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CFD analysis of turbulence in a baffled stirred tank, a three-compartment model M.H. Vakili, M. Nasr Esfahany ∗ Department of Chemical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

A R T I C L E

I N F O

Article history: Received 5 February 2008 Received in revised form 2 July 2008 Accepted 3 October 2008 Available online 5 November 2008 Keywords: CFD Fluid mechanics Mathematical modeling Mixing Simulation Turbulence

A B S T R A C T

Three-compartment model was used to study non-homogeneity of mixing in a fully baffled stirred tank. Multiple reference frame (MRF) technique was used for calculations. Calculations were performed to study the effects of agitator speed, impeller diameter, baffle width and distance of impeller from bottom of the tank on turbulent flow field. Three different zones of the vessel, that were a small zone near the impeller, another zone around the baffles, and a relatively large zone far from the impeller and baffles, named circulation zone, were investigated. Boundaries of these zones were determined using two different methods. The first method used gradient of energy dissipation rate while the other method used cumulative energy dissipation rate to determine the zone boundaries. Zone boundaries determined by both methods were comparable. The turbulent kinetic energy dissipation rate gradient was the preferred method due to its simplicity. Turbulent kinetic energy dissipation rate increased with agitator speed in all zones. Both turbulent kinetic energy dissipation rate and turbulent kinetic energy showed considerable change with impeller diameter at impeller zone, while no remarkable change was observed at baffle and circulation zones. Three-compartment model parameters, impeller and baffle energy dissipation ratios i , b , impeller and baffle volume ratios i , b and impeller and baffle exchange flow rates Qi , Qb were obtained from CFD simulations. Impeller energy dissipation ratio, impeller exchange flow rate and baffle exchange flow rate increased while baffle volume ratio decreased with agitation rate and impeller diameter. Baffle energy dissipation ratio and impeller volume ratio showed no considerable change with agitation rate and impeller diameter. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Stirred vessels are widely used in mineral, metallurgical and chemical process industries to perform a variety of operations such as: homogenization, crystallization, gas dispersion, mechanical flotation, polymerization, heat transfer, etc. Study of turbulent flow and computation of its properties in a stirred vessel is a considerable challenge for existing turbulence models. Factors contributing to this difficulty include the non-isotropic nature of flow in a stirred vessel, the complex geometry of rotating impellers and the large disparity in geometric scales present. Existence of baffles also increases the complexity of the flow field. Analyzing the turbulent flow pattern and its properties in stirred vessels may be a beneficial tool for equipment design, process scale-up, energy conservation and product quality control. Generally, studies of flow can be classified into two parts, namely experimental fluid dynamics (EFD) and computational fluid dynamics (CFD).

∗

Corresponding author. Tel.: +98 3113915631; fax: +98 3113912677. E-mail address: [email protected] (M.N. Esfahany).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.10.037

EFD can provide information for a special flow field. Existing methods used for design of stirred vessels are based on experimental techniques. These methods usually are applied for determination of dimensionless parameters, such as, power number Np and flow number No , that are significant factors in design of mixing tanks. However, neither of these methods give any information about flow pattern, mixing mechanism, flow homogeneity of stirred tanks, whereas each of above parameters play a key role in quality of mixing process. Furthermore, because of some equipment limitations and unsteady nature of turbulent flow, complexity of impeller blade geometry and relative motion among fluid elements, attainment of experimental data in most flow fields, usually is impossible. Of course, experimental results are applicable for validation of computational methods used for simulation of flow field. Therefore, both experimental and computational results abreast are used for design. Use of CFD for simulation of turbulent flow in stirred vessels can generate useful data to study flow behavior, circulation patterns, vortex structures, Reynolds stresses, etc. Transport turbulence models of the averaged Navier–Stokes equations are base of the most CFD simulations which has been reviewed by many investigators (Rodi, 1980; Markatos, 1986).

