Residence Time Distribution in a Multistage Agitated Contactor with

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Ind. Eng. Chem. Res. 2007, 46, 3538-3546

Residence Time Distribution in a Multistage Agitated Contactor with Newtonian Fluids: CFD Prediction and Experimental Validation Lifeng Zhang, Qinmin Pan, and Garry L. Rempel* Department of Chemical Engineering, UniVersity of Waterloo, Waterloo N2L 3G1, Canada

Residence time distributions (RTDs) are investigated in a multistage agitated contactor (MAC) for Newtonian fluids. A suitable computational fluid dynamics (CFD) simulation strategy is presented in this work. The effects of operating conditions, reactor geometries, and a turbulent parameter for the RTD curves are discussed. The resulting mean residence times and variances via the CFD simulation are in good agreement with the experimental data. Various liquids are used to simulate the RTD under different flow conditions. The Reynolds number with respect to the impeller speed ranged from 2600 to 50 000. The simulation results show that a cascade of stirred tanks with a back flow (CTB) model is more suitable to describe the flow behavior in a MAC with few stages than an axial dispersion model. 1. Introduction Multistage agitated contactors (MACs) have great potential applications for single-phase and two-phase systems due to their advantages over single-stage contactors.1,2 However, the application of this promising reactor has been impeded because of a lack of hydrodynamic data with respect to various fluid properties and a lack of the theoretical understanding of the liquid-phase backmixing behavior. In the literature, a few researchers have made contributions to the understanding of MACs.2-11 A number of correlations with respect to mixing behavior, mass transfer, and phase holdup in MACs have been reported, although a unified conclusion has not yet been reached in describing the liquid backmixing in MACs. The lack of a fundamental understanding to interpret backmixing behavior and the complexity of the flow behavior depending on the internal structure and fluid properties give rise to a diversity of correlations for backmixing, and contradictory conclusions have been drawn among those studies. Therefore, before the inception of an application for the MACs, a large amount of hydrodynamic data is required in order to understand the liquid backmixing behavior. However, the extensive experimental approach is usually not so feasible either, due to the cost or due to the time constraints. Computational fluid dynamics (CFD) has become a powerful tool to conduct reactor design and reactor diagnosis and to provide useful detailed information prevailing in the reactors, such as velocity field, concentration distribution, temperature profile, power consumption, and phase holdup distribution. The encouraging achievements of the utilization of CFD tools in stirred vessels have been progressively published, such as flow behavior characterization,12,13 mixing time prediction,14-16 power consumption,12 and residence time distribution.17,18 With the aid of CFD, the flow patterns as well as their influences on mixing performance can be revealed. To the best of our knowledge, the application of CFD simulation to predict mixing behavior in MACs is not available in the literature, although many successful examples on single-stage baffled/unbaffled contactors or reactors with multiple impellers have been published. The function of horizontal baffles is well-known for reducing the * To whom correspondence should be addressed. Tel.: (519) 888-4567, ext 32702. Fax: (519) 746-4979. E-mail: [email protected].

liquid backmixing between consecutive stages. However, contradictory conclusions have been drawn regarding the effect of liquid viscosity on backmixing in MACs. Several studies5,10,19 revealed that the viscosity shows a dampening effect on liquid backmixing, while other researchers8 concluded that with an increase in viscosity the power input under the same stirring speed increases, leading to an increase in turbulence accounting for increased liquid backmixing between stages. The objective of the present study is to investigate the flow behavior in a MAC with the aid of an available commercial CFD package by predicting residence time distribution. The resulting residence time distribution is compared to experimental data to demonstrate the feasibility of using CFD technique to assist in the design of MACs. The flow behavior of a reactor is usually characterized through residence time distribution (RTD) experiments, which allows comparison with ideal reactor models.20 In the present study, experimental RTD is obtained through a traditional pulse input technique. The stirring speed, liquid flow rate, and reactor geometry all could have a profound impact on the flow field in the stirred vessel which, in turn, affects the liquid mixing behavior of a MAC. 2. Experimental Setup and Operating Conditions A schematic diagram of the experimental setup of a multistage agitated contactor is shown in Figure 1. The MAC utilized in this study consists of from two to six stages (H/D ) 1 and D ) 0.1 m) separated by horizontal baffles with a central circular opening of 0.015 m i.d. and a thickness of 2 mm. Each compartment is provided with a standard Rushton turbine attached to a long shaft within the MAC. The diameter of the shaft is 0.008 m. Four vertical baffles with a width of (1/10)D are also supplied and equally spaced. The impeller has a diameter of 0.5D with a disk thickness of 0.0015 m. The impeller is center-positioned. A conductivity cell is installed at the outlet of the setup. A pulse input of a tracer (3 mL of NaCl saturated solution) was quickly injected in the first stage by a syringe. The insertion was completed within about 0.5 s, and it would not influence tracer measurement since the residence time is usually hundreds of seconds. The tracer concentration at the exit was measured with a conductivity probe, and the data were recorded by a data acquisition system. Different liquids were used in order to

