Chemical Engineering Science: X 2 (2019) 100010

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Chemical Engineering Science: X

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CFD based compartment-model for a multiphase loop-reactor

https://doi.org/10.1016/j.cesx.2019.1000102590-1400/� 2019 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

⇑ Corresponding author.E-mail address: andreas.jupke@avt.rwth-aachen.de (A. Jupke).URL: http://www.avt.rwth-aachen.de (A. Jupke).

1 The first two authors equally contributed to this manuscript.

Benedikt Weber 1, Maximilian von Campenhausen 1, Tim Maßmann, Andreas Bednarz,Andreas Jupke ⇑AVT – Fluid Process Engineering, RWTH Aachen University, Forckenbeckstraße 51, D-52074 Aachen, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 July 2018Received in revised form 7 January 2019Accepted 26 January 2019

Keywords:Compartment-modelLoop reactorMultiphase reactorConstant-number Monte-CarloCFD

In multiphase devices, fluid dynamics have a high impact on concentration profiles and mass transferbetween the phases and therefore influence efficiency. Standard models often assume ideally mixed con-ditions or plug flow. The application of such models for multiphase devices with complex flow patternscauses inaccuracies, if the flow deviates from ideally mixed or plug flow conditions. Therefore, for a pre-cise model based design and operation parameter determination of devices with complex flow patterns,the local fluid dynamics should be considered. CFD simulations for multiphase systems including masstransfer, population balance equations for coalescence and breakage as well as reactions are still timeconsuming. Thus, we developed a compartment-model based on prior calculated CFD flow-data. In theCFD simulations, the time consuming population balance equations for coalescence and breakage, masstransfer and reactions are neglected. These phenomena are considered in the compartment-model.Thereby we reduce the overall computing time.This paper presents the CFD based compartment-model applied on a loop-reactor. First, a three-phase

CFD model of the developed multiphase loop-reactor is introduced. Following, the paper presents thecompartment-model and the application of a time-driven constant-number Monte-Carlo approach tosolve population balances. Finally, the compartment-model is applied to the liquid-liquid extraction partof the loop-reactor calculating the drop size distribution and mass transfer based on previously calculatedCFD data.� 2019 The Authors. Published by Elsevier Ltd. This is an openaccess article under the CCBY license (http://

creativecommons.org/licenses/by/4.0/).

1. Introduction For the calculation of multiphase devices, CFD is often coupled

Models assuming plug flow or ideally mixed conditions areoften applied in chemical engineering for model based designand operation parameter optimization of reactors, columns andother apparatus. However, the assumption of plug flow or ideallymixed conditions leads to inaccuracies when the internal flow pat-tern is complex. Especially multiphase devices are characterized bya complex flow pattern, due to the interaction of the differentphases. Such complex flows occur for example in loop reactors,stirred liquid-liquid extraction columns and stirred large-scalemultiphase reactors. For model-based design of such multiphasedevices and the prediction of optimal operation parameters, wedeveloped a CFD based compartment-model to consider the localflow profile. In the following paragraphs, the recent literature con-cerning CFD and compartment-models for multiphase devices issummarized.

with population balance equations (PBE) (Bhole et al., 2008;Amokrane et al., 2014). This enables the calculation of the local res-olution of the particle (drops, bubbles, solid particles) size distribu-tion. Attarakih et al. (2015) and Zhang et al. (2017) additionallyimplemented mass transfer between the phases in the PBEs. Thismethod offers highly accurate results since during the calculationof the fluid flow the effects of changes in drop size distributionand mass transfer are included. However, CFD with PBEs demandshigh computational effort since the PBEs have to be solved for eachgrid cell. Attarakih et al. (2015) for example required for a liquid-liquid extraction column up to 27 days computing time for a 2DCFD simulation coupled with PBE for coalescence and breakageand one transfer component. In addition, the computing time ofthe numerical solution of the CFD-PBE further increases, especiallywhen considering not only one dispersed phase, multi componentmass-transfer and reactions (Bezzo et al., 2004).

Therefore, researchers have started to decouple CFD and PBE byusing compartment-models, also known as cell-models (Bauer andEigenberger, 1999; Rigopoulos and Jones, 2003; Bezzo et al., 2004;Bezzo and Macchietto, 2004; Le Moullec et al., 2010; Irizarry, 2012;Gimbun et al., 2016). Sometimes a simplified CFD simulation

Nomenclature

Symbolsa volume fraction [–]b mass transfer coefficient [m/s]Dm transferred mass [kg]Dt time step [s]q density [kg/m3]l viscosity [Pa s]r interfacial tension [N/m]n coalescence parameter [–]�s stress tensor [kg/(m2 s2)]sc drop statistic lifespan [kg/(m2 s2)]u mass flow fraction [–]CD drag coefficient [–]Ctd constant for turbulent dispersion [–]CIP instability parameter [–]d particle diameter [m]Dt fluid-particulate dispersion tensor [m/s2]D diffusion coefficient [m2/s]Deff effective diffusion coefficient [m2/s]f drag function [–]F!

d drag force [N/m3]F!

td turbulent dispersion force [N/m3]g! gravitation [m/s2]h height [m]H Hamaker coefficient [Nm]Kcd interphase exchange coefficient [kg/(m3 s)]K distribution coefficient [kg/kg]_m mass flow [kg/s]

m mass [kg]n number of compartment nodes [–]N number of representative drops [–]p pressure [Pa]Pr dispersion Prandtl number (definition:Pr ¼

kinematic turbulent viscosityturbulent dispersion coefficient) [–]

