Chemical Engineering Journal

Transcription

1 Chemical Engineering Journal 225 (2013) Contents lists available at SciVerse ScienceDirect Chemical Engineering Journal journal homepage: Euler Euler simulation of bubbly flow in a rectangular bubble column: Experimental validation with Radioactive Particle Tracking Ankur Gupta, Shantanu Roy Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi , India highlights " Exhaustive Euler Euler CFD coupled with population balance, in air water 2D-bubble column. " Equivalence of four different population balance methods shown. " Simulations show excellent agreement with Radioactive Particle Tracking (RPT) experiments. " Systematic investigation of drag and lift forces. article info abstract Article history: Available online 10 November 2012 Keywords: Bubble column Radioactive Particle Tracking Euler Euler CFD Drag force Lift force Virtual mass effect Turbulence models Population balance This work presents an Euler Euler (E E) Computational Fluid Dynamics (CFD) model developed for flow inside a rectangular air water bubble column, in which we have also implemented population balance method (PBM) based evolution of the various bubble size classes using the Homogeneous Discrete method, Inhomogeneous Discrete method, Quadrature Method of Moments (QMOM) and Direct Quadrature Method of Moments (DQMOM) technique. The results are validated with liquid velocity field measurements obtained from the Radioactive Particle Tracking (RPT) technique. The study presents a comparative analysis of the effect of incorporating various interfacial closures like drag force, lift force and virtual mass effect in the E E CFD coupled with PBM framework. The importance of the lift force in the transient three-dimensional simulations for predicting the time-averaged velocity profiles is highlighted. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Bubble column reactors are industrially important reactors with applications ranging from fermentation broths to slurry reactors [1]. Since reaction rates inside these columns are often limited by heat and mass transfer, the knowledge of flow patterns inside these columns becomes crucial in their sizing, troubleshooting and scale-up. To develop an insight into this arguably complex fluid flow phenomena and in particular the liquid velocity field, various experimental techniques like Particle Image Velocimetry (PIV), Laser Doppler Anemometry (LDA) and Radioactive Particle Tracking have evolved in recent decades [2 6]. Apart from experimental studies, numerous CFD studies are available which have attempted to model the flow by using various drag and lift models [7 16] within the Euler Euler framework. Bubble size distribution (BSD) inside conventional bubble columns is often highly dynamic and wide (especially in the so-called Corresponding author. Tel.: ; fax: address: roys@chemical.iitd.ac.in (S. Roy). churn turbulent regime where a typical bubble size can vary from 5 mm to 5 cm), an effect which is not captured in the conventional Euler Euler (E E) description of flow because of its single bubble size assumption. Further, in conventional E E modelling, the dispersed phase (gas) is considered to be a pseudo-continuum hence unless explicitly accounted for, would represent a single bubble size in the closures for various forces such as drag, and would also represent a single advection phase velocity. These issues can be partially addressed by coupling E E CFD with Population Balance Modelling (PBM) wherein an extra conservation equation (population balance equation or PBE) to capture the evolving bubble size distribution (BSD) is solved. PBE is an integro-differential equation [17] and therefore to solve it, special techniques like Discrete Homogenous Method [17 19], Discrete Inhomogeneous Method [20], Quadrature Method of Moments (QMOM) [21] and Direct Quadrature Method of Moments (DQMOM) [22,23] have been developed. A summary of the various works involving coupling of E E CFD with PBM in bubble column flow modelling has been shown in Table 1. Even though common in their philosophical origin, the different PBM approaches are quite distinct from the point /$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.

2 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Nomenclature a(l 1, L 2 ) aggregation Kernel w.r.t. bubble sizes L 1 and L 2 (s 1 ) a(v 1, v 2 ) aggregation Kernel w.r.t. bubble volumes v 1 and v 2 (s 1 ) B a birth term due to aggregation (number of particles/ time) B a,i birth term due to aggregation for the bin of index i (number of particles/time) B b birth term due to breakage (number of particles/time) B b,i birth term due to breakage for the bin of index i (number of particles/time) B a;k integrated birth term due to aggregation after taking kth moment B b;k integrated birth term due to breakage after taking kth moment C f constant in breakage model C 1e constant in turbulence model C 2e constant in turbulence model C l constant in turbulence model C ij convection term in turbulence model C d drag coefficient ( ) C L lift coefficient ( ), constant in Luo model d, d B diameter of the phase considered (m) d i, d j diameter of the bubble represented by an index i or j (m) d 3,2 Sauter mean diameter of the distribution (m) D a death term due to aggregation (number of particles/ time) D a,i death term due to aggregation for the bin of index i (number of particles/time) D b death term due to breakage (number of particles/time) D b,i death term due to breakage for the bin of index i (number of particles/time) D a;k Integrated death term due to aggregation after taking k th moment D b;k integrated death term due to breakage after taking kth moment D T,ij turbulent diffusion term in turbulence model (kg/ms 3 ) D L,ij molecular diffusion term in turbulence model (kg/ms 3 ) Eo Eotvos number ( ) ~ Fk external force per unit volume (N/m 3 ) f v fraction of the diameter into which the parent droplet is breaking ( ) ~ Flift lift force per unit volume (N/m 3 ) ~ F vm virtual mass force per unit volume (N/m 3 ) g(l) breakage Kernel w.r.t bubble size (s 1 ) g(v) breakage Kernel w.r.t bubble volume (s 1 ) H height of column (free surface((m) k Turbulent Kinetic Energy (m 2 /s 2 ), summation constant _m pk Mass transfer from phase p to phase k (kg/m 3 ) P C collision efficiency ( ) I identity matrix K pk momentum exchange coefficient between phases p and q (N s/m 4 ) L bubble size (m) ith abscissa (m) L i m k kth moment of bubble size distribution (m k ) n(v) number size distribution for a given volume of bubble size, location and time p pressure field, shared between all the phases (Pa) P ij stress production term in turbulence model ~ Rpq drag force per unit volume between phases p and q (N/ m 3 ) Re Reynolds number ( ) t time of flow (s) ~u k velocity vector for kth phase(m/s) _u i bubble rise velocity for the bubble representative by index i (m/s) u 0 mi fluctuation in the mixture phase velocity in i coordinate direction (m/s) U g gas superficial velocity (m/s) v volume of bubble ~v m velocity vector