One-Group Reduced Population Balance Model for CFD Simulation of

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Ind. Eng. Chem. Res. 2010, 49, 3442–3451

One-Group Reduced Population Balance Model for CFD Simulation of a Pilot-Plant Extraction Column Christian Drumm,† Menwer Attarakih,†,‡ Mark W. Hlawitschka,† and Hans-Jo¨rg Bart*,† Lehrstuhl fu¨r Thermische Verfahrenstechnik and Center for Mathematical and Computational Modeling, TU Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany, and Chemical Engineering Department, Faculty of Engineering Technology, Al-Balqa Applied UniVersity, P.O. Box 15008, 11134-Amman, Jordan

In this work, a one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM) developed recently by Attarakih et al. (In Proceedings of the 19th European Symposium on Computer Aided Process Engineering, ESCAPE-19, Cracow, Poland, June 14-17, 2009; Jezowski, J., Thullie, J., Eds.; Elsevier: New York, 2009; ISBN-13: 978-0-444-53433-0) is implemented in the commercial computational fluid dynamics (CFD) package FLUENT 6.3 for solving the population balance equation in a combined CFD-population balance model (PBM). The one-group reduced population balance conserves the total number (N) and volume (R) concentrations of the population by directly solving two transport equations for N and R and provides a one-quadrature point for closing the unclosed integrals in the population balance equation. Unlike the published two-equation models, the present method offers accuracy improvement and internal consistency (with respect to the continuous population balance equation) by increasing the number of primary particles (sections). The one-group reduced population balance provides the possibility of a one-equation model for the solution of the PBM in CFD based on the mathematically consistent d30 instead of the classical d32 mean droplet diameter. Droplet breakage and coalescence are considered in the PBM, which is coupled to the fluid dynamics in order to describe real droplet behavior in a stirred liquid-liquid extraction column. As a case study, a full pilot-plant extraction column of a rotating disk contactor (RDC) type consisting of 50 compartments was simulated with the new model. The predicted results for the mean droplet diameter and the dispersed-phase volume fraction (holdup) agree well with literature data. The results show that the new CFD-PBM model is very efficient from a computational point of view (a factor of 2 less than the QMOM and a factor of 5 less than the method of classes). This is because the one-group reduced population balance requires the solution of only one equation (the total number concentration) when coupled to the CFD solver. It is therefore suitable for fast and efficient simulations of small-scale devices and even large-scale industrial processes. 1. Introduction Computational fluid dynamics (CFD) has been used with great success in multiphase flow applications arising in chemical engineering such as bubble and liquid-liquid extraction columns.1-8 This allows for the prediction of detailed hydrodynamics and turbulence characteristics using real equipment geometry. One widely used multiphase model for CFD simulations is the Eulerian multiphase model, which can describe dispersed multiphase flow and accounts for interactions between the dispersed continuous phases.1,7,8 In Euler-Euler multiphase flow, a two-fluid model is usually applied with a constant size for the bubbles, droplets, or particles in the dispersed phase. In reality, a wide particle size distribution can exist in the apparatus due to particle growth, aggregation, or breakage resulting from either mass transfer (particle growth) or interactions between the moving particles and the turbulent continuous phase or/and internal equipment geometry.9-11 Accordingly, this distribution of sizes results in different particle velocities with respect to size. Therefore, the constant-particlesize approach cannot describe the hydrodynamic behavior and, often, the coupled hydrodynamics and mass transfer of the dispersed phase. In fact, when coupling CFD and population balance models, many authors have shown the strong depen* To whom correspondence should be addressed. Tel.: +496312052414. E-mail: [email protected]. † TU Kaiserslautern. ‡ Al-Balqa Applied University.

dence of the bubble and droplet relative velocities on their diameters (e.g., Bhole et al.1 and Drumm et al.3). Moreover, in reality, the basic purpose of bubble or liquid extraction columns is to carry out mass-transfer operations. Because the local masstransfer coefficients (in both the continuous and dispersed phases) depend on the droplet or bubble diameters,11,12 there is no a priori justification for the assumption of equal rise velocities.1 As a possible solution, multifluid models6-8 divide the dispersed phase into classes of different sizes where each class represents one fluid. In this way, the model can predict size-dependent velocities because each class size moves with its own velocity field. Such an approach is more realistic, but unfortunately, it demands a high computational cost and hence a long CPU time. Moreover, it ignores the basic natural phenomena leading to particle size distributions such as breakage and coalescence. On the other hand, population balance modeling (PBM) takes into account particle growth, aggregation, breakage and nucleation and accommodates the dependence of velocity and mass transfer on particle size. This actually gives a full description of the dispersed phase with a result of an infinite number of partial differential equations due to the continuous variation of the particle internal properties (for example, the particles size ranges mathematically from 0 to ∞). Because of this dramatic increase in the computational load imposed by coupling the population balance equation to commercial CFD packages, there exist many numerical methods in the literature as attempts to reduce and solve the PBE in a reasonable computational time.