Re relative Reynolds number [–]r radius [m]t drop lifespan [s]v! velocity [m/s]_V volume flow [m3/s]x concentration in continuous phase [kg/kg]y concentration in dispersed phase [kg/kg]Z coalescence probability [–]

Sub- and superscript1 stationary pointax axiald dispersed phasec continuous phasep general phase pq general phase qrad radial

AbbreviationsCFD Computational Fluid DynamicsPBE Population Balance Equation

2 B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010

without PBE can be used to capture the local flow profile of multi-phase devises. Rigopoulos and Jones (2003) and Gimbun et al.(2016) made the assumption of a constant particle size distribu-tion, because their particle size distribution varied moderately. Insome cases the dispersed phase is even neglected in the CFD model(Laakkonen et al., 2006; Delafosse et al., 2014), when the dispersedphase volume fraction and consequently the dispersed phase’sinfluence on the flow pattern is low. These simplifications makethe PBEs dispensable and thereby reduce the computing time ofthe CFD simulations. All these approaches import the resultingCFD flow profile into the compartment-model to calculate all addi-tional effects like coalescence and breakage of the dispersed phase,mass transfer and reactions with PBE in the compartment-model.The advantages of these compartment-models are the use ofdetailed flow patterns based on CFD simulations and the benefitto reduce the computational effort for multi-phase systems by cal-culating multi-component mass transfer as well as reactionsdecoupled from the momentum balances in a coarser grid com-pared to CFD models.

In literature compartment-models have been used for homoge-neous reactors (Alexopoulos et al., 2002; Wells and Ray, 2005;Guha et al., 2006; Gresch et al., 2009; Le Moullec et al., 2010), bub-ble columns (Bauer and Eigenberger, 1999), airlift reactors(Rigopoulos and Jones, 2003), gas-liquid stirred tank reactors(Vrábel et al., 1999; Laakkonen et al., 2006; Gimbun et al., 2016),crystallizers (Kramer et al., 1996; Kulikov et al., 2005; Irizarry,2008b, 2008a, 2012; Metzger and Kind, 2014) and granulation pro-cesses (Bouffard et al., 2012; Yu et al., 2017). In cooling crystalliza-tion, the formation of crystals, the super-saturation and thetemperature profile strongly influence each other. Therefore,Kulikov et al. (2005) developed an iterative simulation approachwhich transfers information between CFD and compartment-model. Also Bauer and Eigenberger (1999) coupled the two-phaseCFD simulation with their compartment-model, since they

simulated a bubble column where a reaction strongly reducesthe bubble size influencing the flow field. In case of lower reactiv-ity in gas-liquid reactors, researchers only apply a one-way cou-pling (Rigopoulos and Jones, 2003; Laakkonen et al., 2006)neglecting the reaction’s influence on fluid dynamics.

The PBE of the above-named compartment-models are usuallysolved numerically for example with the method of moments. Nev-ertheless, Yu et al. (2017) and Irizarry (2012) used Monte-Carlomethods for crystallization and granulation. The Monte-Carlomethods are usually slower than method of moments but allowthe modeling of multiple effects (e.g. multi-component mass trans-fer and multiple interactions between the dispersed phases) with-out the need for a complex numerical formulation of the problem(Zhao et al., 2005).

This paper presents a compartment-model approach which isapplied for a novel multiphase loop-reactor (Bednarz et al.,2017). The coupling of a multiphase CFD simulation with the pro-posed compartment model is one-way. The velocity profile of thecontinuous phase is calculated with a simplified CFD model. Thisvelocity profile of the continuous phase is transferred to the com-partment model. In the compartment model effects like mass-transfer and coalescence are calculated, which are neglected inthe CFD simulation. At this stage of development, thecompartment-model simulates the liquid-liquid extraction partof the multiphase loop-reactor calculating drop-size distributions,drop motion and mass transfer. This paper begins with an intro-duction of the multiphase loop-reactor. The following section sum-marizes the components’ physicochemical properties used in thecase study. Afterwards the CFD and 2D compartment-model areexplained, followed by an introduction of the CFD simulationresults and a comparison of the drop motion calculated by thecompartment-model and the CFD model. Then the model is usedto quantify the separation efficiency of the liquid-liquid extractionin the multiphase loop-reactor. Finally, a short conclusion is given.

Fig. 2. Dimensions of the multiphase loop-reactor in mm (grey rectangle: simu-lation domain of compartment-model).

B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010 3

2. Materials and methods

2.1. Multiphase loop-reactor

Often fermentations require aeration and are limited in theirproductivity due to product inhibition. The productivity of suchfermentations can be increased by a novel multiphase loop-reactor, which simultaneously supplies oxygen containing gasand removes inhibitory components via liquid-liquid extraction(Bednarz et al., 2016). Fig. 1 shows the set-up of the reactor. Theoxygen rich gas phase enters the reactor in the upcomer andinduces the loop-flow, similar to an airlift reactor. The liquid sol-vent phase performing the extraction meanwhile rises counter-currently in the downcomer.

We improved the already published reactor design (Bednarzet al., 2017) based on results obtained from a three-phase CFD sim-ulation model. In a previous study, the gas was dispersed in theouter cylinder ring and the solvent in the center of the reactor. Thiscaused a large vortex in the downcomer area and reduced the res-idence time of the solvent. Therefore, we improved the reactordesign by changing the dispersion areas. The gaseous phase is dis-persed in the bottom of the center of the reactor and the extractionsolvent rises in the gap between inner cylinder and outer reactorwall as shown in Fig. 1. The top of the reactor is expanded toinclude a gas-liquid separator above the internal cylinder. The finaldimensions of the improved reactor are summarized in Fig. 2. Here,the gray shaded area marks the simulation domain calculated inthe proposed compartment model. CFD simulations were donefor the whole reactor shown in Fig. 2.