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The study of turbulent flow field and its properties in stirred tanks has been subject of numerous theoretical and numerical investigations. Harris et al. (1996) applied both experimental and numerical methods to predict flow field in baffled stirred tank reactors using time-dependent sliding mesh techniques. It was clearly shown that the comparison between the experimental findings and the numerical predictions was generally satisfactory especially in regions far away from the impeller. Brucato et al. (1998) used three alternative modeling approaches, (impeller boundary condition (IBC) method, inner–outer (IO) iterative procedure, sliding-grid (SG) method), to simulate flow field in a baffled mixing tank stirred by a radial impeller and compared simulation results with experimental data. They observed that results obtained by SG approach gave the best agreement with the experimental data. Sahu et al. (1998) have attempted to improve the CFD predictions by means of zonal modeling. They divided the vessel into several zones. In each zone, different sets of values for `k–' model parameters were specified. Based on their work, predictions of `k' were significantly improved by using zonal modeling. Sahu et al. (1999) studied five different designs of axial flow impellers. They extended the zonal modeling concept to predict the flow field generated by these impellers. Montante et al. (2001) simulated the flow and turbulence field in a fully baffled vessel stirred by a turbine, using the sliding-grid and inner–outer methods. They investigated dependency of flow pattern on impeller clearance in stirred vessels. Deglon and Meyer (2006) investigated the effect of grid resolution and discretization scheme on the CFD simulation of fluid flow in a baffled mixing tank stirred by a rushton turbine, using the MRF impeller rotation model and the standard k– turbulence model. Simulations have also been preformed in a rotating frame of reference where the impeller was assumed to be stationary by Alexopoulos et al. (2002). This method is especially suitable for unbaffled vessels, where the outer-wall boundary condition is easily defined. Alexopoulos et al. (2002) applied a two-compartment model, which comprises of two mixing zones. An impeller zone of high local energy dissipation rate and a circulation zone of low energy dissipation rate formed the compartments. Model parameters identified were volume ratio of impeller zone, the ratio of the average turbulent kinetic energy dissipation rates in both compartments and exchange flow rate between the two compartments. They studied the effect of agitation rate, viscosity, impeller diameter and mixing vessel scale on model parameters. They applied their simulation results to determine drop size distribution of non-homogenous liquid–liquid dispersion. Their predictions showed excellent agreement with the experimental results. However, their study was performed in an unbaffled mixing vessel. One of the most important parameters in the liquid–liquid suspension systems is particle size distribution of droplets. Population balance is used to the predict particle size distribution of droplets in such systems. Some parameters of population balance model such as turbulent energy dissipation rate can be obtained from CFD simulation of turbulent flow field. The stirred tank could be divided into three relatively homogenous zones with specific turbulence properties. Then population balance equations could be solved for each zone and PSD for the entire tank could be obtained. In brief, the purpose of three-compartment model is to provide turbulence parameters for different zones to be used by population balance model for accurate prediction of PSD. In this paper, we present CFD simulations for flow field in a fully baffled vessel stirred by a blade impeller. Threecompartment model has been applied. Three mixing zones, namely an impeller zone of high local energy dissipation rate, a baffle zone of relatively high energy dissipation rate and a relatively large circulation zone of low energy dissipation rate were the compartments considered. Multiple reference frame (MRF) method is used for simulations. The model parameters (volume ratios of impeller and baffle zones, ratio of the average turbulent kinetic energy dissipation rate at impeller and baffle zones and the exchange flow rates between

impeller-circulation and baffle-circulation zones) were determined. Turbulent properties and volume density distribution of dissipated energy were studied. The effect of agitation rate, impeller clearance, impeller diameter, and baffle width on above mentioned parameters was investigated. 2. Geometry of the model CFD simulations were conducted in a 30 cm high; 16 cm diameter cylindrical tank that was filled with water up to a height of 28 cm, equipped with a two-blade impeller and four strip baffles as shown in Fig. 1. Baffles 0.5 cm thick and 28 cm high covered all the tank height. Different baffle widths of 1, 1.5 and 2 cm were investigated.

Fig. 1. Model geometry for CFD simulations.

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interface between the inner and outer zones the velocities and velocity gradients set to be the same. 5. MRF technique for CFD simulation

Fig. 2. Tetrahedral meshes of model.

