Operating Strategy for Continuous Multistage Mixed Suspension and

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Operating strategy for continuous multistage MSMPR crystallization processes depending on crystallization kinetic parameters Kiho Park, Do Yeon Kim, and Dae Ryook Yang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b01386 • Publication Date (Web): 11 Jun 2016 Downloaded from http://pubs.acs.org on June 14, 2016

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Industrial & Engineering Chemistry Research

Operating strategy for continuous multistage MSMPR crystallization processes depending on crystallization kinetic parameters Kiho Park, Do Yeon Kim+ and Dae Ryook Yang*

Department of Chemical and Biological Engineering, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea

+Present Address: Imperial College of London, Department of Chemical Engineering, South Kensington, London, SW7 2AZ, UK * [email protected]; phone: +82 2 3290 3298; fax: +82 2 926 6102;

ACS Paragon Plus Environment


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ABSTRACT: Continuous multistage MSMPR crystallization processes are useful for the large scale production of particulate systems. However, the design of operating strategies to meet specific objectives and materials has not been entirely investigated. In this work, the effect of important crystallization kinetic parameters on the optimal operating strategy was examined. The important parameters are the kinetic constants of the primary and secondary nucleation rates, the orders of the nucleation and growth rates, and the number of crystallizer stages. The analyses revealed that a drastic cooling strategy in the primary nucleation dominant region and linear cooling in the secondary nucleation dominant region are best for producing large particle sizes. A stage number of around three is effective in both regions. These results can be utilized to roughly determine the operating strategy for a process, if the crystallization kinetic parameters are already roughly known.

Keywords: Continuous crystallization; Multistage MSMPR; Parameter analysis; Operating strategy; Cooling crystallization


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1. INTRODUCTION Many industrial crystallization processes use three operation types; batch, semi-batch, and continuous.1 Although batch operation offers considerable advantages, such as easy control of crystal size distribution (CSD) and simplicity of process equipment, batch-to-batch inconsistency and high operating costs lead it to small quantity batch production.2-4 Semibatch operation, often called fed-batch, has been applied for maintaining high supersaturation in the crystallizer. Despite difficulty in achieving optimal control, semi-batch operation can result in better CSD than batch operation.5 On the other hand, continuous operation offers lower capital and operating costs and consistent product qualities.4, 6, 7 As these features are very suitable for industrial operations, this operation mode has been favored in large quantity production processes in chemical industry. Recent studies have mentioned that the continuous process is not just limited to large-scale production processes, but is also applicable to smallscale active pharmaceutical ingredient manufacturing processes.8 Thus, the importance of the continuous process has increased considerably. The mixed suspension and mixed product removal (MSMPR) crystallizer has been widely applied as a continuous crystallizer in chemical industry owing to its simplicity of maintenance and control.2 Another advantage of the MSMPR system is its easy conversion from batch to continuous processes through the use of simple technologies.9 Using the mathematical model for MSMPR developed by Randolph and Larson,10 effective design and configuration of the MSMPR crystallizer to improve product quality has been extensively researched.1,

11-15

A number of researchers have reported that the single-stage MSMSR

crystallizer is inefficient because it produces a product with a broader CSD than the multistage MSMPR crystallizer.15, 16 In addition, from the view of process operability, the single-stage MSMPR crystallizer is unfavorable because the temperature controllability is limited owing to its small cooling surface area.17 This results in a higher operating cost than



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for the multistage MSMPR crystallizer. For these reasons, the multistage MSMPR crystallizer has been favored in industrial crystallization. These crystallization systems are usually modeled by utilizing population balance equations (PBE), as a well-designed mathematical framework for describing growth, nucleation, breakage, and agglomeration of crystals.18 The PBE was extended to crystallization by including expressions for the nucleation and growth.19, 20 The PBE have been applied for estimating and modeling various crystallization systems.11, 12, 21-24 However, PBE containing breakage and agglomeration terms, which are commonly expressed as integral functions, are so complex that analytic solution of the PBE could not be obtained.18 Although solutions of the PBE could be obtained by several numerical methods, such as the method of moments,25, 26 the method of characteristics,27 discretized population balances,28-30 and weighted residuals,31 the computation time for model solving is significantly long and it is too difficult to estimate the parameters required for expressing the breakage and agglomeration phenomena. Therefore, breakage and agglomeration is usually neglected to simplify the model and reduce the computation time. The operating strategy for the continuous multistage MSMPR crystallizer depends on the crystallization kinetic parameters of each product material. As both the nucleation and growth rates in the crystallizer depend on the degree of supersaturation and nucleation and crystal growth occur as consuming the degree of supersaturation simultaneously, both rates compete against each other on the degree of supersaturation. Thus, the ratio of nucleation and growth rates significantly affects CSD of the product stream. The ratio is determined by the crystallization kinetic parameters. Therefore, it is very important to determine the crystallization kinetic parameters of the product material for efficient operation of the continuous multistage MSMPR crystallization process.32 There are many investigations regarding the experimental determination of crystallization kinetics.13, 32-34 In addition, there


