multifield computational fluid dynamics model of particulate flow in curved circular tubes

Digital Object Identifier (DOI) 10.1007/s00162-004-0127-3Theoret. Comput. Fluid Dynamics (2004) 18: 205–220

Theoretical and ComputationalFluid Dynamics

Original article

Multifield computational fluid dynamics model of particulate flowin curved circular tubes

Prashant Tiwari1, Steven P. Antal1,3, Andrea Burgoyne4, Georges Belfort2,3, Michael Z. Podowski1,3

1 Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA2 Howard P. Isermann Dept. of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA3 Center for Multiphase Research, Rensselaer Polytechnic Institute, Troy, NY 12180, USA4 Dept. of Chemical and Process Engineering, The University of Sheffield, Western Bank, Sheffield, S10 2TN, UK

Received December 19, 2003 / Accepted May 25, 2004Published online October 19, 2004 – Springer-Verlag 2004Communicated by H.J.S. Fernando

Abstract. Limitations of mass transfer resulting from non-optimized fluid mechanics canseverely affect the performance of synthetic membrane filtration systems. To improve mem-brane efficiency, modern applications of this technology have extensively used curved mem-brane ducts that take advantage of Dean vortices (i.e., curvature-induced secondary flows) tominimize membrane fouling. This paper is concerned with a complete three-dimensional an-alysis of single-phase and two-phase particle/liquid flows around a curved membrane tube.The proposed multidimensional model was implemented in an advanced (next-generation)multiphase computational fluid dynamics (CFD) solver, NPHASE. The results of simulationshave been validated against experimental data and compared against other findings available inthe literature. The consistency and accuracy of the present approach have been demonstrated.The novel aspects of this work include: the demonstration that azimuthal vortices may bi-furcate at Dean numbers lower than previously anticipated, the use of vorticity magnitude asa measure of vortex strength, and the explanation of the role that Dean vortices play to mitigatethe effect of gravity on particle settling. The overall results have direct relevance to syntheticmembrane fouling during filtration of particle suspensions.

Key words: membrane filtration, particle/liquid system, two-fluid model, secondary flows,Dean vortices, vortex bifurcation

Nomenclature

Symbol Definition

A′′′ Interfacial area density (m−1)A Cross-section area for flow (m2)C Model coefficientD Diameter (m)

Correspondence to: M.Z. Podowski (e-mail: [email protected])

206 P. Tiwari et al.

De Dean numberMS

k Shear force per unit volume on field-k (kg/m2/sec2)Fk Total interfacial force per unit volume on field-k (kg/m2/sec2)g Acceleration of gravity vector (m/sec2)p Pressure (kg/m/sec2)R Radius (m)Re Reynolds numbert Time (sec)v̄k Velocity vector of field-k (m/sec)

Greekα Volume fractionρ Fluid density (kg/m3)τ Shear stress (kg/m/sec2)ν Kinematic viscosity (kg/m/sec)δ Radii ratio (Rt/Rc)ω Vorticity vector (sec−1)

Subscriptc Continuous phase; quantity defined on tube’s curved sectiond Dispersed phaset Quantity defined on tube’s curved section

1 Introduction

Limitations of mass transfer resulting from non-optimized fluid mechanics can severely affect the per-formance of synthetic membrane filtration [1–3]. In most modern applications of this technology, the feedsolution, or suspension, flows across the surface of the membrane (in tangential orientation) rather thandirectly toward the membrane. Increased reentrainment of retained dissolved and suspended matter that col-lects on the membrane, back toward the bulk solution, is considered the main advantage of tangential flowfiltration. As new and more demanding applications are sought, this approach has proved inadequate andmore effective methods have been invoked. These include [4–7]: using faster axial flow rates, rotational de-vices in which the membranes are mechanically moved relative to the feed, pulsating axial flow, periodicflow reversal, and the benefits of secondary flow. Two of the most effective depolarizing methods have beenthe use of well-ordered Taylor vortices established in a rotating annular filter module [8, 9] and of multipleDean vortices resulting from flow around a curved duct [5, 10]. In particular, it has been demonstrated [11]that Dean vortices considerably enhance mass transfer and solution mixing.

Although the qualitative aspects of behavior of suspensions during membrane filtration has been de-scribed extensively before [12], only recently have attempts been made to investigate this problem analyt-ically [13]. Numerical analyses of fluid flow in membrane tubes that have been reported in the literature, areprimarily for either single-phase liquids [7] or for particle/liquid flows in straight ducts [14].