2.2. Chemicals and operation parameters

The three-phase system used during the case study consists ofthe EFCE standard test system (Misek et al., 1985) with water ascontinuous phase, toluene as solvent, acetone as solved transfercomponent (between the water and toluene phase) and syntheticair as additional gaseous phase.

Table 1 lists the corresponding chemical properties. The interfa-cial tension r between water and toluene is defined as34:3 � 10�3 N=m and the distribution coefficient K as

Fig. 1. Concept of multiphase loop-reactor.

Table 1Chemical properties.

Density [kg=m3] Dynamic viscosity [Pa � s]Water 998:8 1:029 � 10�3

Toluene 867:5 0:596 � 10�3

Air 1:20 17:10 � 10�6

0:6437 kgsolvent=kgcontinuous phase. The diffusion coefficient D of ace-

tone in toluene is 2:788 � 10�9 m2=s.The multiphase loop-reactor has two inlets: air and solvent. The

reactor is operated with the following volume flows: 0:8 Nm3=hair and 0:0341 m3=h toluene.

2.3. CFD simulation model

In a previous publication we developed a CFD simulation model(CFD Software: Fluent 15.0 from ANSYS, Canonsburg, USA) and val-idated this model with a multiphase loop-reactor in technical scale(Bednarz et al., 2017). Pfleger et al. (1999) concluded during exper-imental and simulative investigations of bubbly flows, that the useof a k-e model is recommended in comparison to laminar assump-tions. According to this study and similar to simulation approachesof airlift reactors (Oey et al., 2003; Talvy et al., 2007; Šimcík et al.,

Fig. 3. Structure of compartment-model.

4 B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010

2011; Ghasemi and Hosseini, 2012; Bednarz et al., 2017; Li et al.,2018) we incorporated a two dimensional axisymmetric Euler-Euler model for further CFD based development and approxima-tion of the velocity profile. For the calculation of turbulentquantities the standard k-e turbulence model (calculating the tur-bulence quantities from the mixture of the phases) was chosenaccording to the above given literature screening on airlift reactors

and since the turbulent Reynolds number (ReT ¼ k2=ðemÞ) in theloop flow inducing section of the reactor, the upcomer, and abovethe gas-liquid separator is in the domain of 3000–6000. For unifi-cation, the k-e turbulence model was applied for the whole reactordomain, even if the Reynolds number in the downcomer is smallerin the domain of 1000–1500. The mass and momentum balance foreach of the three phases q are:

ddt

aqqq

� �þr � aqqq v

!q

� �¼ 0 ð1Þ

ddt

aqqq v!

q

� �þr � aqqq v

!q v!q

� �¼ �aq p

!þ �sq þ aqqq g!þ F

!d;q þ F

!td;q ð2Þ

where aq represents the volume fraction, qq the density, v!q the

velocity, p! the pressure, �sq the stress tensor, g! the gravity, F!

d;q

the drag force and F!

td;q the turbulent dispersion force. Furthermore,for the volume fractions holds:

Paq ¼ 1.

The simulation considers gravitation, drag force and turbulentdispersion force. The model of Schiller and Naumann (1935) isused for the drag force:

F!

d;q ¼ Kcdðv!c � v!dÞ ð3Þ

Kcd ¼ 18adð1� adÞlc

d2 f ð4Þ

f ¼ CDRe24

ð5Þ

CD ¼ 24ð1þ 0:15Re0:687Þ=Re0:44

Re � 1000Re > 1000

(ð6Þ

Re ¼ qc v!

c � v!d

�� ��dlc

ð7Þ

Here Kcd expresses the interphase exchange coefficient, d theparticle diameter, lc the continuous phase viscosity, v!c and v!d

the continuous and dispersed phase velocity, f the drag function,CD the drag coefficient and Re the relative Reynolds number.Šimcík et al. (2011) analyzed an airlift reactor with an high speedcamera and observed a bubble diameter of 5 mm. This particlediameter is used for the air phase. The particle diameter of thedrops is set to 2 mm according to disperser experiments ofGarthe (2006), who observed a drop diameter in the range of 2–2.4 mm. Therefore, a homogenous drop and bubble distribution iscalculated in the CFD simulation, neglecting coalescence andbreakage phenomena.

The turbulent dispersion is calculated with the model ofSimonin and Viollet (1990):

F!

td;c ¼ � F!

td;d ¼ CtdKcdDt;cd

Prcd

rac

acþrad

ad

� �ð8Þ

where Ctd is a constant (set to 0.2), Dt;cd is a fluid-particle turbulentdispersion term and Prcd the dispersion Prandtl number. Both Dt;cd

and Prcd are determined by turbulent quantities from the mixtureof the phases.

The reactor geometry and mesh were designed with the Design-Modeler of ANSYS and ANSYS Meshing. In the previous publication(Bednarz et al., 2017) we presented a complete mesh sensitivityanalysis and compared the results with published mesh sizes.The analysis proved that a mesh size between 1.3 mm and1.8 mm is appropriate.

Air and solvent enter the computational domain of the multi-phase loop-reactor at the dispersers with a velocity inlet. The airand solvent velocity are set to 0.5 m/s and 0.2 m/s, respectivelyas proposed by Bednarz et al. (2016). The inlet area has a porosityof 0.715 for air and respectively 0.53 for toluene.