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are several recent reports about the optimization of the multistage MSMPR crystallizer.8, 9, 3537

However, no general analysis of the effects of various crystallization kinetic parameters on

the effective operating strategy for the crystallization system has been carried out. This kind of analysis is important because it can be a guideline for the appropriate design and operation of a continuous multistage MSMPR crystallization process when the crystallization kinetic data are available. The aim of the present study was to investigate what kind of operating strategy for a continuous multistage MSMPR crystallization process would be effective for various crystallization kinetic parameters. A continuous multistage MSMPR crystallization system was modeled with mass, energy, and population balances containing crystallization kinetics, such as nucleation rate and growth rate. The parameters of the crystallization kinetics were varied depending on the solute and solvent materials. The crystallization kinetic parameters of many materials were collected from the literature and the ranges of the parameters were determined. After selecting the important parameters that can significantly influence the performance of the crystallization process, the effects of each parameter on the crystallization system were investigated by simulation, and then the most efficient operating strategy for the multistage MSMPR crystallizer was determined. This study focuses on offering general operating strategy guidance for the continuous multistage MSMPR crystallization system. Thus when constructing a new continuous MSMPR system with a new material for which the crystallization kinetic parameters are roughly known, the effort required to determine the optimal operating strategy would be reduced.

2. PROCESS AND METHODOLOGY 2.1. Process Description



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A four-stage continuous MSMPR crystallization process was selected as our base case to investigate the optimal operating conditions for the multistage crystallizer. Each stage was assumed to have an identical size and was operated at different temperatures. Figure 1 shows the schematic design of the multistage continuous crystallization process. Based on the actual operating conditions of an industrial process, it was assumed that the amount of mother liquor in each crystallizer is 40 tons and the feed flow rate is 62,000 kg/h. The solution concentration of the feed stream is 0.375 solute kg/suspension kg, the temperature is 235°C, which is high enough to dissolve the entire solid in the feed stream, and the pressure is 4.413 MPa. At each crystallizer, supersaturation is achieved through cooling resulting from the adiabatic flash obtained by adjusting the pressure using vent valves. By decreasing the pressure at each stage, the temperature in each crystallizer is decreased by solvent evaporation. The temperature of the final effluent product stream is fixed at 135°C at a pressure of 1 atm, so the products can be processed by the downstream processes.


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Figure 1.Schematic diagram of the continuous multistage MSMPR crystallization process. F is the flow rate, C is the concentration, T is the temperature, P is the pressure, and µ3 is the third moment of the particles. The subscripts in and out are the inlet and outlet stream, respectively, and v is the vapor stream. The solute for the process is isophthalic acid, which is an organic compound with the molecular formula C6H4(CO2H)2. The solvent for the process is water. After experimentally estimating the crystallization kinetic parameters of these materials, the parameter set of isophthalic acid/water crystallization system was used as base case data of parameter analysis. To achieve the aim of this study, the important parameters which significantly influence the product quality and the other parameters should be categorized. The selected important parameters from the base case would be varied in appropriate ranges to describe the characteristics of many materials unrestricted to isophthalic acid/water crystallization system. However, any other parameters and variables such as solubility and heat of vaporization have been fixed on the value of the base case study during parameter analysis.

2.2. Modeling In this study, a mathematical model for describing the continuous multistage MSMPR crystallization process was developed. The model consists of component mass balances, energy balance and population balance with the following assumptions. I.

The density of the solute is independent of concentration and temperature.

II. The crystallizers are operated at steady-state conditions. III. The crystallizer is an adiabatic system. IV. The heat of crystal formation is neglected. V. Crystal agglomeration and breakage are negligible.



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VI. The crystallizer is assumed as perfect mixing. The population balance in each stage in Figure 1 can be described as11, 38, 39

∂n ∂Gn Fout n − Fin nin + + =0 ∂t ∂L mT

at L ≠ L0

(1) With the steady-state assumption, Eq. (1) becomes

∂Gn Fin nin − Fout n = ∂L mT (2) subject to the boundary condition n(0, t ) =

B (t ) G (0, t )

(3) where G is the growth rate, L is the crystal length, F is the mass flow rate, mT is the mass of suspension in the crystallizer, and n is the number density per unit mass of suspension. The crystal growth rate is defined as38

 −E  γ G( L, t ) = k g 0 exp  g  (1 + kL ) ∆Cα  RT  (4) where kg0 is the growth rate constant, Eg is the activation energy, R is the gas constant, T is the temperature, k and γ are the constants for expressing size dependency of crystal growth, α is a constant denoting growth rate order, and ∆C is the degree of supersaturation. ∆C is defined as