The objective of this paper was to analyze complete three-dimensional single-phase and two-phase(dispersed particles suspended in a continuous liquid) flows around a curved membrane tube. First, the devel-opment is discussed of a consistent numerical method of tracking details of a multidimensional single-phaseliquid flow, including the formation and evolution of Dean vortices, in curved U-bend type geometries, andthe accuracy of results is demonstrated. Next, the liquid-only results are extended to liquid/particle two-phase flows in similar geometrical configurations, and the effect of geometry and flow conditions on localparticle concentration is investigated.

Dean vortices reflect centrifugal instabilities that occur when a viscous fluid flows in a curved duct [15–17]. It has been shown by Berger [18] that two nondimensional parameters characterize such flows, namely

δ = Rt

Rc(1)

Multifield computational fluid dynamics model of particulate flow in curved circular tubes 207

Re = vDt

ν(2)

where, Rt is the tube inner radius, Dt = 2Rt is the tube inner diameter, Rc is the radius of curvature of thecurved/coiled tube, Re is the Reynolds number, v is the average flow velocity and ν is the kinematic viscosityof the fluid.

Combining Eqs. (1) and (2) yields a nondimensional similarity number, the Dean number

De = Re√

δ (3)

The Dean number represents the ratio of the square root of the product of the inertia and centrifugal forces tothe viscous force. Since the secondary flows are induced by the centrifugal force, and their interactions areprimarily with the viscous force, De is a measure of the magnitude of the secondary flow.

The effect of Dean vortices has been analyzed before, but for single-phase flows only. Since most indus-trial filtration operations involve particle/liquid slurries, it is important to understand the effect of geometryof curved tubes and channels on particle distribution across the flow and, in particular, near the membranewall.

In the present work, the governing equations for both single-phase and two-phase flows have been solvedusing a multifield finite-volume-based solver, NPHASE [19, 20], on a Linux platform using an Intel P-IIIworkstation. These equations were numerically discretized using optimized unstructured grids, and appliedto analyze flow patterns in complex geometries.

2 Multidimensional CFD model of particle/liquid flow

2.1 Governing equations

In the multifield ensemble-averaged modeling approch, the governing conservation equations for individ-ual fields are determined with respect to common physical and computational domains, and include variousinterfacial effects between the fields. Specifically, the mass and momentum conservation equations (for adia-batic particle/liquid flows, the energy equation can be ignored), respectively, for each component field ofdispersed particulate flows can be written as [21, 22].

Mass

∂(αkρk)

∂t+∇ · (αkρkvk) = 0 (4)

Momentum

∂(αkρkvk)

∂t+∇ · (αkρkvkvk) = −αk∇ pk +MS

k +αkρkg+Fk (5)

where k is the field indicator, αk is the volumetric concentration of field-k, MSk is the shear force per unit vol-

ume on field-k, Fk is the total interfacial force (also per unit volume) on field-k, and the remaining notation isconventional. For a typical dispersed-particle/liquid two-field model, the index, k, includes: c – continuousliquid field, and d – dispersed particle field.

Several additional conditions, or closure laws, are needed to complete the model given by Eqs. (4)–(5).The form of individual interfacial closure laws depends on the specific physical phenomena. The consistencyand accuracy of the multifield model’s predictions strongly depend on the degree to which these closure laws,determined in terms of ensemble-averaged state variables, are capable of capturing the dominant local massand momentum transfer phenomena. This is particularly important since the averaging procedure normallyintroduces several constraints on the formulation of individual models.


For dispersed (with multiple-size particles) flows, the total interfacial force is given as a superposition ofseveral terms describing various modes of interfacial effects [21, 22]

Fk = FDk +FVM

k +FLk + . . . (6)

where, FDk is the drag force, FVM

k is the virtual mass force, FLk is the lift force, all on field-k.

In the case of dilute dispersed slurry flows containing particles of a fixed size, using a two- field modelproperly represents the liquid (continuous, k = c) and particle (dispersed, k = d) fields. If the particles arevery small, the dominant interfacial force is the drag force, given by [21]

FDc = −FD

d = 1

8CDρc |vd −vc| (vd −vc) A′′′

d (7)

where, A′′′d = 6αd

Dpis the interfacial area density, αd is the volumetric concentration of the dispersed field, Dp

is the particle diameter, and CD is the drag coefficient.An interesting issue is concerned with the shear force. For the continuous phase/field, this force is nor-

mally given by

MSk = ∇ ·αc

(τ

c+ τ Re

c

)(8)

where, αc = 1−αd is the volumetric concentration of the continuous field, τc

is the shear stress due to mo-lecular viscosity, and τ Re

cis the turbulent (Reynolds) stress. Since the flow of dispersed (especially – dilute)

particles is not associated with any physical shear, the shear force for the dispersed field is normally as-sumed to be zero, although it may also be used to reflect the effect of particle-induced turbulence (de facto,an additional interfacial force). Furthermore, it is not uncommon that this term is defined via a superficial“molecular viscosity” of particles. However, it has recently been shown [22, 23] that a consistent formulationof the two-field (or multifield) imposes a constraint in the form of coupling between the continuous and dis-perse fields. Specifically, for a two-field model, the shear force for the dispersed particle field can be deducedfrom the basic model formulation as