The transient simulation model is solved with the pressure-based solver with a coupled algorithm for the pressure-velocitycoupling. The time step size is set to 0.001 s and the simulationabsolute residuals to 5 � 10�5. First, the reactor is calculated for10 s with an air inlet. Then the three-phase loop-reactor is simu-lated for 25 s assuring steady-state operation. The temporal dis-cretization is achieved by first order implicit scheme, the volumefractions with the QUICK scheme (Leonard and Mokhtari, 1990),gradients with a Least Squares Cell-Based method and for all othera second order upwind discretization scheme.

2.4. Two-dimensional compartment-model

The compartment-model is implemented in Matlab Version2016a (The MathWorks, Inc., Natick (MA), USA). In thecompartment-model, the continuous phase is assumed to be ide-ally mixed in each compartment and the convective flux betweenthe compartments is given according to the imported CFD simula-tion results. To capture the effect of the local flow profile like vor-texes on mass transfer (axial dispersion, etc.) the grid has to bealigned with this flow pattern. A time-driven constant-numberMonte-Carlo method calculates a representative number of dis-persed phase particles. The continuous and dispersed phases inter-act with each other according to sub-models describing physicalphenomena like mass transfer, coalescence and drop velocity.Fig. 3 summarizes the structure and the sub-models of thecompartment-model. The following paragraphs first describe thecalculation of the continuous phase’s convective flux. Subse-quently, the implemented sub-models are given and the algo-rithms of the time-driven constant-number Monte-Carlo method.The last section contains the simulation settings of thecompartment-model for the case study.

2.4.1. Convective flux in compartment-modelThe compartment-model uses fluid dynamic data of the contin-

uous phase from the CFD simulations. The extracted data pointsare located at the nodes of the grid used in that CFD simulation.To calculate the flows across the compartment boundaries thenodes are linearly interpolated to obtain a continuous data set.

B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010 5

Eq. (9) determines the resulting volume flow of the continuous

phase over the boundary in axial direction _Vaxc between two radii

r1, r2 at a certain height h:

_Vaxc hð Þ ¼ 2p

Z r2

r1

vaxc r; hð Þ � 1� a r;hð Þð Þrdr ð9Þ

The volume flow in axial direction depends on the local velocityof the continuous phase in axial direction vax

c r;hð Þand the local vol-ume fraction a r;hð Þ. An analogous Eq. (10) provides the volumeflow in radial direction, where v rad

c r;hð Þ describes the continuousphase velocity in radial direction:

_Vradc rð Þ ¼ 2pr

Z h2

h1

v radc r;hð Þ � 1� a r;hð Þð Þdh ð10Þ

The compartment-model calculates the convective fluxbetween the compartments as shown in Fig. 4. The nout leavingmasses miþ1

out;k of the time step iþ 1 are calculated according tothe mass flow ratio of the compartments leaving flow uk, the nin

incoming mass miin;j of the time step i and the transferred mass

between the phases Dmi. This procedure assures conservation ofmass when calculating mass transfer between the phases and cor-rects the flows in case that the calculated mass flows from Eqs. (9)and (10) are faulty due to the interpolation between nodes.

2.4.2. Sub-models for drop motion, mass transfer and coalescenceIn the here presented compartment-model, sub-models are used

for each representative droplet to account for its motion by sedi-mentation, mass transfer and coalescence. Typically used sub-models for liquid-liquid extraction columns are implemented intothe compartment model for the liquid-liquid extraction part. Thesesub-models performed well in a one dimensional PBE model(ReDrop) for different extraction columns (sieve trays, RDC, Kühni,packings) and at various operation points (Henschke, 2004; Kalemet al., 2011; Buchbender et al., 2012; Buchbender, 2013; Grömping,2014; Ayesterán et al., 2015; Kalem, 2015). The sub-models aredescribed in detail in the following paragraphs. A sub-model fordrop breakage is neglected at this stage since the drops can rise

Fig. 4. Calculation of convective flux.

freely in the downcomer without internals or pulsation causingdrop breakage and since the turbulent dissipation rate is low (upco-mer: � � 0:003::0:09 m2=s3 and k � 0:004::0:017 m2=s3; down-comer: � � 0:0002::0:007 m2=s3 and k � 0:0005::0:0016 m2=s3).

The motion of the drops is calculated by superimposing thesingle-drop’s rise-velocity with the continuous phase’s averagedvelocity vector in a specific compartment. The model of Henschke(2004) gives the single-drop steady-state rise-velocity and consid-ers different droplet states (rigid, circulating, oscillating anddeformed). The model has three parameters dum, a15 and a16, whichcan be determined by single-drop rise experiments (Ayesteránet al., 2015). The transition diameter dum describes the transitionof a rigid drop interface to a mobile interface. The parameters a15

and a16 parametrize the drop-oscillation and the correspondingreduction in velocity. The continuous phases’s averaged velocityvector is derived from the CFD simulation result. Swarm effects,which reduce the drops velocity, are neglected, since the volumefraction of drops is low in the downcomer (�2–4% in Fig. 9). Anappropriate sub-model for swarm effects (e.g. by Richardson andZaki, 1954) can easily be included into the model. The drop motionis overlaid by an statistical dispersion via a bivariate Gaussiandensity function (median: 0 m=s; standard deviation:8:165 � 10�5 m=s) in analogy to the turbulent dispersion model ofGosman and Lonnides (1983). This assures a statistical drop pathvariation even when drops are dispersed at the same position, sincethe superposition of the velocities neglects an interaction betweenspecific drops and the influence of turbulence on the drops motion.