for mixture phase (m/s) x k length variable x in coordinate direction k n constant in breakage model n(v) number size distribution for a given volume of bubble size, location and time W width of column (m) w i ith weight ( ) We ij Weber number for coalescence of bubbles with diameter d i and d j z Height above the distributor Greek letters a k volume fraction for kth phase ( ) a g volume fraction for gas phase ( ) e Turbulent Kinetic Energy dissipation rate (m 2 /s 3 ) q k density for kth phase (kg/m 3 ) q l density for liquid phase (kg/m 3 ) q g density for gas phase (kg/m 3 ) q m density for mixture phase (kg/m 3 ) s k stress tensor for kth phase (Pa) l k bulk viscosity for kth phase(kg/ms) l t,m turbulent viscosity for mixture (kg/ms) k k shear viscosity for kth phase (kg/ms) C k turbulent intensity for kth phase (s 1 ) b(l L 0 ) probability distribution function of generating a daughter droplet of diameter L 0 from parent droplet of diameter L # a functional form which represents the number of daughter droplets generated upon breakage of parent droplet r surface tension between the two phases (N/m) r k constant in turbulence model (Eq. (8)) r e constant in turbulence model (Eq. (9)) x c coalescence frequency (s 1 ) / ij pressure strain term in turbulence model (kg/ms 3 ) e ij dissipation term in turbulence model (kg/ms 3 ) of-view of their implementation, hence to perform equivalent PBM implementations is a non-trivial task. Most of these earlier studies have been done in modelling and validation in cylindrical bubble columns operating under churnturbulent conditions. This poses some basic limitations vis-à-vis comparisons with CFD models. First, comparisons are done with radial profiles of velocity (and holdup), which are obtained by first time-averaging and then azimuthally averaging experimental data (in case azimuthally resolved data is obtained; otherwise the data is usually obtained only at various radial location). This completely averages out any azimuthal dependence in the flow, which is clearly visible in any visual observation of a transparent cylindrical bubble column, wherein one sees spiral and helical structures of gas rising around the central axis. Second, at high velocities (churn-turbulent conditions), there are multiple bubble plumes rising from the distributor which all merge and create a rather

3 820 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Table 1 Summary of studies done on bubble columns flow modelling using PBM methods to incorporate coalescence/breakage. Reference Geometry Interfacial closure a PBM b Remarks Wang [24] Frank [25] Zhu [26] Cheung [27] Bhole [28] Bannari [29] Selma [30] Chen [31] Chen [32] Chen [33] Cheung [34] Olmos [35] Olmos [36] Hansen [37] Diaz [38] Sanyal [39] Cylindrical D + L + VM + WL + TD H-D (i) Various coalescence and breakage closures were compared (ii) Good agreement of BSD was obtained for both heterogeneous and homogeneous regimes (iii) Holdup was also compared with the experimental data Cylindrical D + L + TD I-D (i) Velocity and hold up data was compared with experimental data, good agreement was obtained (ii) More detailed investigation into coalescence and breakage closures was suggested Cylindrical D + VM LSM (i) 2-D axisymmetric simulation, velocity and holdup data compared, good agreement obtained Cylindrical D + L + WL + TD ABND (i) Upward bubbly flow was studied (ii) Void fraction, Sauter mean diameter, gas velocity, liquid velocity and interfacial area concentration were compared, good agreement was obtained Cylindrical D + L I-D (i) Axial liquid velocity, Radial Sauter Mean Diameter and Turbulent K.E. were compared, good agreement was obtained (ii) The results improved upon addition of PBM at the operated velocities Rectangular D + L + VM + TD H-D (i) Velocity and hold up data was compared at a single level for different H/W ratios for two interfacial closures Rectangular, agitated reactor D + L + VM DQMOM H-D (ii) Results suggested that PBM marginally improves the results (i) DQMOM and H-D was compared for rectangular bubble column, liquid velocity and hold up data was compared (ii) The model was then extended to a reactor Cylindrical D H-D (i) 2-D axisymmetric simulations were performed, many aggregation breakage closures were compared through holdup, velocity profile, KE and total interfacial area Cylindrical D H-D (i) 2-D axisymmetric simulations were performed, the results showed some improvement in hold up profiles because of PBM Cylindrical D H-D (i) 3-D simulations were performed, follow up study of [32] Cylindrical D + L + WL + TD ABND, H- D (i) ABND and H-D approaches were compared by studying many parameters as done in [27] (ii) Regime transition prediction could not be predicted numerically, reason given as assumption of spherical bubbles Cylindrical D H-D (i) PBM was implemented and results were compared for liquid velocity, overall hold up, local hold up were compared with experimental data, data was taken for two different sparger designs Cylindrical D + L H-D (i) Regime transitions were predicted for two different sparger designs, initial bubble size distribution was carefully calculated which was essential for good agreement Square D + L + VM + TD IACE (i) Systematic analysis was done for various closures, drag + lift force combination was predicted to be very good for flow modelling, virtual mass had little or no effect (ii) Not so good agreement with bubble size was obtained using IACE Rectangular D + L + VM H-D (i) Sauter mean diameter, Plume oscillation period and gas holdup was compared at high gas velocities, good agreement obtained (ii) VM was shown not to have any effect (iii) Lift force led to large errors in results Cylindrical D H-D, QMOM (i) Axisymmetric simulations were done and H-D and QMOM methods were compared, merits of QMOM were shown a b D Drag, L Lift, VM Virtual Mass, TD Turbulent Dispersion, WL Wall Lubrication. H-D Homogeneous Discrete, I-D Inhomogeneous Discrete, QMOM Quadrature Method of Moments, DQMOM Direct Quadrature Method of Moments, LSM Least Square Method, ABND Average Bubble Number Density, IACE Interfacial Area Concentration Equation. complex cocktail of bubble coalescence-and-breakup, bubble-generated turbulence, and multi-scale circulation cells in the liquid, so that the role of individual forces and effects cannot be discerned by simply examining time-averaged velocity profiles. Time-averaged velocity profiles would be instructive only if we were to isolate a single kind of phenomena and keep the flow simple, so that the effect of various forces and effects on the isolated phenomena may be discerned. Finally, the use of distributors like sintered plates and perforated plates (with perforation all across the cross-section), while realistic (vis-à-vis industrial bubble columns), contribute little in isolating a certain kind of governing phenomena in the bubble column (as discussed above): rather they make the flow lot more complex by contributing a large number of interacting gasbubble plumes to the bulk of the bubble column.

