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CFD MODELLING OF MIXING AND HEAT TRANSFERe


www.icheme.org/journals doi: 10.1205/cherd.04080

0263–8762/05/$30.00+0.00 # 2005 Institution of Chemical Engineers Trans IChemE, Part A, January 2005 Chemical Engineering Research and Design, 83(A1): 30–39

CFD MODELLING OF MIXING AND HEAT TRANSFER IN BATCH COOLING CRYSTALLIZERS Aiding the Development of a Hybrid Predictive Compartmental Model
E. KOUGOULOS1? , A. G. JONES1 and M. WOOD-KACZMAR2
2 1 Department of Chemical Engineering, University College London, London, UK Particle Sciences Group, Strategic Technologies, GlaxoSmithKline Medicines R&D Centre, Stevenage, Hertfordshire, UK

A

novel predictive compartmental modelling framework for the dynamic simulation and scale-up of batch cooling suspension crystallization vessels within gPROMS process modelling software based on CFD simulations is presented. The gPROMS compartmental model combines hydrodynamic information obtained from CFD simulations with the full population, mass, concentration and energy balances including the crystallization kinetics within each compartment. The design of the compartmental structure is based on high-resolution simulations of the internal ?ow, particle-solvent phase mixing and heat transfer. CFD simulations of the turbulent ?ow ?eld were carried out using the standard k– 1 turbulence model and particle –solvent phase simulations based on the advanced multi?uid phase model (MFM). A detailed compartmental model is constructed for batch crystallizers equipped with two different impeller con?gurations based on the overall ?ow pattern, local energy dissipation, solids concentration and temperature distribution from CFD simulations. Keywords: mixing; CFD; heat transfer; crystallization; compartmental modelling.

INTRODUCTION Computational ?uid dynamics (CFD) is becoming an increasingly useful and powerful tool for the numerical analysis of systems involving transport processes. This includes the simulation of multiphase ?ow, heat transfer, chemical reactions and particulate processes (Versteeg and Malalasekera, 1995). Recently CFD simulations have been used for the development of compartmental modelling frameworks for the predictive scale-up of crystallization processes. CFD simulations have been used successfully to combine hydrodynamic information to include local energy dissipation rates and ?owrates with crystallization kinetic models using a compartmental modelling approach to predict the crystallization behaviour upon scale-up. Compartmentalization using CFD has been successfully applied to continuous steady-state evaporative crystallizers (Bermingham et al., 1998; Kramer et al., 1999, 2000; ten Cate et al., 2000), reactive semi-continuous and continuous precipitation processes (Zauner and Jones, 2002) and gas –liquid precipitation processes (Rigopoulos and
? Correspondence to: Professor A. G. Jones, Department of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, UK. E-mail: a.jones@ucl.ac.uk

Jones, 2003). The modelling, design and optimization of crystallization processes using CFD requires the implementation of the population balance and crystallization kinetic models within the CFD environment. Crystal population balances have been successfully integrated into CFD codes for the simulation of reactive precipitation processes (Wei and Garside, 1997; Al-Rashed and Jones, 1999; Rousseaux et al., 2001; Jaworski and Nienow, 2003). Liiri et al. (2002) successfully modelled secondary nucleation effects in vessels using CFD simulations. A serious problem involved in directly integrating the crystal population balance within the CFD environment is that simulations are computationally expensive. The technique is also limited by the accuracy of the approximation of the population balance due to the method of moments being used and cannot be used to account for attrition. For predictive modelling of crystallization processes an order of magnitude of days is necessary. Compartmentalization can be introduced to overcome this severe limitation. The effects of particle concentration and size on the degree of mixing are often neglected. Imperfect mixing in crystallizers is common and this lack of homogeneity affects the process performance, product quality and crystal size distribution. Maggiorios et al. (1998) successfully used CFD to model particle size in suspension polymer reactors. Micale et al. (2000a,b) and Montante et al. (2001) have 30

