13.2.2.1 Interphase Forces

13.2.2.1.1 Interphase Drag Force

In stirred reactors, bubbles experience significantly higher turbulence generated by impellers. Unless the influence of this prevailing turbulence on bubble drag coefficient is accounted, the CFD model was not found to predict the pattern of gas holdup distribution adequately. Relatively few attempts (experimental as well as numerical) have been made to understand the influence of prevailing turbulence on drag coefficient (see, for example, Refs. [18, 20–23]). Bakker and van den Akker [20], Brucato et al. [21], and Lane et al. [18] have attempted to relate the influence of turbulence on drag coefficient to the characteristic spatiotemporal scales of prevailing turbulence and therefore seem to be promising. Khopkar and Ranade [17] evaluated the three alternative proposals using a two‐dimensional (2D) CFD‐based model problem. They have observed that the predicted results deviate from the trends estimated by correlation of Lane et al. [18]. However, the predicted results show reasonable agreement with estimation based on correlation Bakker and van den Akker [20] (Eq. 13.12) and Brucato et al. [21] (Eq. 13.13), with 100 times lower correlation constant (K = 6.5 × 10⁻⁶). Interestingly, in both of these correlations, they have used volume‐averaged values of the turbulent viscosity and Kolmogorov scale, respectively:

(13.12)

(13.13)

where

• C_D is the drag coefficient in turbulent liquid.
• C_D0 is the drag coefficient in a stagnant liquid.
• d_b is bubble/particle diameter.
• λ is the Kolmogorov length scale (based on volume‐averaged energy dissipation rate).

The gas–liquid flow in stirred reactor was simulated using the drag coefficients estimated with volume average values of Kolmogorov scale (Eq. 13.13) and turbulent viscosity (Eq. 13.12) for operating conditions of Fl = 0.1114 and Fr = 0.3005. This operating condition represents L33 flow regime (large 33 cavities) in stirred reactor. The quantitative comparison of the predicted gas holdup distribution with the experimental data [19] is shown in Figure 13.2. It can be seen from Figure 13.2a and b that the gas holdup distribution predicted based on Eq. (13.12) shows fairly different gas distribution from the experimental data (shown in Figure 13.2a). The major disagreement was observed in the region below the impeller. The impeller‐generated flow was not sufficient to circulate gas in a lower circulation loop. The computational model has underpredicted total gas holdup (predicted holdup was 2.55% compared to the experimental measurement of 3.3%). The predicted results based on Eq. (13.13) are closer to the experimental data (see Figure 13.2a and c). This model resulted in overprediction of total gas holdup (predicted holdup was 3.97% compared to the experimental measurement of 3.3%). Despite the overprediction, the predicted gas holdup distribution showed better agreement with the data than predicted by Eq. (13.12). Equation (13.13) can therefore be recommended for carrying out gas–liquid flow simulations in stirred tanks.

13.2.2.1.2 Virtual Mass and Lift Force

The other two important interphase forces are virtual mass force and lift force. The virtual mass effect is significant when the secondary phase density is much smaller than the primary phase density. The effect of the virtual mass force was first studied. The predicted gas holdup distributions obtained with and without considering virtual mass force are shown in Figure 13.3a and b. It can be seen from Figure 13.3 that the influence of the virtual mass force on the predicted pattern of gas distribution was significant only in the impeller discharge stream. However, the influence of virtual mass force was not found to be significant in the bulk of the reactor. It should be noted that the value of virtual mass coefficient used in the present study (0.5) is valid for spherical bubble and may not be appropriate for wobbling bubbles. The reported value of virtual mass coefficient is somewhat higher than 0.5 (see, for example, Ref. [24]). However, it should be noted that the predicted results are not very sensitive to the consideration of virtual mass terms. A comparison of the predicted results obtained with values of virtual mass coefficients as 0 and 0.5 did not show any significant differences (see Figure 13.3). Considering this, no specific effort was made to obtain accurate value of virtual mass coefficient.

Similarly, the simulations were carried out with and without considering lift force. The predicted gas holdup distribution obtained with considering lift force is shown in Figure 13.3c. It can be seen from Figure 13.3 that the influence of lift force on the predicted pattern of gas distribution was significant in the impeller discharge stream and below impeller region. The predicted results with lift force predict a lower gas holdup in the region below impeller. In the upper impeller region of the reactor, the influence of lift force was not found to be significant. Therefore, it can be said that the modeling of lift force and virtual mass force may not be essential while simulating gas–liquid flow in stirred vessels.

13.2.2.2 Modeling Bubble Size Distribution

In a gas–liquid stirred reactor, gas bubbles of different sizes coexist. Very fine bubbles are observed in the impeller discharge stream (<1 mm), whereas bubbles of the size of few mm (~5 mm) are observed in the region away from the impeller [25]. The width of the bubble size distribution (BSD) depends upon the turbulence level and prevailing flow regime. Appropriate selection of bubble sizes is very important for the correct prediction of the slip velocity and mass transfer area. Both the slip velocity and mass transfer area can be more accurately estimated by modeling with local BSDs. Local bubble size or gas–liquid mass transfer can be estimated more accurately from local BSDs based on either a population balance [26] or by modeling the bubble number density function [18]. The bubble density function approach [18] is computationally less intensive and requires one additional equation to solve along with the two‐fluid model. This approach predicts the Sauter mean diameter at every grid node point. The predicted results of Lane et al. [18] show reasonable agreement with the experimental data of Barigou and Greaves [25]. Laakkonen et al. [26] simulated the gas–liquid flows in a stirred reactor using population balance modeling. Their simulated results are discussed here to explain the need for modeling the BSD while simulating gas–liquid flow in stirred reactors. The details of the population balance formulation, bubble breakage, and coalescence model are not discussed here and can be found in Laakkonen et al. [26]. The influence of the prevailing turbulence on the interphase drag force was modeled with a slightly modified Bakker and van den Akker [20] correlation. The comparison of the predicted BSD and the mean bubble diameter with the experimental data is shown in Figure 13.4. The following conclusions can be drawn from the comparison between predicted results and experimental data:

• The parameters of the coalescence and breakage models were tuned to fit the experimental measurements. This limits the applicability of the model for different configurations of stirred reactor.
• The tails in the predicted volume BSDs are larger compared to the experimental measurements indicating underprediction of breakage process. The rate of breakage process is dependent on the predicted values of the turbulent energy dissipation rate. The CFD model underpredicts the turbulent kinetic energy dissipation rates and hence led to lower rate of bubble breakage process.
• The enormous requirement of computational requirement for multi‐fluid model does not allow modeler to use fine mesh for simulating the turbulent multiphase flow. The use of a relatively coarse mesh significantly contributes to the underprediction of turbulent properties and hence influences the predicted breakage and coalescence rates.

The present state of understanding of the breakage and coalescence processes and the unavailability of experimental data for different reactor configurations suggest that it may not be advantageous to use population balance‐based multi‐fluid models while simulating industrial gas–liquid stirred reactors. It may be more effective to use effective combination of bubble diameter and interphase drag coefficient to get realistic results.

13.2.2.3 Gas Holdup Distribution in L33, S33, and VC Flow Regimes

Gas–liquid flows generated by the Rushton turbine in a stirred vessel were simulated for other two flow regimes representing S33 (Fl = 0.0788; Fr = 0.6) and VC (Fl = 0.026 267; Fr = 0.6). As discussed previously, Eq. (13.13), based on volume‐averaged dissipation rate and Kolmogorov scale (λ), was used to calculate effective drag coefficients. Comparisons of predicted gas holdup distributions with the experimental results at the mid‐baffle plane are shown in Figure 13.5. It can be seen from these figures that the predicted gas holdup distributions for S33 and VC flow regimes are in reasonably good agreement with the experimental data. However, the computational model overpredicted the values of total gas holdup. The predicted value of total gas holdup (4.85%) was higher than the reported experimental value (4.2%) for the S33 flow regime. Similarly, the predicted value of total gas holdup (2.63%) was higher than the experimental data (2.2%) for the VC flow regime.

Comparisons of axial profiles of radially averaged gas holdup for all three regimes are shown in Figure 13.6. It can be seen from Figure 13.6 that the computational model overpredicts the values of gas holdup in the region above the impeller for all three regimes. The maximum value of predicted radially averaged gas holdup occurs at an axial distance of 0.117 m for L33 and 0.107 m for S33 as well as VC regimes compared with the experimentally observed distance of 0.13 m for L33 and 0.1125 m for S33 as well as VC regimes. The predicted values of gas holdups at this maximum are underpredicted (7.3% for L33, 7.94% for S33, and 3.82 for VC) compared with the experimental value (8.1% for L33, 8.8% for S33, and 4.1% for VC). Quantitative comparisons of angle‐averaged values of predicted gas holdup and experimental data at three different axial locations for all three regimes are shown in Figure 13.7. It can be seen from Figure 13.7 that comparisons of the predicted values of gas holdup and experimental data are reasonably good for all three regimes. The computational model was thus able to simulate all three regimes reasonably well.