∆C = C − Csat (5)


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The nucleation rate may be defined to express the influence of primary and secondary nucleation simultaneously

B = ( kb + kb 2 µ 2 ) ∆C β (6) where kb is the primary nucleation rate constant, kb2 is the secondary nucleation rate constant, µ2 is the second moment of crystal, and β is the nucleation rate order. For the mass balance of the solute, the total amount of solute in the feed is the same as that in the product stream because the crystallizer is operated at steady state. Therefore, the mass balance can be expressed as

Fin (Cin + ρc kv µ3,in ) = Fout (C + ρc kv µ3 )

(7)

where C is the mass solute per mass of suspension, ρc is the density of crystal, kv is the volume shape factor, and µ3 is the third moment of crystal. Assuming that the heat of crystal formation can be neglected, the energy balance does not influence the crystallization process directly. However, the temperature control of each crystallizer is achieved by an adiabatic flash, and the energy balance equation should be solved to estimate the amount of evaporated solvent. In this process, the energy balance is given as FinTin C p , w + ( Cin + ρc kv µ3,in ) ⋅ C p , s 

= FoutT C p , w + ( C + ρc kv µ3 ) ⋅ C p , s  + Fv ⋅ (C p ,wT + ∆H v )

(8)

where Cp,w and Cp,s are the heat capacities of water and the solute, respectively, Fv is the amount of solvent vaporization, and ∆Hv is the heat of vaporization of the solvent. Using Eq. (7), Eq. (8) becomes Fv =

(Tin − T )  FinC p ,w + C p ,s Fin ( Cin + ρc kv µ3,in ) ∆H v


(9)


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Using Eq. (9), the amount of solvent evaporated at each stage can be estimated. Because the energy balance does not require any information about the solution concentration in each stage and is not coupled with the other balance equations, it could be calculated easily, just by defining the exit temperature for each stage. Thus, the change in solute concentration at each stage owing to solvent evaporation can be obtained. The algorithm for solving the continuous multi-staged MSMPR crystallization system is shown in Figure S1. In this study, the simulation were conducted by dynamic simulation approach. From initial condition, the amount of solvent evaporated at each stage were estimated at first. Then, the mass and population balance equations were solved. In the dynamic simulation approach, the population balance equation is partial differential equation. Thus, the crystal size were discretized using finite difference method, and fourth-order Runge-Kutta method developed as a programming tool which was named as ode45 in MATLAB was used for solving the population balance equation on the time domain. The mass balance equation was also solved simultaneously. After performing the dynamic simulation until predetermined final time, the steady-state was checked under a criteria which is the normalized concentration differences between present concentrations at each time and the concentration at the final time should be lower than 0.001, and the criteria should be satisfied more than 60% in the time range. If the steady-state condition were satisfied, the dynamic simulation was finished. If not, new dynamic simulation were performed within the newly determined time range from the state at the final time of previous dynamic simulation.

2.3. Parameter Analysis 2.3.1. Base Parameter Estimation As described in section 2.1, the base parameters should be decided for the parameter analysis. In this study, the base parameters were obtained from the industrial data for the


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continuous four-stage MSMPR crystallization process with isophthalic acid/water system. The heat of vaporization of water required to calculate the energy balance was modeled as a fourth-order polynomial with coefficients obtained from the literature.40 Even though the industrial data could not be open because the data are proprietary information, the estimated base parameters are summarized in Table S1.

2.3.2. Selection of Important Parameters For the sake of simplicity and generality of the analysis, the crucial parameters should be selected to sufficiently explain what happens in the crystallizer. In batch and reaction crystallization processes, nucleation occurs predominantly by primary mechanisms, not secondary nucleation.34 In the industrial case, however, secondary nucleation is very significant because of the huge number of crystals in the crystallizer.41 Therefore, the primary and secondary nucleation constants are very important for describing the various outcomes of the crystallizer. The growth and nucleation rate orders are also important because the dependency on the degree of supersaturation is significantly different for each material.13 Therefore, the nucleation and growth rate orders should be categorized as important parameters to describe the characteristics of different cases. In addition, as the target process is continuous multistage MSMPR crystallization, the stage number is also regarded as an important parameter. The parameter analysis was conducted by varying each selected parameter within the ranges of variation while the other parameters were fixed. The base values and the ranges of variation of the important parameters are listed in Table S2. The base values and the ranges of variation of the primary and secondary nucleation rate constants were determined from the estimated crystallization parameters in Table S1. These parameter values are secondary nucleation dominant compared with those for primary nucleation. As this parameter set can



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represent the secondary nucleation dominant case, the parameter values for different situations, i.e., primary nucleation dominant case, were selected by increasing the primary nucleation rate constant and decreasing the secondary nucleation rate constant. The ranges of variation for the growth and nucleation rate orders were selected to incorporate the experimentally estimated kinetic parameters from the literature.11,