10.1021/ie060567+ CCC: $37.00 © 2007 American Chemical Society Published on Web 04/19/2007

Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3539

a rotating domain. The interface between the rotating domain and stationary domain was set to half the distance between the impeller tip and the tank wall. In the present study, the meshes used were unstructured. Additional mesh refinements near the walls of the vessel and near the stirrer were furnished so that the accuracy in the zone of high velocity gradients was improved. The impeller domain contained a mesh scheme of around 100 000-600 000 cells, while the tank domain consisted of a mesh scheme of around 200 000-1 200 000 cells in the contactors with different stages. A commercial package, CFX 5.7.1,23 was used for modeling and simulations in this work. The standard k- model was used for dealing with fully turbulent regions, while a low Reynolds number turbulent model, RNG k-, was also investigated for early turbulent regions for purposes of comparison. The conservation of mass and momentum were determined using the Reynolds averaged Navier-Stokes eqs 1 and 2.23

∂F + ∇‚(FU) ) 0 ∂t

(1)

∂ FU + ∇‚(FU X U) ) -∇p′ + ∇‚(µeff(∇U + (∇U)T)) + B ∂t (2) where F is the liquid density, B is the sum of body forces, and U is the mean velocity vector. In eq 2, µeff is the effective viscosity and evaluated using the following equation: Figure 1. Schematic description of the continuous setup.

µeff ) µ + µt

(3)

Table 1. Physical Properties of the Liquid Phase

water 25% sugar 50% sugar

density, g/cm3

viscosity, mPa‚s (at 25 °C)

surface tension, N/m (at 25 °C)

1.0 1.09 1.21

1.0 4.2 12.5

0.073 0.068 0.059

understand backmixing behavior in different flow regimes in the MACs. The physical properties of liquids employed are listed in Table 1. The experimental procedure was repeated for various liquid flow rates and stirring speeds. 3. CFD Simulations In analysis of stirred vessels using CFD, a challenging issue is to model rotating impellers. In the literature, various approaches have been employed to model rotating impellers, namely, (1) black box approach, (2) multiple frames of reference approach (MFR), (3) inner-outer approach (IO), and (4) sliding mesh approach (SM).16,21 The black box approach requires impeller boundary conditions as input, which need to be determined experimentally; it is successful in prediction of the bulk flow. The sliding mesh approach was introduced by Luo et al.,22 where transient simulations are conducted step after step for each relative position of the stirrer and baffles. However, this approach is very computation-intensive. Both IO and MFR approaches simplify the simulation by assuming suitable steadystate conditions for the velocity field simulation, resulting in substantial savings of computational demand compared to the SM approach. In the present study, an MFR approach is adopted. Due to the unsymmetrical fluid input, the whole 2π azimuthal extent of the MAC was considered in the present study. The meshes employed for the simulations consisted of 300 000 cells for the contactor of two stages and up to 1 800 000 cells for the contactor of six stages. In an MFR approach, the tank bulk is set to a stationary domain while the impeller is embedded in

The turbulent viscosity, µt, is given by

µt ) CµF

k2

(4)

where the model constant Cµ is set to 0.09; k is the kinetic energy and is a measure of velocity fluctuation. is the turbulent eddy energy dissipation. The equations of continuity and motion were solved first to attain velocity profiles, kinetic energy, eddy energy, and eddy diffusivity. Subsequently, the information of the flow field and turbulent field obtained was used to solve a nonreacting scalar transport equation, based on the assumption that the tracer is dispersed in the vessel by convection and diffusion as follows:

(

µt ∂FΦ + ∇FUΦ ) ∇‚ FDA∇Φ + ∇F ∂t σt

)