Henschke and Pfennig (1999) developed an experimentally val-idated model for single-drop mass transfer. The instability factorCIP correcting the physical diffusion coefficient of the dispersedphase is determined by standardized single-drop mass transferexperiments. The model was successfully included into the popula-tion balance model ReDrop (Henschke, 2004). Therefore, it is alsoused in the compartment model. In the following, the model is pre-sented in more detail. The mass-transfer into the drop is describedas

Dm ¼ Dt � pd2 � 11

qdbdþ K

qcbc

� y� Kxð Þ; ð11Þ

where Dt is the time step, d the drop diameter, qd and qc the dis-perse and continuous phase’s density, bd and bc the correspondingmass-transfer coefficients, K the distribution coefficient and y andx the concentrations of the disperse and continuous phase. Themass-transfer resistance in the continuous phase can be neglectedwhen the drops have a residence time greater than 5 s (Henschkeand Pfennig, 1999). Assuming no counter-current flow in the down-comer and no swarm effects, which would increase the residencetime, toluene drops with a diameter of 2 mm have a residence timeof 6.2 s (rise velocity �60 mm/s (Henschke, 2004)). Therefore, themass-transfer resistance in the continuous phase is neglected andthe mass-transfer coefficient is set to bc ¼ 1. The mass-transfercoefficient of the dispersed phase is defined by

bd ¼ Deff

d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4d2

pDeff tþ p4

s; ð12Þ

where Deff represents an effective diffusion coefficient and t thedrop age (=actual simulation time – simulation time of dispersion)(Henschke and Pfennig, 1999; Henschke, 2004). The formulaincludes an enhancement of the drop’s diffusion due to convectiveflux (inner circulation) and is determined by

Deff ¼ Dd þ v1d

CIP 1þ ldlc

� � ; ð13Þ

6 B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010

where Dd is the diffusion coefficient of the dispersed phase, v1 therelative velocity between drop and continuous phase, CIP the insta-bility parameter, which considers mass transfer acceleration due toinstabilities at the drop interface (Henschke and Pfennig, 1999), andmd and mc the dispersed and continuous phase’s viscosities.

The applied coalescence model derived from Henschke (2004)calculates for every drop a statistic lifespan sc until the drop coa-lesces with another drop. sc is defined by

sc ¼ nlcd

1=3

ar1=3H1=6cd Dqgð Þ1=2

; ð14Þ

where n is the coalescence parameter, a the volume fraction calcu-lated in the compartment model, Hcd the Hamaker coefficient (fororganic-aqueous systems 10�20 Nm), Dq the density differenceand g the constant of gravitation. The probability Zc of coalescenceof a drop during a time step is then calculated with

Zc ¼ Dtsc

: ð15Þ

The drops are marked for coalescence when an uniformly dis-tributed random number s (between 0 and 1) is smaller than orequals Zc. When two or more drops in one compartment aremarked for coalescence, pairs of two form a new bigger drop.

2.4.3. Constant-number Monte-Carlo method in compartment-modelThe here used constant-number Monte-Carlo approach is a sta-

tistical approach considering only a fixed number of representativedrops Nmax drops in each compartment, which is chosen at thebeginning of the calculation. If more drops are in the compartmentvolume, each considered drop represents a multiple of itself. Thisapproach is advantageous when many particles are formed in ashort time, since the calculated number of drops Nmax drops staysconstant. Consequently, this method is often used to calculatethe nucleation in crystallization where the number of particles(nuclei) increases drastically (Irizarry, 2012). Other applications,where the constant-number Monte-Carlo approach is advanta-geous, are liquid-liquid systems with a high drop breakage rate,e.g. aqueous two phase systems in stirred extraction columns.

The previously presented sub-models for coalescence, dropmotion and dispersion at the reactor inlet influence the drop num-ber Ndrops in a compartment. Consequently, the number of consid-ered drops can change due to these sub-models, so that the dropnumber Ndrops does not equal the previously fixed number of rep-resentative drops Nmax drops. Therefore, additional drops have to beadded into or deleted from the compartment so that Ndrops equalsNmax drops again. By adding and deleting representative drops, thedistribution of properties, as size, concentration and position, mustnot change. Thus, we implemented an algorithm depicted in Fig. 5,which considers three different cases depending on the number ofthe actual representative drops Ndrops:

� Case 1: The number of the actual representative drops Ndrops

equals the previously fixed number of representative drops,Nmax drops.

� Case 2: The number of the actual representative drops Ndrops isbigger than the previously fixed number of representativedrops, Nmax drops.

� Case 3: The number of the actual representative drops Ndrops issmaller than the previously fixed number of representativedrops, Nmax drops.

If the number of representative drops, after calculating the sub-models, equals the maximum number of representative drops, thenumber of tracked drops remains constant (case 1). The samenumber of drops is used in the next time step. When the number

of representative drops Ndrops exceeds Nmax drops (case 2), for exam-ple due to drop breakage, drops are randomly deleted until Ndrops

equals Nmax drops. If less representative drops than the maximumnumber of representative drops exist in a compartment (case 3),new representative drops will be chosen to keep the number con-stant. The question arises where to place these new drops and howto determine their properties. Since drops could mainly be locatedin a certain area of a compartment, a random choice of drop posi-tions would lead to an unphysical dispersion of the drops over thecompartments area. In addition, drop distribution properties likesize and composition change with the height in the compartment.For example, when drops enter at the bottom of one compartmentand need several time steps to cross the compartment, the dropsize will increase due to coalescence and the amount of transfercomponent will increase inside the drop through mass-transfer.Additionally, large drops have a low specific interfacial area anda small residence time, which in turn causes differences in compo-sition between the different drop sizes. Consequently, there is agradient over the height in a compartment and drop size. There-fore, we developed a method where the locations of these dropsare randomly determined according to the compartments localdrop distribution-density and the properties from the surroundingdrops in this area, so that the gradients of drop size and composi-tion over the location in the compartment are included. Conse-quently, the algorithm takes multivariate connections of theproperties of the representative drops into account.