33, 42-49

By varying the

orders, the CSD in the product effluent stream can be determined when each order is higher or lower than 1. The range of variation of the stage number was one to ten, even though the stage number would generally not be more than six. Within these parameter ranges, steady-stage simulations were carried out to obtain the final CSD, average size of the product, and process yield for each diverse parameter set. Based on the magnitudes of the nucleation rates, the parameter ranges can be categorized into several featured regions. Depending on the nucleation rate constants, there can be three featured regions, with one dominated by primary nucleation, the second dominated by secondary nucleation, and the third with comparable magnitudes for both nucleation rates. With these featured regions, simulations were performed by applying different operating strategies to investigate which operating strategy would be the most effective in each featured region.

2.4. Operating Strategy As the main operating variable for each crystallizer is its temperature, the temperatures of each stage, except the last were selected with different strategies, while the feed conditions and final temperature of the last crystallizer were the same for all simulations. Depending on the temperature profiles of the intermediate stages, the final product qualities, which are the CSD, average size of the product, and product yield, were investigated. As shown in Table S3 and Figure 2, the operating strategies are designed to have five cases. Case 1, which can be called extreme controlled cooling manner, adopts a strategy where the temperatures of the


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crystallizers are slightly decreased until the 3rd crystallizer, and then the temperature is dropped rapidly to the final temperature in the 4th crystallizer. Meanwhile, case 5 can be called extreme natural cooling manner. In case 5, the temperature decrease occurs only in the 1st crystallizer, and the temperatures of the following crystallizers are maintained at the final temperature for ageing toward the equilibrium. Cases 2-4 are intermediate operating strategies for changing the temperature. The temperatures in cases 2-4 were designed based on the solubility data. For case 2, about 10% of the total amount of solute obtainable throughout the whole process is crystallized in the 1st, 2nd, and 3rd crystallizers, and the remaining 70% is crystallized in the 4th crystallizer based on the equilibrium concentration. Cases 3 and 4 are designed in same manner, but adopt different percentages of solute crystallized in each crystallizer. Figure 2 shows the all of operating strategies of each stage to solubility.



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Figure 2. Solubility and operating strategies for the continuous four-stage MSMPR crystallization process. The degrees of supersaturation in each crystallizer are denoted with its crystallizer number. (1) Cases 1, 2, and 3, (2) Cases 3, 4, and 5.


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3. RESULTS AND DISCUSSION 3.1. Effect of Nucleation Rate Constant 3.1.1. Primary Nucleation Dominant Region As described in section 2.3, the featured region can be characterized by the dominant nucleation mechanism. In some cases, such as reactive crystallization, the primary nucleation mechanism is dominant for generating crystal nuclei. The featured region in which primary nucleation is very significant compared with secondary nucleation is defined as the primary nucleation dominant region (PNDR). The base parameter values applied in this region are kb = 4.8534 × 108, kb2 = 4.7299 × 10-8, α = 1, β = 1, and N = 4. To investigate the optimal operating strategy in this region, simulations were performed for five cases as described in the previous section. The results of these simulations are illustrated in Figure 3, which shows the crystal number density, and volume fraction of the final products. Among these cases, case 5 is the best operating strategy in the PNDR for particles with large average sizes. The number of nuclei generated by primary nucleation is dependent on the operating strategy. If most of the nuclei were generated in the 1st crystallizer, as in case 5, there would be enough time for the particle size to grow in the following crystallizers. Although the nuclei were largely generated in the 1st stage, a high degree of supersaturation would also influence the growth rate to produce large particles. Thus, the operating strategy of decreasing the temperature as much as possible in the 1st crystallizer is the most effective for obtaining larger particles with the continuous multistage MSMPR crystallization system.



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Figure 3. Simulation results with changing operating strategies in the PNDR; kb = 4.8534 × 108, kb2 = 4.7299 × 10-8, α = 1, β = 1, and N = 4. (a) Crystal number density, (b) Crystal volume fraction. In contrast, if the objective of the process is to obtain a product with a narrow CSD, case 2 is the most effective operating strategy. The simulation of case 3, which is called linear cooling manner, also generates a relatively narrow size distribution, which is almost the same as that for case 2. As shown in Figure 3 and Table S4, there is an inverse correlation between the coefficient of variation (CV) and the average particle size. As a moderate degree of supersaturation, which is smaller than that in case 5, is maintained throughout the crystallizers, a relatively low crystal growth rate is obtained. Thus, the average size of the product decreases and the CSD becomes narrower. The process yield which is defined as the amount of actual crystallized particles at the crystallizer over the amount of maximum obtainable particles on the solubility line, as shown in Table S4, tends to increase as the operating strategy is changed from case 1 to case 5. For higher yields, the final crystallizer should be close to the equilibrium. Thus, using case 5, which has most of the temperature drop at early stages, the solution can approach equilibrium during the following stages. Meanwhile, case 1 showed the lowest process yield because


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most of the temperature drop occurred at the last stage. A large temperature drop at the final crystallizer should be avoided for high process yields. In summary, if the objective of the process is to obtain the largest particle size and the highest process yield, case 5 is the best strategy. However, if the objective is to generate a narrow CSD and maximize the yield, then the best strategy is case 3. Although the CV of case 2 is lower than that in case 3, there is a significant difference in the process yields of these cases.