MISd = αd

αcMS

c = αd

αc

[∇ ·αc

(τ

c+ τ Re

c

)](9)

2.2 Boundary conditions

Because of the elliptic nature of the Navier–Stokes type of equations, the associated boundary conditionsmust be defined at all domain boundaries. For the geometries analyzed in this paper, the boundaries includethe inlet and outlet of the module (for pressure, velocity, and concentration) and the tube wall. Typically,a reference pressure is assigned at the exit boundary, either uniform (if there is no effect of gravity acrossthe flow) or given by a hydrostatic pressure distribution according to the orientation of the outlet area. Atthe inlet, a section of straight pipe is included to allow the flow to fully develop before entering the curvedgeometry.

Since for all the cases investigated here the Reynolds number was well below 2300, the flow was treatedas laminar.

2.3 Computational approach

The model presented above has been encoded using the NPHASE computer code [19, 20]. NPHASE isa segregated, nominally pressure-based finite-volume CFD code. Individual scalar transport equations aresolved for the momentum (in general, also for the energy) and turbulence quantities for each field. The mix-ture and field continuity equations are solved in coupled or uncoupled fashions, using frozen coefficientlinearizations. The code is fully unstructured and utilizes second-order accurate convection and diffusiondiscretizations. A key feature of NPHASE is that from the outset the software design has focused on the de-velopment of a reliable solver for multiphase flows. The discretization of the geometry was accomplished


Fig. 1. Computational mesh across a U-bend

using unstructured grids generated by the GRIDGEN [24] software. The overall computational model wasused to simulate single and two-phase flows in the straight pipe and U-bend geometries. Furthermore, themodel was parametrically tested for convergence and accuracy.

3 Model testing and validation

3.1 Testing for numerical accuracy and Grid-independence

A grid optimization study was performed using a U-bend geometry as reference [25]. The effect of grid sizeand geometry was studied using meshes consisting of two-parts: a near-orthogonal grid section in the near-wall region of the tube, and an equally-spaced unstructured grid in the inner region. A sample mesh is shownin Fig. 1.

An example of the axial nodalization of the U-bend is shown in Fig. 2. The total number of elementsfor each of the individual grids tested were: 16 458, 29 016, 52 572 and 102 960, respectively. The numericaltesting was performed by running the NPHASE code using each grid for identical single-phase flow condi-tions. Typical secondary flow patterns at the mid section of the U-bend, calculated using four different gridswith increasing number of cross-sectional cells, are shown in Fig. 3. A thorough examination of these resultsclearly indicates that the grids shown in Figs. 3c and d are very similar and that both are capable of properlycapturing the secondary flows in the central region of the curved section of the U-bend tube. Therefore, thegrid in Fig. 3c was chosen as adequate for future validations and applications.

In addition to the cross-sectional nodalization, another factor that may affect the accuracy of predictionsfor developing flows is concerned with the number and size of the axial nodes. Testing was performed usingtwo grids: one with 52 572 nodes (ref. grid in Fig. 3c), the other with 72 792 total nodes. The number of axialnodes in the curved U-bend section of the second grid was doubled, since the most interesting phenomenaoccurs in that tube region, whereas the number of nodes in the straight pipe sections, before and after theU-bend, was unchanged. The axial nodes in the curved U-bend section were equally-spaced.

Figure 4 shows a comparison of the streamwise velocity along the transverse axis (A-A) at the mid-section of the U-bend. The predicted axial velocities correspond to the coarse and fine axial grids. As can beseen, the difference between the predicted axial velocities for both cases did not exceed 4%. Hence, it wasconcluded that the grid in Fig. 3c was sufficiently accurate, and this grid was used in the remaining part ofthe analysis.