10.1021/ie901411e  2010 American Chemical Society Published on Web 02/23/2010

Ind. Eng. Chem. Res., Vol. 49, No. 7, 2010 13

Reviews of various methods are available in Ramkrishna, Marchisio et al.,14 Attarakih et al.,15-17 and Drumm et al.3 The common methods are briefly introduced here. Classical solution methods of the PBM that have been used for CFD-PBM coupling in the literature are the classes methods (CM) based on the fixed-pivot technique of Kumar and Ramkrishna18 and the quadrature method of moments19 (QMOM), the direct quadrature method of moments20 (DQMOM), and the sectional quadrature method of moments17 (SQMOM). In the CM, the particle size is discretized into a finite number of contiguous classes as a simple attempt to truncate the infinite particle size domain. The population in each class (or section) is considered to behave as a single fixed particle irrespective of the shape of the distribution function contained in the section. This results in a discontinuous reconstruction of the population density function over a truncated domain. On the other hand, in the method of moments, the distribution is destroyed, and only a small amount of information is retained in terms of its moments. In this way, a few scalar (transport) equations are written in terms of the moments of the distribution. The classical moment methods suffer from closure problems in the sense that integrals over the unknown distribution cannot be written closely in terms of low-order moments (see Ramkrishna13). The QMOM is able to resolve the closure problem with excellent accuracy by applying an adaptive Gauss-like quadrature. The nodes and the weights of this quadrature are obtained by solving an eigenvalue problem iteratively using the product-difference algorithm (PDA).19 It is well-known that the problem becomes illconditioned when large numbers of quadrature points are used or when the distribution becomes very sharp.17 In the limit as the distribution tends to the Dirac delta function, the PDA fails. A variation of the QMOM is the direct quadrature method of moments (DQMOM),20 in which the quadrature nodes and weights are tracked directly through the solution of a system of transport equations coupled to an algebraic linear system. The algebraic linear system has the same numerical difficulties as the eigenvalue problem associated with the PDA used in the QMOM. However, the obvious advantage of the DQMOM is its extension to bivariate population balances. Here, the problem of choosing a suitable number of low-order moments to avoid a singular linear system is unfortunately problem-dependent and is based on the average quantities to be tracked. In trying to alleviate this problem, Fox21 demonstrated that poor choices of the moment set can lead to nonunique abscissas and even negative weights. In his work, empirical optimal moment sets are sought when the multivariate population balance equation is solved using the DQMOM. Recently, the sectional quadrature method of moments (SQMOM) was introduced by Attarakih et al.17 to avoid the solution of large eigenvalue problems or the inversion of illconditioned linear systems. In mono- and multivariate cases, the SQMOM assures the production of uniform positive analytical weights and physically meaningful abscissas.17,22,23 The SQMOM is an adaptive method that combines the advantages of the CM and the QMOM and minimizes their drawbacks. The SQMOM is based on the concept of primary and secondary particles, where the primary particles are responsible for the distribution reconstruction (classes), whereas the secondary ones (method of moments) are responsible for breakage and coalescence events and carry detailed information about the distribution. The method can theoretically track any set of low-order moments with the ability to reconstruct the shape of the distribution. If one primary particle is used, the QMOM or, equivalently, the DQMOM is obtained. On the other

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hand, if one secondary particle is used, the moving-pivot technique is recovered in a more numerically efficient form than the original one.13 Other methods appearing in the literature (e.g., Bove et al.,24 Alopaeus et al.25) are merely special cases of the general SQMOM as shown by Attarakih et al.17,22 Confining ourselves to CFD and the population balance literature, coupled CFD-PBM models have been widely investigated.1,3,5,26,27 One can choose between different approaches for the coupling: a two-fluid or multifluid model in CFD and a whole set of models for the solution of the PBM (CM, QMOM, DQMOM, SQMOM). Therefore, the existing coupling approaches vary according to accuracy, efficiency, and recovered information about the distribution. The standard approach so far is to use the two-fluid model together with CM or QMOM. Lo28 was the first to implement a multiple-size-group (MUSIG) model for the solution of the population balance equations in the commercial CFD code CFX. Several authors have applied such an approach for the solution of multiphase flows.3-5,27,29-33 The information about the distribution can be increased by increasing the number of classes in the CM, but this is at the expense of a high computational load because of the resulting increase in the scalar equations imposed by the discrete form of the population balance equation. The CM is particularly useful when the particle sizes do not vary widely. On the other hand, one limitation is its inability to predict accurately integral quantities (low-order moments as a special case) associated with populations that have sharp shapes.13,16 The necessary number of classes is dependent on the range of particle sizes and the shape of the distribution. Usually, 20 and more classes are necessary to track sharply evolving distributions. The use of the QMOM reduces the number of tracked scalars in CFD, which is an obvious advantage. Usually, four or six moments are sufficient to conserve the moments of the distribution and even allow for an empirical reconstruction of the distribution.3 Nevertheless, the shape of the distribution is not available, and it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities.20 The QMOM was first implemented in CFD by Marchisio et al.26 and applied by many researchers in a two-fluid model.3,5,34,35 Krepper et al.,36 Sha et al.,8 and Bhole et al.1 developed a multifluid multisize group model based on the CM in which each fluid is represented by one class or a group of classes. Depending on the number of fluids and the number of classes in the PBM, the accuracy can be increased at the expense of increased CPU time. The computing time is always high because of the CM, and one has to find an effective tradeoff between accuracy, furnished information, and computing time. On the other hand, other coupled multifluid models are based on the DQMOM7,37 or on the SQMOM.38 In the DQMOM framework, each node of the quadrature approximation has its own velocity field in CFD. The DQMOM was carried out for two, three, or four nodes by Fan and Fox.37 However, the DQMOM is limited to this small number of fluids because the inversion of an ill-conditioned linear system is encountered as the number of quadrature nodes increases. The DQMOM also fails when a monosize distribution is encountered, and specialized ad hoc algorithms are needed to prevent the collapse of the method.20 Because of its dual nature (in the sense of it being a combination between finite-difference and QMOM schemes), the SQMOM represents a perfect basis to couple CFD and PBM in a multifluid model. The multifluid model accounts for the proper description of the fluid dynamics, and the SQMOM allows for an accurate description of the particle size distribution