(5)

where Φ is the tracer concentration, DA is the molecular diffusivity, and σt is the turbulent Schmidt number, indicating the ratio between the rate of momentum transport and passive scalars. The molecular diffusivity was assumed to be equal to 10-9 m2/s, a typical value for liquids. The required boundary conditions for the system are complex. In the stationary domain, the tank walls, vertical baffles, and horizontal baffles were assumed as standard wall boundary conditions with no-slip flow while the shaft was set to a rotating wall with a speed of 5-20 rps. The velocity at the inlet was set uniform with a value reflecting a superficial liquid velocity from 0.5 to 2 cm/s under investigation, whereas at the outlet, pressure was set as being uniform across the outlet area. In the rotating domain, the rotating speed was set to the same as that of the impeller and blades were set to wall boundary conditions. For the initial conditions at the walls, a zero flux was imposed and zero tracer concentration was set to computational domains.

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Table 2. Effect of Mesh Size on Simulation (uL ) 0.001 m/s and N ) 10 rps in Water System; Re ) 2.5 × 104) two-stage contactor number of cells power number, Np deviation, %

296 113 4.30

(0.35

four-stage contactor

631 936 4.33

592 245 4.30

The solution of the resulting set of equations is a threedimensional time-dependent map of tracer concentrations. At a given instant, the tracer concentration at the outlet could be obtained by taking the value of Φ at the same location from the transient results. Thus, the residence time distribution is evaluated as follows:

C(t) ) Φoutlet(t) E(t) )

dC(t) C0 dt

(6) (7)

The average residence time is calculated as follows:

τ)

∫0∞E(t)t dt

(8)

The second moment of the residence time distribution can be evaluated as follows:

σ2 )

∫(t - τ)2E(t) dt

(9)

and the normalized variance is given by the following equation:

σθ2 )

σ2 τ2

(10)

4. Results and Discussion The quality of meshes is essential for CFD simulations, and the size of meshes should be optimized in order to obtain accurate results within a reasonable calculation time. The maximum residue (MAX) < 10-4 was chosen to ensure a tight convergence of all the variables to be solved. The velocity is not measured in this work, and the power number, Np, is used as a criterion to determine the mesh quality, which is a common approach largely adopted in the literature.15,24 The power number is evaluated from the torque exerted on the blades and the shaft from the simulations as follows:

Np )

P 2πTN ) 3 5 FN dI nFN3dI5

(11)

where P is the power input and T is the total torque exerted on blades and the shaft. n is the number of impellers. The effect of the size of meshes on the power number is shown in Table 2. In Table 2, it can be seen that when the size of meshes increases approximately 3-fold, the predicted power number only changes within about 0.4%. Therefore, it can be concluded that the effect of the mesh size is not significant over this range of mesh sizes. In addition, the predicted power number is in fairly good agreement with the value of Np ) 4.2 predicted by the equation proposed by Bujalski et al.,25 which relates the power number to the minor dimensions of the impeller such as the disk thickness as well as the scale of reactors for Reynolds numbers g2 × 104. The slight difference exists due to the position of the impeller adopted in the present study. The impeller is half the diameter of the contactor and is positioned at the center of each stage, and the deviation from the standard reactor combination possibly causes the slightly higher power