4 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Fig. 1. Geometry modelled in present study, taken from Upadhyay [27], (a) front, top and side view of the geometry. (b) Enlarged view of the distributor with 8 apertures of diameter 0.8 mm in a square pitch of 6 mm. (c) Modelling the equivalent distributor as a rectangular patch of 24 mm 12 mm. In light of the above discussion, what may be more suitable for comparison with CFD codes are more sanitized flow conditions in bubble columns: namely low velocity flow, well-defined gas distributor which should release a single plume of bubbles, and perhaps isolated rise of these bubbles (free from interactions with other plumes), so that the effect of various governing forces may be discerned. Indeed, the above list of desirables is well-realised in a suitably designed rectangular bubble columns (with a thin gap for gas liquid flow like shown in Fig. 1, the so-called two-dimensional bubble column, even though we refrain from using this term in this paper since the flow is really completely three-dimensional) and a localised inlet for gas. The bubble column depicted in Fig. 1 is inspired by the work of Pfleger [2], whose work, in turn was preceded by the work of Becker [63]. This kind of system provides an easy benchmarking option since bubble trajectories can be discerned clearly against a background of almost completely quiescent liquid. Several researchers have used rectangular bubble columns for validating CFD codes and models ([2,8,10 12]). Thus, in this work, we have restricted our attention for model validation in high resolution velocity data collected in flat, rectangular bubble columns. Further, in addition to a detailed treatment of various drag, lift and virtual mass force closures, in this work we have used different population balance implementations, through Homogenous Discrete Method, Inhomogeneous Discrete Methiod, QMOM and DQMOM methods, and validated against high-resolution Radioactive Particle Tracking experiments (which has never been reported in a flat rectangular bubble column in quiescent conditions, thus far). The validation is thus attempted with a perspective for distinguishing the various controlling effects in terms of their prediction of the time-averaged liquid velocity profile. Thus, we have the benefits of the rectangular bubble column as listed above, and we also have complete three dimensional liquid velocity profile information in that systems. Further, the velocity profiles have been compared at three different representative levels in the column, something which has not been done before inside a rectangular bubble column (all prior comparisons were at a single level or element within the vessel). This was possible because of the rich velocity profile data that has been obtained by Upadhyay [40] recently, using Radioactive Particle Tracking (RPT), the details of which have been described later in this study. Rather than directly using all interfacial closures and bubble polydispersity effects together, we have attempted to systematically analyse the relative importance of interfacial closures (i.e. drag, lift and virtual mass) and the entire need and approach towards Population Balance Modelling (PBM). 2. Mathematical model Simulations in this study were performed using commercial CFD software package ANSYS Fluent Ò 14 as the base platform. Governing equations of Euler Euler multiphase model and PBM have been discussed in brief in this section Euler Euler multiphase model Eulerian description of fluid flow is based on the notion of pseudo-continuum, i.e. the approach defines a point volume fraction for each phase which represents the probability of a particular phase to be present at that point in multiple realizations of flow [41]. Same pressure field is shared between all the phases. The force interaction between phases is incorporated through various effective volumetric force functions, such drag force, lift force and virtual mass effect (defined as net force between phases per unit volume). The conservation equations for the Euler Euler multiphase model are as follows: Continuity equation (solved for ða ƒ! X kq k Þþrða k q n k u k Þ¼ _m pk _m kp p¼1;p k Momentum equation (solved for each ƒ! ƒ! ƒ! a k k u k þ r ða k q k u k uk Þ ¼ a k rp þ r:s k þ a k q!! k g þ F k þ X n ƒ! _m pk v pk _mkp ƒ! v kp p¼1;p k þ Xn p¼1;p k ƒ! ƒ! ƒ! R pk þ Flift þ Fvm ð1þ ð2þ

5 822 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) where ƒ! s k ¼ a k l k r u k þ r ƒ! T uk þ a k k k 2 3 l k r ðu k IÞ The term ~ R pk in Eq. (2) represents the drag force between the phases p and k. The mathematical description of drag force is given as follows: ƒ! ƒ! ƒ! R pk ¼ Kpk ð u p uk Þ ð4þ where K pk ¼ 3 4 a ka p l k C d Re d 2 p K pk is the drag momentum exchange coefficient and C d is the single particle (or single bubble or droplet) drag coefficient between the two phases. Essentially, Eq. (5) corrects for the fact that in fact all particles/droplets see a modified drag from the continuous phase owing to the presence of other neighbouring dispersed phase entities. Some of the drag coefficient models used and compared in this study have been summarized in Table 2. Note that these are all single particle (bubble or droplet) drag models and are all to be modified in a similar fashion, as in Eq. (5). Note also that all models for drag coefficient listed in Table 2, when modified for presence of other particles of bubbles in the neighbourhood, leads to a diameter sensitivity that varies significantly. A detailed study regarding the same can be looked up in Tabib [46]. The term ~ F lift in Eq. (2) represents the lift force acting on secondary phase (k) because of primary phase (p). The mathematical description of lift force is [47]: ƒ! ƒ! ƒ! F lift ¼ CL q k a k ð u k up Þðr ƒ! uk Þ ð6þ C L is known the lift coefficient between the two phases. Since the lift force expression involves the curl of the bubble velocity, in vector product with