This approach is illustrated for an exemplary drop distributiongiven in Fig. 6. The drop distribution-density varies in radial andaxial direction. The resulting two-dimensional probability densityfunction is multivariate. Thus, the density distribution in one direc-tion depends on the other direction. For example, the probabilitydensity function in radial direction for zone B is another than forzone D. In the proposed algorithm, the compartment is discretizedinto five radial and axial zones resulting in a resolution of 25 zonesper compartment.

Before determining the coordinates of the new representativedrop, the algorithm determines the radial (r-axis) and the axial(y-axis) zone according to the position probability distribution ofa drop in the compartment. The property density function in radialdirection over all axial zones is represented in Fig. 7. Afterwardsthis discrete function is cumulated as shown in Fig. 8 and normal-ized. Via a random number between zero and one (in this example:0.625) the zone for the drop is chosen as shown in Fig. 8 (in thisexample: zone 3).

To determine the axial zone of the new representative drop inthe compartment, the same algorithm is repeated with the differ-ence that only the drops in the radial zone chosen in the stepbefore (here zone 3) are evaluated to create the property densityfunction in axial direction. A uniform distribution within this zoneprovides the coordinates of the new representative drop. An imple-mented vectorization of the algorithm assures a fast calculation.From this randomly chosen position, the drop properties as size,concentration and drop age are set by a linear interpolation ofthe three surrounding drops properties, determined by a Delaunaytriangulation (Delaunay, 1934).

With this algorithm drops are added and deleted to have a con-stant number of representative drops. The deletion necessarilyleads to a change in the property distributions of the dispersephase and their integral properties as the Sauter diameter. In orderto minimize that change in integral properties caused by the algo-rithm and to resolve peaks in property distributions in an adequatedetail, the maximal representative drops in the compartment Nmax

drops must be high enough (Smith and Matsoukas, 1998). For exam-ple considering a uniform distribution of nine drops with a diame-ter of 100 mm and an additional drop with a diameter of 1000 mm.Only a single drop would picture the peak in the size distribution

Fig. 5. Algorithm for the calculation of representative drops for one time step.

B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010 7

and would be cancelled out with the probability of its number frac-tion (here 10%). Setting the maximal representative drops to 50instead of ten, 5 drops would represent the 1000 mm drop size peakand the probability of deleting this peak with the presented algo-rithm would shrink to only about 4:7 � 10�5%, which would pre-serve the drop size distribution and therefore its integralproperties too.

Some sub-models require the calculation of the volume fractionof the dispersed phase a. In every compartment the quotient ofsimulated representative drop mass mdrops to the total drop massmall drops is stored. Consequently the total mass of solvent per com-partment can be calculated and therefore the volume fraction.

2.4.4. Simulation settings of the compartment-modelThe liquid-liquid extraction part of the reactor (marked with a

grey rectangle in Fig. 2) is modelled with the two-dimensionalcompartment-model calculating drop motion, drop size distribu-

tions and mass transfer. The velocity profile of the continuousphase is loaded from the previously performed three phase CFDsimulation as described in Section 2.4.1. The three phase CFD sim-ulation is required, since the gas phase induces the loop flow in thereactor, which is decelerated by the counter-currently rising dropsin the downcomer.

The compartments have to be chosen so that the principal flowis captured well. Otherwise, the calculation of the drop movementand axial dispersion of the continuous phase can be incorrect. Atthis stage of development, we chose three compartments in radialdirection and eight in axial direction to capture the vortex in theupper part of the downcomer. In further investigations, differentcompartment grid resolutions have to be analyzed in more detail,for example by comparing tracer simulations. Other methods forautomatically choosing the grid resolution are given by Bezzoand Macchietto (2004), Bezzo et al. (2004) and Wells and Ray(2005).

Fig. 6. Exemplary distribution of representative drops in a compartment.

Fig. 7. Distribution of representative drops in radial zones 1–5 from Fig. 6.

Fig. 8. Cumulated probability density of representative drops in radial zones 1–5from Fig. 6.

8 B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010

At the top of the domain, water loaded with 0:025 kg=kgoftransfer component (acetone) flows into the extraction area. Theunloaded dispersed phase enters at the lower domain at a radiusof 0.08 m with the assumption of a Gaussian drop diameter distri-bution with a median of 2 mm, a standard deviation of 0.5 mm, aminimal diameter of 0.2 mm and a maximal diameter of 5 mm.Mass transfer of the exemplary transfer component, acetone, ismodeled between water and toluene.