(0.12

1 032 987 4.31

six-stage contactor 888 657 4.31

(0.23

1 807 809 4.33

number. Therefore, the simulated results can be considered reasonably accurate. Typical flow patterns are shown in Figure 2, and it can be seen that the velocity profile around each impeller is similar, where the liquid is pumped outward from the impeller and pushed back when approaching the tank wall. Therefore, two circulation loops are formed above and below the impeller, which is consistent with the experimental observation from other researchers.26 The flow patterns could vary significantly with varying geometric structures. One circulation would occur when the position of the impeller is very close to the bottom of the tank.27 Due to the application of horizontal baffles and a relatively long distance between each impeller, the velocity field can be considered to be developed independently. The time evolution of the tracer concentration can be obtained by setting up a suitable time interval by solving eq 5. In eq 5, different turbulent Schmidt number values have been adopted by the various authors, ranging from 1 to 0.1, and different recommendations have been given depending on applications.16,17 Montante et al.16 suggested a value of 0.1-0.2 for the forecasts of mixing time in stirred vessels with six pitched blades. In this work, the default value of 0.9 has been used down to 0.1 in order to examine the influence of the turbulent Schmidt number on RTD prediction. Another critical factor considered in the present study is the time interval adopted. A time interval of 5 s was used, which is small enough compared to the average residence time studied (the latter is usually at least several hundred seconds), and its effect on RTD is explored by performing additional simulations with a time step of either 1 order of magnitude larger or smaller than this time interval. It can be seen from Figure 3 that when the time step is reduced to 5 s no further improvement is observed. Therefore, a time interval of 5 s is used in the present study. As discussed earlier, different values for the Schmidt number have been selected by various authors, ranging from 1 to 0.1. However, in the present work, it is found that the Schmidt number has no significant impact on RTD predictions as shown in Figure 4 when the Schmidt number varies from 0.9 to 0.1. This is not consistent with the effect of the turbulent Schmidt number on homogenization curves and the mixing time in stirred vessels with multiple turbines reported by Montante et al.16 The independent behavior of the RTD simulations on the Schmidt number in the present work is due to the fact that the mixing performance within one stage is completely dominated by the convection flow and each stage has reached the maximal mixing, which can be validated by the experimental phenomena that the mixing time (usually around 3 s) is fairly small compared to the mean residence time (hundreds of seconds) for the MACs studied in this work. Therefore, the default value of 0.9 is used in all simulations later on. In Figure 4, the mean residence time and normalized variance predicted are 216 s and 0.53, respectively, while the corresponding experimental values are 222 s and 0.55. The space time under the same operating conditions, i.e., the reactor volume over liquid flow rate, is 217 s. A small deviation of 3% indicates that the residence time distribution in the contactor can be satisfactorily predicted by the CFD simulation and


Figure 2. Typical flow patterns in a four-stage contactor with water system (N ) 15 rps, uL ) 0.001 m/s, and Re ) 3.75 × 104).

Figure 4. Effect of value of turbulent Schmidt number on RTD in a twostage contactor (N ) 10 rps, uL ) 0.001 m/s, and Re ) 2.5 × 105). Figure 3. Dependence of RTD on time interval selected in a two-stage contactor (turbulent Schmidt number ) 0.9, N ) 10 rps, Re ) 2.5 × 104, and uL ) 0.001 m/s).

therefore the liquid flow behavior in MACs can be understood by CFD simulation. In the simulations, the transient concentration profile can be obtained and a typical concentration distribution at an instant is shown in Figure 5. In addition, a concentration profile taken in the middle height of each stage is also presented in the right column. Figure 5 shows that there is no concentration gradient observed within one stage, indicating that each stage is perfectly mixed, which further confirms the above analysis regarding the effect of the Schmidt number. In the literature, there are two commonly used models to describe the flow behavior in MACs. One is the axial dispersion (AD) model, and the other is a cascade of stirred tank reactors with back flow (CTB). The AD

model assumes that the extent of liquid mixing is constant anywhere in the column. This is not consistent with the current knowledge of flow patterns in stirred vessels. The CTB model assumes perfect mixing at each stage, which is more realistic. In addition, the CTB model possesses mathematical simplicity over the AD model. They can accommodate each other when the number of stages is high.28 However, with a small number of stages, the CTB model is more suitable than the AD model since there is less physical meaning of the AD model in a MAC at a relatively low number of stages.8,19 Hence, the simulation provides a line of evidence that the AD model is not suitable to describe the flow behavior in MACs with few stages. 4.1. Effect of Stirring Speed on RTD. Figure 6 illustrates the effect of stirring speed on the normalized variance of the RTD (σθ2) predicted by CFD. With an increase in stirring speed, the spread of residence time distribution increases as reflected

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Figure 5. Typical transient tracer concentrations at different stages in a six-stage contactor (t ) 250 s, N ) 10 rps, uL ) 0.001 m/s, and Re ) 2.5 × 104).

Figure 6. Effect of stirring speed on residence time distribution in a fourstage contactor (uL ) 0.001 m/s and water system: (1) N ) 10 rps; (2) N ) 15 rps; (3) N ) 20 rps).