13.2.2.4 Gross Characteristics

Predicted influence of the gas flow rate on gross characteristics, power, and pumping numbers is also of interest. Pumping and power numbers were calculated from simulated results as

(13.14)

(13.15)

where

• B is blade height.
• D_i is impeller diameter.
• N is impeller speed.
• r_i is impeller radius.
• U_r is radial velocity.

The calculated values of pumping and power number from the simulated results are listed in Table 13.3. As the gas flow rate increases, impeller pumping as well as power dissipation decreases. The extent of decrease increases with an increase in the gas flow rate (or in other words, as flow regime changes from VC to S33 and further to L33). Bombac et al. [19] have not reported their experimental values of power dissipation or pumping number. In the absence of such data, the predicted values were compared with the estimates of empirical correlations proposed by Calderbank [27], Hughmark [28], and Cui et al. [29]. While demonstrating the qualitative trend, the CFD model underpredicts the decrease in power dissipation in the presence of gas compared to the estimates of these correlations. CFD model, however, could correctly capture the overall gas holdup distribution and can therefore simulate different flow regimes of gas–liquid flow in stirred reactors.

13.2.3.1 Interphase Drag Force

In stirred reactors, particles experience significantly higher turbulence generated by impellers. Similar to the discussion included in previous subsection, unless the influence of this prevailing turbulence on particle drag coefficient is accounted, the CFD model will not predict the solid suspension adequately. Khopkar et al. [37] evaluated the two alternative proposals [21, 39] using a 2D CFD‐based model problem. They have observed that the predicted results deviate from the trends estimated by correlation of Pinelli et al. [39]. However, the predicted results show reasonable agreement with estimation based on correlation by Brucato et al. [21]. They correlated the predicted results by considering the sole dependence on d_p/λ for a range of solid holdup values (5 < α < 25%). They observed that the predicted results require ten times lower proportionality constant (K = 8.76 × 10⁻⁵) in Eq. (13.13) as compared with that proposed by Brucato et al. [21].

Solid–liquid flow generated by the Rushton turbine has been simulated for a solid volume fraction equal to 10.0%, d_p = 264 μm, and at an impeller rotation speed N = 20 rps. Both the formulation of drag coefficient and actual Brucato et al. [21] (K = 8.76 × 10⁻⁴) and the modified Brucato correlation (K = 8.76 × 10⁻⁵) were used for the evaluation of the interphase drag force formulation. The value of dispersion Prandtl number, σ_pq, has been set to the default value of 0.75. The predicted solid holdup distributions by using both drag coefficient formulation at the mid‐baffle plane are shown in Figure 13.10a and b. It can be seen from Figure 13.10a that with actual Brucato et al. [21] correlation, the computational model has predicted almost complete suspension of the solid particles. However, the simulated solid holdup distribution using modified Brucato et al. [21] did not capture the complete suspension of solid particles in stirred reactor (see Figure 13.10b). The simulated solid holdup distribution shows the presence of solid accumulation at the bottom and near the axis of the reactor. For quantitative comparison the predicted solid concentrations/holdups were compared with the experimental data of Yamazaki et al. [38]. The quantitative comparison of the azimuthally averaged axial profile of solid holdup at a radial location (r/T = 0.35) is shown in Figure 13.10c. It can be seen from Figure 13.10c that the computational model with drag coefficient formulation of Brucato et al. [21] has overpredicted the solid suspension height. However, the suspension height predicted by the modified Brucato correlation is in good agreement with the experimental data. It can also be seen from Figure 13.10 that solid holdup distribution predicted with the use of modified Brucato et al. [21] correlation has captured the presence of higher solid concentration in the impeller discharge stream (a bell shaped in the concentration profile), which is a characteristic of the solid–liquid flow generated by Rushton turbine. However the prediction with Brucato et al. [21] correlation does not show any such characteristics. Overall, it can be said that the modified Brucato et al. [21] correlation predicted solid holdup distribution in stirred vessel more accurately.

13.2.3.2 Turbulent Dispersion Force

The developed computational model was then extended to study the influence of the turbulent dispersion force on the suspension quality in the stirred reactor. The magnitude of the turbulent dispersion force was varied by varying the value of the dispersion Prandtl number, σ_pq, in the range of 0.0375–3.75. Figure 13.11 shows the comparison of the predicted solid concentration distribution in the stirred reactor with the experimental data of Yamazaki et al. [38] with and without turbulent dispersion force. It can be seen from Figure 13.11 that the turbulent dispersion has a significant effect on the predicted suspension quality in the stirred reactor. The computational model predicted a more uniform suspension with a decrease in the value of dispersion Prandtl number. This is expected as the drift velocity (or turbulent dispersion force) is inversely proportional to the dispersion Prandtl number (see Eq. (13.3)). Decreasing the latter means increase in the turbulent dispersion force, which consequently results in more dispersion of the particles, resulting in more uniform suspension. Overall, it can be said that the simulations carried out with σ_pq = 0.375 and 0.0375 have overpredicted the suspension quality. However, for σ_pq = 3.75 the computational model has underpredicted the suspension quality. Therefore, CFD simulation of solid–liquid stirred reactor needs to be carried out with σ_pq = 0.75 for adequate prediction of suspension quality.

13.2.4.1 Solid Suspension in an Aerated Stirred Vessel

Minimum speed for JS is a very important hydrodynamic parameter for designing gas–liquid–solid stirred reactor. Experimental studies so far on gas–liquid–solid suspensions have clearly indicated the requirement of increased suspension speed, thereby more power input, on the introduction of gas [40–43]. This is because of a decrease in impeller pumping efficiency and power draw due to the formation of ventilated cavities behind the impeller blades on gassing [46]. Recently, Zhu and Wu [47] carried out experimental measurements in a three‐phase stirred reactor to determine the JS speed for a variety of solid sizes, solids loading, impeller sizes, and tank sizes. They suggested the possibility of relating relative just off‐bottom suspension speed (RJSS) with just suspension aeration number (based on just suspension speed for solid–liquid system). They also observed that the proposed relation was independent of impeller size, solid size, solids loading, and tank size and can be used to scale up laboratory data to full‐scale vessel. The same definition (Eq. 13.16) was used in the present study to identify the JS speed for different gas flow rates:

(13.16)

where

• m and n are constants.

For the Rushton turbine, the values of m and n are 2.6 and 0.7, respectively. The simulations were carried out for three just suspension aeration numbers: 0, 0.025, and 0.05. The impeller rotational speeds (N_jsg) for the three aeration numbers are 9.30, 11.16, and 12.27 rps, respectively.

The predicted solid holdup distribution at mid‐baffle plane for all three aeration numbers is shown in Figure 13.13. It can be seen from Figure 13.13 that for three‐phase system the computational model has predicted more accumulation of solids at the bottom of reactor near the central axis in comparison with the two‐phase system. The predicted cloud height values were also found to drop in the presence of gas. The predicted solid volume fraction values were then used to describe the suspension quality in the reactor. The criterion based on the standard deviation value, calculated using Eq. (13.17), was used to describe suspension quality for all three cases. It was observed that the computational model predicted standard deviation value (σ) equal to 0.45 for two‐phase flow. However, for three‐phase flow, computational model predicted σ equal to 0.82 and 0.90 for Na_js equal to 0.025 and 0.05, respectively. Overall, it can be said that the computational model has predicted JS condition for two‐phase flow (σ < 0.8), but incomplete suspension for three‐phase system (σ > 0.8):

(13.17)

The predicted gas holdup distribution at mid‐baffle plane for two suspension aeration numbers is shown in Figure 13.14. It can be seen from Figure 13.14 that for both conditions, the computational model has predicted higher values of gas holdup in both circulation loops of flow. This indicates that the computational model predicted complete dispersion condition of gas phase in the vessel. These predicted results also support the experimental observations made by Dyalag and Talaga [40] on quality of gas dispersion.