CFD MODELLING OF MIXING AND HEAT TRANSFER recently developed the multi?uid model (MFM) to model dilute particle suspensions with monodisperse particle sizes in agitated vessels. Size-dependent classi?cation effects within crystallizers are also not considered. Sha et al. (2001) have used CFD simulations to model sizedependent classi?cation in suspension crystallizations but are also computationally expensive. Knowledge of internal classi?cation effects and solid phase distribution within crystallizers is necessary for process design and scale-up. Most predictive batch cooling crystallization processes have not considered the effects of mixing, hydrodynamics and heat transfer on the process performance upon scaleup. The majority of CFD simulations used for the development of existing compartmental models described consider a ‘single’ phase only. They also assume a well-mixed crystallizer and that no particle segregation occurs, which can potentially lead to poor predictions. In this contribution, a novel compartmental modelling framework is presented for the predictive scale-up of seeded batch cooling suspension crystallization processes for an organic ?ne chemical using a hybrid gPROMS-CFD technique (Bezzo et al., 2004). CFD simulations were carried out to simulate particle –solvent phase mixing, local energy dissipation rates and heat transfer to aid the development of the compartmental modelling framework. The presence of particles is considered in the development of the compartmental modelling framework. The predictive gPROMS compartmental model and coupling with CFD hydrodynamic information is also described. COMPARTMENTAL MODELLING FRAMEWORK DEVELOPMENT The compartmental modelling framework is constructed based on CFD simulations. The overall ?ow pattern, volumetric ?owrates, local energy dissipation, solids concentration and temperature distribution pro?les produced in batch cooling crystallizers are used to develop the compartments. The generalized compartmental model is a projective mapping of the CFD grid into a coarser network of fully mixed compartments. The criterion used for selecting the compartments is that negligible gradients should exist with respect to the local energy dissipation rate, temperature distribution and solids concentration within each compartment. The compartments are physically connected via input and output streams. The interconnecting ?ows can be estimated using the mean velocity information yielded from CFD codes entering or leaving a compartment and cross-sectional areas of each compartment. The number, size, location and volume of each compartment can be estimated from the CFD grid crystallizer dimensions. In order to avoid re-compartmentalization for different operating conditions, i.e. impeller velocity, the compartmental model is derived for the lowest impeller frequency that would produce the largest spatial non-uniformity. From CFD simulations, a batch cooling crystallizer has four distinct regions: a bulk zone (main body), cooling zone, impeller zone and a high solids concentration zone. These zones are split into a network of compartments. Upon scale-up, the number and location of compartments remain unchanged as the overall ?ow pattern does not change. However, the interconnecting volumetric and mass ?owrates between compartments, local energy dissipation

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rates, volume and cross-sectional areas for each compartment do change. The great advantage of using a compartmental modelling approach is that CFD hydrodynamic information can be coupled with the crystallization kinetics, solubility and the distributed population balance to produce a computationally ef?cient method of predicting the crystallization behaviour upon scale-up without the need for high spatial resolutions. Each compartment within the modelling framework contains a population, mass, concentration and heat balance including crystallization kinetic models for sizedependent growth, primary and secondary nucleation, and attrition. Physical models describing particle – impeller interactions to model attrition can also be introduced into the compartmental population balance model (Gahn and Mersmann, 1999). The population balance can also include a size-dependent classi?cation function to model classi?cation in the exiting stream from each compartment. These conservation equations and crystallization kinetic models (Kougoulos et al., 2004) cannot be implemented directly into a CFD environment due to the excessive computational requirements as discussed above. The CFD hydrodynamic information used within the predictive compartmental modelling framework includes the interconnecting volumetric and mass ?owrates between compartments and local energy dissipation rates estimated from the standard k– 1 turbulence model. Each compartment is assumed to be a well-mixed and homogeneous compartment. The compartmental population balance for the lth compartment contains a size-dependent classi?cation function, interconnecting volumetric ?owrates between compartments determined from CFD simulations, including source and sink terms for nucleation and attrition as follows: @?n(L, t)V(t)? @?n(L, t)G(L, t)? ? @t @L NI X ? QV,in, j (t)nin (L, t)
j?1

?

NO X k?1

QV,out,k (t)nout (L, t)hout (L, t) (1)

? f? (L, t)V(t) ? f+ (L, t)V(t) nuc attr

The compartmental solute concentration (mass) balance for the lth compartment consists of mass ?uxes for nucleation and growth with interconnecting volumetric ?owrates and is given as follows:
NI @?c(t) ? V(t)? X ? QV,in, j (t)cin, j (t) @t j?1

?

NO X k?1

QV,out,k (t)cout,k (t) (2)

? fm,nuc (t) ? fm,growth (t)

The compartmental heat balance for the lth compartment consists of interconnecting mass ?owrates determined

Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A1): 30–39

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KOUGOULOS et al. model (Launder and Spalding, 1974) and is solved using the interphase slip algorithm (IPSA). The model solves the continuity and momentum equations for a generic multiphase system and determines separate ?ow ?elds for each phase simultaneously. In the case of two-phase ?ow, the particle phase is treated as a separate ?uid and occupies disconnected regions of space in the continuous liquid phase. The different inertia of the liquid and solid phase is taken into account when using the MFM and will be closer to reality when compared with the settling velocity model (Micale et al., 2000a, b). In addition, the effect of the solid phase concentration on the liquid ?ow ?eld can be considered. The standard MFM model used in this contribution suffers from limitations in that the second-order effects relating to the in?uence of particles on the turbulence structure (particle– liquid) and particle – particle interactions (two-way or four-way coupling) are not taken into account (Micale et al., 2000a,b). Kuipers and Swaaj (1997), however, consider two-way and four-way coupling effects and could be used for further work. Furthermore, the MFM needs to be coupled with the crystal population balance so that the growth of initial seed crystals in a batch cooling crystallizer can be modelled using CFD. The prediction of the MFM based on monodisperse particle sizes also becomes less realistic as the particle concentration increases. The multiple frames of reference (MFR) technique was coupled with the MFM to model impeller rotation. The standard k –1 turbulence model was chosen to estimate local energy dissipation and turbulent kinetic energy rates. A homogeneous approach was employed whereby both phases share the same values for k (turbulence kinetic energy) and 1 (turbulent energy dissipation) and the interphase turbulence transfer is not considered. The standard k –1 assumes that turbulence is isotropic, i.e. has no favoured direction. For all the CFD simulations the turbulent Schmidt number was taken as equal to 0.8, according to suggestions by Micale et al. (2000a, b) for the liquid phase. The numerical solution of the resulting system equations was achieved using the ?nite volume method, together with the Rhie and Chow algorithm to prevent ‘chequer-board’ oscillations. The SIMPLEC algorithm was used to couple pressure and velocity. The hybrid-upwind discretization scheme was used for convective terms and a linear – logarithmic ‘wall function’ used on the crystallizer walls. Estimations of the particulate drag coef?cient are important in order to correctly model the momentum transfer. The particle drag coef?cient depends only on the particle Reynolds number and can be determined experimentally for ?uids at rest as follows: Rep ?

from CFD simulations and a heat removal term as follows: " NI X @T(t) ? Cp (t) rL (t)V(t)Cp (t) Qm,in, j (t)Tin, j (t) @t j?1 # NO X ? Qm,out,k (t)Tout,k (t)
k?1

? F(t)A?T ? Tc (t)?

(3)

The size-dependent growth rate model within the compartmental population balance is de?ned as follows: G(L, t) ? kg s(1 ? exp??a(L ? c)?) (4)

The source term for secondary nucleation is calculated is de?nes as follows:

f? nuc (L, t) ? d(L ? Lo )BN n,

(5)

The secondary nucleation kinetic model is de?ned as follows:
i BN ? kb MT Gjav 1k loc

(6)

The source and sink terms for attrition is determined as follows, ?1 0 ? fn,attr ? bd S (Li , Lj )n(Li )n(Lj ) dL (7)
Lj

?f? ? bd n(L) n,attr The attrition kinetic model is de?ned as follows:

(8)

bd ? kd 1loc s n

(9)

The local energy dissipation rates estimated from CFD simulations are introduced into the secondary nucleation and attrition rate functions describing the compartmental population balance. The initial boundary condition for the compartmental population balance describes a seed distribution. The compartmental modelling framework developed describes a complete description for the batch cooling suspension crystallization process. The generalized compartmental model contains a combination of differential, partial differential, integral and algebraic equations. The model is to be implemented into gPROMS process modelling software developed by Process Systems Enterprise Limited. This software is speci?cally designed to allow the direct speci?cation of all of the above classes of equations in a high level language and has powerful numerical methods for their solution and will be the subject of further communications. CFD SIMULATION THEORY Numerical simulations of particle concentration distribution inside baf?ed agitated crystallizers were carried out using an advanced model available in CFX 4.4 code (AEA technology, Harwell): the multi?uid model (MFM), which belongs to the class of Euler – Euler methods. The standard MFM is coupled with the standard k –1 turbulence

r ut dp m

(10)

For different particle Reynolds numbers the particle drag coef?cient is given as follows: 18:5 for 0:3 , Rep , 103 Re0:6 p 24 for Rep , 0:3 CD ? Rep CD ? (11) (12)

Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A1): 30–39

CFD MODELLING OF MIXING AND HEAT TRANSFER Brucato et al. (1998) proposed a new correlation for predicting the drag coef?cient in turbulent velocity whereby a correlation between particle size and Kolmogorov length scale of dissipative eddies was used. This correlation is suf?cient for the batch cooling crystallization of the organic ?ne chemical as dilute suspensions of up to 5% v/v are only considered.  3 CD ? CO ?4 dp ? 8:76 ? 10 (13) CD l The Kolmogorov length scale is given as follows:  3 1=4 n l? 1

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model gas – liquid-phase CFD simulations. In this method, two coupled frames are adopted: a rotating frame of reference surrounding the impeller, where the impeller blades appear stationary, and a normal inertial frame of reference in the rest of the crystallizer. Hence, the ?nite volume grid remains ?xed in position and the equations can be solved in steady-state mode, leading to a much more ef?cient computation. This method has been adopted here to simulate solid –liquid ?ow. CFD SIMULATION CONDITIONS

(14)