Figure 7. Effect of liquid flow rate on RTD curves in a four-stage contactors (N ) 10 rps: (1) uL ) 0.002 m/s; (2) uL ) 0.001 m/s; uL ) 0.0005 m/s; water system).

in an increase in σθ2 values. When the stirring speed increases from 10 to 20 rps, σθ2 increases from 0.262 to 0.43, indicating that more back flow is created by enhancing the agitation level. This observation is consistent with many previous studies.8,10,29,30 It is believed that the turbulence around the central opening causes back flow. Therefore, turbulence parameters can be employed to relate the extent of backmixing. The discussion of a relation between backmixing and turbulence is presented in section 4.5. In addition, the peaks on the RTD curves occur earlier at a higher stirring speed, as indicated in Figure 6. The peak appears at 206 s when N ) 20 rps, while it occurs at 254 s when N ) 10 rps. Since a high stirring speed leads to earlier occurrence of the RTD peaks and a broader distribution,

employing high agitation intensity is not preferred if the purpose is to suppress the backmixing. 4.2. Effect of Liquid Flow Rates on RTD Curves. It is shown in Figures 7 and 8 that increasing liquid flow rate decreases the average residence time as expected. In addition, the spread of the RTD becomes narrower as indicated from the normalized variances when liquid flow rates are increased. This can be explained by the suppression effect of the bulk flow at the central opening between stages. However, it is found from the simulation results that the flow rate plays a marginal role in the averaged kinetic energy, which is mainly determined by the power input. In a four-stage contactor, the corresponding experimental variances are 0.253, 0.3, and 0.39 for liquid flow


Figure 8. Effect of liquid flow rate on RTD curves in a two-stage contactor (N ) 10 rps: (1) uL ) 0.002 m/s; (2) uL ) 0.001 m/s; (3) uL ) 0.0005 m/s; water system).

Figure 9. Effect of horizontal baffles on RTD curves predicted via CFD simulation in a two-stage contactor (N ) 10 rps and uL ) 0.001 m/s).

rates of 0.002, 0.001, and 0.0005 m/s, respectively. In a twostage contactor the corresponding experimental variances are 0.503, 0.58, and 0.69, respectively. Compared to the values from the CFD simulation, the simulated results with the experimental data are in reasonably good agreement. At a lower liquid flow rate, a higher σθ2 value is obtained; namely, a lower stage efficiency results. At uL ) 0.0005 m/s, the performance of a four-stage contactor is equivalent to 2.85 CSTRs when an n CSTRs in series model is used and the performance of a twostage contactor is equivalent to 1.5 CSTRs under the same operating conditions. The stage efficiency, defined as the equivalent number of CSTRs divided by the actual number of stages in a MAC, reduces to as low as 0.72, indicating that decreasing the liquid flow rate to achieve a longer residence time is not desirable. 4.3. Effect of Horizontal Baffles on the RTD Curves. It is known that the application of horizontal baffles is helpful in reducing back flow between consecutive stages. The influence of the opening area on the residence time distribution is predicted as shown in Figures 9 and 10. In Figures 9 and 10, it can be seen that the horizontal baffles profoundly change the RTD curves. The peak of the RTD curve appears earlier and the tail of RTD becomes longer when the central opening is larger. For instance, it appears at 200, 400, and 600 s in a MAC of six stages for three openings, respectively, as illustrated in Figure 10. Theoretically, it will occur at a time of zero when the back flow between stages approaches infinity and the overall performance of the whole reactor is equal to one CSTR with

Figure 10. Effect of horizontal baffles on RTD curves in a six-stage contactor (N ) 10 rps and uL ) 0.001 m/s).

Figure 11. Effect of two different turbulent models on RTD curves in a two-stage MAC (25% sugar solution, N ) 15 rps, uL ) 0.001 m/s, and Re ) 9167; 50% sugar solution, N ) 10 rps, uL ) 0.001 m/s, and Re ) 2600).

the same volume, where the RTD curve exponentially decays from t ) 0. In addition, the resultant variances show that the stage efficiency decreases from 0.9 to 0.45 in a six-stage reactor while it decreases from 0.91 to 0.66 in a two-stage contactor when the central opening area to cross section of the column ratio increases from 0.0161 to 0.0836. A larger central opening allows more back flow between stages, which is confirmed by many experimental findings.10,29,30 4.4. Effect of Liquid Viscosity on RTD. CFD simulations were also performed in sugar solutions with different concentrations in order to investigate the effect of liquid viscosity on RTD curves. When dealing with a viscous system, the Reynolds number with respect to the impeller falls into early turbulent regions (2000 < Re < 10 000). Therefore, the dependence of RTD predictions on different turbulent models was investigated. The utilization of the standard k- model and RNG k- model was investigated at Re ) 9167 and Re ) 2600. There is no significant difference between the predictions by the two models in two cases, as shown in Figure 11. The RNG k- model predicts a power number about 5% higher than that obtained from the standard k- model at Re ) 9167. The values of τ for the k- and RNG k- models are 191.8 and 192.0 s, respectively and their corresponding σθ2 values are 0.779 and 0.781. Similarly, the RNG k- model estimates a 4.5% higher power number than that from the k- model at Re ) 2600. The values of τ for the k- and the RNG k- models are 192.7 and 192.8 s, respectively. Compared to Re ) 9167, lower normalized variances at Re ) 2600 are obtained, 0.75 for the