13.2.4.2 Gross Characteristics

Predicted influence of the gas flow rate on gross characteristics, power number, gas holdup, and suspension quality is also of interest. The calculated values of power number, gas holdup, and standard deviation from the simulated results are listed in Table 13.4. Few conclusions can be drawn from Table 13.4. First, the computational model has predicted the drop in impeller power number value in the presence of gas. While demonstrating the qualitative trend, the CFD model has underpredicted the actual power number value (predicted value of power number for single‐phase flow equal to 3.85). Second, the CFD model was able to predict just suspension condition for solid–liquid flows. However, the model failed to predict just suspension condition in the presence of gas. The standard deviation value (describing suspension quality) increases with an increase in gas flow rate. For lower aeration rate the model has predicted the total gas holdup values reasonably well. However, model has underpredicted total gas holdup value for higher aeration rate. Overall, CFD model with the presented modeling approach could reasonably predict gas–liquid–solid flow in a stirred reactor at low aeration rates. Further work is needed to develop adequately accurate model capable of simulating gas–liquid–solid flow in stirred reactor at higher aeration rates.

13.3.1.1 Shortstop of Runaway Reaction Through Reaction Inhibition

Classical reactor analysis and design engineering commonly assume one of two idealized flow patterns: plug flow or completely back‐mixed flow. Real reactors may approach one of these; however, it is often the nonidealities and their interaction with chemical kinetics that lead to poor reactor design and performance [63]. For well‐designed batch stirred tank with simple reaction schemes and kinetics that are slow relative to the mixing time, the completely mixed flow (CMF) works well. But in many practical situations, the kinetics is relatively faster than the mixing process. In such situations, addition location of reactant or additive and prevailing fluid dynamics plays very important role. Validated computational models play a very important role not only in elucidating the implications of reaction kinetics, mixing time, and additive addition or inlet/outlet locations but also in designing effective operating protocols or desired design modifications for meeting the process requirement. To explain the influence of addition location and amount of additive on the mixing‐controlled reaction performance, we reproduce some results obtained by Dakshinamoorthy et al. [64].

Dakshinamoorthy et al. [64] used CFD‐based model to understand the role of imperfect mixing on shortstopping of a runaway reaction in a fully baffled stirred reactor. The propylene oxide (PO) polymerization runaway reaction in a stirred vessel was considered as a model problem. The polymerization of PO is catalyzed by potassium hydroxide. The reaction rate is proportional to temperature and concentrations of the monomer and catalyst. Adding an acid to neutralize the basic catalyst inhibits the runaway reaction.

A CMF model is conventionally used in practice to study the runaway and inhibition of runaway reactions. The application of a CMF model for developing operating protocols may be adequate when the characteristic runaway time is much greater than the mixing time of the reactor. When the characteristic runaway time is smaller or comparable to the mixing time, the CMF model cannot give reliable and useful information on the shortstopping process. To demonstrate this, they simulated a case with an operational constraint such as late detection of the runaway and hence delayed addition of inhibitor. The predicted temperatures from the CMF model for this case indicated that the late detection of the runaway at an average reactor temperature 450 K allows a design engineer only 60 seconds to control the reactor by inhibiting the runaway reaction. The time available (60 seconds) compared with the vessel mixing time of 450 seconds is very small and makes it more difficult and challenging to shortstop, due to imperfect mixing of the inhibitor present in the reactor. In such situations, the addition location of inhibitor and its quantity play important role.

Dakshinamoorthy et al. [64] investigated the influence of three different addition locations on the performance. Figure 13.16 shows the inhibitor addition locations considered in their study. For the first inhibitor addition location, the inhibitor was added just below the top surface of the liquid (see Figure 13.16a). For second addition location, the inhibitor was added in the impeller discharge stream to facilitate faster mixing of the inhibitor (Figure 13.16b). The selection of the third location was chosen after the analysis of the predicted temperature distributions of the first two addition locations. In this case, the inhibitor is distributed to both of the addition location: 75% of the inhibitor is added in the impeller discharge stream, and the remaining 25% is added at the top surface (see Figure 13.16c). The total quantity of inhibitor added in the reactor was first kept constant for all three additions and was added in the reactor when the reactor temperature reached to 450 K (delayed addition) due to runaway reaction.

The computational model was then used to simulate the inhibition of the runaway reaction for all three addition locations. They recorded the average reactor temperature for all the three cases and compared with the average reactor temperature predicted using CMF model (see Figure 13.17). They observed that the addition location has significant impact on the predicted reactor temperature and the average reactor temperature approaches to the CMF model predictions for the third addition location.

The simulated contour plot of temperature for all the three cases was then used for identifying the implications of addition location. Figure 13.18 shows the comparison of the predicted temperature distributions. They observed quite different distributions of the high temperature zones for the first and second inhibitor addition locations. The first location showed a high temperature zone in the bulk volume of the vessel, but no hotspots were predicted near the impeller shaft. Also the highest temperature predicted near the top surface was found to be lower than the temperature predicted for the second location. For the third inhibitor addition location, the final reactor temperature (475 K) was found to be slightly higher than the CMF results (470 K). The predicted temperature distribution at the mid‐baffle plane shows the advantage of adding the inhibitor at both locations with substantial reduction in the hotspot volumes compared with the addition at a single location.

The performance of the inhibition process was substantially improved for the third addition location. However, some part of reactor was still showing high temperature regions with temperatures greater than 500 K. In order to avoid such high temperature regions, they simulated an additional case of excess inhibitor addition for third addition location. The predicted average reactor temperature, as the reaction proceeds, with an inhibitor amount corresponding to two and half times that of the initial simulations is shown in Figure 13.17. For this addition they found out that the reactor behaved almost like a completely mixed reactor with a final average temperature of 465 K similar to the temperature predicted by CMF model. The predicted temperature distribution shown in Figure 13.18d demonstrated almost uniform temperature distribution within the vessel.

13.3.1.2 Continuous Flow Stirred Vessel

In many practical applications, continuous flow system is used. Measurement of RTD is the only mixing tool covered in the undergraduate studies. It is a well‐known method for assessing the nonideality of continuous process equipment. RTD is a concept first developed by Danckwerts in his classic 1953 paper [65]. In RTD analysis, a tracer is injected into the flow, and the concentration of tracer in the outlet line is recorded over time. From the concentration history, the distribution of fluid residence times in the vessel can be extracted. Information on the mixing nonidealities such as channeling, bypassing, and dead zones are then extracted from the measured RTD curve.

Usually, continuous operation of a stirred vessel is considered as almost ideal when the ratio of residence time to mixing time is about 10. But in many instances, the performance of stirred vessel deviates from the ideal. The mixing performance of the vessel is function of complex interaction between hardware, operating conditions, and the prevailing fluid dynamics. Experimental testing of different configurations of these interactions will be time consuming and expensive. In addition to that, the other weakness of experimental measurement of RTD analysis is that from the diagnostic perspective, an RTD study is based on the injection of a single tracer feed, whereas real reactors often employ the injection of multiple feed streams. In real reactors the mixing of separate feed streams can have a profound influence on the reaction. It is possible to avoid such limitations by using computational models. The computational models provide an opportunity to mathematically simulate mixing of multiple species entering into the reactor through multiple inlets. Such simulations can also help in simulating inlet plume interaction and their influence on the process and product yield.

Khopkar et al. [66] used CFD‐based model for simulating RTD in a continuous flow stirred vessel equipped with Mixel TT impeller. The simulated geometry is shown in Figure 13.19. They simulated the flow in a continuous flow reactor having residence time to mixing time ratio equal to 9.6 (N = 360 rpm and Q_l = 2.016 67 × 10⁻⁴ m³/s) closer to the standard value of 10 commonly used in practice for determining the ideal performance behavior. The predicted exit age distribution is shown in Figure 13.20. From the overshoot in tracer concentration observed at the outlet, it appears that the high velocity inlet jet may be interacting directly with the outlet. This was further confirmed by the lower slope of predicted RTD curve compared with that of ideal CSTR. The combination of overshoot at the beginning and lower slope at later stage indicates that part of the incoming fluid bypasses stirred vessel and flows effectively through a small volume plug flow reactor and the remaining part of the incoming fluid flows through a stirred vessel with much larger effective residence time than that calculated from the total incoming flow (Figure 13.20b). The nature of predicted exit age distribution they modeled was a combination of ideal stirred reactor and plug flow reactor operating in parallel. The analysis indicated that the effective residence time of the ideal CSTR part is about 68 seconds (this means only about 36% of the incoming liquid flows through a vessel and about 64% of the incoming fluid short‐circuits to the outlet). They further studied the influence of impeller speed on the exit age distribution. They found significant influence of impeller speed on the RTD. Even at very high ratio of residence time to mixing time (=19.2), they found 31% of feed getting bypassed in the selected vessel configuration.

Khopkar et al. [66] then numerically simulated various inlet and outlet configurations of the same vessel. Particle streak lines were simulated by releasing neutrally buoyant tracer particles from the inlet pipe. The simulated particle streak lines (for a flow time of five seconds) for different configurations are shown in Figure 13.21. Particle streak lines shown in Figure 13.21 help in understanding the complex interaction of incoming feed with impeller‐generated flow and outlet.