The drag coef?cient of particles whose size is comparable with that of the turbulent eddies is practically unaffected by the free-turbulence stream. For larger particles, the ?uid –mechanical interaction with turbulent eddies becomes more signi?cant and leads to an increase in the particle drag. This increase becomes larger as the ratio of the particle size to l increases. Batch settling experiments were performed using different particle size ranges in order to determine terminal settling velocities allowing for the particle Reynolds number and subsequently the modi?ed drag coef?cients to be estimated. CFD simulations were also used to model heat transfer (Yang et al., 2000) based on estimated liquid side heat transfer coef?cients on various geometrically similar crystallizer scales and operating conditions, combined with the implementation of a linear cooling pro?le within the CFD environment. CFD is used to model heat transfer in order to determine if ‘cooling zones’ or surfaces exist within batch cooling crystallizers. Heat transfer is directly associated with the mixing intensity, cooling pro?le and impeller con?guration used. The existence of cooling zones results in the production of temperature and supersaturation gradients within agitated crystallization vessels. The importance of supersaturation is that it is the driving force for a crystallization process and affects the crystallization kinetics, particle size distribution and product performance. MODELLING IMPELLER ROTATION A signi?cant problem arises in modelling of baf?ed stirred tanks due to the relative motion between impeller and baf?es since there is no single frame of reference for computation. The sliding grid technique is commonly used. In this method, the section of the grid surrounding the impeller is allowed to rotate stepwise, and the ?ow ?eld is recalculated for each time step. However, this method is computationally demanding. Two-phase simulations require even more computational time since the number of equations used increases and obtaining a steady-state particle distribution is a slow procedure. If the sliding grid technique is used, a time dependent calculation must be carried out using a large number of small time steps, which leads to large computational requirements. An alternative method, the multiple frames of reference (MFM) technique is used in this contribution. Lane and Schwarz (1999) successfully used the MFM technique to

A three-dimensional ?nite volume grid and geometry for the crystallizers was generated using a mesh generating software package called CFX-Promixus. The crystallizer scale, type of impeller used, crystallizer dimensions and number of hexahedral grid cells adopted for each CFD simulation is shown in Table 1. The crystallizers used exhibit geometrical similarity, are cylindrical with a ?at bottom and are ?tted with four baf?es. In the case of using a 458 pitch blade with four blades and a Rushton turbine impeller with six blades, the computational domain was limited to p and p/2, respectively, as periodic boundary conditions in the azimuthal direction was imposed. No slip conditions were speci?ed on the crystallizer walls, except for the surface, which is located at the top of the surface and a horizontal ?ow was allowed due to turbulent ?ow. Re?nement of the grid cells was not carried out in order for the converged solution to be grid-independent due to time constraints. It is assumed that the grid cells adopted in the CFD simulations is ?ne enough to ensure a fully converged solution. Each simulation used 8000 iterations achieving absolute mass residuals between 1 ? 1026 and 1 ? 1027. Underrelaxation factors were also applied to the velocity components equal to 0.3. CFD simulations to determine the hydrodynamics and mixing were carried out using the MFR technique until a steady-state solution was obtained. Computations were done on twin Intel Xeon 3 GHz processors with 1 GB RAM. Scale-up CFD simulations were carried out using constant speci?c power input per unit mass and impeller frequencies between 300 and 500 rpm. The Reynolds numbers studied cover a range of 1.87 ? 104 – 2.5 ? 105, producing fully turbulent conditions in each CFD simulation case, allowing the use of the k– 1 turbulence model. Transient heat transfer simulations were carried out using the steady-state solutions for the ?ow ?elds and local energy dissipation obtained from previous simulations that used the MFR technique. The heat transfer simulations therefore only consider the enthalpy balance.

Table 1. Crystallizer scales, dimensions and number of grids cells. Batch cooling crystallizer dimensions Grid cells using different impellers Rushton turbine 135,648 114,588 114,588 458 pitch blade 62,217 53,294 53,294

Scale 1l 5l 25 l

d (m) 0.040 0.065 0.110

D (m) 0.105 0.180 0.300

h (m) 0.035 0.060 0.100

H (m) 0.110 0.190 0.320

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KOUGOULOS et al. CFD SIMULATION RESULTS AND DISCUSSION velocity when compared with the ?ow in the impeller zone. In the bulk zone or ‘inner core’ of the crystallizer, axial ?ow in the downward direction back into the crystallizer impeller zone dominates. As the z/H ratio increases (Figure 1b), the ?ow along the baf?es and walls becomes signi?cantly reduced as energy is dissipated further away from the impeller along the walls. The overall ?ow pattern produced using a pitch blade impeller con?guration, is vastly different from that when using the Rushton turbine. The impeller zone is situated in a different position due to the downward pumping of the solid-liquid phase as a result of the inclined blades (Figure 2a). Between the impeller and baf?es, recirculation zones lead to an increase in the axial ?ow within the crystallizer. Below the impeller, however, a dead zone exists with the ?ow velocities being signi?cantly reduced where the particle and solvent phase is stagnant. The baf?es aid