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using two, three, or four moments in each section (class) or fluid. Any desired number of additional fluids or primary particles can also be easily added, as only the computational power sets the limit.38 Drumm et al.38 implemented the SQMOM in FLUENT in a three-fluid model using two primary particles in the SQMOM, whereas Tiwari et al.39 used five primary particles in a two-fluid model using a finite-pointset method (FPM) as a CFD solver. The efficiency of the method seems to be best for a small number of classes or fluids, because two, three, or four additional moments have to be tracked for each fluid. To summarize, many attempts have been made to improve the accuracy of coupled CFD-PBM simulations at the price of a higher computing time. One can chose between various methods that guarantee a high accuracy for the fluid dynamics and the information about the distribution. For a multifluid multisize group model, up to N additional transport equations for the N classes in addition to the equations for the number of M fluids have to be solved. The most efficient method at the moment seems to be the QMOM in a two-fluid model, as only four or six additional transport equations for the moments are tracked. That is why coupled CFD-PBM models are computationally expensive and still limited. Authors often help themselves by two-dimensional or symmetry assumptions for a three-dimensional geometry or consider only laboratory-scale devices.1,3,7,27,36 Moreover, because of the tremendous computational load, parameter fitting is hardly feasible.32 Therefore, it is usually of great industrial importance to simulate full-size equipment instead of simulating small segments. This is to gain insight into the overall equipment behavior due to different inlet disturbances and to the wide variation of the operating conditions encountered in actual industrial equipment. In the present article, instead of increasing the accuracy and the computational load, the opposite approach is followed by implementing a one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM)22,40 in the commercial CFD package FLUENT 6.3. The OPOSPM is a special case of the sectional quadrature method of moments17 in which only one primary particle and one secondary particle are used. This is indeed equivalent to a one-quadrature-point approximation in the QMOM framework. The reduced population balance model consists of only two transport equations, namely, the total number and total volume concentrations. The derivation and comprehensive test of the method were presented in Attarakih et al.,22,40 where the model was found to produce exact solutions for many popular breakage and coalescence kernels (frequencies). Because the volume concentration (fraction) is nothing more than the continuity equation of the dispersed phase, this results in a one-equation PBM in CFD, as the continuity equation is already implemented in a two-fluid Euler-Euler model. In addition, the source terms are very simple and allow fast and easy implementation of coalescence and breakage models. The implementation of the new method by user-defined scalars in the commercial CFD package FLUENT 6.3 is explained in detail in section 2. The coupled model can find many applications in the engineering fields where rapid calculations or full equipment geometry are needed. Here, we focus on one of the most important separation processes (after distillation): liquid-liquid extraction in agitated columns. In previous work, because of the limited computational resources and the demands placed on the solver by the full population balance model, only a small five-compartment section of a rotating disk contactor (RDC) (type of an agitated extraction

column) was investigated.2,3,39 The fluid dynamics in the column was simulated using constant droplet diameters in an Euler-Euler model and agreed well with the experimental particle image velocimetry results.2 Different closures for coalescence and breakage were implemented in CFD and applied in a combined CFD-PBM model using the CM and the QMOM.3 In the same work, a modified model of Luo and Svendsen41 for coalescence and breakage was found to result in reliable predictions of the experimental drop size distribution. On the other hand, Drumm et al.38 and Tiwari et al.39 implemented the SQMOM in two independent commercial CFD packages (FLUENT 6.3 and the free-mesh CFD solver of the Fraunhofer Institute) using up to five primary particles (five classes). The two solvers produced comparable results and good agreement with the builtin population balance solvers (in FLUENT 6.3) based on the QMOM. These results established the basis of the present reduced model in which only one primary particle and one secondary particle are used. It is worthwhile to state here that, by using only one primary particle and one secondary particle, the SQMOM is reduced to the one-group inconsistent equation (with respect to the zero moment) described by several researchers for coupling the population balance equation with CFD solvers such as CFX4.46,47 These authors, led by the work of Ishii et al.,48 derived the onegroup equation with no further consideration of the quadrature approximation of the unclosed integrals or how the accuracy of the method could be improved by increasing the number of either primary or secondary particles. Moreover, the total number concentration derived by Lane et al.46 was inconsistent with respect to the zero moment derived from the continuous population balance equation because of double-counting of the number of collisions between the same type of particles. In the present work, because of the reduced computational load imposed by the population balance solver, CFD simulations were carried out for a whole pilot-plant RDC extraction column consisting of 50 compartments using the OPOSPM together with the model of Luo and Svendsen.41 The one-equation PBM in CFD makes the simulation of the full pilot-plant column possible, and the results of the mean drop size and volume fractions are compared to the available experimental data.42 This article is structured as follows: The one-group reduced population balance model (or OPOSPM) and the computational model are briefly described. A test case is presented to compare the one-group reduced population balance to the QMOM. Finally, simulation results for the whole pilot-plant RDC extraction column are reported. A summary and conclusions are provided at the end of the article. 2. Reduced CFD-PBM Model In this section, the reduced CFD-population balance model is presented based on the idea of the sectional quadrature method of moments (SQMOM).17 Here, the CFD framework for coupling is introduced, followed by the OPOSPM as a special case of the SQMOM. These two techniques and the kernels of Luo and Svendsen41 for drop breakage and coalescence complete the mathematical model. 2.1. CFD Framework. A two-fluid model in FLUENT 6.3 based on the Eulerian approach, where all phases are treated mathematically as interpenetrating continua, is applied for the coupling. The conservation equations are solved for each phase. The continuity equation for the phase i is ∂(RiFi) + ∇ · (RiFiui) ) 0 ∂t

