Euler-Lagrange modeling of a gas-liquid stirred reactor with consideration. of bubble breakage and coalescence

Transcription

1 Euler-Lagrange modeling of a gas-liquid stirred reactor with consideration of bubble breakage and coalescence R. Sungkorn Institute for Process and Particle Engineering, Graz University of Technology, A-8010 Graz, Austria J. J. Derksen Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6 Canada J.G. Khinast* Research Center Pharmaceutical Engineering, A-8010 Graz, Austria Abstract Simulations of a gas-liquid stirred reactor including bubble breakage and coalescence were erformed. The filtered conservation equations for the liquid hase were discretized using a lattice Boltzmann scheme. A Lagrangian aroach with a bubble arcel concet was used for the disersed gas hase. Bubble breakage and coalescence were modeled as stochastic events. Additional assumtions for bubble breaku modeling in an Euler-Lagrange framework were roosed. The action of the reactor comonents on the liquid flow field was described using an immersed boundary condition. The redicted number-based mean diameter and long-term averaged liquid velocity comonents agree qualitatively and quantitatively well with exerimental data for a laboratory-scale gas-liquid stirred reactor with dilute disersion. Effects of the resence of bubbles, as well as the increase of the gas flow rate, on the hydrodynamics were numerically studied. The modeling technique offers an alternative engineering tool to gain detailed insights into comlex industrial-scale gas-liquid stirred reactors. *Corresonding author. address: khinast@tugraz.at Keywords Comutational fluid dynamics, lattice-botlzmann method, immersed boundary condition, Lagrangian article tracking, gas-liquid stirred reactor, bubble breaku and coalescence 1

2 Introduction Stirred tank reactors are among the most widely used reactor tyes in a large variety of industrial rocesses involving multihase flows. Tyical examles include industrial hydrogenations or oxidations, as well as aerobic fermentation rocesses in which gas bubbles are disersed in turbulent fluid flow induced by one or more imellers. In biotechnological rocesses, the activity and the growth of microorganisms (e.g., bacterial or fungal systems) or cells are sensitive to a number of arameters, such as dissolved oxygen content, substrate concentration and H-level. A major challenge in these rocesses is to rovide adequate liquid mixing and to generate a large interfacial contact area, while avoiding shear damage of microorganisms and cells caused by hydrodynamic effects. 1,2 The flow structures in a single liquid-hase stirred reactor are known to be highly comlex associated with time-deendent, three-dimensional henomena covering a wide range of satial and temoral scales. 3 The comlexity increases drastically when a gas hase is introduced. Additional effects include the interaction between hases in terms of mass, energy and momentum exchange, the interaction between the second hase and the imeller, and bubble breakage and coalescence. Since the reactor erformance is a comlex function of the underlying henomena, a detailed knowledge regarding the hydrodynamics and the evolution of the disersed hase are essential for the engineering of a high erformance reactor. Traditionally, engineering of gas-liquid stirred reactors is based on emirical correlations derived from exeriments. The information obtained from this aroach is usually described in global arametric form and alicable within a narrow window of geometry configurations and oerating conditions. Recently, the use of comutational fluid dynamics (CFD) has gained some oularity among researchers and ractitioners for the engineering of stirred reactors. Significant rogress has been made over the last decades in the fields of turbulence and multihase-flow modeling, numerical methods as well as comuter hardware. Consequently, time-deendent, three-dimensional simulations of gas-liquid stirred reactors with a sohisticated level of detail and accuracy are feasible today. 2

3 Various CFD modeling techniques for gas-liquid flows, where the gas hase is disersed in the liquid hase, have been reorted in the literature. These techniques can be grossly categorized based on their treatment of the disersed hase into Euler-Euler (EE) and Euler-Lagrange (EL) aroaches. 4,5 In the EE aroach, both hases are treated as interenetrating continua. The interactions between hases are modeled via the hase interaction terms that aear in the conservation equations describing the dynamics of the system. The EE aroach assuming monodiserse sheres has been used in the work of Deen et al., 6 Khokar et al. 7 and Zhang et al. 8 for simulations of gas-liquid stirred reactors. A more sohisticated EE aroach, where the local bubble size distribution (BSD) is comuted by solving oulation balance equations (PBEs), has been reorted by Venneker et al., 9 Laakkonen et al. 10 and Montante et al. 11,12 In the EL aroach, each individual bubble or a arcel of bubbles is reresented by a single oint. The linear motion of the bubbles is governed by Newton s law of motion. Rotational momentum balances are tyically not considered. The EL aroach requires closure relations to account for the inter-hase forces, which can be obtained from emirical relations or from simulations with higher level of detail, e.g., a front tracking (FT) aroach 13 or volume of fluid (VOF) methods. Closures for gas-liquid systems which take into account the effect of high gas-hase loadings are still an area of ongoing research (e.g., Behzadi et al. 14 ). The EL aroach is comutationally exensive. However, its advantages are the high flexibility with resect to incororating microscoic and bubble-level henomena, such as bubblebubble interactions, bubble-wall interactions, breaku and coalescence of bubbles. Simulations of gasliquid stirred reactors using the EL aroach are relatively rare in the literature. Some examles include the work of Wu et al. 15 and Arlov et al. 1 for simulations of reactors with mono-diserse sheres and the work of Nemoda & Zivkovic 16 for the simulation of a reactor with bubble breakage. The goal of this work is to assess the detailed modeling of a gas-liquid stirred reactor by an EL aroach. The simulations are restricted to laboratory-scale reactors (with a volume in the order of 10 liters) with dilute disersion (i.e., a global gas hase of u to 2% volume fraction). The reason for simulating these systems is that, in this work, we mainly focus on the validation of the modeling technique resented in the next section. Detailed validation requires highly resolved exerimental information concerning liquid flow field and local bubble size distributions (BSD) which hard (if not 3

4 imossible) to obtain in dense systems. Disersed-hase volume fraction effects in the conservation equations of the continuous hase, and high-frequency collisions drastically increase the comlexity in denser systems, also from a comutational oint of view. Being able to numerically deal with denser systems is the subject of (our) current research. In engineering ractice, the gas hase fraction and the size of the reactor are tyically much larger. However, since the resented modeling technique is based on elementary hysical rinciles which also lay a role at the full-scale, the understanding of the underlying henomena obtained in this work makes it worthwhile for researchers and ractitioners. The main elements of the modeling technique used in our study are: The continuous liquid hase is modeled using a variation of the lattice-boltzmann (LB) scheme due to Somers. 17 The LB scheme is used to solve the large-scale motions of the turbulent flow using the filtered conservation equations. The Smagorinsky subgrid-scale model is alied to model the effects of the sub-filter scales. 18 It has been demonstrated that the scheme can accurately redict turbulent hydrodynamics in single hase system 3 as well as in multihase systems. 19,20,21 An adative force-field rocedure, 22 also known as an immersed boundary method, is used for describing the action of the reactor comonents (i.e., the imeller, tank wall, baffles and internals) on the liquid flow field. The motion of the individual bubbles are comuted by Lagrangian tracking taking into account the sum of forces due to stress gradients, net gravity, drag, lift and added mass. The momentum transfer between hases, i.e., the two-way couling, is achieved by the maing function with the virtual diameter concet introduced by Deen et al. 24 The imact of turbulence on the motion of the bubbles, i.e., the fluctuations of the sub-filter or residual fluid velocity along the bubble trajectory, is comuted using the Langevin equation model introduced by Sommerfeld et al. 25 Collisions of bubbles are governed by the so-called stochastic inter-article collision. 26 Based on the stochastic model, coalescence of bubbles is determined by comaring the film drainage time with the bubble contact time. 27 Breaku of bubbles is accounted for using a theoretical 4

