CFD simulation of bubble columns incorporating population balance modeling

Transcription

1 Chemical Engineering Science 63 (8) CFD simulation of bubble columns incorporating population balance modeling M.R. Bhole a, J.B. Joshi a, D. Ramkrishna b, a Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 4 9, India b School of Chemical Engineering, Purdue University, West Lafayette, IN , USA Received 3 August 7; received in revised form 9 December 7; accepted January 8 Available online January 8 Abstract A computational fluid dynamics (CFD)-code has been developed using finite volume method in Eulerian framework for the simulation of axisymmetric steady state flows in bubble columns. The population balance equation for bubble number density has been included in the CFD code. The fixed pivot method of Kumar and Ramkrishna [996. On the solution of population balance equations by discretization I. A fixed pivot technique. Chemical Engineering Science 5, 3 33] has been used to discretize the population balance equation. The turbulence in the liquid phase has been modeled by a k.ε model. The novel feature of the framework is that it includes the size-specific bubble velocities obtained by assuming mechanical equilibrium for each bubble and hence it is a generalized multi-fluid model. With appropriate closures for the drag and lift forces, it allows for different velocities for bubbles of different sizes and hence the proper spatial distributions of bubbles are predicted. Accordingly the proper distributions of gas hold-up, liquid circulation velocities and turbulence intensities in the column are predicted. A survey of the literature shows that the algebraic manipulations of either bubble coalescence or break-up rate were mainly guided by the need to obtain the equilibrium bubble size distributions in the column. The model of Prince and Blanch [99. Bubble coalescence and break-up in air-sparged bubble columns. A.I.Ch.E. Journal 36, ] is known to overpredict the bubble collision frequencies in bubble columns. It has been modified to incorporate the effect of gas phase dispersion number. The predictions of the model are in good agreement with the experimental data of Bhole et al. [6. Laser Doppler anemometer measurements in bubble column: effect of sparger. Industrial & Engineering Chemistry Research 45, 9 97] obtained using Laser Doppler anemometry. Comparison of simulation results with the experimental measurements of Sanyal et al. [999. Numerical simulation of gas liquid dynamics in cylindrical bubble column reactors. Chemical Engineering Science 54, ] and Olmos et al. [. Numerical simulation of multiphase flow in bubble column reactors: influence of bubble coalescence and breakup. Chemical Engineering Science 56, ] also show a good agreement for liquid velocity and gas hold-up profiles. 8 Elsevier Ltd. All rights reserved. Keywords: Bubble column; CFD; Population balance model; Coalescence; Break-up. Introduction Bubble column reactors are widely used in chemical and allied industry for gas liquid and gas liquid solid contacting or chemical reactions because of their simple construction and ease of operation. There exist a large number of correlations for the prediction of gas hold-up, heat and mass transfer coefficients, interfacial area in bubble columns. However, these correlations are specific only to a particular set of operating Corresponding author. address: ramkrish@ecn.purdue.edu (D. Ramkrishna). conditions for which they are developed. Hence, the predictions for a new gas liquid systems based on existing correlations are difficult and can lead to poor design of bubble columns. For a more rational design and better operation, the hydrodynamics of flows in bubble columns must be understood and the relationship between the flow pattern and design objectives must be established (Joshi, ; Ghadge et al., 5). In a majority of cases, the multiphase flows in bubble columns are turbulent in nature. Computational fluid dynamics (CFD) has clearly emerged as a promising tool for the simulation of local hydrodynamics in bubble columns. This is an approach based on 9-59/$ - see front matter 8 Elsevier Ltd. All rights reserved. doi:.6/j.ces.8..3

2 68 M.R. Bhole et al. / Chemical Engineering Science 63 (8) 67 8 first principles in which the governing equations of continuity, momentum and energy for the gas and liquid phases in bubble columns are solved. Detailed information about local flow pattern, turbulence, interfacial area density can be used for the design objectives. CFD simulations of gas liquid two phase flows in bubble columns started about two decades ago. This subject has been reviewed by Jakobsen et al. (997), Joshi () and Sokolichin et al. (4). Most of the studies are based on two-fluid models which assume the gas and liquid phases to be interpenetrating continua. This is equivalent to Euler Euler approach. On the other hand, the Euler Lagrange involves tracking of individual bubbles in the liquid phase and it was followed by researchers like Sokolichin et al. (997) and Delnoij et al. (997). From computational considerations, the Euler Euler approach is more economical and hence more popular. A two-fluid model with Euler Euler approach uses a constant bubble size for the calculation of interfacial forces in the governing equations of CFD. In majority of the cases, the bubble columns operate in the heterogeneous regime. In this case a wide bubble size distribution exists in the column due to bubble coalescence and break-up phenomena. Further, the bubble diameter can vary depending upon the operating conditions such as superficial gas velocity and pressure, the physical properties such as density, viscosity and surface tension of liquid phase and the details of sparging device such as free area, hole diameter etc. The mechanics of gas liquid flow is significantly dependent on the bubble size. Depending upon the bubble size, their rise velocities can vary from 5 to 5 mm/s. Accordingly, the local flow pattern, turbulence and the gas hold-up can vary significantly in the column. The complexities of gas liquid flows due to bubble shapes and bubble wakes are also dependent on bubble size. The bubble size also directly affects the interfacial area available for gas liquid mass transfer. Owing to the fundamental importance of bubble size in gas liquid