3.1.2. Secondary Nucleation Dominant Region The secondary nucleation dominant region (SNDR) can be defined as the featured region in which the secondary nucleation is dominant compared with primary nucleation. The base parameter values applied in this region are kb = 1.8534 × 104, kb2 = 4.7299 × 10-4, α = 1, β = 1, and N = 4. The analyses were performed in the same manner as in the PNDR. The results are illustrated in Figure 4. In the SNDR, as shown in Table S4, case 3 is the best operating strategy for obtaining particles with larger average sizes, but the difference between the cases with the minimum and maximum average size is only 4%. This difference is very small compared with that of 66.8% in the PNDR. Thus, the operating strategy does not significantly influence the quality of the product in the SNDR. Despite the small differences, the maximum average particle size was observed in case 3, and the minimum size was obtained in cases 1 and 5. These results show an opposite tendency to those for the PNDR. This is because the large number of nuclei caused by the high temperature drop have a large surface area, which leads to a high nucleation rate due to the secondary nucleation mechanism. However, when the temperature decreases linearly, as in case 3, some degree of crystal growth occurs at each crystallizer. The overall surface area of the crystals in case 3 is relatively small compared with that for cases



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with drastic cooling manner, such as case 1 or 5. Thus, the nucleation rate is minimized and the average size of the product is maximized, although this effect is not substantial. If the process objective is to obtain a high process yield, case 5 is the best strategy, as shown in Table S4. This is the same trend as in the PNDR. In addition, the trend of the process yield depending on the operating strategy is almost same as that in the PNDR, except for case 1. As the operating strategy does not have a significant effect on the average particle size in the SNDR, case 5 is the best operating strategy when considering both process yield and average size.

Figure 4. Simulation results with changing operating strategies in the SNDR; kb = 1.8534 × 104, kb2 = 4.7299 × 10-4, α = 1, β = 1, and N = 4. (a) Crystal number density, (b) Crystal volume fraction.

3.1.3. Both Nucleation Comparable Region In the previous sections, the extreme situations, PNDR and SNDR, were investigated. In many cases in industrial crystallization, the primary and secondary nucleation mechanisms have comparable contributions to nucleation, which is defined as the both nucleation comparable region (BNCR).


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The parameter values were selected not only to describe the behavior of the BNCR, but also to maintain an order of magnitude of the overall nucleation rate that was similar to the other cases. The range of the primary nucleation rate constant was selected as 4.8534 × 1074.8534 × 108 and that of the secondary nucleation rate was 4.7299 × 10-4-4.7299 × 10-5. The growth rate and nucleation rate orders were fixed at 1. In the BNCR, the parameter values, which have a relatively higher primary nucleation rate than secondary nucleation rate even though these rates are comparable, were kb = 4.8534 × 108 and kb2 = 4.7299 × 10-5. On the other hand, the parameter values when the effect of secondary nucleation is relatively larger were kb = 4.8534 × 107 and kb2 = 4.7299 × 10-4. To achieve a smooth transition from PNDR to SNDR, the parameter values were changed, first from kb2 = 4.7299 × 10-5 to kb2 = 4.7299 × 10-4 while maintaining kb = 4.8534 × 108, and then, kb was decreased from 4.8534 × 108 to 4.8534 × 107 while maintaining kb2 = 4.7299 × 10-4. The nucleation rates obtained by changing these nucleation rate constants are shown in Figure 5, just for representative cases, and in Figure S2 for all cases. The detailed nucleation rate constants of all cases are shown in Table S5. Case (a) and (e) are both extremes cases, and case (c) is a median case in the BNCR. The simulation results illustrated in Figure 6 show the volume fractions for different operating strategies when the nucleation rate constants are changed, as described above. In all cases, most of the nuclei are generated at the earlier crystallization stages by primary nucleation. However, as the process progresses toward the last stage, the nucleation rate is governed by secondary nucleation owing to the increased number of crystal particles obtained at the earlier stages. Therefore, in this region, both primary and secondary nucleation influence the overall nucleation rate. As the parameter values change from (a) to (e) in Figs. 5 and 6, the secondary nucleation rate increases relative to the primary nucleation rate in the BNCR. As the performance gap for the different operating strategies decreases, the best



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operating strategy for large average particle sizes changes from case 5 in (a) to case 3 in (e). This behavior revealed that there is a smooth transition between when the parameters for PDNR are changed toward those for SNDR.