Fig. 2. U-bend geometry used in the present study: (a) sample mesh on the U-bend wall, (b) characteristic dimensions

Fig. 3. Transverse velocity plot at mid-section of the U-bend geometry of referencefor four different grid sizes. Total num-ber of nodes on outer edge of the tube:(a) 30, (b) 40 (c) 50, (d) 80. I, denotes theinner bend, O, denotes the outer bend

3.2 Model validation against experimental data

To validate the present model, single-phase (water only) calculations have been performed using the ex-perimental conditions of Chung et al. [26]. In these experiments, the local velocity distribution in a U-bend


Fig. 4. Comparison of streamwise velocity profiles for the coarse and fine axial grids. The plotted velocities are along the A-Aaxis at the mid-section of the U-bend

geometry was measured using magnetic resonance flow imaging. The U-bend was made of a 3.9 mm internaldiameter tube having a radius of curvature of 25.4 mm. The flow of liquid water inside the tube correspondedto an average velocity of 4.2 cm/sec. The corresponding Reynolds number was 328, so that the flow waslaminar.

Since the inlet (straight) section of the tube was sufficiently long to obtain fully-developed flow condi-tions at the inlet to the curved section that changed flow direction by 180◦, a similar straight section wasincluded in the simulations.

A comparison between the measured and predicted axial velocity contours at two different locationsalong the U-bend is shown in Fig. 5. The velocity magnitude was normalized using the average axial vel-ocity. An examination of both, the results of simulations and the experimental data, at the entrance to thecurved section of the U-bend, shown in Fig. 5a, confirms that the assumption of a fully developed vel-ocity profile at that location was justified. Also, it can be seen that the predictions agree very well with themeasurements (the error does not exceed 1–2%).

Figure 5b shows a comparison between the measured and predicted axial velocity contours at a plane5 mm after the end of the curved section of the U-bend. As can be seen, both the shape and magnitude ofthe predicted velocity fields agree well with the measurements. The predicted flow pattern indicates the for-mation of Dean vortices similar to those observed in the experiments. A detailed comparison shows thataccuracy of simulations was within a 5% error. Interestingly, the experimental velocity distribution expe-riences some degree of asymmetry with respect to the horizontal axis, whereas the numerical results shownearly perfect axial symmetry. The observed differences may be explained by the presence of small-scalerandom effects and flow disturbances in the experimental data, that have not been observed in the present(deterministic) numerical simulations. It should be mentioned here that the minor departure from symmetryof the numerical results is mainly due to the combined effects of discretization (the use of unstructured grids)and graphical representation.


Fig. 5. Contours of streamwise velocitiesat two locations along the U-bend usedby Chung et al. [26]: (A) at the entranceto the curved section, (B) at the exit ofthe curved section of the U-bend; (A1) &(B1) experimental data [26], (A2) & (B2)predictions using the present model

Table 1. Geometry definitions and flow parameters used in the calculations

Radius of Ratio of AverageTube Radius Curvature the radii velocity

Case (Rt , mm) (Rc, mm) (Rt/Rc) (m/sec) Re De

U-bend-1 8.02 25.4 0.316 0.154 2470 1388U-bend-2 1.6 10 0.16 0.1 320 128

4 Analysis of flow patterns in circular U-bend tube

In order to investigate the effect of geometry on flow conditions, simulations were performed for two U-bendgeometries. In each case, the U-bend was positioned in such a way that the y-axis (along the main flow di-rection in the straight section) was horizontal, the z-axis was vertical, and the x-axis was horizontal in theplane normal to the main flow direction in the straight section of the tube (see Fig. 2). Unlike most previ-ous works [27–29], where flow symmetry across the U-bend was assumed (and, thus, half grids were used),the present simulations were performed using a full 3-D geometry of the U-bend that was discretized usingfully unstructured grids. This, in turn, allowed a complete investigation of Dean vortex bifurcation and of theeffect of gravity on flow patterns (particularly important for two-phase particle flows, as shown next).

Table 1 lists the geometrical parameters and flow conditions that were used in the simulations.In order to demonstrate the development of Dean vortices in the curved section of each U-bend, the sec-

ondary flow patterns have been plotted at every 30◦ section of the U-bends, as shown in Fig. 2. The resultsfor the U-bend-1 case are shown in Fig. 6. As the fluid enters the curved section of the U-bend, the flow splitsand each of the two resultant streams moves radially outwards. This, in turn, results in symmetric vortexformation of a bean-shaped structure. Initially, the center of the vortices is at the center of the semicircularregion. As the fluid moves forward (see the 60◦ section in Fig. 6b), the center of Dean vortices starts moving


Fig. 6. Development of Dean vortices in U-bend-1. The plots are shown at: (a) 30◦, (b) 60◦, (c) 90◦, (d) 120◦, (e) 150◦, (f) 180◦sections of the U-bend

toward the inner bend of the tube (i.e., toward Ri in Fig. 2b, or downwards in Fig. 6). Interestingly, a similartrend was also observed by Mallubhotla et al. [25].