A 5 l batch cooling crystallizer equipped with a Rushton turbine and pitch blade impeller operating at 300 rpm will be used as a case study throughout this contribution, for the development of the compartmental model framework using CFD simulations. Overall Flow Pattern Based on the overall ?ow pattern produced from CFD simulations, a classic double vortex pattern above and below the impeller zone is obtained when using a Rushton turbine. The highest velocities and gradients occur in the impeller zone both in the radial and tangential direction (Figure 1a). The baf?es have the effect of guiding the particles and solvent along the walls into the upper region of the crystallizer. The ?ow in the bulk zone of the crystallizer has less step gradients and a more reduced and uniform

Figure 1. Velocity pro?le (m s21) and overall ?ow pattern for liquid phase (Rushton): (a) vertical plane; (b) horizontal planes.

Figure 2. Velocity pro?le (m s21) and overall ?ow pattern for liquid phase (pitch blade): (a) vertical plane; (b) horizontal planes.

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CFD MODELLING OF MIXING AND HEAT TRANSFER the ?ow into the upper region, but due to the lower speci?c power input per unit volume, the velocity pro?les are less than that when using a Rushton turbine impeller at the same impeller frequency (Figure 2b). Hence the upper region is virtually stagnant as the mixing intensity is signi?cantly reduced. The ?ow again is directed back into the bulk zone or ‘inner core’. The overall ?ow pattern can initially be used to construct the compartmental modelling framework for the batch cooling crystallizer. However, this is rather arbitrary and is subject to further re?nement using energy dissipation rates, solids concentration and temperature distribution pro?les. The bulk zone is split into two compartments for both impeller types to take into account the main ?ow circulation ?ow pattern and the lower velocities being evident in the upper region of crystallizer. It should be noted that the solid phase used to produce the following ?ow patterns consists of seed particles. CFD simulation validation

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based on the power number (ten Cate et al., 2000) is satisfactory for compartmental modelling framework development. CFD has an in built function to predict the power number based on the power input and this can be compared to experimental power number values based on the Rushton turbine and pitch blade impellers, respectively. The Po number was calculated from CFD to be 4.56 and 1.25 and is in excellent agreement with the literature values of 4.5 and 1.27 for the Rushton and pitch blade impeller con?gurations. Local Energy Dissipation Using CFD simulations, the local energy dissipation rates in different regions of the crystallizer can be estimated. As expected, the highest local energy dissipation rates are observed in the impeller zone, whereas in the bulk

Figure 3. Local energy dissipation pro?les (W kg21) in a vertical plane for the liquid phase: (a) Rushton turbine; (b) pitch blade.

Figure 4. Seed concentration pro?le (volume fraction) in a vertical plane: (a) Rushton; (b) pitch blade.

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KOUGOULOS et al.

zone, signi?cantly lower and uniform energy dissipation rates are observed. In the impeller zone large gradients in local energy dissipation are evident (Figure 3a and b). In order to correctly capture the energy dissipation gradients, the impeller zone is broken down into three compartments for both the Rushton turbine and pitch blade impellers. The size, volume and location of compartments in the impeller zone for the batch cooling crystallizer modelling framework are different for the Rushton turbine and pitch blade impeller con?gurations. The importance of the local energy dissipation, in crystallization processes, is that crystallization kinetic models such as secondary nucleation and attrition are strongly dependent on the local energy dissipation.

Solids Concentration In this work, a predictive compartmental crystallizer model is developed according to the distribution of crystalline particles. Based on a modi?ed drag coef?cient, the particle concentration for a mean seed particle size of 3 mm was estimated. The volume fraction of seed used for batch cooling crystallizations is 0.05% v/v. Figure 4(a, b) shows the seed volume fraction distribution using a Rushton turbine and pitch blade impeller, respectively, upon seeding in the batch cooling crystallizer. The seed is homogeneously distributed throughout the bulk in both cases. A high concentration of seed particles is evident under the impellers at the bottom of the tank for both impeller con?gurations studied. This is a result of a dead zone present, due to reduced velocities and mixing intensity in this region producing internal classi?cation effects. This causes unwanted particle deposition and the formation of a high solids concentration zone. Upon addition of seed particles, the overall ?ow pattern throughout the vessel bulk remains unchanged when compared with ?ow ?elds using only the liquid phase. Figure 5(a) shows the velocity pro?les for both the liquid phase (IPA) and solid phase (API) having the same velocity at different distances in the radial direction from the impeller shaft when seed particles are used. Therefore the seed particles are carried in the liquid phase streamlines and negligible particle settling occurs. Owing to CFD limitations, it is not yet possible to model the growth of seed particles, size-dependent classi?cation, evolution of the crystal size distribution and increase in the solids concentration as the batch crystallization process proceeds. This requires the full population balance to be implemented into CFD as discussed previously. Hence CFD simulations were carried out to determine the solid concentration pro?le that is expected to develop towards the end of a batch cooling crystallization process. It is assumed that a 5% v/v solids concentration with a mean particle size of 200 mm is produced towards the end of a batch crystallization process. Figure 6 infers that, as the seed particles grow and the solids concentration increases, the mixing becomes less homogeneous. This leads to the formation of a clear solvent layer in the upper region of the crystallizer. The high solids concentration zone at the bottom of the crystallizer is still evident. A limitation of the CFD predictions is that it is based on a monodisperse particle size. In reality, a crystal size