flows, the predictions of bubble size distribution become very important for the understanding of the hydrodynamics of bubble columns. When bubble coalescence and breakup are insignificant, the bubble size distribution and hence the mean bubble size are governed by the sizes of the bubbles generated at the gas sparging device and their velocities through the column. This is typically the situation for the homogeneous regime of operation of bubble columns. For the semi-batch operation of bubble column operating in the homogeneous regime, the time-averaged local liquid velocities are zero everywhere in the column. Thus, the description of liquid flow pattern is not essential and the population balance equation is decoupled from the equations of motion of liquid phase. This case was studied by Bhole et al. (7). It is especially relevant for tall columns where the variation in bubble sizes owing to the variations in hydrostatic head from sparger to the top are significant. On the other hand, when bubble coalescence and break-up are significant, the evolution of the bubble size distribution is also governed by the relative magnitudes of bubble coalescence and break-up rates. This is typically the case of transition or heterogeneous regime of operation of bubble columns. In this case, there exists a wide variation in the gas hold-up, bubble sizes, liquid velocities, turbulent kinetic energy and energy dissipation rates across a column cross-section. There exists a liquid circulation pattern in the column which is dependent on the profiles of gas hold-up and bubble sizes. Further, the bubble size at any location in the column is dependent on local turbulence intensities as they directly affect the bubble coalescence and break-up phenomena. Clearly, the equations of motion for the flow and the population balance equation for bubble number density are coupled to each other. A combined CFD PBM (population balance modeling) model for bubble columns is a better approach than a two-fluid model with constant bubble size. The evolution of bubble size is directly obtained from bubble population balance equation and its impact on flow pattern is obtained from CFD simulation. It gives a better understanding of complexities of two phase flows in bubble columns.. Literature review The investigations of coalescence and break-up of bubbles and the CFD simulations of bubble columns in the framework of two-fluid model have occurred practically independent of each other. Their combination in the form of combined CFD PBM modeling for bubble columns is a relatively new research area which began toward the end of last decade. Lo (996) was the first to report the results of CFD PBM simulations for bubbly flows. He developed the multiple size group (MUSIG) model for the same. According to this model, the following equations can be written to describe the hydrodynamics of bubbly flows: t (ρ kɛ k ) + (ρ k ɛ k u k ) =, k = L, G, () t (ρ kɛ k u k ) + (ρ k ɛ k u k u k ) = ε k p + ρ k ɛ k g.(ɛ k τ k ) ± M Ik, () ρ G ɛ G f i t + (u G ρ G ɛ G f i ) = ρ G S i, i =,,...,M. (3) The first two equations are the equations of continuity and motion written for both gas and liquid. The last equation is a population balance equation written for volume fraction f i of the bubbles of size group i. The term S i is then appropriate source term to include bubble coalescence and break-up. The bubble coalescence model of Prince and Blanch (99) and the bubble break-up model of Luo and Svendsen (996) were used. All the bubbles of M discrete size groups were assumed to travel with the same velocity u G. This simplification is necessary to reduce the gas phase momentum equations to a single equation with a velocity field u G as in the case of a standard two-fluid model. With this framework, one needs to solve two continuity, two momentum equations and (M ) population balance equations and the two additional equations for k and ε for the liquid phase turbulence. If one considers a general framework of multi-fluid model and allows the gas bubbles of different sizes to move with different velocities, then one must

3 M.R. Bhole et al. / Chemical Engineering Science 63 (8) Terminal rise velocity (m/s) Bubble diameter, d B (mm) Fig.. Variation of the terminal rise velocity with bubble diameter. The lower and upper dotted lines are the experimental data of Clift et al. (978) for clean and contaminated water. The continuous line is the prediction from the drag law (Eq. (4)) used in this work. solve the momentum equations for each of the M gas phases considered in the model. Since, this is a computationally expensive problem; the assumption of equal bubble velocity for all size groups has been made in the MUSIG model. However, this is not a realistic simplification. The dynamics of bubbly flows is significantly affected by bubble sizes. As seen from Fig., the bubbles of different sizes move with different rise velocities. Although this result is for the isolated bubbles in the infinite stagnant liquid and the velocities of bubbles may vary significantly in a swarm, there is no a priori justification for the assumption of equal rise velocities for the bubbles of all sizes. Further, there is a preferential radially inward motion of large sized bubbles (d B > 6mm) and radially outward motion of smaller sized bubbles under the influence of the lift force in a shear field typical of bubble columns as shown by Tomiyama et al. (). Clearly the assumption of equal velocity for all bubble sizes is not justified. This is perhaps the most severe limitation of the MUSIG model. It has remained in all the subsequent investigations employing the MUSIG model developed by Lo (996). Fortunately, the remedy is simple. One can indeed relax the assumption of equal velocity for all size groups and still carry out the simulations with a moderate computational expense, if one considers the simplified gas phase momentum balances. An order of magnitude analysis of the gas phase momentum equation reveals that it is predominantly the balance between interface forces and the pressure gradient. The inertial and stress terms can be neglected. This reduces the momentum balance for the gas phase to an algebraic equation. The resulting model is referred to as algebraic slip model and it yields the solution for the gas phase velocity in one