the relative velocity between phases, its action is to move the bubble or the dispersed phase in a direction that is transverse to the mean flow direction of the continuous phase. The lift coefficient models used in this study have been summarized in Table 3. The term ~ F vm in Eq. (2) represents the virtual mass force which increases the inertia of the secondary phase whenever the secondary phase accelerated with respect to the primary phase. The mathematical description of virtual mass force is as follows [64]:! ƒ! ƒ! Du F vm ¼ 0:5qk a k k Du ƒ! p ð7þ Dt Dt 2.2. Turbulence model Table 2 Various models for drag coefficient in literature (Eq. (5)). Drag law (Reference) Expression Schiller Naumann [42] C d ¼ 24 Re ; Re :44; Re P 1000 Ishii Zuber [43] C d ¼ 2 3 Eo1 2 Tomiyama [44] 8 Eoð1 E 2 Þ 3 f ðeþ 2 E 2 3Eoþ16ð1 E 2 ÞE 4 3 C d ¼ 1 E ¼ p1þ0:163eo ffiffiffiffiffiffiffiffi p 0:757 ffiffiffiffiffiffiffiffi sin 1 1 E 2 E 1 E 2 1 E 2 Zhang and Vanderheyden [45] C d ¼ 0:44 þ 24 Re þ 6 p ffiffiffiffi 1þ Re ð3þ ð5þ Table 3 Various models for lift coefficient in literature (Eq. (6)). Reference Expression for lift coefficient Auton [47] C L = 0.5 Legendre and C L ¼ 1þ16 Re 1þ Magnaudet [48] 29 Re Tomiyama [49] C L ¼ minð0:288 tanhðreþ; f ðeoþ f ðeoþ ¼0:00105Eo 3 0:0159Eo 2 0:204Eo þ 0:474 Eo ¼ gðq q l g Þd2 B r The turbulence was modelled for the mixture i.e. for the mixture of primary and secondary phase(s). Turbulence model used in the present study was RNG k e model [50] for most of the simulations because of its merit for system where Reynolds number is low compared to the conventional k e model [51]. RNG k e model has a similar mathematical description as the k e model [52], however it is perhaps not as extensive because of its assumption of isotropy of Reynolds stresses. Of course, this assumption can be relaxed in the RSM model. Anyhow, the overall low Reynolds number of the flow justifies the use of the RNG k e model. To check the effect of different turbulence closures, simulations with simple k e and RSM models (with linear pressure strain closure, see [52]) were also done. The governing equations for simple k e model and RSM have been tabulated in Table 4. Since, RNG k e model has very similar equations as the k e model, they are not described in Table 4 and the details can be referred to in [65] Population Balance Modelling (PBM) In Eqs. (1) (7), the value of secondary phase diameter is used only in the expression of K pq (i.e., C D ) and C L. Therefore, as mentioned before, in conventional Euler Euler multiphase description of flow (in which the dispersed gas phase is treated as a pseudocontinuum), only a single value of diameter can be used, by assumption, which is held constant throughout the domain. Clearly, this may seem like a gross oversimplification, at least at higher gas velocities, since the whole dynamics and liquid flow in the bubble column is driven by the circulation induced by bubbles in their sojourn from the inlet distributor to the exit of the bubble column. As they rise, bubbles coalesce, break, re-disperse and hence undergo size and shape change owing to their traverse into varying pressure fields. All these factors contribute to bubble size evolution, which in turn has an important role to play in the type and velocity pattern of liquid circulation (note that the advection velocity of each bubble size, in general, should also be sizedependent). The restriction of an assumed single bubble size can be relaxed by using population balance where we solve indirectly for secondary phase diameter at every cell and at every time step. Population balance modelling involves writing an extra conservation equation for the bubble size distribution that varies in time or space. In the present context, implementing population balance would means writing an extra conservation equation for bubble size distribution (BSD). Population balance equation (PBE) relevant for the present system has been described below, essentially based on the development presented by ðnðvþþ þ r ð~u gnðvþþ ¼ B a D a þ B b D b B a ¼ 1 Z v aðv v 0 ;v 0 Þnðv v 0 Þnðv 0 Þdv Z 1 D a ¼ aðv; v 0 ÞnðvÞnðv 0 Þdv 00 0 Z 1 B b ¼ tgðv 0 Þbðvjv 0 Þnðv 0 Þdv 0 v D b ¼ gðvþnðvþ ð8þ ð9þ ð10þ ð11þ ð12þ Eq. (7) is the number conservation equation with right hand side being the source term. The source term is obtained through combi-

6 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Table 4 Description of turbulence models. Simple k e model m kþþrðq ƒ! l m um kþ¼rð t;m r k rkþþg k;m q m u0 m;i u0 m;j ÞþC ij ¼ D T;ij þ D L;ij P ij þ u ij e ij meþþrðq m u m eþ¼rð l t;m r k reþþ e k ðc 1eG k;m C 2e q m eþ C k ðq m u m;k u 0 m;i u m;jþ k l t;m ¼ q m C 2 l e D k ðq m u 0 m;i u0 m;j u0 m;k þ pðd kju 0 m;i þ d kju 0 m;j Þ ƒ! ƒ! G k;m ¼ l t;m ðrv ;m þ v T ƒ! ƒ! ;mþ : rv;m P ij ¼ qðu ƒ! m;j m;i u0 þ u m;i m;k m;j u0 m;k q m ¼ P n i¼1a i q i P n ƒ! v m ¼ a! i¼1 iq i u i q m u ij ¼ k i Þ e ij 0 m;i k k nation of four terms (Eqs. (9) (12)) representing birth and death due to coalescence (aggregation) and breakage, respectively. Since the general population balance equation is an integro-differential equation, analytical solutions exist for very few special cases. Therefore, numerical methods are used to solve this equation. The closures used for PBM in the following study is the closure for aggregation and breakage by Luo and Svendsen [54,55] model. The coalescence model for Luo and Svendsen [53] considers only the coalescence induced by turbulence. Other works in literature, for instance the work of Wang [56], describes alternative coalescence closures that may be used. The mathematical description of the closures (for Eqs. (9) and (10)) is as follows: aðd i ; d j Þ¼x c ðd i ; d j ÞP c ðd i ; d j Þ x c ¼ p pffiffiffi 1 2 e3ðdi þ d j Þ 2 ðd 2 3 i þ d 2 3 j 4 Þ1 2 8 h >< 0:75ð1 þ 1 2 ij Þð1 þ 13 ij Þ1 2 P C ¼ exp C L ð q g >: q þ 0:5Þð1 þ l 1 ij Þ 3 i1 2 9 >= >; We1 2 ij ð13þ ð14þ ð15þ where 1 ij ¼ d i d j ; u ij ¼ð_u i þ _u j Þ 2. The