Figure 5. Comparison between primary and secondary nucleation rates in each crystallizer with different operating strategies (case 3 and 5) in the BNCR. Parameter sets (a) kb = 4.8534 × 108, kb2 = 4.7299 × 10-5, (c) kb = 4.8534 × 108, kb2 = 4.7299 × 10-4, (e) kb = 4.8534 × 107, kb2 = 4.7299 × 10-4. Graph (1) and (2) show each nucleation rate in the BNCR for cases 3 and 5, respectively.


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Figure 6. Distribution of the crystal volume fraction for different operating strategies in the BNCR. (a) kb = 4.8534 × 108, kb2 = 4.7299 × 10-5, (b) kb = 4.8534 × 108, kb2 = 9.7299 × 10-5, (c) kb = 4.8534 × 108, kb2 = 4.7299 × 10-4, (d) kb = 9.8534 × 107, kb2 = 4.7299 × 10-4, and (e) kb = 4.8534 × 107, kb2 = 4.7299 × 10-4.



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3.2. Effect of Growth and Nucleation Rate Orders In this section, the effect of growth and nucleation rate orders on the product quality was studied. As no distinctive characteristics were discovered in the region between the PNDR and SNDR, this study focused only on the PNDR and SNDR, not on the BNCR. The range of the order parameters was selected as 0.5-1.5. First, the growth rate order was fixed at 1 while the nucleation rate order was changed from 0.5 to 1.5 to investigate the effect of changing the nucleation rate order. When changing the order parameter, the nucleation rate constants were adjusted to obtain a similar average size of the final product for case 5 in the PNDR and for case 3 in the SNDR. This adjustment is required to isolate the order effects from those of the nucleation and growth rates. Then, the same analyses were performed for the growth rate order changes. The results of the simulations with different operating strategies are summarized in Tables S6 and S7. In both the PNDR and SNDR, the most effective operating strategy was not different even with different growth and nucleation rate orders. In the PNDR, case 5 is the most effective operating strategy to obtain larger crystals, while cases 2 and 3 are the worst. In terms of the process yield, case 5 is also the best strategy, while case 1 is the worst over the entire range of the order parameter changes. In the SNDR, the largest crystal size is obtained in case 3, and case 5 is the best operating strategy for a higher process yield. This result is the same as those in the previous sections; therefore, the growth and nucleation rate orders do not significantly affect the choice of operating strategy. However, the differences in the process yields for different strategies are noticeably affected by the nucleation and growth rate orders, as shown in Figure 7. The difference in process yields for the worst and best strategies increases as both the nucleation and growth rate orders increase in the PNDR, whereas this difference does not change considerably in the


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SNDR. This tendency is clearly noticeable in Table S6. As the process yield is closely related to the degree of supersaturation in the final crystallizer, the reason for the change in process yield with different strategies could be related to the degree of supersaturation, as shown in Figure 8. In the PNDR, the degrees of supersaturation for case 1 increase as both the nucleation and growth rate orders increase, while those for case 5 show a smaller change. In the SNDR, despite changing the order parameters, the degrees of supersaturation are comparable for cases 1 and 5. This is likely the reason for the increased difference between the process yields of the best and worst cases in the PNDR when the nucleation and growth rate orders are increased. In the SNDR, no significant differences were observed. These results show that in the PNDR, the selection of appropriate operating strategy becomes more important as the nucleation and growth rate orders increase. However, in the SNDR, the importance of selection of operating strategy is not closely related to the order parameters. The changes in the average particle size for different strategies reveal a different tendency. In the PNDR, as the order of the growth rate increases, the difference between the best and worst strategies becomes more pronounced, which implies that the selection of the operating strategy becomes more important. Meanwhile, increasing the order of the nucleation rate decreases the difference between the best and worst strategies. However, in the SNDR, this difference is not changed significantly and is less than 5% over the whole range of growth and nucleation rate orders. As the order parameters increase, the nucleation and growth rates are predominately affected by the degree of supersaturation. Thus, the differences between the minimum and maximum nucleation and growth rates increases depending on the operating strategy which determines the degree of supersaturation in each crystallizer. Therefore, increasing the order of the growth rate, the increased difference in the growth rate caused by the operating strategy causes the difference in average size to increase significantly. Meanwhile, as the decrease in the nucleation rate effectively increases the



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growth rate, the change in the nucleation order showed the opposite trend. Thus, in the PNDR, the appropriate selection of operating strategy becomes much more important as the order of the growth rate increases and the order of the nucleation rate decreases. However, in the SNDR, the operating strategy does not influence the average size of the product significantly, as the change in the secondary nucleation rate with negligible primary nucleation is closely related to the surface area the of crystals, which, in turn, is a function of the average size of the crystals.