step (i.e. non-iterative) when the converged liquid velocity field is known. This idea has been used in the model developed in this work, which is explained subsequently. Another approach taken by Frank et al. (5) involves the division of bubble sizes into N velocity groups, and subdivision of each velocity group into M groups each having identical velocity. This is a multi-fluid approach in which N M population balance equations are solved. However, the number of gas phase momentum equations is only N. Typically 3 5 velocity groups may be sufficient to capture the dynamics of bubble columns and hence the computational burden is moderate. This has been referred to as inhomogeneous MUSIG model by Frank et al. (5). Thus, the original MUSIG model developed by Lo (996) is now referred to as homogeneous MUSIG model. Olmos et al. () have carried out CFD simulations of a bubble column of diameter. m and height.35 m operating in the transition regime. They have concluded that the simulated results of liquid velocity profiles and the average gas hold-up in the column are not substantially affected by bubble coalescence and break-up functions. However, they have scaled down the coalescence and break-up rates by a factor of.75. This leads to an interesting conclusion that the CFD PBM simulations are capable of capturing the mean flow pattern correctly but not the evolution of bubble sizes. Similar are the observations of the Chen et al. (4, 5). Chen et al. (5) have studied the effect of bubble coalescence and break-up models on the predictions of CFD simulation. They considered the models of Prince and Blanch (99), Luo and Svendsen (996) for bubble coalescence and the models of Luo and Svendsen (996), Martinez-Bazan et al. (999) for bubble break-up and concluded that the choice of these models does not substantially affect the simulation results as long as the magnitude of break-up rate is increased tenfold. Dhanasekharan et al. (5) have simulated the external loop air-lift reactor to obtain the oxygen transfer rate. They have used the framework provided by Fluent 6.. Their idea was to obtain the improved estimates of interfacial area based on the PBM. The bubble size affects the interfacial area per unit volume. In case of the bio-reactors, where the mass transfer is controlling step, an improved estimate of interfacial area based on the PBM is expected to improve the design of such systems. Wang et al. (6) have simulated the bubble columns and upward cocurrent gas liquid pipe flows. They obtained the core-peaking gas hold-up profile in case of bubble columns and wall-peaking gas hold-up profiles in case of cocurrent upward pipe flows. The difference between the two cases was attributed to lift coefficient formulation of Tomiyama et al. () which can capture the radial segregation of bubbles depending on their size. Almost all the models from the literature use a k.ε model for turbulence in CFD PBM framework. Although, this is sufficient for the engineering calculations of turbulent flows, the Reynolds stress model (RSM) is expected to simulate the turbulence in a better way by taking into account the inherent anisotropy of the turbulence. The first attempt in this regard is by Ekambara et al. (8). They have simulated a bubble column of.5 m diameter and.9 m height. The experimental data of Kulkarni et al. (7) was used for the validation of models. An extensive comparison of the local mean velocity, bubble sizes, gas hold-up and Reynolds stresses was presented.

4 7 M.R. Bhole et al. / Chemical Engineering Science 63 (8) 67 8 Analysis of CFD PBM studies from the literature shows that the critical requirement at present is of reasonably good coalescence and break-up models themselves. The requirement of huge computational power can be tackled with the modern computers. As far as the issues related to interface forces are concerned, the experience gained from the conventional CFD modeling of bubble columns (without PBM) can be used with some confidence. However, to obtain the evolution of bubble size distribution and its impact on the mean flow as well as turbulence, robust models for the bubble coalescence and break-up are required. It is instructive to see the state of art in this regard. Various investigators have concluded that the bubble coalescence rate is overpredicted by the model of Prince and Blanch (99). Hence, to adjust the simulation results with the experimental data, either coalescence rate is arbitrarily multiplied by some constant less than (Lo, 996; Olmos et al., ) or break-up rate is arbitrarily multiplied by some constant greater than (Chen et al., 4, 5). In the absence of independent confirmation for the validity of coalescence or break-up models, it is not known whether one should reduce the coalescence rate or increase the break-up rate or do both. Most importantly, in doing so, the predictive capabilities of the models are lost. The models can be tuned for the gas liquid systems under a restricted set of operating conditions where the experimental data are available. For unknown gas liquid systems, the predictions become difficult. Another issue is of gas liquid interface. The bubble coalescence rate is substantially lower in electrolyte solutions. Similarly, the bubble break-up rate is also affected by the presence of surfactants in the liquid phase. The present models for bubble coalescence and break-up cannot account for these observations satisfactorily. At present, this is the central problem for the CFD PBM modeling of bubble columns. We now turn our attention to the CFD PBM framework developed in this work. Section 3 describes the formulation of model. The approach taken here is to give a proper perspective of the combination of CFD with PBM. We also describe the interface forces and the bubble coalescence and break-up functions used in this work. The rationale for the modification of turbulent collision frequency function by Prince and Blanch (99) is explained. Section 4 briefly describes the method of solution and it is followed by results and discussions in Section Mathematical model 3.. CFD PBM model formulation With reference to MUSIG model, we had observed that the assumption of equal velocity for all bubble size groups was motivated by computational economy rather than by the actual physics. In an attempt to relax this