(1)


with the constraint R1 + R2 ) 1

(2)

The conservation of the momentum of phase i is expressed as ∂(RiFiui) + ∇ · (RiFiuiui) - ∇ · τi ) -Ri∇p + RiFig + Fi ∂t (3) Only the drag force was taken into account for the interphase forces Fi between the continuous and dispersed phases. The drag force is represented by Fc,d )

3FcRcRdCD |ud - uc |(ud - uc) 4dd

(4)

The model of Schiller and Naumann43 was taken for the drag coefficient. The model can describe terminal velocities of drops in a liquid-liquid system.3 For turbulence modeling, the standard k-ε turbulence model was applied, together with standard wall functions and the FLUENT mixture turbulence model. First-order upwind differencing schemes were used to discretize the convection terms initially, and QUICK schemes were used for the final solution. PRESTO was applied as the discretization method for pressure. The first-order implicit scheme was used for time advancing, and the pressure-velocity coupling was done using the SIMPLE algorithm in two-phase flow. 2.2. One-Group Reduced Population Balance Model Based on the OPOSPM. The general population balance equation in terms of a number concentration function n is written as ∂ [F n(V, t)] + ∇e · [uFdn(V, t)] + ∇i · [GFdn(V, t)] ) FdS(V, t) ∂t d (5) where V is the drop volume and S(V,t) is the source term that accounts for the net number of drops (per unit volume) generated by breakage and coalescence due to birth and death rates, B and D, respectively. The symbols ∇e and ∇i are the gradients with respect to external and internal coordinates, respectively, and G is the particle growth rate (e.g., due to interphase mass transfer). The birth terms of drop breakage and coalescence are due to the interactions among the drops themselves or between drops and the turbulent continuous phase. The birth of drops by breakage and coalescence is accompanied by drop death, and the source term can be further expanded as S(V, t) ) BC(V, t) - DC(V, t) + BB(V, t) - DB(V, t)

(6)

The breakage and coalescence kernels (frequencies) are usually functions of the droplet size, the system physical properties (the most important of which are the viscosity and surface tension), and the turbulence energy dissipation.10 These source terms involve linear (breakage) and nonlinear (coalescence) integrals,13 and hence, eq 5 is an integro-partial differential equation. The general solution of eq 5 does not exist, and hence, numerical approximations are needed. Attarakih et al.17 showed that many related sectional and quadrature methods appearing in the literature for the numerical solution of the population balance equation can be considered as special cases of the SQMOM. In the SQMOM framework of discretization, the particle size (V) is divided into Npp (number of primary particles) contiguous sections. These primary particles are responsible for the distribution reconstruction. If the shape of the distribution itself is not of engineering interest, then a single

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primary particle can be used. Because each primary particle is associated with Nsp secondary particles, these particles can estimate any number of low-order moments belonging to the underlying distribution. Actually, each secondary particle can selectively represent two low-order moments and provides a quadrature node for estimating an unclosed integral. When the number of secondary particles is equal to 1, a sectional method of one-node integration quadrature is recovered with one primary particle for each section. On the other hand, when the number of primary particles (sections) is equal to 1, the classical QMOM is recovered, with the secondary particles representing exactly the nodes of the integration quadrature (see Attarakih et al.17). On the other hand, a large number of primary particles in the classical sectional methods is required, not only to reconstruct the shape of the distribution, but also to estimate the desired integral quantities associated with the distribution, which are normally predicted with low accuracy. The SQMOM is able to overcome this fundamental problem of the sectional methods by using Nsp secondary particles in each section, where 2Nsp low-order moments can be reproduced. The one-group reduced population balance model based on the one primary and one secondary particle method22 (OPOSPM) is introduced as a special case of the SQMOM by using only one primary particle and one secondary particle. Therefore, the OPOSPM is the simplest form of the SQMOM in which two low-order moments can be conserved where the secondary particle can be imagined as a Lagrangian fluid particle carrying information about the droplet population through its low-order moments. Although the selection of these moments is arbitrary, the total number and volume concentrations are the most natural candidates for the conservation of the total number and mass of droplets. The idea is briefly summarized here, whereas the complete derivation can be found in the original works of Attarakih et al.17,22,40 In the OPOSPM, the secondary particle coincides exactly with the primary particle. Because the total number and volume concentrations are conserved, the population density is represented by a single particle (assumed to have a spherical shape) whose position (size) is given by d30 )