In the compartments, a representative population of the totalamount of drops is calculated with the developed constant-number Monte-Carlo simulation approach. The number of repre-sentative drops is chosen according to a one-dimensional Monte-Carlo Simulation for liquid-liquid extraction columns (ReDrop),which requires 500–1000 representative drops per meter extrac-tion column (Kalem et al., 2011). In the compartment model,approximately four times as much representative drops per meterare chosen (50 drops per compartment and 1200 for 29.2 cmdowncomer). The time step size is set to Dt ¼ 0:1 s and the reactoris simulated until steady state for 1500 s. For the sub-models, fol-lowing parameters are chosen according to Henschke (2004):instability parameter CIP ¼ 9445, single-drop velocity parameters:

dum ¼ 7:1 mm, a15 ¼ 1:52 and a16 ¼ 4:5. In this study the coales-cence parameter was set to n ¼ 150 according to a parameter fit-ting of liquid-liquid extraction column experiments of Garthe(2006) with the ReDrop population balance model.

3. Results & discussion

This section presents the results of the CFD simulation of theimproved reactor geometry. Subsequently, the calculated dropmotion in the compartment-model is compared with CFD simula-tions and the calculated drop size distribution in the compartmentmodel is shown. First simulation results of the separation effi-ciency in the loop-reactors downcomer are given and the comput-ing times of CFD and the compartment-model are compared.

3.1. CFD simulation

The multiphase loop-reactor of Bednarz et al. (2017) was fur-ther developed based on CFD simulations. The volume fractionsof air and solvent as well as the water velocity profile are presentedin Fig. 9. The air rises in the center of the reactor, induces the loopflow and leaves the aqueous phase at the gas-liquid separator. Thesolvent rises counter-currently to the aqueous phase withoutentrainment into the aerated area. Therefore, with this setup thedispersed phases leave the reactor in different locations. Therefore,no further phase separation is required.

3.2. Fluid dynamic calculations in the compartment-model

The liquid-liquid extraction part of the multiphase loop reactoris simulated with the two-dimensional compartment-model (Sec-tion 2.4). The performed CFD simulation of the loop-reactor (Sec-tion 2.1) supplies the compartment-model with the detailed flowfield. Fig. 10 shows the velocity profile for the water phase of theCFD simulation and the compartment grid. The vertical watervelocity vectors are added at the boundaries of the compartments.The two left hand columns of compartments contain the down-ward directed flow and the right hand column is able to describethe vortex flow in the upper part of the downcomer. Consequently,Fig. 10 depicts that the principal flow is captured well with thechosen grid.

Fig. 9. CFD simulation results of multiphase loop-reactor – left: volume fractions (air on the left and solvent on the right); right: water velocity.

B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010 9

The distribution of the dispersed phase of the CFD simulationand compartment-model are compared in Fig. 11. The left side pre-sents the compartment grid with drops marked as circles. The sizesof the circles represent the drop diameter. The color of the com-partments displays the averaged volume fraction calculated in

Fig. 10. Compartments in the downcomer with velocity vectors at their horizontalboundaries and velocity profile of the continuous phase from CFD simulation.

the compartment and uses the same color code as the shown vol-ume fraction profile calculated in the CFD simulation on the righthand side.

Considering the drop motion, the general flow characteristic ofthe multiphase flow is captured well by the two-dimensionalcompartment-model. During their rise, they move to the outer wallof the reactor. This is also shown in the CFD simulation on the rightside of Fig. 11. Looking at the color code of the compartments, thedispersed phase volume fraction is lower in the upper part of thedowncomer in both simulations. This behavior results from achange in the overlapped flow of the continuous phase fromcounter-current in the downcomer’s lower part to co-current in

Fig. 11. Distribution of the dispersed phase in the downcomer (on the left:representative drops in compartment-model, on the right: volume fraction profilein CFD model).

10 B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010

the area of the vortex. The main deviation between thecompartment-model and the CFD model can be found in the lowerpart of the downcomer. In the CFD simulation, the drops movenearly horizontally to the wall bevor they move upwards, whereas

0.08m < height < 0.15m

0 1 2 3 4diameter [mm]

0

0.2

0.4

volu

me

fract

ion

0.15m < height < 0.23m

0 1 2 3 4diameter [mm]

0

0.2

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volu

me

fract

ion

0.23m < height < 0.30m

0 1 2 3 4diameter [mm]

0

0.2

0.4

volu

me

fract

ion

Fig. 12. Volumetric drop size distribution in three different height sections withSauter diameter, indicated by the dashed line.

0.06 0.08 0.1radius [m]

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

heig

ht [m

]

4

6

8

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14

mas

s fra

ctio

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eton

e [k

g/kg

]

10-3

00.08

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0.16

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heig

ht [m

]

Fig. 13. Concentration profile of the liquid disperse phase in each drop (left) and average1000 and 1500 s.

in the compartment-model they move directly upwards with aslight motion towards the wall. This is caused by averaging thevelocity field of the water phase in each compartment. The impactof averaging is especially high above the disperser where high localhorizontal water velocity vectors (found at the lowest compart-ment boundary in Fig. 10) are reduced in the compartment-model. To capture the radial movement of the drops, observed inthe CFD simulation, a finer local grid resolution is recommendedin the compartment-model.

The evolution of the volumetric drop size distribution in thedowncomer is plotted in Fig. 12. The distributions are similar tothe normal size distribution injected at the disperser (mean diam-eter = 2 mm). The Sauter diameter slightly increases from 1.95 mmat the lowest measuring area to 2.27 mm in the highest, due to coa-lescence. Therefore the drops are in the range of ideal drop diam-eters for liquid-liquid extraction columns reported by Goedecke(2006) (between 1.5 mm and 2.5 mm).