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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 Table 3. Comparison of Back Flow Ratio Predicted from CFD Simulation with Experimental Data back flow ratio N, reactorsa rps

Figure 12. Comparison of RTD prediction by CFD simulation with experimental data in a six-stage reactor with 50% sugar solution (N ) 15 rps, uL ) 0.001 m/s, and Re ) 3000).

6 6 6 6 6 4 4 4 4 4 2 2 2 2 2 6* 6* 6* 6* 6† 6† 6† a

Figure 13. Effect of liquid viscosity on RTD curves in a six-stage reactor (N ) 15 rps and uL ) 0.001 m/s).

standard k- models and 0.755 for the RNG k- models, indicating that backmixing is less at a lower Reynolds number. From the comparison, it can be seen that the choice of either model does not considerably affect the prediction of RTD curves at early turbulent regions (2600 < Re < 10 000). Therefore, all simulations in sugar solutions were performed with the standard k- model otherwise stated differently. A comparison of simulation with experimental data in a MAC of six stages was conducted, and the results are shown in Figure 12. The mean residence time and variance of RTD from the CFD simulation in this case are 634 s and 0.200, respectively, while the corresponding experimental values are 618 s and 0.215 under the same operating conditions. Deviations of 2.5% and 7.5% for the mean residence time and normalized variance respectively are obtained, indicating that the RTD curve for a viscous system also can be predicted by the current approach. Varying sugar concentrations results in different RTD curves, as shown in Figure 13. It is shown in Figure 13 that, with an increase in sugar concentration, σθ2 decreases from 0.23 to 0.2, indicating that the back flow between stages in a more viscous system becomes less at the same operating conditions. The observation is in good agreement with previous studies.5,10 The peaks of the RTD curves appear slightly earlier for a less viscous system. In a water system (Re ) 3.7 × 104), it occurs at 350 s, while it appears at 365 and 375 s for 25% (Re ) 9.4 × 103) and 50% sugar (Re ) 3.0 × 103) solutions, respectively. 4.5. Backmixing Characterization. When the CTB model is utilized to characterize the liquid mixing in a MAC, a back flow ratio, back flow rate between consecutive stages over liquid bulk flow rate, is usually employed to characterize the back-

10 15 20 10 10 10 15 20 10 10 10 15 20 10 10 10 15 20 15 10 15 20

uL, m/s

turbulent kinetic energy, J/kg

calcd from CFD

exptl

0.001 0.001 0.001 0.002 0.0005 0.001 0.001 0.001 0.002 0.0005 0.001 0.001 0.001 0.002 0.0005 0.001 0.001 0.001 0.002 0.001 0.001 0.001

0.016 0.035 0.065 0.015 0.016 0.015 0.033 0.062 0.015 0.014 0.017 0.037 0.065 0.017 0.016 0.015 0.035 0.057 0.033 0.02 0.041 0.068

0.12 0.25 0.45 0.03 0.33 0.11 0.23 0.5 0 0.5 0.087 0.25 0.49 0 0.47 0.08 0.19 0.35 0.04 0.05 0.12 0.3

0.17 0.3 0.5 0.04 0.4 0.14 0.29 0.55 0.05 0.6 0.12 0.28 0.53 0.04 0.6 0.12 0.24 0.4 0.05 0.07 0.19 0.35

Magelli Xu et al.5 et al.10 0.198 0.35 0.71 0.031 0.45

0.1 0.24 0.495 0.03 0.47

0.2 0.3 0.6 0.06 0.14 0.22 0.41

0.09 0.22 0.46 0.034 0.05 0.17 0.4

/, 25% sugar system; †, 50% sugar system; others for water system.

