model is used to compute the unclosed turbulent terms. In order to apply the numerical fully compressible formulation to incompressible flows, a preconditioning technique is used. The preconditioned system of the Navier–Stokes equations, in compact vector form, is

( ) SPflux viscousPxF

PtQ

j

j ⋅=+

∂∂

+∂∂ (1)

where Q represents the vector of conservative variables ( )Ti v~,E,u,Q ρρρρ= , Fj the corresponding inviscid fluxes and S the source term vector. The preconditioning matrix is given by 1

modMMP −⋅= . M represents the Jacobian matrix of the vector Q with respect to the vector of the so-called viscous-primitive variables ( )Tiv v~,T,u,pQ = . Mmod represents a modified version of M. All matrices are given in reference [1].

2.1 Numerical method

Equations are integrated with a cell-centered Finite-Volume method on block-structured meshes. Convective inviscid fluxes are computed by a second order Roe’s scheme [1]. In Finite-Volume and semi discrete form, the system (1) becomes

RESMMRESPtQ 1

mod ⋅⋅−≡⋅−=∂∂ −Ω (2)

where RES represents the vector of residuals and Ω the cell volume. Updating is done in terms of the viscous primitive variables Qv. If an implicit numerical scheme is used to discretize the time derivative and after linearization:

RESQQ

RESt

M v

old

vmod =

∂∂

+ ∆∆Ω (3)

This linear system is solved with an iterative red–black relaxation scheme [1].

3 Representation of the dispersed phase

3.1 Monte Carlo technique

Monte Carlo technique consists in calculating the characteristics of a system by generating a certain number of random events enough to catch its behaviour.



Assuming that the evolution of the physical system can be described by a pdfs, the Monte Carlo simulation can proceed by sampling from these pdfs. This requires a fast and effective way to generate a random numbers uniformly distributed on the interval [0, 1]. This method is used both to compute initial diameter, position and velocity of every droplet injected and to model turbulence effects by means of the “eddy interaction model”. It is well known, from experimental studies, that in general cases the particles diameters are distributed according to a so-called Rosin Rammler distribution. The initial position of the particle is determined by assuming the particles generated from an annular section with a constant density distribution over the surface. The initial velocity is computed in the same way. Currently the velocity pdfs are assumed to be Gaussian [2].

3.2 Velocities and trajectories computing

The Basset–Boussinesq–Oseen equation for forces balance on the droplet is [3]

( )

( )dt

dmmv1g

tvutd

ttvu1

29

dtdu

23vuf

dtdv

211

d

dd

ct

0r

21

d

c

d

c

rd

c

−

−+

−+′

′−−

++−=

+

∫ ρρ

τ

ρρ

πρρ

τρρ

(4)

where ( )c2

dr 18D µρτ = and uutu

dtdu

∇⋅+∂∂

= for a stationary case.

Equation (4) is simplified neglecting the Basset force (in air 3dc 10 −≈ρρ ).

So (4) becomes:

( ) CvuBdtdvA +−= (5)

For a spherical particle f is given by a correlation proposed by Clift and Gauvin [4]. When a droplet distortion occurs f must be corrected. Equation (5) is integrated by using a Crank-Nicholson scheme:

C2

vvuBt

vvAn1n

*n1n

+

+−=

− ++

∆ (6)

where n represents the time iterations number and

( ) tuvuu;uu21u nnn1n1nn* ∆∇⋅+=+= ++ (7)

The trajectory is obtained directly by integrating the velocity fields. During each integration step coefficient B is updated to take account for evaporation and break-up.

3.3 Break-up model

The break-up model used in this work is the Taylor analogy break-up model (TAB) [2]. This model is suited for low Weber numbers sprays (less than 100). It is based upon an analogy between an oscillating and distorting particle and a spring mass system.



In a non-dimensionalized form, by setting ( )rCXY b= , the equation of the forced oscillator becomes:

dtdY

rCY

rC

ru

CC

dtYd

2d

dd3

d

k2

2

d

c

B

F2

2

ρµ

ρσ

ρρ

−−= (8)

where CB is a constant equal to 0.5. Break-up occurs for Y > 1. The coefficients of this equation are taken from Taylor’s analogy. The numerical integration of (8) is done by considering its coefficients constant. To check whether break-up occurs, one estimates the amplitude of the oscillation, assuming no damping [2]. Anyway if break-up doesn’t occur the mean YM is calculated. Instead, if break-up occurs we calculate the Sauter mean diameter of the child droplets and consequently the child droplets number and velocities.

3.4 Evaporating model

During its trajectory each particle can evaporate or condensate according to its surrounding conditions. The rate of change of droplet mass is [3]:

Dww

DDShdt

dm ,As,Avc

2d ∞−= ρπ (9)

The vapour mass fraction at the droplet surface wA,s can be evaluated if the droplet temperature is known. For a dilute spray wA,∞ can be assumed to be equal to 0. From (9) and by integration it is quite straightforward to obtain:

tDD 20

2 λ−= (10) with

( )∞−= ,As,Ad

vc wwDSh4ρρλ (11)

This is the so-called D2-law. It is necessary to write the energy equation for the droplet to compute the energy exchanged by the droplet with the continuum.

3.5 Turbulence effects

To determine the droplet transport due to turbulence, the “eddy interaction model” is used [5]. This approach models the turbulence flow as a set of random eddies each characterized of a certain length le and a certain lifetime Te. The former dependent on a Eulerian flow scale l and the latter on the Lagrangian time scale lτ . So the continuum velocity at a given point is iii uUu ′+= where the turbulent fluctuation iu′ is constant inside the eddy “radius” le. Moreover, thanks to the central limit theorem, iu′ will have a Gaussian pdf and the standard

deviation can be taken equal to 3k2 . The Eulerian length scale and the Lagrangian time scale can be determined by a two-equation turbulence model. When using Spalart Allmaras turbulence model only a single turbulence quantity is solved. A second turbulence quantity must be available in order to assemble a turbulent kinetic energy and its dissipation rate [6]:



ε

23

e

k166.0l = ; ε

τk135.0l ≈

The time a droplet takes to cross an eddy of 2 le long, assuming stokesian drag force, is

−−−=

uvl21lnt

r

ercross τ

τ (12)

The lower between tcross and eddy lifetime ( )l2τ is the time a droplet actually interacts with the eddy. The trajectory of the droplet is calculated for a period equal to the interaction time.

3.6 Source terms computation

The source terms are calculated by adding to cach local cell counters the mass, the momentum and the energy exchanged by each droplet during its path. The 5 source terms are computed for each time step (and added to the scoring computed so far) as follows. Mass source term:

( ) ffiM SMMS −= (13)

where ( )fi MM − is the droplet mass evaporated during the time step and Sf is a scale factor to account the actual mass flow rate of the spray from the droplets total mass injected. Momentum source terms:

( ) fmffiiP StgMvMvMS ∆+−= (14) Where Mm is the mean mass during the time step. Energy source term:

( ) ( )( )[ ]cfdd,pfimffiifE eeTcMMtQXgMeMeMSS −+−+−⋅+−= ∆δ (15) where ei and ef are respectively the initial and the final droplet kinetic energy, δX is the droplet displacement, and ec is the kinetic energy of the continuum in the current position. During each droplet trajectory two more scorings are updated in order to estimate the droplets volume fraction and the mean droplets diameter in each cell [2].


4.1 Turbulence dispersion test

To verify the accuracy of the code in the turbulence dispersion case in homogenous isotropic stationary turbulence (HIST), the results are compared to those provided by Graham and James [5]. The test consists in injection of a high number of droplets with zero initial velocity in a HIST flow with zero mean velocity. The mean squared displacement, for t→∞, opportunely scaled, i.e. ( )l

2'2d

2d tu2XX τ= , is monitored while varying lr ττ and el lu2 τα ′= .



For a cylindrical mesh, 2dX can be viewed either as the squared displacement

along z or half the squared radial displacement. For this and for all the following cases the cylindrical grid is an angular sector (angle π/32) with periodic boundary conditions. The results are resumed in Table 1.

Table 1: Results HIST.

2dX [5] 2

dX (along z) 2dX radial)

lr ττ =0.01; α=1.2

lr ττ =10; α=1

lr ττ =10; α=2

0.99

0.98 0.83

0.99

0.98 0.78

0.99

0.98 0.79

Figure 1: Volume fraction maps in laminar (left) and turbulent case (right).

4.2 Qualitative results for one way coupling

Now we depicted some qualitative results for a one way coupling. The effects of breakup and turbulence dispersion are also shown. In all these cases the continuum velocity is uniform and directed towards the positive z axis with magnitude equal to 1m/s; the initial droplet velocity is equal to 3m/s. The droplet size is supposed to follow a Rosin Rammler pdf, while the cone angle of the atomizer is supposed to follow a Gaussian distribution; knowing the module of velocity (assumed constant in this case) it will be possible to determine the velocity of the droplet if it is assumed no swirl at the atomizer exit. Two test cases are performed. The first assuming no turbulence in the carrier flow and the second assuming homogenous turbulence. From Figure 1, one can notice that turbulence tends to homogenize the volume fraction distribution. For the mean droplet diameter in each cell, the influence of turbulence dispersion is shown in Figure 2. The heaviest droplets are on the spray periphery in laminar case. In turbulent case the heaviest droplets are also on the spray periphery which



is nearly spread in all the computational volume. Figure 3 shows the effect of breakup on the energy source terms. Breakup generates smaller child particles with a slower rτ which will exchange energy more rapidly. In these cases the number of particles used is 500.000; in order to obtain source terms fields enough “smooth”.

Figure 2: Mean droplet diameter maps in laminar (left) and turbulent case (right).

Figure 3: Energy source terms, without breakup (left) and with breakup (right).

4.3 Qualitative results for two-way coupling

In this case a complete coupling between continuum and water droplets in laminar case is assumed. Two cases have been performed: the first is a surface injection parallel to the inflow continuum velocity; the second is a conic point injection with injection angle equal to 15 degrees. For both of them the



continuum inlet temperature is equal to 320 K, and the droplets temperature is equal to 300 K. The continuum velocity is 1m/s and the droplets velocity is 10m/s. The droplet diameters are assumed to be constant. Results are compared with the ones provided by the Fluent code (Figure 4). Here breakup model has not been used since Fluent doesn’t allow breakup for stationary cases. The local and total mass and energy conservation have been verified at the end of every McSpray and Karalis run. The number of particle used for surface injection is about 106; a high number of particles is necessary to have a homogeneous emission from the surface (in this case we used a big area of emission, so to obtain a good statistical distribution it is necessary to use a great number of particles). For the cone injection the number of particle used is 104, this relatively small number of particles is due to the small surface of injection (nearly a point).

4.3.1 Surface injection For the first test, two continuum-spray mass flow rate ratios have been tested, respectively equal to 5 and 20 (ratio 5 and ratio 20). The results are shown in Figure 4 and Figure 5.

Figure 4: Continuum temperature map for ratio 5. McSpray (left) and Fluent (right).

4.3.2 Cone injection For the cone injection the injection point is on the cylinder axes. The continuum/droplet mass flow rate ratio is equal to 20. The results are shown in Figure 6.

5 Conclusion

The Eulerian–Lagrangian simulation of two phase flow with breakup and evaporation is performed. The effect of turbulence on spays properties (evaporation and breakup) is depicted. Good results have been found comparing



with those provided by Graham and James [5] for HIST flow in one way coupling and by the commercial code Fluent for two-way coupling in laminar case. As future work, the McSpray code will be extended to account for wall and inter-particle collisions.

Figure 5: Continuum temperature map for ratio 20. McSpray (left) and Fluent (right).

Figure 6: McSpray–Karalis results (left) and Fluent results (right).

Nomenclature

D particle diameter, m r particle radius, m md droplet mass, kg v droplet velocity, m.s-1 u continuum velocity, m.s-1



We Weber number, σρ ruWe 2c=

X droplet equator displacement, m Y droplet equator displacement Dv vapor diffusion coefficient, m2.s-1 Sh Sherwood number κ turbulent kinetic energy m2.s-2 Greek symbols ε turbulent Dissipation, m2.s-3 µ dynamic viscosity kg.m-1.s-1 ν cinematic viscosity m2.s-1 σ droplet surface tension N. m-1

velocity response time, s Indices and exponents c continuum d droplet s vapor

References

[1] Mulas, M., Chibbaro, S., Delussu, G., Di Piazza, I. & Talice, M., Efficient parallel computations of flows of arbitrary fluids for all regimes of Reynolds, Mach and Grashof numbers, Int. J. of Numerical Methods for Heat and Fluid Flow, Vol.12 No.6, pp. 637-657, 2002.

[2] Fluent User’s guide [3] Crowe, C., Sommerfield, M. & Tsuji, Y., Multiphase flows with droplets

and particles, CRC Press LLC, ISBN 0-8493-9469-4, 1998. [4] Clift, R. & Gauvin, W.H., The motion of particles in turbulent gas

streams, Proc. CHEMECA ‘70, pp. 14-28, 1970. [5] Graham, D. I. & James, P. W., Turbulent dispersion of particles using

eddy interaction models, Int. J. Multiphase Flow Vol.22, N.1, pp. 157-175, 1996,

[6] Mulas, M. & Talice, M., Fully compressible simulation of low speed premixed reactive flows, AIAA paper 4253-2003, The 33rd Fluid Dynamics Conference, (23-26 June 2003, Orlando).



Dynamic hydraulic jumps in oscillating containers

P. J. Disimile1, J. M. Pyles2 & N. Toy2 1US Air Force, Survivability and Safety Flight, Ohio, USA 2Engineering and Scientific Innovations Inc, Batesville, Indiana, USA

Abstract

When the liquid in a tank undergoes sudden movement, as in the case of a fuel tank in an aircraft or in a marine vessel, it may be subjected to as many as 6 degrees of freedom. Three of these are in rotation; yaw, pitch and roll, and three in translation; sway, surge, and heave. Work is currently being conducted on simulating the effects of the liquid motion under roll conditions in a rectangular tank of dimensions 1.9 x 0.94 x 1.2 m3 located on a 6 degree of freedom simulator capable of mimicking the movements typical of an aircrafts performance. At present, water is being used to investigate the fluid motion when subjected to oscillating roll frequency of 0.35 Hz and oscillation amplitudes of 2.420, 3.500, and 4.710 for different liquid depths. It has been found that under such motions, typical of those obtained within the flight envelope of military, private and commercial aircraft, a dynamic hydraulic jump can occur. This jump is out of phase with the roll motion and is produced as the fluid abruptly changes direction within the tank. As the tank reaches its lowest rotational position in the roll manoeuvre the fluid level at this point of the tank increases rapidly against the end wall causing splashing, resulting in bubble formation and a fine spray. This change in direction increases the fluid depth and this has to move against the residual oncoming fluid that is at a much lower depth, resulting in a very dynamic, moving, wave that breaks and forms into a hydraulic jump comprised of air and liquid mixing. This preliminary investigation into the characterization of this phenomenon using water shows that the spatial characteristics of the hydraulic jump and the dynamic range of the resultant spray are affected by the amplitude of the tank oscillation. Keywords: hydraulic jump, multiphase flow, particle image velocimetry.



doi:10.2495/MPF070271

1 Introduction

Liquid dynamics in moving containers is of special interest to air, marine and ground vehicles due to the movement of the liquid, or slosh, in the container and the resulting impact forces it produces on the walls of the container. The role that liquid dynamics play in the stability of an aircraft has been extensively examined by several researchers. Slosh can influence the stability and control systems of air vehicles and care must be taken when designing a tank so that the sloshing impact forces can be controlled, [1, 2]. Furthermore, sloshing in partially filled liquid containers can significantly alter the motion of cargo ships [4, 5]. Each of these studies notes the formation of a non-linear, multiphase event, characterized by an air/liquid turbulent region, when the vessel is oscillated at the resonance frequency of the liquid in the container. This phenomenon is known as a hydraulic jump. Hydraulic jumps are of special interest in dynamic container flows because they represent a transition between two flow states: subcritical and supercritical. Waves travelling faster than the wave celerity of the liquid depth must dissipate energy in the form of a hydraulic jump. The jump is characterized by a rapid change in liquid depth with a turbulent region between the two depths. These jumps can be stationary or moving, [6]. Previous research conducted by Chanson [7] has shown distinct flow regimes and spray regions for stationary hydraulic jumps in channel flows. His research showed that typical hydraulic jumps are comprised of an entrained air shear layer that forms at the base of the jump. Above this shear layer are multiple recirculation regions with air bubble entrainment. At the top of the jump, three distinct spray regions are produced: an aerosol/fog region, a spray/mist region, and large droplet region. However, the research of Chanson [7] utilized intrusive measurements which are not practical for dynamic environments. The goal of the present research is to investigate the characteristics of dynamic hydraulic jumps that form under near resonance conditions. The spatial characteristics of the jump will be explored. Furthermore, a preliminary study of using optically based techniques to measure the spray distribution formed by the jump will be performed. This will provide insight into this multiphase phenomenon.

2 Theory

Two coordinate systems were setup to provide a theoretical basis for characterizing the liquid free surface in the tank coupled with the motion of the test fixture. These coordinate systems will also be referred to when discussing shallow water flow theory. The coordinate systems, shown in figure 1, contains the stationary coordinate system of O – x0 y0 and the moving coordinate system of G – x y, which rotates about the origin O. The moving coordinate system moves with the tank with its origin, G, located at the center point on the base of the tank. The incline angle of the tank is given as θ, the wave height normal to



the bottom of the tank and measured from the rest depth of the liquid is represented as η. The rest height of the bottom of the tank as H, and the rest height of the liquid is denoted by h0.

x0

y0

O

B

θ

h0

η

y

x

G H

x0

y0

O

B

θ

h0

η

y

x

G H

Figure 1: Coordinate system of dynamic tank.

The liquid dynamics within an oscillating tank have been extensively studied and modelled [3, 4, 9, 10]. It has been shown in these references that the resonance frequencies for a rectangular tank experiencing roll oscillations are given as:

1/ 2

01 tanh 1, 2,...,2n

g n n hf nB Bπ π

π = = ∞

(1)

For shallow liquid depths, the fundamental resonance frequency (n = 1) can occur under small perturbations such as turbulence for aircraft or general water waves for marine vessels. As a result, an out of phase hydraulic jump is observed to travel back and forth between the end walls of the tank [8, 10]. According to a linearized theory called “shallow water wave theory” [11], the theoretical phase difference between the tank oscillation and hydraulic jump formation approaches 90° as the tank oscillation approaches the resonance frequency of the liquid depth. This implies that the jump forms at the centre of the tank for liquid depths where h0 << B [10].

3 Methods and materials

A testing facility at Wright-Patterson Air Force Base was utilized for an investigation on the effect of low frequency, small amplitude roll oscillations on the liquid dynamics in a rectangular tank. The test facility consisted of a state-of-the-art motion simulator and a clear, rectangular tank. To replicate generic roll aircraft dynamics, a hydraulically activated Sarnicola Hexad AIES Six-Degree-of-Freedom motion simulator was employed. This simulator has base dimensions of 3.28 m (129’’) x 2.03 m (80’’) at a rest height



of 1.73 m (68’’), and it is capable of carrying an 11,364 kg (25,000 lb) payload. This motion simulator has 6 degrees of freedom provided by six hydraulic cylinders or actuators and is controlled through proprietary computer software called HexTest. The hydraulic cylinders are arranged in a hexapod configuration that allows for maximum rotational excursions. A clear tank provides optical access to three spatial planes through the top, side, and front of the tank by means of 25.4 mm (1’’) thick walls of Lexan with the other walls fabricated from 6.35 mm (0.25’’) thick steel. An interior steel frame was constructed to provide additional reinforcement for the Lexan sides. The clear tank frame supports a tank with overall internal dimensions of 1.9 m (73’’) x 1.2 m (48’’) x 0.94 m (37’’), yielding a maximum capacity of 2.1 m3 (563 gal), and will provide a larger scale study than any previous experiments discussed in the previous section. Lexan walls with a thickness of 12.7 mm (0.5’’) were added on the outside of the open sides of the tank shell to provide optical access in each spatial plane. The steel sides of the frame provided the internal tank walls for the remaining sides. The generic tank was positioned inside of the steel shell constructed of 6.35 mm (0.25 in) thick steel tubing and placed on the motion simulator. Steel tubing located on the top and two steel bars along the clear side of the tank provide additional support for the Lexan walls. The total motion simulator and tank setup is shown in figure 2.

Figure 2: Overall simulator and slosh tank setup.

3.1 Data acquisition

A high-speed digital Nanosense XS-3 CMOS camera from IDT with a resolution of 1280 x 1024 pixels was utilized to capture the full field liquid dynamics within the generic clear tank. The camera was positioned on a tripod at a height of 1.93 m (76’’) at a distance of 2.24 m (88’’) from the front wall of the tank fixture. A 20 mm Nikon lens was attached to the camera and captured the full field of view of the test apparatus. The digital imaging system acquired video at 75 Hz for a total of 15 seconds at the start of the oscillation. All images were stored in the onboard camera memory and then transferred to the computer via the USB 2.0 connection upon completion of each test for analysis. The full field visualization setup is shown in figure 3.



To investigate the spray region above the hydraulic jump suggested by Chanson [7], a particle image velocimetry (PIV) system was used. The system is composed of a two-dimensional light fan created from a Nd:YAG laser and a high speed digital imaging system. An XS-4 camera with a 512 x 512 pixel resolution and attached 60 mm Nikon lens was used to image the spray region. This resulted in a field of view of 167 mm x 167 mm. The camera and laser are synchronized by a personal computer (PC) to pulse simultaneously. Several images of the spray above the hydraulic jump region were captured and downloaded to the PC for image analysis. The PIV system setup is diagrammatically shown in figure 4.

Figure 3: Full field setup.

Figure 4: PIV setup.

3.2 Testing conditions

A liquid depth of 0.265 m (approximately 25% of the volume of the tank) was selected for the examination of multiphase flows in oscillating containers. This depth has a characteristic resonance frequency of 0.42 Hz based on eqn (1). The tank was oscillated at a near resonance frequency of 0.35 Hz and oscillation amplitudes of 2.420, 3.500, and 4.710 according to the expression in eqn (2). A slight offset, A, had to be introduced due to a bias in the motion simulator when the centre of gravity (CG) is shifted from its manufactured position. The asymmetry of the materials of each tank wall caused this shift in the CG. Att +−= )sin()( 0 φωθθ (2)

3.3 Data analysis

All post processing analysis was performed in the X-Vision software from IDT for the full field digital videos. The software was utilized to time stamp and measure spatial characteristics of the hydraulic jump formation. Each event was enlarged and its significant pixel locations noted. Also, the pixel locations of the bottom corners of the tank were recorded to provide insight into the angle of the tank as well as relative jump location with respect to the bottom and sides of the tank. A calibrated scale was positioned on the tank to determine a calibration coefficient, which was used to calculate the physical dimensions and locations of the hydraulic jump from the pixel locations determined previously.

Camera

Lights

Simulator

Clear TankCamera

Lights

Simulator

Clear Tank



For the PIV images of the spray region, the image analysis was performed in the Flowmanager software from Dantec Dynamics. This software implements a method termed “shadow sizing” to measure the equivalent diameter of the droplets. The equivalent diameter is the spherical diameter of a droplet that gives similar geometric and optical properties for the examined droplet. This method assumes each particle reflects light with a Gaussian intensity distribution with the greatest light intensity at the center of the droplet. A light intensity threshold is set by the user and employed by the software to find the edges of individual droplets.

4 Results

Each 15 second video was analyzed for hydraulic jump formations and the spatial characteristics of each jump were measured. A typical hydraulic jump in the 0.265 m depth is shown in figure 5 below. This phenomenon is characterized by the multiphase, turbulent region of air/water mixing coupled with a rapid change in liquid depth as evident in the encircled region in the figure.

Droplet Separation

Fully Developed Hydraulic Jump

h2h1

Droplet Separation

Fully Developed Hydraulic Jump

h2h1

Figure 5: Fully developed hydraulic jump.

4.1 Hydraulic jump formation

The formation of each hydraulic jump was recorded and the tank angle at which the jump occurred was measured. The hydraulic jump formation points are presented below in figure 6. The data points represent the observation of a hydraulic jump and the solid and dashed lines represent the calculated tank angle based on eqn (2). Figure 6 shows the hydraulic jump formation for all three oscillation amplitudes at the 0.35 Hz oscillation frequency. Left to right hydraulic jumps formed at lower tank angles than right to left travelling hydraulic jumps. Each of these jumps formed at the frequency of the simulator.



-6-5-4-3-2-101234567

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time (s)

Tank

Rol

l Ang

le (d

eg)

Hydraulic Jump(2.42 deg) Simulator(2.42 deg)Hydraulic Jump(3.50 deg) Simulator(3.50 deg)Hydraulic Jump(4.71 deg) Simulator(4.71 deg)

Figure 6: Hydraulic jump formation for three amplitudes.

75

100

125

150

175

200

225

250

275

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time (s)

Hyd

raul

ic H

eigh

t (m

m)

2.42 deg 3.50 deg 4.71 deg

Figure 7: Hydraulic height for 0.265 m at 0.35 Hz.

4.2 Hydraulic height

To provide insight into the physical nature of each of the hydraulic jumps throughout the oscillation, the height of each jump at the time of formation was measured. Increased hydraulic height suggests greater energy dissipation and an increased amount of spray from an end wall impact. Furthermore, larger jumps suggest greater wave speeds and near resonance conditions. In figure 7, the hydraulic height variation of each observed jump throughout the oscillation is plotted. Increasing the amplitude of the tank oscillation increased the height of



the hydraulic jump due to the greater gravitational force. The height changes with time due to the spray from the previous hydraulic jump impacting the end wall. Therefore, a tank oscillation produces a range of hydraulic heights throughout the oscillation.

4.3 Spray formation

The PIV system was used to image the spray region that forms above the hydraulic jump for the three oscillation amplitude conditions. Each image was processed by the acquisition software for the droplet size distribution in the image. These results are part of a preliminary investigation on the feasibility of using the PIV system to characterize the spray region. A typical PIV image from the 3.50° configuration is shown in figure 8.

Figure 8: Typical PIV image.

Each image was analysed by the PIV software and a distribution of droplet sizes directly above the hydraulic jump for each oscillation amplitude was produced. Figure 9 shows the typical distributions for a given image at each of the oscillation amplitudes. The 3.50° and 4.71° amplitude appeared to produce a large amount of droplets compared to the 2.42° amplitude. Furthermore, the distribution shifted to larger droplets as the oscillation amplitude increased.