3

πRd ) 6Nd

3

m3 m0

(7)

where N and R are the total number and volume concentrations, respectively. These are related to the zeroth (m0) and third (m3) moments of the distribution. Note that d30 is the only adaptive (with respect to space and time) integration quadrature node and the weight is nothing more than Nd. d30 represents the particle mean mass diameter. This mean diameter is widely used in the literature on particulate solids44 and bubbly flows as well.46-48 Now, the transport equation for R is derived from the population balance equations (eqs 5 and 6) by mathematically representing the number density function by a single Dirac delta function centered at V(d30), namely, n ) Nδ[V - V(d30)]; multiplying eqs 5 and 6 by V; and integrating both sides with respect to V from 0 to ∞ to obtain ∂(RdFd) + ∇ · [RdFdud(d30)] ) FdG(d30)Nd ∂t

(8)

where G is the drop growth rate accounting for interphase mass transfer (it is assumed to be negligible in this work). On the other hand, the number concentration transport equation is derived by setting n ) Nδ[V - V(d30)] and integrating both sides of eqs 5 and 6 with respect to V from 0 to ∞ to obtain

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∂ (F N ) + ∇ · [Fdud(d30)Nd] ) FdS ∂t d d

(9)

The source term appearing in eq 9 accounts for droplets’ breakage and coalescence and is given by Attarakih et al. as22 1 S ) [nd(d30) - 1]g(d30)Nd - a(d30, d30)Nd2 2

(10)

Here, nd is the mean number of daughter particles due to breakage and g and a are the breakage and coalescence kernels (frequencies), respectively. Note that the source term in the OPOSPM is very simple in comparison to those in common solution methods (e.g., QMOM, SQMOM, MC). The source terms in the CM and QMOM normally consist of integrals, sums, or vector and matrix multiplications (e.g., see refs 13, 17, and 26). Here, two single terms account for the breakage and coalescence of the droplets. Because only the total number of droplets is of interest, functions for the daughter droplet distribution are implicitly implied and reflected by nd, which is, in general, a function of the mother droplet size. The same could be said about the aggregation kernel, where only collisions of equal-sized droplets are considered a(d30,d30). Despite the simplicity of the source term (eq 10), it emphasizes the nature of the first-order linear breakage process and the second-order nonlinear droplet coalescence. This is the exact information drawn from the continuous population balance equation. Note that eqs 9 and 10 are similar to the equations appearing in the work of Lane et al.,46 Kerdouss et al.,47 and Ishii et al.48 However, those authors were concerned only in deriving oneequation (or two-equation) models to predict the interfacial area concentration or the mean droplet diameter without paying attention to the quadrature approximation of the unclosed integrals appearing in the convective and source terms of the population balance equation. They did not introduce a general numerical framework (such as the SQMOM framework) to improve the accuracy of the reduced model or to derive the multifluid model from the continuous population balance equation. Failure to recognize the connection between the quadrature method of moments and the closure of the unclosed integrals led to an inconsistent form of the coalescence source term in the number balance equation (see, for example, eq 28 in Lane et al.,46 in which the coalescence events between two similar groups of particles are counted twice). This will result in serious errors when the fitted parameters in their equations are used in general population balance equations. The population balance equation is now represented by two transport equations conserving both the total number and volume of the whole droplet population. The accuracy of the method can be improved by increasing either the number of primary particles (sections) or the number of secondary particles (quadrature nodes),17 and here lies the fundamental difference between the present method and that derived by Ishii et al.48 and Lane et al.46 Additionally, the OPOSPM can be extended to a multiple-primary-particle and one-secondary-particle method to improve the accuracy of the adaptive integration quadrature and starts reconstructing the distribution in a natural way. For liquid-liquid problems without considering interphase mass transfer, eq 8 is identical to eq 1 in the two-fluid model. Therefore, for the solution of the PBE in CFD, only one additional transport equation for the total number concentration (eq 9) is needed. This equation is introduced in FLUENT 6.3 as a user-defined scalar. The simple source term is written as a user-defined function and added to the transport equation. For the purpose of two-way coupling with the fluid dynamics, the standard approach in the literature is to calculate d32 (surface

Table 1. Dimensions of the RDC Extraction Column parameter

symbol

value

column diameter (mm) shaft diameter (mm) compartment height (mm) height stirrer/stator (mm) diameter stirrer (mm) stator inner diameter (mm)

DK Dshaft HC HR DR Ds

150 54 30 1 90 105

mean diameter) in every time step and return it back to the drag force (eq 5).3,28 Because the second moment (droplet surface area) is not available, we have a different coupling based on the available d30. This seems mathematically more logical than using d32, because d30 is a natural quadrature node whereas d32 is not. This agrees with the approach of Lane et al.46 and Kerdouss et al.47 When the third moment is normalized to 1, d30 is simplified to d30 )

3

1 m0

(11)