Figure 7. The min-max differences in the process yield and average size for different operating strategies as the order of the crystallization kinetic equations is varied. (a) Process yield with order of growth rate (b) Process yield with order of nucleation rate (c) Average size with order of growth rate (d) Average size with order of nucleation rate.


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Figure 8. Degree of supersaturation in the final crystallizer for each case in each featured region as the order parameters are varied. In case 5 with α = 0.5 and β = 1, the degree of supersaturation is almost zero because equilibrium is reached in the final crystallizer.

3.3. Effect of Stage Number As the process analyzed in this study is multistage MSMPR crystallization, the number of stages is an important parameter. In the previous sections, the stage number was fixed at 4. To investigate the effects of the stage number, in this section, the number of stages was varied from 1 to 10, while maintaining a constant total volume of the crystallizers. These analyses were performed in two different featured regions, as in the previous sections, with the orders of the nucleation and growth rates fixed at 1. For the sake of simplicity, the operating strategy used in these analyses was fixed as the best operating condition found in



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the previous sections for large average particle sizes. Thus, case 5 was selected and analyzed in the PNDR and case 3 in the SNDR.

Figure 9. Effect of the number of stages on (a) average particle size, and (b) process yield. The results of the simulation showed different behavior depending on the characteristics of the featured regions. As shown in Figure 9, the average size of the particles in the PNDR decreases as the number of stages increases, whereas the average size in the SNDR increases slightly. The reason of the difference can be explained by the analysis results in sections 3.1.1 and 3.1.2. Generally, supersaturation is affected by the temperature and residence time of each crystallizer. If a single-stage crystallizer is used, a high degree of supersaturation due to the large temperature drop is sustained in the whole volume of the well-mixed crystallizer, and the particle size increases over the entire residence time owing to the influence of high supersaturation. However, when the stage number is increased to more than one, very high degree of supersaturation is sustained only in the early stage crystallizers resulting in the generation of a large number of nuclei, whereas in the later crystallizers, the degree of supersaturation is reduced rapidly and the particle size grows owing to the relatively low degree of supersaturation. Thus, the average particle size is smaller than that obtained with a single-stage crystallizer in the PNDR, while the yield is increased. This result is similar to


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that reported in a previous study.50 The difference in the CSD with changing the number of stages is not as big as that reported by Couper et. al., as shown in Figure 10. In this previous study, nucleation was assumed to occur only in the 1st crystallizer. However, if nucleation occurs in all of the crystallizers, the difference in the CSD will be reduced.50 In the SNDR, however, the average particle size increases as the number of stages increases. With a stage number of more than five, the average size no longer increases. Linear cooling manner is the best strategy for large average particle sizes in the SNDR, as determined in section 3.1.2. As the number of stage increases, the temperature between each stage decreases. Because of this, the shape of the temperature drop curve approaches linearity. Thus, larger particles can be obtained as the number of stages increases.

Figure 10. Dependence of CSD on the stage number (a) in this paper, and (b) from the reference 50. The analysis of the process yield according to stage number showed a different tendency than that obtained for average size. The process yield in the PNDR increases when the stage number is increased, whereas the process yield in the SNDR decreases. As the number of stages increases for case 5 in the PNDR, the volume of crystallizers in which the final temperature is maintained and there is no temperature difference between the inlet stream and



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outlet stream increases. The residence time is sufficient to reach equilibrium, so the process yield increases. On the other hand, if a linear cooling strategy, such as case 3, is applied in this crystallization system, the volume of the crystallizer maintained at the final temperature decreases, so the process yield in the SNDR is reduced slightly. According to these results, the single-stage crystallizer system is the best in terms of average size in the PNDR, even though it may not be possible to apply a large temperature drop in one stage due to physical limitations. Considering both average size and process yield, however, it is best to choose a stage number of three or four. In the SNDR, although the process yield decreases with increasing number of stages, a stage number of two or three is appropriate for both average size and process yield. However, there are some limitations on applying this result directly to industrial systems. There is a critical assumption of perfect mixing, which cannot be easily achieved during industrial crystallization. The non-ideality of mixing in the crystallizer is greater with lower numbers of stages, which, in turn, increases the volume of each crystallizer. Therefore, these results should be used with care for designing industrial crystallizers.