5 model derived from the theory of isotroic turbulence. 28 It is assumed that breaku is caused mainly by the interaction of bubbles with turbulent eddies. Although various elements of our aroach have been reorted in literature, the resent modeling technique, for the first time, assesses the feasibility of using the LB scheme with the Lagrangian article tracking (LPT) model with consideration of bubble breakage and coalescence for simulations of gas-liquid stirred reactors. The simulations rovide a detailed insight into gas-liquid stirred reactors with a high level of accuracy along with reasonable comutational requirement. In the next section, the modeling technique for turbulent bubbly flows in the EL framework will be briefly introduced. A detailed discussion of this aroach can be found in our revious work. 21 Additional models for the simulation of gas-liquid stirred reactors, e.g., the treatment of reactor comonents, bubble breaku and coalescence, will be discussed in detail. In the subsequent section, the model validation, ranging from a bubble column to a gas-liquid stirred reactor, will be resented and discussed in detail. The conclusion will be summarized in the final section. Numerical Modeling Asects Liquid hase hydrodynamics In this work, the lattice-boltzmann (LB) scheme is used to model the turbulent liquid flow. The LB scheme is based on a simle form of the Boltzmann kinetic equation, which can be used to recover the macroscoic hydrodynamic behavior of fluids. 29 The basic idea is that fluid flow, which is governed by conservation laws, can be simulated by a many-articles system obeying the same laws. A set of (fictitious) articles residing on a lattice moves to neighbor sites and exchanges momentum (i.e., collide) with articles coming from other directions. The collision rules and the toology of the lattice are defined such that the Navier-Stokes equations are recovered. 30 The secific LB scheme emloyed here is due to Somers 17 (see Eggels & Somers 31 and Derksen & Van den Akker 3 ) with a cubic and uniform lattice. The scheme was chosen because of its robustness for turbulence simulations. This is due to the exlicit treatment of the high-order terms which results in enhanced stability at low viscosities and, thus, allows simulations of high Reynolds 5

6 numbers. In the LB scheme, the arithmetic oerations are local, i.e., data required for udating the flow in a grid oint are obtained from its next neighbors and, secifically, the stress tensor is exlicitly obtained from the data stored in a single node. Therefore, arallelization through domain decomosition requires only communication of subdomain boundary values, resulting in efficient arallel algorithms. Gas-liquid flows in a bubble column or in a stirred reactor, are normally turbulent, even at lab scale and with gas volume fraction as low as 1%. Considering the liquid motions induced by disersed gas bubbles, the turbulent stress can be divided into two comonents; one due to bubble buoyancy leading to liquid velocities above the turbulence onset and one art due to the so-called seudoturbulence caused by the fluctuation of the bubbles, i.e., the zig-zagging motion of bubbles relative to the fluid, resulting in turbulent-like flows due to vortex shedding and interaction henomena. Direct numerical simulation (DNS) of these flows is not feasible due to limitation in comutational resources, as the resolution of all length and time scales requires enormous amounts of grid cells and time stes. In order to overcome this limitation, only the evolution of the large-scale motions is resolved by alying a filtering rocess to the conservation equations of the liquid hase. The resolved flow can be interreted as a low-ass filtered reresentation of the real flow. The imact of the residual motion that resides at scales smaller than the filter width is modeled using the sub-grid scale (SGS) model due to Smagorinsky. 18 In this model, the SGS motion is considered to be urely diffusive and the model only drains energy from the resolved motions without feed-back. A larger fraction of the eddies and more of the energy residing in the flow are resolved with a finer grid sacing, i.e., a higher resolution. However, the choice of grid sacing in the EL aroach is also restricted by the size of the bubbles, as will be discussed in detail in the next section. Furthermore, Hu & Celik 32 ointed out that the residual motion of the seudo-turbulence, which osses a universal energy sectrum different from the classical -5/3 decay in single hase turbulence, could (in rincile) be catured using a dedicated SGS model. Such a reliable and accurate SGS model for multihase flows is not available. Therefore, the SGS model used in this work is adoted directly from the single-hase SGS model. This is justified by 6

7 our favorable results in relation to exerimental data as will be shown later in this work. The eddy viscosity t concet is used to reresent the imact of the SGS motion as: with the Smagorinsky constant t 2 2 C S, (1) S C S, the filter width (equal to the cubic grid cell size h ) and the resolved deformation rate 2 S. The value of C S is ket constant at 0.10 throughout this work. It has been demonstrated that the variation of C S has only marginal effect on the redicted flow field. 21,33 Since the simulations discussed here are restricted to dilute disersions, i.e., global gas volume fractions of u to 2%, it can be assumed that the void fraction term in the momentum conservation equations for the liquid hase has a relatively small effect on the flow. The filtered conservation equations (with source terms reresenting forces exert by the bubbles and the reactor comonents) for single hase flow are aroximately valid. The formulation and detailed discussion for the conservations equations emloyed in this work can be found in Eggels & Somers 31 and Derksen & Van den Akker. 3 This assumtion has been successfully used for simulations of multihase flows within the dilute disersion limit by several researchers (e.g. Derksen, 19 Hu & Celik 32 and Sungkorn et al. 21 ). Imeller and tank wall treatment Stirred tanks tyically consist of a cylindrical vessel equied with one or more imellers, baffles, and, otionally, other internals. They can be divided, into static (i.e., tank wall, baffles, and internals) and moving comonents (i.e., imellers and shaft). In this work, an adative force-field technique, also known as an immersed boundary method, 22,34 is emloyed to describe the action of the reactor comonents on the liquid flow field. The method mimics these comonents by a set of control oints on their surface. It comutes forces on the flow such that the flow field has a rescribed velocities at the control oints within the domain, i.e., equal to zero at static comonents and equal to the surface velocity for the moving comonents. The deviation from the rescribed velocities is estimated by a second-order interolation and imosed back on the lattice sites during the collision ste. 3 7

8 In the vicinity of walls, the turbulence becomes anisotroic, i.e., fluctuations in the wallnormal direction are suressed. Consequently, the SGS Reynolds stresses should become zero. These effects are accounted for by emloying the Van Driest wall daming function 35 with the universal velocity rofiles. 36 Bubble dynamics The disersed hase (i.e., the bubble hase) is tracked in a Lagrangian manner. A oint-volume (also known as oint-force) assumtion is emloyed. In this assumtion, the bubble is assumed to have a sherical shae and the bubble surface effect on the continuous fluid is neglected. 5 Each oint reresenting a arcel of bubbles with identical roerties (i.e., osition, velocity and diameter) is tracked simultaneously in the time-deendent, three-dimensional flow field. It is noted that the number of bubbles in a arcel is a real number. Its trajectory is comuted based on Newton s law of motion. All relevant forces, such as net gravity force, forces due to stress gradient, drag, net transverse lift, and added mass force are considered. Thus, the following set of equations will be solved for the motion of bubbles in a arcel: with V d u t dt x u, (2) 1 2 l Vg lvd tu CDld u u u u 8 C V u uu C V D u D u x being the centroid osition of the arcel, bubble volume with the diameter L l A l u the velocity, d, g the gravitational acceleration, t t, (3) the bubble density, V the l the liquid density, and u the liquid velocity at x. The drag C D and lift C L coefficients deend on the bubble Reynolds number Re 2 u u d and the Eötvös number Eo g d. The added mass force l l coefficient C A is assumed to be constant at 0.5. Exressions for the forces acting on a bubble and its coefficients are summarized in the Aendix. Note that, since the buoyancy force has been already included in the net gravity force, the forces due to stress gradient in the fluid include only the dynamic ressure and the deviatoric stress gradient. Hence, the forces due to the fluid stress gradient can be 8