5 Conclusions

The spatial properties of multiphase flows in an oscillating tank environment have been examined. It has been shown that near resonance conditions, the formation of a hydraulic jump characterized by liquid/air mixing is observed.



The formation location of this event is dependent on the frequency of the tank oscillation. Furthermore, the angle of the tank at which the hydraulic jump formed appeared to be independent of the oscillation amplitude of the tank.

0.62

0.79

0.95

1.07

1.19

1.33

1.44

1.56

1.68

1.82

1.94

2.04

2.18

2.32

2.45

2.55

2.74

2.95

3.23

3.46

3.73

3.97

4.37

4.98

5.58

2.42

deg

3.50

deg

4.71

deg

0

5

10

15

20

25

Cou

nt

Equivalent Diameter (mm)

2.42 deg3.50 deg4.71 deg

Figure 9: Equivalent diameter distribution.

The height of the hydraulic jump for the tested frequency is affected by the amplitude of the oscillation and the spray from the previous jump impact on the end wall. It is suggested that the increased gravitational force due to the greater incline angle of the tank increases the speed of the wave. As a result, a larger jump forms to dissipate the increased energy. All oscillation amplitudes of the tank exhibited a range of hydraulic heights produced throughout the oscillation. Droplet separation was observed using the PIV system and preliminary results suggest that increased oscillation amplitude increases the size of the droplets produced. It is believed that the increased amplitude delivers additional kinetic energy to the hydraulic jump, allowing larger droplets to be ejected from the surface. Difficulties with the intensity threshold were noted and future tests are expected to provide a characterization of the droplet dynamic range as a function of height above the hydraulic jump.

Acknowledgements

The authors would like to thank the funding support from the Department of Defense, the Joint Aircraft Survivability (JAS) program office, and Marty Lentz of the USAF 46th Test Wing, Aerospace Survivability & Safety Flight.



References

[1] Abramson, H. N. (ed.), “The Dynamic Behavior of Liquids in Moving Containers,” NASA, NASA SP-106, Washington D. C., 1966.

[2] Fontenot, L. L., “Dynamic Stability of Space Vehicles: Volume VII – The Dynamics of Liquids in Fixed and Moving Containers,” NASA, NASA CR-941, Washington D. C., March 1968.

[3] Graham, E. W. and Rodriguez, A. M., “The Characteristics of Fuel Motion Which Affect Airplane Dynamics,” Journal of Applied Mechanics, 123(9), pp. 381-388, September 1952.

[4] Armenio, V., La Rocca, M., “On the Analysis of Sloshing of Water in Rectangular Containers: Numerical Study and Experimental Validation,” Ocean Engineering, 23(8), pp. 705-739, 1996.

[5] Journee, J. M. J., “Fluid Tanks and Ship Motions,” Delft University of Technology, Report 1237, the Netherlands, October 2000.

[6] Chaudhry, M. H., Open Channel Flow, Prentice Hall, Englewood Cliffs, N. J., 1993.

[7] Chanson, H., “Bubble Entrainment, Spray and Splashing at Hydraulic Jumps,” Journal of Zhejiang University Science A, 7(8), pp. 1396-1405, 2006.

[8] Lee, T., Zhou, Z, and Cao, Y, “Numerical Simulations of Hydraulic Jumps in Water Sloshing and Water Impacting,” Journal of Fluids Engineering, 124, pp. 215-226, March 2002.

[9] Bauer, H. F. and Eidel, W., “Hydroelastic vibrations in a two-dimensional rectangular container filled with frictionless liquid and a partly elastically covered free surface,” Journal of Fluids and Structures, 19, pp. 209-220, 2004.

[10] Verhagen, J. H. G., and van Wijngaarden, L., “Non-Linear Oscillations of Fluid in a Container,” Journal of Fluid Mechanics, 22(4), pp. 737-751, 1965.

[11] Stoker, J. J., Water Waves, Interscience, New York, 1957.



Section 5 Suspensions

treatment plants

B. Zajdela1,2, A. Hribernik2 & M. Hribersek2 1Regional Development Agency Mura Ltd, Slovenia 2Faculty of Mechanical engineering, University of Maribor, Slovenia

Abstract

This contribution deals with the movement of flocs in suspensions, as they appear in biological water treatment (BWT) plants. The first part of the work deals with the description of a model BWT plant, followed by the definition of a problem. The greatest attention is given to the geometrical and sedimentation characteristics of solid flocs, key parameters for developing a fast numerical procedure for simulation of flocs’ movements. An extensive analysis is presented of floc size distribution and their shapes. Based on the results of experimental investigations, the main geometrical parameters of the flocs are defined and calculated. The second part of the work deals with calculating floc density, based on free settling sedimentation tests and known empirical correlations for drag coefficient. Keywords: biological treatment plants, sedimentation, image analysis, size distribution, drag coefficient, density.

1 Introduction

Today, wastewater treatment with the activated sludge is the most widespread process for removing dissoluble substances, small insoluble substances and colloidal organic pollutants from wastewater. The efficiency of the process primarily depends on growing biological biocenoze, consuming and consequently removing unwanted substances, and secondarily on accumulating the activated sludge into sludge flocs. The sludge flocs are subjected to circulation inside the wastewater processing tank and in the final phase of the flocs’ sedimentation process, separating the cleaned water from the accumulated



doi:10.2495/MPF070281

of flocs in suspensions of biological water Experimental investigations of sedimentation

impurities in the sludge [1]. The rate of sedimentation depends on the properties of the flocs, such as size, shape, density, permeability, number density and type of wastewater (industrial, municipal). In the presented paper, the greatest attention is therefore given to determining the size distribution and main geometrical parameters of the sludge flocs, by means of image analysis. Additionally, free settling tests in connection with empirical models for determining of the drag coefficient (CD) are used, in order to evaluate the densities of the flocs.

2 Materials and methods

2.1 Biological treatment plant

The performed analysis of the sludge flocs concentrated on the data, obtained from the wastewater samples of the Water treatment plant in Lendava, Slovenia. The wastewater samples, taken from the plant, were analysed within a time period of 1-3 days. The samples were diluted and kept at 20oC. The analysed wastewater was a mixture of technological and municipal waste water in the ratio of 75:25. Several analyse of various waste water samples were needed in order to obtain accurate and representative results.

2.2 Image analysis system

Wastewater samples were investigated using a Nikon stereoscopic microscope, which allowed us to see the samples at 20-126x magnification. Pictures were taken with a high-resolution Sony CCD videocamera, computer controlled using the Lucia M (Nikon, Japan) software package. Lucia M is a colour-image analysis software system, handling and analyzing High Color (3x5 bits for RGB components) digital images of typical 752x524 pixels resolution. Lucia M basically recognizes two types of images – binary and colour. Binary images have two possible values, 0 for background and a maximum of 62 for objects and structures. Binary images are produced by segmentation functions such as threshold and are often referred to as segmented images.

Figure 1: Colour and binary images of sludge floc.



Once the threshold value is defined the object can be clearly distinguished from the background, such segmented images (fig. 1) being used for automated shape and size measurements (area, diameter, and perimeter).

2.3 Preparation of samples

The sludge sample must be sufficiently diluted to avoid saturation of the image and the liquid layer thickness must be small enough to enable the use of relatively high magnifications (80x), focusing on a thin plane. By experimenting with different levels of dilution we discovered that some areas overlap even at the dilution ratio 1:15. Further diluting increased the isolation of the flocs and at a dilution of 1:20 there was no floc overlap. As quoted in Grijspeerdt and Verstraete [2], it is inherent to image analysis that smaller objects produce larger measurement errors. Objects are approximated by square pixels, resulting in a less accurate measurement of smaller objects. This problem could be solved by imposing a lower limit on the number of pixels in an object. Russ noted that 0.1% of objects comprising a number of pixels smaller than 0.1% of the total number of pixels of an image should not be taken into account [3]. In our case the size of the picture was 752x524 pixels and so 0.1% of the picture represented 394 pixels. In the case of 80x magnification this means that 0.1% of the picture was within an area of 0.001646 mm2, therefore, flocs smaller that this area were not analysed.

2.4 Drag on non-spherical particles

In general, the force balance for a floc moving steadily in an infinite medium can be described as follows [4]:

2

)(431 v

dgCe

fwp

Dw

wp

wf

ρρρ

ρρρρ

−Ω

=−=−− (1)

ρf is floc density, ρw is the density of the water and ρP is the density of the primary particles comprising the floc, CD is the drag coefficient, g is gravitational acceleration, v is the terminal velocity, df is the diameter of the floc, and e is porosity of the floc. In most studies, ρP is assumed to be equal to ρS, the dried solid density [5, 6]. Ω is the ratio of the resistance experienced by a porous floc to that of an equivalent solid sphere. Floc porosity is neglected in many studies [5, 7] and Ω is simply set to 1. The ρf is difficult to measure experimentally, therefore we decided to compute it based on data (velocity) obtained from the sedimentation properties of the flocs, as follows:

wf

Dwf dg

vCρ

ρρ +

⋅⋅⋅⋅⋅⋅

=)4(

)13( 2 (2)

In order to evaluate the right hand side in eqn. (2) geometrical data and data

on the CD of the flocs must be known. In the present work, the CD correlations



were used of different authors, who have performed comparison studies. The diameters and volumes of the flocs were calculated based on image analysis. Sedimentation tests were performed for terminal velocity.

In general, CD is a function of Reynolds number (Re) and floc sphericity. An equal volume sphere diameter is needed as the characteristic linear dimension in order to evaluate the CD. As a consequence, the sphericity (ψ) is commonly introduced in order to quantify the extent of the particle’s deviation from the spherical-shaped (Wadell [8]):

p

p

p

p

p

p

s AV

AV

AV

d32

32

31

2 84.4)6(6

==

= πψ (3)

where Vp is volume of the floc and Ap is the surface area of the floc. Corey’s shape factor (β) can be used as an alternative to sphericity, with a>b>c the lengths of the three principal axes of the floc:

β=c/(ab)1/2 (4)

When dealing with the sedimentation of particles in an infinite medium, the particle Reynolds number is defined as:

ηρψ fwdv

=Re (5)

where v is the terminal velocity of the floc in the medium of dynamic viscosity η which, due to the incorporation of sphericity, differs from the classical equation for spheres:

ηρ fwdv

=Re (6)

The starting point for derivation of a suitable expression for the CD of a non-

spherical particle is the well-known expression for the sphere in a laminar flow:

Re24

=DC (7)

In the case of non-spherical flocs it is most appealing to derive a correction to the expression (7). Haider and Levenspiel [9] developed an expression for CD for nonporous spherical and non-spherical flocs in incompressible media:

Re1

)Re1(Re24

DCAC B

D

+++= (8)



with the values of A, B, C and D as model constants and the equal volume sphere diameter used in the definition of Re. This method is applicable for Re < 2.6 x 105, uncertainty is predicted to be 15-20%. They also proposed the following equation for flocs of Ψ > 0.67:

[ ]

)2122.6(

)0748.5(

)5565.00964.0()0655.4(

378.5Re69.73

Re*1716.81Re24

ψ

ψ

ψψ

ee

eCD

⋅+⋅

+

⋅+=

−

+−

(9)

Ganser [10] assumed that every floc experiences a Stokes’s regime where

drag is linear in velocity and a Newton’s regime where drag is proportional to the square of velocity. He thus introduced two shape factors K1 and K2 applicable in both the Stokes and Newton regimes, respectively, in the following drag correlation:

( ) 21

6567.021

212

Re33051

4305.0Re1118.01Re

24

KK

KKKKK

CD

+++=

(10)

with Re based on the equal volume sphere diameter, and K1 and K2 as unique functions of sphericity:

( )[ ] 15.0

1 )3/2(3/ −−+= ψfn ddK (11)

5743.0)log(8148.12 10 ψ−=K (12)

where dn is the equally projected area of the circles diameter.

Swamee and Ojha [11] employed the equal volume sphere diameter and used the so called Corey shape factor (β) in the following drag expression:

( )

+

++

++

= 8.018

32.0

64.08.035.0 05.11

100100ReRe

Re5.415.48

ββββDC (13)

which was stated to be applicable within the range 0.3 < β <1 and 1 < Re <10.000.

Finally, Chien [12] proposed the following simple expression for drag, originating from petroleum engineering literature:

)03.5(289.67Re)/30( ψ−⋅+= eCD (14)

for 0.2 ≤ ψ ≤1 and Re< ~ 5000, where the equal volume sphere diameter was used.



2.5 Floc size distribution

The presented image analysis system was applied for floc size distribution measurements. Automatic processing was performed by allowing an appropriate threshold for the transformation of pictures into binary images. The results obtained were morphological characteristics of the flocs. From the standard deviation σ analysis the minimum number of flocs in a representative sample was determined to be n>400. Approximately 18 photos were needed to reach the minimum size for a representative sample. The basic sample consisted of 729 flocs with the flocs surface areas within the range from 0.0000042 mm2 to 0.0759 mm2. According to the “< 0,1%” rule, we excluded all those flocs whose areas were smaller than 0.001646 mm2. 72 flocs remained and were separated into 7 size classes. The details are presented in fig. 2, showing flocs’ frequency distribution and cumulative flocs’ surface area distribution. Following from our observations, the average floc’s equivalent diameter is 0.086837 mm and the average floc’s surface area is 0.007845 mm2.

0

10

20

30

40

50

0,063 0,097 0,128 0,166 0,203 0,297

Mean diameter (mm)

Num

ber o

f par

icle

020406080100120

% a

rea

Numberof particle% area

Figure 2: Flocs frequency distribution and cumulative flocs surface area

distribution.

2.6 Main geometrical parameters

Only observation of floc’s plane projection is possible sing floc image analysis. No information on a floc’s volume is necessary for the calculation of floc sphericity, CD and density. Due to the different shapes of flocs it is very difficult to determine a representative three dimensional floc shape and volume. A large number of flocs were, therefore, analysed. Each individual floc was rotated in order obtain floc images in three orthogonal planes (fig. 3) and then an attempt was made to create a 3D shape of a floc using its 3 orthogonal 2D images.

The 3D floc’s shape was approximated by an equivalent cuboid whose projections on the orthogonal planes were equal to those of the floc. The transformation can be written using a simple model as shown in fig. 3.

14 randomly selected flocs from the diluted sample (1:20) of waste water were processed in this way. Three replications were made for each individual floc and the proportions APCAPBAPA ⋅=⋅=⋅= 3,2,1 with P1,P2 and P3 as weighting factors, were estimated. Weighting factors P1 and P2 deviated only slightly from their average values, while the scatter of weighting factor P3



was much bigger. In spite of this, the average values of weighting factors were used for the sake of simplicity and the ratio of the equivalent cuboid surfaces was found to be 69.0:89.0:13:2:1:: == PPPCBA .

Experiments showed that the flocs always set themselves up in a way that the projection surface in the horizontal plane which corresponds to the projection A (fig. 3) was the largest. Therefore, it is possible to estimate the sizes of projection planes B and C by measuring the projection of flocs in plane A and considering the ratio A : B : C. The edges of the equivalent cuboids a, b and c are then determined and the volume of each floc is estimated as V=a·b·c. Finally the shape factors ψ and β, necessary in empirical equations for CD, are calculated. Since the constant ratio A : B : C was applied, the shape factors were equal for all flocs.

Eqn. (3) was applied for the prediction of floc sphericity ψ, which was found to be ψ = 0.802 for the sampled wastewater. Corey factor β of flocs was calculated from eqn. (4) as β =0.86.

Figure 3: Equivalent cuboid; A, B, C - projection areas.

2.7 Drag coefficient (CD)

A large number of free settling tests were performed and terminal settling velocities of flocs were measured in order to estimate the suitability of the presented CD models given by eqns. (7) – (14)

A 50 ml glass cylinder was used and the time of travel of a single floc on a chosen distance was measured. Three replications of each settling test were made and the average terminal settling velocities was calculated. Only larger flocs were selected for tests in order to improve visualisation quality. The results are shown in fig. 4. The terminal settling velocity of the flocs varied between 2.5 and 4.7 mm/s. It increases with the flocs diameter (size), which agrees with the results of other authors [13].

A

C

BACBcbaA ⋅

=→= *

CBAbcbB ⋅

=→= *

BCAacaC ⋅

=→= *



0,00

1,00

2,00

3,00

4,00

5,00

0,00 1,00 2,00 3,00

Equal volume sphere diameter (mm)

Term

inal

vel

ocity

(mm

/s)

0

2

4

6

8

10

Rey

nold

s nu

mbe

r

Equal volume spherediameter vs terminalvelocityEqual volume spherediameter vs Reynoldsnumber

Figure 4: Terminal velocity and Reynolds number of flocs.

Figure 5: Comparison between different CD models. Data obtained from the free settling tests were used to predict the CD, fig. 5.

The differences between individual models were as high as 300%. Models proposed by Chien (eqn. (14)) and Haider (eqns. (8) and (9)) result in higher CD values than the perfect sphere model (eqn. (7)), while the models by Ganser (eqns. (10)-(12)) and Swamee (eqn. (13)) predict lower CD values.

The CD decreases with Re and the differences between the individual models become smaller. However, it should not be neglected that only the largest flocs were studied by free settling tests. An average floc and its Re are much smaller, thus its CD is substantially higher than those presented in fig. 5, and an appropriate CD model is essential. It is evident, that models proposed by Ganser and Swamee under-predict CD value (it should not be under the CD value of the sphere). Models by Haider and Chien predict more reliable CD values which are very close together (± 10 %). Because of its simplicity and a wide application range the Chian model (eqn. (14)) was chosen for application in our model.

2.8 A floc’s density

A floc’s density plays an important role in the sedimentation process. The sedimentation of suspended matter in the water is possible only if its density is higher than the water density. Otherwise it floats.

02468

101214

0 2 4 6 8 10

Re

CD

CD GanserCD ChienCD Haider1 CD Haider2 CD laminarCD Swamee in Ojha



Using eqn. (2) the flocs density can be predicted if the floc’s terminal velocity, CD and equivalent diameter are known. The densities of the flocs used in the free settling tests were estimated in this way. The measured terminal velocities and the floc’s equivalent diameter were applied and CD predicted by the Chien model was used. The results are presented in fig. 6, where the difference in the floc’s and water densities is plotted against the floc’s diameter. As can be seen the floc’s density decreases with the floc’s size. Flocs are formed from a large number of primary flocs with the in-between free space filled by water. Therefore, the floc density decreases with its size, which was also proven by other authors [13].

0123456789

0 0,5 1 1,5 2 2,5 3

Equal volume sphere diameter (mm)

∆ρ (k

g/m

3 )

Figure 6: Difference of flocs and water density against flocs diameter.

3 Conclusion

The properties of the activated sludge flocs were examined. An image-analysis system was applied to determine floc size distribution. A representative waste water sample comprising 729 flocs was acquired. Those flocs larger than 0.1% of the whole image area were selected from this sample and analyzed. The 3D floc shape was approximated by an equivalent cuboid whose projections on the orthogonal planes were equal to those of the floc. The ratio of the equivalent cuboid surfaces was determined and used to predict floc volume and shape factors by measuring only the horizontal projection areas of the flocs. Mean floc shape factors were found to be ψ = 0.802 and β =0.86.

A large number of free settling tests were performed and the obtained results were applied in six different CD models. The model proposed by Chien was selected as the most appropriate and applied in the floc-free settling model for floc density computation. A floc’s density was found to vary with its size. The density of the smallest analysed floc was 1% higher and the density of the largest analysed floc was only 0.3 % higher than density of the water.

The obtained results present a starting point for numerical simulation of a floc’s motion in water, where the density of the floc, its size and its drag coefficient are key parameters for particle path computations.



References

[1] Roš, M., Biološko čiščenje odpadne vode, GV založba, (2001), (in slovene).

[2] Grijspeerdt, K. & Verstraete, W., Image analysis to estimate the settleability and concentration of activated sludge, Wat. Res. 31, pp. 1126-1134, 1997.

[3] Russ, J. C., Computer – Assisted Microscopy: the Measurement and Analysis of Images, Plenum Press, New York, 1990.

[4] Huang, H., Porosity – size relationship of drilling mud flocs: fractal structure, Clay Clay Miner 41, pp. 373 – 379, 1993.

[5] Tambo, N. & Watanabe, Y., Physical characteristics of flocs. 1. The floc density function and aluminum floc, Wat. Res. 13, pp. 409 – 419, 1979.

[6] Li, D.-H. & Ganczarczyk, J. J., Stroboscopic determination of settling velocity, size and porosity of activated sludge flocs, Wat. Res. 21, pp. 257 – 262, 1987.

[7] Concha, F. & Almerdra, E. R., Settling velocities of particulate system, 1. Settling velocities of individual spherical particles, Int. J. Min. Process. 5, pp. 349 – 367, 1979.

[8] Wadell, H., Sphericity and roundness of rock particles, J. Geol 41, pp. 310 – 331, 1933.

[9] Haider, A. M., Levenspiel, O., Drag coefficient and terminal velocity of spherical and nonspherical particles, Power Technol. 58, pp.63 – 70, 1989.

[10] Ganser, G. H., A rational approach to drag prediction of spherical and nonspherical particles, Power Technol. 77, pp. 143 – 152, 1993.

[11] Swamee, P. K. & Ojha, C. P., Drag coefficient and fall velocity of nonspherical particles, J. Hydraul. Eng. 117, pp. 660 - 667, 1991.

[12] Chien, S. F., Settling velocity of irregularly shaped particles, SPE Drilling and Completion, pp. 281 – 289, 1994.

[13] Lee, D. J., Chen, G. W. & Hsieh, C. C., On the fre-settling test for estimating activated sludge floc density, Wat. Res. 30, pp. 541 – 550, 1996.



Modelling molecular gas suspension diffusionand saturation processes in liquid media

R. GrollCenter of Applied Space Technology and Microgravity,University of Bremen, Germany

Abstract

A model describing the suspension diffusion process of gas molecules in liquidmedia is presented in this paper. This process is not yet solved by a satisfactorymodel for micro-scale applications at this time. The new approach allows thesimulation of diffusion processes in continuous media considering the molecularmass flux in a suspension/carrier phase mixture. Modelling the diffusion of gassuspensions in liquid media the saturation mass ratio is reached near the liquid/gassurface very quickly. The increase of gas concentration in the liquid domaindepends on the elapsed time and the physical properties of gas and liquid media.The molecular gas velocity is described by a Maxwell probability density function.Modelling the gas species diffusion the molecular convection is considered.Modelling the mass flux of the molecular gas suspension characteristic time scalesare developed describing the completion level of the saturation progress based onnon-dimensional formulations of the molecular convection equation. The presentmodel is implemented in a CFD code and validated by a family of parametricsimulation results depending on the saturation mass ratio of the suspended gasphase. This simulation result array shows the dependency of saturation time andsaturation mass ratio of the suspended gas molecules. Based on this relationmacroscopic diffusion processes in micromixers and microchannels are describedwith this model and without an extra solution of molecule trajectories or spectralfields of molecule velocity.Keywords: two-phase flow, molecular diffusion, gas/liquid, dispersion, saturation.



doi:10.2495/MPF070291

1 Introduction

The phase interaction of multiphase systems is a topic which is investigatedfor the last century [6, 11, 12, 15] without giving clear answers modelingthe diffusion process of a suspended disperse phase in a continuous carrierphase [19, 20]. Modelling the transport of suspended gas molecules in a liquidcarrier phase can not be described by discrete models [3, 5, 10, 16]. The motionof suspended molecules follows diffusion characteristics [1, 4, 7, 13, 18] asdescribed in temperature or diffusive combustion processes. The major boundarycondition modelling the gas suspension process is the start concentration of thesuspension at the phase interface. The stability of numerical simulation dependson the sensitivity of such a boundary condition [9, 17]. Modelling the masstransport at this boundary condition defines the global mass balance and is aresignificant parameter for the quantitative results [2]. Finally basic approximationsof molecular motions are necessary to give better values to be considered forconstruction and development of future advanced mixing devices.

2 Continuum mechanical approach

In Fig. 1 the mass transport procedure at the phase interface is shown. During thissuspension process the liquid phase of the species (k = F ) absorbs gas moleculesof the species (k = G). The partial densities of both phases ρk are defined by theproduct of the molecule mass of liquid phase molecules or gas phase moleculesmk and the number density nk, which means the molecule number per volume ina given mixture volume V .

ρF = mF NFV

V= mF nF (1)

ρG = mG NGV

V= mGnG (2)

Supposing that suspended gas molecules replace no fluid molecule the efficientmixture density ρ∗ is defined by the sum of the partial densities of gas and liquid:

ρ∗ = ρF + ρG (3)

The concentration ck of the species k is defined by the ratio of partial density andmixture density:

ck =ρk

ρ∗(4)

Based on the concentration definition of fluid and gas phase media the partialdensity ρkis defined on concentration ck and the efficient micture densityρ∗:

cF + cG = 1 (5)

ρ∗ck = ρk (6)



Because of the continuity of both species the corresponding equations for thepartial densities are defined depending on the partial velocities of of the liquidand gaseous phase:

∂

∂tρF +

∂

∂xj

(ρF uF

j

)= 0 (7)

∂

∂tρG +

∂

∂xj

(ρGuG

j

)= 0 (8)

Summarizing these equations the global mass balance results:

0 =∂

∂t

ρF + ρG︸︷︷︸

ρ∗

+

∂

∂xj

(ρF uF

j + ρGuGj

)

=∂

∂tρ∗ +

∂

∂xj

ρ∗

ρF uFj + ρGuG

j

ρF + ρG︸︷︷︸=:U∗

j

=∂

∂tρ∗ +

∂

∂xj

(ρ∗U∗

j

)(9)

This equation is the continuity equation of the liquid/gas mixture of the species Gand F . The global mass flow results from the effective mixture density ρ∗ and theeffective mixture velocity U∗.

Figure 1: Distributed phases: G (gas) is suspended in phase F (fluid) with the massflow dm/dt at the phases interphase.