breakage model of Luo [54] considers breakage only due to isotropic turbulence. This model has been very widely used because of its ability to give the daughter distribution function as well the breakage function simultaneously [39,56]. The mathematical description of the closures (for Eqs. (11) and (12)) is as follows: bðf v ; dþ ¼0:923ð1 a g Þn e d 2 Z 0:5 gðvþ ¼ bðf v jdþdf v 0 bðf v jdþ ¼ 2bðf v; dþ R 1 bðf 0 vjdþdf v 1 3 Z 1 n min 2 ð1 þ nþ n 11 3! exp 12c f r dn bq l n 11 8 e 2 3d 5 3 ð16þ ð17þ ð18þ There are various methods that are available for solving PBE such as the Homogeneous Discrete method, Inhomogeneous Discrete method, Quadrature Method of Moments (QMOM) and Direct Quadrature Method of Moments (DQMOM). Generally speaking, these methods can be classified into two types: ones in which we directly discretize the governing equations and solve directly for the number density function, and the other category being in which we integrate out the internal coordinate and solve for the moments using some numerical approximation. Naturally, the computational expense for the methods based on moments will be less as compared to direct solving of the distribution because of the less number of the variables involved, since we solve transport equation for moments only and not the entire distributions. The idea for finding moment based methods stems from the fact that for all practical purposes, we do not need the entire distribution and are generally more focussed on obtaining information about valuable variables like Sauter-mean diameter, which can be obtained from the lower order moments. It is important to note that Homogeneous Discrete method and Quadrature Method of Moments assume that the secondary phase moves with the same velocity even for different bubble sizes, whereas this crucial assumption is relaxed in Inhomogeneous Discrete method and Direct Quadrature Method of Moments. A pertinent question that arises is whether these models are equivalent, and if they are, does having a difference advection velocity for different bubble sizes make any significant difference? In principle, they should be equivalent but numerical implementation and approximations may render them quite different in implementation. While there have been few attempts to compare different PBM approaches coupled with Euler Euler CFD, thus far no published work has compared all the different approaches together, and also benchmarked the results against high-resolution experimental data. The next section briefly describes these four different population balance modelling methods Discrete Homogenous and Discrete Inhomogeneous Methods Discrete method is one of the few numerical methods which were developed to solve this equation [17 19]. In this method, the PBE is solved for specific bubble diameters (also referred to as bins, or classes) and it is assumed that bubble diameters can only take up the values given to different bins or classes. This method is generally less preferred because it is computationally expensive [39] and requires a prior knowledge of expected bubble size distribution (BSD) for proper bin size selection. However, it has an advantage that it directly gives us the number density function, which has to be otherwise built from the moments (in other methods like QMOM and DQMOM). The homogenous discrete method assumes the same velocity for all the bins and only one dispersed phase is assumed. This method can be described by the following equations, details of which are presented in þ r ðu ƒ! g ni Þ¼S i ð19þ S i ¼ B a;i D a;i þ B b;i D b;i ð20þ X jpk B a;i ¼ 1 1 v i 1 6v j þv k 6v 2 d jk hðv j þ v k Þaðv j; v k Þn j n k i ð21þ D a;i ¼ XM j¼1 aðv i ;v j Þn i n j B b;i ¼ XM th ij gðv i Þn i j¼1 D b;i ¼ gðv i Þn i v iþ1 ðv j þv k Þ v h ¼ iþ1 v i ðv j þv k Þ v i 1 h ij ¼ Z viþ1 v i ;v i6 ðv j þ v k Þ 6 v iþ1 ð22þ ð23þ ð24þ ð25þ v i v i 1 ;v i 16 ðv j þ v k Þ 6 v i Z v iþ1 v viþ1 v v bðv;v v iþ1 v j Þdv i 1 þ bðv;v i v i v j Þdv ð26þ i 1 v i

7 824 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) The Inhomogeneous Discrete method [20] relaxes the assumption (to some extent) of all the bins moving with same velocity by assuming multiple phases for the dispersed phase, allowing each phase to have its own velocity. For instance, if we divide a dispersed phase into M phases and each phase has N classes, then every phase (and its N classes) will move with one velocity whereas each phase can have its own velocity. Hence, for homogenous discrete model, M =1. For this method, continuity equation becomes slightly more involved because of the presence of mass transfer between the multiple phases (within a single dispersed phase). Also, number density equation will have a different dispersed phase velocity depending upon the phase under which the bin is assigned. Since the equations are very similar to the homogenous method, they are not described here (the reader is advised to refer to [20] for details) Quadrature Method of Moments (QMOMs) and Direct Quadrature Method of Moments (DQMOMs) QMOM solves the population balance equations by solving conservation equations for moments (and not number density, hence no notion of bins and classes) by using the quadrature point approximation [21]. Generally three quadrature points (or six moments) are sufficient for most flow problems of interest [21]. Since only six moments are used, this method is computationally less expensive than the discrete method. Also it does not require a prior knowledge of expected bubble size distribution. However, because QMOM solves directly for moments, number distribution, if needed, has to be extracted from moments and is not readily available as in discrete method. Also, QMOM assumed that all bubble sizes move with same velocity and hence is not able to capture segregation [23]. m k ¼ XN w i L k ðm kþþrðu! m k Þ¼S k S k ¼ B a;k D a;k þ B b;k D b;k B a;k ¼ 1 2 X N i¼1 D a;k ¼ XN L k i w i i¼1 X N w i w j ðl 3 i þ L 3 j Þk 3 aðli ; L