9 reformulated in term of the fluid acceleration as shown in Table A1. 23,32 The liquid velocity u at the centroid of the arcel in Eq. (3) consists of the resolved liquid velocity u ~ and a (residual) liquid fluctuating comonent u. The latter comonent is used to mimic the imact of turbulence on the motion of the bubble, i.e., the fluctuations of the sub-filter (residual) liquid velocity along the bubble trajectory. The interolation of the liquid roerties on the Eulerian grid nodes to the centroid of the arcel on the Lagrangian reference frame (and vice versa) is achieved using a chea clied fourthorder olynomial maing technique roosed by Deen et al. 24 (see Aendix for the formulation). In rincial, the maing function evaluates a roerty, such as the liquid velocity at the arcel centroid, by the integration of the liquid velocity at the Eulerian grid nodes that is located inside a redefined influence diameter (set to 2d in this work). Forces exerted by a arcel on the Eulerian grid nodes are accounted for via the inter-hase force terms in the conservation equations F l, i.e., the back couling. The force terms consist of the drag F D, lift F L, and added mass forces F A as a function of the maing function and the number of bubbles in the arcel volume V cell, j, the forces exerted by a arcel i can be exressed as: F n F F F n. At a grid node j with j,i l, j D L A. (4) Vcell, j As mentioned earlier, a larger fraction of the eddies (and more of the energy residing in the flow) can be resolved using a higher grid resolution. Thus, a more accurate rediction of turbulent flow hydrodynamics requires finer grid sacing. In the framework of the EL aroach, it is, however, suggested that the grid sacing ratio to the bubble diameter h d should be greater than unity to maintain the validity of the oint-volume aroach. Therefore, the h d value emloyed here should comromise between a sufficiently fine grid resolution to cature the most energetic eddies and a sufficiently coarse grid resolution to kee the oint-volume assumtion valid. However, we relax this restriction by allowing the h d value to be lower than unity but greater than 0.5. This allows us to include large bubbles (with a diameter greater than the grid sacing) which are resent mostly at the 9

10 sarger and close to the imeller shaft. Note that the volume of arcels with large bubble is lower than 10% of the total bubble volume (and the number of large-bubble arcels is less than 1% of the total number of bubbles) having only a minor effect on the resolved flow field. The alicability of a h d value less than unity has been demonstrated by Darmana et al. 37 and is justified by favorable results obtained in this work. Collisions Collisions between bubbles can be analyzed using direct collision models, e.g., soft shere 38 and hard shere collision model, 39 or statistics-based collision models, e.g., the stochastic inter-article collision model. 26 In this stochastic model, no direct collisions, where a large amount of information from surrounding bubbles is required, are considered. Instead, only a fictitious collision artner (i.e., arcel) is assumed and a collision robability according to kinetic theory is established for each arcel at each time ste of the trajectory calculation. The bubble size and velocity of the fictitious arcel are randomly generated based on the statistic information (with Gaussian distribution) regarding bubble size distribution and velocity collected at Eulerian grid nodes. The fictitious arcel consists of bubbles with identical roerties and similar number as in the considering arcel. Hence, the bubbles in the fictitious arcel can be considered as a reresentative for the surrounding bubbles. The collision robability P coll is calculated based on the roerties of the arcel (and its collision artner) and the local fluid roerties as: P coll 4 2 d d u n t fict u, (5) where the subscrit fict reresents roerties of the bubbles in the fictitious arcel. The occurrence of a collision between a arcel and a fictitious arcel is determined by a comarison between the robability with a uniform random number in the interval [0,1]. The collision can result in momentum exchange (bouncing) or coalescence between the bubbles in the arcels. In case of a bouncing collision, the imact oint is statistically determined on a collision cylinder where the fictitious arcel is stationary. A detailed descrition of the stochastic inter-article collision rocedure can be found in fict 10

11 the work of Sommerfeld. 26 The rocedure to determine coalescence of bubbles will be described in the next section. Collisions between a arcel and the surfaces (tank, baffles and imeller) are considered to be elastic and frictionless. Since the motion of the moving comonents (e.g., imeller shaft, blades, and disc) is rotational, a collision with these comonents adds momentum to the bubble resulting in a change of the bubble tangential velocity: 19 u u 2r, (6),,out,,in where u and u,, in reresent the tangential velocity after and before the collision, resectively.,, out The local velocity at the imact oint r is the roduct of the angular velocity 2N (with N being the imeller rotational seed) and the distance from the center axis r. At imact the distance between a arcel and a solid comonent is calculated based on the effective radius r eff of a arcelbubble as r eff 1 3 3n V 4, (7) with n being the number of bubbles in the arcel and V being the bubble volume. Coalescence model Following a collision between a arcel and a fictitious arcel described in the revious section, coalescence between the bubbles in arcel and the bubbles in the fictitious arcel is determined using the aroach due to Prince & Blanch. 40 In this aroach, coalescence will take lace when the bubble contact time ij is greater than the film drainage time t ij. Otherwise, a rebound occurs. It is assumed that, in the frame of the considered Lagrangian collision model, the contact time can be exressed by: CCR ij ij, (8) u n with the equivalent bubble radius R ij 11

12 where R ij , (9) d d fict u n is the relative aroaching velocity in normal direction, C C is the deformation distance as a fraction of the effective bubble radius; its value of 0.25 gives the best agreement with the exerimental data 27 and is used throughout this work. Neglecting the effects due to surfactants and Hamaker forces, the film drainage time can be exressed as: 3 Rij h 0 t ij ln, (10) 16 hf with the initial film thickness h 0 for air-water set to 0.1 [mm], the final film thickness before ruture h f set to 0.01[μm], 40 and the surface tension. The roerties of the new bubble after coalescence are calculated from a mass and momentum balance. The new bubble diameter after coalescence is calculated as d,new 3 3 d d 1 3. (11),old Since the total volume of the arcel must be conserved, the number of bubbles in a arcel after coalescence is exressed as fict n d 3,old,new n,old. (12) d,new Based on the collision cylinder where the fictitious arcel is stationary, only the bubble normal velocity to the bubble in the fictitious arcel is changed, while the other velocity comonents remain unchanged. Thus, the normal velocity after coalescence 12 u,n, new can be exressed as 3 d,old u,n,new u,n,old. 3 3 (13) d,old dfict It is imortant to emhasize here that a coalescence of bubbles takes lace between a (real) arcel and a fictitious arcel, not between two real arcels. In the case of bouncing collisions (the contact time less than the drainage time), the normal velocity after collision is a function of the coefficient of restitution (set to 0.90 in this work) and can be described by:

13 3 3 d dfict u,n,new u,n,old. 3 3 d dfict (14) Breaku model A breaku model due to Luo & Svendsen 28 is emloyed in our work. The model was derived from the theories of isotroic turbulence and contains no adjustable arameters. The bubble interaction with turbulent eddies is assumed to be the dominant breaku mechanism. It is further assumed that only the eddies of length scale smaller than or equal to the bubble diameter articiate in the breaku mechanism. Larger eddies simly transort the bubble without causing breaku. The breaku rate of bubbles with volume V into volumes of V and 1 f BV V when being in contact with f BV turbulent eddies in the size range of where B 1 1 V : Vf BV n 2 g d min to d can be exressed as: 1 3 min ex 2 3 2c d f d, (15) g is the gas hase volume fraction, the energy dissiation at the centroid of the arcel, and min min d. The breakage volume fraction f BV is calculated from: RN 0.5 f BV tanh10, (16) where RN is an uniform random number with the interval [0,1]. The resulting volume fraction of the daughter bubbles has a U-shaed distribution, i.e., the breaku into equal size has the lowest robability, while the breaku into infinitesimal volumes has the highest robability. In this work, we limit the range of breakage volume fraction in the interval [0.2,0.8]. Accordingly, the increase coefficient of surface area c f is exressed as: 3 1 f c f 2. (17) f BV Bv 13