mixing extent between consecutive stages. From the resulting normalized variance, a back flow ratio can be evaluated using the method reported in previous studies.2,29 A comparison of the back flow ratio from the CFD simulation with experimental values was also conducted, as shown in Table 3. In addition, the values predicted from the correlations proposed by Magelli et al.5 and Xu et al.10 are only compared in a six-stage contactor as shown in Table 3, since the number of stages are assumed to play a marginal role in the back flow ratio between stages in their correlations. In different contactors with different stages, either the back flow ratio from the CFD simulation or the experimental values are almost the same at the same stirring speed and liquid velocity, which is consistent with the theoretical foundation of the CTB model. With respect to the effect of viscosity on the back flow ratio, both the simulation and the experimental results show the same trend that viscosity plays a negative role in back flow, which agrees with previous studies.5,10,29 The data predicted using the relation provided by Magelli et al.5 is generally 30% higher than that provided in the current work, probably due to the fact that the central opening area over the cross-sectional area is only 1.5% in the present work while the correlations are generated based on reactors with large central openings (greater than 11%). In contrast, the correlations reported by Xu et al.10 are in relatively good agreement with the present study since the effect of central opening on backmixing has been reflected by the effect of liquid forward velocities through the opening adopted in their work. From Table 3, it is shown that the back flow ratio predicted from the CFD simulations generally is lower than the corresponding experimental value with a deviation of less than 20%. A relatively large deviation usually occurs at a low back flow ratio since it is difficult to get accurate experimental values. In this table, a global turbulence parameter, volumetric average kinetic energy, is also given under various operating conditions in the stirred vessels. In three liquid systems, the turbulent kinetic energy is significantly influenced by stirring speed and it is proportional to N2. The back flow ratio is also profoundly influenced by the stirring speed, and it has a similar relation to that of turbulent kinetic energy with stirring speed. It is apparent


that the back flow ratio holds a linear relationship with kinetic energy at the same liquid flow rate since the liquid flow rate does not show any impact on k. In contrast, the superficial liquid velocity significantly decreases the back flow ratio by its suppression effect. In addition, it is found that, under the same stirring speed and liquid flow rate, the volumetric average kinetic energy is almost the same in the three MACs of two stages, four stages, and six stages employed. As indicated in Table 3, the back flow ratio decreases from the water system to sugar solutions under the same stirring speed and liquid velocity. However, the turbulent kinetic energy does not exhibit the same behavior. The kinetic energy decreases from the water system (Re 25 000-50 000) to 25% sugar solution (Re 6470-13 000) while it increases again in 50% sugar solutions (Re 2600-5200). It is apparent that the damping effect of viscosity on turbulence is not sufficiently captured by the current two models and an advanced turbulence model is sought for prediction improvement. 5. Conclusions The CFD technique is employed to predict residence time distribution (RTD) curves for MACs with Newtonian liquids. The resulting RTD curves have been compared with experimental data. The observed mean residence time and normalized variance are in reasonably good agreement with the experimental data. The concentration profile obtained from simulations provides a line of evidence that the cascade of stirred tanks with a back flow model is more suitable to describe the flow behavior in MACs with few stages. Both simulation and experimental work show that the increase in the viscosity reduces the backmixing. However, the current k- models cannot sufficiently capture this damping effect. A more advanced turbulence model is suggested to improve CFD prediction in MACs. Acknowledgment Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and LANXESS is gratefully acknowledged. Nomenclature AD ) axial dispersion model B ) sum of body forces, N C(t) ) outlet tracer concentration at an instant, m3/s Cµ ) constant of the k- model CSTR ) continuous stirred tank reactor CTB ) cascade of stirred tanks with back flow dI ) impeller diameter, m dhb ) diameter of central opening between stages, m D ) tank diameter, m DA ) molecular diffusivity, m2/s E(t) ) residence time distribution, s-1 E(θ) ) dimensionless residence time distribution H ) height of each stage, m k ) turbulent kinetic energy, J/kg n ) number of impellers N ) stirring speed, rps Np ) power number of impeller P ) power input, W p′ ) modified pressure, Pa Re ) Reynolds number with respect to impeller, FNdI2/µ uL ) superficial liquid velocity, m/s U ) mean velocity vector, m/s T ) residence time, s