Figure 5. Velocity pro?le (horizontal plane) below impeller: (a) 3 mm seed particles (API) and liquid phase (IPA); (b) 200 mm particles (API) and liquid phase (IPA).

distribution of particles exists within the vessel and hence will contain ?ner particles that may be suspended in the clear solvent region. Thus, CFD is not yet capable of providing a complete picture. Figure 5b shows that, at a particle size of 200 mm, the liquid phase is moving at a higher velocity than the solid phase due to gravitational effects. The reverse is true when the ?ow direction is from above, into the impeller zone. The difference between the two velocities is known as the particle slip velocity and results in some particle segregation. Using different monodisperse particle sizes, it was observed that negligible particle settling occurs up to particle sizes of 50 mm. The reason for this is that the extent of particle settling velocity not only depends on the size but also on the density difference between the solid and liquid phase. For the organic ?ne chemical crystallization system, the density difference is not signi?cant. Therefore,

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CFD MODELLING OF MIXING AND HEAT TRANSFER limited particle settling is expected to occur with an increase in particle size. The most signi?cant size-dependent classi?cation area is in the high solids concentration zone and can modelled by introducing a classi?cation function into the compartmental population balance modelling framework. By increasing the agitator speed to 500 rpm, the suspension can be homogeneously mixed with respect to solids concentration and particle size throughout the batch cooling crystallization process. At this impeller frequency, the need for a classi?cation function within the compartmental modelling framework is eliminated and hence reduces the predictive modelling complexity of the batch cooling crystallization process. Based on these two-phase CFD predictions we can infer that the velocity pro?les and hence the interconnecting mass and volumetric ?owrates of the solid and liquid phases does not change signi?cantly with an increase in particle size and solids concentration. This is due to the dilute solid concentrations being studied and the small density difference between the solid and liquid phase. The predictive compartmental modelling framework assumes that the interconnecting volumetric and mass ?owrates between the compartments for the population, mass and heat balance will remain unchanged during the batch crystallization process. In other crystallization systems, this assumption may not be valid, especially where high solids concentrations are used with large particle sizes and density differences, and signi?cant particle segregation occurs. Hence, the change in the velocity pro?les must be taken into account within the modelling framework.

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to lower temperatures. Figure 7(a) shows the temperature distribution in a vertical plane within the batch cooling crystallizer after 360 s simulation equipped with a Rushton turbine impeller. The temperature distribution is uniform in the bulk zone, which can be explained by the ef?cient turbulent heat transfer. Similarly, no temperature gradients exist in the impeller zone as well. However, cooling zones exist in the upper region of the crystallizer where temperatures are the lowest and gradients exist. This can be explained by the effect of the less turbulent ?ow in this part of the crystallizer and also by the cooling wall temperature on the wall boundary. The temperature difference between the bulk and near the wall of the cooling zones is approximately 0.2 8C. On an industrial scale of, say, 1500 l, however, the temperature gradient between the bulk and the cooling zone may be expected to be much more signi?cant. This could affect process performance signi?cantly as a result of encrustation occurring and large supersaturation gradients being realized. Figure 7(b) shows the temperature pro?le after 360 s numerical simulation using a pitch blade impeller con?guration. The bulk zone has a uniform temperature pro?le and no temperature gradients exist. However, the size and temperature gradient within the cooling zone when compared with the Rushton turbine impeller is larger. A temperature difference of approximately 0.4 8C is observed between the bulk and temperature near the wall in the cooling zone. This is a result of less ef?cient turbulent heat transfer using the pitch blade impeller. CONCLUSION A novel compartmental modelling framework has been successfully developed using CFD simulations to determine the effects of mixing with the presence of particles and heat transfer on a seeded batch cooling crystallization process. The compartmental structure is based on hydrodynamic information obtained from CFD simulations to include the overall ?ow pattern, velocity, local energy dissipation rates, solids concentration and temperature distribution. Figures 8 and 9 show the comprehensive compartmental structure produced using a Rushton turbine and pitch blade impeller, respectively. The compartmental model developed will be used to predict the crystallization behaviour upon scale-up, whereby the population balance is coupled with ?uid dynamic information and crystallization kinetic models. The hydrodynamic information estimated from CFD simulations includes the local energy dissipation rates and interconnecting volumetric and mass ?owrates, which are combined with the population, mass, energy balance including crystallization kinetic models within each compartment. The size, location and volume of each compartment are estimated from CFD using the physical crystallizer dimensions. CFD suffers from serious shortcomings when considering a batch cooling crystallization process due to the transient nature of the system. Modelling of sizedependent classi?cation, evolution of the crystal size distribution and implementation of the ‘full’ population balance within the CFD environment is not possible yet. This can be overcome using the compartmental modelling framework as presented in this contribution. A compartmental