14 The minimum size of eddies in the inertial subrange of isotroic turbulence roortional to the length of the Kolmogorov-scale eddies with min. 4 ms ms : min is assumed to be 11, (18) ms, (19) where is the liquid hase kinematic viscosity. In this work, the breaku rate is considered to be a stochastic value determined from a randomly generated breakage volume fraction Eq. (16). Thus, the breaku of a arcel within a certain interval is decided in the similar manner as for the collision of bubbles, i.e., by comarison with a uniform random number in the interval [0,1]. Furthermore, it is assumed that the breaku only takes lace within the arcel and will not result in a new arcel. Instead, it will result in a arcel with a new bubble diameter calculated according to the breakage volume fraction. The number of bubbles in a arcel after breaku is calculated using the Eq. (12). Regardless of the success of the breaku, the arcel is assumed to interact with certain eddies in a certain interval characterized by the article-eddy interaction time t e. The next estimation of the breaku rate will be carried out after t e [s], see Aendix for the formulations. This assumtion is very imortant for a system where a high level of turbulence is resent, e.g., in a stirred reactor. It revents unhysical consecutive breakus of bubbles and rovides a time ste resolution-indeendent solution. Results and Discussion Bubble column with mono-diserse bubbles Disersed gas-liquid flows in three-dimensional (non-stirred) bubble columns have been studied exerimentally by several grous, including Deen et al. 41 and Van den Hengel et al. 42 Coalescence of bubbles was inhibited by adding salt solution resulting in a bubble column with (aroximately) uniform bubble size. As their data are used to validate our model, we neglected breaku and 14

15 coalescence of bubbles in this art of the study. Fluid flow was induced mainly by bubbles where the motion of bubbles relative to the liquid resulted in turbulent-like flow. The bubble lume fluctuated only weakly at the lower art of the column and was meandering around the column at the uer art. This fluctuation was caused by various mechanisms including, most imortantly, bubble-bubble collisions and turbulence. Modeling of these bubble columns was reorted by Sungkorn et al. 21 Their redicted mean and fluctuating liquid velocity comonents were in good agreement with the exerimental data of Deen et al. 41 cited above. Sensitivity to grid size over bubble diameter ratio h d was also studied by erforming simulations with a h d -value of 1.10, 1.25, and The study concluded that a finer grid (i.e., a lower h d value) rovides a better agreement with the exeriment, while the coarsest grid was not sufficient to correctly cature the features of the flow field. In the resent work, we study the sensitivity of the redictions when a h d -value lower than unity is used. It is worth to emhasize that in the EL aroach considered here, a oint-volume assumtion is used and that the bubbles surface effect on the continuous fluid flow is neglected. We relax the assumtion by further assuming that a h d -value less than unity (but greater than 0.5) can be used and will not drastically violate the oint-volume assumtion. This additional assumtion will be justified by the favorable results obtained here. The bubble column considered has a square cross-section with a width, deth and height of 0.15, 0.15 and 0.45 [m], resectively. Air bubbles are introduced at the bottom-center lane with an area of [m 2 ] with a suerficial gas velocity of 4.6 [mm/s]. A bubble mean diameter of the order of 4 [mm] was observed in the exeriments. 41 Bubbles were assumed to have uniform size in this work. In order to verify the validity of a model with a h d -value less than unity, a simulation with a h d value of 0.75 was carried out and comared with the simulation results reorted in Sungkorn et al. 21 for a h d value greater than unity. The domain was discretized by a uniform cubic grid of lattices in width, deth, and height, resectively. This resulted in a bubble size of 1.5 times the lattice distance. A no-sli 15

16 boundary condition was alied at the walls, excet for the to where a free-sli boundary condition was alied. Bubble arcels were injected at the bottom of the column and left the simulation domain once they touched the to surface. In this case, due to the low gas hase fraction, one arcel contained only one bubble. The calculation started with a quiescent liquid and roceeded for 150 [s] with a time ste for the liquid hase of 10 [μs]. A sub-time ste of 1 [μs] was used for the calculation of the bubble motion. In order to obtain statistically meaningful data, the averaged quantities were comuted from 20 to 150 [s]. FIGURE 1 A comarison between the simulations with various h d values and measured data is shown in Figure 1. As can be seen from the to figure, the averaged vertical velocity rofile is accurately reroduced in all cases excet for a h d value of Similarly, the fluctuating comonents of the resolved flow field were correctly catured by all simulations excet, again, for a h d value of This might be attributed to an insufficient resolution when a grid too coarse is emloyed. As can be seen, the use of a h d value of 0.75 imroves the redictions, esecially at the region near the wall. This is because, with a finer grid resolution, a larger fraction of the eddies and, consequently, more of the energy residing in the flow are resolved. These favorable results show that the use of a h d value lower than unity in this case will not deteriorate the simulation, at least in the rediction of the flow field. However, it is imortant to note that the refinement of the grid also results in an increase of comutational cost: reducing the (uniform) grid sacing by a factor q increases the grid size with a factor q 3 and (due to exlicit time steing) the comutational effort by q 4. Bubble column with inclusion of breakage and coalescence A seudo-two-dimensional bubble column has been used to study the effect of suerficial gas velocity on the bubble size distribution (BSD) by Van den Hengel et al. 42 The underlying henomena are similar to the bubble column discussed earlier with the additional comlexity due to bubble coalescence. The dimensions of the bubble column are 0.20, 0.03 and 1.40 [m], in width, deth, and height, resectively. Air bubbles were injected at the mid-bottom from a nozzle with a diameter of 16

17 0.02 [m] with a gas suerficial velocity of , , and [m/s]. It was reorted in their work that the bubbles at the nozzle had a size distribution around 3 [mm]. In our study, the fluid domain was discretized on a uniform cubic grid of lattices in width, deth, and height, resectively. The other simulation settings were similar to the revious simulation case. Gaussian-shae BSDs with a mean diameter of 2.5 and 3 [mm] (corresonding to a bubble size of 0.5 and 0.6 times the lattice distance, resectively) and a variance of 0.25 [mm] were generated at the nozzle. Breaku and coalescence of bubbles were taken into account. Due to the restriction of the h d value discussed earlier, coalescence will only take lace with a bubble smaller than 2.0 times the lattice sacing and, similarly, breaku will only take lace with a bubble larger than 0.1 times the lattice sacing. The long-term averaged results were based on results between 20 and 150 [s]. FIGURE 2 Figure 2 shows the measured and redicted long-term averaged BSD at various heights. It can be observed from the exerimental data that the mean bubble diameter and the BSD slightly shifted to the right side, i.e., to bigger diameters, caused by bubble coalescences. A similar behavior is obtained in the simulations. Although the agreement is quite good, there exist, however, some deviations in the BSD rofiles. Similar deviations were also obtained in the simulations of Van den Hengel et al. 42 One reason for this observation may be the resolution of the measurement and/or the lack of accurate BSD data at the nozzle (i.e., at the air inlet). The latter exlanation is also suorted by the fact that in the exeriment a bi-modal BSD is observed. FIGURE 3 Figure 3 shows the number mean diameter at various heights with different suerficial gas velocities. The sensitivity of the BSD with resect to the sarger (initial) BSD was studied by using initial BSDs with a mean diameter of 2.5 and 3 [mm]. At low suerficial gas velocities (i.e., and [m/s]), good agreements for the number mean diameter d 10 along the axial direction were obtained with the initial BSD of 2.5 [mm]. However, the simulation with the initial BSD of 3.0 [mm] rovides a good agreement with the exeriment at a higher suerficial gas velocity (i.e