t0 ) time interval, s T ) torque, N‚m Greek Symbols ) dissipation rate of k, m2/s3 θ ) dimensionless residence time µ ) viscosity, Pa‚s µeff ) effective viscosity, Pa‚s µt ) turbulent viscosity, Pa‚s F ) density, kg/m3 σt ) turbulent Schmidt number, ratio of rate of momentum transport and mass scalar σθ2 ) normalized variance of residence time distribution τ ) mean residence time, s Φ ) tracer concentration, kg/m3 Literature Cited (1) Joshi, J. B.; Pandit, A. B.; Sharmar, M. M. Mechanically Agitated Gas-liquid Reactors. Chem. Eng. Sci. 1982, 37, 813. (2) Zhang, L. F.; Pan, Q. M.; Rempel, G. L. Liquid Backmixing and Phase Hold-up in a Gas Liquid Multistage Agitated Contactor. Ind. Eng. Chem. Res. 2005, 44, 5304. (3) Lelli, U.; Magelli, F.; Sama, C. Backmixing in a Multistage Mixer Column. Chem. Eng. Sci. 1972, 27, 1109. (4) Meister, D.; Post, T.; Dunn, I. J.; Bourne, J. R. Design and Characterization of a Multistage Mechanically Stirred Column Absorber. Chem. Eng. Sci. 1979, 34, 1367. (5) Magelli, F.; Pasquali, G.; Lelli, U. Backmixing in Multistage Mixer ColumnssII. Additional Results. Chem. Eng. Sci. 1982, 37, 141. (6) Breman, B. B.; Beenackers, A. A. C. M.; Bouma, M. J. Flow Regimes, Gas Hold-up and Axial Gas Mixing in the Gas-Liquid Multistage Agitated Contactor. Chem. Eng. Sci. 1995, 50, 2963. (7) Breman, B. B.; Beenackers, A. A. C. M.; Bouma, M. J.; Van Der Werf, M. H. The Gas-liquid Mass Transfer Coefficient (kLa) in the Gas Liquid Multistage Agitated Contactor (MAC). Chem. Eng. Res. Des. 1996, 74(A), 872. (8) Breman, B. B.; Beenackers, A. A. C. M.; Bouma, M. J.; Van Der Werf, M. H. Axial Liquid Mixing in a Gas-liquid Multistage Agitated Contactor. Chem. Eng. Res. Des. 1996, 74(A), 669. (9) Takriff, M. B.; Penny, W. R.; Fasano, J. B. Insterstage Backmixing of an Aerated Multistage Mechanically-Agitated Compartmented Column. Can. J. Chem. Eng. 1998, 76, 365. (10) Xu, B. C.; Penney, W. R.; Fasano, J. B. Interstage Backmixing for Single-Phase Systems in Compartmented Agitated Columns: Design Correlations. Ind. Eng. Chem. Res. 2005, 44, 6103. (11) Zhang, L. F.; Pan, Q. M.; Rempel, G. L. A Hydrodynamic Study and Mass Transfer Coefficient on a Multistage Agitated Contactor with Cocurrent Gas Liquid Upflow. Presented at the Symposium of 7th World Congress of Chemical Engineering, Glasgow, Scotland, July, 2005, Paper No. 05-003. (12) Ranade, V. V. An Efficient Computational Model for Simulating Flow in Stirred Vessels: A Case of Rushton turbine. Chem. Eng. Sci. 1997, 52, 4473. (13) Patwardhan, A. W. Prediction of Flow Charateristics and Energy Balance for a Variety of Downflow Impeller. Ind. Eng. Chem. Res. 2001, 40, 3806. (14) Sahu, A. K.; Kumar, P.; Patwardhan, A. W.; Joshi, J. B. CFD Modeling and Mixing in Stirred Tanks. Chem. Eng. Sci. 1999, 54, 2285. (15) Bujalski, J. M.; Jaworski, Z.; Bujalski, W.; Nienow, A. W. The Influence of the Addition Position of a Tracer on CFD Simulated Mixing Times in a Vessel Agitated by a Rushton Turbine. Chem. Eng. Sci. 2002, 80, 824. (16) Montante, G.; Mostek, M.; Jahoda, M.; Magelli, F. CFD Simulations and Experimental Validation of Homogenization Curves and Mixing Time in Stirred Newtonian and Pseudoplastic Liquids. Chem. Eng. Sci. 2005, 60, 2427. (17) Patwardhan, A. W. Prediction of Residence Time Distribution of Stirred Reactors. Ind. Eng. Chem. Res. 2001, 40, 5686. (18) Chio, B. S.; Wan, B.; Philyaw, S.; Dhanasekharan, K.; Ring, T. A. Residence Time Distribution in a Stirred Tank: Comparison of CFD Predictions with Experiment. Ind. Eng. Chem. Res. 2004, 43, 6548.

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ReceiVed for reView May 6, 2006 ReVised manuscript receiVed February 27, 2007 Accepted March 15, 2007 IE060567+

Residence Time Distribution in a Multistage Agitated Contactor with

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