Further downstream, at the 90◦ section (see Fig. 6c), the transverse velocity plot shows an interesting phe-nomenon. Namely, a stagnation zone is formed in the middle of the U-bend. This is due to flow reversal asthe fluid changes direction in the other half of the U-bend. The transverse velocity plot at the 120◦ section,shown in Fig. 6d, indicates that the bean-shaped structures split to form four vortices, which are again sym-metric in nature. These vortices are clearly visible in Fig. 6e at the 150◦ section of the U-bend. As shown inFig. 6f, as the fluid approaches the exit section of the U-bend, the four-vortex structure starts diminishing,and it disappears completely in the straight exit section. It is clear that the secondary-flow vortices arise dueto the centrifugally-induced pressure gradient in the curved section of the tube. So, as the fluid approachesthe straight pipe section, the associated centrifugal instabilities cease to exist. Hence, the vortices graduallydisappear and the flow pattern returns to the Hagen-Poiseuille flow in a straight pipe.

To extend these observations to other geometrical and flow conditions, calculations were also performedfor another U-bend geometry (see Table 1: case U-bend-2). The Rt/Rc ratio of the U-bend-2 was about halfof that for U-bend-1, whereas the Reynolds and Dean numbers, respectively, were: Re = 320, De = 128. Theresults of simulations are shown in Fig. 7. Figure 7a presents the secondary flow behavior at the 30◦ sectionof the tube. As it can be seen there, two symmetric vortices form, with the eye of each vortex practically atthe middle of the respective half-tube. The vortices gradually move towards the inner bend of the tube as thedistance from the inlet to the curved section increases. These results show good agreement with the data re-ported in the literature for low and intermediate Dean number flows [18]. The start of vortex splitting can beseen in Fig. 7d. However, the instability is not distinct enough to support the four-vortices as the flow exitsthe U-bend.

The splitting of Dean vortices (or vortex bifurcation, understood as a flow instability resulting on doub-ling the number of vortices) and a four-vortex pattern are reported in the literature [28, 30, 31]. Dennis andNg [28] first reported the four vortex solution in curved tubes of circular cross-section. They also concludedthat the four-vortex pattern is a part of dual solution behavior which may exist in such geometries for Deannumbers higher than a critical value of 956 (as defined in [28], which converts to Dec = 169 according tothe definition given by Eq. (3)). Yanase [31] reconfirmed this behavior, and also concluded that the four-vortex patterns are unstable to asymmetric disturbances, whereas the two-vortex pattern is normally stablein response to any small disturbances. It is interesting to notice that the present results (see Fig. 7d) suggest


Fig. 7. Development of Dean vortices for the U-bend-2 case. The plots are shown at: (a) 30◦, (b) 60◦, (c) 90◦, (d) 120◦, (e) 150◦and (f) 180◦ sections of the U-bend

that the tendency to form a four-vortex pattern may start appearing at even lower Dean numbers. Mallub-hotla et al. [25] have also shown that vortex bifurcation (first from two to four and, then, to six vortices) couldoccur at Dean numbers lower than the value suggested by Dennis and Ng [28].

5 Quantification of the effect of curvature on secondary flows

An analysis of the results discussed in the previous Sections indicates that the magnitude of flow vorticity canbe used to quantify the strength of Dean vortices in curved tubes. The vorticity magnitude has been used be-fore to detect vortex production in vector fields, and it is an excellent measure of the secondary flow strength.Also, since a complete velocity field (including gradients) is normally fully determined as a result of CFDsimulations, the vorticity distribution can be readily and accurately extracted from typical data generated bythe output files of codes such as NPHASE [19, 20].

Using the definition of vorticity

ω = ∇ ⊗v (u, ν,w) (10)

the individual components of the vorticity vector in the Cartesian coordinates (used for any geometry in thecomputations based on unstructured grids) are

ω = (Wx, Wy, Wz

) ={(

∂w

∂y− ∂ν

∂z

),

(∂w

∂x− ∂u

∂z

),

(∂ν

∂x− ∂u

∂y

)}(11)

Equation (11) can be readily used to evaluate the local distribution of the magnitude of vorticity, a parameterthat proves particularly useful as a measure of the strength of Dean vortices.

Figure 8 shows the vorticity magnitude at the outer and inner sections of the U-bends listed in Table 1.For the U-bend-1 case, it is seen that the vorticity magnitude along the outer bend first increases from theinlet to the 20◦ section, then decreases till 30◦, and increases again to reach a maximum at about 65◦ alongthe bend. For comparison, the vorticity in a straight pipe is also shown in Fig. 8, for reference.