18 3 [m/s]). This behavior may be exlained by the well-known fact that the initial bubble size at the sarger increases with increasing suerficial velocity. 10 Furthermore, also immediate coalescence at the sarger occurs. This, again, highlights the imortance of knowing the initial BSD at the sarger for an accurate rediction. Although the initial bubble size can be roughly estimated using, for examle, the model roosed by Geary & Rice, 43 detailed information concerning the sarger is essential, yet rarely available in literature. Gas-liquid stirred reactor Detailed exerimental investigations of a gas-liquid stirred reactor were reorted by Montante et al. 11,12 and were used for validation of our method. In their work image analysis was emloyed to collect data concerning liquid flow field and the BSD at the mid-lane between the baffles. The reactor had a standard configuration consisting of a cylindrical, flat-bottomed, baffled tank with diameter T = 23.6 [cm] and the liquid filled-level H T. The reactor was equied with a Rushton turbine with a diameter D T 3, at the center of the reactor C T 2. The geometry of the reactor is deicted in Figure 4. The working fluid was water with a viscosity of [m 2 /s] and density of [kg/m 3 ]. The imeller rotational seed N was fixed at 450 [rm] throughout the study, corresonding to a blade ti seed of 1.85 [m/s] and a Reynolds number, defined as 2 Re ND, of aroximately 46,000. Air bubbles were injected into the system via a sarger made of a tube of 3.3 [mm] diameter with a orous membrane on to. The sarger had a distance of T 4 from the bottom. The gas flow rate was varied from 0.02, 0.05, to 0.07 [vvm]. The reactor oerated in the comlete disersion regime. 11 FIGURE 4 For the simulations a cubic comutational grid of 85 3 lattice cells was defined. A no-sli boundary condition was emloyed at all faces excet for the to surface, where a free-sli boundary condition was defined to reresent the free surface. Sets of control oints, reresenting the cylindrical wall, the baffles, the imeller, the imeller shaft, and the sarger tube, were generated inside the comutational domain according to the forcing algorithm introduced earlier. In the simulations, a grid 18

19 sacing h equal to [m] was emloyed. The diameter of the imeller had a size of 27 times the grid sacing. The distance between two control oints at the imeller surface was 0.7h. Similar strategy was used to reresent the other arts. The total number of control oints in the domain was 33,000. The simulations started with the reactor at rest and roceeded with a time ste for the liquid hase of 20 [μs] and a sub-time ste of 4 [μs] for the calculation of the bubble motion. Thus, the imeller comletes a full revolution in 6667 time stes. At any given moment, for examle, in case 5, the simulation has aroximately 1,300 arcels, corresonds to 90,000 bubbles and global gas holdu of 0.4 % (also Montante et al. 12 reort gas holdus below 1%). After 30 imeller revolutions, the data of the liquid flow field and the BSD were collected for the following 60 revolutions and statistically analyzed. The wall-clock time for one imeller revolution was aroximately 1.5 hours when 4 Intel Xeon E5540 (at 2.53 GHz) rocessors were used. This averaging eriod was shown to be sufficiently long for generating statistically meaningful results. An overview of the simulation cases is shown in Table 1. Note that the initial bubble diameter used in the simulations was greater than the grid sacing, i.e., a h d value less than unity. For examle, an initial diameter of 4 [mm] results in a h d value of As will be shown later, the significant bubble breaku occurs in the imeller region, resulting in much smaller bubbles. Hence, only 1% of the bubbles or less in the reactor have a diameter greater than the grid sacing. TABLE 1 First, the influence of the initial bubble diameter on the simulation results was studied. Two simulations were carried out: one with an initial bubble diameter of 4 [mm] (case 1) and another one with a Gaussian BSD distribution with a mean diameter of 3.5 [mm] and variance of 0.5 [mm] (case 2). The gas flow rate was set to 0.02 [vvm] in both cases. Figure 5 shows the measured and redicted long-term averaged local number-mean diameter d 10 in the reactor. Note that the redicted d 10 shown here is a angle-averaged value. The trend and magnitude of the redicted d 10 in both cases agrees fairly well with the measured data. In case 1, the simulations slightly over-redicted d 10 in the imeller regime and slightly under-redicted the diameter in the rest of the reactor. In contrast, the 19

20 redicted d 10 in case 2 agrees well with the measured data in most arts, excet in the imeller regime where the d 10 is over-redicted. This, again, highlights the imortance of initial bubble size accuracy of the simulation. In order to reduce the arameters that will influence the rediction and for the simlicity of the study, a uniform initial bubble size of 4 [mm] was used for the rest of the study. FIGURE 5 FIGURE 6 A comarison between the redicted (case 1) and the measured cumulative size distribution (CSD) of bubble at the lower and uer art is shown in Figure 6. The redicted CSD was calculated by dividing the diameter between 0.0 and 4.5 [mm] into 15 classes and counting the frequency of bubbles in each class. As can be seen, the redicted CSD distribution agrees very well with the measured data. Deviation can be observed in the regime of small bubbles. This could be due to many reasons, for examle, in the break u model, where the daughter size distribution is governed by the U-shae distribution Eq. (16) in which the robability of very small and very large daughter sizes is the highest. Thus, other daughter size distributions, such as an M-shae distribution, 44 may be used to imrove the rediction. It is also imortant to note that the smallest detectable bubble in the exeriment of Montante et al. 12 was 0.3 [mm] whereas, in the simulation, the smallest allowable bubble was aroximately 0.25 [mm] and was included in the first cumulative class. Based on the good agreement between exeriment and simulation for the disersed hase, we further examined the quality of the rediction of the liquid flow field. Predicted and exerimental axial and radial long-term averaged liquid velocity comonents along the radial direction at various heights are shown in Figure 7. As can be seen, a quite good agreement between the redicted and measured data is achieved at all considered heights. The radial velocity rofiles changes only slightly along the radial direction. A significant change in magnitude can be seen at the height of the imeller where a strong outflow exists (z/t = 0.49) as the configuration corresonds to a radially discharging imeller. In contrast, the axial velocity rofiles change in magnitude and sign along the radial direction at all elevations. Also, a very good agreement is achieved for the long-term averaged velocity comonents along the axial direction at various radial ositions, see Figure 8. A jet-like radial outflow has the 20

21 highest magnitude close to the imeller and decreases with further distance from the imeller. The axial velocity comonent changes its magnitude along the axial direction and radial osition according to flow recirculation generated by the Rushton turbine. Slight deviations can be noticed for the radial velocity comonent at the eak of the imeller jet. FIGURE 7 FIGURE 8 In Figure 9, the long-term averaged root mean square (RMS) radial and axial velocity comonents at various elevations are shown. The simulation and the exeriment have the maximum magnitude of the RMS values at the height of the imeller (z/t = 0.49). While the agreement between exeriments and simulations is not as good as for the averaged velocities, the redicted RMS values have similar order of magnitude as in the exeriment. However, deviations can be observed, esecially near the imeller. Clearly, these deviations were due to an insufficient grid resolution used in this work. However, it is worth to reeat here that the grid resolution is restricted by the size of bubbles resent in the system. That is, the h d value has to be a comromise between a sufficiently fine grid resolution to cature the most energetic eddies and a sufficiently coarse grid resolution to kee the oint-volume assumtion valid. FIGURE 9 The redicted instantaneous and long-term averaged velocity vector fields are shown in Figure 10. Several small and large vortices induced by a jet-like outflow from the Rushton imeller can be observed in the lot of the instantaneous flow field. These vortices interact with each other and change their size, shae and osition randomly with time. The long-term averaged velocity vector field reveals two rimary liquid recirculation zones at the uer and lower art of the reactor. Two small recirculations at the uer and lower corners can be also noticed. Snashots of the redicted bubble disersion attern from the front and to view are shown in Figure 11 and Figure 12, resectively. As can be seen, the bubbles introduced at the sarger fluctuate within a small range before they are drawn into the imeller regime. Consequently, the bubbles collide with the imeller, exchange their momentum and velocity direction and are drawn into the vortex behind the imeller blades. Most of 21