j Þ j¼1 X N j¼1 w j aðl i ; L j Þ Z B b;k ¼ XN 1 w i L k gðl i ÞbðLjL i ÞdL i¼1 D b;k ¼ XN w i L k i gðl iþ i¼1 d 3;2 ¼ m 3 m 2 o ð27þ ð28þ ð29þ ð30þ ð31þ ð32þ ð33þ ð34þ In Direct Quadrature Method of Moments (DQMOM), rather than solving for moments (m k ),transport equation of abscissas (L i ) and weights (w i ) are directly solved which allows us to specify a different velocity at each quadrature point, unlike QMOM [23]. Hence, DQMOM is able to capture the effect that different bubble sizes move with different velocities without compromising the advantages mentioned earlier for QMOM. The governing equations for DQMOM are presented below. Detailed derivation of DQMOM is presented in þ r ðu! i wi Þ¼a i i L i! þ r ðu i wi L i Þ¼b ð36þ where a i and b i can be solved numerically by applying Gauss-Siedel method to the system of equations described below: X N i¼1 ð1 kþl k i a i þ kl k 1 i b i ¼ S k ð37þ where S k is same as described in Eqs. (29) (34). For tracking six moments in QMOM and DQMOM, N =3ork varies from 0 to 5 and i varies from 1 to Experimental The experimental data for the present study has been taken from Upadhyay [40] where velocity and kinetic energy profiles were obtained for an air water rectangular bubble column using Radioactive Particle Tracking (RPT). In the Radioactive Particle Tracking technique [4,57,58] as applied to the present system, a single, small-sized radioactive tracer was designed to be neutrally buoyant with respect to the liquid water phase, and its motion was scanned with an arrangement of (eight in number in this case) of NaI/Tl scintillation detectors positioned around the rectangular bubble column (Fig. 1). The system was operated for a long time (over 8 h or so), so that over multiple realizations of flow, the tracer particle visited every location in the column many times. The position time series was reconstructed using an established algorithm [40,57] for this purpose, followed by data processing to get the Lagrangian velocity time series and from that the Eulerian velocity field. Upadhyay [59] have shown that the RPT results obtained in this geometry agree very well with the LDA results obtained in the same geometry by Pfleger [2], but as a bonus, RPT results could be obtained with greater detail and at conditions at which LDA would not provide any meaningfully accurate results. Thus, Upadhyay s work [40] on the rectangular bubble column provides a rich database of information for CFD validation, and has been correspondingly used as the main database for the present simulation work. 4. Results and discussion The geometry considered in present study is an air water rectangular bubble column of dimensions cm (Fig. 1a). The gas distributor of the column consists of eight holes of 0.8 mm diameter each, arranged in two centrally positioned rows, and with a pitch of 6 mm (Fig. 1b). Similar experimental setup has also been used in Pfleger [2], Buwa and Ranade [10,11], and Diaz [38]. In the grids generated to model the present geometry, gas distributor (inlet) was modelled as a rectangular patch of size 24 mm 12 mm (Fig. 1c). For the numerical simulations, four grid densities were generated to first check for grid independence. The details of various grids have been described in Table 5. Numerical diffusion was checked by using three different discretization schemes, i.e. QUICK, Second order upwind, and Third Order MUSCL. Grid independence and numerical diffusion were tested using only drag as the interfacial force, with constant bubble size of 5 mm for a gas superficial velocity of 1.33 mm/s. The results of time-averaged axial liquid velocity profile (at a height of 13 cm from distributor) for different Table 5 Description of various grids generated. Grid name Description Grid-1 Full domain , distributor 1 2 Grid-2 Full domain , distributor 1 2 Grid-3 Full domain , distributor 4 2 Grid-4 Full domain , distributor 3 4

8 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) grids have been compiled in Fig. 2. From Fig. 2a, it can be clearly seen that all the grids are practically converging to the same solution (in the rest of the paper, Grid-4 was used). It should be noted that different grids have different cell aspect ratios as well, and hence the results are independent of this parameter. From Fig. 2b, it is clear that different discretization schemes are also converging to the same solution and hence it can be concluded that numerical diffusion is not playing a prominent role in these simulations. In the rest of the paper, QUICK scheme was used as the discretization scheme because it converges fastest among the three schemes. It may be noted that in several papers available in the open literature related to bubble column flow simulations, the grid convergence of the experimental results has not been rigorously implemented, so that several of the trends observed have been erroneously ascribed to physical effects, when in reality they were merely artefacts of numerical diffusion. Therefore, an a priori elimination of mesh refinement issues was essential to continue with exploring the effects of various closures for forces and the population balance. Initial value of bubble diameter specified was 5 mm, which is the value estimated by Buwa and Ranade [10] at the distributor through experiments (in conventional E E CFD, 5 mm value has been specified throughout the column). Also, single bubble formation models like the widely used Vogelphohl Gaddis model [60] also predicts nearly a value of 5 mm in these conditions. The boundary conditions and important properties used for different liquid velocity vectors at the central plane have been shown using only drag as the interfacial simulation cases used have been described in Table 6. Table 7 reports the conditions and parameters of a base case simulation. As the various forces and parameters were varied in the many simulations whose results are reported below, all other parameter values and models were kept