2.1 Concentration of the suspension

To get a transport equation for the concentration, the concentration dependingdensity ρk (eq. (6)) is substituted in the continuity equation (eq.(7)):

∂

∂t

(ρ∗cF

)+

∂

∂xj

(ρ∗cF uF

j

)= 0 , (10)

For this following transport equation with a mixture velocity depending convectiveterm results:

∂

∂t

(ρ∗cF

)+

∂

∂xj

(ρ∗cF U∗

j

)

=∂

∂xj

[ρ∗cF

(U∗

j − uFj

)]

=∂

∂xj

cF

ρ∗U∗

j︸︷︷︸=ρF uF

j +ρGuGj

−ρ∗uFj

=∂

∂xj

cF

(ρGuG

j

)+(cF − 1

)︸︷︷︸

=−cG

(ρF uF

j

) (11)

Based on a substitution with one of the both concentration terms (eq. (5)) thetransport equation of the species concentration results:

∂

∂t

(ρ∗cF

)+

∂

∂xj

(ρ∗cF U∗

j

)=

∂

∂xj

[cF(ρGuG

j

)− cG(ρF uF

j

)](12)

Here a transport equation depending on the mixture density ρ∗ and the mixturevelocity U∗ analogous to the transport equation of other physical values of thismixture process is given.

2.2 Modelling the diffusion approach

Based on the consideration, that the developed diffusion term is modelable by thelinear approximation, the transport follows from Fick’s law:

∂

∂t

(ρ∗ck

)+

∂

∂xj

(ρ∗ckU∗

j

)=

∂

∂xj

(ρ∗D

∂ck

∂xj

)(13)



and following equivalence from eq. (12):

ρ∗D∂cF

∂xj= cF

(ρGuG

j

)− cG(ρF uF

j

)+ Aj (14)

with a constant vector A.It is considered with the hypothesis |uF | = 0 that the motion of the liquid

medium is negligible during the diffusion process of the gaseous molecules. Bythe upper result (eq. (14)) following relation of the concentration gradient results:

ρ∗D∂cF

∂xj= cF

(ρGuG

j

)+ Aj (15)

At the wall for all species k it is given:

∂ck

∂xj

∣∣∣∣W

= 0 , ukj

∣∣W

= 0 (16)

From eq. (15) the constant vector A = 0 results and considering eq. (6) the mixturevelocity is given:

uGj =

ρ∗

cF ρGD

∂cF

∂xj

=D

cF cG

∂cF

∂xj(17)

By this definition the global relation between diffusion rate and concentrationgradient is shown.

2.3 Mass flow rate at the phase interface

Based on the assumption that the suspended gas phase reaches the saturationconcentration at the phase interface (PI)

cG∣∣PI

= cGmax (18)

the interfacial mass flow is given in the following way:

uGj

∣∣PI

=D

(1 − cGmax)cG

max

∂cF

∂xj

∣∣∣∣PI

=D

(cGmax − 1)cG

max

∂cG

∂xj

∣∣∣∣PI

(19)

m|PI =(ρ∗cGuG · ez

)PI

=ρ∗D

cGmax − 1

∂cG

∂z

∣∣∣∣PI

(20)

Because of the dependence of the mixture density ρ∗ from the partial suspensiondensity ρG and therefore from the suspension concentration it is substituted by



ρ∗ = ρF /cF . So the following explicit form of the interfacial mass flow results:

m|PI = − ρF D

(1 − cGmax)

2︸︷︷︸=const.

∂cG

∂z

∣∣∣∣PI

(21)

With this form the interfacial mass flow is defined by the initial density of theliquid phase (without suspension) ρF , the saturation concentration cG

max, the binarydiffusion coefficient D and the local concentration gradient of the suspendedspecies.

3 Transport equations

Inducing the resulting suspension velocity (eq. (17)) the continuity equation of thesuspended gaseous phase is given by following equation:

∂

∂tρG +

∂

∂xj

(ρGD

cF cG

∂cF

∂xj

)= 0 (22)

By eq. (6) following transport equation results independent from the suspensionvelocity:

∂

∂t

(ρ∗cG

)=

∂

∂xj

(ρ∗D

1 − cG

∂cG

∂xj

)

with ρ∗ =ρF

1 − cG(23)

The implicit form of this equation is unsuitable for linear solvers:

∂

∂t

(cG

1 − cG

)=

∂

∂xj

(D

(1 − cG)2∂cG

∂xj

)

=∂

∂xj

[D

∂cG

∂xj

(cG

1 − cG

)](24)

In analogy to the exact solution of the implicit formulation the ratio cG/(1 − cG)is the exact solution of a mathematical heat equation (eq. (24)) in a non-restricteddomain:

cG

1 − cG=

cGmax

1 − cGmax

[1 − erf

(x

2√

Dt

)](25)



After a corresponding conversion the explicit formulation of the suspensionconcentration cG results depending on spatial position and time.

1cG

− 1 =1

cGmax

− 1

1 − erf(

x2√

Dt

) (26)

cG =

1 +

1cG

max− 1

1 − erf(

x2√

Dt

)−1

=

1 +

1 − cGmax

cGmax

(1 − erf

(x

2√

Dt

))−1

(27)

This exact approach is valid for a non-restricted domain only, where the suspensiondiffuses.

4 Simulation results and verification

In a simulation the proposed mass flow inlet boundary condition (eq. (21)) isverified. The input/start requirement concentration is zero over complete domainat t = 0. This is the initial condition for the diffusion process simulation. At thephase interface only (x = 0) the concentration of the suspended gas is equivalentwith the saturation concentration cmax.

Based on this initial condition two simulations are performed. Verifying themass flow model at the phase interface the simulation results of the boundarycondition with a given concentration at the interface c|x=0 = cmax (Inlet 1)are compared with the proposed mass flow boundary condition (Inlet 2). Forthe comparing simulations as the exact solution cmax = 0, 2, L = 0.1m andD = 10−3 m2

s are used.

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1

cG

x(m)

t = 0.05s

Inlet 1

Inlet 2

erf

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1

cG

x(m)

t = 0.1s

Inlet 1

Inlet 2

erf

Figure 2: Computational simulations of the concentration distribution (Inlet 1,Inlet 2) are conform with the error function (erf) away from a wall.



Now it is shown that during the time, when the concentration at the wall isnegligible, the influence of the wall is also negligible and the simulation result ofboth inlet conditions (Fig. 2) are equivalent with the exact solution of a diffusionprocess in an open domain (erf, eq .26).

After reaching the wall of the higher concentrations (Fig. 3), simulation resultsof eq. (26) diverge from the open domain solution, because this exact solutionis not defined in restricted volumes. But because of the equivalence of bothsimulation results with the initial and the modelled mass flow boundary condition,the mass flow model at the phase interface is verified for open and restricteddomains.

5 Saturation of a restricted domain

Describing the time-dependent saturation eq. (23) is dedimensioned. All variablevalues are normalised with the constants L, ρF and D, so that the normalizedvalues are non-dimensional. For this transformation the equation is defineddepending on:

t+ =tD

L2, x+ =

x

L, ρ+ =

ρ∗

ρF(28)

by multiplying eq. (23) with the ratio L2/(DρF ). L is the characteristic length ofthe restricted domain:

∂

∂(

tDL2

) ( ρ∗

ρFcG

)=

∂

∂(xj

L

)(

ρ∗

ρF

11 − cG

∂cG

∂(xj

L

))

⇒ ∂

∂t+

(ρ+cG

)=

∂

∂x+j

(ρ+

1 − cG

∂cG

∂x+j

)(29)

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1

cG

x(m)

t = 1s

Inlet 1

Inlet 2

erf

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08 0.1

cG

x(m)

t = 5s

Inlet 1

Inlet 2

erf

Figure 3: Computational solution (Inlet 1, Inlet 2) diverge from the exact opendomain solution when the concentration near the wall increase (erf,without wall), but they are equivalent among each other.



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

c +

x+

cmax=0.01

t+=0.001t+=0.005t+=0.010t+=0.050t+=0.100t+=0.500t+=1.000t+=2.500

Figure 4: Time-dependent change ofthe spatial distribution of thelocal saturation level betweent+ = 0, 001 and t+ = 2, 5with cG

max = 0.01.

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t +

cmax

99,5%99,0%98,0%95,0%90,0%

2.698-0.415*log(1+11.0*x)1.980-0.415*log(1+1.90*x)1.585-0.415*log(1+0.72*x)1.158-0.415*log(1+0.24*x)0.861-0.415*log(1+0.10*x)

Figure 5: Isolines of the mean satu-ration level in a restricteddomain over cG

max × t+ andits modelling result.

The unsteady derive of the relative saturation level c+ = cG/cGmax is defined with

following equation:

∂

∂t+

(ρ+cG

+

)=

∂

∂x+j

(ρ+

1 − cGmaxcG

+

∂cG+

∂x+j

)(30)

As shown this time-dependent change of saturation level depends on the saturationconcentration cG

max.To stop an unsteady simulation with cG

max = 0, 01 the void time-scale t+ = 2, 5seems to be great enough, because the mean saturation level has reached themaximum and the diffusion process is nearly completed. Analysing the isolinesof the mean saturation level (Fig. 5), it is shown that the wanted time scale t+ hasto be much higher to define the saturation process as completed (mean saturationlevel > 99%) for low saturation concentrations cG

max.The estimation of a time T , for what a domain with the length L is

approximately saturated, has to be between t+ = 2, 0 and t+ = 2, 7. A proposefor an over-all approximation would be t+ = 2, 5 or:

TD

L2= t+ ≈ 2, 5 ⇒ T ≈ 5L2

2D(31)

A finer approximation considering the saturation concentration cGmax would be:

T =[2, 7 − 0.415 ln

(1 + 11 cG

max

)] L2

D< 2, 7

L2

D(32)

For this model more than 99.5% of the maximum suspendible gas mass would beabsorbed in the carrier phase.



6 Conclusion

Modelling the diffusion process of a suspension in a liquid carrier phase by eq. (23)is possible with two kinds of boundary conditions:

1. Given concentration value at the phase interface c|x=0 = cmax

2. Modelled mass flow at the interface by eq. (21)The simulation results of both boundary condition types are equivalent. If thediffusion process is executed far away from a domain restricting wall thesimulation results are equivalent the exact solution

cG =

1 +

1 − cGmax

cGmax

(1 − erf

(x

2√

Dt

))−1

of a heat equation (eq. (24)) additionally.The characteristic time of an approximately completed diffusion process in a

restricted domain, after what the mean local saturation level has nearly reachedthe maximum ( > 99.5%) is restricted by T < 2, 7 L2

D and is given by eq. (32)depending on the domain length L, the diffusion parameter D and the saturationconcentration cG

max.

References

[1] R. V. Calabrese & S. Middleman , The Dispersion of Discrete Particles in aTurbulent Fluid Field; AIChE Journal, Vol. 25(6), S. 1025-1035, 1979.

[2] R. Clift, J. R. Grace & M. E. Weber , Bubbles, Drops and Particles; AcademicPress, New York, 1978

[3] C. T. Crowe, M. Sommerfeld & Y. Tsuji , Multiphase Flows with Dropletsand Particles; CRC Press LLC, 1998.

[4] G. T. Csanady , Turbulent Diffusion of Heavy Particles in the Atmosphere; J.Atm. Sc., Vol.20, S.201-208, 1963.

[5] D. A. Drew , Mathematical Modelling of Two-Phase Flow; Ann. rev. FluidMech., Vol. 15, S. 261-291, Ann. Rev. Inc, 1983.

[6] H. Grad , On the Kinetic Theory on Rarefied Gases; Communications onPure and Applied Mathematics, Vol.2, Nr.4, S. 331-407, 1949.

[7] R. I. Issa & P. J. Oliveira , Accounting for Non-Equilibrium TurbulentFluctuations in the Eulerian Two-Fluid Model by Means of the Notion ofIntroduction Period; The Third International Conference on Multiphase FlowICMF’98, Lyon 1998.

[8] J. T. Jenkins & M. W. Richman , Grad’s 13-Moment-System for a Dense Gasof Inelastic Spheres; Arch. Ration. Mech. Anal., Vol.87, S. 355-177, 1985.

[9] P. K. Khosla & S. G. Rubin , A Diagonally Dominant Second-Order AccurateImplicit Scheme; Computers Fluids, Vol. 2, S. 207-209, 1974.

[10] M. R. Maxey & J. J. Riley , Equation of Motion for a Small Rigid Sphere ina Nonuniform Flow; Physics of Fluids, Vol. 26, S. 883-889, 1983.



[11] R. A. Milikan , The General Law of Fall of a Small Particle through a Gas,and its Bearing upon the Nature of Moleculare Reflection from Surfaces;Phys. Rev., Vol. 22, 1923.

[12] F. Odar & W. S. Hamilton , Forces on a Sphere accelerating in a viscous fluid,Vol. 18, S. 302 ff., 1964.

[13] P. J. Oliveira , Computer Modeling of Multidimensional Multiphase Flow andApplication to T-Junctions; PhD Thesis, Empirial College London, 1992.

[14] W. A. Sirignano , Fluid Dynamics and Transport of Droplets and Sprays;Cambridge University Press, 1999.

[15] S. L. Soo , Fluid Dynamics of Multiphase Systems; University of Illinois,Blaisdell Publishing Company, 1967.

[16] C.-M. Tchen , Mean Value and Correlation Problems Connected with theMotion of Small Particles Suspended in a Turbulent Fluid; Martinus Nijhoff,Den Haag 1947.

[17] C. Vit, I. Flour & O. Simonin , Modelling of Confined Bluff Body FlowLaden with Polydispersed solid particles; Two-Phase Flow Modelling andExperimentation 1999, Edizioni ETS Pisa, 1999

[18] L.-P. Wang & S. E. Stock , Dispersion of Heavy Particles by TurbulentMotion; J. Atmos. Science, Vol.50, S. 1897-1913, 1993.

[19] L. P. Yarin & G. Hetsroni , Turbulence Intensity in Dilute Two-Phase Flows1-3; Int. J. Multiphase Flow, Vol. 20., No. 1, S.1-44, Elsevier Science Ltd.,1994.

[20] D. Z. Zhang & A. Prosperetti , Averaged Equations for Inviscid DisperseTwo-Phase Flow; J. Fluid Mech., Vol. 267, S. 185-219, Cambridge UniversityPress, 1994.



Analysis of two- and three-particle motion in aCouette cell

M. Popova, P. Vorobieff & M. IngberDepartment of Mechanical Engineering,The University of New Mexico, Albuquerque, New Mexico, USA

Abstract

We present an experimental investigation of the irreversibility of two and threespherical particle interactions in shear flow. The experiment is performed in astratified two-dimensional fluid inside a Couette cell at a very low Reynoldsnumber. The particles are placed into the cell in well-characterized initial positions.Their motion is driven by the inner wall of the cell that is repeatedly rotatedby the same angle clockwise and then counterclockwise. Nominally the flowis completely reversible (if the particles do not come close to each other, theyreturn to their initial positions). Three types of particles are used with differentsurface roughnesses. In two-particle interactions, the degree of irreversibility onthe macroscopic scale is found to be correlated with the average microscopicroughness of the particles. Subsequently, we investigate three-particle interactionand find an appreciably different behaviour, suggesting that forces in a multi-particle system cannot be reduced to force-pairs between individual particles.

1 Introduction

Addition of a suspended phase (gas bubbles, particles, or droplets) to a fluid flowappreciably complicates the challenge of predicting the flow, either theoreticallyor numerically. In many modern applications, especially in the emerging areasof micro- and nanoscale processing, the flow regime of most interest isviscous, nonlinear shear suspension flow. It may occur during composite andceramic processing, production of semiconductors and magnetic storage media,and encapsulation of electronic components. Viscous shear flows with particlesuspensions are also important in such earth, environmental, and planetary scienceproblems as transport of sediments, contaminants, and slurries, and secondary oilrecovery by hydraulic fracturing.



doi:10.2495/MPF070301

The main challenge in addressing these applied problems lies in the necessity torelate between the physical phenomena on the scale of individual particles and theresulting macroscopic behaviour of the flow, and so far, development of rheologicalmodels predicting suspension flows has been only partially successful, despitemajor advancements in the field in the last two decades (see [1–5] and paperscited therein). One of the important macroscopic characteristics of suspensionflows is the particle concentration profile. Existing models can predict such profileswell only for steady states, which are rarely achieved in many suspension flowapplications, where the flows remain transient. A common example of such anapplication is fluid moving in the annular region between two rotating cylinders(Couette flow). For simplicity, many researchers consider the particles seeding theflow to be spheres of uniform size. For a constant concentration of such spheres,dimensional considerations used in rheological models lead to a prediction thatthe rate of particle migration in Couette flow should scale as the square of thesphere radius. However, experiments [6] reveal that the actual scaling is closer tothe cube of that radius. Moreover, if the current models are used to predict the timerequired for a Couette flow to achieve steady state, the model prediction and thetime measured in experiment can differ by orders of magnitude.

Arguably the main reason for this discrepancy is the highly non-conservativenature of the forces in multi-particle suspension systems, causing such systemsto exhibit strongly irreversible behaviour. In contrast, the governing equations forthe pure fluid flow in many viscous shear-flow systems are the Stokes equations,which are reversible in time. To reconcile the models with reality, the physicalmechanisms of irreversibility must be understood and accounted for.

One of the proposed mechanisms of irreversibility is due to microscopic surfaceroughness, originally suggested by Arp and Mason [7]. Interaction betweenmicroscale features on particle surfaces causes the particles to “stick” together,thus producing irreversibility. The same kind of roughness effect would alsomake the process of a heavy particle settling towards the bottom of a fluid-filledcontainer irreversible (in this case, due to the particle “sticking” to the bottom),as the experiments of Smart and Leighton [8] confirm. This study also founda qualitative agreement between the hydrodynamic roughness as the measure ofirreversibility of the particle interaction with the container wall and the actualsurface roughness of the particle according to profilometry measurements. In thesetwo studies, roughness effects were assumed to be the sole source of irreversibility.A recent investigation [9] of two-particle interaction in Couette flow confirmsthis notion, establishing a quantitative correlation between irreversibility in theflow and average surface roughness of the particles as measured with a scanningelectron microscope (SEM).

But can the interaction between multiple particles in a suspension be reducedto particle-pair interactions? It has been almost universally assumed in previousworks that the most important interactions are those in particle pairs, withthe force balance for a multi-particle system obtained as the sum of binaryforces. This notion, however, was recently shown to be theoretically incorrectfor a dense configuration of particles [10]. Thus another source of irreversibility



due to nonlinear multi-particle interactions may exist in suspension flows.Its existence could account for the chaotic behaviour of the trajectories ofperfectly smooth particles in numerical simulations of three-particle interactions innominally reversible viscous flow [11, 12], where any surface roughness–relatedirreversibility is eliminated in the numerical formulation of the problem.

In this paper, we present an experimental study of two- and three-particleinteractions in a viscous Couette cell flow attempting to elucidate the influenceof the number of particles involved in the interaction on the irreversible behaviourmanifested by the flow.

2 Experimental arrangement

Our experimental apparatus (Fig. 1) is a wide-gap Couette cell with the outer(stationary) cylinder diameter Do = 20 cm and the inner (rotating) cylinderdiameter Di = 10 cm. The outer cylinder is machined from acrylic polymertogether with the stationary bottom plate, while the inner cylinder can be rotatedby an electric stepper motor (“Compumotor,” Fig. 1) via a belt drive. An arbitrarymotion profile for the rotating cylinder can be downloaded to the computerinterface controlling the motor (“Compumotor controller,” Fig. 1). The cell is 4 cmdeep and filled with a vertically stratified viscous fluid (water solution of ZnCl2and commercial water-miscible solvent Triton X-100). The refraction index of thefluid (1.49) matches that of the outer cylinder of the cell, to facilitate imaging ofthe side views of the cell and minimize refraction off boundaries. The density andstratification of the fluid render the PMMA (polymethyl methacrylate) particlesused in experiment neutrally buoyant at a depth of 1 cm. The specific densityof PMMA at this depth is 1.05, and the overall density difference from top tobottom of the cell due to stable stratification is about 1in fluid viscosity dueto stratification. We conducted rheometry measurements to obtain the kinematicviscosity of the fluid ν = 146 cm2/s. The rheometry measurements also showeda linear relationship between torque and strain with no memory effects. Thus thefluid can be considered Newtonian (i.e., shear stress in it is linearly proportional tothe velocity gradient).

The PMMA particles used in the experiments have been individually mappedusing a scanning electron microscope. Thus for each individual particle, the rmsaverage and the peak values of surface non-uniformities are known. During theexperiments, two or three PMMA particles of nominal diameter dp=0.635 cm wereplaced into the cell using a template – a flat acrylic plate with holes positioned atthe same location with respect to the outer cylinder, ensuring consistency of theinitial positions of the particles in the pair or triplet. After the particles settledto their neutral buoyancy depth, the inner cylinder was rotated 750 forward(clockwise if viewed from above the cell). Then the direction of the rotation wasreversed, and the inner cylinder was rotated 750 backward (clockwise) at thesame angular velocity. The characteristic dimensionless parameter describing the



Figure 1: Experimental setup. Top: photograph of the actual arrangement withindividual components labeled. Bottom: schematic of the partial viewof the cell from above with dimensions.

relative importance of dissipation with respect to inertia in the flow is the large-scale Reynolds number

Re = (πfDi) (Do − Di)ν,

where f is the rotation frequency. In the experiments described here, Re ∼ 0.01,indicating that the flow is laminar and should be nominally consistent with theStokes approximation (Re << 1) for viscous time-reversible flow.

The experiment was visualized by a ten-bit grayscale digital camera with apixel resolution of 1536 by 1024. The camera was positioned directly above theCouette cell (“Top view camera,” Fig. 1). The top view of the cell was augmented



by side views captured with another camera through the flat side of the acrylicblock and through mirrors at about 45 to the adjacent sides. These side viewswe used to verify the two-dimensionality of the particle interactions. During theexperiments, the image capture rate was maintained at 0.25 Hz, thus yielding animage sequence of 72 captures per experimental run. The camera spatial resolutionwas 0.20 mm/pixel, thus the characteristic particle size was 31.2 pixels. Prior toeach experimental run, an image of the experiment without particles was captured.In postprocessing, this image was subtracted from the image of the cell withparticles, thus making it possible to enhance the image contrast. Then centroidsof the particles were found using a standard algorithm [13] and converted to polarcoordinates r, θ with the center corresponding to the center of rotation of theneutral buoyancy plane (i.e., the axis of rotation of the inner cylinder).

In the experiments described in the following section, the smoothest particles atour disposal were used. Their characteristic roughness is 250 nm, or about 10−4

of the particle radius. In relative terms, this roughness is appreciably lower thanthat of the particles used in the experiments of Arp and Mason [7] and Smart andLeighton [8].

3 Observations and analysis

Here we consider the Couette flow with either two or three particles in the fluid.The initial conditions produced by the template as described in the previous sectionhave the particles nominally separated by 10 in the tangential (θ) direction. Thedifferences in the radial distances are nominally 0.025 of the inner cylinder radiusri between two particles in a pair. For the case of three particles, the initial radialdistance increases by the same amount (0.025 ri) between each particle. Theactual initial conditions may differ by some small amount from the nominal initialconditions set by the template. These deviations develop during the settling ofthe particles to the neutral buoyancy depth. The velocity imparted by the rotationof the Couette cell to the particle closest to the inner cylinder is the highest, soit “catches up” to the next particle. This results in what we refer to as “particleinteraction,” with the particles in close proximity to each other and moving as apair. The resolution of our acquisition system is not sufficient to tell whether themicro-roughness features on the surface of the particles are actually in contact,thus we chose to use the term “interaction” to leave a certain ambiguity.

Figure 2 shows a superposition of images of a two-particle experiment, withthe image on the left corresponding to the forward (clockwise) rotation of theinner cylinder and the image on the right showing the reverse (counterclockwise)part of the cycle. Particle pairs in each sequence are labeled using the angle φof the rotation of the inner cylinder, φ = 0 corresponding to the initial and thefinal conditions. The “trailing” particle in the pair, initially positioned closer tothe rotating cylinder, is marked with a white dot. In the second (counterclockwiserotation) picture, initial positions of the two particles are marked with contours.

From the image sequence, it is clear that the close interaction of the two particlesbrings irreversibility to the flow. The particles do not return to the initial positions.



Figure 2: Forward (left) and reverse (right) parts of an experimental run with twoparticles. One of the particles is marked with a white dot for ease ofidentification (the dot is not physically present during the experiments).Values of the angle of rotation of the inner cylinder φ are shown foreach particle pair. Dashed contours in the right image show the initialpositions of the particles.

Moreover, the interactions themselves take place at different ranges of φ for theforward and reverse parts of the cycle: 3.5 < φ < 6.2 during clockwise rotation ofthe inner cylinder and 5.0 > φ > 2.2 during the counterclockwise rotation. Thisproperty of the interaction becomes even more apparent if the trajectories of theparticles are plotted in the plane of polar coordinates (r, θ). Figure 3 shows suchtrajectories for two realizations of the experiment with different initial conditions,with the plot on the left corresponding to the image sequence of Fig. 2.

Despite the differences in the initial conditions, both experimental realizationsmanifest themselves similarly in the (r, θ) plane. It is also apparent from thetrajectories in Fig. 3 that the center of gravity of the particle pair moves awayfrom the rotating cylinder, in good agreement with earlier experimental results [9].While the particles do not return to their initial positions, their respective positionsare retained (the “trailing” particle marked by the white dot in Fig. 2 returns to its“trailing” place counterclockwise of the unmarked particle).

How does the addition of a third particle change the behaviour of the system?Figure 4 shows a characteristic image sequence constructed in the same way as thesequence of Fig. 2, but with artificial markers now identifying the “middle” andthe “trailing” particle, and the distance from the inner cylinder of the Couette cell



Figure 3: Polar coordinates (r, θ) of two particles during the forward-reverse cycleof the inner cylinder rotation. Plot on the left corresponds to Fig. 2,plot on the right was obtained from different initial conditions (greaterinitial distance from the inner cylinder). Representative error bars arein the upper right corner of each plot. Letters “I” and “F” denote theinitial and final positions of the particles. Radius r is normalized by theinner cylinder radius ri, angle θ is in radians, measured in the clockwisedirection from a horizontal axis passing through the center of rotation.

increasing in the initial conditions from the “trailing” to the “middle” and to the“leading” (unmarked) particle.