assumption without excessively increasing the computational burden, the gas phase momentum equation (Eq. ()) is simplified. The density of gas phase is negligible compared to that of the liquid phase. An order of magnitude estimate for Eq. () shows that the inertial terms on the left-hand side and body force term on the right-hand side which are proportional to the gas density can be neglected in comparison with the pressure gradient and interface force terms which scale with the liquid density. Further, the stress terms in Eq. () are negligible since the turbulence is considered only in the liquid phase for bubble columns. Thus, the simplified momentum balance for the gas phase is = ɛ G p M IG. (4) Eq. (4) states that the momentum balance for the gas phase is simply the balance between the interface forces and pressure gradient terms. Since Eq. (4) is an algebraic equation, the resulting model is called as algebraic slip model. Clearly, this is a simplified Eulerian momentum equation. In doing so, we have made no unrealistic assumption per se. Since our interest is in the steady state simulation, we do not resolve the complexities of flow around individual bubble with a fine grid and time step. Had it been the case, one must solve the full momentum equation for the gas phase. But our interest is in mean flow pattern on a much larger scale (with the Euler Euler framework) and hence this simplifying assumption commensurate with the level of detail we seek from the simulation. Eq. (4) can also be obtained directly from the steady state force balance on a single bubble which demands that the force due to pressure gradient must balance with the interface forces acting on a bubble. (The weight of the bubble is neglected on dimensional ground.) Hence, the assumption of steady state bubble motion is consistent with the algebraic slip model. Although the complexities of flow field in bubble columns (for example, bubble plume oscillations) can only be described by unsteady three-dimensional simulations, the two-dimensional axisymmetric simulations are sufficient for the prediction of time-averaged hold-up, velocity and bubble size profiles. The advantage with Eq. (4) is that it is an algebraic equation. Hence, the calculation of gas phase velocity is a non-iterative step. During the iterative calculation procedure, at a given iteration, if one has converged flow field solution for the liquid phase, it is a one step calculation for the gas phase velocities. Hence, we can rather solve all M discrete (size-specific) gas phase momentum equations without any significant computational burden. The combination of Eq. (4) with Eq. () leads to the following equation of motion for the liquid phase: t (ρ Lɛ L u L ) + (ρ L ɛ L u L u L ) = p + ρ L ɛ L g.(ɛ L τ L ). (5) This is another advantage with this approach. The liquid momentum equation is quite similar to its single phase counterpart and can be readily solved by the robust computational procedures which have been developed for the single phase flows. The source term of interface momentum exchange has been completely eliminated from the momentum equation of the liquid phase. It is now concealed in the pressure gradient term. This approach has been developed by Sokolichin et al. (4) for the simulation of bubble columns using a two-fluid model. It has been extended here for the case of multi-fluid model. The population balance equation is a balance equation for the number density (N i ) of the bubbles of the ith size range.

5 M.R. Bhole et al. / Chemical Engineering Science 63 (8) During the iterative calculation of number density, the roundoff errors can occur due to very large numerical values of the number density and hence, the population balance equation for the number density is multiplied by bubble volume to obtain the equation for size-specific gas hold-up. We use the following identities: N i x i = ɛ Gi, (6) M ɛ Gi = ɛ G = ɛ L. (7) i= The balance equations for the size-specific gas hold-up can be written as follows: ρ G ɛ Gi + (ρ t G u Gi ɛ Gi ) = ρ G S i, i =,,...,M. (8) We again note the important difference between Eqs. (3) and (8). In Eq. (8), the velocity of individual bubble size group is retained. On the other hand, Eq. (3) of the MUSIG model considers the same velocity for all sizes. The population balance equation (Eq. (8)) can be regarded as the size-specific continuity equation neglecting any changes in bubble sizes as a result of compressibility. Thus, the combined CFD PBM model for the bubble column is a multi-fluid model (i.e. M + fluid model) wherein continuity and momentum equations for liquid phase are solved together with the continuity equations for M gas phases (population balance equations) and momentum equations for M gas phases (algebraic slip equations viz. Eq. (4)). In writing Eq. (8), the discrete equations in the domain of bubble volume, the fixed pivot technique of Kumar and Ramkrishna (996) has been used. The source term for Eq. (8) can be written as S i = where δ j,k = j k j,k x i (x j +x k ) x i+ ɛ Gi M k= {, j = k,, j = k, Q i,k ɛ Gk x k + [ ] δ j,k ηq j,k ɛ Gj ɛ Gk M k=i xi+ x n i,k = i+ v β(v, x k ) dv + x i x i+ x i x i+ v, x i v<x i+, x η = i+ x i v x i, x i v<x i, x i x i x i x j x k n i,k Γ k ɛ Gk x i x k Γ i ɛ Gi, (9) xi x i v x i x i x i β(v, x k ) dv, where Q j,k is the coalescence frequency between the bubbles of size group j and k, Γ i is the break-up frequency of bubbles of size group i. β(v, x k ) is the number of bubbles of volume v formed from the breakage of the bubble of volume x k. The details of the derivation can be found in Kumar and Ramkrishna (996) or Ramkrishna (). We note here in passing that the sink and source terms for coalescence and breakage have been included in the population balance equation as additive. That this is not strictly permissible has been pointed out by Ramkrishna () except under specific circumstances. The specifics of this issue to the current context are addressed in a forthcoming publication by the authors. In the present paper, however, we retain the assumption of additive combination of the rates of two processes. The closure for the eddy viscosity in the momentum equation for the liquid phase is achieved by a k.