Impeller blades 1.5 cm high and 0.5 cm thick with different diameters of 5, 7, and 10 cm were studied. Distance of impeller from bottom of tank was also varied (2, 4 and 8 cm). 3. Grid generation Due to the use of MRF method for simulation, volume of tank should be divided into two cylindrical zones. Inner zone comprises of impeller and outer zone includes tank walls and baffles. A nonuniformed grid with tetrahedral element consisting of 31×144×100 nodes in the r–z– directions, respectively (total ∼451 000 nodes), was used in this study as shown in Fig. 2. The grids were refined near the impeller blade and baffle. A finer grid, (44×192×128) was used to establish mesh independency. Negligible differences were observed between predictions obtained from the two mesh sizes. 4. Governing equations Governing equations for incompressible fluid flow are: Continuity equation:

ju¯ j =0 jxj

(1)

Motion equation: −

j j jP¯ (u¯ j u¯ i ) − (uj ui ) + ∇ 2 u¯ i − + gi = 0 jxj jxj jxi

(2)

In Eq. (2) −ui uj , Reynolds stress, can be modeled by semi empirical

relations. Two-equation k– model was used in this work. Governing equations for turbulent kinetic energy and kinetic energy dissipation rate are (Patel et al., 1985): ¯i jk j (t) k (t) ju u¯ j = ij − + + (3) jxj jxj jxj k jxj

u¯ j

2 j (t) ju¯ i j + = CE1 ij − CE2 jxj jxj jxj k k

+

(t)

j jxj

(4)

where CE1 = 1.44;

CE2 = 1.92;

C = 0.09;

k = 1.0;

= 1.3

For liquid in contact with solid surfaces no-slip boundary condition was used. Zero shear stress was used for free surface. At the

Modeling problems that involve both stationary (e.g., baffle) and moving (e.g., impeller) zones may be approached in three ways, the multiple reference frame (MRF) method, the mixing plane method and the sliding mesh method. The MRF method is the simplest of the three. This approach is appropriate when flows at the boundary between inner and outer zones are nearly uniform. Since the impeller–baffle interactions are relatively weak, in mixing tanks, large-scale transient effects are not present and MRF method can be used. Fluent 6.0.12 software was used to perform calculations. First order upwind discretization scheme was used for the convection term of momentum, turbulent kinetic energy and energy dissipation rate equations. Relative velocity formulation and k– turbulence model were used for simulating a tank containing water. To use MRF approach, a rotating coordinate system has been adopted for the inner zone named fluid 1, which its rotating rate was set equal to impeller agitation rate, and a non-moving coordinate system has been defined for outer zone including baffles named fluid 2. Angular velocity of impeller has been set zero with respect to the rotating coordinate system. To start computations initial guesses should be specified for velocity, pressure and turbulence parameters. The angular velocity of rotating reference frame at the first step of calculations has been set as 10% of the case under study. After about 1000 time steps, results were saved and used as initial guess. This procedure prevents solution from being diverged. Convergence was achieved when residuals on continuity, velocities, kinetic energy and energy dissipation rate all become less than 10−5 . 6. Model validation To validate the model calculations for an unbaffled stirred tank with specifications given by Alexopoulos et al. (2002) was performed. Simulation was conducted in a 28 cm high, 15.4 cm diameter tank that was filled with water up to a height of 25.2 cm, and equipped with a two-blade impeller. Other parameters are given in Fig. 3. Fig. 4(a) shows the contours of turbulent kinetic energy dissipation rate and velocity vectors, obtained from Alexopoulos et al. work. Fig. 4(b) shows the contours of energy dissipation rate and velocity vectors obtained from the present CFD simulations. Our results are comparable with Alexopoulos et al. work. Density of the turbulent kinetic energy dissipation rate contours around the impeller tip is very large relative to farther points. It shows that energy dissipation rate gradient near the impeller tip is very large. Velocity vectors show that velocity magnitude around the impeller tip is much greater than the rest of the fluid. 7. Results Flow field in a fully baffled mixing tank stirred by a two-blade impeller as shown in Fig. 1 containing water, was simulated. Simulations were performed in nine different runs. The specification of each run is given in Table 1. Fig. 5 shows the energy dissipation rate contours (Figs. 5(a) and (b) axial profile and radial profile, respectively) for run 1. From Fig. 5(a) it is seen that the turbulent kinetic energy dissipation rate at the impeller tip side that force the fluid to flow is approximately two times the dissipation rate in the wake of the blade. The average value of energy dissipation rate at the side that force the fluid to flow is 0.127 m2 /s3 while it is 0.069 m2 /s3

354

M.H. Vakili, M.N. Esfahany / Chemical Engineering Science 64 (2009) 351 -- 362 Table 1 Runs' specifications.

Fig. 3. Model geometry of Alexopoulos et al. (2002) work.