Khopkar et al. [66] used quasi‐steady‐state approach with multiple snapshots for simulating the flow in an industrial continuous flow stirred tank reactor. They observed that multiple numbers of snapshots are needed to adequately represent the flow generated by an impeller and the interaction of the incoming liquid jet with the impeller would depend on the design of the impeller. They further recommended that after establishing the flow, detailed species transport equations with any one of the snapshots could be used for simulating RTD at the outlet. However, the mixing simulation with quasi‐steady‐state approach does not accurately capture the interaction of the incoming jet with the rotating impeller, but the obtained results are found to be sufficient for the “a priori” suggestion for the evaluation of the different design configuration.

13.3.1.3 Viscous Mixing

Formulations are widely used in pharmaceutical, paint, food, polymer, and personal care products. Some of the most difficult mixing problems do get encountered in making these formulations. In many situations these formulations exhibit highly viscous, non‐Newtonian, and viscoelastic fluids. With all non‐Newtonian fluids, the potential exists that a portion of a tank will remain unmixed because of inadequate fluid motion. Because of both the high viscosity and non‐Newtonian behavior, special equipment is often required for mixing.

Mixing can be brought about in viscous systems only by mechanical action or by the forced shear or by elongational flow of the matrix. Mixing achieved through the history of deformation imparted to the fluid is called distributive mixing. In distributive mixing the homogeneity of the mixture can be quantified through the scale of segregation [67], whereas in dispersive mixing a consequence of the history of the fluid mechanical stresses is imposed on the mixture for breakup of agglomerates and drops. A dispersive mixing index can be quantified through strength of the pure elongational flow. The interrelationship between dispersive and distributive mixing is illustrated in Figure 13.22. In general, viscous mixing operations require some combination of dispersive and distributive actions. Before explaining some examples of viscous mixing, basic definitions of distributive and dispersive mixing are explained in the following subsection. These definitions will help in quantifying mixing in the process equipment.

13.3.1.3.1 Measures of Distributive and Dispersive Mixing

As said earlier the purpose of the distributive mixing is to create uniform macrostructure. It is created due to the presence of chaotic motion in the deterministic laminar flows [68]. A variety of tools have been developed to examine and characterize chaotic flows including Poincare sections, periodic point analysis [69], and stretching distributions [70]. These tools do provide rich insight into the nature of chaotic flows and reveal the mechanism of chaotic mixing, but do not provide tools to quantify the distributive mixing behavior. Tucker III and Peters [71] numerically simulated mixing in a 2D time‐periodic flows in a rectangular cavity with upper and lower moving walls (see Figure 13.23). The mixing pattern was simulated using mapping method, which has a finite spatial resolution and certain amount of numerical diffusion. A variety of protocols involving time‐periodic sliding motions of the upper and lower cavity surfaces were studied. For quantifying distributive mixing, two measures of distributive mixing were proposed and examined: the standard deviation among samples σ and the maximum sample error E. Both these measures are closely related to the distributive mixing and small value of either means that the cell concentrations are nearly uniform:

(13.18)

(13.19)

(13.20)

where

• V_avg is the averaging volume.
• c_avg is the final uniform concentration.
• L is the length scale of the averaging volume.
• M is the number of points in the mixer separated by the distance L.

Tucker III and Peters [71] observed that E and σ both provide the ability to compare the mixture patterns that may be quite different in appearance and to say which mixture is better distributed.

Dispersive mixing is usually more difficult to achieve than the distributive mixing. Dispersive mixing involves breakup of agglomerates and droplets in flow, caused by stresses large enough to overcome the cohesive or interfacial forces that tend to keep the agglomerates or the droplets intact. Two main mechanisms are responsible for dispersive mixing, viz. shear flow and elongational or extensional flow. Quantitative studies of droplet breakup in simple shear and pure elongational flows have shown that the elongational flow is more effective than the simple shear flow especially in the case of high viscosity ratios and low interfacial tensions. This is been clearly explained in Figure 13.24 where the Weber or capillary number is plotted again the viscosity ratio. Based on Figure 13.24, the minimum dispersed phase drop radius can be achieved where the viscosity p is close to unity, but the dispersion by shear flow is not possible if p exceeds 4. This limit could be different for viscoelastic fluids.

Accurate estimation of dispersive mixing efficiency would involve tracking of the dispersed phase during their entire residence time in the equipment and following the dynamics of their breakup and coalescence. However, such approach is numerically very expensive. A simpler global approach is required, which will help in discriminating between various designs and processing conditions for effective mixing equipment selection. A flow field characteristic relevant to dispersive mixing is known as flow strength [73, 74]. The flow strength is function of rate of deformation tensor and Jaumann time derivative of rate of deformation tensor. The flow strength parameter ranges from zero for pure rotational fluid to infinity for pure elongational flow; its value is unity for simple shear flow. However, numerically determining flow strength for complex geometry is difficult. Yang and Manas‐Zloczower [75] defined dispersive mixing index λ that quantifies the relative strength of the pure elongational flow component:

(13.21)

where

• |D| and |ω| are the magnitudes of the rate of strain and vorticity tensors, respectively.

The above parameter assumes values between 0 for pure rotation and 1 for pure elongation, with a value of 0.5 for simple shear. This mixing index is not frame invariant, but still can be used as first approximation to discriminate between various equipment designs and processing conditions [75, 76].

13.3.1.3.2 Examples of Viscous Mixing

It has been well established that the chaos is necessary for achieving efficient mixing in laminar stirred vessel [77]. Chaotic mixing is characterized by the exponential rate of stretching and folding of fluid elements. To illustrate this, Arratia et al. [78] used the planar laser‐induced fluorescence (PLIF) snapshot through the axis of the mixing tank filled with Newtonian fluid and stirred with concentric disks and Rushton turbines. Figure 13.25a demonstrates that the fluid does not mix down to a very small length scale. No sign of chaotic mixing was observed. Flow was characterized by a linear rate of stretching. Later, they induced chaos in the system by replacing the disk by Rushton turbines. They found out that the passage of blades periodically perturbed the flow and triggered the stretching and folding of fluid material, which helped in achieving efficient mixing (Figure 13.25b).

Zalc et al. [79, 80] used the computational model for simulating the chaotic mixing in the stirred vessel equipped with multiple Rushton turbines. They computed flow field using the ORCA software (Fujitsu, Campbell, CA). Extensive comparison of simulated results was done with the PIV and PLIF measurements. Figure 13.26 shows the comparison of experimentally measured flow field with numerical results. CFD results accurately predicted size and location of the poorly mixed regions in the stirred vessel. Zalc et al. [79, 80] carried out quantitative measure of mixing intensities in chaotic flows by computing the accumulated stretching of small fluid filaments. These simulations were performed by placing small vectors in the flow. The deformation of each infinitesimal vector by the instantaneous velocity gradient along its trajectory while being convected throughout the flow domain was calculated for quantifying stretching value (λ). Figure 13.27 illustrates the stretching field for the different Reynolds number values 20, 40, and 160 after 20 revolutions. The contour maps reveal the heterogeneity of stretching and subsequently help in identifying poor mixing regions in the stirred vessel. The contour plot also shows the change in the poor mixing region structure with impeller Reynolds number. A priori knowledge of such stretching distribution pattern not only helps in identifying the poor mixing regions but also provides guideline in deciding the injection point for additives to achieve optimum distribution.

In order to improve the chaotic mixing, few approaches were suggested in literature, viz. using variable agitation speed instead of constant agitation by Szalai et al. [81], using eccentrically mounted impellers [77, 78, 82], and using dual shaft mixers [83]. Cabaret et al. [83] experimentally studied the mixing kinetics in a laminar flow stirred vessel equipped with multiple Rushton turbines. They used color discoloration method for quantifying the mixing in vessel. They observed that the mixing in a dual shaft stirred vessel is more efficient in comparison with centrally mounted shaft and off‐centered shaft systems (see Figure 13.28 for details). Dual shaft system also provides an additional degree of freedom, i.e. direction of rotation for influencing prevailing fluid dynamics. Cabaret et al. [83] observed better mixing in counterrotating mode as compared with corotating mode of operation.

Nowadays, the industry needs impellers that can work in laminar, transitional, or turbulent regimes with minimum modifications. Standard agitators like close‐clearance and open impellers exhibit some limitations with this aspect. On one hand, close‐clearance impellers such as helical ribbons have a good distributive mixing performance in laminar regime. However, this situation is completely reversed when the condition changes from laminar to transitional or turbulent. Also, they do not generate sufficient dispersive mixing that is essential for improving mixing on smaller scale. On the other hand, open impellers like the Rushton turbine or pitched blade turbine are known to be very efficient at high Reynolds number, but in laminar regime, segregated zones are produced. The situation becomes critical if along the process time the phases to be mixed develop non‐Newtonian rheological properties such as shear thinning.