There are several appreciable differences between the two- and three-particlecases. First, while the particles in a pair have a close interaction once during theforward and reverse parts of the cycle, particles in a triplet may have a differentnumber of such encounters. In Fig. 4, the forward part of the cycle shows onesuch interaction beginning at φ = 8.3. This interaction causes the “middle” andthe “leading” particles to exchange their order with respect to the direction ofrotation. A similar exchange takes place in Fig. 2. However, during the reversepart of the three-particle experiment, two close approaches happen. The first one,occurring around φ = 8.8, involves the “middle” and the “leading” particle again,with their initial order restoring itself . Then another close encounter happens,this time involving the “middle” and the “trailing” particle around φ = 3.9. Asthe result, these two particles reverse their order in the θ coordinate direction.The initial order of the particles was “leading,” “middle,” “trailing.” In Fig. 4,the final order of the particles is “leading,” “trailing,” “middle.” Without detailedknowledge of the particle trajectories, it would not be possible to tell which oneis which! This behaviour bears an uncanny resemblance to the chaotic “dance”of three sedimenting particles in the original experiments of Jayaweera et al.[14]



Figure 4: Forward (left) and reverse (right) parts of an experimental run with threeparticles. One particle (initially closest to the rotating cylinder) is markedwith a white dot for ease of identification. The particle in the middle ismarked with a “+” sign. The particle initially farthest from the rotatingcylinder is unmarked. The marks are not physically present during theexperiments. Values of the angle of rotation of the inner cylinder φ areshown for each particle triplet. Dashed contours in the right image showthe initial positions of the particles.

and the numerical experiments of Janosi et al. [12]. For additional elucidationof these features, Fig. 5 shows the particle trajectories in the (r, θ) plane for theexperimental run depicted in Fig. 4.

4 Conclusion

Our experimental study shows that the behaviour of a system of three particlesin viscous, stably stratified radial shear flow is appreciably different from thebehaviour of a particle pair under similar conditions. The behaviour of the particlepair is irreversible in the sense that the particles do not return to their initialcondition upon reversal of the flow. This irreversibility is also present in the three-particle case. There also exists a shared trend for the “center of mass” of theparticle system to move away from the inner, rotating cylinder of the Couettecell towards the lower shear rate region of the flow field. However, the three-particle system manifests a greater degree of unpredictability in the sense that,without the knowledge of the particle trajectories, it is impossible to tell which



Figure 5: Polar coordinates (r, θ) of three particles during the forward-reversecycle of the inner cylinder rotation. This plot corresponds to Fig. 4.Refer to the text for the description of the initial positions of the leading,middle, and trailing particle. Radius r is normalized by the inner cylinderradius ri, angle θ is in radians, measured in the clockwise direction froma horizontal axis passing through the center of rotation.

particle corresponds to which when comparing their initial and final positions.The presence of a third particle greatly increases the system sensitivity to smallfluctuations in the initial conditions. These behaviours are observed for fairlysmooth (relative roughness ∼ 10−4) spheres.

This work was partially supported by the US Department of Energy (DOE) grantDE-FG02-05ER25705. The financial support does not constitute an endorsementby the DOE of the views expressed in this paper.



Acknowledgement

References

[1] Phillips, R., Armstrong, R., Brown, R., Graham, A., & Abbott, J., Numericalanalysis of normal stress in non-Newtonian boundary layer flow. Physics ofFluids A, 4(1), pp. 30–40, 1992.

[2] Buyevich, I., Particle distribution in suspension shear flow. ChemicalEngineering Science, 51(4), pp. 635–647, 1995.

[3] Morris, J. & Brady, J.F., Pressure-driven flow of a suspension: Buoyancyeffect. International Journal of Multiphase Flow, 24(1), pp. 105–130, 1998.

[4] Fang, Z., Mammoli, A., Brady, J., Ingber, M., Mondy, L. & Graham, A.,Pressure-driven flow of a suspension: Buoyancy effect. International Journalof Multiphase Flow, 28(1), pp. 137–166, 2002.

[5] Pozarnik, M. & Skerget, L., Boundary element method numerical modelbased on mixture theory of two-phase flow. Computational Methodsin Multiphase Flow II, eds. A. Mammoli & C. Brebbia, WIT Press:Southampton, UK, pp. 3–12, 2003.

[6] Tetlow, N., Graham, A., Ingber, M., Subia, S., Mondy, L. & Altobelli, S.,Particle migration in a couette apparatus: Experiment and modeling. Journalof Rheology, 42(2), pp. 307–327, 1998.

[7] Arp, P. & Mason, S., Kinetics of flowing dispersions. 9. Doublets of rigidspheres (Experimental). Journal of Colloid and Interface Science, 61(1),pp. 44–61, 1977.

[8] Smart, J. & Leighton, D., Measurement of the hydrodynamic surface–roughness of noncolloidal spheres. Physics of Fluids A – Fluid Dynamics,1(1), pp. 52–60, 1989.

[9] Popova, M., Vorobieff, P., Ingber, M. & Graham, A., Interaction of twoparticles in a shear flow. Physical Review E, 2007. Submitted for publication.

[10] Putkaradze, V., Holm, D. & Weidman, P., 2007. Preprint.[11] Wang, Y., Mauri, R. & Acrivos, A., The transverse shear-induced liquid and

particle tracer diffusivities of a dilute suspension of spheres undergoing asimple shear flow. Journal of Fluid Mechanics, 327, pp. 255–272, 19996.

[12] Janosi, I., Tel, T., Wolf, D. & Gallas, J., Chaotic particle dynamics inviscous flows: The three-particle Stokeslet problem. Physical Review E,56(3), pp. 2858–2868, 1997.

[13] Prasad, A., Adrian, R., Landreth, C. & Offutt, P., Effect of resolution on thespeed and accuracy of particle image velocimetry interrogation. Experimentsin Fluids, 13(2–3), pp. 105–116, 1992.

[14] Jayaweera, K., Mason, B. & Slack, G., The behaviour of clusters ofspheres falling through a viscous fluid. Part 1. Experiment. Journal of FluidMechanics, 20(1), pp. 121–128, 1964.



Micropolar fluid flow modelling using the boundary element method

M. Zadravec, M. Hriberšek & L. Škerget University of Maribor, Faculty of Mechanical Engineering, Slovenia

Abstract

Flows in nature are very complex and express different behaviour under different conditions. Therefore we are interested in using proper numerical models to describe the physical behaviour of such fluid flows. The micropolar fluid flow theory enables accurate computation of flows in a scale, where questions arise on the accuracy of the Navier–Stokes equation. In the present paper, the micropolar fluid flow theory is incorporated into the framework of velocity-vorticity formulation of Navier–Stokes equations. Governing equations are derived in differential as well as integral form, resulting from the application of boundary element method (BEM). Keywords: micropolar fluid, boundary element method, numerical modelling.

1 Introduction

Micropolar fluid theory was developed by Eringen [2] forty years ago and has gain attention of researchers in recent years. Lukaszewicz [7] presented in his book mathematical aspects of the theory of micropolar fluids. Many of researchers worked on natural convection of micropolar fluid in rectangular enclosure (Eringen [2], Hsu et al [5], Hsu and Wang [6]). In this work a parametric study of the effect of microstructure on the flow and heat transfer in comparison with Newtonian fluid is undertaken. The results show that dependence of the microrotation term and heat transfer on microstructure parameters is significant.

In the last few years there has been significant progress on micromachining technology. Scientists argue that flows on the microscale are different from that on macroscale, described by classical Navier–Stokes equations. Papautsky et al [9] described microchannel fluid behaviour with a numerical model based on



doi:10.2495/MPF070311

micropolar fluid theory and experimentally verified the model. Results showed that micropolar fluid theory present better agreement with experiment than use of classical Navier–Stokes theory. Applicability of the theory of the micropolar fluids in microchannels depends on the geometrical dimension of the flow field (Pietal [8]).

The micropolar fluid model describes the flow of fluids where the flow behaviour of microstructures affects entire flow. This model is derived from the Navier–Stokes model and takes into account rotation of particles (molecules) independently of the fluid flow and its local vorticity field. Some examples of fluids with microstructures are animal blood carrying deformable particles (platelets), clouds with smoke, suspensions, slurries and liquid crystals.

Among different approximation methods for solving problems of fluid flow BEM is increasingly gaining attention. Here, we will focus on the development of BEM for velocity-vorticity formulation of Navier–Stokes equations Škerget et al [10, 11], Hriberšek et al [12], and show how to incorporate the micropolar fluid theory into the BEM framework.

2 Mathematical formulation

Conservation laws, which define the micropolar flow, are: conservation of mass, eqn. (1), conservation of momentum, eqn. (2), conservation of microinertia eqn. (3) and conservation of energy (4):

(1)

(2)

(3)

(4)

where is ρ density, v velocity vector, N microrotation vector, j microinertia, P modified pressure ( )rgpP ⋅−= ρ , p thermodynamic pressure, g gravity aceleration vector, µ dynamic viscosity, λ second order viscosity coefficient, kV vortex viscosity coefficient, α , β and γ spin gradient viscosity coefficients, f body force per unit mass, s body torque per unit mass, a diffusivity coefficient and DtD /)(⋅ material derivative. If assuming that 0==== γβαVk and

vanishing f and s , microrotation N becomes zero and the eqn. (2) reduces to the classical Navier–Stokes equation.

( ) 0=⋅∇+∂∂ v

tρρ

( ) ( ) fNkvkvkpDt

vDVVV ρµµλρ +×∇+×∇×∇+−∆+++∇−= 2

( ) svkNkNNDt

NDj VV ργγβαρ +×∇+−×∇×∇−∆++= 2

TaDtDT

∆=



If we assume that fluid is incompressible with constant material properties we can rewrite eqn. (1)-(4) in the following form for planar flow:

(5)

(6)

(7)

(8)

In momentum equation (eqn. (6)) buoyancy effect is modelled by the Boussinesq approximation included in body force gFf B= . Function BF can be formulated by equation FB=-βT (T-To), where βT is thermal volume expansion coefficient.

By taking a curl of eqn. (6) we eliminate pressure term and use velocity-vorticity formulation instead of eqn. (5) and (6) with suppose that the vorticity vector representing curl of the velocity field:

(9)

(10)

(11)

(12)

where we assume that Nv ,,ω are solenoidal vectors. Eqn. (9) is elliptic differential equation and represents the kinematics of fluid motion, expressing the compatibility and restriction condition between velocity and vorticity field functions. The kinetic part is governed by parabolic diffusion-convection vorticity transport equation eqn. (10) in which the second term on the right-hand side is an additional term to classical vorticity equation and represents connection between vorticity and microrotation flow field. Eqn. (10) is written for planar flows in which twisting and stretching term is identical to zero ( ) 0=∇⋅ vω . In microrotation conservation equation eqn. (11) appears microinertia j which is defined as the length scale:

02 =×∇+∇ ωv

Nj

kj

kNjx

NvtN VV

j

j

ρω

ρργ 2

−+∆=∂

∂+

∂∂

0=⋅∇ v

( ) gFNkvkPDt

vDBVV ρµρ +×∇+∆++∇−=

NkvkNDt

NDj VV 2−×∇+∆= γρ

TaDtDT

∆=

( ) TaTvtT

xTv

tT

j

j ∆=∇⋅+∂∂

=∂

∂+

∂∂

( )gFNkkx

vt B

VV

j

j ×∇+∆−∆

+=

∂

∂+

∂∂

ρω

ρµωω



( ) ( ) ∫∫∫ΩΓΓ

Ω∂∂

−Γ∂∂

=Γ∂∂

+ dxued

nuved

nuvvc

jij

tjijii

***ωξξ

(13)

Spin gradient viscosity coefficient γ is proposed by Ahmadi [1] in form

(14)

2.1 Integral representations

Advantage of boundary domain integral method originates from the application of Green’s fundamental solutions as particular weighting functions. Different conservation models can be written in the form with an appropriate selection of a linear differential operator L[·] in the following general form

(15) where the operator L[·] can be either elliptic or parabolic, u(rj,t) is an arbitrary field function, and the nonhomogenous term b(rj,t) is applied for non-linear transport effects or pseudo body forces.

2.1.1 Integral representation for flow kinematics Procedure for integral representation for flow kinematics, presented with elliptic Poisson partial differential equation eqn. (9) is given in Škerget et al [10]

(16)

2.1.2 Integral representation for flow kinetics To apply integral representation for flow vorticity, microrotation and energy eqn. (10-12), the non-homogenous velocity field ( )rv must be decomposed into the constant ov and a variable or perturbated part ( )rv

~ , so that the diffusion-convective equation with first order chemical reaction has the form

(17)

where ao and co are constant transport material properties, β is the reactor rate constant, while Io stands for known source term. This equation can be stated as

(18)

where L[·] is linear differential operator, and b stands for pseudo body force term. If we suppose that we know the fundamental solution u*(ξ,s) satisfying

2Lj =

jkV

+=2

µγ

0][ =+ buL

0~

02

=+∂

∂−−

∂∂∂

=∂

∂+

∂∂

oj

j

jjo

j

oj

cI

xuv

uxxua

xuv

tu β

0][2

=+−∂∂

−∂∂

∂=+ bu

xuv

xxuabuL

joj

jjo β



(19)

where L*[·] denotes the adjoint operator to L[·]. By applying Green’s theorem s for scalar field function, the following boundary-domain integral representation can be formulated as

(20) The pseudo body source term b includes the convection for the perturbated velocity field only, source term and initial condition

(21)

Rendering the eqn. (18) by applying Gauss theorem to the domain integral of pseudo body source term we can write

(22)

The fundamental solution of the diffusion-convective equation with first order reaction term is represented as

(23) Parameter β is defined as sum time increment parameter and χ which accounts other reaction terms

(24) The fundamental solution u* and his normal derivative are expressed as Škerget et al [11]

(25)

(26)

( ) 0,][ ** =+ suL ξδ

( ) ( ) Ω+Γ−Γ∂∂

=Γ∂∂

+ ∫∫∫∫ΩΓΓΓ

dubduvudunuad

nuuauc onoo

****

ξξ

tu

cI

xuv

b F

o

o

j

j

∆++

∂

∂−= −1

~

( ) ( )

∫∫

∫∫∫∫

Ω−

Ω

ΩΓΓΓ

Ω∆

+Ω+

Ω∂∂

+Γ−Γ∂∂

=Γ∂∂

+

duut

duIc

dxuuvduvudu

nuad

nuuauc

Foo

jjnoo

*1

*

***

*

11

~ξξ

( )( ) ( )

( )( ) ( ) ( ) 0,,,,2

=+−∂

∂+

∂∂∂ ∗

∗∗

ssusx

suvsxsx

suaj

ojjj

o ξδξβξξ

χβ +∆

=t

1

( )

=∗

o

jojo

oo a

rvrK

aau

2exp

21 ςπ

( ) ( )

−=

∂∂ ∗

o

jojojo

oj

o

jj

j arv

vrKarrrrK

ar

nn

xu

2exp

22

2

12 ςςςπ



where the parameter ς is defined as

(27) Ko and K1 are the modified Bessel function of the second kind, rj(ξ,s) is the vector from the source point ξ to the reference point s, while r is its magnitude r=|rj| and vo

2=vojvoj.

Table 1: Parameters for different conservation equations.

u ao co χ Io

Vorticity ω

+ρ

µ Vk 1 0 ( )gFNkB

V ×∇+∆−ρ

Microrotation N

jρ

γ 1 jkV

ρ2 ω

ρ jkV

Energy T a 0 0 0

Parameters given in Table 1, together with eqn. (22) and eqn (23), give integral forms for flow vorticity eqn. (28), microrotation eqn. (29) and energy eqn. (30).

(28)

(29)

(30)

( ) ( )

Ω∆

+Ω∂∂

−Γ+

+Ω∂∂

∂∂

+Γ

∂∂

−Ω

∂∂

+

+Γ−Γ∂∂

+=Γ

∂∂

++

∫∫∫

∫∫∫

∫∫∫

Ω−

ΩΓ

ΩΓΩ

ΓΓΓ

dut

dxuFgeduFgne

dxN

xukd

nNukd

xuv

duvdun

kdn

ukc

Fi

BjijBjiij

jj

VV

jj

nVV

*1

**

**

*

***

1

~

ω

ρρω

ωωρ

µωρ

µξωξ

( ) ( )

Ω∆

+Ω

+Ω

∂∂

+

+Γ−Γ∂∂

=Γ

∂∂

+

∫∫∫

∫∫∫

Ω−

ΩΩ

ΓΓΓ

duNt

duj

kdxuNv

duvNdunN

jd

nuN

jNc

FV

jj

n

*1

**

***

1~ ωρ

ργ

ργξξ

( ) ( )

Ω∆

+

+Ω∂∂

+Γ−Γ∂∂

=Γ∂∂

+

∫

∫∫∫∫

Ω−

ΩΓΓΓ

duTt

dxuTvduvTdu

nTad

nuTaTc

F

jjn

*1

***

*

1

ξξ

βς +

=

22

2 o

o

av



2.2 Numerical algorithm

The derived integral equations contain several new terms, composed with the classical approach (Škerget et al [10], Hriberšek and Škerget [12]) and which can therefore be easily included into the existing numerical scheme. This scheme is presented in short in figure 1.

Figure 1: Flowchart scheme of micropolar theory extended BEM.

3 Conclusions

The paper presented the derivation of integral equations for numerical simulation of fluid flow with micropolar fluid theory. The derivation showed that the derived equations include several additional terms, compared with the set of

Time loop

F=F+1

Initial vorticity and

initial microrotation F=0

KINEMATIC VORTICITY

Calculating boundary

values ω , vt , vn

Calculating vx , vy

Calculating

ijε

Calculating ω

Vorticity normal derivatives

Vorticity underelaxation

ENERGY

Calculating T

Temperature normal

derivatives

Convergence error

MICROROTATION

Calculating N

Microrotation normal derivatives

Convergence error

Time loop END

C ≥ error

C<error

C ≥ error

F ≥ NT F < NT

C<error



equations from Navier–Stokes equations. The model will be incorporated into the BEM numerical code and tested on several benchmark problems, including flow in microchanels.

References

[1] Ahmadi, G., Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite flat plate. Int. J. Engng. Sci., 14, pp. 639-646 1976.

[2] Eringen, A.C., Theory of Micropolar Fluids. J. Math. Mech., 16, pp. 1-18, 1966.

[3] Eringen, A.C., Microcontinuum Field Theories. II. Fluent media, Springer Verlag, New York, 2001.

[4] Hsu, T.H. & Chen, C.K., Natural convection of micropolar fluids in rectangular enclosure. Int.J.Engng.Sci., 34 (4), pp. 407-415, 1996.

[5] Hsu, T.H., Hsu, P.T. & Tsai, S.Y., Natural convection of micropolar fluids in an enclosure with heat sources. Int. J. Heat and Mass Transfer, 40 (17), pp. 4239-4249, 1997.

[6] Hsu, T.H. & Wang, S.G., Mixed convection of micropolar fluids in a cavity. Int. J. Heat and Mass Transfer, 43 (9), pp. 1563-1572, 2000.

[7] Lukaszewicz, G., Micropolar Fluids: Theory and Application, Birkhäuser, Boston, 1999.

[8] Pietal, A.K., Microchannels flow modelling with the micropolar fluid theory. Bulletin of the Polish Academy of Sciences, 52 (3), pp. 209-214, 2004.

[9] Papautsky, I., Brazzle, J., Ameel, T. & Frazier, A.B., Laminar fluid behaviour in microchannels using micropolar fluid theory. Sensors and actuators, 73 (1-2), pp. 101-108, 1999.

[10] Škerget, L., Hriberšek, M., & Žunič, Z., Natural convection flows in complex cavities by BEM. Int. J. Numer. Methods Heat Fluid Flow, 13 (5/6), pp. 720-736, 2003.

[11] Škerget, L., Hriberšek, M., & Kuhn, G., Computational fluid dynamics by boundary-domain integral method. Int. J. Numer. Met. Engng., 46, pp. 1291-1311, 1999.

[12] Hriberšek, M., & Škerget, L., Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries. Comput. Methods Appl. Mech. Engrg., 194, pp. 4196-4220, 2005.



Numerical modelling of colloidal fluid in a viscous micropump

H. El-Sadi & N. Esmail Department of Mechanical and Industrial Engineering, Concordia University, Quebec, Canada

Abstract

Non-Newtonian fluid can be encountered in many applications of Microdevices. In this study, two-dimensional non-Newtonian simulations of a viscous micropump were performed. The viscous micropump consists of a rotating cylinder located eccentrically inside a microchannel. When the cylinder rotates, a net force is transferred to the fluid due to the unequal shear stresses on the upper and lower surfaces of the cylinder, thus causing the fluid to displace. Non-Newtonian fluid is predicted by Navier Stokes equations and proposed by a modified Bingham model to describe the fluid flow Keywords: micropump, non-Newtonian, Bingham model, eccentricity, bulk velocity, microchannel, Navier Stokes equation, shear stress.

1 Introduction

Increasing efforts are being directed towards applying the technologies of microfluidic, to the development of micro-devices for a wide range of applications such as medical, biological and related technologies. The main advantage of MEMS, in addition to their small size, is the fact that the manufacturing costs are remarkably lower when compared to their bigger counterparts, due to the mass fabrication methods used to produce them. Micropumps are between the most developed of all MEMS devices, and have been executed into the mainstream (Voigt et al [1], Schomburg and Goll [2]). Micropumps are imperative components for distributing fluid and samples in microanalysis system. Positive displacement pumping is the most widespread method used in micropumps, on the other hand the actuation of the reciprocating diaphragm can be achieved by different principles such as piezoelectric,



doi:10.2495/MPF070321

pneumatic, electrostatic etc. (Shoji and Esashi [3], Gravesen et al [4]). However, various pumping ideas were proposed to overcome the valve problem correlated with positive displacement pumps. Our interests in this work are to study the effect of the height of viscous micropump and dynamic parameters on the flow behavior of colloidal liquid in a viscous micropump.

2 Problem description

2.1 Pump geometry

The pump geometry is shown in figure 1. The dimensionless parameters, which are associated to the geometry, are channel height (S) and cylinder eccentricity (ε), defined as:

dhS =

(1)

dhYh c

−−

=)2(

ε (2)

Where d is the cylinder diameter and h is the channel height, Yc the distance between the lower wall of the channel and the center of the cylinder. The cylinder rotates with an angular velocity ω and is placed at different positions between the upper and lower plates of the channel based on the eccentricity. The pressure is exerted on the inlet and outlet of the channel.

2.2 Mathematical equations and boundary conditions

For incompressible and steady state non-Newtonian fluid, the continuity equation and the equation of motion are:

0 =⋅∇ V (3) 0).( =∇+∇+∇ τpVV (4)

with: γητ . ∇−=∇ and

TVV )(∇+∇=γ (5)

where τ is the extra stress tensor, γ the rate of strain tensor, V the tangential velocity and η the fluid viscosity. Since non-Newtonian fluid respond to the imposed flows. A different constitutive equation is needed, material

information).


The rotor eccentricity has a crucial effect on the performance of micro viscous pump. For colloidal fluid, the shear stress increases with increasing the rate of strain. Figures 3 shows the curves τ = ƒ (ε) respectively, where S (channel height) is maintained constant. It is concluded that the pumping performance of colloidal fluid is decreasing with the channel height.

,,( VVf ∇=τ



Y = b + a

d = 2a

b

h

L

ω

Figure 1: Problem geometry and variable velocity distribution.

x

y

Figure 2: An example of mesh generation.

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

0.8 0.9 1 1.1 1.2

ε

P*=0.5

P*=40

P*=150τ (pa)

Figure 3: Changes of the stress as a function of eccentricity. For p*=0.5, 40 and 150.



References

[1] Voigt P. Schrag G. and Wachutka G. “Electrofluidic full-system modeling of a flap valve micropump based on Kirchhoffian network theory, sensors and actuators” A66, 9-14(1998).

[2] Schomburg W. K. and Goll C. “Design optimization of bistable microdiaphragm valves, sensors and actuators” A 64,259-264 (1998).

[3] Shoji S. and Esashi M., “Microflow devices and systems”, J. Micromech. Microeng. 4, 157(1994).

[4] Gravesen P., Branebjerg J. and Jensen O.S., “Microfluidics-a Review”, J. Micromech. Microeng. 3, 168-182 (1993).



Section 6 Turbulent flow

Computational and experimental analyses of a liquid film flowing down a vertical surface

S. Sinkunas1, J. Gylys1 & A. Kiela2 1Department of Thermal and Nuclear Energy, Kaunas University of Technology, Lithuania 2Department of Technical Sciences, Kaunas College, Lithuania

Abstract

The calculations evaluating velocity distribution across water and transformer oil film, and correspondingly the thickness of the film, are performed in this paper. Laminar plane film downward flow, and the film flowing on different convex surfaces of vertical tubes are explored. Equations for the calculation of local velocities in the film, with respect to cross curvature of wetted surface, were established. An evaluation of cross curvature influence on the film thickness is presented. An experimental study of velocity profiles for turbulent liquid film flow in the entrance region is performed as well. Analysis of profiles allowed estimating the length of stabilization for turbulent film flow under different initial velocities. Keywords: velocity distribution, thickness of the film, laminar film, cross curvature, turbulent film, entrance region.

1 Introduction

The determination of hydromechanical parameters of liquid in falling film, emerging from a slit, is an interest in many applications of chemical, environmental and power engineering. The precise prediction of internal or external film flow plays an important role in the design of heat exchangers.

The minimum total energy criteria for determining the minimum wetting rate, and the minimum thickness of an isothermal thin liquid film flowing down a vertical adiabatic surface, was examined in [1]. An analytical expression of the profile of a stable liquid rivulet and of two-dimensional velocity distribution in the rivulet was developed.



doi:10.2495/MPF070331

Instantaneous velocity profiles across a wavy laminar film and axial profiles of thickness were measured simultaneously and statistically analysed in [2]. Experimental data indicated that the time-averaged mean and maximum velocity data are significantly over-predicted by Nusselt’s theory, while the time-averaged film thickness data are slightly under-predicted.

The velocity distribution of the falling film was investigated [3]. The cylindrical model appeared to be more appropriate over the Cartesian model when the film thickness to tube diameter ratio is large. The study showed that the wave characteristics depend on the parameters such as dimensionless wave velocity, tube radius and Reynolds number.

Experimental data on falling film characteristics have been obtained in [4]. These data suggest that the tube diameter strongly affects film flow development, possibly promoting wave interaction and damping. At small liquid Reynolds numbers, critical flooding velocities decrease with increasing liquid rate.

Dimensionless empirical models for the average thickness and radial velocity of wavy films based on thickness measurements on a rotating cone surface were developed [5]. The proposed models express the film thickness and radial velocity as functions of cone geometrical and operating parameters.