The zero moment is included as a user-defined scalar in FLUENT. The governing values in the source terms for coalescence and breakage (e.g., the turbulent energy dissipation ε or the volume fraction R) are returned for each cell from the FLUENT CFD solver. The velocity ud in eqs 8 and 9 is calculated from the solution of Navier-Stokes equations; thus, a complete two-way coupling between CFD and PBM is assured. The model of Luo and Svendsen41 describing the breakage and coalescence of the droplets was used in the previous work of Drumm et al.3 The model was investigated in a fivecompartment section of the present pilot-plant column. The model gave good results in terms of the droplet size distribution and the Sauter mean diameter (d32), when a constant scaling factor of 0.1 for the coalescence kernel was introduced a(d30, d30) ) 0.1aLuo and Svendsen(d30, d30)

(12)

The same factor was applied in the present work to investigate the validity for the full pilot-plant column. This constant factor (0.1) is more an engineering estimate than an adjustment and was also used by Chen et al.30 One could obtain a “better” comparison against experimental data by adjusting this factor but would lose the predictive nature of CFD.30 The governing equations of this model can be found in the original works of Luo and Svendsen41 or Drumm et al.3 3. Simulations of a Pilot-Plant Extraction Column Simulations were carried out for a pilot-plant RDC extraction column with 50 RDC compartments and 150-mm inner diameter (Table 1 and Figure 1). The total length exceeds 2 m. A water-toluene system was considered, with toluene as the dispersed phase introduced at the bottom of the column while the continuous phase (water) flows countercurrently from the top. The literature data for the drop size distribution and the holdup are available for this system at different throughputs and stirrer speeds.42 The drop size at the inlet (d30,in) was calculated from the drop size distribution. The RDC column was operated at throughputs of 100 and 112 L/h for the aqueous and organic phases, respectively. The stirrer speed was set to 250 and 300 rpm. The computational model was a twodimensional axisymmetric geometry consisting of 85000 quadrilateral cells with 1-mm grid spacing. Velocity-inlet and pressure-outlet boundary conditions were applied at the top and


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Figure 1. Sketch of a rotating disk contactor compartment.

Figure 3. Test case: comparison of OPOSPM (d30) and QMOM (d30 and d32).

Figure 2. Details of the computational grid.

bottom of the computational grid (see Figure 2). A time step of 0.05 s size was chosen, and a duration of more than 200 s was simulated so that a steady-state solution is assured. 4. Results and Discussion 4.1. Test Case: Five-Compartment RDC Segment. As a test case, a five-compartment segment of the pilot-plant column was simulated as in the previous work of Drumm et al.3 The purpose of this test case was the correct implementation of the one-group reduced population balance model based on the OPOSPM and the influence of the different droplet diameters (d32 and d30) on the two-way coupling. The predictions of the one-equation model were compared to the results of the wellknown QMOM. First, the predicted d30 values in the two models were compared. Because both models can conserve the tracked moments, the resulting d30 values should be approximately equal. Consequently, any differences can be attributed to the accuracy of our present one-node integration quadrature and the multinode QMOM quadrature. Second, differences were investigated when d30 was used for coupling in eq 4 instead of the Sauter mean diameter (d32). The simulations were carried out for the system toluene-water and constant source terms for breakage and coalescence given by eq 10. A monomodal drop size distribution at the dispered-phase inlet was assumed. Therefore, the two diameters d30 and d32 were equal at the bottom of the section (d30 ) d32 ) 2.66). A uniform daughter droplet distribution was assumed for the QMOM, whereas the one-group reduced

population balance model does not need assumptions for the daughter droplet distribution. Laminar flow was considered in this test case, because the aim was not the correct prediction of the fluid dynamics but the validation of the present new model. In Figure 3, results of the predicted mean diameters are shown for a constant breakage kernel (g ) 0.2). The results show that the values of diameter d30 are approximately equal in the two methods, as expected. This is because the one-group reduced population balance model can conserve the zeroth and third moments of the distribution similarly to the QMOM. Furthermore, the Sauter mean diameter (d32) is different in the QMOM, as expected, and therefore, the coupling to the fluid dynamics with the present new model is different. The influence on the fluid dynamics due to the different coupling in eq 4 is depicted in Figure 4. The volume fraction of the dispersed phase is slightly higher with the one-group reduced population balance model (d30) because d30 is less than d32. According to eq 4, the drag force is approximately inversely proportional to the mean droplet diameter, which results in a larger drag force (in the case of using d30) and, hence, higher dispersed-phase holdup. Actually, the holdup in the five sections is 5.2% using the onegroup reduced population balance model instead of 4.4% when the QMOM (d32) is used. Although the coupling with d30 is different from the common coupling in CFD-PBM models using d32, it is not necessarily worse, but one should keep in mind the differences and the correct mathematical coupling based on d30 instead of d32. 4.2. Full Pilot-Plant-Column Simulation. A full pilot-plant extraction column of a rotating disk contactor (RDC) type was simulated using FLUENT 6.3 as a CFD solver and the onegroup reduced population balance model as the population balance model. The liquid-liquid flow was modeled using the standard k-ε turbulence model in conjunction with the Eulerian two-fluid equations. For the solution of the PBM, the one-group reduced population balance model was introduced in FLUENT together with the modified model of Luo and Svendsen.41 The number of daughter droplets in eq 10 was taken as 2.5 according to a daughter droplet distribution given by Schmidt et al.10 In the following section, the simulation results obtained using the system toluene-water where the RDC was operated at stirrer

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Figure 6. Comparison between predicted and experimental mean mass diameter (d30) at the outlet at a stirrer speed of 250 rpm.