Recently, several innovative strategies have been proposed to tackle this problem. Multi‐shaft mixers with bulk flow impeller mounted on one shaft and open impeller or high‐shear mixer mounted on second shaft provide freedom in achieving desired level of mixing (see, for example, Refs. [84–87] and so on). The main idea is simple – association of different classes of agitators rotating at different speeds. In this way it is possible to create a mixer that achieves the process objectives – “blending” the capabilities of several agitators. At the end, a dynamic mixing unit that adapts with the process necessities is obtained. Scaling up of these equipments is still a challenge as not sufficient experimental data is available in the literature. Also, the available data does not cover all the possible degrees of freedom available with the designs of the multi‐shaft mixers. Validated CFD model can help us in filling these gaps with the experiments. Rivera et al. [86] simulated flow in a coaxial mixer using POLY3D^TM (Rheosoft, Inc.). They simulated the flow generated in a coaxial mixer equipped with anchor impeller on one shaft and Rushton turbine on second shaft. They correlated the quality of distributive and dispersive mixing with the ratio of flow number N_q and head number N_h. Their results suggest that the corotating mode has better distributive mixing than counterrotating mode for both Newtonian and non‐Newtonian fluids (see Figure 13.29). They also observed that the predicted value of ratio N_h/N_q is smaller for corotating mode as compared with counterrotating mode (see Ref. [86] for more details).

13.3.2.1 Crystallization in Stirred Vessel

In pharmaceutical industry, crystallization and precipitation are widely used processes in manufacturing of various drug molecules. Precipitation and crystallization refer to unit operations that generate a solid from a supersaturated solution. The nonequilibrium supersaturated condition can be induced in a variety of ways such as removal of solvent by evaporation, addition of another solvent, changes of temperature or pressure, addition of other solutes, oxidation–reduction reactions, or even combinations of these. The distinction between precipitation and crystallization is quite often based on the speed of the process and the size of the solid particles produced. The term precipitation commonly refers to a process that results in rapid solid formation that can give small crystals that may not appear crystalline to the eye, but still may give very distinct X‐ray diffraction peaks. Amorphous solids (at least as indicated by X‐ray diffraction) may also be produced through precipitation. The term precipitation also tends to be applied to a relatively irreversible reaction between an added reagent and other species in solution, whereas crystallization products can usually be redissolved using simple means such as heating or dilution. Precipitation processes usually begin at high supersaturation where rapid nucleation and growth of solid phases occur. In both precipitation and crystallization processes, the same basic steps occur: supersaturation, nucleation, and growth.

Supersaturation affects both crystal growth and nucleation rates, which in turn impact the particle size distribution. A higher level of nucleation leads to smaller particles and vice versa. Also, a high degree of nucleation rate over crystal growth rate due to a high degree of supersaturation can lead to poorer rejection of impurities [88]. Nucleation does not necessarily begin immediately on reaching a supersaturated condition, except at very high supersaturation, and there may be an induction period before detection of the first crystals or solid particles.

Tung [88] used these time scales with the liquid‐phase mixing time for explaining complexity of the process. He defined two dimensionless numbers, viz. Da_nucleation and Da_{crystallization}:

(13.22)

(13.23)

where

• τ_mixing is the mixing time.
• τ_induction is the induction time for primary nucleation.
• τ_{release of supersaturation} is the time required to release the supersaturation.

Da_nucleation ≪ 1 means a complete mixing before the nucleation is achieved. Similarly, Da_{crystallization} ≪ 1 means a complete mixing is achieved before the supersaturation is released. This would be the case for crystallization with relatively slow crystallization kinetics in releasing the supersaturation. Since the order of mixing time in a crystallizer is generally available, it is straightforward to learn if the crystallization system could be sensitive to mixing by comparing the induction time for nucleation or time for release of supersaturation. This suggests that the liquid‐phase mixing time plays a very important role in the crystallization.

It is important to recognize the influence of mixing on the product characteristics. In general mixing is defined at three scales, viz. macromixing, mesomixing, and micromixing. Macromixing is defined on the scale of the vessel, mesomixing is defined in context of dispersion of the antisolvent plume at the feed point, and micromixing determines the mixing at the molecular level and influences the induction time. Micromixing is influenced by the impeller type and speed plus relative location of the antisolvent feed pipe with respect to impeller. Thus, feed pipe location, pipe diameter, and antisolvent flow can impact both micromixing and mesomixing times. For example, when the antisolvent feed rate is faster than the local mixing rate, resulting in a plume of highly concentrated antisolvent that is not mixed at the molecular level. This can yield a high localized nucleation rate. This will present scale‐up difficulties, requiring a thorough engineering analysis for success.

The modeling of well‐mixed crystallizers involves the computation of the PBE together with the material balance equations for each species in solution. Numerous numerical techniques that compute the full crystal size distribution (CSD) have been used to model well‐mixed batch, semi‐batch, or continuous crystallizers. In most of the studies, the actual prevailing mixing condition was not simulated and is approximated with ideal mixing behaviors. For a more realistic simulation, it is necessary to solve the standard momentum and mass transport equations together with the PBE coupled flow simulation. Rielly and Marquis [89] provided an explanation on the pivotal role of fluid dynamics on the kinetics of crystallization and resulting CSD. Figure 13.30 explains how different scales of crystallizer fluid dynamics influence the crystallization process. Rielly and Marquis [89] built a computational model where particle or crystal motion was simulated in Lagrangian framework. They observed that the crystal particle experiences region with very different micro‐flow properties and it can be explained through the variation in the distribution of instantaneous slip velocities. They also concluded that the distribution of the variation in slip velocities experienced by the crystals is strongly dependent on the particle microscale and macroscale Stokes number defined by the two flow time scales, viz. lifetime of turbulent eddy and circulation time or mixing time. They also observed that neglecting the effects of microscale flow properties significantly reduces the variance of the macroscale time scale.

In all the studies available on the CFD modeling of crystallizer, the presence of solids is modeled by treating the slurry as a pseudo‐homogeneous fluid with a spatial distribution of effective viscosity that depends on the local solid fraction. Thus, they do not exactly simulate the flow essential for estimating accurate microscale flow properties that will further influence the macroscale flow time scales. There are multiple reasons for not yet development of a comprehensive computational model for simulating whole crystallization process. First, the computational model for simulating hydrodynamics of slurry is not yet fully validated. Second, although certain progress has been made in simulating micromixing effect on the crystallization kinetics, other mechanisms such as secondary nucleation, agglomeration, and breakage are yet to be modeled and validated. Third, most important reason is huge computational requirement for coupling algorithms of multiphase flow and crystallization kinetics. It is therefore necessary to validate these models separately and develop an innovative approach for combining the results of these models for effective design of crystallization process.

13.3.2.2 CFD Simulation of Solid Suspension in Stirred Vessel

In the previous section a detailed discussion on the computational modeling of stirred slurry reactor is presented. The developed model was evaluated by comparing results for solid volume fraction profiles averaged over reactor cross section. A close look at the prevailing fluid dynamics in the crystallizer shows very complex behavior. The average density and viscosity continuously changes due to formation and growth of solids. There is a possibility of solid suspension mechanism demonstrating various transient phenomena in crystallizer. These transients will definitely influence the transport processes. It is therefore necessary that the computational model must predict the transients associated with the slurry suspension and its implications on the liquid‐phase mixing. Sardeshpande et al. [90] observed the hysteresis in the suspension quality with respect to impeller speed. Figure 13.31 shows the visual observation of hysteresis observed by Sardeshpande et al. [90]. Their data also suggest that the observed hysteresis is more profound at higher solids loading.

Sardeshpande et al. [90] used computational model with different interphase drag force formulations for simulating the sudden hysteresis observed in cloud or suspension height. Figure 13.32 shows the comparison of the cloud height predicted using Brucato et al. [21] drag law and modified Brucato's drag law recommended by Khopkar et al. [38]. They observed that for both increasing and decreasing impeller rotational speed, the computational model with Brucato's drag law overpredicted the cloud height. However, the computational model recommended by Khopkar et al. [37] reasonably captured the cloud height for both increasing and decreasing impeller rotational speed conditions. Ability of computational model to predict hysteresis enhances the capability of process engineer for a priori controlling the performance of slurry reactor, where small deviation in the suspension quality could influence the final product properties through different rates of transport and mixing rates as well as different particle–particle interaction.