Experiments [6] were performed to investigate the flow and surface structure in laminar wavy films over Reynolds numbers from 27 to 200. Measurements of the velocity and film thickness by a fluorescence technique enabled detailed information on the transient conditions within the three-dimensional wavy flow to be gained.

Film velocity and thickness measurements were made in wavy laminar films falling on the exterior of a vertical column [7]. Examination of these conditions revealed that large waves behave as lumps of liquid sliding over a continuous substrate. Velocity fields within substrates displayed reduced sensitivity to the large waves with increases in substrates.

Analytical and experimental study of velocity profiles for laminar and turbulent liquid film flow on vertical surface was the purpose of the present work.

2 Analytical method

2.1 Influence of a wetted surface cross curvature on the thickness of laminar film

Let’s consider a stabilized laminar film flow on the external surface of a vertical tube and on a vertical plane surface respectively. A stabilized gravity film flow is assumed when the film thickness and average velocity do not change in flow direction. In this case the thickness of the boundary layer is equal to the film thickness.

The major contribution to the analysis of falling down laminar film is Nusselt’s solution of the motion equation



012

2

=

−+

ρρ

νgg

dywd (1)

with boundary conditions

,0=w for 0=y ; ,0=dydw for δ=y . (2)

In the case of stabilized gravity film flow often is gρ << ρ , so the member

ρρ g−1 can be ignored. Solving eqn (1), one can obtain parabolic equation of velocity for laminar film

−=δν

δ ygyw 5.01 . (3)

By integrating eqn (3) within the limits from 0 to δ , we obtain the expression of mean velocity for laminar film flow on vertical plane surface

31

22

483

== Reggw ν

νδ . (4)

Multiplication of eqn (4) by δρ leads to the formula defining the thickness of laminar film

31231

433

=

= Re

ggΓ

plν

ρνδ . (5)

Equation of motion for the liquid film in cylindrical coordinates is as follows

012

2

=++νg

drdw

rdrwd . (6)

By working out eqn (6) with the following boundary conditions

,0=w for Rr = ; ,0=drdw for δ+= Rr , (7)

we obtain velocity distribution across the film



( ) ( )222

42Rrg

RrlnRgw −−

+=

ννδ (8)

or

( )

−−+= 15.01

2 2

22

2

Rr

RrlngRw Rεν

. (9)

The mean velocity for film flow on the external tube surface we can express as

∫

∫+

+

=δ

δ

R

R

R

R

rdr

wdr

w (10)

and identify the following relation

( ) ( ) ( )[ ] ( ) 25.0175.0115.014

243

−++−+++

= RRRR

lngRw εεεδεν

. (11)

Mass flow rate of liquid film is defined as ( )RwRfwG ερδπρ 5.012 +== (12) and by substituting eqn (11) for eqn (12), we get

( ) ( )[ ] ( ) 25.0175.0112

244

−++−++= RRR lngRG εεεν

πρ . (13)

Defining Reynolds number for film flow as

ρνπR

GRe2

4= (14)

and substitution of eqn (13) by eqn (14) leads to the following expression

( ) ( )[ ] ( ) 25.0175.011 243

3

−++−++= RRRR

lngRe εεενεδ . (15)



From eqn (15) thickness of the film may be evaluated as follows

( ) ( )[ ] ( ) 3124312

25.0175.011 −−++−++

= RRRR lnRe

gεεενεδ . (16)

By solving a set of eqns (5) and (16), we obtain the expression defining the thickness of the film flowing down a convex surface

( ) ( )( ) ( )[ ] 3124 250175011750 ..ln. RRR

Rpl

−++−++=

εεε

εδδ , (17)

where curvature coefficient which corrects film thickness for cross curvature of the film can be defined as

( ) ( )( ) ( )[ ] 3124 25.0175.01ln175.0 −++−++

=RRR

RRC

εεε

εδ . (18)

Finally, the thickness of laminar gravitational liquid film on vertical convex surface can be calculated as

δνδ RCReg

31

43

= . (19)

Eqn (18) for the calculation of curvature correction coefficient is sufficiently complicated, so the following equation of a simple form has been developed ( ) 13.01 −+= RRC εδ . (20) Practically the case with values or relative cross curvature exceeding 1 is not significant. Therefore, eqn (19) with sufficient accuracy can be simplified

R

pl

εδ

δ3.01+

= . (21)

In engineering calculations the cross curvature of wetted surface is usually known and the thickness of the film flowing down a vertical plane surface one can estimate from eqn (5). Then, the equation describing the film flowing down the convex surface results from eqn (21) ( )12.1167.1 −+= RR plδδ . (22)



By substituting eqn (5) for eqn (22) we obtain

( )

−+= 109.1167.1 31

RGaReRδ . (23)

As it was mentioned above, in practice the relative cross curvature of the film is

Rε < 1, so it is more convenient to estimate the relative cross curvature from the following equation

( )

−+= 109.1167.1 31

RR GaReε (24)

and then the film thickness RRεδ = . (25)

The above-discussed method was applied for the calculations of velocity distribution in transformer oil film. The kinematic viscosity of transformer oil was equal ν=27.8×10-6 m2/s at 20°C temperature. Three values of Reynolds numbers were chosen for the computation. The calculations were performed for the liquid film flowing down a plane and convex vertical surfaces correspondingly with different outside diameters of vertical tubes (3 and 30 mm diameters).

The results of calculations are shown in figure 1. As we can see from figure 1, the influence of cross curvature for the liquid film of high viscosity is noticeable.

Figure 1: Local velocity profiles of transformer oil film flowing down a pane and convex surfaces.

w, m/s

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0 3.0 4.0D=30 mm; Re=100 Plane; Re=100 D=30 mm; Re=500

D=30 mm; Re=1000 Plane; Re=1000

y, mm

Plane; Re=500



3 Experimental method

3.1 Experimental set-up

In order to measure velocity profiles of turbulent film a special device (figure 2) was applied. Velocity profiles in water film were measured by a needle probe, which was 0.3 mm in outside diameter and 0.18 mm in inside diameter. A coordinating mechanism carried out the movement of the needle probe and fixed its zero position with respect to the wall. The needle probe was connected to a vertical glass tube through the flexible tube. Dynamic pressure in the needle probe was determined by the height of the water column in the glass tube. In order to increase the accuracy of readings, a clock-faced indicator was fixed to the coordinating mechanism.

The location of the velocity measuring point in the water film was determined by the following formula dyy n 1.0+= . (26)

Figure 2: Device to measure velocity profiles of water film: 1 – wetted tube; 2 – water film; 3 – needle probe; 4 – coordinating mechanism; 5 – flexible tube; 6 – clock-faced indicator; 7 – scale; 8 – glass tube.

3.2 Velocity profiles in the entrance region of a turbulent film

The entrance region of a film flow is fixed when the film average thickness and average velocity becomes stable. The entrance region can be determined by the function ( )ηϕ f= provided that dynamic velocity ∗v is calculated the regularities of stabilized flow and local velocity of the film is real. In that way,

1

2

3

4

5

6

7

8



defined velocity profiles take up a position above or below profiles of stabilized flow depending on film velocity in the entrance region of film flow. It evidently is seen from figures 3 and 4, where measured profiles in the entrance region of turbulent film flow are presented. As we can see from figures 3 and 4, the initial length of film flow depends on parameter stabd ww=ε and wetting density (Re). In this case the film’s mean velocity in the liquid distributor was calculated as follows ρdd fGw = . (27) The film mean velocity for stabilized flow has been calculated from the following equation ( ) 1253185.1 Regwstab ν= . (28)

Figure 3: Velocity profiles of water film in the entrance region of film flow when Recr < Re < Ret: a) 1 – Re = 5.7·103, ε = 1.46, x = 0.08; 2 – Re = 8.16·103, ε = 1.2, x = 0.08; 3 – Re = 5.84·103, ε = 0.89, x = 0.08; 4 – Re = 4.87·103, ε = 1.48, x = 0.28; 5 – Re = 5.25·103, ε = 0.91, x = 0.28; 6 – theoretical calculation, stabilized flow, Re = 5.05·103; b) 1 – Re = 6.2·103, ε = 1.44, x = 0.48; 2 – Re = 7.76·103, ε = 1.21, x = 0.48; 3 – Re = 6.43·103, ε = 0.87, x = 0.48; 4 – Re = 5.43·103, ε = 1.46, x = 1.07; 5 – Re = 6.19·103, ε = 1.28, x = 1.07; 6 – Re = 5.5·103, ε = 0.90, x = 1.07; 7 – theoretical calculation, stabilized flow, Re = 5.05·103.

6 6 4 2 101 102

2

10

14

6 18

14

10

6

η

ϕ 1

2

3

4

5

1 2 3 4

5 6

6

7

(b)

(a)



Figure 4: Velocity profiles of water film in the entrance region of film flow when Re >Ret : a) 1 – Re = 15.8·103, ε = 1.14, x = 0.08; 2 – Re = 13.2·103, ε = 0.94, x = 0.08; 3 – Re = 16·103, ε = 0.69, x = 0.08; 4 – Re = 41.8·103, ε = 0.78, x = 0.08; 5 – Re = 15.8 ·103, ε = 1.14, x = 0.28; 6 – Re = 13.9·103, ε = 0.72, x = 0.28; 7 – theoretical calculation, stabilized flow, Re = 35·103; b) 1 – Re = 15.9·103, ε = 1.14, x = 0.48; 2 – Re = 16.3·103, ε = 1.0, x = 0.48; 3 – Re = 15·103, ε = 0.69, x = 0.48; 4 – Re = 37·103, ε =0.82, x = 0.48; 5 – Re = 14.6·103, ε = 1.16, x = 1.07; 6 – Re = 13.7·103, ε = 1.04, x = 1.07; 7 – Re = 13.8·103, ε = 0.71, x = 1.07; 8 – Re = 36.8·103, ε = 0.82, x = 0.71; 9 – theoretical calculation, stabilized flow, Re = 35·103.

4 Conclusions

It is evident that under the influence of convex surface curvature the film thickness decreases changing local velocity distribution simultaneously. An equation for the calculation of local velocities in the film with a wetted surface curvature has been established.

Physical properties of the liquid, especially its viscosity, have greater influence on hydromechanical parameters of the film than cross curvature of the wetted surface.

The initial velocity and wetting density has a significant influence on the length of stabilization for turbulent film flow. It is obtained that film flow stabilization takes place at the distance 0.5 m when 6·E3 <=Re <= 7·E3 and at 1.0 m when Re < 4·E4.

1 2 3 4 5 6

(b)

(a)

1

2

3

4

5

7 6

7 8

9

ϕ

14

10

6

2

18

14

10

6

6 101 2 4 6 102 η



Nomenclature

D – tube diameter, m; d – needle probe outside diameter, m; f – cross sectional area of film flow, m2; G – liquid mass flow rate, kg/s; g – acceleration of gravity, m/s2; RGa – Galileo number, 23 νgR ; R – tube external radius, m; r – variable radius, m; Re – Reynolds number of liquid film, ( )ρνΓ4 ; ∗v – dynamic velocity, ( ) 21ρτ w , m/s; w – local velocities of stabilized film, m/s; w – average velocities of stabilized film, m/s; y – distance from wetted surface, m; ny - distance from needle probe centre to the wall; Γ – wetting density, ( )mskg ; δ – liquid film thickness, m; ε – relative film velocity, stabd ww ; Rε – relative cross curvature of the film, Rδ ; η – dimensionless distance from the wetted surface, δy ; ν – kinematic viscosity, m2/s; ρ – liquid density, kg/m3; ϕ – dimensionless film velocity,

∗vw ; wτ – shear stress at the wall, Pa; Subscripts: cr – critical; d – distributor; g – gas or vapour; pl – plane surface; stab – stabilized flow; t – turbulent; w – wetted surface.

References

[1] El-Genk, M.S. & Saber, H.H., Minimum thickness of a flowing down liquid film on a vertical surface. International Journal of Heat and Mass Transfer, 44(15), pp. 2809-2825, 2001.

[2] Moran, K., Inumaru, J. & Kawaji, M., Instantaneous hydrodynamics of a laminar wavy liquid film. International Journal of Multiphase Flow, 28(5), pp. 731-755, 2002.

[3] Kil, S.H., Kim, S.S. & Lee, S.K., Wave characteristics of falling film on a vertical circular tube. International Journal of Refrigeration, 24(6), pp. 500-509, 2001.

[4] Mouza, A.A., Paras, S.V. & Karabelas, A.J., The influence of small tube diameter on falling film and flooding phenomena. International Journal of Multiphase Flow, 28(8), pp. 1311-1331, 2002.

[5] Makarytchev, S.V., Langrish, T.A.G. & Prince, R.G.H., Thickness and velocity of wavy liquid films on rotating conical surfaces. Chemical Engineering Science, 56(1), pp. 77-87, 2001.

[6] Adomeit, P. & Renz, U., Hydrodynamics of three-dimensional waves in laminar falling films. International Journal of Multiphase Flow, 26(7), pp. 1183-1208, 2000.

[7] Mudawar, I. & Houpt, R.A., Measurement of mass and momentum transport in wavy-laminar falling liquid films. International Journal of Heat and Mass Transfer, 36(17), pp. 4151-4162, 1993.



The transition of an in-line vortex to slug flow:correlating pressure and reaction forcemeasurements with high-speed video

B. J. de Witt & R. J. HugoDepartment of Mechanical and Manufacturing Engineering,University of Calgary, Calgary, Alberta, Canada

Abstract

In this work, the performance of a Slug Flow Expander (SFE) was investigated.Multiphase flow consisting of air and water was injected into the SFE. Critical flowrates of air and water were identified that caused transition to the slug flow regime.It was observed that at water flow rates of 1.00 L/s and air flow rates above 3.00 L/s,the flow regime in the SFE transitioned to slug flow. A high-speed digital camerawas used to visualize the flow inside the SFE. Wall pressure and reaction forcemeasurements were synchronized with the high-speed camera. Autocorrelation,cross-correlation, and power spectral density functions were performed on reactionforce, wall pressure, and air-core diameter. Power spectra of reaction force andwall pressure revealed strong energy peaks at low, middle, and high frequencies.Dominant low frequency energy (2.9 Hz and 4.4 Hz) was found to be attributedto the rotation rate of the liquid component in the SFE. The dominant middlefrequency spikes (39.5 Hz, 31.3 Hz, 24.4 Hz for reaction force; 20 Hz, 15.6 Hzfor wall pressure) are believed to be attributed to the formation, convection,and exiting of the liquid slugs from the SFE. It is speculated that the dominanthigh-frequency spikes are attributed to both mechanical vibrations and pressurefluctuations caused by the flow facility pump.Keywords: Slug Flow Expander, high-speed digital video, pressure and reactionforce correlation, transition to slug flow regime.



doi:10.2495/MPF070341

1 Introduction

Many important industrial processes involve simultaneous flow of multiple phasesthat can include liquid, gas, and solids. A two-phase flow pattern called slug flowis encountered in horizontal pipe flow when a concentrated mass of liquid flowsintermittently with a gas phase over certain ranges of flow rates. In a verticallyoriented pipe, slug flow occurs when a sequence of liquid slugs successivelyfollow long bubbles [1]. Understanding slug flow is made difficult by the multi-dimensional fluid dynamic processes that govern it.

An innovative mechanism by which slug flow can be initiated involves aSlug Flow Expander (SFE), shown in Fig. 1(a). Based on previous experimentalresults, slug flow in a vertical section of pipe was initiated by introducing liquidtangentially into a cylindrical cavity, causing a swirl flow to develop [3–5].This results in annular swirl flow, with the central air-core surrounded by aliquid coaxial shell. When the air flow rate is increased, the diameter of the air-core decreases, resulting in the amplification of liquid-gas instability waves thateventually grow to subtend the entire pipe. This is shown in section D-D’ inFig. 1(b). Collapse of the air-core initiates the slug flow regime.

Inlet Liquid

Flow

Inlet Air

Flow

Inlet

Water

Flow

Slug Flow

B

C

B‘

C‘

E‘

Section B - B’

Section C - C’

Section D - D’

Section E - E’

Transition

to

Slug Flow

Gas

Core

Water

Inlet

Water

Flow

Uθ

Gas

Core

C

E

Inlet Air Flow

D D‘

(a) (b)

Figure 1: Slug Flow Expander (SFE): (a) 3-D rendering of SFE, (b) transition tothe slug flow regime.

The central focus of this investigation was to study the transition to the slugflow regime and to investigate flow conditions that impart maximum momentumto the liquid. Flow rates of water and air that maintained a stable slug flow regimewithin the SFE were identified. High-speed video was then used to correlatepressure fluctuations inside the SFE cavity with the reaction force caused byaccelerating liquid slugs on the entire mechanical unit. The paper begins with



a description of the experimental apparatus used to complete the investigation,followed by a section that discusses high-speed video data synchronized withpressure fluctuations inside the SFE cavity and reaction force of liquid slugs. Thenext section performs statistical analysis of reaction force, wall pressure, and air-core diameter data. A conclusions section follows.

2 Experimental apparatus

The experimental apparatus consisted of two main parts: (1) the Slug FlowExpander with associated flow piping and (2) a high-speed digital video camera.

2.1 SFE

Figure 2 depicts the design of the SFE section. It was similar to models usedby previous researchers [2–5, 7]. It included a tangential inlet tube, cylindricalcavity, end caps, and a downstream tube. The inlet tube was constructed of castacrylic tubing. This section introduced water tangentially into the cylindricalcavity, inducing a swirl component. Transparent plexiglass sheets were used forboth the top and bottom end caps of the cylindrical cavity, with a 1

4

′′(6.35 mm)

NPT hole tapped at the center of the top end cap to allow air to flow into thecylindrical cavity. Figure 2 also illustrates a wall pressure port, allowing staticpressure measurements to be made near the inside wall.

Water

Inlet

Acrylic End Cap

Acrylic End Cap

Acrylic Cylindrical Cavity

Downstream Acrylic Tube

Ø 1“

(25.4 mm)

Air

Inlet

Ø 8“

(203.2 mm)

8“

(203.2 mm)Wall pressure port

Load

Cell

4.5“ (114.3 mm)

Figure 2: Schematic of the SFE.

2.2 Instrumentation

There were several measurement devices used in this experiment. The componentsincluded a Photron Ultima 1024 Digital Camera, a pressure transducer thatmonitored wall pressure, and a load cell that monitored reaction force of the entireSFE unit.



2.2.1 High-speed digital cameraA high-speed digital camera (Photron-Ultima 1024) was used to interrogate theflow inside the SFE. Images were acquired at 1000 fps at 512 x 1024 resolution.The analog voltage output from the wall pressure and reaction force transducerswere synchronized with the high-speed camera using a Multi-Channel-Data-Link(MCDL). The A/D conversion resolution of the MCDL was 11 bits, with asampling rate of 10 kHz. A total of 1.024 s of high-speed video and MCDL datawas acquired for each flow condition.

2.2.2 Pressure measurementA Validyne DP15 differential pressure transducer was used to perform pressurefluctuation measurements close to the wall of the cylindrical cavity. The ValidyneDP15 had a frequency response of 1 kHz and featured interchangeable diaphragmsenabling the range of the transducer to be adjusted. The material connecting thetransducer to the SFE was 1

4

′′(6.35 mm) Nyaflow tubing, a semi-rigid plastic

material. This reduced the amount of mechanical vibration transferred to thepressure transducer, while not degrading the pressure signal through the expansionand contraction of the tubing.

2.2.3 Reaction force measurementThe reaction force of the entire SFE unit was monitored using an OMEGADLC101 Dynamic Force Sensor (load cell). During experiments, the load cell waskept in slight compression with a 1

4

′′(6.35 mm) steel bolt. The load cell had a

frequency response from 0.08 Hz to 25 kHz, calibrated by the manufacturer.

3 High-speed camera measurements and results

For the data presented in this work, the water flow rate was fixed at 1.0 L/s andthe air flow rate was varied from 0 L/s to 8.0 L/s. High-speed video data wascollected for water flow rates of 1.0 L/s with varying flow rates of air. Air flowrates varied from 0 L/s, 1.25 L/s, 4.0 L/s, and 8.0 L/s. Additionally, a PC with aPowerDAQ PD2MF 16-333/16H data acquisition card was used to collect 30 sof data (sampling at 1 kHz) for wall pressure, and reaction force fluctuationsfor each component flow rate. This card had a 16-bit A/D converter capableof sampling 16 channels. This additional data set supplemented the 1.024 s ofdata collected using the MCDL, enabling better statistical representation. All datacollected in this investigation were sent through low-pass anti-alias filters set toreject frequency components above 100 Hz. Three different flow regimes wereobserved: (1) annular swirl flow (ASF) regime; (2) transition to slug flow (T)regime; and (3) slug flow (SF) regime.



3.1 Flow regimes observed in SFE

The ASF regime was observed at water flow rates of 1.00 L/s (air flow 0 L/s), asshown in Fig. 3(a). This flow regime was characterized by an air-core surroundedby an annulus of liquid in the larger cylindrical cavity. In the downstream acrylictube, the flow appeared as a liquid film coating the outer edges of the acrylic tube.The high-speed video data set for this water flow rate (1.00 L/s) was comprised of1024 images. Figure 3(a) shows frame 100 out of 1024 images. Reaction force andwall pressure measurements for the 1.024 s of data are also shown.

Figure 3: High-speed video data. (a) ASF, (b) T, and (c) SF regimes.

The T flow regime was observed when injecting air into the SFE at 1.25L/s, as shown in Fig. 3(b). In this flow regime the diameter of the air-core was



observed to vary in size, but did not fully collapse. Reaction force and wallpressure fluctuations are displayed in Fig. 3(b), illustrating the instabilities ofthis flow regime. The time-history for reaction force exhibited a high-frequencyoscillation of approximately 70 Hz (0.014 s), and low-frequency oscillation of 10Hz. The time-history for the wall pressure exhibited a high-frequency oscillationof approximately 50 Hz (0.02 s), and a low-frequency (“beat”) oscillation of 5 Hz(0.2 s). Also included in Figs. 3(a) and 3(b) are the diameters of the air-core foreach frame, and this was measured by importing images into Adobe Photoshop.The number of pixels spanning the air-liquid interface at the axial location of thewall pressure port was calibrated to a known length in the image. In the ASFregime, the air-core diameter remained a constant size. In the T flow regime, thediameter of the air-core was shown to vary in size.

Slug flow, shown in Fig. 3(c) was observed for water flow rates of 1.00 L/scombined with air flow rates above 3.00 L/s. The SF regime was characterizedby large concentrated masses of liquid accelerating through the SFE at regularintervals. The air-core would collapse at regular intervals, and the SFE would shakeviolently due to the accelerating liquid slugs. Figure 3(c) was frame 639 out ofthis data set. Reaction force and wall pressure are shown accompanying Fig. 3(c).These measurements were more pronounced in the SF regime when compared tothe ASF and T flow regimes. The dotted vertical lines designate a region of interestwhere a liquid slug was observed to develop, shown in greater detail within Fig. 4.Figure 4(a) shows 6 images, with wall pressure and reaction force displayed aboveeach frame. Figure 4(b) displays accompanying reaction force, wall pressure andair-core diameter measurements.

The wall pressure of the cylindrical cavity was observed to rise due to thefollowing: (1) localized pressure events triggered by liquid slugs passing by thewall pressure port and (2) large-scale pressure events triggered by liquid slugsexiting from the cylindrical cavity. Frame #349 in Fig. 4(a) illustrates the beginningof the slug formation, marked by the sudden “necking” of the air-core diameternear the top acrylic end-cap. This “necking” was shown to accelerate downward inFrame #355. Wall pressure continued to rise to 38 kPag in Frame #361. Frame#374 featured two events: (1) a liquid slug being purged from the cylindricalcavity into the downstream tube and (2) “necking” near the top acrylic end-cap.The wall pressure dropped to 4.7 kPag in this frame, but increased significantly to94 kPag in Frame #380 as the liquid slug exited the cavity. When examining the 30second time history for wall pressure, the high-frequency pressure “spikes” wereassociated with liquid slugs exiting the cylindrical cavity. Frame #385 illustratesthe liquid slug completely exiting the SFE unit and the “necking” of the air-corediameter near the acrylic end-cap.

The diameter of the air-core for each frame is shown in Fig. 4(b). From the high-speed video, the air-core was found to flatten and stretch in the slug flow regime.There was uncertainty in this measurement because it was assumed the air-corewas cylindrical.



Figure 4: High-speed video data – slug flow (SF) regime: (a) image sequence,(b) reaction force, wall pressure, air-core diameter – frames #340 to #390of data set shown in Fig. 3(c).

3.2 Data analysis

Correlation between reaction force, wall pressure, and air-core diameter isexamined in greater detail in this subsection. The aforementioned signals areanalyzed by computing autocorrelation coefficient , cross-correlation coefficient,and power spectral density functions.

The 30 s of data described at the beginning of Section 3 was used to calculateautocorrelation coefficient functions for reaction force and wall pressure. They areshown in Figs. 5(a) and 5(b). Examining autocorrelation coefficients of reactionforce for air flow rates of 0 L/s, a middle-frequency component of 0.016 s (62.5Hz) was discovered. The wall pressure autocorrelation revealed a high-frequencyoscillation of 0.011 s (91 Hz). For air flow rates of 1.25 L/s, frequency oscillationsof 0.014 s (71 Hz) and 0.012 s (83 Hz) were resolved for reaction force and wallpressure, respectively. For air flow rates of 4.00 L/s frequency components werepresent for reaction force and wall pressure at 0.014 s (71 Hz) and 0.011 s (91 Hz),respectively. Low-frequency (“beat”) oscillations were measured at 0.306 s (3.26Hz) for both reaction force and wall pressure. This “beat” oscillation shifted tolower frequencies as air flow was increased to 8.0 L/s. Reaction force experienceda “beat” of 0.313 s (3.19 Hz), while wall pressure revealed these effects at 0.320 s



(3.125 Hz). This “beat” frequency was attributed to the angular rotation rate of theliquid component in the SFE.