Run Agitation number rate (rpm)

Impeller diameter (cm)

Impeller clearance (cm)

Baffle width (cm)

Reynolds number

1 2 3 4 5 6 7 8 9

5 5 5 7 10 5 5 5 5

4 4 4 4 4 2 8 4 4

1 1 1 1 1 1 1 1.5 2

13 52 104 25 52 13 13 13 13

100 400 800 100 100 100 100 100 100

066 265 531 610 265 066 066 066 066

dissipation rate at 4.6 cm distance from the bottom of the tank. Two maximum values of are observed in the plot. One is located at 2.6 cm of radial coordinate near the impeller tip and the other at 6.9 cm near the baffle edge. Mass-weighted average of the turbulent kinetic energy dissipation rate around the impeller tip, the baffle and bulk of the fluid are equal to 0.145, 0.0901 and 0.0106 m2 /s3 , respectively. Therefore, dividing the mixing vessel into three compartments seems to be justified. The first compartment is a small region around the impeller characterized by very large turbulent kinetic energy dissipation rate. The second compartment is a bigger region around the baffles characterized by relatively large turbulent kinetic energy dissipation rate. The third compartment is a very large circulation region between the impeller and baffles, where the turbulent flow field is nearly homogenous and the turbulent kinetic energy dissipation rate is relatively small. Zone boundaries have been specified based on a characteristic non-homogenous property of the turbulent flow. The most characteristic property of turbulent flow is the turbulent kinetic energy dissipation rate, . The energy dissipation rate affects micro-mixing and the rates of heat and mass transfer, crucial in reactive systems. Large localized energy dissipation rates can cause cellular damage and reduce yield in bio-reactors. Furthermore, two-phase process characteristics such as drop size distribution, interfacial area, and gas-holdup are strongly dependent on the distribution of the energy dissipation rate. Zone boundaries have been specified based on the turbulent kinetic energy dissipation rate gradient. Impeller zone begin from impeller tip to point where energy dissipation rate gradient begin to decrease. Baffle zone also begin from baffle edge to point where the energy dissipation rate gradient begin to decrease. The remaining fluid is in the circulation zone. The distance between energy dissipation rate contours around the impeller tip and baffle edge is relatively small. The first contour that has higher distance from former contours, specify the boundary of impeller and baffle zones. Another way for determination of zone boundaries is use of cumulative energy dissipation rate. Cumulative energy dissipation is total energy dissipation between min and i which can be defined over the discretized range of dissipation rates as (Alexopoulos et al., 2002): Ei =

i

j nj

(5)

j=1 Fig. 4. Energy dissipation rate contours and velocity vectors: (a) Alexopoulos et al. (2002) work, (b) this work.

in the wake. It is seen from Fig. 5(b) that energy dissipation rate gradient around the baffle and impeller tip are large while in the bulk of the fluid is very small. Energy dissipation rate in circulation zone is relatively constant. Fig. 6 shows radial variation of energy

In the MRF method used in this work, volume of tank is divided into two parts, an inner part named fluid 1 and outer part named fluid 2. The boundary between impeller and circulation zones is located at fluid 1 and boundary between baffle and circulation zones is located at fluid 2. Energy dissipation rate can be divided into several equal discrete ranges in fluid 1 and fluid 2. Then the cumulative energy dissipation rate curves were drawn in both fluid 1 and 2. Fig. 7 shows cumulative dissipation energy rate curves for run 3, at fluid 1

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355

Fig. 5. Energy dissipation rate contours: (a) axial, (b) radial.

Fig. 6. Radial variation of energy dissipation rate.

and fluid 2. A clear break was seen in slope of each of these curves, named energy dissipation rate cut-off, cut that reveal the boundary of compartments. Obtained values of energy dissipation rate cut off from both energy dissipation rate gradient and cumulative energy dissipation rate methods for all runs are shown in Table 2. It is seen that average (maximum) difference between cut obtained from energy dissipation rate gradient and cut obtained from cumulative energy dissipation rate is about 12 (30.1) percent at fluid 1 and 8.7