The strong reversal in vorticity magnitude between the 20◦ and 30◦ sections is the result of a recircula-tion zone that forms in the axial direction. This recirculation zone is present only in the larger Rt/Rc case


Fig. 8. Vorticity magnitude along the outer and inner curved U-bend sections; 0◦ corresponds to the U-bend inlet, 180◦ – to theU-bend exit: (a) U-bend-1; (b) U-bend-2

(Rt/Rc = 0.316), and such effect is not observed in the U-bend-2 case. Another important observation is thatthe vorticity magnitude plots for both geometries in Fig. 8 show some similarity (except near the recircula-tion zone in the U-bend-1 case), although changes are more dramatic in the case of larger Rt/Rc ratio. Thevorticity magnitude in both cases reaches a maximum value between the 50◦ and 70◦ sections of the tubeat the outer bend, and around the 100◦ section on the inner bend. The maximum strength of vortices on theouter bend is similar for both geometries. The observation that Dean vortices reach a maximum strength inthe region between 50◦ and 100◦ angles along the U-bends of different geometries, could be important fordesigning membrane modules using helical circular tubes.

6 Liquid-particle two-phase flow in curved U-bend geometries

In order to understand the behavior of dispersed particles in the liquid flow field, a study was performed sim-ulating silica-particles/water flows in a U-bend tube corresponding to the geometry of the U-bend-2 case


Fig. 9. Effect of gravity on particle flow for the U-bend-2 case: (a), (b) – secondary flows at 90◦; (c), (d) – particle concentrationdistributions at 90◦; (e), (f) – plots of particle concentration at three different diagonals at mid-section of the U-bend; (a), (c), (e)– without the effect of gravity; (b), (d), (f) including the effect of gravity

(see Table 1). The two-field version of the multifield model discussed before was used for this purpose.Table 2 lists the physical properties of both the particles and the liquid, used in the study.

Since the particles are heavier than water, the buoyancy force acts to settle the particles on the bottom halfof the tube. In order to quantify this effect, two series of simulations were performed, one without, the otherwith, the effect of gravity. In the latter case, the effect of gravity was applied perpendicular to the U-bend, asshown in Fig. 2.


Table 2. Physical properties of particle-liquid system

Property Liquid (water) Particles (silica)

Density (ρ, kg/m3) 1000.0 2200.0Viscosity (µ, Pa-sec) 10−3 –Characteristic Diameter (D, m) – 20×10−6

Volumetric concentration (α) 0.9 0.1

Fig. 10. Effect of particle inertia on the concentration distribution of particles at the 90◦ section of U-bend-2 for the case withoutgravity: (a) particle-to-liquid density ratio of 2.2, (b) neutrally-buoyant particles. I, denotes the inner bend; O, denotes the outerbend

A comparison of the results of simulations for the cases without and with gravity is shown in Fig. 9. Ascan be seen in Fig. 9a, in the absence of gravity, the secondary flow in the curved section is still symmetric,similar to the single-phase flow case. In contrast, Fig. 9b clearly demonstrates that gravity causes an asym-metry of the pair of Dean vortices formed inside the U-bend. It also causes particle settling at the lower rightregion of the curved U-bend section, as one can see by comparing Figs. 9c and 9d. This effect is shown quan-titatively in Figs. 9e and 9f. Specifically, it can be noticed that Dean vortices limit the concentration buildupdue to the influence of the secondary flows on particle drag. Without this secondary flow, nearly all of theparticles would settle in the bottom half of the tube.

As a consequence of the color scale used in Fig. 9, the result in Fig. 9c seems to indicate that the par-ticle distribution in the absence of gravity is nearly uniform. Actually, particle inertia may also affect localconcentration of particles in the curved section of the U-bend. This is shown in Fig. 10, where the resultsfor the particles analyzed before (the density of which was approximately twice the density of water, asshown in Table 2) were compared against those for neutrally-buoyant particles, both without the effect ofgravity. Using a finer scale, one can readily notice that the distribution of neutrally-buoyant particles is prac-tically uniform (the observed slight differences reflect the numerical accuracy of calculations), whereas inthe heavy particle case, the concentration changes over a range of about 12% of the average value, αd = 0.1.In the latter case, the maximum concentration of particles can be seen near the inner bend of the tube, whereDean vortices cause the formation of a slight stagnation zone. The particles trapped in that region will staythere until they reach the end of the curved section of the U-bend. This finding may be significant for mem-brane filtration, in particular in long helical tubes, since it indicates that Dean vortices tend to move particlesazimuthally, thus reducing their relative concentration over a large fraction of the near-wall region. Further-more, for the particle-to-liquid density ratios less than one, the centrifugal Dean vortices are expected to trapsome of the particles inside the vortex region and, thus, keep them away from the walls.