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.4−1

−0.5

0

0.5

1

Water flow 1.00 L/s

Air flow 1.25 L/s

Water flow 1.00 L/s

Air flow 8.0 L/s

Water flow 1.00 L/s

Air flow 4.0 L/s

Water flow 1.00 L/s

Air flow 0 L/s

τ (sec) τ (sec)τ (sec)τ (sec)

(a)

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.40.4−1

−0.5

0

0.5

1

τ (sec) τ (sec)τ (sec)τ (sec)

Water flow 1.00 L/s

Air flow 1.25 L/s

Water flow 1.00 L/s

Air flow 8.0 L/s

Water flow 1.00 L/s

Air flow 4.0 L/s

Water flow 1.00 L/s

Air flow 0 L/s

(b)

Figure 5: Auto-correlation coefficients: (a) reaction force, (b) wall pressure.

Cross-correlation coefficient functions were computed for the following: (1)wall pressure and reaction force; and (2) wall pressure and air-core diameter. Theresults displayed in Fig. 6(a) were for wall pressure and reaction force, using the30 s of data described at the beginning of Section 3. Cross-correlation coefficientfunctions between wall pressure and reaction force were shown to be below 0.4for air flow rates of 0 L/s and 1.25 L/s. As the air flow rate was increased to 4.00L/s, cross-correlation coefficients peaked to 0.1, signifying that the two signalswere somewhat correlated. When air flow rates of 8.00 L/s were injected into theSFE, cross-correlation coefficients were calculated to be 0.4, indicating that wallpressure and reaction force signals were well correlated and in phase. Figure 6(b)shows cross-correlation coefficients for wall pressure and air-core diameter, usingonly the 1.024 s of data synchronized with high-speed video. Cross-correlationcoefficients for air flow rates of 0 L/s and 1.25 L/s signified nearly no correlation.Cross-correlation coefficients for air flow rates of 4.00 L/s and 8.0 L/s show thatwall pressure and air-core diameter are slightly out of phase.

By computing power spectral density (PSD) functions of reaction force and wallpressure signals, dominant spectral components were identified. Power spectraldensity functions of wall pressure and reaction force signals were estimated usingWelch’s method [6], based on a 2048 point FFT using a Hanning window and50% data overlap. Power spectral density function estimates are referred to as“wall pressure spectra” and “reaction force spectra,” respectively. Figure 7 showswall pressure spectra and reaction force spectra. These spectra are shown withincreasing air flow rate, shifted incrementally in the vertical direction. Dominantspectral components at low, middle, and high frequencies are indicated in Figs. 7(a)and 7(b).



−0.04−0.02 0 0.02 0.040.04−1

−0.5

0

0.5

1

−0.04−0.02 0 0.02 0.040.04−1

−0.5

0

0.5

1

−0.04−0.02 0 0.02 0.040.04−1

−0.5

0

0.5

1

−0.04−0.02 0 0.02 0.040.04−1

−0.5

0

0.5

1

Water flow 1.00 L/s

Air flow 1.25 L/s

Water flow1.00 L/s

Air flow 4.0 L/s

Water flow 1.00 L/s

Air flow 8.0 L/s

Water flow 1.00 L/s

Air flow 0 L/s

τ (sec) τ (sec)τ (sec) τ (sec)

(a)

τ (sec) τ (sec)τ (sec) τ (sec)

Water flow 1.00 L/s

Air flow 1.25 L/s

Water flow 1.00 L/s

Air flow 4.0 L/s

Water flow 1.00 L/s

Air flow 8.0 L/s

Water flow 1.00 L/s

Air flow 0 L/s

−0.02 −0.01 0 0.01 0.020.02−1

−0.5

0

0.5

1

−0.02 −0.01 0 0.01 0.02−1

−0.5

0

0.5

1

−0.02 −0.01 0 0.01 0.020.02−1

−0.5

0

0.5

1

−0.02 −0.01 0 0.01 0.020.02−1

−0.5

0

0.5

1

(b)

Figure 6: Cross-correlation coefficients (SF regime): (a) wall pressure and reactionforce, (b) wall pressure and air-core diameter.

100

101

102

−500

−450

−400

−350

−300

−250

−200

−150

−100

−50

0

50

100

150

200

250

300

350

400

450

500

Frequency ( Hz )

Ma

g (

dB

)

Air flow 0 L/s

Air flow 1.25 L/s

Air flow 4.0 L/s

Air flow 8.0 L/s

500

No Water Flow

No Air Flow

2.9 Hz58.6 Hz

73.2 Hz

4.4 Hz

2.9 Hz

24.4 Hz

71.8 Hz

39.5 Hz

31.3 Hz 71.2 Hz

2.9 Hz

8.8 Hz

8.8 Hz

70.8 Hz

100

101

102

−500

−450

−400

−350

−300

−250

−200

−150

−100

−50

0

50

100

150

200

250

300

350

400

450

500

Frequency ( Hz )

Ma

g (

dB

)

Air flow 0 L/s

Air flow 1.25 L/s

Air flow 4.0 L/s

Air flow 8.0 L/s

500

No Water Flow

No Air Flow

2.9 Hz

2.9 Hz

4.4 Hz

2.9 Hz

15.6 Hz 93.8 Hz

20 Hz88.3 Hz

87.4 Hz

58.6 Hz

8.8 Hz90.3 Hz

(a) (b)

Figure 7: Power spectral density functions.

The dominant low frequency spikes were found to occur at identical frequenciesfor both reaction force and wall pressure spectra. As mentioned earlier, byexamining high-speed video with reaction force and wall pressure time-historydata sets, the low frequency energy was found to be attributed to the rotationrate of the liquid component in the SFE. It is believed that the dominant high-frequency spikes (approximately 70 Hz and 90 Hz for reaction force and wallpressure, respectively) are attributed to both mechanical vibrations and pressurefluctuations caused by the liquid pump. This claim is made based on the fact thatthese spikes only appear when the pump is running, and do not seem to changesignificantly with changing air flow rate.



The dominant middle frequency spikes (39.5 Hz, 31.3 Hz, 24.4 Hz for reactionforce; 20 Hz, 15.6 Hz for wall pressure) are believed to be attributed to theformation, convection, and exiting of the liquid slugs in the SFE. This was based onexamining high-speed video with its associated time-history data set in conjunctionwith the 30 second time-history data set taken independent of the high-speedvideo. The middle frequency energy for each air flow rate was found to be slightlydifferent between reaction force and wall pressure. The reason for this is believedto be due to the fact that reaction force is an integral measure of liquid beingaccelerated in the SFE, whereas wall pressure fluctuation is related to eitherpressure build-up in the SFE, or a change in air-core diameter at the location ofthe wall pressure port. Pressure build-up in the SFE is associated with a “necking”of the air-core diameter at the inlet, and with liquid slugs exiting the cylindricalcavity.

4 Future work and conclusions

From this preliminary investigation, it is clear that future work must be done in twoareas. The first will be to collect high-speed video with different geometries of SFEdesign. The second area will include developing a simple analytical model relatingthe wall pressure and air-core pressure to the diameter of the air-core in the SFE.Conditions for the failure of this rudimentary model may connote a transition to theSF regime. In conclusion, an experimental apparatus suitable for analysis of a SFEwas designed in the course of this investigation. The autocorrelation and cross-correlation functions of pressure and reaction force signals aided in understandingthe transition to the slug flow regime. Power spectral density functions of reactionand wall pressure for the SF regime were shown to have energy at low, middle, andhigh frequency components.

References

[1] C.E. Brennen. Fundamentals of Multiphase Flow. Cambridge UniversityPress, 2005.

[2] R.C. Chanaud. Observations of oscillatory motion in certain swirling flows.Journal of Fluid Mechanics, 21:111127, 1965.

[3] R.J. Hugo and C. Veer. An experimental investigation of vortex breakdown inmultiphase pipe flow. AIAA-2003-3602, 2003.

[4] H. Sato and K. Watanabe. Experimental study on the use of a vortex whistleas a flowmeter. IEEE Transactions on Instrumentation and Measurement,49:200205, 1954.

[5] B. Vonnegut. A vortex whistle. Journal of the Acoustical Society of America,26:1820, 1954.

[6] P.D. Welch. The use of fast fourier transform for the estimation of powerspectra: A method based on time averaging over short, modified periodograms.IEEE Trans. Audio Electroacoust., pages 7073, 1967.

[7] Z. Zhang and R. J. Hugo. Stereo particle image velocimetry applied to a vortexpipe flow. Experiments in Fluids, 40, 2006.



A DNS approach to stability study about a supersonic mixing layer flow

F. Guan, Q. Wang, N. Zhu, Z. Li & Q. Shen CAAA (China Academy of Aerospace Aerodynamic), Beijing, China

Abstract

In this paper linear and nonlinear issues of a mixing layer at Mc=1.2 are studied with a DNS method. Navier–Stokes equations in perturbation form are solved with a finite difference method of third-order accuracy. An approximated boundary condition treatment of small disturbance along the outside boundary is proposed on flow characteristics. This boundary condition is verified to be valid in the numerical case. Linear issue of a mixing layer at Mc=1.2 is simulated. Three modes of instability in the mixing layer have been simulated: Slow-Mode, Fast-Mode and Mix-Mode. Nonlinear issues of the mixing layer at Mc=1.2 are also studied. The mode transition of the mixing layer instability is simulated. At low-frequency disturbance, simple mode instability develops in the mixing layer upstream, which is Slow-Mode instability. In the middle of the mixing layer, the Mix-Mode instability develops, and Mix-Mode instability are shifting to Fast-Mode in the downstream. Keywords: mixing layer, DNS, boundary condition, acoustic radiation vortex mod, mode transition.

1 Introduction

Instability of a supersonic mixing layer is a fundamental phenomenon leading to turbulent flow. It is important to study the production and evolution of two-dimensional and three-dimensional unstable structures in a planar supersonic mixing layer to understand the mechanism of transition. It is known that there are several ways leading to the development of three-dimensional structures. Secondary instability is a natural way to make three-dimensional structures and then to transition.



doi:10.2495/MPF070351

DNS (direct numerical simulation) is a useful tool to study the instability of the mixing layer. Boundary condition’s treatment is an important technique for DNS to resolve the compressible N-S equations in the study of supersonic mixing layer instability and transition. Inadequate boundary condition’s treatment will introduce the non-physics fluctuations from the boundary of the flow field, and affect the natural instability of supersonic shear layer. The present work is focused on the secondary instability and the boundary condition’s treatment in DNS of the supersonic mixing layer.

In decades, many studies about shear layer instability have been made. It is well known that the supersonic shear layer instabilities are Kelvin–Helmholtz instabilities from the incompressible flow that is involved with vortex roll-ups and pairing, and which is also a kind of Tollmien-Schlichting wave.

In 1984, W.S. Saric, V.V. Kozlov and V.Y. Levchenko studied the development of three-dimensional disturbance waves in incompressible boundary layers by visualization. They found that the two-dimensional T-S wave grows into three-dimensional structures in evolvement processes and the new three-dimensional perturbation wave is a periodic “λ” vortex in streamwise and lateral direction. The three-dimensional instability waves are cataloged into three types: C-type [1], H-type [2] and K-type [3].

D. Papamochou and A. Roshoko [4] defined the convective Mach number, which is the parameter of the compressibility in the supersonic shear layer and proved that by experiment in 1986. The Mc has been found to be a key parameter to describe the compressibility of the flow.

In 1989, S. K. Lele [5] studied the compressible free shear flow with the DNS method. In his simulations on supersonic mixing layer flows of Mc<1.0, vortex roll-up and pairing dominate the instability process. He also found shock-lets may emerge with vortex structure for Mc>0.7.

In the year 2000, Q. Shen and H. Zhang [6] studied a spatially developed two-dimensional supersonic mixing layer at Mc=0.5 with the DNS method, and found the intermittency takes place during the vortex unstable evolvement. The intermittent structure comes from nonlinear vortex roll-up and paring process that was analyzed by J. R. Luo et al [7]. The Lyapunov index and fractal dimension were calculated, and show a typical nonlinear growth.

In 2006, the secondary instability of a planar supersonic shear layer at Mc=0.5 is simulated. The secondary instability is a nature transition phenomenon by Q. Shen et al [8]. In 2006, F.M. Guan and Q. Shen [9] made a DNS study on secondary instability in a planar supersonic shear lay at Mc=0.5 and found the secondary instability take place along with the T-S wave evolution and the two-dimensional vortex pairing are intermittent in the supersonic shear layer. In 2007, F.M. Guan et al [10] studied the K-type and H-type instability in a supersonic mixing layer, and found the secondary instability could be obviously hampered by vortex pairing.

In 2007, Q. Shen et al [11] had studied the mixing layer at Mc=1.2 on LST. They found that there are three modes of the mixing layer instability, which are Slow-Mode instability led by low frequency disturbance, Fast-Mode instability



led by high frequency disturbance and Mix-Mode led by middle frequency disturbance.

In the present work, a boundary condition’s treatment is studied to approach the supersonic shear layer instability in resolving the perturbation form N-S equations. Instability of a planar two-dimensional supersonic shear layer at Mc=1.2 has been studied with this boundary condition’s treatment, and the acoustic radiation vortex mode has been found. All of the numerical results of the supersonic shear layer instability indicate that the boundary conditions treatment is adequate.

2 Governing equations and numerical method

The compressible two-dimensional unsteady Navier–Stokes equations are:

ηξηξ ∂∂

+∂∂

=∂∂

+∂∂

+∂∂ vv FEFE

tQ ˆˆˆˆˆ

(1)

where

=

evu

JQ

ρρρ

1ˆ ,

( )

+++

=

UpepUvpUu

U

JE

y

x

ξρξρ

ρ1ˆ ,

( )

+++

=

VpepVvpVu

V

JF

y

x

ηρηρ

ρ1ˆ ,

( ) ( )

−++−+++

=

yxyyxxyxxx

yyyxyx

xyyxxxv

qyyvuqvuJ

E

ττξττξτξτξτξτξ

0

Re1ˆ ,

( ) ( )

−++−+++

=

yyyxyyxxyxxx

yyyxyx

xyyxxxv

qvuqvuJ

F

ττηττητητητητη

0

Re1ˆ ,

( ) pvue1

121 22

−++=γ

ρ ,∞

∞∞=µ

ρ LuRe ,

vuU yx ξξ += , vuV yx ηη +=

∂∂

−∂∂

=yv

xu

xx 232 µτ ,

∂∂

−∂∂

=xv

yu

xy µτ ,

∂∂

−∂∂

=xv

yu

yy 232 µτ ,



( ) xT

Mqx ∂

∂−

−=∞ Pr1 2γ

µ, ( ) y

TM

qy ∂∂

−−=

∞ Pr1 2γµ

ργ pMT 2

∞= ,TCCT++

=12

3

µ and∞

=T

KC 4.110.

In the study of supersonic shear layer instability, the flow field is computed in two processes that first one is the study undisturbed flow field computation and the second one is unsteady disturbed flow field computation. Assuming undisturbed flow is steady state, as 0

ˆˆ EE = , 0ˆ FF = , 0

ˆˆvv EE = , vov FF ˆˆ = ,

which satisfy steady Navier–Stokes equations as

ηξηξ ∂∂

+∂∂

=∂∂

+∂∂ 0000

ˆˆˆˆvv FEFE

(2)

where subscript “0” is denoted for steady state i.e. base flow, the flow can be taken as the combination of the base flow and the perturbation flow. Denoting the perturbation variables with prime, then it can be taken that QQQ ′+= ˆˆˆ

0 ,

EEE ′+= ˆˆˆ0 , FFF ′+= ˆˆˆ

0 , vvV EEE ′+= ˆˆˆ0 , vvv FFF ′+= ˆˆˆ

0 , and

ηξηξ ∂′∂

+∂′∂

=∂′∂

+∂′∂

+∂′∂ FEFE

tQ v

ˆˆˆˆˆ (3)

where

( )( )

′

′

′′

=′

evu

JQ

ρρρ

1ˆ ,

( )( ) ( )( ) ( )

( )( ) ( )

′++′+′+′′+′+′+′

′+′+′+′

′+′+′

=′

UpeUUpepUUvUvpUUuUu

UUU

JE

y

x

000

000

000

00

1ˆξρρξρρ

ρρ

,

( )( ) ( )( ) ( )

( )( ) ( )

′++′+′+′′+′+′+′

′+′+′+′

′+′+′

=′

VpeVVpepVVvVvpVVuVu

VVV

JF

y

x

000

000

000

00

1ˆηρρηρρ

ρρ

,

( ) ( ) ( ) ( )

′−′+′+

′−′+′

′+′′+′

=′

yyyxyyxxyxxx

yyyxyx

xyyxxx

v

qvuqvuJ

E

ττξττξ

τξτξτξτξ

0

Re1ˆ ,



( ) ( ) ( ) ( )

′−′+′+

′−′+′

′+′′+′

=′

yyyxyyxxyxxx

yyyxyx

xyyxxx

v

qvuqvuJ

F

ττηττη

τητητητη

0

Re1ˆ ,

( ) ( )[ ] ( ) ( )[ ]12

12221 2

02

0000 −′

+′++′+′+′′++′′+=′γ

ρρ pvvuuvvvuuue

( )

∂∂

−∂∂′+

∂′∂

−∂′∂′+=′

yv

xu

yv

xu

xx00

0 2232 µµµτ ,

( )

∂∂

−∂∂′+

∂′∂

−∂′∂′+=′

xv

yu

xv

yu

xy00

0 µµµτ ,

( )

∂∂

−∂∂′+

∂′∂

−∂′∂′+=′

xu

yv

xu

yv

yy00

0 2232 µµµτ ,

( ) ( )

∂′∂′+

∂′∂′+

−−=′

∞ xT

xT

Mqx µµµ

γ 02 Pr11

,

( ) ( )

∂′∂′+

∂′∂′+

−−=′

∞ yT

yT

Mqy µµµ

γ 02 Pr11

,

′−

′=′

000 ρ

ρppTT , T

TCT′

+

−=′00

01

23µµ ,

( ) ( ) ( )[ ] 00000 xxxxxxxx uuuuu τµττµµτ ′+′+′′+′+=′ ,

( ) ( ) ( )[ ] 00000 xyxyxyxy uuuuu τµττµµτ ′+′+′′+′+=′ ,

( ) ( ) ( )[ ] 00000 xyxyxyxy vvvvv τµττµµτ ′+′+′′+′+=′ ,

( ) ( ) ( )[ ] 00000 yyyyyyyy vvvvv τµττµµτ ′+′+′′+′+=′ ,

Once Q′ is solved from eqn (3), the flow variables can be deciphered as

followings ( )

ρρρρ′+′−′

=′0

0uuu ,( )

ρρρρ′+′−′

=′0

0vvv . Eqn (3) are the governing

equations, which define the perturbation waves travelling within the base flow. These equations are solved using a finite difference method [8] which is the

third-order accuracy in spatial and in temporal direction.



3 Boundary conditions

The boundary conditions treatment in compressible Navier–Stokes equations had been mature. The boundary conditions treatment is according to the direction of disturbance propagation in the compressible Navier–Stokes equations. In two-dimensional unsteady compressible Navier–Stokes equations, there are four

eigenvalues of QEˆˆ

∂∂

in ξ direction which are U== 21 λλ , cU +=3λ ,

cU −=4λ , and four eigenvalues of QFˆˆ

∂∂

in η direction which are

V== 21 λλ , cV +=3λ , cV −=4λ . c is the acoustic speed, ρ

γ pc = .

In the supersonic flow field, the four eigenvalues on stream wise are all positive and one eigenvalue on flow normal direction is positive. There three eigenvalues on stream wise are positive and one eigenvalue on flow normal direction is positive in subsonic flow.

The schematic of a mixing layer is shown in fig. 1. On the inflow boundary, the flow is supersonic and flow variables are constants and equal to incoming flow parameters. In supersonic, extrapolation is usually used to get the flow variables on the outflow boundary. The top and bottom boundaries are free flow conditions. For the undisturbed base flow, 021 === Vλλ ,

03 >+= cVλ and 04 <−= cVλ .

Figure 1: Schematic of a mixing layer.

For the perturbation flow, the disturbance assumes to travel from free boundary to the outside. Outside the mixing layer, the amplitude of the disturbance wave is quite small. These small disturbance waves are assumed as adiabatic. With this assumption u′ , v′ and p′ are obtained by extrapolations. ρ′ is obtained from adiabatic relation as:

pdpd ρ

γρ 1= (4)



By this approximation, the non-reflecting boundary condition is obtained. This relation is to be verified in the next computational case.

4 Acoustic radiation vortex mode in two-dimensional supersonic shear layer at Mc=1.2

A computational case is set which is a supersonic free mixing layer at Mc=1.2. In present computational cases, M1=3.5, M2=1.1 are taken for two flow streams. Other flow conditions are T∞=1200.0.0K for flow temperature, ρ∞=1.0 for non-dimensional flow density, and ReL=1.0E+6 with reference length L.

The undisturbed steady flow field is obtained by resolving the steady Navier–Stokes equation with a second order accurate NND scheme. The base flow used for DNS computation is cut from this flow field. The profile of base flow at x/L=1.0 which gives u, ρ and p by y is shown in fig.2.

u,p,ρ

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.01

-0.0075

-0.005

-0.0025

0

0.0025

0.005

0.0075

0.01 upρ

Figure 2: Steady mean flow profiles at x=1.0.

4.1 Linear simulation

One case is set which is used to simulate linear development of a disturbance wave in the mixing layer. The computation region is 1.0<x<2.5 in stream wise and -0.188<y<0.188 in lateral direction. In the present computation, 205×201 grids in x, y directions are used, shown in fig. 1. A fundamental frequency disturbance wave in lateral velocity is introduced in the central point of the inflow boundary, which is:

( )tv ϖε sin=′ (5)

There, the amplitude of the disturbance wave is 510−=ε , and the circular frequency of the disturbance wave is ϖ . Three circular frequencies have been chosen in present computational case, which are 50,40,20=ϖ . The results are shown in fig. 3.



(a) 0.20=ϖ

(b) 0.40=ϖ

(c) 0.50=ϖ

Figure 3: Contours of perturbation pressure p′ .

In the downstream of the mixing layer, three disturbance waves with different frequency all lead the expansion/compression wave’s structures. This flow structure is a radiation characteristic, and will suppress the rate of mixing layer growth. Especially, three flow structures are obvious different radiation characteristic, that the flow structure led by low frequency disturbance wave radiates in the region of high velocity, the flow structure led by high frequency disturbance wave radiates in the region of low velocity and the flow structure led by middle frequency disturbance wave radiates in both high velocity region and low velocity region. It is indicated that the response of the supersonic mixing layer to disturbance wave depends on the frequency of disturbance wave. We name the flow structure led by low frequency disturbance Slow-Mode, the flow structure led by high frequency disturbance Fast-Mode and the flow structure led by middle frequency disturbance Mix-Mode.

4.2 Nonlinear simulation

The other case is set which is used to simulate the nonlinear development of the disturbance wave in the mixing layer. The computation region is 1.0<x<5.0 in stream wise and -0.25<y<0.25 in lateral direction. In the present computation, 4001×201 grids in x, y directions are used, shown in fig. 1. A fundamental



frequency disturbance wave in lateral velocity is introduced in the central point of the inflow boundary, which is:

( )tv ϖε sin=′ (6)

There, the amplitude of the disturbance wave is 310−=ε , and the circular frequency of the disturbance wave is 30＝ϖ . The results are shown in fig. 4.

Figure 4: Disturbance pressure p′ contours.

In the linear simulation, the low frequency of disturbance wave will lead the Slow-Mode. In the nonlinear computation, the thickness of the mixing layer grows in stream wise, and the response frequency to simple frequency rises in stream wise. So, the low frequency disturbance wave leads the Slow-Mode in inflow region of the mixing layer, the Mix-Mode in the middle region of the mixing layer. In the end of the mixing layer, the flow structure is that which the Fast-Mode dominates, shown in fig. 4.

Otherwise, the far field boundary conditions treatment in the present is verified in this computation, shown in fig. 5. Firstly, the phase of p′ is the same as the phase of ρ′ . Secondly, flow variables of a point arbitrary selected from the mixing layer flow field, which is far from the centre of mixing layer and far from the far free boundary, is studied. In this case, a point studied at (3.145,-0.159) in the flow field is selected and the flow variables on it are:

407.7 −=′ ep , 0258.0=p , 458.8 −=′ eρ , 0.1=ρ

It can be found from these values that 4.1/ =′′ρρ

pp

. This result indicates that

it is an adiabatic process and the boundary condition given by eqn. (4) is valid.

Figure 5: Perturbation pressure and density outside the mixing layer (y=-0.159).



5 Conclusion

An approximated boundary condition treatment of small disturbance outside the boundary is proposed based on flow characteristic.

Linear issues of a mixing layer at Mc=1.2 are simulated on the DNS method. Three modes of instability in the mixing layer are simulated, which are Slow-Mode, Fast-Mode and Mix-Mode.

Nonlinear issues of the mixing layer at Mc=1.2 are also studied. The instability mode transition of the mixing layer is found. At low frequency disturbance, simple instability mode develops in the mixing layer upstream, which is Slow-Mode instability. In the middle of the mixing layer, the Mix-Mode instability develops, and Mix-Mode instability are shifting to Fast-Mode in the downstream.

References

[1] Craik A D D. “Non-liner resonant instability in bound layer,” JFM, Vol. 40, 1971, pp. 393-413.

[2] Herbert Th. “Secondary instability of plane channel flow to subharmonic three-dimensional disturbances,” Physic of Fluid, Vol. 63, 1983, pp. 871-974.

[3] Kachanov Yu S, “On the resonant nature of the breakdown of laminar boundary layer,” JFM, Vol. 184. 1987, pp., 43-74.

[4] D. Papamoschou and A. Roshko, “Observation of supersonic free shear layers,” AIAA Paper 86-0162, 1986.

[5] Lele, S. K., “Direct numerical simulation of compressible free shear flows,” AIAA Paper 89-0374, 1989.

[6] Q. Shen, and H.X. Zhang, “Numerical simulation of intermittent structure in two-dimensional supersonic mixing-layer flow (Mc=0.5),” Proceedings of the Fourth Asia Computational Fluid Dynamics Conference, edited by H. X. Zhang, UEST press, Chengdu (China), 2000, pp. 272-277.

[7] J. R. Luo, Q. Shen, and H. X. Zhang, “Analysis for nonlinear process of two-dimensional supersonic shear flow,” Acta Aerodynamica Sinica, Vol. 20, No. 3, 2002 (in Chinese), pp. 282-288.

[8] Q. Shen, F.G. Zhuang, and F.M. Guan, “Numerical Simulation on a Planar Supersonic Free Shear Layer Secondary Instability”, AIAA Paper 2006-3351, 2006.