Summing up, the fluid dynamic calculations of the preliminaryCFD simulation has a higher resolution in space but neglects effectslike coalescence and mass transfer. These additional effects aremodeled in the compartment-model, where the special resolutionof the continuous phase velocity is reduced. However, the principalflow characteristic of the dispersed phase is captured well in thedowncomer of the multiphase loop-reactor.

3.3. Separation efficiency

With the compartment-model, we can determine the separa-tion efficiency of the liquid-liquid extraction part in the reactorconsidering the local flow field for the first time. During dropmotion, the transfer component enriches in the drops. Fig. 13 onthe left hand side depicts the concentration of each drop. On theright hand side, Fig. 13 gives the mass fraction of the transfer com-ponent in the dispersed phase averaged in the compartments atthe same height and averaged over the time span from 1000 to1500 s. The error bars represent the standard deviation during thistime. In the simulation, the continuous phase enters the simulationdomain loaded with 0.0250 kg/kg transfer component and leaves

0.002 0.004 0.006 0.008 0.01 0.012 0.014mass fraction acetone [kg/kg]

d in the compartments (right). Error bars represent the standard deviation between

B. Weber et al. / Chemical Engineering Science: X 2 (2019) 100010 11

the simulation domain with 0.0246 kg/kg transfer component. Thedispersed phase is at 85.1% equilibrium (0.0161 kg/kg) when leav-ing the simulation domain. Therefore, 26 g/(Lh) of transfer compo-nent is extracted from the reactor (15.5 L). According to theequation of Kremser (Goedecke, 2006), the downcomer hasapproximately 0.48 numbers of equilibrium stages. For higher sep-aration efficiency, internals could be included to increase dropbreakage and the residence time and therefore mass transfer.

3.4. Computing time of CFD and compartment-model

This compartment model is an approach, which breaks downthe computing effort for a device with complex multi-phase flowstill keeping a reasonable degree of detail. Therefore, a CFD modelcalculates the simplified fluid dynamics, which is afterwardsloaded into the compartment model to separately calculate popu-lation balances and mass transfer. Decoupling this phenomenaresults in 111.8 h over all simulation time for the downcomer partof the here presented device until steady state. Most of that com-puting time is required by the CFD simulation of the multiphaseloop-reactor (106 h on four cores Intel Nehalem EX X7550 ofRWTH compute-cluster) simulating 35 s until fluid dynamic steadystate neglecting effects like mass-transfer, coalescence and break-age of the drops. The compartment model calculates these effectsand reaches steady state after 1500 s. This requires 5.8 h comput-ing time on a standard desktop computer with Intel XenonES-2630. Including PBEs and mass transport into CFD, we expecta significant increase of computing times than calculating onlythe simplified fluid dynamic. For example Attarakih et al. (2015)simulated a liquid-liquid extraction column in technical scale(DN80, height 2.95 m) with a 2D CFD simulation coupled withPBE and one transfer component and consumed 27 days (648 h)computing time (2 AMD Opteron (4184) processors; each 6 cores).

Summing up, one would require even more computing time topredict a process intensified device like the loop reactor with PBE-CFD only, where coalescence, breakage and mass transport formultiple disperse phases (bubbles, drops) and reactions in the con-tinuous phase need to be taken into account. Exporting somephenomena in the compartment model promises a significantreduction in computing time. Moreover, the compartment modelstill has potential to be accelerated by parallelization since thecompartments are calculated sequentially.

4. Conclusion

We developed a CFD based compartment-model for multiphasedevices with complex flow pattern, which shall be used to designdevice dimensions and to determine operating parameters (e.g.gas and solvent flows). The developed compartment-model wasexemplarily applied for the multiphase flow in the liquid-liquidextraction part of a multiphase-loop reactor (Bednarz et al.,2016). In the compartment-model, we calculated the drop motion,drop-size distribution, mass transfer and resulting separation effi-ciency according to the local flow profile. The flow profile wasobtained by a multiphase CFD simulation with an Euler-Eulermodel developed by Bednarz et al. (2016). The population balancekernels of the drops in the compartment-model are solved with atime-driven constant-number Monte-Carlo method. The calculateddispersed phase distribution in the compartment model is in goodagreement with the previously validated CFD simulation model.

In comparison to a full PBE-CFD model, the compartment-model approach substantially reduces the computing time, sinceit decouples effects like mass transfer between the phases, dropbreakage and coalescence from the fluid dynamics calculation inCFD. Another advantage is that additional effects like multiple

component mass-transfer, multiple dispersed phases and reactionsare easily included into the model. The decoupling of these addi-tional effects from the CFD simulation is especially valid, whenthe hold-up is low and the mass transfer has a minor impact on vis-cosity, density and interfacial tension. Summing up, concerningaccuracy and computing time, the compartment model could beclassified in between CFD-PBE models and simpler one dimen-sional (in space) population balance models (Henschke, 2004;Attarakih et al., 2015), which neglect the local flow pattern.

In further investigations, models will be added to describe thewhole reactor including an additional gas phase (air) and a bio-catalytic reaction including the microorganism as solid phase dis-solved in the aqueous medium, so that the whole device can bemodelled and evaluated with the compartment model. Thecompartment-model will be used to improve the design and oper-ational parameters of the multiphase-loop reactor. In addition, thesensitivity of the number of representative drops per compartmentinfluencing the particle size distribution and of the time step sizehas to be determined. Furthermore, we will investigate the influ-ence of the assumption of a constant drop diameter in the CFDsimulation.

Conflict of interest

The authors declared that there is no conflict of interest.

Acknowledgments

We would like to thank the IT-Center of the RWTH AachenUniversity for the computational resources.

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