Figure 11 shows particle concentration distributions along the curved section of the U-bend, in the pres-ence of gravity. It can be seen that at the entrance to the U-bend the particle concentration is symmetric.


Fig. 11. Particle concentration plots in U-bend-2 with gravity. Plots are shown at the: (a) 0◦(entry), (b) 30◦, (c) 60◦, (d) 90◦,(e) 120◦, (f) 150◦ and (g) 180◦ sections of the U-bend. I denotes the inner bend; O denotes the outer bend

Then, a significant flow asymmetry starts developing, and a nonuniform particle concentration can be clearlyseen in Figs. 11b through 11f.

As can be seen in Fig. 11, the highest concentration is initially located at the bottom of the tube and, as thefluid moves along the curved section, the effect of Dean vortices shifts the particle settling zone by as muchas 45◦. The maximum particle concentration observed in this case is, αd,max ≈ 0.5, compared to the aver-age particle concentration, αd = 0.1. As expected, this concentration buildup gets augmented in the straightsection of the tube upstream of the U-bend, since the particles tend to settle and no Dean vortices are present.

7 Conclusions

A consistent theoretical and numerical method has been developed for tracking details of multidimensionalsingle-phase and two-phase particulate flows in curved U-bend geometries. Extensive numerical testing ofthe new CFD model has been performed, showing that the results are practically grid-independent. The mainconclusions are:

• A full 3-D analysis of single-phase flows in U-bend curved tubes was validated against experimental re-sults using the magnetic resonance flow imaging technique. The results of numerical simulations agreedvery well with the measured data.


• Secondary flow behavior was investigated in curved U-bend tubes at low and intermediate Dean num-bers. It has been observed that the original Dean vortices tend to bifurcate at Dean numbers lower thanpreviously anticipated, although the secondary pair becomes stable only if the Dean number increases.

• The effect of Dean vortices has been quantified using the vorticity magnitude as a measure of vortexstrength. It has been shown that the peak in Dean Vortices occurs at the intermediate section of U-bendsof various geometries (between 50◦ and 100◦ along the 180◦ curve).

• It has been demonstrated that Dean vortices play a significant role mitigating the effect of gravity onparticle settling for laminar particulate two-phase flows in thin long tubes. Also, the maximum concen-tration of particles moves by about 45◦ from the bottom of the tube.

Whereas the primary objective of the present work was to investigate fundamental phenomena governingthe flow field and particle distribution in curved U-bend geometries, the results obtained to date providea very promising starting point for future investigations aimed at studying two-phase particle/liquid flowsin permeable membranes of helical and other geometrical shapes.

References

1. Belfort, G., Nagata, N.: Fluid mechanics and cross-flow filtration: Some thoughts, Desalination. 53, 57–79 (1985)2. Belfort, G.: Fluid mechanics in membrane filtration: recent developments. Jr. Mem. Sci. 40, 123–147 (1989)3. Schnabel, S., Moulin, P., Nguyen, Q.T., Roizard, D., Aptel, P.: Removal of volatile organic components (VOC) from water

by pervaporation: separation improvement by Dean Vortices. Jr. Mem. Sci. 142, 129–141 (1998)4. Mores, W.D., Davis, R.H.: Yeast fouling effects in crossflow microfiltration with periodic reverse filtration. I&EC Res. 42,

130–139 (2003)5. Winzeler, H.B., Belfort, G.: Enhanced performance for pressure-driven membrane processes: The argument for fluid insta-

bilities. Jr. Mem. Sci. 80, 35–47 (1993)6. Gehlert, G., Luque, S., Belfort, G.: Comparison of ultra and microfiltration in the presence and absence of secondary flow

with polysaccharides, protein and yeast suspensions. Biotech. Prog. 14, 931–942 (1998)7. Moulin, Ph., Veyret, D., Charbit, F.: Dean vortices: comparison of numerical simulation of shear stress and improvement of

mass transfer in membrane processes at low permeation fluxes. Jr. Mem. Sci. 183, 149–162 (2001)8. Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A. 233–239 (1923)9. Belfort, G., Mikulasek, P., Pimbley, J.M., Chung, K.-Y.: Diagnosis of membrane fouling using a rotating annular filter,

2. Dilute particle suspensions of known particle size. Jr. Mem. Sci. 77, 23–39 (1993)10. Dean, W.R.: Fluid motion in a curved channel. Proc. Roy. Soc. Lon. 121(787), 402–420 (1928)11. Gelfgat, A.Y., Yarin, A.L., Bar-Yoseph, P.Z.: Dean vortices-induced enhancement of mass transfer through an interface

separating two immiscible liquids. Physics of Fluids 15(2), 330–347 (2003)12. Belfort, G., Davis, R., Zydney, A.: The behavior of suspensions and macromolecular solutions in crossflow microfiltration.