[9] Guan F.M., Shen Q., “Direct numerical simulation on secondary instability in a planar supersonic shear lay at Mc=0.5,” The Eleventh Asian Congress of Fluid Mechanics, May, 2006. Malaysia.

[10] Guan F.M., Shen Q., Zhuang F.G., “Three-Dimension numerical simulation on controlled stability of a planar supersonic free shear layer”, AIAA Paper 2007-1311.

[11] Q. Shen, Q. Wang, F.G. Zhuang. “Numerical analysis of acoustic radiation vortical modes in a spatial evolving supersonic plane shear layer”, Chinese Journal of Theoretical and Applied Mechanics, Vol. 39, 2007, pp.7-14.



Hydrodynamic transmission operating with two-phase flow

M. Bărglăzan, C. Velescu, T. Miloş, A. Manea, E. Dobândă & C. Stroiţă “Politehnica” University of Timişoara, Romania

Abstract

Heavy automotive vehicles and automobiles are, almost exclusively, equipped with automatic hydrodynamic transmissions. This article is devoted to one of the possibilities to control the operation of hydrodynamic transmissions through partially filling the torque converter with liquid. The investigation was centred on torque converters with two-phase flows, namely oil–air. There are proposed theoretical, numerical models and an experimental facility, testing rig, was erected.

The obtained results are in the hydrodynamic field (velocities and pressures) in the torus and the characteristic curves of the two-phase flow hydrodynamic transmission. Keywords: hydrodynamic transmission, torque converter, two-phase flow, numerical models, experimental facility, characteristic curves.

1 Introduction

Torque converters are now commonly used in a wide variety of applications requiring smooth torque transmission, most notably in automobiles. They usually consist of an input shaft that drives a pump impeller and one or more added set stator vanes, and between them the turbines stage runners that transmit the torque to an output shaft coaxial with the input shaft.

All this, inside the hydrodynamic circuit, forms a torus filled with fluid, which is usually hydraulic oil. The device is normally equipped with a cooling system to dissipate the generated heat.



doi:10.2495/MPF070361

In the present paper we present a Lysholm-Smith type torque converter and we investigated, both theoretically and with experiments, its behaviour with partial filling of the torus, so we have a two-phase flow oil–air in the hydraulic transmission.

2 Numerical simulation of the flow in the CHC-350 torque converter

The torque converter CHC-350 has the shape and size of the hydraulic circuit (torus) given in figure 1.

r

z

r

z

arc_1

arc_2

arc_3 arc_4

arc_5 arc_6

(a) (b)

Figure 1: Shape and size of the hydraulic circuit of the torque converter: (a) complete section, (b) superior half section versus rotating axis.

The flow in the meridian plane was simulated trough the FEM. It was considered the axial-symmetry and half of the circuit is presented in figure 1 (b). The modality to solve the problem of the hydrodynamic closed circuit proposed two serial domains of flow connected with reciprocal influence between them.

The partial derivate equation of the flow function ψ is a Helmholtz equation of an elliptical type in cylindrical coordinates:

012

2

2

2

=∂∂

−∂∂

+∂∂

rrrzψψψ

(1)

The meridian speed modulo is calculated for every element with the relation:

( ) ( )22 er

ez

em vvv += (2)



The pressure coefficient Cp is given by:

( )2

20

0 1

2

mp vvppCp −=

−==

ρ (3)

where =mv vm / v0.. The flow and equipotential lines are given in figure 2.

(a) (b)

Figure 2: Flow and equipotential lines of the hydrodynamic spectrum: (a) left-side extended domain, (b) right-side extended domain.

Solving this problem, here, was obtained the flow-map presented in figures 3 and 4. Figures 3 and 4 shows the velocities and pressures occurring inside the torus during the rated operation of the torque converter.

3 Experimental investigations

The testing facility of hydraulic transmissions is located in the Laboratory of Hydraulic Machinery from “Politehnica” University of Timişoara (LMHT). The facility is presented in figure 5.



(a)

(b)

Figure 3: Variation of the speeds along flow lines: (a) left domain, (b) right domain.

The facility consists from o testing rig with the following components: • OG – oil group

P – pump OR – oil reservoir CR – cooler radiator



• EM 1, 2, 3 – electrical motors • TC – torque converter • T_t – torque transducer

TB - tensometric bridge • EG 1, 2, 3 – electrical generators • ERG – electrical regenerative group

RE - variable resistor

(a)

(b)

Figure 4: Variation of the pressure coefficient Cp along flow lines: (a) left domain, (b) right domain.



Figure 5: The CHC-350 torque converter testing rig.

Figure 6: Two-phase flow-map after [2].

• FC – frequency converter • Rs_1, Rs_2 – rotational speed transducers

EC – electronic counter Vm – voltmeter

• Tt – temperature transducer • P_t – pressure transducer • C – computer

Knowing the speed of rotation range investigated between 600 and 1200

rev/min- and the filling degree – between 100% and 70 % oil – it was possible to establish the flow regime of the gas/liquid mixture. Introducing the



approximately calculated parameters from the above given data, quantitative results could be introduced in the two-phase flow map given in [2]. The marked zone (domain) in figure 5 was obtained. The first hand conclusion is that in quite all regimes of flow inside the torque converter’s torus it develops a homogeneous disperse flow.

To study the influence of heat developed during the torque converter operation the cooling hydraulic circuit was closed. The temperature rise trough the torque converter operation in different regimes of filling degree and braking capacity was strictly monitored and one of the obtained set of curves is given in figure 7.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50Time [min]

Tem

pera

ture

[deg

ree

C]

Figure 7: Oil temperature versus time inside the CHC-350 torque converter.

The partial conclusion is that it is advisable to measure the mean parameters of the torque converter somewhere between 33 and 50 minutes from the start corresponding to a temperature range of 70–90 degrees Celsius. Testing the CHC-350 torque converter the experiment results were obtained in the shape of plotted primary characteristics, such as figure 8 and the comprehensive universal characteristics sets of curves in figures 9 and 10. The used formulas are:

1

nni 2= (4)

for the ratio of turbine to pump speed and

11

22

1

2MnMn

PP

⋅⋅

==η (5)

the torque converter efficiency (eta) as a function of speeds n1 and n2 and the turbine, M2, and pump, M1, torques. All the measurements were made for different volumetric degrees of filling Хu (chiu).



n1=975 rot/min

0

20

40

60

80

100

120

140

0 100 200 300 400

n2 [rot/min]

M1,

M2

[Nm

], et

a [%

]

M2, Chiu=100%

eta,Chiu=100%

M2, Chiu=97.5%

eta, Chiu=97.5%

M2, Chiu=95%

eta, Chiu=95%

M2, Chiu=92.5%

eta, Chiu=92.5%

M2, Chiu=90%

eta, Chiu=90%

M2, Chiu=87.5%

eta, Chiu=87.5%

M2, Chiu=85%

eta, Chiu=85%

M1

Figure 8: Primary characteristics of the CHC-350 torque converter with constant filling degree.

Chiu=90%

0

10

20

30

40

50

60

70

0 100 200 300 400

n2[rot/min],i=n2/n1[-]

M2[

Nm

]

n1=1200 rot/minn1=1100 rot/minn1=1000 rot/minn1=900 rot/minn1=800 rot/minn1=700 rot/minn1=600 rot/mineta=20%eta=20%eta=17.5%eta=17.5%eta=15%eta=15%eta=12.5%eta=12.5%eta=10%eta=10%eta=7.5%eta=7.5%eta=5%eta=5%

Figure 9: The universal characteristics of the CHC-350 torque converter with constant filling degree.



n1=975 [rot/min]

0

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5

i=n2/n1[-]

M2[

Nm

]

Chiu100%Chiu97.5%Chiu95%Chiu92.5%Chiu90%Chiu87.5%Chiu85%eta63%eta63%eta60%eta60%eta50%eta50%eta 40%eta40%eta30%eta30%eta25%eta25%eta20%eta20%eta15%eta15%eta10%eta10%eta5%eta5%

Figure 10: Universal characteristics of the CHC-350 torque converter with

4 Conclusions

The torque converter type CHC-350 operating with two-phase flow, namely oil–air showed the following features: - A quite linear influence on the performance of the torque converter with different degrees of filling in the range of 80% to 100% oil in the transmission’s torus. - Rated parameter decreasing of the hydraulic transmission with the reducing of filling degree (chiu). - The utilizable degree of filling is between 100% and 70% oil - The primary characteristic curves, figure 8, give the possibility to estimate the torque, speed and efficiency of the transmission. - The universal characteristic set of curves, figure 9 and figure 10, shows the optimum parameters of the torque converter. - The possibility to control the torque converter’s performances through the variable degree of filling. - The two-phase flow structure inside the torque converter torus is very various depending on the torus geometry, rotational speed, degree of filling and fluid pressure.



variable filling degree.

References

[1] M. Bărglăzan, C. Velescu, Cuplaje, transformatoare şi frâne hidrodinamice, ed. Politehnica: Timişoara, pp. 68-72, 2006.

[2] C. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press: Cambridge, pp. 165-181, 2005.

[3] T. Miloş, Computer Aided Design Optimization of the Centrifugal Pump Runner, Proc. of the Workshop on Numerical Simulation of Fluid Mechanics, ed. Orizonturi Universitare: Timişoara, pp. 69-78, 2001.

[4] E, Dobândă, The Dynamic Behaviour of Torque Converters, XIth International Conference “Man in knowledge based organization”: Sibiu, pp. 126-132, 2006.

[5] R. R. By, B. Lakshminarayana, Measurement and Analysis of Static Pressure Field in a Torque Converter Pump, ASME Journal of Fluids Engineering, no. 1, pp.109-115, 1995.

[6] B. A. Gavrilenco, V. A. Minin, Ghidrodinamiceschie mufti, Oboronghiz: Moscva, pp. 116-171, 1959.

[7] N. Peligrad, Cuplaje hidraulice şi convertizoare hidraulice de cuplu, ed. Tehnică: Bucureşti, pp. 246-250, 1985.

[8] A. Bărglăzan, V. Dobândă, Turbotransmisiile hidraulice, ed. Tehnică: Bucureşti, pp. 267-283, 1957.

[9] Iu. F. Ponomarenco, Ispîtanie ghidroperedaci, Izdat. Maşinostroenie: Moscva, pp. 97-116, 1969.



A note on crossing-trajectory effects ingas-particle turbulent flows

B. OesterleLEMTA, Nancy-University, CNRS, ESSTIN, France.

Abstract

In the frame of Lagrangian stochastic dispersion models used to predict thebehaviour of small inertial particles moving in a turbulent flow, crossing-trajectoryeffects are generally accounted for by modifying the integral time scales accordingto the famous analysis of Csanady (J. Atmos. Sci., 20, pp. 201-208, 1963). Here,an alternative theoretical analysis of the time correlation of the fluid velocityfluctuations along a particle trajectory is presented. Analytical expressions of theintegral time scales of the fluid seen by the particles in isotropic turbulence arefirst derived in the asymptotic limit where the mean relative velocity is much largerthan the particle velocity fluctuations, then a correction is proposed to extend theirapplicability over the whole range of mean relative velocity. These expressionsdo not depend on the presumed shape of the two-point fluid velocity correlations.The predicted time scales in the transverse direction differ from some availableproposals in the literature, but are in agreement with other analyses based on anassumed functional form of the turbulence spectrum, at least in the limit of largemean relative velocity. Additionally, some comparisons with numerical predictionsobtained in a synthetic Gaussian turbulence show that the present theoreticalresults are in good agreement with the computations in a large range of meanrelative velocity and particle inertia.Keywords: gas-particle flow, particle dispersion, crossing-trajectory, turbulence.

1 Introduction

As is well known, the behaviour of small inertial particles moving in a turbulentflow can be described by means of Lagrangian stochastic models that consist inbuilding a proper stochastic process to predict the instantaneous velocity of thefluid seen by a discrete particle. The so-called crossing-trajectory effect is observedwhen the fluid and particle mean velocities differ due to some external force



doi:10.2495/MPF070371

field, leading to significant decorrelation of the fluid velocity fluctuations alongthe discrete particle path. Due to continuity requirements, the decorrelation effectis larger in the directions perpendicular to the mean relative velocity. Crossing-trajectory effects are generally accounted for by modifying the integral time scalesaccording to the famous analysis of Csanady [1]. Alternate formulations to expressthe non isotropic decrease of the correlation time scales were derived by Wang andStock [2], Mei et al [3] and Derevich [4].

Here, we present a theoretical analysis of the time correlation of the fluidvelocity fluctuations along a particle trajectory, under the assumption of isotropicturbulence which was also made in the above cited papers. Expressions of theintegral time scales of the fluid seen are derived in the asymptotic limit wherethe mean relative velocity is much larger than the particle velocity fluctuations.A correction is then proposed to make these expressions valid in the wholerange of mean relative velocity, taking the inertia effect into consideration. Theresults are compared with the various available expressions in [1-4] and withnumerical predictions achieved through particle trajectory computations in asynthetic Gaussian turbulence as well.

2 Analysis

The autocovariance tensor of the fluid velocity fluctuations along the discreteparticle path (fluctuations of the fluid seen) is defined by

Q∗ij(x, t; τ) = 〈u′∗

i (t)u′∗j (t + τ)〉 (1)

where u∗(t) = uf (xp(t), t) is the instantaneous velocity of the fluid at theparticle location xp(t), the prime denoting the fluctuating part of the velocity, andx = xp(t). Under the assumption of homogeneous stationary turbulence, wherethere is no dependence on position or time, we will use the shortened notationQ∗

ij(τ) = 〈u′∗i (t)u′∗

j (t + τ)〉 ∀x, t.As pointed out by Derevich [4, 5], Q∗

ij can be expressed in terms of the Euleriantwo-point two-time velocity covariance tensor of the fluid measured in a referenceframe moving with the mean fluid velocity,

Qij (r, τ) = 〈u′i (x, t) u′

j (x + r, t + τ)〉 , (2)

and of the particle transition probability Ψ(r, τ), which is the probability densityfunction (PDF) of a particle having a displacement r during time τ . Theautocovariances of the fluid seen can be written as

Q∗ij(τ) =

∫Qij(r, τ)Ψ(r, τ) dr , (3)

where it is assumed that the particle displacement PDF is independent of theinstantaneous velocity fluctuation of the fluid phase (independence approximation[6]), which means in particular that possible effects of preferential concentrationare not considered here.



In what follows, we are mainly interested by the integral time scales of the fluidseen by the particles in isotropic turbulence, under the effect of a constant meanrelative velocity V between the fluid and the particles due to some external forcefield (equivalently, V is the mean particle velocity in a reference frame movingwith the mean fluid velocity). Provided one axis of the co-ordinate system isaligned with this mean relative velocity, the study can be restricted to the diagonalcomponents of the integral time scale matrix, defined by (no summation overGreek indices)

T ∗αα = 〈u′2〉−1

∫ ∞

0

Q∗αα(τ) dτ , (4)

where it is assumed that there is no bias between the velocity variance of the fluidseen and the Eulerian one 〈u′2〉 (the subscript is omitted according to the assumedisotropy of the fluid turbulence). Without loss of generality, we will set V = V e1.

2.1 Approximate expressions of the space-time correlations

In order to take advantage of eqn. (3), we have first to assess the space-time correlations by means of some expressions which fulfill the asymptoticrequirements for r −→ 0 and τ −→ 0. To this purpose, we know that Csanady[1] suggested that the lines of constant space-time covariance for longitudinalseparation r and time lag τ are ellipses obeying the equation

r2

L2f

+τ2

T 2E

= constant , (5)

where Lf , TE are the longitudinal integral length scale and the Eulerian integraltime scale, respectively, assuming isotropic turbulence. In slight contrast withCsanady’s formulation, here we use TE for sake of consistency with the limitr −→ 0, keeping in mind that TE is the Eulerian time scale measured in areference frame moving with the mean fluid velocity (“moving Eulerian timescale”). Denoting the longitudinal two-point two-time correlation by f(r, τ), i.e.

f(r, τ) = 〈u′2〉−1Qαα(r eα, τ) ∀α (6)

(no summation over α), Csanady’s hypothesis comes down to assume that f(r, τ)is a function of the only variable

ξ(r, τ) =

(r2

L2f

+τ2

T 2E

)1/2

, (7)

that isf(r, τ) = ϕ(ξ). (8)

From eqns. (7)-(8) and from the definition of the integral length and time scales,namely Lf =

∫∞0

f(r, 0)dr and TE =∫∞0

f(0, τ)dτ , it may be noticed that the



function ϕ(ξ) obeys ∫ ∞

0

ϕ(ξ)dξ = 1 , (9)

a property that will be used later.To assess the two-point two-time covariances Qαα(r, τ) for any direction of the

separation vector, we make use of the relationship between the longitudinal andtransverse correlation functions f(r, τ) and g(r, τ) in isotropic turbulence,

g(r, τ) = f(r, τ) +r

2∂f

∂r(r, τ) , (10)

to express the three diagonal components of the space-time covariance tensor asfollows :

Qαα(r, τ) = 〈u′2〉(

(f(r, τ) − g(r, τ))r2α

r2+ g(r, τ)

)

= f(r, τ) +r2−r2

α

2r

∂f

∂r(r, τ) (11)

where rα denotes the component of the separation vector r in direction α, andr2 =

∑α=3α=1 r2

α = riri. According to Csanady’s hypothesis, f(r, τ) is assumed toobey eqn. (8), thus

∂f

∂r(r, τ) =

∂ξ

∂rϕ′(ξ) =

r

L2fξ

ϕ′(ξ) . (12)

From eqns. (11)-(12), the diagonal space-time covariances for any direction ofthe separation vector can finally be written

Qαα(r, τ) = 〈u′2〉(

ϕ(ξ) +r2 − r2

α

2L2fξ

ϕ′(ξ)

). (13)

2.2 Particle transition probability and the asymptotic case V ⟨u′2⟩1/2

The main difficulty in using eqn. (3) on taking particle velocity fluctuationsinto account lies in the dependence of the particle displacement PDF upon thecovariances Q∗

ij(τ), thus making the problem highly non linear. The analyticalresults arising from such approaches, due to Mei et al [3] and later by Derevich[4], can be summarized by the following expressions of the covariances of the fluidseen in the directions parallel (Q∗

11) and perpendicular (Q∗22) to the mean relative

velocity :

Q∗11(τ) =

⟨u′2⟩µ

−1/21 µ2

2 exp[−π

4τ2

T 2E

(1 +

γ2

µ1

)], (14)

Q∗22(τ) =

12Q∗

11

[1 +

µ2

µ1

(1 − π

2τ2

T 2E

γ2

µ1

)], (15)



where

γ =V TE

Lfand µα = 1 +

σ2α(τ)4

. (16)

Unfortunately, eqns. (14)-(15) are clearly unclosed because their right handsides involve the r.m.s. particle displacement σα which depends on Q∗

αα asmentioned just above. Therefore the only means of determining the integral timescales of the fluid seen from such expressions would be through some numericaliterative procedure. As our objective is to find the integral time scales T ∗

11 andT ∗

22 under the form of analytical expressions that can be introduced in stochasticdispersion models, we will first investigate the simple asymptotic limit where themean relative velocity is much larger than the particle velocity fluctuations, i.e.

V ⟨u′2⟩1/2

(or γ 1) and then we will try to extend the obtained expressionsin order that they remain valid in the opposite limit where only the inertia effect ispresent.

In the case γ 1 the r.m.s. particle displacement can be neglected compared tothe mean particle displacement during time τ which is 〈r〉 = Vτ . This assumptionleads to a drastic simplification of the particle displacement PDF, which can beexpressed as

Ψ(r, τ) = δ(r − Vτ) , (17)

and so the corresponding covariance of the fluid seen, hereafter denoted byQ∗

αα(τ), obeys

Q∗αα(τ) =

∫Qαα(r, τ) δ(r − Vτ) dr = Qαα(Vτ, τ) . (18)

Using eqn. (13) to express Qαα(Vτ, τ), the expressions of the time scales T ∗αα

can be obtained by integration (the tilde stands for the case γ 1). In the directionparallel to the mean relative velocity (direction 1), eqns. (13), (18)and (9) yield

T ∗11 =

∫ ∞

0

ϕ(ξ)dτ =TE√1 + γ2

∫ ∞

0

ϕ(ξ)dξ =TE√1 + γ2

(19)

where we used

ξ =

√V 2τ2

L2f

+τ2

T 2E

=τ

TE

√1 + γ2 .

In the directions perpendicular to V (for example α = 2), the continuity effectis taken into account by using eqn. (13) with r = r1 = V τ , r2 = 0 :

Q22(Vτ, τ)=〈u′2〉[ϕ(ξ)+

V 2τ2

2L2fξ

ϕ′(ξ)

]= 〈u′2〉

[ϕ(ξ) +

γ2ξϕ′(ξ)2(1 + γ2)

], (20)

hence, after integration by parts :

T ∗22 =

TE√1 + γ2

[1+

γ2

2(1 + γ2)

∫ ∞

0

ξϕ′(ξ)dξ

]=

TE√1 + γ2

[1− γ2

2(1 + γ2)

]. (21)



It is worth noticing that the derivation of expressions (19) and (21) does notrequire to prescribe the shape of the double velocity correlations of the fluid,except that Csanady’s assumption implies that the longitudinal two-point one-timecorrelation and the one-point two-time correlation have similar shapes.

2.3 Extension to smaller mean relative velocity

Let us introduce the integral time scale of the fluid seen for V = 0, denoted byT ∗

0 , which depends on the Stokes number StE = τP/TE where τ

Pis the particle

relaxation time. As is well known, for tracer particles (StE 1) T ∗0 is equal to

the Lagrangian integral time scale TL, whereas for high inertia particles, such thatStE 1, T ∗

0 tends to the moving Eulerian time scale TE (> TL) : this is the so-called inertia effect [7]. The time scale T ∗

0 can be estimated in terms of the Stokesnumber from the semi-empirical correlation proposed by Wang and Stock [2] orfrom the analytical formula derived by Derevich [5], for example.

The main objective here is to introduce the Stokes number dependence into theevaluation of the integral time scales when crossing trajectory effects are present.Any extension of the analytical expressions of T ∗

αα obtained at large mean relativevelocity must fulfill the condition that T ∗

αα → T ∗0 for γ → 0. In order to meet

this requirement, the simple method suggested here is to replace TE by T ∗0 in the

results derived for large γ, i.e. in eqns. (19)-(21). Therefore the proposed formulaeare

T ∗11 =

T ∗0√

1 + γ∗2 , (22)

T ∗22 =

T ∗0√

1 + γ∗2

[1 − γ∗2

2(1 + γ∗2)

], (23)

where

γ∗ =V T ∗

0

Lf. (24)

3 Discussion

Our expressions of the integral time scales of the fluid seen in the case γ 1, eqns.(19) and (21), are in agreement with the results of Mei et al. [3] and Derevich [4],as can be proved by integrating eqns. (14)-(15) with µ1 → 1, µ2 → 1 (according

to the hypothesis V ⟨u′2⟩1/2

). Let us emphasize, however, that no assumptionhas been made here concerning the velocity correlation functions or the spectrumtensor of the fluid, whereas Mei et al. [3] and Derevich [4] had to prescribe afunctional form of the spectrum tensor (namely the Kraichnan’s model spectrum).Unfortunately, no comparison can be made with these works in the general casewhere the velocity fluctuations are not small compared to V , due to the alreadymentioned unclosedness of the correlation expressions (14)-(15).



Further comparison can be made with the proposals by Csanady [1] and byWang and Stock [2]. Csanady [1] assumed exponential forms of the longitudinaltwo-point correlation f(r, 0) and of the one-point Eulerian time correlationf(0, τ), and considered that the moving Eulerian time scale TE equals theLagrangian time scale TL. Therefore the integral time scale of the fluid seen bya heavy particle for V = 0 is equal to the fluid Lagrangian time scale TL whateverthe particle inertia, and the formula derived by Csanady in the asymptotic caseγ 1 remains valid in the opposite case of very small relative velocity. Thereforethe above expression (22) of the integral time scale of the fluid seen in the directionof the mean relative velocity is in line with Csanady’s analysis.

For the particle dispersion in the lateral directions, Csanady suggested anempirical formula so as to obtain the correct asymptotic behaviour for V −→ ∞,i.e. T ∗

22 = T ∗11/2. Extending his proposal using T ∗

0 as explained in section 2.3(keeping in mind that TE = TL was assumed in Csanady’s work), the Csanady-like formulation of the transverse integral time scales of the fluid seen would be :

T ∗22,C =

T ∗0√

1 + 4γ∗2 . (25)

Later, Wang and Stock [2] developed a theoretical analysis including the effectof particle inertia. Still using exponential functions for f(r, 0) and f(0, τ), they

10-1 100 1010

0.2

0.4

0.6

0.8

1

eqn. (22)St E = 0.036St E = 0.072St E = 0.36St E = 0.72St E = 1.44

T11∗ /T0

∗

γ∗

Figure 1: Integral time scale ratio T ∗11/T ∗

0 (direction of the mean relative velocity)as a function of the parameter γ∗.



10-1 100 1010

0.2

0.4

0.6

0.8

1

eqn. (23)eqn. (25)eqn. (26)St E = 0.036St E = 0.072St E = 0.36St E = 0.72St E = 1.44

T22∗ /T0

∗

γ∗

Figure 2: Integral time scale ratio T ∗22/T ∗

0 (direction perpendicular to the meanrelative velocity) as a function of the parameter γ∗.

extended the idea of Csanady in assuming that the combined influence of τ , T ∗0

and γ∗ in the correlation of the fluid seen Q∗11 can only appear through the variable

(τ/T ∗0 )√

1 + γ∗2. Whereas their result is the same as ours in the direction of V,the following expression was obtained in the transverse directions :

T ∗22,WS =

T ∗0√

1 + γ∗2

[1 − γ∗

2√

1 + γ∗2

]. (26)

To sum up, regarding the decorrelation of the fluid seen in the direction of themean relative velocity it is found that there is no difference between our resultand the available proposals in the literature. In contrast, some discrepancies canbe found in the expressions of the time scale of the fluid seen in the transversedirections, see eqns. (23), (25) and (26). In Fig. 1 and 2, these various formulaeare compared with some numerical predictions issued from particle trajectorycomputations in a synthetic Gaussian turbulence whose properties are as follows(see [8]) : Von Karman spectrum, semi-Gaussian Eulerian time correlation, TE =1.39 and TL = 0.44 (arbitrary units), ReL ≈ 180 (turbulence Reynolds numberbased on integral length scale Lf ). For each value of the Stokes number StE , 105

particles were tracked.As can be observed in Fig. 1 which displays the time scale ratio T ∗

11/T ∗0 (i.e.

in the direction of V) as a function of γ∗, the analytical prediction compares very



well with the numerical simulation whatever the Stokes number. As regards thetime scale of the fluid seen in the transverse directions, comparison of the threeexpressions (23), (25) and (26) is provided by Fig. 2, which shows slight butsignificant discrepancies between the analytical predictions. Equation (23) can beseen to lead to the best agreement with the numerical predictions.