13.3.2.3 Solid Suspension and Mixing in Stirred Reactor

Liquid‐phase mixing is quite important in many solid–liquid reactions as well. It not only affects the selectivity of reactions but also controls the temperature distribution inside the reactor for exothermic reactions. In many cases, stirred reactors are operated with higher solids loading (solid volume fraction >5.0%). In such situations, the liquid‐phase mixing process was found to show a complex interaction with the suspension quality (for example, Ref. [91]). A computational model, which is able to predict suspension quality and its influence of liquid‐phase mixing, will definitely help reactor engineers to obtain optimum performance of stirred slurry reactors. Recently, Kasat et al. [92] simulated liquid‐phase mixing in a stirred reactor for different operating conditions. Their simulations are reproduced here to explain the liquid‐phase mixing in a stirred slurry reactor.

The simulations are carried out in the experimental setup of Yamazaki et al. [38] with solid volume fraction equal to 10.0% and particle diameter equal to 264 μm. The simulations are carried out for 10 different impeller rotational speeds starting from 2 to 40 rps. The completely converged solid–liquid flow simulations were used to simulate liquid‐phase mixing. Mixing simulations were carried out with 1.0% (by volume) of tracer, having same physical properties of liquid in the vessel. The tracer history was recorded at eight different locations. In stirred slurry reactor, delayed mixing was usually observed near the top surface of the liquid. Therefore, tracer history was recorded at four different locations close to top surface (Ref. [92] for more details). The mixing time in the present work is defined as the time required for the tracer concentration at these locations to lie within ±5.0% of the final concentration (C_∞).

It will be very helpful to first shed light on the predicted suspension quality before discussing the influence of suspension quality on the mixing process. In a stirred slurry reactor, the critical impeller speed for complete off‐bottom suspension Nc_s and complete suspension N_s are two very important design parameters. The concepts of critical impeller speed were introduced more than 40 years ago and are primary design parameters used even today by reactor engineers for scale‐up and design of stirred slurry reactor. The predicted suspension quality was analyzed to estimate the Nc_s and N_s. Several criteria are available in the literature to determine the values of Nc_s and N_s. However, those criteria are applicable for experimental measurements and cannot be extended directly to the CFD simulations with the EE approach. Bohnet and Niesmak [93] have proposed alternative criteria based on the standard deviation σ of solid concentration to describe the suspension quality (see Eq. (13.17)). The same criterion is used in the present work to describe the suspension quality. The decrease in standard deviation value is manifested as an increase in the quality of the suspension. Based on the range of the standard deviation, the quality of the suspension is broadly divided into three regimes: homogeneous suspension, where the value of the standard deviation is smaller than 0.2 (σ < 0.2); complete off‐bottom suspension, where the value of the standard deviation lies between 0.2 and 0.8 (0.2 < σ < 0.8); and incomplete suspension, where the standard deviation value was found to be higher than 0.8 (σ > 0.8). This criterion enables the prediction of Nc_s and N_s and also gives the information on quality of suspension prevailing in the vessel.

The standard deviation values were estimated from the predicted solid volume fraction for all the 10 simulations carried out at different impeller rotational speeds. It must be noted that solid volume fraction values at all computational cells were used to estimate the standard deviation value. The predicted variation of standard deviation values with respect to impeller speed is shown in Figure 13.33. It can be seen from Figure 13.33 that three distinctly different suspension conditions, viz. incomplete suspension, complete off‐bottom suspension, and homogeneous suspension, can be identified in the vessel. At a lower impeller speed, the computational model predicted very high values of the standard deviation (σ > 0.8), indicating incomplete suspension in the vessel. It is also observed that the standard deviation values drop sharply with an increase in the impeller rotational speed until complete off‐bottom condition is achieved. The computational model predicted standard deviation value equal to 0.7 for 15 rps. This indicates the presence of a critical impeller speed for complete off‐bottom suspension (σ = 0.8) close to 15 rps. This is in good agreement with the Nc_s (= 13.4 rps) estimated using correlation proposed by Zwietering [94] for the experimental setup of Yamazaki et al. [38]. With further increase in the impeller rotational speed, the values of standard deviation drop slowly till the system achieves homogeneous suspension condition. The predicted results suggest that the homogeneous suspension condition for the experimental condition of Yamazaki et al. [38] is achieved at impeller speed N_s equal to 40 rps (σ = 0.17).

The species transport simulations are then carried out to understand the mixing process in the experimental setup of Yamazaki et al. [38]. The variation of predicted dimensionless mixing time (Nt_mix) with impeller rotational speed is shown in Figure 13.34. It can be seen from Figure 13.34 that the dimensionless mixing time first increases sharply with increase in the impeller rotational speed and then drops slowly with further increase in impeller speed. Figure 13.34 shows minimum value of dimensionless mixing time for lowest impeller speed (2 rps). The predicted liquid velocity vector plot was studied to understand the possible reason behind the observance of a minimum mixing time. The predicted flow characteristics for an impeller rotational speed equal to 2 rps are shown in Figure 13.35. It can be seen from Figure 13.35a that the computational model predicted single‐loop velocity pattern in the vessel. It is possible that for such a low impeller speed, impeller action is not sufficient to lift the particles from the vessel bottom (see Figure 13.35b). The solid bed present at the bottom of the reactor might offer apparent low off‐bottom clearance to the impeller‐generated flow and therefore lead to single‐loop flow pattern for Rushton turbine. In such a scenario, all the energy dissipated by the impeller becomes available for generating liquid circulations in the vessel and for fluid mixing. Therefore, it is possible to have faster mixing in the reactor at low impeller rotational speed.

With increase in impeller rotational speed, the dimensionless mixing time increases, reaches maxima, and then drops slowly (see Figure 13.34). In the present work, the maximum in the mixing time was found to happen at around 5 rps. At 5 rps the impeller‐generated flow becomes sufficient to start suspending solids into the bulk volume of the reactor. The energy dissipated by the impeller is now distributed for generating liquid circulations and fluid mixing and suspending the solids. The single‐loop flow pattern changes to the classical two‐loop structure for the Rushton turbine. The part of the energy dissipation for solid suspension and the rate of exchange between the two loops contribute to the slower mixing in the vessel for the 5 rps condition. A further increase in the impeller rotational speed leads to a reduction in the mixing time (increase in the mixing efficiency). This observed reduction in the mixing time continues till the system achieves the complete off‐bottom suspension condition (i.e. Nc_s = 15 rps). The operating conditions (impeller rotational speed) after Nc_s show a gradual decrease in the mixing time with an increase in impeller rotational speed. The present simulations also supported the operating range at which maxima of mixing time occur, i.e. N = Nc_s/3 [91].

The simulated results are further analyzed to understand the mixing in the stirred slurry reactor. The predicted tracer histories in the bulk volume of reactor (more close to impeller) and near the top surface of vessel were compared. The comparison of predicted tracer histories is shown in Figure 13.36. It can be seen from Figure 13.36 that the homogenization process is much faster in the region close to impeller compared with the region near the top surface. It was also observed that the difference between the top region and the impeller region is strongly dependent on the suspension quality present in the vessel. Figure 13.36 shows that in incomplete suspension and in complete off‐bottom suspension conditions, the time required for the homogenization near the top surface is significantly high compared with the time required for homogenization close to the impeller. The difference decreases as the system approaches the homogeneous suspension condition. Kasat et al. [92] showed that the lower liquid velocities present in the clear liquid layer above the solid cloud is responsible for the slower mixing process.

Preceding subsections discuss mixing issues in gas–liquid and solid–liquid systems. Similar approach can be used for addressing issues in other systems like liquid–liquid or gas–liquid–solid stirred vessels. Apart from predicting dispersion or suspension quality and mixing time, CFD models allow estimation of circulation time distribution, different zones in stirred vessels with different prevailing shear rates, interaction of impeller stream with inlet and outlet nozzles, and so on. It is possible to creatively use information obtainable from CFD to gain better insight and support engineering decision making. For example, information on different shear zones and RTD in these different zones often provides used clues for quantifying influence of scale on “breakup” and “coalescence” dominated zones. It is not possible to discuss actual industrial cases here for the sake of protecting confidential information. It is however hoped that information provided here will allow resourceful engineer to develop appropriate computational flow models and use the simulated results for addressing practical design and scale‐up issues.

13.3.2.4 CFD Simulation of Antisolvent Crystallization

Pharmaceutical and fine chemical makers frequently rely on antisolvent crystallization, also known as precipitation for generating solid from a solution in which the product has high solubility. This technique is used for a variety of applications such as polymorph control, purification from a reaction mixture, and yield improvement. Antisolvent crystallization achieves supersaturation and solidification by exposing a solution of the product to another solvent (or multiple ones) in which the product is sparingly soluble. Although this technique has the potential to achieve a controlled and scalable size distribution, it is not without problems. The product requires purification or separation steps to remove the antisolvent(s).