A review for the North American Membrane Society’s Annual Reviews and for the Jr. Mem. Sci., 96, 1–58 (1994)13. Baruah G.L., Belfort. G.: A predictive aggregate transport model for microfiltration of combined macromolecular solutions

and poly-disperse duspensions: model development. Biotech Progr., 19(5), 1524–1532 (2003)14. Frey, J.M., Schmitz, P.: Particle transport and capture at the membrane surface in cross-flow microfltration. Chem. Engg.

Sci. 55, 4053–4065 (2000)15. Tanishita K., Richardson P.D., Galletti P.M.: Tightly wound coils of microporous tubing: Progress with secondary-flow

blood oxygenator design. Trans. Am. Soc. Artif. Int. Organs. 21, 216–223 (1975)16. Manno, P., Moulin, P., Rouch, J.C., Aptel, P.: Mass transfer improvement in helically wound hollow-fiber ultrafiltration

modules. Yeast Suspensions, Sep. Purification Tech. 14, 175–182 (1998)17. Moulin, P., Serra, C., Rouch, M.J., Clifton, M.J., Aptel, P.: Mass transfer improvement by secondary flows – Dean vortices

in coiled tubular membranes. Jr. Mem. Sci. 114, 235–244 (1996)18. Berger, S.A., Talbot, L., Yao, L.-S.: Flow in curved pipes. Ann. Rev. Fluid Mech. 15, 461–512 (1983)19. Antal, S.P., Ettorre, S.M., Kunz, R.F., Podowski, M.Z.: Development of a next generation computer code for the prediction

of multicomponent multiphase flows. International Meeting on Trends in Numerical and Physical Modeling for IndustrialMultiphase Flow. Cargese, France (2000).

20. Kunz, R.F., Yu, W.S., Antal, S.P., Ettorre, S.M.: An unstructured two-fluid method based on the coupled phasic exchangealgorithm. AIAA-2001-2672 (2001)

21. Drew, D.A.: Analytical modeling of multiphase flows. In: Boiling Heat Transfer; Modern Developments and Advances.Lahey, R.T. (ed.), Elsevier Publishers, New York (1992)

22. Podowski, M.Z.: Modeling of two-phase flow and heat transfer; remarks on recent advancements and future needs. Proc.3rd Int. Conf. on Transport Phenomena in Multiphase Systems. Kielce, Poland (2002)

23. Tiwari, P., Antal S.P., Podowski M.Z.: On the modeling of dispersed particulate flows using a multifield model. Computa-tional Fluid and Solid Mechanics 2003, Bathe, K.J. (ed.) 1, 1160–1165 (2003)

24. GRIDGEN 14 User Manual. Pointwise Inc. (2002)


25. Mallubhotla, H., Belfort, G., Edelstein, W.A., Early, T.A.: Dean vortex stability using magnetic resonance flow imaging andnumerical analysis. AIChE Jr. 47(5), 1126–1140 (2001)

26. Chung, K.Y., Belfort, G., Edelstein, W.A., Li, W.: Dean vortices in curved tube flow: 5. 3-D MRI and numerical analysis ofthe velocity field. AIChE Jr. 39(10), 1592–1602 (1993)

27. Collins, W.M., Dennis, S.C.R.: The steady motion of a viscous fluid in a curved tube. Q. Jr. Mech. Appl. Math. 28, 133(1975)

28. Dennis, S.C.R., Ng, M.: Dual solutions for steady laminar flow through a curved tube. Q. Jr. Mech. Appl. Math. 35, 305–324(1982)

29. Daskopoulos, P., Lenhoff, A.M.: Flow in curved ducts: bifurcation structure for stationary ducts. Jr. Fluid. Mech. 203, 125(1989)

30. Nandakumar, K., Masliyah, J.H.: Bifurcation in steady laminar flow through curved tubes. Jr. Fluid Mech. 119, 475–490(1982)

31. Yanase, S., Goto, N., Yamamoto, K.: Dual solutions of the flow through a curved tube. Fluid Dynamics Research 5, 191–201(1989)

multifield computational fluid dynamics model of particulate flow in curved circular tubes

Documents