Heat Transfer The Nusselt number for heat transfer is a function of the impeller type used, Reynolds number, Prandtl number and viscosity as follows:   m (15) Nu ? f Re, Pr, b mw During the simulation, the initial solution, saturated at 353 K, was cooled down to 293 K over a period of 1 h. The heat transfer coef?cient was estimated using an impeller frequency used of 300 rpm for both the Rushton turbine and pitch blade impeller, respectively. During the simulation of the batch cooling crystallization, the temperature distribution was recorded at 60 s intervals corresponding to a cooling rate of 21 K min21. The transient CFD simulations were performed using the steady-state hydrodynamic information and hence only the cooling pro?le, enthalpy balance and ?lm heat transfer coef?cient were implemented into the CFD model. CFD was used to identify any ‘cooling zones’ or surfaces where steep gradients in the temperature distribution were likely to occur. Gradients in the temperature distribution affect the supersaturation pro?les within a batch cooling crystallization vessel. Supersaturation is the driving force for a crystallization process, and affects both the crystallization kinetics and the ?nal product performance. In comparison with the temperature distribution, higher levels of supersaturation correspond

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KOUGOULOS et al.

Figure 8. Compartmental structure showing (a) the location of compartments and (b) the connectivity diagram using a Rushton turbine impeller. Figure 6. Solid concentration pro?le (volume fraction) using an average solids concentration of 5% v/v (Rushton).

Figure 9. Compartmental structure showing (a) the location of compartments and (b) the connectivity diagram using a pitch blade impeller.

model allows the spatial resolution to be drastically reduced when compared to that used in the CFD environment. This allows estimations of process behaviour to be obtained over a period of a few days. NOMENCLATURE
A BN CD CO Cp c d dp D F h H Gav kb Figure 7. Temperature distribution pro?le (K) after 360 s: (a) Rushton turbine; (b) pitch blade. kg kd heat transfer area, m2 nucleation rate, no. m23 s21 drag coef?cient modi?ed drag coef?cient speci?c heat capacity, J kg21 K21 concentration, kg m23 impeller diameter, m particle size, m tank diameter, m heat transfer coef?cient, W m22 K21 impeller clearance, m tank height, m average growth rate, m s21 nucleation rate coef?cient, no. [m3 s(m s21)i(kg m23) j (rev s21)k]21 growth rate coef?cient, m s21 disruption kernel

Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A1): 30–39

CFD MODELLING OF MIXING AND HEAT TRANSFER
L Lj Li Lo MT N n(L) n P Q T t u V Z Greek symbols d bd r m 1 l n s Subscripts c b t w loc m v Dimensionless Nu Po Pr Re particle size, m particle size of class j, m particle size of class i, m critical nuclei size, m solid density, kg m23 agitator speed, rps population density, no. m24 supersaturation order power, W ?owrate, m3 s21 temperature, K time, s velocity, m s21 compartmental volume, m3 tank diameter, m

39

Dirac function disruption rate, s21 density, kg m23 viscosity, N s m22 energy dissipation per unit mass, W kg21 Kolmogorov length scale, m kinematic viscosity, Ns m kg21 supersaturation

cooling bulk terminal wall local mass volumetric

Nusselt number, Nu ? fZ/l Power number, Po ? P/rN 3d 5 Prandtl number, Pr ? mCp/l Reynolds number, Re ? rNd 2/m

REFERENCES
Al-Rashed, M.H. and Jones, A.G., 1999, CFD modelling of gas–liquid reactive precipitation, Chem Eng Sci., 54(21): 4770–4784. Bermingham, S.K., Kramer, H.J.M. and van Rosmalen, G.M., 1998, Towards on-scale crystallizer design using compartmental models, Comput Chem Eng, 22: S355–362. Bezzo, F., Macchietto, S. and Pantilides, C.C., 2004, A general methodology for hybrid multizonal (CFI) models, Computers and Chemical Engineering, 28: 501–525. Brucato, A., Grisa?, F. and Montante, G., 1998, Particle drag coef?cients in turbulent ?uids, Chem Eng Sci, 53(18), 3295–3314. Gahn, C. and Mersmann, A., 1999, Brittle fracture in crystallization processes Part A. Attrition and abrasion of brittle solids, Chem Eng Sci, 54(9): 1273–1282. Jaworski, Z. and Nienow, A.W., 2003, CFD modelling of continuous precipitation of barium sulphate in a stirred tank, Chem Eng J, 2–3: 167– 174. Kougoulos, E., Jones, A.G. and Wood-Kaczmar, M.W., 2004, Estimation of crystallization kinetics for an organic ?ne chemical using a modi?ed