22 bubble breaku takes lace in this region due to the resence of high turbulence activity. The bubbles are then disersed following the jet-like outflow. Small bubbles tend to follow the recirculations resulting in a long residence time, while large bubbles tend to rise near the wall or move into a region with low turbulence activity, i.e., near the imeller shaft. These underlying henomena and much more, such as bubble trailing and rolling of bubble swarm, can be observed in three-dimensional and anaglyhical animated results made available in the journal s website. FIGURE 10 FIGURE 11 FIGURE 12 A contour lot of the long-term averaged resolved liquid hase turbulent kinetic energy (TKE) at the mid-lane between baffles is shown in Figure 13. A region of high turbulent activity, i.e., high TKE, is found to emanate from the blades region and has a maximum of TKE at the midway between the imeller and the tank wall. The TKE rofile is qualitatively similar to that for single hase flows - see for examle Derksen & Van den Akker. 3 It can be concluded that the TKE is only weakly modified by the resence of bubbles due to the relatively low gas flow rate investigated here. FIGURE 13 A simulation of a single-hase stirred tank (case 3) was carried out to study the effect of bubbles on the liquid flow field. The redicted single-hase long-term averaged velocity vector field is shown on the left side of Figure 14. For the investigated range of the gas flow rate, no significant changes in liquid flow field are observed. More details can, however, be deduced by subtracting the averaged vector field obtained from the simulation with gassing (case 1) from the field without gassing (case 3). The result is shown on the right side of Figure 14. As can be seen, most of the differences take lace in the outflow of the imeller region, where the liquid motion is modified by the bubbles. Our observation agrees qualitatively well with the result obtained from the exeriment of Montante et al. 11 It should be noted however, that the magnitude of the difference is aroximately one order of magnitude lower than the averaged flow field. FIGURE 14 22

23 In another set of simulations the gas flow rate was increased from 0.02 to 0.05 and 0.07 [vvm] in case 4 and 5, resectively, to investigate the effect of gassing on the gas-liquid hydrodynamics. The initial bubble diameter was assumed to 4 [mm] in all cases. Figure 15 shows the redicted local number mean diameter obtained from the simulations with various gas flow rates. At all gas flow rates, the evolution of the bubble size follows a similar trend as discussed in the revious section. However, it can be noticed that the increase of the gas flow rate results in a smaller mean bubble diameter in most arts of the reactor. Significant changes occured when the gas flow rate was increased from 0.02 to 0.05 [vvm]. The difference between the flow rate of 0.05 and 0.07 [vvm] was minor. An exlanation can be rovided by the hel of the contour lot of the hase-averaged gas volume fraction shown in Figure 16. (It should be noted that the lot of the gas fraction was made by maing the bubble s volume based on its osition on the Lagrangian frame of reference to its nearest Eulerian grid node). A significant increase in the concentration and the disersion area of the gas hase can be observed by raising the flow rate from 0.02 to 0.05 [vvm]. In the first case, the disersion attern, i.e., the gas volume fraction contour, is divided into three regimes: imeller outflow, uer, and lower recirculation zones. This is because the bubbles are only disersed from the imeller but the recirculations are not strong enough to draw the bubbles back in the imeller regime. In contrast, the disersion attern for 0.05 [vvm] are comletely connected. Thus, the bubbles are recirculating into the imeller outflow region. Consequently, more breaku takes lace, resulting in smaller bubble diameters. An increase of the flow rate from 0.05 to 0.07 [vvm] does not qualitatively change the icture and results mainly in an increase of the concentration of the gas hase, esecially in the regime near the imeller. The mean diameter of bubble changes only significantly in the regimes near the imeller shaft, where a higher rate of bubble coalescence occurs. Note, that a different trend of the the mean diameter will be obtained in exeriments. This is because, as discussed in the revious section, the BSD at the sarger increases significantly with the gas flow rate and, consequently, the BSD in the reactor. FIGURE 15 FIGURE 16 23

24 Conclusions A modeling technique for the simulations of gas-liquid stirred tank reactors based to an EL aroach has been resented. The turbulent flow field was established using the filtered conservation equations. A variation of the LB scheme roosed by Somers 17 was used to discretize the equations. The bubble arcel concet was used to reresent a grou of bubbles with identical roerties. A oint-volume concet was used to track the trajectory of the bubble, i.e., the arcel. A set of aroriate correlations for the inter-hase closure was carefully chosen from the literature. The immersed boundary condition method 22 was emloyed to describe the action of the moving comonents and the tank walls. The restriction regarding the grid size over bubble diameter ratio, h d, was relaxed. It was demonstrated that the use of a h d value less than unity (but greater than 0.5) can be emloyed in EL simulations. This aroach can be envisioned as a distributed bubble aroach where bubbles are allowed to be slightly bigger than the grid size and reresented with a more satially distributed forces. However, the h d value emloyed in a simulation should be a comromise between a sufficiently fine grid resolution to cature the most energetic eddies and a sufficiently coarse grid to kee the oint-volume assumtion valid. It should be noted that, this resolution limitation will largely disaear when a simulation of an industrial-scale reactor is considered. Since an increase of the reactor size will tyically result in only slightly larger bubbles, finer grids (relative to the reactor size) will have h d ratios greater than one. Collisions and coalescence of bubbles was modeled based on the stochastic inter-article collision model as used by Sommerfeld et al. 27 Using the coalescence model, a good agreement between the redicted and measured BSD in a bubble column has been obtained. Simulations with various initial bubble sizes and gas flow rates showed that the accuracy of the rediction is highly sensitive to the initial bubble size at the sarger. Therefore, detailed information concerning bubble size (or air inlet/sarger) is essential for an accurate rediction. The breaku model of Luo & Svendsen 28 was emloyed. In this work, breaku of bubble was treated as a stochastic event. A daughter size was randomly selected from a U-shae distribution. 24

25 Accordingly, breaku of bubbles was decided by comaring a breaku frequency with a uniform random number. Additionally, in this work we roosed that this event should be bounded by an involved time scale which is a function of the flow field, i.e., the article-eddy interaction time, to obtain time resolution-indeendent solution and to avoid unhysical consecutive breakus. Our modeling technique was then used to simulate gas-liquid flow in a stirred tank reactor following the exeriments of Montante et al. 11,12 The simulations were able to reroduce the trends and the magnitude of the local BSD from the exeriment. Small bubbles were over-redicted. This could be imroved by emloying alternative daughter bubble-size distributions. Also, desite a coarse grid sacing used in this work, the simulations rovided a good agreement with measured data for the longterm averaged velocity comonents. This is because the scale of large energy-containing eddies in a stirred reactor is of similar size as the imeller diameter. Therefore, the largest art of the energy in the liquid flow field was resolved. However, due to the restriction of the choice of the grid sacing, the redicted second-order statistics (RMS of velocity comonents) agree only by order of magnitude with the measured data. In order to imrove the rediction of the second-order statistics, an alternative aroach, such as very-large-eddy simulations (VLES), may be emloyed. The VLES aroach uses a satial-filtering rocess with a turbulence model (similar to that for unsteady Reynolds-averaged Navier-Stokes (URANS) aroach) on a coarse grid to resolve a minor art of the turbulence sectrum and to model the rest. 45,46 It should be also ointed out that, in a gas-liquid reactor with dense disersion regime (i.e., a global gas hase above 10% volume fraction), the gas void fraction has a significant effect on the flow field and should be included in the conservation equations. Additionally, in the dense disersion regime, large amount of bubbles tend to accumulate and coalesce in the lowressure region behind the imeller blades resulting in ventilated cavity. 47 In order to accurately model a reactor within this regime, the hase-averaged conservation equations and a hybrid aroach for the modeling of the disersed hase, e.g., the hybrid two-fluid/discrete element method introduced by Sun et al., 48 may be emloyed. We are working on these imrovements in our current research. Bearing in mind the caabilities and limitations of the resented modeling technique, the effects of the bubble hase and the gas flow on the gas-liquid were studied numerically. It can be concluded that the resence of bubbles, in the investigated range of oerating conditions, slightly 25