Figure 4. Test case: d30 vs d32 influences on the fluid dynamics.

Figure 5. Inlet and outlet drop size distributions and calculated mean droplet diameters d30 and d32 at a stirrer speed of 250 rpm.

speeds of 250 and 300 rpm are shown and compared to the experimental data of Modes.42 The volumetric flow rates in all simulations were set to 100 and 112 L/h for the aqueous and organic phases, respectively. The experimental inlet and outlet drop size distributions, as well as the calculated d30 and d32 values, are shown in Figure 5 at a stirrer speed of 250 rpm. Because of the relatively high surface tension of the chemical system (toluene-water), coalescence is believed to be hindered, and hence, droplet breakage dominates at 250 rpm. This results in a smaller mean droplet diameter (d30) at the outlet compared to that at the inlet. The experimental mean droplet diameter at the inlet (d30,in) was also taken as the inlet boundary condition in the simulation. The predicted outlet mean droplet diameter is shown in Figure 6 and compared to the experimental one at the outlet. The mean droplet diameter decreases from d30,in ) 4.28 mm to around d30,out ) 3.03 mm in the simulation and agrees well with the experimental outlet mean droplet diameter of d30,out ) 2.95 mm. The PBM source terms given by eq 10 were turned on in the volumes of the 50 compartments, whereas no source terms were used at the top and bottom volumes near the inlets and outlets of the phases. When the source terms were turned on in these regions, the coalescence was too strong and led to higher drop sizes. The contours of the mean droplet diameter (d30) inside the compartments are shown in Figure 7. It is clear that the droplets break up near the tip of the stirrer

Figure 7. Contours of predicted mean mass droplet diameter d30, inside one compartment at a stirrer speed of 250 rpm.

(due to the shearing force: see Schmidt9) resulting in small mean droplet diameters of around 2 mm. Moreover, breakage and coalescence are strong functions of the turbulent energy dissipation, which reaches the highest values at the tip of the stirrer. On the contrary, the largest mean droplet diameters are found under the stators, where the droplets accumulate in the stagnant regions under the stirrers. A comparison of the predicted local droplet mean mass diameter as shown in Figure 7 with the experimental data is not possible because measurements of the local droplet size distributions (from which the local mean diameters were calculated) are not available in the published literature. The model probably cannot describe the real local size distribution with a high precision, but the predicted behavior is reasonable and encouraging. The contours of the volume fraction (holdup) of the droplets are shown in Figure 8. It is obvious that the droplets accumulate under the stators and move mainly through the middle of the compartments. In summary, the model can reproduce the wellknown droplet behavior inside the RDC compartment, where droplets accumulate under stators and break up at the stirrer tip. This behavior was also experimentally observed by many researchers.2,9,10,42 The mean holdup calculated in one compartment was around 8%, whereas the experimental measured holdup was around 10% at a stirrer speed of 250 rpm. The deviations could be partially


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Table 2. Experimental and Predicted Mean Droplet Diameters at the Top and Bottom of the RDC Column stirrer speed (rpm)

d30,in (mm) d30,out,exp (mm) d30,out,sim (mm) holdup (%) experimental simulated

250

300

4.28 2.95 3.03

4.11 2.32 2.70

8 7.4

10.7 8.6

Table 3. Comparison between the QMOM, CM, and Present Model (One-Group Reduced Population Balance Model Based on the OPOSPM) in Terms of Computational Time

QMOM, 6 moments CM, 30 classes OPOSPM without PBM Figure 8. Contours of predicted dispersed-phase volume fraction (holdup) at a stirrer speed of 250 rpm.

Figure 9. Comparison between predicted and experimental mean mass diameter (d30) at a stirrer speed of 300 rpm.

explained by the way the holdup was measured. In the literature,10,42 the holdup was measured by taking 500 mL samples of the column content through sample ports. Because the droplets are not dispersed in a homogeneous way everywhere (they move mainly through the middle of the compartment and accumulate under the stirrers and stators), it is questionable whether such measurements can reflect the real holdup accurately. Depending on the position of the sample ports, the samples could contain more or less dispersed phase. Future phase-Doppler anemometer measurements could provide the local drop size, volume fraction, and droplet velocity in one compartment for a better comparison and a further improvement of the models. Figure 9 compares the predicted and experimental d30 mean droplet diameters at 300 rpm. The overall deviation between the experimental and predicted mean droplet diameters is now higher but is still below 20%. The predicted mean droplet diameter (d30) at the outlet is around 2.7 mm, whereas the experimental value is 2.32 mm. Finally, the simulated and experimental mean droplet diameter and dispersed-phase holdup at stirrer speeds of 250 and 300 rpm are summarized in Table 2. As expected, the remarkable efficiency of the present reduced population balance model is the dramatic reduction of the computational time. The computational times of the OPOSPM, QMOM (6 moments), and CM (30 classes) using a single 3