4. CONCLUSIONS A series of parameter analyses were performed in this study to investigate the optimal operating strategy for a continuous multistage MSMPR crystallization. Based on the estimated base parameters obtained from industrial data, several parameter sets were designed to investigate the effects of operating strategies on the final product quality. As the nucleation rate is governed by primary and secondary nucleation, the nucleation rate constants can be categorized into three featured regions; PNDR, SNDR, and BNCR. In addition, the effects of the orders of the crystallization kinetics and the stage number were observed in terms of the average particle size and the process yield. Case 5 is revealed as the


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best operating strategy in the PNDR for both a large average size and high process yield. In the SNDR, case 3 is the best for a large average size, whereas case 5 is best for a high process yield. There was no noticeable difference in the BNCR, with a smooth change from the PNDR to the SNDR. This tendency was maintained, even with different nucleation and growth rate orders. In addition a stage number of around three is best in both the PNDR and SNDR. Therefore, the most important parameters in continuous multistage MSMPR crystallization systems are the primary and secondary nucleation rate constants, which should be carefully considered when designing and operating multistage MSMPR crystallizers. For simplicity, the model developed in this study assumed perfect mixing and no breakage or agglomeration. However, in industrial crystallization, there are some cases that conflict with these assumptions. Thus, future work should focus on incorporating non-ideal mixing behavior and crystal breakage and agglomeration into the model.

Supporting information Figure S1. Numerical solving algorithm for steady-state continuous multi-staged MSMPR crystallizer system Figure S2. Comparison between primary and secondary nucleation rates in each crystallizer with different operating strategies (cases 3 and 5) in the BNCR. Table S1. Estimated base crystallization kinetic parameters. Table S2. Base values and variation ranges of the important parameters. Table S3. Operating strategies at each crystallizera Table S4. Average particle size, coefficient of variation and process yield of the product at each operating strategy and each featured region



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Table S5. Primary and secondary nucleation rate constants of parameter sets (a) ~ (e) in the BNCR region. The parameter set (a) is the closest to the PNDR, and the parameter set (e) is the closest to the SNDR Table S6. Process yields with changing nucleation and growth rate order at each operating strategya. Table S7. Average sizes (㎛) of crystals with changing nucleation and growth rate order at each operating strategya.

This information is available free of charge via the Internet at http: //pubs.acs.org.

ACKNOWLEDGEMENTS This work was financially supported by the Human Resources Development program (No. 20134010200600) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy, LOTTE CHEMICAL CORPORATION and Korea University grant.

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List of figure captions Figure 1. Schematic diagram of the continuous multistage MSMPR crystallization process. F is the flow rate, C is the concentration, T is the temperature, P is the pressure, and µ3 is the third moment of the particles. The subscripts in and out are the inlet and outlet stream, respectively, and v is the vapor stream.

Figure 2. Solubility and operating strategies for the continuous four-stage MSMPR crystallization process. The degrees of supersaturation in each crystallizer are denoted with its crystallizer number. (1) Cases 1, 2, and 3, (2) Cases 3, 4, and 5.

Figure 3. Simulation results with changing operating strategies in the PNDR; kb = 4.8534 × 108, kb2 = 4.7299 × 10-8, α = 1, β = 1, and N = 4. (a) Crystal number density, (b) Crystal volume fraction.

Figure 4. Simulation results with changing operating strategies in the SNDR; kb = 1.8534 × 104, kb2 = 4.7299 × 10-4, α = 1, β = 1, and N = 4. (a) Crystal number density, (b) Crystal volume fraction.

Figure 5. Comparison between primary and secondary nucleation rates in each crystallizer with different operating strategies (case 3 and 5) in the BNCR. Parameter sets (a) kb = 4.8534 × 108, kb2 = 4.7299 × 10-5, (c) kb = 4.8534 × 108, kb2 = 4.7299 × 10-4, (e) kb = 4.8534 × 107,


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kb2 = 4.7299 × 10-4. Graph (1) and (2) show each nucleation rate in the BNCR for cases 3 and 5, respectively.

Figure 6. Distribution of the crystal volume fraction for different operating strategies in the BNCR. (a) kb = 4.8534 × 108, kb2 = 4.7299 × 10-5, (b) kb = 4.8534 × 108, kb2 = 9.7299 × 10-5, (c) kb = 4.8534 × 108, kb2 = 4.7299 × 10-4, (d) kb = 9.8534 × 107, kb2 = 4.7299 × 10-4, and (e) kb = 4.8534 × 107, kb2 = 4.7299 × 10-4.

Figure 7. The min-max differences in the process yield and average size for different operating strategies as the order of the crystallization kinetic equations is varied. (a) Process yield with order of growth rate (b) Process yield with order of nucleation rate (c) Average size with order of growth rate (d) Average size with order of nucleation rate.

Figure 8. Degree of supersaturation in the final crystallizer for each case in each featured region as the order parameters are varied. In case 5 with α = 0.5 and β = 1, the degree of supersaturation is almost zero because equilibrium is reached in the final crystallizer.

Figure 9. Effect of the number of stages on (a) average particle size, and (b) process yield.

Figure 10. Dependence of CSD on the stage number (a) in this paper, and (b) from the reference 50.



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