(22.8) percent at fluid 2, respectively. cut obtained from energy dissipation rate gradient was used to analyze CFD simulation data. Fig. 8 shows cut for different (a) agitation rates, (b) impeller diameter, (c) impeller clearance, and (d) baffle width. cut increases by agitation rate, impeller diameter, impeller clearance, and baffle width for fluid 1 (impeller-circulation boundary). For fluid 2 (circulationbaffle boundary) the cut increase was seen for agitation rate, impeller diameter, and impeller clearance with a smaller slope than fluid 1, while cut decrease was observed with baffle width. Fig. 9 shows location of three different compartments in the tank for run 2. The baffle zone consists of four individual volumes around baffles as shown. Turbulent kinetic energy is another important parameter of turbulent flow field. Agitation with relatively high speed leads to turbulence generation. Turbulent kinetic energy spreads in the fluid by convection and diffusion. Due to the intense stir of the fluid in the vessel, turbulent kinetic energy does not show a significant change in different zones. Mass-weighted average of turbulent kinetic energy in impeller, baffle and circulation zones are 0.007, 0.0063 and 0.0058 m2 /s2 , respectively.

8. Agitation rate effect Simulations were performed for run 1, 2 and 3 with impeller agitation rate of 100, 400 and 800 rpm, respectively. Fig. 10(a) shows the energy mass-weighted average dissipation rate variations with

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Fig. 7. Cumulative energy dissipation curves for run 3 at (a) impeller and (b) baffle zones.

Table 2 Energy dissipation rate cut off obtained from gradient method and cumulative method. Run number

(cut-gradient )

1 2 3 4 5 6 7 8 9

0.207 13.04 107.2 0.486 3.48 0.16 0.468 0.244 0.31

fluid 1

(cut-cumulative ) 0.19 12.6 92.04 0.5 3.1 0.12 0.45 0.24 0.36

fluid 1

(cut-gradient ) 0.0693 3.22 27.5 0.122 0.303 0.0728 0.1 0.052 0.0141

fluid 2

(cut-cumulative )

fluid 2

0.066 3.08 22.4 0.112 0.3 0.072 0.086 0.046 0.02

Fig. 8. Energy dissipation rate cut off variation with (a) agitation rate (run 1,2,3), (b) impeller diameter (run 1,4,5), (c) clearance (run 1,6,7) and (d) baffle width (run 1,8,9).

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357

Fig. 9. Location of three different zones by axial and radial view: (a) impeller zone, (b) baffle zone and (c) circulation zone.

agitation rate in three different zones. It is observed that energy dissipation rate increases with agitation rate in all three zones with smaller slope in the circulation zone. Results show that doubling the agitation rate from 400 to 800 increases the energy dissipation rates 8 folds in the impeller region while an increase of 6 folds is seen in baffle region. Indeed energy dissipation rate at the impeller zone is proportional to N3 . The power input to the tank is P = Np N3 D5 , where is density of fluid, Np is power number, N is agitation rate of impeller and D is impeller diameter. Considering that flow is fully turbulent, power number Np is constant; therefore power input to the tank is proportional to N3 . Fig. 10(b) shows turbulent kinetic energy (mass-weighted average) variations with agitation rate in three zones. Increasing agitation rate leads to increase in kinetic energy everywhere in the tank. The slopes of turbulent kinetic energy in baffle and circulation zones are alike. The slope is greater in impeller zone. Since all the energy given to the fluid is converted to turbulent kinetic energy, it is proportional to the square of linear velocity. By doubling the agitator speed from 400 to 800, turbulent kinetic energy increase almost 4.5 folds in the impeller region and an increase of 4 folds is seen in both baffle and circulation zones. Finally Fig. 10(c) shows turbulent viscosity (mass-weighted average) variations with agitation rate in three zones. There is increase in turbulent viscosity in all zones with agitation rate. However, it is higher in circulation zone. In circulation zone, eddies are the major contributor to scalar transfer. Therefore, turbulent viscosity is greater than other zones. Fig. 11 shows effect of agitation rate on volume density distribution of energy dissipation rate. It is seen that increasing the agitation rate leads to shift in the peak of volume density distribution toward the greater values considerably. With increasing agitation rate from 100 to 800 rpm, volume density distribution peak shifts from 0.008 to 2.5 m2 /s3 (i.e., about third order of magnitude). Length scale of eddies is related to turbulence properties (Brucato et al.,1998): L∝

k2/3

where L is length scale of eddies. As mentioned above doubling the agitation rate leads to at least 6 folds increase in energy dissipation rate, while turbulent kinetic energy increases by 4 folds. Therefore length scale of eddies decrease almost by 0.5 folds. Thus increasing agitation rate results in decrease in length scale of eddies as

Fig. 10. Effect of agitation rate: (a) energy dissipation rate, (b) turbulent kinetic energy and (c) turbulent viscosity.