26 modifies the flow at the imeller outflow region. Furthermore, it has been found that the increase of the gas flow rates triggers a change in the disersion attern and, consequently, the BSD. While the study resented here has been carried out for laboratory scale reactors with dilute disersions, the resent modeling technique consists mainly of models based on elementary hysical rinciles which are also valid for a larger scale. The models contain not many adjustable arameters; only some rooms exists for adoting different closures and for using different theoretical constants. It should be also stressed, that all elements of the resent modeling technique rovide high efficiency for arallel comuting, as has been demonstrated by Derksen & Van den Akker 3 and Sungkorn et al. 21 The maximum benefit of the resent modeling technique can be achieved when industrial large-scale simulations are realized. 26

27 Notation C = clearance between the imeller disc and the reactor bottom, m c f = increase coefficient of surface area calculated in Eq. (17) C S = Smagorinsky constant (set to 0.10) d = diameter, m D = imeller diameter, m F = force, N f BV = breakage volume fraction calculated in Eq. (16) g = gravitational acceleration (set to 9.82), m/s 2 h = grid cell size, m H = liquid-filled level, m h 0 = initial film thickness to be used in Eq. (10), m h f = final film thickness before ruture to be used in Eq. (10), m N = imeller rotational seed, rev/s r,r = radius, m S = rate of deformation tensor to be used in Eq. (1), 1/s T = reactor diameter, m t,τ = characteristic time, s u = velocity, m/s V = volume, m 3 x = osition α = coefficient of restitution (set to 0.90) Δ = filter width to be used in Eq. (1), m ε = energy dissiation rate, m 2 /s 3 λ = eddies length scale, m ν = kinematic viscosity, m 2 /s ρ = density, kg/m 3 ς = maing function calculated from Eq. (A1) 27

28 σ = surface tension, N/m Ω = rotational seed, rad/s Ω B = breaku rate calculated in Eq. (15), 1/s Subscrits eff = effective fict = fictitious l = continuous liquid hase n = normal direction new = roerties after collision, coalescence, or breaku old = roerties before collision, coalescence, or breaku = disersed bubble hase t = turbulence involved roerties θ = tangential direction Dimensionless number 2 Eo = Eötvös number = g d Re = article Reynolds number = u ud l l 28

29 References 1. Arlov D, Revstedt J, Fuchs L. Numerical Simulation of Gas-Liquid Rushton Stirred Reactor LES and LPT. Comuters & Fluids. 2008;37, Gogate, PR, Beenackers AACM, Pandit AB. Multile-Imeller Systems with a Secial Emhasis on Bioreactor: A Critical Review. Biochem Eng J. 2000;6, Derksen JJ, Van den Akker HEA. Large Eddy Simulations on the Flow Driven by a Rushton Turbine. AIChE J. 1999;45, Crowe C, Sommerfeld M, Tsuji Y. Multihase Flows with Drolets and Particles. Boca Raton: CRC Press; Loth E. Numerical Aroaches for Motion of Disersed Particles, Drolets and Bubbles. Prog Ener Combust. 2000;26, Deen NG, Solberg T, Hjertager BH. Flow Generated by an Aerated Rushton Imeller: Two-Phase PIV Exeriments and Numerical Simulations. Can J Chem Eng. 2002;80, Khokar AR, Rammohan AR, Ranade VV, Dudukovic MP. Gas-Liquid Flow Generated by a Rushton Turbine in Stirred Vessel: CARPT/CT Measurements and CFD Simulations. Chem Eng Sci. 2005;60, Zhang Y, Yang C, Mao Z-S. Large Eddy Simulation of the Gas-Liquid Flow in a Stirred Tank. AIChE J. 2008;54, Venneker BCH, Derksen JJ, Van den Akker HEA. Poulation Balance Modeling of Aerated Stirred Vessels Based on CFD. AIChE J. 2002;48, Laakkonen M, Moilanen P, Aloaeus V, Aittamaa J. Modelling Local Bubble Size Distribution in Agitated Vessels. Chem Eng Sci. 2007;62, Montante G, Paglianti A, Magelli F. Exerimental Analysis and Comutational Modelling of Gas- Liquid Stirred Vessels. Tran IChemE. 2007;85, Montante G, Horn D, Paglianti A. Gas-Liquid Flow and Bubble Size Distribution in Stirred Tanks. Chem Eng Sci. 2008;63, Unverdi SO, Trygvvason G. A Front-Tracking Method for Viscous, Incomressible Multi-Fluid Flow. J Comut Phys. 1992;100,

30 14. Behzadi A, Issa RI, Rusche H. Modelling of Disersed Bubble and Drolet Flow at High Phase Fractions. Chem Eng Sci. 2004;59, Wu Z, Revstedt J, Fuchs L. LES of a Two-Phase Turbulent Stirred Reactor. In: Lindborg E. Turbulence and Shear Flow Phenomena 2. Stockholm: KTH; 2001: Nemoda SDJ, Zivkovic GS. Modelling of Bubble Break-U in Stirred Tanks. Thermal Science. 2004;8, Somers JA. Direct Simulation of Fluid Flow with Cellular Automata and the Lattice-Boltzmann Equation. J Al Sci Res. 1993;51, Smagorinsky J. General Circulation Exeriments with the Primitive Equations: 1. The Basic Exeriment. Mon Weather Rev. 1963;91, Derksen JJ. Numerical Simulation of Solid Susension in a Stirred Tank. AIChE J. 2003;49, Derksen JJ, Van den Akker HEA, Sundaresan S. Two-Way Couled Large-Eddy Simulations of the Gas-Solid Flow in Cyclone Searators. AIChE J. 2008;54, Sungkorn R, Derksen JJ, Khinast JG. Modeling of Turbulent Gas-Liquid Bubble Flows Using Stochastic Lagrangian Model and Lattice-Boltzmann Scheme. Chem Eng Sci (acceted). 22. Derksen JJ, Kooman JL, Van den Akker HEA. Parallel Fluid Flow Simulation by Means of a Lattice-Boltzmann Scheme. Lect Notes Comut Sci. 1997;125, Sommerfeld M. Theoretical and exerimental modeling of articulate flows. Lecture Series von Karman Institute for Fluid Dynamics. 3 rd -7 th Aril Deen N, Van Sint Annaland M, Kuiers JAM. Multi-Scale Modeling of Disersed Gas-Liquid Two-Phase Flow. Chem Eng Sci. 2004;59, Sommerfeld M. Kohnen G, Rueger M. Some Oen Questions and Inconsistencies of Lagrangian Particle Disersion Models. In: Proceedings of the Ninth Symosium on Turbulent Shear Flows. Kyoto Jaan, 1993; Paer No Sommerfeld M. Validation of a Stochastic Lagrangian Modeling Aroach for Inter-Particle Collisions in Homogeneous Isotroic Turbulence. Int J Multihas Flow. 2001;27,

31 27. Sommerfeld M, Bourloutski E, Broeder D. Euler/Lagrange Calculations of Bubbly Flows with Consideration of Bubble Coalescence. Can J Chem Eng. 2003;81, Luo H, Svendsen HF. Theoretical Model for Dro and Bubble Breaku in Turbulent Disersions. AIChE J. 1996;42, Succi S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford: Clarondon Press; Chen S, Doolen GD. Lattice Boltzmann Method for Fluid Flows. Annu Rev Fluid Mech. 1998;30, Eggels JGM, Somers JA. Numerical Simulations of Free Convective Flow Using the Lattice- Boltzmann Scheme. Int J Heat Fluid Flow. 1995;16, Hu G, Celik I. Eulerian-Lagrangian based Large-Eddy Simulation of a Partially Aerated Flat Bubble Column. Chem Eng Sci. 2008;63, Derksen JJ. Assessment of Large Eddy Simulations for Agitated Flows. Trans IChemE. 2001;79A, Goldstein D, Handler R, Sirovich L. Modeling a No-Sli Flow Boundary with an External Force Field. J Comut Phys. 1993;105, Van Driest ER. On Turbulent Flow Near a Wall. J Aeronautical Science. 1956;23, Salding DB. A Single Formula for the Law of the Wall. J Al Mech. 1961;28, Darmana D, Deen NG, Kuiers JAM. Numerical Study of Homogeneous Bubbly Flow: Influence of the Inlet Conditions to the Hydrodynamic Behavior. Int J Multihas Flow. 2009;35, Tsuji Y, Tanaka T, Ishida T. Lagrangian Numerical Simulation of Plug Flow of Cohesionless Particles in a Horizontal Pie. Powder Technol. 1992;71, Hoomans BPB, Kuiers JAM, Briels WJ, Van Swaaij WPM. Discrete Particle Simulation of Bubble and Slug Formation in a Two-Dimensional Gas-Fluidised Bed: A Hard-Shere Aroach. Chem Eng Sci. 1996;51, Prince MJ, Blanch HW. Bubble Coalescence and Break-U in Air-Sarged Bubble Columns. AIChE J. 1990;36,