fixed at the conditions mentioned in this table. In Fig. 3, time-averaged axial liquid velocity results obtained after incorporating various PBM with E E CFD model has been shown. The results have been compared at three different representative heights of 13 cm (z/h = 0.29), 25 cm (z/h = 0.56) and 37 cm (z/h = 0.82) for gas superficial velocity of 1.33 mm/s (even though RPT data is available at other heights as well, in fact everywhere in the column). Schiller Naumann drag model was used as the interfacial force and lift force was turned off for the present set of simulations. It can be seen from Fig. 3 that though velocity profile are slightly improved because of the addition of PBM in the model, results at all levels have not changed significantly for all Fig. 2. (a) Test simulations for grid independence (Table 5). (b) Test simulations for various discretization schemes. (Gas superficial velocity = 1.33 mm/s, Schiller Naumann drag, bubble diameter = 5 mm, RNG k e turbulence model, no PBM.)

9 826 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Table 6 Summary of the boundary conditions and simulations setup. Boundary conditions Inlet Outlet Others Velocity specified Population balance variables specified for mono-dispersed BSD for gas Turbulent Kinetic Energy specified as unity Turbulent Kinetic Energy dissipation rate specified as unity Pressure specified at atmospheric pressure Gradient of population balance variables specified zero Turbulent Kinetic Energy specified as unity Turbulent Kinetic Energy dissipation rate specified as zero Stationary walls Initial conditions Initial height of water 45 cm or aspect ratio = 2.25 Time step and under relaxation factors Time step 0.01 s, implicit scheme used Under relaxation factors Pressure 0.6 Momentum 0.4 Volume fraction 0.2 Turbulence variables 1.0 Population balance variable 0.5 Setup for homogenous discrete model Number of bins 9 Geometric ratio exponent 1.4 (ratio of two consecutive bubble diameters) Minimum bubble diameter 1 mm Setup for Inhomogeneous Discrete model Number of secondary phases 3 Number of bins/phase 3 Value of bin diameters Same of homogenous but first three bins in one phase, next three bins in second phase and last three bins in third phase Setup for QMOM Number of moments 6 Setup for DQMOM Number of secondary phases 3 the PBM models and at all levels. Also, it can be clearly seen that results are not in particularly good agreement with experimental data. An important observation in this regard is that all these profiles at lower levels (z =13cm(z/H = 0.29), z =25cm(z/H = 0.56)) deviate from the experimental RPT data largely in the central region (which, by the way, is the region of the rising bubble plume) than the off-centre region. This might be because of incorrect or incomplete description of the coupling of two phases because deviation is more in central region where gas-holdup is relatively higher. In other words, solution seems to agree well with data away from central region where there is practically no hold up and hence we are solving a single phase flow equation. Since it was observed that the deviation of the model predictions seem from experimental RPT data to be higher in the region of larger holdup, the next step was to improve the coupling between the two phases. A small analysis was done where lift force was turned on with constant lift coefficient of value +0.5 and virtual mass effect was also included. Then, PBM was incorporated into the model. Fig. 4 depicts the time-averaged profiles for all the simulations after incorporation of the lift force and virtual mass effect. Observing the results of Drag and Drag + Virtual Mass shown in Fig. 4, one can clearly see that virtual mass effect marginally improves the answers near the centre whereas not much effect is seen near the wall. However, only slight improvement in seen when Drag + Lift is compared to Drag + Lift + Virtual Mass. Hence, though virtual mass does improve the results very marginally, its incorporation slows down the convergence; hence it will not be considered in the remaining simulations reported in this paper. One also observes in Fig. 4 that after the addition of lift force to the system, velocity profiles agree much better with the experimental data (both in the Drag + Lift comparison, as well as the Drag + Lift + Virtual Mass). In fact, agreement of the simulated profiles with experimental RPT data is remarkably for z = 13 cm (z/ Table 7 Properties and settings of base case simulation. Drag model Schiller Naumann [42] Lift model On (unless mentioned), C L = 0.5 (unless mentioned) Virtual mass effect Off (unless mentioned) Initial diameter (for PBM cases) 5 mm (using approximation from [60]) Diameter (without PBM cases) 5 mm (using approximation from [60]) Turbulence model RNG k e model [50] H/W 2.25 Gas superficial velocity 1.33 mm/s H = 0.29) and z =25cm (z/h = 0.56) whereas a minor deviation is seen at z =37cm(z/H = 0.82). However, even this deviation is not large and might also be removed if we try to model the flow more extensively, suggestions for which have been explained later in the paper. Be that as it may, to understand more deeply effect of lift force, we tracked the axial liquid velocity vectors at the central plane with respect to time and few key snapshots have been shown in Fig. 5. In Fig. 5a, instantaneous and time-averaged snapshots of liquid velocities have been shown for the D and D + L case of Fig. 4. These plots clearly suggest that when drag is used as the interfacial force, the central region is the region of action. In other words, the gas simply zips through the centre without any lateral movement. This however is not the case in reality where plume oscillations are observed. It may also be noted that velocity vectors remain the practically the same when population balance is also added to system of equations (figure not shown) and hence insignificant effect of addition of population balance was observed. This is perhaps to be expected, since the gas velocities involved are very small and hence the actual rates of bubble coalescence are not significant