ε model for the turbulence adapted for two-phase flows. These equations are as follows: μ T = ρ L C μ k (ρ L ɛ L k) t ε, () + (ρ Lɛ L u j k) x j = ɛ L τ ij u i x j ɛ L ρ L ɛ + (ρ L ɛ L ε) t = ɛ L ε k + x j + (ρ Lɛ L u j ε) x [ j u i C ε τ ij x j ] C ε ρ x L ε j ) ε [ɛ L ( μ + μ T σ ε x j ( [ɛ L μ + μ ) ] T k, () σ k x j ], () where C μ =.9, C ε =.44 and C ε =.9 are the constants. Generally bubble induced turbulence is accounted in a k.ε model by additional source terms. At present, the formulation of these terms for polydispersed bubbly flows is not known. They are not included in the present work. The two-fluid model simulations by Hillmer et al. (994) and Ranade (997) have shown that these terms are not significant. Ekambara et al. (5) have shown over a wide range of V G that about 5 3% of the bubble generated turbulence participates in momentum transfer. Clearly more studies are needed in this area. 3.. Interface forces The interface forces with a clear physical significance in case of bubbly flows are drag force and lift force. The virtual mass force is neglected. This is because the steady state motion is assumed. Further it is assumed that the bubbles attain their rise velocity immediately at the sparger (z = ) and also they disengage the liquid at the top (z = H) with their terminal rise velocity. Hence, the acceleration effects associated with the bubble motion at the entrance and the exit in the column are neglected. The drag force per unit volume for the bubbles of size group i is obtained as follows: M Di = 3C Di 4d Bi ρ L ɛ Gi u Gi u L (u Gi u L ). (3)

6 7 M.R. Bhole et al. / Chemical Engineering Science 63 (8) 67 8 The drag coefficient is a function of bubble Reynolds number and Eotvos number and the following correlation which has also been utilized by Pan et al. (999) has been used: C D = MAX [ 4 Re ( +.5Re.687 ), 8 3 ( Eo Eo + 4 )]. (4) The lift force arises due to interactions between the bubble and the vorticity in the liquid and it is given by M Li = ρ L C Li ɛ Gi (u Gi u L ) ( u L ). (5) The expression for the lift coefficient given by Tomiyama et al. () has been used. It is given as { min[.88 tanh(.re), f (Eod )], Eo C L = d < 4, f(eo d ), 4 Eo d.7, (6) where f(eo d ) =.5Eo 3 d.59eo d.4eo d The lift coefficient in Eq. (6) depends upon the modified Eotvos number given by Eo d = g(ρ L ρ G )dh, (7) σ where d H is the maximum horizontal dimension of a bubble which can be obtained knowing the mean bubble diameter and the bubble aspect ratio. Eq. (6) predicts the lift coefficient in range <C L.88 for small bubbles of diameter less than 6 mm whereas for the bubbles of higher diameter (> 6mm) it predicts the negative lift coefficients in the range of.88 <C L <. It must be noted that the correlation of Tomiyama et al. () for lift coefficient is empirical and is based on the measured trajectories of single bubbles in simple shear flows. The negative lift coefficient leads to preferential migration of bubbles toward the center of the column in a typical bubble column whereas the opposite is true for the positive lift coefficient. This has been shown by Sokolichin et al. (4) and Wang et al. (6). In a typical bubble column operating in the heterogeneous regime, the hold-up of the large-sized bubbles is higher at the center whereas the hold-up of smaller sized bubbles is higher toward the wall. Hence, an appropriate correlation for the lift coefficient is essential to capture the appropriate radial profiles of the bubble diameter Bubble coalescence model Coalescence between the two bubbles in a turbulent environment occurs in three steps. Bubbles collide with each other, trapping a small amount of liquid between them. The liquid then drains until the liquid film separating the two bubbles reaches a critical thickness. At this point, the film ruptures and the bubbles coalesce. There are various mechanisms by which bubbles can come closer to each other and collide. Prince and Blanch (99) have considered three mechanisms of bubble collisions. Bubbles can collide due to random motion in a turbulent flow field. Two bubbles of different rise velocities can approach each other and it is called as buoyancy driven collision rate. Further, the bubbles can collide due to mean shear in the flow field. Out of these three mechanisms, in this work, we consider only the collision mechanism due to turbulence. In a turbulent bubbly flow, the liquid eddies impart a fluctuating motion to bubbles also. Turbulent collisions can be treated analogous to collisions of the molecules in a Maxwellian gas. The collision frequency is given as θ T jk = N j N k S jk (u Bj + u Bk )/. (8) In Eq. (8), N j and N k are the number densities of bubbles of diameter d Bj and d Bk, respectively, u B is the root mean square (RMS) fluctuating velocity of the bubble and S jk is the collision cross-sectional area of the bubbles defined by S jk = π 4 (d Bj + d Bk ). (9) In the original model by Prince and Blanch (99) the RMS fluctuating velocity of a bubble is assumed to be same as that of a liquid eddy of size, d B. Since, most of the bubbles in a turbulent bubble column have sizes that fall in the inertial subrange of the turbulence; the following expression was obtained for the fluctuating bubble velocity: u Bj = u fi = (εd Bi ) /3, () where ε is energy dissipation rate in the liquid. However, the bubbles do not follow the liquid eddy motion faithfully. On account of their slip velocity, they tend to escape from the eddy in which they are entrapped. Thus, the fluctuating velocity of the bubble is less than the fluctuating velocity of the eddy of same size. According to Kruis and Kusters (997), the RMS velocity of a dispersed phase particle is related to the eddy velocity by the following expression: u Bj u fi = σ f = + St, () where σ f is bubble dispersion number and St is Stokes number given as St = τ B. () τ f In Eq. (), τ B is the bubble relaxation time and τ f is the integral time scale for fluid motion. They are given by the following expressions (Loth, ): τ B = 4C V d B 3C D V B, (3) τ f = Cμ 3/4 k ε. (4) The bubble relaxation time, τ B increases with an increase in the bubble diameter and hence the RMS velocity of bubble fluctuation decreases compared to the RMS velocity of the eddy of the same size. The gas phase dispersion number used in the bubble collision frequency model also appears in the time-averaged continuity equations of the two-phase flows (Shirolkar et al., 996).