Woo et al. [95] discussed the model development in the framework of CFD for simulating the effect of mixing on the antisolvent crystallization process. In the CFD model they included PBE for the evolution of the CSD and the PDF of the local turbulent fluctuations. In addition to that, they used separate sub‐models for nucleation and growth kinetics and Einstein equation for modeling effect of solid concentration on the rheology. For detail model equations and its implementation, please refer to Woo et al. [95]. Computational model was then used to predict the evolution of volume‐averaged antisolvent, mass percent, supersaturation, nucleation rate, and mean growth rate. While comparing with the earlier published experimental observations, Woo et al. [95] found that the computational model was able to simulate temporal evolution of nucleation and growth rates at the inlet. They also predicted higher growth rates at the impeller region due to higher turbulence (Figure 13.37). They attributed this to improved mass transfer. This improved mass transfer at higher impeller rotational speed consequently led to faster desupersaturation, which resulted in lowering overall nucleation rate. Its impact on the overall crystallization process is explained through predicted final CSD. Figure 13.38 shows the predicted CSD at the end of one hour for three different rotational speeds. The predicted results show that fewer and slightly larger crystals were obtained for higher agitation rate. These results are qualitatively in agreement with the experimental findings.

Woo et al. [95] then used the computational model for simulating antisolvent crystallization with reverse mode of addition. Figure 13.39a shows the predicted final volume‐averaged CSD. As anticipated, the computational model predicted finer particle size with reverse mode of addition in comparison with the normal mode of addition. They also used computational model for predicting influence of scale of the equipment on the CSD. Figure 13.39b shows the comparison of the predicted CSD for different scale and different scale‐up criteria. The computational model predicted nonsignificant impact of scale of operation on the predicted CSD for constant power per unit volume that is consistent with the experimental findings by Torbacke and Rasmuson [96].

13.3.2.5 Process Innovation in Crystallization

It is been known that by reducing drug particle size to the absolute minimum may lead to significantly improving drugs' wettability and bioavailability. This will require a small mean crystal size and a narrow size distribution. Normally, pharmaceutical powders are polydisperse, i.e. consisting particles of different sizes. Polydisperse powders create considerable difficulties in the production of dosage forms. Particles of monosize (equal size) may be ideal for pharmaceutical purposes. In practice, powders with narrow range of size distribution can obviate the problems in processing them further. Making such products demand continuous processing either through an in‐line mixer or via conducting milling on finished product [97]. At times milling technique can cause negative results such as dusting, caking, electrostatic charges, and in some times a polymorphic transformation. Process intensification by combining impinging jet reactor with stirred vessel (see Figure 13.40) provides an opportunity to control the final product particle size. It is been observed that uniform or nearly homogeneous nucleation happens in the impinging jet reactor [99]. These uniformly formed nucleates then feed into a stirred vessel where crystal growth or ripening will happen. Validated computational model not only will help in designing new innovative such systems but also will help in optimizing the operating parameters and mixing and transport processes for achieving desired process and product requirements.

13.3.2.6 Solid Suspension in Viscous Medium

There are several pharmaceutical suspensions available in the market. In these suspensions therapeutically active ingredients are uniformly dispersed in the medium. The solid particles used in these suspensions are smaller than 5 μm but have a tendency to settle. These suspensions are broadly differentiated into dilute suspensions (2–10% w/w) or concentrated suspensions (up to 50% w/w). It is imperative from the perspective of product specification to have uniform suspension in order to have the administration of a measured dose. Formulation experts avoid or minimize the settling either by reducing particle size or by increasing the viscosity of the continuous phase. Increasing the viscosity of the continuous phase is more predominantly used in formulations of these suspensions. The influence of rheology on solid suspension has received relatively less attention in the literature. Ibrahim and Nienow [100] showed that at higher viscosity, the Zwietering equation is likely to fail, with an error as large as 90%. The possible reason could be noninclusion of complex rheology behavior in the equation. It is known that in laminar condition, the presence of fine solids in the liquid produces shear‐thinning non‐Newtonian behavior [101].

Wu et al. [102] experimentally studied the influence of non‐Newtonian rheology on the characteristics of solid suspension in a mechanically agitated vessel. They studied the suspension of glass beads in water and Carbopol solution. Their experimental results are presented in Figure 13.41. Very interesting observations were made by Wu et al. [102]. They observed that adding 0.04% Carbopol in water resulted in the reduction in the just suspension speed of impeller (N_js) by 15%. They attributed this drop to the reduction in the settling velocity of the solids. They also observed that above 0.09% of Carbopol in water, the N_js sharply reduced to zero (Figure 13.41b). To explain this, the variation of settling velocity is plotted in Figure 13.41c for different viscosity values for glass beads having 100 μm diameter. It is seen that the settling velocity approaches to the few tens of μm/s for 0.09% Carbopol solution and hence will not see settling for longer time. In such situations, the definition of N_js becomes irrelevant in defining impeller operating conditions, and definition of effective dispersion becomes more relevant.

Handling solid dispersion in viscous medium is not an easy task. A complication that arises is that the fine particles (microscopic in size) maintain electrical and molecular attraction. These fine particles tend to lump together and form agglomerates that no amount of mixing will break. An aggregate (or agglomerate) is composed of a group of particles that are strongly adherent and can be broken down only by the application of relatively strong mechanical forces. In the days before the advancement of disperser, different mills were used in practice for reducing the size of the product. However, the processing was very time consuming and led to very long batch time. With the advent of the disperser, the de‐agglomeration process could be accomplished much more rapidly within the same vessel while mixing, resulting in a smoother, more uniform end product. The disperser is an impeller having thin disk with carefully designed teeth distributed radially about the circumference. Its action tears particles apart and disperses them uniformly throughout the product. This work is done with two actions: firstly, particles hitting the impeller are broken apart, or de‐agglomerated, and secondly, the intense turbulence surrounding the impeller causes particles to hit each other with great momentum and inertia. The energy of this impact physically breaks apart agglomerates. Figure 13.42 shows the commonly used disperser in the industry.

Disperser has very limited pumping capacity, and hence in a larger mixing vessel, it is imperative to use other flow impeller to continuously feed disperser for achieving better product results. In many industrial applications having high viscosity mixing, multi‐shaft impellers are widely used. In these systems, a close‐clearance impeller such as Anchor, Paravisc, or helical ribbon is mounted on the one shaft for facilitating distributive mixing, and single or multiple dispersers are mounted on the other shafts for dispersive mixing. Barar Pour et al. [103] studied the slurry blending in a dual shaft impeller system having continuous phase viscosity of 1 Pa·s and solid holdup of 10%. Figure 13.43 shows experimental setup of Barar Pour et al. [103]. They observed that the final particle size distribution is function of the disperser speed (see Figure 13.43b).

Empirical approach based on the experimental observation is still used in the designing of the mixing system for dispersing powdered solids in the viscous fluids. The dispersion process is itself very complicated and involves several stages – wetting, incorporation, agglomeration, and rupture. The current level of understanding on each of these stages is not sufficient. The experimental results of Barar Pour et al. [103] show different behavior of torque experienced by Paravisc impeller for different solids. The actual reason for this variation is not very clear. Also, the current level of understanding on agglomerate formation and its breakup in a complex flow is not up to the level of understanding of bubble and/or droplet coalescence and breakage. There are few studies reported in literature mainly by Manas‐Zloczower and coworkers [104–106], and Hansen et al. [107] presented models for simulating agglomerate breakup. Zeidan et al. [108] simulated the evolution of aggregates subjected to simple shear flow using a combined continuum and discrete model (CCDM). They solved the motion of discrete particles using Newton's second law of motion (DEM), and the continuum fluid flow was solved using locally averaged Navier–Stokes equations (CFD). This allowed Zeidan et al. [108] to study fully coupled aggregate deformation and breakup in a simple shear flow. They studied the breakage for aggregate having different cohesive strength. Figure 13.44 shows the simulated aggregate breakage for two different cohesive strengths. It can be seen that two different aggregate mechanisms exist. Weak aggregates rupture and at higher shear rate produce smaller and more uniform‐sized aggregates. However, erosion mechanism is dominant in breakage of strong aggregates, and it results in wider distribution.