Fig. 11. Volume density distribution of the energy dissipation rate: effect of agitation rate.

we expected. Indeed smaller values of length scale of eddies relate to greater values of turbulent intensity and energy dissipation rate. Therefore with increasing agitation rate, energy dissipation rate values shift to higher values as mentioned above.

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Fig. 13. Volume density distribution of the energy dissipation rate: effect of impeller diameter.

diameter from 7 to 10 cm increases energy dissipation rate 5.5 folds in impeller zone, that shows energy dissipation rate is proportional to D4.8 . Power induced to the fluid by impeller is P = Np N3 D5 . For fully turbulent regime where Np is constant, power transfer to the fluid by impeller is proportional to D5 which is in agreement with proportionality of dissipation rate with D at impeller zone. Increasing power input to the tank also increases turbulent intensity around the impeller or turbulent kinetic energy in impeller zone. Fig. 12(c) shows impeller diameter effect on turbulent viscosity (mass-weighted averaged values) in different zones. It is seen that increasing impeller diameter has no considerable effect on turbulent viscosity in impeller zone. However, in baffle and circulation zones increasing impeller diameter slightly decreases turbulent viscosity. Fig. 13 shows volume density distribution of energy dissipation rate for various impeller diameters. It is seen that there is no considerable variation of volume density distribution peak with impeller diameter. 11. Baffle width effect

Fig. 12. Effect of impeller diameter: (a) energy dissipation rate, (b) turbulent kinetic energy, (c) turbulent viscosity.

9. Effect of impeller clearance from bottom of the tank Simulations were performed for run 1, 6 and 7 with clearances 4, 2 and 8 cm, respectively. Impeller clearance has no considerable effect on turbulent properties in three different zones. 10. The effect of impeller diameter To study impeller diameter effect, simulations were performed for run 1, 4 and 5 with impeller diameters of 5, 7 and 10 cm, respectively. Figs. 12(a), (b) and (c) show effects of impeller diameter on energy dissipation rate, kinetic energy and turbulent viscosity (massweighted averaged values) at different zones. Both energy dissipation rate and turbulent kinetic energy show considerable increase with impeller diameter at impeller zone, but no remarkable effect was seen in baffle and circulation zones. Increasing the impeller

Simulations were performed for run 1, 8 and 9 with baffle widths of 1, 1.5 and 2 cm, respectively. Figs. 14(a), (b) and (c) illustrate the effect of baffle width on energy dissipation rate, kinetic energy and turbulent viscosity (mass-weighted averaged values) at three different zones, respectively. Fig. 14(a) shows that increasing baffle width decreases energy dissipation rate in baffle zone significantly. Dissipation rate variations in circulation and impeller zones are relatively low. Most turbulent kinetic energy dissipation in baffle zone occurs around the baffle edge. Increasing the baffle width lead to baffle edge nearing the axis of tank where the tangential velocity is lower and fluid flow impacts to the baffle edge with lower strength. Therefore energy dissipation rate decreases at baffle zone. Fig. 14(b) shows that baffle width increase leads to decrease in the turbulent kinetic energy in baffle and circulation zones and has no considerable effect in impeller zone. Increasing the baffle width causes increase in dead zone between baffle base and tank wall that leads to decrease in velocity magnitude in this dead zone and hence decrease turbulent kinetic energy in baffle and circulation zones. Fig. 14(c) shows baffle width effect on turbulent viscosity (mass-weighted averaged values) in different zones. It is seen that increasing baffle width leads to decrease in turbulent viscosity in circulation zone, but no considerable effect on impeller and baffle zones was seen. This is confirmed with this fact that turbulent kinetic energy at circulation zone decreases with baffle width while energy dissipation rate shows no considerable change.

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Fig. 15. Volume density distribution of the energy dissipation rate: effect of baffle width.