7 M.R. Bhole et al. / Chemical Engineering Science 63 (8) E-4 Bubble dispersion number, σ f (-) Collision frequency (m 3 /s).e-4.6e-4.e-4 8.E-5 4.E-5 d Bk =.5 m Bubble diameter, d B (m) Fig.. Bubble dispersion number as a function of bubble diameter for a typical case of k =. m /s and ε =.m /s 3..E Bubble diameter, d Bj (m) Fig. 3. Turbulent collision frequency as a function of bubble diameter for ε=.m /s 3. () Model by Prince and Blanch (99); () the model suggested in this work. The gas phase dispersion can be included in the time-averaged continuity equation for the gas phase as follows: ( ) t (ρ μt Gɛ G ) + (ρ G ɛ G u G ) = ɛ G. (5) σ f The term on the right-hand side of Eq. (5) accounts for the bubble diffusion along the hold-up gradient. Joshi () has observed that σ f can vary from to as the bubble size increases progressively. The small-sized bubbles which follow the eddy motion faithfully have dispersion number closer to unity. On the other hand, the large-sized bubbles, on account of their significant slip velocity tend to escape the eddy in which they are entrained and hence they have dispersion number much greater than unity or in other words, the smaller (turbulent) diffusion coefficient. For the typical case of bubble column with k =. m /s and ε =.m /s 3, the variation of the gas phase dispersion number with the bubble diameter is shown in Fig.. In this case, the drag coefficient given by Eq. (4) and the virtual mass coefficient formulation given by Sankaranarayanan et al. () is used for the calculation of dispersion number. The model developed in this work gives a smaller collision frequency compared to the collision frequency given by Prince and Blanch (99). This is shown in Fig. 3. Chen et al. (5) have pointed out that the model of Prince and Blanch (99) overestimates the collision frequency. The reason for the same is the overestimation of the RMS velocity of the bubble motion. Not all the collisions between the bubbles lead to coalescence. When the two bubbles are sufficiently close to each other so that the liquid film trapped between them starts draining, there is a possibility that the bubbles may be taken away from each other by a turbulent eddy. Thus, it is customary to consider efficiency of the process leading to coalescence. Bubble coalescence frequency is then given by Q jk = θ jk λ jk, (6) where λ jk is termed as the collision efficiency. The expression for collision efficiency given by Prince and Blanch (99) has been retained in this work Bubble break-up model We use the bubble break-up model of Luo and Svendsen (996). This model is based on the idea that the break-up occurs when the eddy of sufficient energy collides with the bubble. Hence, the break-up frequency is related to the frequency of collision of eddies with the bubble and is given by ( ) /3 Γ(v j : v i ) ε ( + ξ) =.93 ( ɛ G )N i dbi ζ min ξ /3 ( ) c f σ exp ρ L ε /3 d 5/3 dξ. (7) Bi ξ/3 Eq. (7) gives the break-up frequency for the breakage of bubble of volume v i into a bubble of volume v j. The fractional increase in the surface area upon break-up is given by c f = f /3 BV + ( f BV ) /3, (8) where f BV = v j /v i and ξ is the dimensionless eddy size (the ratio of eddy size to bubble diameter). The details of the model can be found in Luo and Svendsen (996). The break-up frequency for the bubble of volume v i is given by Γ i = 4. Method of solution Γ(f BV v i : v i ) df BV. (9) The CFD PBM equations for bubble columns are written in cylindrical coordinate system for the steady state simulation using FORTRAN-9. Axisymmetry has been assumed so that

8 74 M.R. Bhole et al. / Chemical Engineering Science 63 (8) 67 8 Table Diameter and rise velocity of bubble classes included in CFD PBM simulation Bubble class Diameter d B (m) Rise velocity V B (m/s) the derivative of any dependent variable in the tangential direction is zero. Further, it is assumed that the flows are without any swirl, which means that the tangential velocity itself is also zero. The simulations have been carried out from the center of the column r = up to the wall r = R and from the bottom z = up to the top z = H. The equations are discretized using the finite volume method described by Patankar (98). The convection diffusion is handled by the power law discretization scheme. The framework of SIMPLER algorithm has been used to handle the pressure velocity coupling (Patankar, 98). All the discretized equations are solved sequentially using tridiagonal matrix algorithm (TDMA). To avoid the divergence of the iterative scheme, the under-relaxation factor of.7 is used. For pressure and pressure correction equations, no underrelaxation is employed. For the population balance equations, the under-relaxation factor is set to.5. An outer iteration of SIMPLER consists of the following steps. First, the pressure is calculated and it is followed by the calculation of axial liquid velocity, radial liquid velocity, pressure correction and velocity correction in that order. The next step is the solution of the equations for k.ε model. It is followed by the calculation of gas phase velocities by solving the algebraic slip model described earlier. Finally, the size-specific gas hold-up is calculated by solving discretized population balance equations. Each of these steps requires 5 internal iterations for the convergence. The solution for all the variables once constitutes one outer iteration of SIMPLER algorithm and is repeated till convergence. The dimensionless mass residue of 3 is maintained as the convergence criterion. Typically outer iterations are required for convergence depending upon the grid size. For the column of diameter, D =.5 m and height, H =.9 m, a grid of Δr =.5 m and Δz =.6 m is maintained. The bubble volume grid consists of 3 bubble classes with the diameter ranging from to 5 mm as shown in Table. 