1 time step, 25 iterations

4000 time steps at a time-step size of 0.05 s

64 s 160 s 34 s 32 s

71 h 178 h 38 h 36 h

GHz CPU performing a single task are compared in Table 3. It is clear that the solution of the PBE using the one-group reduced population balance model is just slightly higher than that without the PBM. This is because only one transport equation is defined in FLUENT (eq 9) in addition to the user-defined functions for breakage and coalescence kernels. On the other hand, the QMOM using 6 moments doubles the computing time, whereas the CPU time using 30 classes in the CM increases by a factor of 5. The last column gives the overall computing time for 4000 time steps at a time-step size of 0.05 s (200 s of real time). To recapitulate, despite the one quadrature nature of the onegroup reduced population balance model, it still allows fast and efficient CFD-PBM simulations and has very simple source terms when compared to the QMOM or CM, where expensive (from a computational point of view) integrals have to be numerically evaluated. On the other hand, it seems “useless” to use accurate (and, of course, computationally expensive) PBM when the currently available models for coalescence and breakage are not of high precision and can hardly account for the prediction of the drop size distribution without adjustable parameters. In addition, theses models are strong functions of the multiphase turbulence models and especially the turbulent energy dissipation. In this connection, it is well-known that the current available multiphase models still demand improvement.29 Therefore, by using the coupled CFD-one-group reduced population balance model, the computational time does not increase appreciably, and the combined model is still capable of reproducing the experimental data. For the simulation of an entire pilot plant (or even industrial-scale columns), the computational time becomes crucial, and hence, the one-group reduced population balance model is offered to bridge this gap in simulation between small- and large-scale devices. From the scaleup point of view, the scaling factor used in the model of Luo and Svendsen, which was estimated in a five-compartment section of the column, is still valid for the whole pilot-plant column. Thus, the one-group reduced population balance model coupled to CFD solvers provides a feasible tool for checking the scaleup of extraction columns. As a possible future approach, one could estimate parameters in the models for coalescence and breakage in laboratory-scale or small column parts and then use them for large-scale devices. This approach provided reasonable results for scaling up extraction columns and was applied successfully in stand-alone PBM codes such as LLECMOD.45 LLECMOD was used by Schmidt et al.9,10 to

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scale up RDC and Ku¨hni extraction columns using small laboratory-scale devices to carry out the experimental work. 5. Summary and Conclusions In this work, the one-group reduced population balance model based on the one primary and one secondary particle method (OPOSPM) was implemented in the commercial CFD code FLUENT 6.3 for solving the population balance equation in a combined CFD-population balance model. The one-group reduced population balance model offers the possibility of a one-equation model for the solution of the PBM in CFD based on the d30 mean droplet diameter as a means of two-way coupling. Unlike the published two-equation models, the onegroup reduced population balance model based on the OPOSPM (being a special case of the SQMOM) establishes the basis of multifluid models with an internal consistency with the continuous population balance equation. The method was compared to the QMOM for a test case, and the two methods produced comparable results. Differences in the coupling due to the different mean droplet diameters (d30 or d32) were shown and discussed. It can be concluded that using d30 as the mean droplet diameter for coupling has a more mathematical basis as a one quadrature node than does using the classical d32 mean droplet diameter. The results show that the new CFD-PBM model is very efficient from a computational point of view with the conservation of total droplet mass and number. The computational time of the present model was found to be two times less than that of the QMOM and five times less than that of the CM. Droplet breakage and coalescence were considered in the CFD-PBM in order to describe real droplet behavior in a stirred liquid-liquid extraction column of the RDC type. The model of Luo and Svendsen was incorporated to account for coalescence and breakage of droplets. A scaling factor for the coalescence kernel was estimated in previous work in a fivecompartment section of the column and also applied for the simulation of the pilot plant. For realization of the CFD-PBM, a full pilot-plant extraction column of RDC type was simulated. The predicted results for the mean droplet diameter and volume fraction were shown to be in good agreement with the published literature data. Hence, one can conclude that the one-group reduced population balance model coupled to CFD solvers is suitable for fast and efficient simulations of small laboratory devices or even complete processes. Acknowledgment The authors acknowledge the Deutsche Forschungsgemeinschaft (DFG) for the financial support. Nomenclature a ) coalescence kernel, 1/(m3 s) B ) birth rate, 1/s CD ) drag coefficient d ) droplet diameter, m D ) loss rate, 1/s F ) interaction force, N G ) growth rate with respect to internal coordinates, m/s g ) breakage frequency, 1/(m3 s) g ) gravitational constant, m/s2 mk ) kth moment, mk-3 N ) number of droplets n ) stirrer revolutions, rpm nd ) number daughter droplets

p ) pressure, kg/(m s2) t ) time, s u ) velocity, m/s V ) volume, m3 X ) coordinate, m Greek Letters R ) volume fraction υ ) kinematic viscosity, m2/s F ) density, kg/m3 τ ) stress-strain tensor, N/m2 AbbreViations CFD ) computational fluid dynamics CM ) classes method DQMOM ) direct quadrature method of moments OPOSPM ) one primary one secondary particle method PBM ) population balance model QMOM ) quadrature method of moments RDC ) rotating disk contactor rpm ) revolutions per minute SQMOM ) sectional quadrature method of moments

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ReceiVed for reView September 9, 2009 ReVised manuscript receiVed February 2, 2010 Accepted February 8, 2010 IE901411E

One-Group Reduced Population Balance Model for CFD Simulation of

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