4 Conclusion

The theoretical expressions derived here for the integral time scales of the fluidseen by the particles are in line with the predictions of Mei et al. [3] and Derevich[4] in the limit of large mean relative velocity. These expressions have beenextended to make them consistent with the presence of inertia effect whateverthe mean relative velocity may be. The main result is that the predicted timescales in the transverse direction differ from both proposals by Csanady [1] andby Wang and Stock [2]. Comparison of the various available expressions withsome numerical predictions obtained in a synthetic Gaussian turbulence, in a largerange of mean relative velocity and particle inertia, shows that the transversedecorrelation due to crossing-trajectory effect is better predicted by the expressionarising from the present analysis.

References

[1] Csanady, G.T., Turbulent diffusion of heavy particles in the atmosphere. JAtmos Sci, 20, pp. 201–208, 1963.

[2] Wang, L.P. & Stock, D.E., Dispersion of heavy particles by turbulent motion.J Atmosph Sci, 50, pp. 1897–1913, 1993.

[3] Mei, R., Adrian, R.J. & Hanratty, T.J., Particle dispersion in isotropicturbulence under Stokes drag and Basset force with gravitational settling. JFluid Mech, 225, pp. 481–495, 1991.

[4] Derevich, I.V., Influence of internal turbulent structure on intensity of velocityand temperature fluctuations of particles. Int J Heat & Mass Transfer, 44,pp. 4505–4521, 2001.

[5] Derevich, I.V., Statistical modelling of mass transfer in turbulent two-phasedispersed flows - 1. Model development. Int J Heat & Mass Transfer, 43,pp. 3709–3723, 2000.

[6] Weinstock, J., Lagrangian-Eulerian relation and the independence approxima-tion. Phys Fluids, 19, pp. 1702–1711, 1976.

[7] Reeks, M., On the dispersion of small particles suspended in an isotropicturbulent field. J Fluid Mech, 83, pp. 529–546, 1977.

[8] Thomas, L. & Oesterle, B., An investigation of crossing trajectory effectsin turbulent shear flows. Proc. ASME FED Summer Meeting, FEDSM’05,Houston, Texas, 2005. Paper No FEDSM2005-77303.



Large eddy simulation and the filtered equation of a contaminant

F. Gallerano, L. Melilla & G. Cannata Dipartimento di Idraulica, Trasporti e Strade, Università “La Sapienza”, Roma, Italia

Abstract

A new model of LES is proposed together with a new methodology for the simulation of concentration fields of contaminant coherent with the LES methodology. In this paper a new LES model is proposed. The closure relation for the generalised SGS turbulent stress tensor: a) complies with the principle of turbulent frame indifference; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy. In the proposed model: a) the closure coefficient which appears in the closure relation for the generalised SGS turbulent stress tensor is theoretically and uniquely determined without adopting Germano’s dynamic procedure; b) the generalised SGS turbulent stress tensor is related exclusively to the generalised SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor. The calculation of the viscous dissipation is carried out by integrating its exact balance equation. In this paper the form invariance and frame dependence of the above mentioned equation of the viscous dissipation transport is shown. The concentration field is simulated by the spatially filtered equation of the concentration. In this equation the first-order tensor (produced by the correlation between the velocity and the concentration) is related to the gradient of the resolved concentration according to an original dynamic Germano procedure. Keywords: concentration, contaminant, LES, turbulence.

1 A new LES model

In order to remove the assumption of alignment between the unresolved part of the generalised SGS turbulent stress tensor and the resolved strain rate tensor and



doi:10.2495/MPF070381

to take into account the anisotropy of the unresolved scales of turbulence, the generalised SGS turbulent stress tensor is expressed in the following form

2mmnij ij ijmnL Sτ ν= − (1)

in which mnS is the resolved strain rate tensor and mi jL is the modified Leonard

tensor. The eddy viscosity is expressed in the above-mentioned equation by a fourth-order tensor proportional to the product of a second-order tensor, bij, which represents the turbulence velocity scales, and a second-order tensor, dmn, which represents the turbulence length scales, according to the following equation:

ijmn ij mnCb dν = where ij jib b= , =mn nmd d . (2)

The expression of the eddy viscosity in terms of a fourth-order tensor enables the anisotropic character of the turbulence to be fully represented, since it does not either assume the existence of a single turbulence velocity scale and or a single turbulence length scale, as is found in models in which the viscosity is expressed as a scalar. The second-order tensor which represents the turbulence velocity scales is defined as

=mij

ij mkk

Lb E

L (3)

where E is the SGS turbulent kinetic energy. The second-order tensor dmn is defined as

( )13

1 2 3/= ∆ ∆ ∆ ∆ ∆mn m nd

in which ∆i is the vector of which the components are the filter dimensions in the three coordinate directions. In this manner it is assumed that the anisotropy of the unresolved turbulence velocity scales, expressed by tensor bij, is equal to the anisotropy of the smallest resolved scales, associated with the modified Leonard tensor. This assumption is based on scale similarity, according to which the scales that are contiguous in the wavenumber space have strict dynamic analogies related to the energy exchange processes which occur between them. Introducing eqn (2) and (3) into (1) gives:

1 mij ij( r )Lτ = + where 2 mn mn m

kk

Er Cd SL

= − (4)

In order to close the equations governing the turbulent flows, it is necessary to determine the coefficient C which appears in eqn (4.a) or else, the coefficient r which appears in eqn (4.b). In this paper the coefficient r is determined without Germano’s dynamic procedure, the inconsistencies of which are fundamentally linked to the fact that



in the wall region the dimensions of the filters used in the dynamic procedure are larger than those of the largest eddies governing the energy and momentum transfer. The coefficient r is uniquely and theoretically determined by using the relation between the generalised SGS turbulent kinetic energy and the generalised SGS turbulent stress tensor. In fact, by definition, the generalised SGS turbulent kinetic energy is equal to half the trace of the generalised SGS turbulent stress tensor.

2 1 mkk kkE ( r )Lτ = = + i.e.

2 mkk

mkk

E Lr

L−

= (5)

introducing (5) into (4) gives

2 21m

m mkkij ij ijm m

kk kk

E L EL LL L

τ −

= + =

(6)

The closure relation (6) is obtained without any assumption of local balance between the production and dissipation of generalised SGS turbulent kinetic energy and may thus be considered applicable to LES with the filter width falling into the range of wave numbers greater than the wave number corresponding to the maximum turbulent kinetic energy. The closure relation for the generalised SGS turbulent stress tensor (6): a) complies with the principle of turbulent frame indifference given that it relates only objective tensors [1]; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) assumes scale similarity in the definition of the second-order tensor representing the turbulent velocity scales; d) guarantees an adequate energy drain from the grid scales to the subgrid scales and guarantees backscatter; e) overcomes the inconsistencies linked to the dynamic calculation of the closure coefficient used in the modelling of the generalised SGS turbulent stress tensor. The generalised SGS turbulent kinetic energy, E, is calculated by solving its balance equation, defined by the following equation:

( ) ( )21

2mk k m k k k

mk Ok km m m m m m m

p,u(u ,u ,u ) u u uDE E F ,u ,Dt x x x x x x x

τττ ν τ ντ

∂ ∂ ∂ ∂ ∂∂= − − − + + −

∂ ∂ ∂ ∂ ∂ ∂ ∂ (7)

Equation (7) is form-invariant under Euclidean transformations of the frame and frame-indifferent [5]. The last term on the right of (7) is defined SGS viscous dissipation:

i i

j j

u u,

x xε ντ

∂ ∂= ∂ ∂

(8)

In the proposed LES model a further balance equation is introduced for the subgrid viscous dissipation ε. This equation, expressed in terms of the generalized central moments, takes the form:



εε εν ν τ ν τ

τν τ ν ντ

ν τ

∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ + − + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂

+ − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂+

∂ ∂ ∂

2

2

2 2 2

2

k i i i ik k

k k k k j j k j j

k i ik i k i

k i i k j j k j j

k i i

j k

u u u u uu , , u ,t x x x x x x x x x

pu u u u u, , ,x x x x x x x x x

u u u,x x

ν τ ν τ

τν ν τ ντ

∂ ∂ ∂ ∂ ∂ ∂+ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2 22 0

2 2

2 2 2 0

i k i i i k

j k j j j k j

ik i i i i i

j j k j k j k j j

u u u u u u, ,x x x x x x x

u u u u F, ,x x x x x x x x x

(9)

This equation is form-invariant under Euclidean transformations of the frame but is not frame-indifferent [5]. In the balance equation of the SGS kinetic energy (7) and in the balance equation of the SGS viscous dissipation (9) some tensors are unknown. These unknown tensor are calculated by means of the closure relations. These relations define the above mentioned unknown tensors as a function of known quantities by coefficients. The calculation of these coefficients is carried out adopting the Germano’s dynamic procedure.

2 The spatially filtered concentration equation

The filtered equation of the concentration of suspended solids is

( ) 0ii i

i i

u CC u C u Ct x x

∂∂ ∂+ + − =

∂ ∂ ∂ (10)

where C represents the spatially filtered concentration field. The last term of equation (10) is a first-order tensor given by the generalised central moment relative to the velocity vector and the concentration and is defined as follows

( ) ( )τ= = −i i i ia u ,C uC uC (11)

The above tensor can be split in terms of generalised central moments = + +C C C

i i i ia L C R (12) where

( )τ= = −Ci i i iL u ,C u C u C (13)

( ) ( ) ( )C ' ' ' ' ' 'i i i i i iC u ,C u ,C u C u C u C u Cτ τ= + = − + − (14)

( )τ= = −C ' ' ' ' ' '

i i i iR u ,C u C u C (15)



The sum of the second and third term on the right hand side of eqn (12) represents an unknown first- order tensor. In this paper a new closure relation for this unknown tensor is proposed. The closure relation between this unresolved tensor and the gradient of the filtered concentration is defined by a second-order tensor which is assumed proportional to the subgrid turbulent velocity scale and the subgrid turbulence length scale:

Cij c ijc Edν = (16)

The above closure relation is:

∂+ = +

∂c ci i c in

n

CC R c Edx

(17)

Let ( )iT u ,C be the generalised central moment at the test level related to the velocity vector and the concentration, the following first-order tensor is defined

( )= = − ii i iA T u ,C u C u C (18)

where the symbol ( ). indicates the filter operation at the test level. In terms of generalised central moments iA can be split as:

= + +T T TC C C

i i i iA L C R (19)

where:

( )TCi i iiL T u ,C u C u C= = − (20)

( ) ( ) = + = − + −

TC ' ' ' ' ' 'i i ii i iC T u ,C T u ,C u C u C u C u C (21)

( )TC ' ' ' ' ' 'i i i iR T u ,C u C u C= = − (22)

The sum of the last two terms on the right hand side of eqn (19) is an unknown quantity that is modelled with an expression analogous to equation (17):

∂+ =

∂T TC C T T

i i c inn

CC R c E dx

(23)

The calculation of the coefficient cc is carried out by using the following identity:

i i i iA a u C u C− = − (24)

Equation (24), with the use of the closure relations expressed by eqns (17) and (23), becomes



∂ ∂− = − − + + −

∂ ∂T T

r in r in i i i i i in n

C Cc Ed c E d uC u C u C u C uC u Cx x

(25)

where ET indicates the SGS kinetic energy relative to the test-filter and dTin

indicates the second-order tensor associated with the subgrid turbulence length scales relative to the test-filter. Equation (25) allows the dynamic calculation of the coefficient cc. The numerical integration of eqn (10) for the simulation of the filtered concentration field of suspended solids may be carried out once the boundary conditions have been defined. The plane in proximity to the bottom, which defines the boundary for the concentration field, is placed immediately above the viscous sublayer, inside the buffer layer: the “Reference Concentration”, Cr is imposed on that plane as a boundary condition. This reference concentration is calculated as function of the resolved tangential stress at the bottom, and the critical stress representing the threshold beyond which the movement of solid particles from the bottom is produced. In particular, in the proposed model, the value of local and instantaneous reference concentration, Cr, is related to the resolved velocity field at the bottom by means of the formula proposed by Van Rijn [9]. The aforesaid Van Rijn’s formula reads:

,p

r ,*

D TC ,a D

=

1 5

0 30 015 (26)

in which

a: distance from the bottom at which Cr is calculated

Dp: diameter of the solid particles

/

* p( )g

D Dδν− =

1 31 (27)

δ : relative density of the particle

ν : kinetic viscosity

2 2* *

2*

crit

crit

u uTu−

= (28)

and the quantity 2*u which appears in eqn (28) is linked to the tangential stress at

the bottom, τf , produced at each time step by the tangent resolved velocity component at the bottom, u , by means of the following relation:

2* =

τν

ρ∂

=∂

f uuy

(29)




The proposed model is used to simulate the phenomenon of the re-suspension (from the bottom) of solid particles in a lid-driven cavity. The values of the diameter and the relative density of the solid particles are, respectively, equal to 100 µm and 1.65. The simulation of the lid driven cavity flow is performed at Reynolds number Re = 72,000 which is based on the lid velocity Ulid and the cavity length B. The Ulid is directed towards the growing x axis.

Figure 1: Vortex identification with λ2 method, x-z plane.

In fig. 1 is shown, with the λ2 visualisation method (Joeng & Hussain, [8]), an instantaneous longitudinal section of the turbulent resolved velocity field inside the lid driven cavity at Re = 72,000. The near wall vortex structures (inside the turbulent boundary layer) are clearly identified: the dimensions of the spatial discretisation steps allow the optimal simulation of the above mentioned vortex structures that govern the transport, the production and the dissipation of the turbulent kinetic energy. As can be seen in fig. 1, starting from the top right hand corner and going towards the bottom, vortex structures of small dimensions following the main flow (which moves in a clockwise direction) are formed along the right vertical wall and grow in dimension. In correspondence with the abscissa x = 2, the main clockwise moving vortex structure is at a minimum distance from the bottom, and from there it moves away transporting towards the centre of the domain the small secondary vortexes that form on the bottom. At the same time, in proximity to the bottom right corner, a vortex structure (that moves in a clockwise direction and is confined to the area between the main clockwise vortex and the walls) is formed and destroyed. In fig. 2-5 is shown a sequence of longitudinal sections of instantaneous concentration fields inside the lid driven cavity produced by a re-suspension



from the bottom of the solid particles caused by a turbulent flow characterised by Re = 72,000. In this case the maximum tangential stresses at the bottom are produced in correspondence with the clockwise moving vortex structure and with the appearance of the anti-clockwise vortex structure confined to the bottom right hand corner. The particles, that are re-suspended from the bottom in correspondence with x<2, are transported upwards by the main clockwise vortex rapidly scattered in the whole calculation domain; the particles, that are re-suspended from the bottom in correspondence with x>2, are caught by the anti-clockwise vortex structure confined to the bottom right hand corner where they tend to precipitate. The visualized concentration values, expressed as a percentage of volume fraction, are inclusive among 1x10-4 and 2.4x10-1.

Figure 2: Lid Driven Cavity: filtered concentration field.






4 Conclusions

In this paper a new LES model is presented together with a new methodology for the simulation of concentration fields of solids in suspension which is coherent with the LES methodology. The proposed closure relation for the generalized SGS turbulent stress tensor: a) complies with the principle of turbulent frame indifference; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales. The concentration field is simulated by the spatially filtered equation of the concentration. In this equation the first order tensor (produced by the correlation between velocity and concentration) is related to the gradient of the concentrations resolved according to an original dynamic Germano procedure. By the proposed model the phenomenon of re-suspension, from the bottom, of solid particles in a lid driven cavity at Re = 72.000 is simulated.



References

[1] Hutter, K, Jonk, K, Coontinum methods of Physical of Modelling, Springer, 2004.

[2] Germano M, Piomelli U, Moin P, Cabot WH, A dynamic subgrid scale eddy viscosity model. Phys. Fluids A3, 1760-1765¸ 1991.

[3] Meneveau C, Lund TS, Cabot WH, A Lagrangian dynamic subgrid-scale model of turbulence, J. Fluid Mech. 319, 353-385.

[4] Ghosal S, Lund TS, Moin P, Aksevoll K, A dynamic localisation model for large-eddy simulation of turbulent flows, J. Fluid Mech. 286, 229-255, 1995.

[5] Gallerano F., Pasero E., Cannata G., A dynamic two-equation Sub Grid Scale model. Continuum Mech. Thermodyn. 17, 101-123, 2005.

[6] Chong, M.S. Perry, A.E. Cantwell, B.J, A general classification of three dimensional flow field, Phys. Fluids. A 2, 765, 1990.

[7] Hunt. J.C.R., Vassilicos, J.C. & Kevlahan, N.K.R., ‘Turbulence-. A state of nature or a collection of phenomena?’, in Branover, H. & Unger, Y. (Eds) Progress in turbulence Research, 7th Beer Sheva Int. Sem. on MHD flows and turbulence, Beer eb. 1993, Progress in Astronautics Series, AIAA Sheva, Israel, Feb. 1993.

[8] Joeng, J. Hussain, F, On the identification of a vortex, J.Fluid Mech. 285, 69-94, 1995.

[9] Van Rijn L.C., Sediment transport, Delft Hydraulic Laborator. Vol. 334, 1985.



Author Index

Aguillón O. ............................... 39 Ahmad M................................... 75 Ahn K. H.................................. 215 Alam K. ....................................... 3 Alvarez I. D. ................................ 9 Alvarez J. T.................................. 9 Ames R. G. ................................ 29 Arora R. ................................... 205 Asuaje M.................................... 39 Baicar T. .................................. 249 Bărglăzan M. ........................... 369 Bello O. O.................................. 97 Belmrabet T. ............................ 271 Berthoud G. ............................... 75 Cannata G. ............................... 389 Carneiro J. N. E. ...................... 185 Chung C................................... 215 Cordazzo J. .............................. 133 Couput J. P............................... 195 Dayton P. A. ............................ 261 de Castro J. A........................... 163 de Witt B. J. ............................. 349 Disimile P. J............................. 281 Dobândă E. .............................. 369 Doinikov A. A. ................ 239, 261 Eisenreich N. ............................. 19 Ellis N. ....................................... 51 El-Sadi H. ................................ 333 Esmail N. ................................. 333 Falcone G................................... 97 Fanelli M.................................. 205 Fourmigue J. F. .......................... 75 Francisco A. S.......................... 163 Gajan P. ................................... 195 Gallerano F. ............................. 389 Giudici M................................. 153 Glass A. ................................... 205

Groll R. .................................... 303 Guan F. .................................... 359 Gylys J. .................................... 339 Hamilaki E............................... 173 Hanchi S. ................................. 271 Hernández G. ............................. 39 Hevia B. G. .................................. 9 Hočevar M. .............................. 249 Hribernik A.............................. 293 Hriberšek M..................... 293, 325 Hugo R. J. ................................ 349 Hulsen M. A. ........................... 215 Ingber M. ................................. 315 Kelessidis V. C. ....................... 173 Kenyery F. ................................. 39 Kiela A..................................... 339 Klasinc R. ................................ 249 Lee S. J. ................................... 215 Li Z. ......................................... 359 Litt R........................................ 205 Lougedo S. T. .............................. 9 Mahinpey N. .............................. 51 Maliska C. R. ........................... 133 Manea A. ................................. 369 Mao D........................................ 61 Martino R................................... 87 Melilla L. ................................. 389 Mercier P. .................................. 75 Miloş T. ................................... 369 Miloshevich H. .......................... 19 Misychenko N. I. ..................... 239 Mls J. ............................... 115, 125 Mulas M................................... 271 Murphy M. J. ............................. 29 Nieckele A. O. ......................... 185 Oesterlé B. ............................... 379


Papa M. N. ................................. 87 Pasadakis N.............................. 173 Pasic H......................................... 3 Polák M.................................... 125 Popova M................................. 315 Pyles J. M. ............................... 281 Qiu D. ...................................... 205 Repouskou E............................ 173 Rudak L. V. ............................. 239 Russo R.................................... 271 Rychkov A................................. 19 Salque G. ................................. 195 Sarno L. ..................................... 87 Shen Q. .................................... 359 Shokin Yu. ................................. 19 Silva A. F. C. ........................... 133 Silva L...................................... 205 Singh B. B................................ 143 Sinkunas S. .............................. 339 Širok B..................................... 249 Škerget L.................................. 325 Stroiţă C................................... 369 Strzelecki A. ............................ 195

Telenta M..................................... 3 Teodoriu C................................. 97 Teterev A. V. ........................... 239 Tirtowidjojo M. ......................... 61 Tonkovich A. L........................ 205 Toy N....................................... 281 Tremante A. ............................... 39 Tsamantaki C........................... 173 Vassena C. ............................... 153 Vejahati F. ................................. 51 Velescu C................................. 369 Vidal A. ..................................... 39 Vorobieff P. ............................. 315 Wang Q.................................... 359 Weidert D. ............................... 205 Weiser V.................................... 19 Yan Y. Y.................................. 227 Zadravec M.............................. 325 Zajdela B. ................................ 293 Zhu N....................................... 359 Zu Y. Q.................................... 227


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Fluid StructureInteraction andMoving BoundaryProblems IVEdited by: S.K CHAKRABARTI, OffshoreStructure Analysis Inc. USA, C.A.BREBBIA, Wessex Institute of Technology,UK, J.D. BAUM, Science ApplicationsInternational Corporation, USA and M.GILTRUD, Defence Threat ReductionAgency, USA

Publishing papers presented at the FourthInternational Conference on Fluid StructureInteraction, this book features contributionsfrom experts specialising in this field on newideas and the latest techniques. A valuableaddition to this successful series and will beof great interest to mechanical and structuralengineers, offshore engineers, earthquakeengineers, naval engineers and any otherexperts invlovedin topics related to fluidstructure interaction. Topics covered include:Hydrodynamic Forces; Response ofStructures including Fluid Dynamics;Offshore Structure and Ship Dynamics; FluidPipeline Interactions; Structure response toserve Shock and Blast Loading; VortexShedding and Flow Induced Vibrations;Cavitation Effects in Turbo Machines andPumps; Wind effects on bridges and tallstructures; Mechanics of Cables, Risers; andMoorings; Biofluids and biological tissueinteraction; Computational Methods;Advances in Interaction problems in CFD;Experimental Studies and Validation;Vibrations and Noise; Free Surface Flowsand Moving Boundary Problems.

WIT Transactions on the BuiltEnvironment, Vol 92

ISBN: 978-1-84564-072-9 2007 368pp£125.00/US$245.00/€187.50

Solitary Wavesin FluidsEdited by: R.H.J. GRIMSHAW,Loughborough University, UK

After the initial observation by John ScottRussell of a solitary wave in a canal, hisinsightful laboratory experiments and thesubsequent theoretical work of Boussinesq,Rayleigh and Korteweg and de Vries, interestin solitary waves in fluids lapsed until themid 1960’s with the seminal paper ofZabusky and Kruskal describing thediscovery of the soliton. This was followedby the rapid development of the theory ofsolitons and integrable systems. At the sametime came the realization that solitary wavesoccur naturally in many physical systems,and play a fundamental role in manycircumstances.The aim of this text is to describe the rolethat soliton theory plays in fluids in severalcontexts. After an historical introduction, thebook is divided into five chapters coveringthe basic theory of the Korteweg-de Vriesequation, and the subsequent application tofree-surface solitary waves in water, internalsolitary waves in coastal ocean and theatmospheric boundary layer, solitary wavesin rotating flows, and to planetary solitarywaves with applications to the ocean andatmosphere; The remaining chapterexamines the theory and application ofenvelope solitary waves and the nonlinearSchrödinger equation to water waves.

Series: Advances in Fluid Mechanics,Vol 47

ISBN: 978-1-84564-157-3 2007 208pp£70.00/US$130.00/€105.00

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Atmosphere-OceanInteractionsVolume 2Edited by: W. PERRIE, Bedford Instituteof Oceanography, Canada

The recent increase in levels of populationand human development in coastal areashas led to a greater importance ofunderstanding atmosphere-oceaninteractions. Human activities that dependon the oceans require improvements inoperational forecasts for marine weather andocean conditions, and associated marineclimate. This second volume onatmosphere-ocean interactions aims topresent several of the key mechanisms thatare important for the development of marinestorms.The book consists of eight chapters, eachpresenting separate topics that arepredominantly self-contained. The first fivechapters are concerned with marineobservations and understanding theirparameterizations as they relate toatmosphere-ocean systems. The subsequentthree chapters consider some of theimplications of these parameterizations, asrelated to applications in coupledatmosphere, ocean, and wave modelsystems.


ISBN: 1-85312-929-1 2006 240pp£79.00/US$142.00/€118.50

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Vorticity andTurbulence Effects inFluid StructureInteractionAn Application to HydraulicStructure DesignEdited by: M. BROCCHINI, University ofGenoa, Italy and F. TRIVELLATO,University of Trento, Italy

This book contains a collection of 11research and review papers devoted to thetopic of fluid-structure interaction.The subject matter is divided into chapterscovering a wide spectrum of recognizedareas of research, such as: wall boundedturbulence; quasi 2-D turbulence; canopyturbulence; large eddy simulation; lakehydrodynamics; hydraulic hysteresis; liquidimpacts; flow induced vibrations; sloshingflows; transient pipe flow and airentrainment in dropshaft.The purpose of each chapter is to summarizethe main results obtained by the individualresearch unit through a year-long activityon a specific issue of the above list. Themain feature of the book is to bring state ofthe art research on fluid structure interactionto the attention of the broad internationalcommunity.This book is primarily aimed at fluidmechanics scientists, but it will also be ofvalue to postgraduate students andpractitioners in the field of fluid structureinteraction.


ISBN: 1-84564-052-7 2006 304pp£98.00/US$178.00/€147.00

computational methods in multiphase flow iv

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