5. Results and discussion The details of the gas sparger are not considered explicitly in CFD PBM simulations. The superficial gas velocity, bubble diameter and gas hold-up at the sparger are specified. Most Mean bubble diameter, d B (m) Dimensionless axial coordinate, z/d (-) Fig. 4. Axial evolution of sauter mean bubble diameter in a bubble column. Bubble diameter at the sparger is 5 mm. The break-up model of Luo and Svendsen (996) is used. V G = mm/s. () Coalescence model of Prince and Blanch (99); () coalescence model suggested in this work. of the simulations are carried out for superficial gas velocity of mm/s unless specified otherwise. It is assumed that the gas comes from the entire cross-section at z =. A flat gas hold-up profile is assumed to prevail at the sparger. A constant bubble diameter is specified at the sparger. This is clearly a simplification. In reality, there can be a bubble size distribution emerging from the sparger and it can have a significant impact on simulation results. However, the bubble size distribution at the sparger is generally not available as it is very difficult to measure. Hence, a constant bubble diameter at the sparger is specified as per the correlation of Miyahara et al. (983). Fig. 4 shows the evolution of sauter mean bubble diameter along the column height. In this case, the break-up model of Luo and Svendsen (996) is used. When the coalescence model of Prince and Blanch (99) is employed, the sauter mean bubble diameter is seen to increase from 5 mm at the sparger to about 3 mm at the top. Thus, the axial profile of sauter mean diameter indicates the equilibrium between coalescence or break-up is not achieved. This is probably due to overestimation of coalescence frequency (or underestimation of break-up frequency). On the other hand, when the model of Prince and Blanch (99) is modified as discussed in the previous section, the sauter mean bubble diameter increases from 5 mm at the sparger to about 7 mm at the top. In this case, the equilibrium between the coalescence and break-up is attained at about z/d = 4 as indicated by the flatness of axial profile of sauter mean bubble diameter in the column. Depending upon superficial gas velocity and bubble size at the sparger, an equilibrium bubble size is obtained at an appropriate axial location in a bubble column as observed by Joshi () and Kulkarni et al. (7). It must be noted that the most of the researchers in the past have manipulated coalescence or break-up frequency with a multiplicative constant in order to attain the equilibrium bubble size.

9 M.R. Bhole et al. / Chemical Engineering Science 63 (8) We have seen that the bubble velocity variation with respect to size cannot be included in CFD simulations with the homogeneous MUSIG model. A single velocity field for the gas phase corresponding to local sauter mean bubble diameter is used in the MUSIG model whereas the present work retains the velocities of individual bubble size groups in CFD PBM simulations. It is instructive to see the differences between these two cases. Fig. 5 shows the radial profiles of sauter mean bubble diameter, gas hold-up and axial liquid velocity at z/d = 4. In case of MUSIG model, the bubble diameter profile is relatively flat as all bubble size groups have identical velocity. In this case, the radial segregation of bubbles is possible only due to differences in local coalescence and break-up rates. On the other hand, with the present model, the radial segregation of bubbles is possible on account of their inherent velocity differences also. It is apparent from relatively steep bubble diameter profile shown in Fig. 5. Similarly, the differences in the gas hold-up and liquid velocity profiles can be explained. Thus, we get different hydrodynamic behavior with the two different models. Clearly, one cannot tell which model is superior based on simulation results such as those shown in Fig. 5. Experimental data on local bubble size distribution, gas hold-up, liquid velocities etc. at various locations in bubble columns are required for model validation. The comparison of simulation results with the available experimental data are discussed subsequently. At present, we note that the model developed in this work is expected to be superior as it takes into account a bubble velocity distribution more appropriate than uniform. This is necessary to obtain the proper spatial distribution of bubbles which in turn determines the gas hold-up and liquid circulation profiles in the column. Obviously to obtain the proper bubble velocity distribution, accurate information on interface forces is required. The closures for interface drag and lift forces such as Eqs. (4) and (6) are generally based on the single bubble experiments. However, in bubble columns the complexities of multiphase flows such as hindrance, wake effects, reduced pressure gradients and turbulent dispersion are present and hence, the closures for interface forces are difficult to specify. We now show the effect of different bubble velocity distributions on hydrodynamics of bubble columns. This is accomplished by considering three different formulations for lift coefficient viz. the lift coefficient formulation of Tomiyama et al. (), C L = for all bubble sizes and C L =. for all bubble sizes. The CFD PBM simulations were carried out for these three cases. The typical radial velocities of bubbles as a function of size at r/r =.5 and z/d = 4 for these three cases are shown in Fig. 6. With the correlation of Tomiyama et al. (), the radial bubble velocities are positive up to d B = 6 mm and they are negative for higher bubble diameter. With C L =, the radial force balance is simply the balance between the radial drag force and radial pressure gradient. Since the radial pressure gradients are negligible, the radial bubble velocities are practically zero in this case. With C L =., the radial bubble velocities are positive for all sizes at the location considered in the column. Axial slip velocities of bubbles are not affected significantly with lift force formulation as the axial motion is predominantly governed by balance between drag Sauter mean bubble diameter, d B (m) Gas hold-up (-) Axial liquid velocity (m/s) Dimensionless radial coordinate, Dimensionless radial coordinate, Dimensionless radial coordinate, Fig. 5. Radial profiles of sauter mean diameter, gas hold-up and axial liquid velocity at z/d = 4 for V G = mm/s. () Simulations based on present model; () simulations based on MUSIG model. and buoyancy. These steady state bubble velocities form the input to population balance equations which decides the spatial distribution of bubbles in the column. The differences in the