Cong and Gupta [109] used PELDOM^TM software for simulating solid dispersion in a corotating twin‐screw extruder. The screws were rotated at 60 rpm, and 200 blue and 200 red segregated agglomerates were placed in the two halves of the twin‐screw extruder entrance. Quality of distributive mixing in a corotating twin‐screw extruder was evaluated by finding the change in the spatial distribution of initially segregated particles. Shannon entropy of mixture was first estimated, and then it was used for estimating color homogeneity index (CHI). CHI value lies between 0 and 1. Zero corresponds to completely segregated system, whereas one corresponds to completely mixed system. For the same twin‐screw extruder, quality of dispersive mixing was determined by using the erosion model of Scurati et al. [105]. In this model the rate of agglomerate size reduction is proportional to the shear rate and the difference between hydrodynamic (F_h) and cohesive (F_c) forces:

(13.24)

Please refer to Cong and Gupta [109] for more details on the definition of hydrodynamic and cohesive forces. Particle tracing scheme is used for the simulating particle motion. The change in the spatial distribution was determined by following the particle path lines to the desired axial location of the extruder. The predicted CHI for distributed and dispersive mixing is shown in Figure 13.45. It can be seen from Figure 13.45a that starting with completely segregated red and blue particles in each lobes, at z = 30 mm, the distribution of the particles in both the lobes is nonhomogeneous and the homogeneity improves with distance. At z = 90 mm, the particle distribution in the twin‐screw extruder is quite uniform. Initial guess of agglomerate and fragment radii determines the number of fragments erodes from the single agglomerate. For 0.5 and 0.11 mm for agglomerate and fragment radii, respectively, Alemaskin et al. [110] observed 93 fragments from a single agglomerate. Very high number of particles negatively influences the computational efficiency, and hence Cong and Gupta [109] in their simulations used 0.3 and 0.1 mm as the agglomerate and fragment radii, respectively. This led to 26 number of fragments resulting from erosion of single agglomerate. Figure 13.45b shows the spatial distribution of agglomerates and eroded fragments at three different locations of twin‐screw extruder. It shows that with only 400 agglomerates at entrance, the population of the fragments increases substantially with extruder length. They also observed that the CHI, which determines the mixing efficiency, improves significantly due to dispersive mixing. This suggests that alone simulating distributive mixing will not be sufficient for quantifying mixing efficiency in such applications.

13.3.2.7 Solid Drawdown in a Stirred Vessel

In many practical applications, low density solids, more particularly powders, are added into the liquid. Powder addition is usually accompanied with a variety of problems, irrespective of whether powder is soluble or insoluble. Typical changes do happen during hydration of solids in liquid and are explained schematically in Figure 13.46. The first case is simpler case compared with rest of the three. The challenge in first case is to wet the surface of solids and incorporate them in the liquid if required without entraining air. However, in the remaining three cases, rheological challenges do crop up. For example, if the rate of dissolution is faster than the rate of surface wetting, then there is a possibility of having a high viscosity layer of liquid near the top surface. This layer will eventually cover the dry solids and may delay the further dissolution. If the liquid develops Bingham behavior, i.e. a yield stress, wetting will stop completely. Swelling of particles always results in a slower rate of wetting, which may even approach zero. Many food ingredients like starch and some proteins tend to swell in water; in general the particles may swell and/or dissolve in the liquid to various degrees (Figure 13.46). It is impossible to predict in general whether or not such behavior leads to a faster or slower rate of wetting, as compared with an unchanged bulk material.

The rate of addition and surface motion can either worsen or improve powder addition. Many powders need to be added slowly enough that they do get sufficient time for wetting their surface and their incorporation into the liquid. Some hydrating thickeners such as cellulosic polymers need to be added quickly, while the fluid is still low viscosity and prevailing turbulence is there to provide aid in the addition and dispersion of the powder. This suggests that there is a specific rate of addition exists based on the wetting characteristics of the powders. Different mechanisms are recommended in the open literature for incorporating low density solids into the liquid in turbulent condition. Khazam and Kresta [112] identified three mechanisms of solid drawdown in stirred tanks: (1) Formation of stable single vortex (with no baffles or single baffle system) causes downward axial velocities at the surface responsible for drawdown. (2) Turbulent fluctuations form mesoscale eddies/vortices on the surface, which intermittently pull particles in the liquid. (3) Mean drag produced by the liquid circulation loops draw particles into the liquid where the downward axial velocities are greater than the particle slip velocity.

Most commonly used mechanism is generating a strong single vortex for incorporating solids (mechanism 1). The strong vortex can be generated using either no or partial baffling. Joosten et al. [113] used single baffle, Hemrajani et al. [114] used four baffles of width 1/50 tank diameter, and Siddiqui [115] recommended three partially immersed baffles 90° apart. Edwards and Ellis [116] found three‐blade marine propeller without any baffles to be the most energy‐efficient design. In this mechanism, during operation, headspace gas/air also gets entrained in the liquid. It is possible that in many practical applications, gas entrainment in the liquid needs to be avoided. This definitely puts challenge on the process engineer in designing the vessel and its operating conditions for efficient drawdown of the solids.

Several works have been reported in the past explaining the different baffling concepts and its implications on the drawdown of solids. It is well known that baffling suppresses the stable surface vortex formation and increases the intensity of mean drag and turbulence at the surface. This generates strong top‐to‐bottom liquid circulation. However, this circulation rapidly brings particles back to the surface. Hence researchers have recommended the partial or nonstandard baffling. Khazam and Kresta [112] experimentally and computationally studied the drawdown of the floating solids in a stirred vessel for two different baffle designs, viz. half baffles and surface baffles. Schematics of the different baffling configurations used by them are shown in Figure 13.47. The objective was to maintain a high level of turbulence at the surface while reducing the return circulation from the bottom of the tank. The performance of the baffle configurations was compared using the just drawdown speed, N_jd, and cloud depth, CD, for the PBTD, PBTU, and A340 impellers. CFD simulations and measurements of the power number for the fully baffled and the surface baffled configurations were also reported. They found out that the drawdown speed for all the impellers was very similar for full and half baffles (see Figure 13.48). But with surface baffles, they found advantage in significant reduction in the drawdown speed N_jd. They also observed that a more robust performance at large submergences and a better distribution of solids was obtained for the surface baffles.

Atibeni et al. [117] used the PIV technique for elucidating the effect of baffle design on the drawdown of the floating particles. They observed that for the standard baffle system, just drawdown speed reduces by placing impeller off‐center. They observed more than 50% reduction in impeller power by this modification. They also found that around 50% impeller power reduces by modifying standard baffles with down triangle baffles. Hsu et al. [118] used both visual observations and CFD model for simulating the effect of impeller clearance and baffle design on the drawdown of the floating particles. They also observed positive impact of new baffle designs on the drawdown process over the standard baffles.

13.3.2.8 Solid Dissolution in Stirred Vessel

Solid dissolution is a common process unit operation during liquid mixing in the pharmaceutical industry. Incomplete dissolution can result in subpotent batches, resulting in batch failure. Competing elements such as the time required for dissolution versus the potential of microbial contamination over an extended mixing period may also be in play in a manufacturing setting. Dissolution in a stirred tank is influenced by many parameters. Khinast and coworkers [119, 120] provided an Ishikawa diagram (Figure 13.49) for explaining the various parameters that influence the operation. Many of these parameters cannot be changed in a pharmaceutical manufacturing setting. For example, most of the physicochemical properties of a raw material are fixed, although variations do occur due to the batch‐to‐batch variability of APIs. The process engineer has to work with process parameters, human environment, and equipment for improving the dissolution rate or time required for achieving complete dissolution. Other than the particle properties, the dissolution kinetics depend on the local mass transfer coefficient (which in turn is a function of suspension state and the local turbulence level), on the thermodynamics of the crystal, and on the equilibrium solubility at the fluid temperature.

A key fluid dynamic parameter that will govern the dissolution time is the solid–liquid mass transfer coefficient. The rate of mass transfer is defined as

(13.25)

where

• k is the solid–liquid mass transfer coefficient.
• a is interfacial area.
• C^* is the concentration at the interface that is saturation concentration.
• C is the concentration in the bulk liquid phase.

Thus by definition, achieving a constant mass transfer coefficient across different scales, one should achieve similar dissolution rates and hence similar dissolution times. In order to achieve constant mass transfer coefficients across different scales, it is important to know how the mass transfer coefficient is affected by different operating parameters.

Koganti et al. developed CFD model for simulating the dissolution of propylparaben in water for two different scales, viz. 2 L lab scale and 4000 L commercial scale. Their aim was to study the influence of scale‐up procedure on the dissolution process. They maintained the geometric similarity for both the scales, and operating conditions were scaled based on uniform power per volume. The predicted evolution of dissolution for both lab scale and commercial scale is shown in Figure 13.50. It shows that the difference between the laboratory scale and commercial scale is reasonably small and in order of 75–88% confidence for various operating conditions. It must be noted that the CFD simulation of dissolution process is only limited to the mass transfer where the model assumes perfectly wet condition for particles from the initial time. This is particularly because CFD models have not yet reached to the level where wetting by capillary action can be modeled with reasonable accuracy. However, they can be used for a priori predictions for evaluating influence of different design and process parameters.