10 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Fig. 3. Simulated liquid velocity profile with different PBM methods at gas superficial velocity of 1.33 mm/; Schiller Naumann drag, 5 mm initial bubble diameter, RNG k e turbulence model: (a) z = 13 cm(z/h = 0.29), (b) z = 25 cm(z/h = 0.56) and (c) z = 37 cm(z/h = 0.82). for the effect of bubble polydispersity to affect the time-averaged liquid velocity profile. However, as soon as we add lift to the system, as shown in Fig. 5b, the bubble plumes start to oscillate about the centre. This phenomenon is observed because lift force acts in the direction perpendicular to the flow and therefore takes gas away from the centre. Since now the velocity vectors also have a lateral component, the axial liquid velocity in the centre drops significantly. Though it was observed that the simulation results are more or less insensitive to use of any of the PBM techniques (as discussed above, owing arguably to the low gas superficial velocity), when Drag was the only interfacial force (Fig. 3), a set of simulations

11 828 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Fig. 4. Simulated liquid velocity profile with combination of drag (Schiller Naumann) and lift (constant Auton lift) and virtual mass, but no PBM (at gas superficial velocity of 1.33 mm/s, 5 mm bubble diameter, RNG k e turbulence model): (a) z = 13 cm(z/h = 0.29), (b) z = 25cm(z/H = 0.56) and (c) z = 37 cm(z/h = 0.82). were performed to reflect whether there is some effect of PBM when it is used with Drag + Lift being the interfacial forces. The results have been compiled in Fig. 6. The results clearly suggest that as in Fig. 3, all the PBM methods are not adding significant difference to the velocity profiles, though slight improvements are present. This is not very unexpected because we are in deep bubbly flow where the bubble size distribution is narrow and hence the impact of population balance will not be very high. Therefore, the importance of these methods at higher velocities needs to be rigorously tested both in cylindrical and rectangular geometry. Both in Figs. 3 and 6, the results are almost same for all the four PBM model implementations. This is important as this established the equivalence between the four models. To compare the equivalence of all the PB models even further, average Sauter-mean diam-

12 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Fig. 5. (a) Instantaneous and time-averaged snapshots of liquid velocity vectors at central plane with only Schiller Naumann drag as interfacial force and (b) instantaneous and time-averaged snapshots of liquid velocity vectors at central plane with Schiller Naumann drag and constant lift coefficient (Auton) lift force as interfacial force. eter was plotted for against the height of the column. The results have been shown in Fig. 7 for the cases shown in Fig. 3. It is clear from Fig. 7 that all the four models predict similar trend of bubble sizes and hence it is safe to say that all the four models are equivalent, at least in for the simulations presented here. Though DQMOM and Inhomogeneous discrete method are arguably superior as compared to other methods because they can take into account that the different bubble sizes can move with different advective velocities, in the present case, no significant difference has been observed between all the methods. It is notable that in a previous reference (Selma [30]), the homogenous discrete method was compared with DQMOM wherein both methods were also shown to have similar results. It should also be noted that the trend of increasing average Sauter-mean diameter with increase in axial coordinate is consistent with reality because coalescence rate is higher than breakage rate as we move away from the distributor. It should also be noted that variation of the Sauter-mean diameter is not wide, ranging from 5 mm to 1 cm and hence velocity profiles are relatively insensitive to PBM (as pointed out above). This insensitivity might be due to the fact that upon implementing PBM (and as compared to the case of a single prescribed bubble size), sizewise gas fraction (and corresponding interfacial area) simply redistributes as a result of the PBM implementation itself but overall value of gas volume fraction remains the same (assuming no consumption or production of the gas phase). Hence, the net momentum exchange between liquid and gas phase remains practically the same even after implementation of PBM. In reality, however, it is entirely possible that the average size of the bubble population in the column may be quite different from the bubble size at the distributor (and the frequency with which the bubbles

13 830 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) Fig. 6. Simulated liquid velocity profile with many PBM methods at gas superficial velocity of 1.33 mm/with Schiller Naumann drag, constant lift coefficient force, 5 mm initial bubble diameter and RNG k e turbulence model: (a) z = 13 cm(z/h = 0.29), (b) z = 25 cm(z/h = 0.56) and (c) z = 37 cm(z/h = 0.82). are formed and released at the distributor). In that case we expect the incorporation of the population balance model to make a significant impact. However, at this point such an argument is speculative, particularly in the context of CFD validation with a sophisticated technique such as RPT, since at least with the current set of results one cannot make any generic statements about the role of PBM at higher velocities. In any case, the story is quite different for the mass transfer problem, since in that case, many small bubbles of high interfacial area would lead to much more efficient mass transfer than that from a few large bubbles. Thus, even with the current results, we expect the PBM to play an important role in modelling mass transfer in bubble columns.