32 41. Deen NG, Solberg T, Hjertager BH. Large Eddy Simulation of the Gas-Liquid Flow in a Square Cross-Sectioned Bubble Column. Chem Eng Sci. 2001;56, Van den Hengel EIV, Deen NG, Kuiers JAM. Alication of Coalescence and Breaku Models in a Discrete Bubble Model for Bubble Columns. Ind Eng Chem Res. 2005;44, Geary NW, Rice RG. Bubble Size Prediction for Rigid and Flexible Sargers. AIChE J. 1991;37, Lehr F, Millies M, Mewes D. Bubble-Size Distribution and Flow Fields in Bubble Columns. AIChE J. 2002;48, Rurecht A, Helmrich T, Buntic I. Very large eddy simulation for the rediction of unsteady vortex motion. In: Conference on Modelling Fluid Flow. Budaest Hungary, Fares E. Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann aroach. Comut Fluids. 2006, 35, Middleton JC, Smith JM. Gas-liquid mixing in turbulent systems. In: Paul EL, Atiemo-Obeng VA, Kresta SM. Handbook of Industrial Mixing: Science and Practice. New Jersey: Wiley-Interscience, 2004: Sun J, Battaglia F, Subramaniam S. Hybrid two-fluid DEM simulation of gas-solid fluidized beds. Trans ASME. 2007;129: Hennick EA, Lightstone MF. A Comarison of Stochastic Searated Flow Models for Particle Disersion in Turbulent Flows. Energ Fuel. 2000;14,

33 Radial fluctuating velocity [m/s] Axial fluctuating velocity [m/s] Axial liquid velocity [m/s] Figure 1. Comarison of the redicted and exerimental long-term averaged liquid velocity and fluctuating velocity comonents with various grid size to bubble diameter ratios 0.28 [m] and a deth of [m]. h d at a height of x [m] 0.25 exeriment h/d_ = 1.25 h/d_ = 1.50 h/d_ = 1.10 h/d_ = 0.75 exeriment h/d_ = 1.25 h/d_ = 1.50 h/d_ = 1.10 h/d_ = x [m] exeriment h/d_ = 1.25 h/d_ = 1.50 h/d_ = 1.10 h/d_ = x [m] 33

34 Frequency [-] Figure 2. Comarison of the exerimental (to) and the redicted (bottom) bubble size distribution at various heights with the suerficial gas velocity of [m/s]. The simulation was erformed with a BSD with a mean diameter of 2.5 [mm] at the nozzle [m] 0.25 [m] 0.50 [m] 0.75 [m] V s = [m/s] d [mm] 34

35 Number mean diameter [mm] Number mean diameter [mm] Number mean diameter [mm] Figure 3. Comarison of redicted and exerimental number-mean diameters at various heights using different initial bubble diameters and gas suerficial velocities. (All diagrams have the same y-axis scale) Simulation, 2.5 [mm] Simulation, 3.0 [mm] Exeriment 3.0 V s = [m/s] Simulation, [mm] Simulation, 3.0 h [m] [mm] Exeriment 3.0 V s = [m/s] Simulation, 2.5 h [m] [mm] Simulation, 3.0 [mm] 4.0 Exeriment 3.0 V s = [m/s] h [m] 35

36 Figure 4. Geometry of the stirred reactor with a Rushton turbine and a tube sarger. H=T D=T/3 C=T/2 S=T/4 T 36

37 Figure 5. Local number mean diameter [mm] in the reactor with N = 450 [rm] and Q = 0.02 [vvm]. Measured data are underlined following with the redicted values from the case 1 and 2, resectively / /

38 Cumulative number [-] Cumulative number [-] Figure 6. Cumulative distribution of bubble equivalent diameter in the uer half and lower half of the reactor with N = 450 [rm] and Q = 0.02 [vvm]. The rediction from the simulation case 1 is shown Exeriment Simulation Uer half Bubble diameter [mm] 0.5 Exeriment Simulation 0.0 Lower half Bubble diameter [mm] 38

39 V mean / V ti [-] V mean / V ti [-] V mean / V ti [-] Figure 7. Exerimental and redicted (case 1) long-term averaged axial and radial liquid velocity comonents at three heights Axial (ex.) Axial (sim.) Radial (ex.) Radial (sim.) z/t = z/t = 0.49 r/t [-] Axial (ex.) Radial (ex.) r/t [-] 0.15 Axial (sim.) Radial (sim.) z/t = Axial (ex.) Axial (sim.) Radial (ex.) Radial (sim.) r/t [-] 39

40 V mean / V ti [-] V mean / V ti [-] Figure 8. Exerimental and redicted (case 1) long-term averaged axial and radial liquid velocity comonents at two radial distances Axial (ex.) Radial (ex.) Axial (sim.) Radial (sim.) r/t = z/t [-] 0.2 r/t = Axial (ex.) Axial (sim.) Radial (ex.) Radial (sim.) z/t [-] 40

41 V RMS / V ti [-] V RMS / V ti [-] V RMS / V ti [-] Figure 9. Exerimental and redicted (case 1) long-term averaged axial and radial fluctuating comonents at three heights z/t = 0.30 Axial (ex.) Radial (ex.) 0.10 Axial (sim.) Radial (sim.) r/t [-] z/t = r/t [-] 0.10 Axial (ex.) Radial (ex.) Axial (sim.) z/t = 0.71 Axial (ex.) Radial (sim.) Radial (ex.) Axial (sim.) Radial (sim.) r/t [-] 41

42 Figure 10. Predicted instantaneous (left) and long-term averaged (right) liquid velocity vector field at the mid-lane between baffles obtained from case 1. V / V ti [-] 42

43 Figure 11. Snashot of bubble disersion attern. The reactor oerates at N = 450 [rm] and Q = 0.02 [vvm]. Note that the imeller and the sarger geometry are only an interolated contour lot. The baffles and tank wall are excluded for visualization urose. Bubble diameter [mm]

44 Figure 12. Snashot of bubble disersion attern and the liquid velocity vector field at cross section below the imeller. The reactor oerates at N = 450 [rm] and Q = 0.02 [vvm]. (Only bubbles below the imeller are shown) V / V ti [-] Bubble diameter [mm]

45 Figure 13. Contour lot of the redicted long-term averaged turbulent kinetic energy TKE for case 1. TKE / V ti 2 [-] 45

46 Figure 14. Vector lot of long-term averaged velocity field for ungassed conditions (case 3) (left) and the difference between the liquid velocity for gassed (case 1) and ungassed conditions (case 3) (right). V / V ti [-] V / V ti [-] 46

47 Figure 15. Local number-mean diameter [mm] in the reactor at the gas flow rate of (from to to bottom) 0.02, 0.05, and 0.07 [vvm], resectively. The imeller rotational seed was fixed at 450 [rm] / /

48 Figure 16. Long-term averaged gas volume fraction at the mid lane between baffles at gas flow rate of 0.02, 0.05, and 0.07 [vvm], i.e. case 1, 4, and 5, resectively. Gas fraction [-] 0.02 [vvm] 0.05 [vvm] 0.07 [vvm] 48