Table 3 Three-compartment parameters obtained from CFD simulations. Run number

i

b

i

b

Qi×104 (m3 /s)

Qb×104 (m3 /s)

1 2 3 4 5 6 7 8 9

13.7 25.6 25.8 32.3 127.1 15.85 26.3 22 27.9

8.5 7.5 7.5 7.4 13.35 8.1 7.2 4.13 2.55

0.00157 0.00108 0.00108 0.00148 0.00117 0.00121 0.00184 0.00198 0.00226

0.0130 0.0114 0.0110 0.0123 0.0048 0.0110 0.0183 0.0139 0.0166

1.88 7.52 15.04 4.97 14.97 1.83 1.87 1.93 2.12

5.62 22.48 44.96 9.92 20.1 5.62 5.28 5.13 3.00

i and b are energy dissipation rate ratios that are ratios of average energy dissipation rate in impeller zone i and average energy dissipation rate in baffle zone b to average energy dissipation rate in the entire tank , respectively. i and b are volume ratios that are

Fig. 14. Effect of baffle width: (a) energy dissipation rate, (b) turbulent kinetic energy, (c) turbulent viscosity.

Fig. 15 shows volume density distribution of energy dissipation rate for various baffle widths. It is seen that variation of baffle width from 1 to 2 cm leads to shift in volume density distribution peak from 0.0015 to 0.008 m2 /s3 (i.e., about 5 folds). Therefore baffle width has relatively strong effect on volume density distribution of energy dissipation rate. 12. Compartment model parameters Due to the existence of three different zones in the fully baffled stirred tank, three-compartment model was used to study this system. It is essential to determine the model parameters. We can describe compartment model parameters as follows:

i =

i ;

b =

b

(6)

i =

Vi ; V

b =

Vb V

(7)

ratios of volume of impeller zone Vi and volume of baffle zone Vb to the entire volume of the tank V, respectively. Qi and Qb are other model parameters that are exchange flow rates between impeller and circulation zones and baffle and circulation zones, respectively. Compartment parameters in circulation zone are dependent on the parameters in the two other zones. Table 3 shows threecompartment parameters obtained from CFD simulations for run 1 to run 9. Comparison of our results with Alexopoulos et al. (2002) work that implemented two-compartment model in unbaffled stirred tank shows that compartment energy dissipation ratios and exchange flow rate parameters have the same order of magnitude. But this is not the case for volume ratio; it may be because of difference between model geometries. In our work ratio of impeller diameter to the tank diameter is 0.312 while at Alexopoulos et al. model this ratio is about 0.825. The impeller in their model sweeps a big volume, thus impeller zone is much larger compared to our model. Fig. 16 shows variation of energy dissipation ratio with agitation rate, impeller diameter, clearance and baffle width. It is seen from Fig. 16(a) that impeller energy dissipation ratio, i increases with agitation rate considerably, for the low value of agitation rate. Increasing agitation rate from 100 to 400 rpm causes an increase in impeller energy dissipation ratio by about 87%. However, increasing agitation rate beyond 400 does not show considerable effect on impeller energy dissipation ratio. The baffle energy dissipation ratio b is almost independent of agitation rate. It is confirmed by Fig. 10(a) showing that rate of change of energy dissipation rate with agitation rate at low agitation rate (i.e., low Reynolds number) in impeller zone is relatively high compared to other zones.

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Fig. 16. Energy dissipation ratio variation with (a) agitation rate, (b) impeller diameter, (c) clearance and (d) baffle width.

It is seen from Fig. 16(b) that impeller diameter has a great effect on i and relatively low effect on b. Increasing impeller diameter from 5 to 10 cm leads to increase in i almost 8 folds and b about 1.5 folds. It is seen from Fig. 16(c) that impeller energy dissipation ratio i increases with clearance. Clearance has no considerable effect on baffle energy dissipation rate ratio b . It is seen from Fig. 16(d) that impeller energy dissipation rate ratio increases and baffle energy dissipation rate ratio decreases when baffle width increases. When baffle width increases from 1 to 2 cm, i increases about 2 folds and b decreases about 2.4 folds. It is confirmed by Fig. 14(a) showing that energy dissipation rate shows no considerable change with baffle width in impeller zone but decrease in baffle zone and with a smaller slope in circulation zone. Fig. 17(a) shows that baffle volume ratio b decreases with agitation rate, while impeller volume ratio i show no remarkable change. Baffle volume ratio decreases with impeller diameter but impeller volume ratio shows no considerable change as can be seen from Fig. 17(b). It is seen from Figs. 17(c), (d) that baffle volume ratio increases with impeller clearance and baffle width and there is no considerable change in impeller volume ratio with both impeller clearance and baffle width. It is seen from Fig. 18(a) that both baffle exchange flow rate Qb and impeller exchange flow rate Qi variation with agitation rate N is linear. It is seen from Fig. 18(b) that impeller exchange flow rate is proportional to D3 (D is impeller diameter) and baffle exchange flow rate is proportional to D2 . It is comparable with relation Q = ND3 (1.165(r/ri )−0.379) where r is radial coordinate and ri is impeller radius (Bourne and Yu, 1994). Both impeller and baffle exchange flow rates show no remarkable change with clearance as it can be seen in Fig. 18(c). It is seen from Fig. 18(d) that impeller exchange flow rate shows no considerable change with baffle width, while baffle exchange flow rate decreases with baffle width.