10 76 M.R. Bhole et al. / Chemical Engineering Science 63 (8) 67 8 Radial bubble velcoity (m/s) Bubble diameter, d B (mm) Fractional gas hold-up (-) Fig. 6. Radial bubble velocity as a function of bubble diameter at r/r =.5 and z/d=4 for V G = mm/s. () Lift coefficient by correlation of Tomiyama et al. (); () C L = ; (3) C L = Dimensionless radial coordinate, Fig. 8. Radial variation of fractional gas hold-up at z/d=4 for V G = mm/s. Bubble diameter at the sparger is 5 mm. () Lift coefficient correlation of Tomiyama et al. (); () C L = ; (3) C L =.. Sauter mean bubble diameter, d B (m) Axial liquid velcoity (m/s) Dimensionless radial coordinate, Fig. 7. Radial variation of bubble diameter at z/d = 4 for V G = mm/s. Bubble diameter at the sparger is 5 mm. () Lift coefficient correlation of Tomiyama et al. (); () C L = ; (3) C L =.. hydrodynamic behavior in these three cases are shown in Figs. 7. Fig. 7 shows the radial profiles of mean bubble diameter at z/d = 4 in the column. It can be seen that the lift coefficient formulation of Tomiyama creates the radial segregation of bubbles with larger bubbles preferably toward the center and the smaller bubbles preferably toward the wall. When C L = is used, the radial profile of bubble diameter becomes relatively flat and with C L =., the profile becomes still flatter as seen from Fig. 7. The gas hold-up profile is steep with the lift coefficient formulation of Tomiyama et al. () and the profile becomes flatter in going from C L = to. as seen in Fig. 8. The radial gas hold-up profile creates the driving force for liquid circulation in the column. Hence, the liquid circulation velocities are highest with the lift coefficient Dimensionless radial coordinate, Fig. 9. Radial variation of axial liquid velocity at z/d =4 for V G = mm/s. Bubble diameter at the sparger is 5 mm. () Lift coefficient correlation of Tomiyama et al. (); () C L = ; (3) C L =.. formulation of Tomiyama et al. () whereas the circulation gets dampened and the centerline axial velocities tend to decrease in going from C L = to.. This behavior is clearly apparent from Fig. 9. The production of turbulent kinetic energy (source term in k equation) is predominantly determined by the radial gradient of axial liquid velocity which forms the main source of shear in the bubble column. Clearly, the higher the liquid velocity gradient, the higher is the value of turbulent kinetic energy. This explains the observed behavior of turbulent kinetic energy in Fig.. With the C L formulation of Tomiyama et al. (), the shear in the column is highest and hence the turbulent kinetic energy is also the highest.

11 M.R. Bhole et al. / Chemical Engineering Science 63 (8) Turbulent kinetic energy (m /s ) Bubble diameter, d B (m) z/d = Dimensionless radial coordinate, Fig.. Radial variation of turbulent kinetic energy at z/d = 4 for V G = mm/s. Bubble diameter at the sparger is 5 mm. () Lift coefficient correlation of Tomiyama et al. (); () C L = ; (3) C L =.. Size-specific gas hold-up, (-) Bubble diameter, d B (m) Fig.. Size-specific gas hold-up distribution at z/d = 4 for V G = mm/s. Bubble diameter at the sparger is 5 mm. () r/r = ; () r/r =.5; (3) r/r =.97. Thus, the lift force plays an important role in CFD simulations of bubble column. As indicated by Sokolichin et al. (4), the magnitude of lift coefficient can be adjusted to tune the simulation results with the experimental data. While this is true for two-fluid model with constant bubble size, it becomes even more important for CFD simulation with PBM. Most importantly, the steady state formulation for bubble motion under mechanical equilibrium of forces allows for the proper incorporation of bubble velocities due to lift force. We note that the correlation of Tomiyama et al. () is for a single bubble in well-defined liquid shear. In case of turbulent multiphase dispersion in bubble columns, the lift coefficient can be significantly affected by the presence of neighboring bubbles and clearly the more experimental information on the lift force is desired. An attempt in this Bubble diameter, d B (m) Bubble diameter, d B (m) z/d = z/d = Fig.. Comparison of simulation results with the experimental data ( ) of Bhole et al. (6) for the radial profiles of bubble diameter in the bubble column with perforated plate sparger at V G = mm/s.

12 78 M.R. Bhole et al. / Chemical Engineering Science 63 (8) z/d =. z/d = Gas hold-up (-) Axial Liquid Velocity (m/s) Gas hold-up (-) z/d = 3 Axial Liquid Velocity (m/s) z/d = z/d = 4 Gas hold-up (-) z/d = 4 Axial Liquid Velocity (m/s) Fig. 3. Comparison of simulation results with the experimental data ( ) of Bhole et al. (6) for the radial profiles of gas hold-up in the bubble column with perforated plate sparger at V G = mm/s. direction has been made by Kulkarni (3). The importance of lift force in the simulation of bubble columns has also been shown by Lain and Sommerfeld (4) and Reddy Vanga et al. (4). Fig. 4. Comparison of simulation results with the experimental data ( ) of Bhole et al. (6) for the radial profiles of axial liquid velocity in the bubble column with perforated plate sparger at V G = mm/s. The distributions of gas hold-up according to bubble sizes at z/d = 4 for various radial locations are shown in Fig.. These are the size-specific gas hold-up at r/r =,.5,.97. At all these radial locations, unimodal distributions are seen. The fact that the bubble size decreases from center to wall is also apparent from Fig.. The correlation of Tomiyama et al. () for lift coefficient has been used in this case.