continuous cooling mixed suspension mixed product removal (MSMPR) crystallizer, J Crystal Growth (in press) Kramer, H.J.M., Bermingham, S.K. and Van Rosmalen, G.M., 1999, Design of industrial crystallizers for a given product quality, J Crystal Growth, 198/199: 729–737. Kramer, H.J.M., Dijkstra, J.W., Verheijden, P.J.T and Van Rosmalen, G.M., 2000, Modelling of industrial crystallizers for control and design purposes, Powder Technol, 108: 185–191. Kuipers, J.A.M. and Swaaij, W.P.M., 1997, Application of computational ?uid dynamics to chemical reaction engineering, in Reviews in Chemical Engineering, Amundson, N. (ed.) (Luss and Marmur), pp. 1–118. Lane, G.L. and Schwarz, M.P., 1999, CFD Modelling of gas–liquid ?ow in stirred tank, in Chemeca ’99, Newcastle, Australia. Launder, B.E. and Spalding, D.B., 1974, The numerical computation of turbulent ?ows, Comput Meth Appl Mech Eng, 3: 269 –289. Liiri, M., Koiranen, Tuomas, A. and Aittamaa, J., 2002, Secondary nucleation due to crystal-impeller and crystal-vessel collisions by population balances in CFD-modelling, J Crystal Growth, 237–239; 2188–2193. Maggiorios, D., Goulas, A., Alexopoulos, A.H., Chatzi, E.G. and Kiparissides, C., 1998, Use of CFD in prediction of particle size distribution in suspension polymer reactors, Comput Chem Eng, 22(1): S315–S322. Micale, G., Montante, G., Grisa?, F., Brucato, A. and Godfrey, J., 2000a, CFD simulation of particle distribution in stirred vessels, Trans IChemE, 78: 435 –444. Micale, G., Montante, G., Magelli, F. and Brucato, A, 2000b, CFD simulation of particle distribution in a multi-impeller high aspect ratio stirred vessel, in 10th European Conference on Mixing, Delft, The Netherlands, pp 125–132. Montante, G., Micale, G., Magelli, F. and Brucato, A., 2001, Experiments and CFD predictions of solid particle size distribution in a vessel agitated with four pitch blade turbines, Trans IChemE, Part A, Chem Eng Res Des, 79(A8): 1005–1010. Rigopoulos, S., and Jones, A.G., 2003, A hybrid CFD-reaction engineering framework for multiphase reactor modelling: basic concept and application to bubble column reactors, Chem Eng Sci, 58: 3077–3089. Rousseaux, J.M., Vial, C., Muhr, H. and Plasari, E., 2001, CFD modelling of precipitation in the sliding-surface mixing device, Chem Eng Sci, 56(4): 1677–1685. Sha, Z., Oinas, P., Louhi-Kultanen, M., Yang, G. and Palosaari, S., 2001, Application of CFD simulation to suspension crystallization—factors affecting size-dependent classi?cation, Powder Technol, 121(1): 20– 25. ten Cate, A., Bermingham, S.B., Derksen, J.J and Kramer, H.M.J., 2000, Compartmental modelling of an 1100L DTB crystallizer based on Large Eddy ?ow simulation, in 10th European Conference on Mixing, Delft, The Netherlands, pp. 255–264. Versteeg, H.K. and Malalasekera, W., 1995, An Introduction to Computational Fluid Dynamics. The Finite Volume Method, 1st ed (Longman, Harlow, UK). Wei, H. and Garside, J., 1997, Application of CFD modelling to precipitation systems, Chem Eng Res Des, 75(A2): 219–227. Yang, G., Louhi-Kultanen, M. and Kallas, J., 2002, The CFD simulation of programmed batch cooling crystallization, Chem Eng Trans, 1: 83 –88. Zauner, R. and Jones, A.G., 2002, On the in?uence of mixing on crystal precipitation processes-application of the segregated feed model, Chem Eng Sci, 57(5): 821 –831.

ACKNOWLEDGEMENTS
The authors would like to thank GlaxoSmithKline pharmaceuticals for ?nancial support via an EPSRC Industrial case award. The manuscript was received 26 March 2004 and accepted for publication after revision 7 October 2004.

Trans IChemE, Part A, Chemical Engineering Research and Design, 2005, 83(A1): 30–39



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