13. Conclusions Results from CFD simulations show that three compartments in the baffled stirred tank may be considered. A small zone around the impeller that turbulent kinetic energy dissipation rate is very large. A zone around the baffles that energy dissipation rate is relatively large, less than impeller zone. And finally a relatively, homogenous large zone in fluid bulk, named circulation zone with small kinetic energy dissipation rate. Therefore, the three-compartment model was used to describe non-homogeneity of mixing in a 30 cm high; 16 cm diameter cylindrical tank, which was filled with water up to a height of 28 cm and equipped with a two-blade impeller and four strip baffles. Zone boundaries were determined with an energy dissipation rate value named energy dissipation rate cut off cut which was specified by two different methods. The first method was based on energy dissipation rate gradient and the second was based on cumulative energy dissipation rate. Values of cut obtained by both methods were comparable. Three-compartment model parameter (i , b , i , b , Qi and Qb ) were determined from CFD results. Impeller energy dissipation ratio i increases with impeller diameter and agitation rate. Baffle energy dissipation rate b increases with baffle width, decreases with impeller diameter and shows no considerable change with agitation rate. Impeller volume ratio i shows no remarkable change with agitation rate, impeller diameter and baffle width. Baffle volume ratio b decreases with agitation rate and impeller diameter and increases with baffle width. Both Impeller exchange flow rate Qi and baffle exchange flow rate Qb increase with agitation rate and impeller diameter. Impeller exchange flow rate shows no considerable change with baffle width while baffle exchange flow rate decreases with baffle width.

M.H. Vakili, M.N. Esfahany / Chemical Engineering Science 64 (2009) 351 -- 362

Fig. 17. Volume ratio variation with (a) agitation rate, (b) impeller diameter, (c) clearance and (d) baffle width.

Fig. 18. Volume flow rate variation with (a) agitation rate, (b) impeller diameter (c) clearance and (d) baffle width.

Turbulent kinetic energy and turbulent kinetic energy dissipation rate increase with agitation rate in all three zones. Both energy dissipation rate and turbulent kinetic energy show considerable changes with impeller diameter at impeller zone, but at baffle and circulation zones show no remarkable change. Both turbulent kinetic energy and turbulent kinetic energy dissipation rate decrease with baffle width in baffle zone.

Notation CE1 CE2 D Ei G

experimental coefficient in equation of experimental coefficient in equation of impeller diameter, m cumulative dissipated energy, m2 /s3 gravity, m/s2

361

362

k L nj N Np P P¯ Q Qb Qi r ri t u u¯ u V Vb Vi

M.H. Vakili, M.N. Esfahany / Chemical Engineering Science 64 (2009) 351 -- 362

turbulent kinetic energy, m2 /s2 length scale of eddies, m volume density distribution of energy dissipation rate impeller agitation rate, s−1 impeller power number impeller power average pressure, Pa exchange flow rate exchange flow rate between baffle and circulation zones, m3 /s exchange flow rate between impeller and circulation zones, m3 /s radial coordinate impeller radius, m time variable, s velocity, m/s average velocity in x direction, m/s fluctuation velocity, m/s volume volume of baffle zone, m3 volume of impeller zone, m3

Greek letters

b cut i b i (t)

turbulent energy dissipation rate, m2 /s3 mass-weighted average of energy dissipation rate in baffle zone, m2 /s3 energy dissipation rate cut-off, m2 /s3 mass-weighted average of energy dissipation rate in impeller zone, m2 /s3 baffle energy dissipation ratio impeller energy dissipation ratio viscosity, kg/m s turbulent viscosity, kg/m s

b i k ij (t)

baffle volume ratio impeller volume ratio density, kg/m3 experimental constant in equation k experimental constant in equation turbulent momentum flux (Reynolds stress)

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