Multiobjective Optimization of Unseeded and Seeded Batch Cooling

Article pubs.acs.org/IECR

Multiobjective Optimization of Unseeded and Seeded Batch Cooling Crystallization Processes K. Hemalatha†,‡ and K. Yamuna Rani*,†,‡ †

Process Dynamics and Control group, Chemical Engineering Department & ‡Academy of Scientific and Innovative Research (AcSIR), CSIR-Indian Institute of Chemical Technology, Hyderabad 500007, India ABSTRACT: Multiobjective optimization (MOO) of batch cooling crystallization is carried out for the development of optimal operating recipes for unseeded and seeded crystallization processes. Mean size and coefficient of variation (CV) are the two objectives considered for unseeded batch cooling crystallization of paracetamol, whereas mean size, CV, and nucleated mass are considered as objectives for the seeded batch cooling crystallization of potassium nitrate. In this work, along with finding the optimal temperature trajectories, the effect of choice of objectives on the final achievable Pareto front is analyzed using two different objective functions, namely number mean size and surface-weighted mean size. Further, the capability of MOO is also exploited to determine the feasible ranges of combinations of seed properties like initial seed mass and seed mean size for better crystal size distribution (CSD).

1. INTRODUCTION Batch cooling crystallization is one of the most common operations in industries especially in small scale production of high value added chemicals, due to its ability to provide high purity separations and flexibility of operation using different recipes.1−3 Optimization of such an operation is very important to achieve desired crystal size distribution as it influences the product quality and efficiency of downstream processes.2−4 The operating conditions of a crystallizer directly influence the product characteristics such as mean crystal size, crystal size distribution, yield, purity, and morphology,5 which in turn influence the downstream operations such as filtration, drying, and ultimately the quality of the finished product. Thus, in order to obtain the finished product of desired quality, it is necessary to achieve large crystal size, uniform distribution, high yield and purity, and suitable morphology during crystallization operation. In case of batch cooling crystallization, the optimal operating recipe can be determined by manipulation of temperature or supersaturation profile alone for an unseeded process3 and manipulation of seed properties along with temperature profile for a seeded process.4,6,7 Significant research has been directed toward determining the optimal temperature trajectory for a batch cooling crystallization process considering a single objective of maximizing weight mean size (WMS) or minimizing coefficient of variation or minimizing the ratio of a moment of nucleation to the moment of seed.2−6 There has been an increased interest in application of multi objective optimization to various crystallization processes during the past decade.7−11 However, research concerning the application of multi objective optimization to batch cooling crystallization processes is sparse. A multi objective framework for a seeded batch cooling crystallizer has been presented7 through definition of different © XXXX American Chemical Society

combinations of objectives in terms of CSD along with their Pareto optimal solutions for potassium sulfate. Recently, a multiobjective optimization framework has been employed considering crystal size and aspect ratio as the objectives and the resulting optimal temperature profiles are implemented11 experimentally for paracetamol and potassium dihydrogen phosphate. Paracetamol and potassium nitrate are some of the industrially important chemicals whose crystallization from aqueous solutions has been of academic interest.11−17 For the cooling crystallization of paracetamol from the aqueous solution, single objective optimization (SOO) to maximize weight mean size has been attempted to generate optimal temperature and concentration profiles, which have been implemented through feedback control.14 For this system, a complex multidimensional population balance model has been developed11 and employed for multiobjective optimization with mean crystal size and aspect ratio as size and shape objectives through experimental and simulation studies, and it was concluded that variation of temperature profiles has no significant effect on the achievable aspect ratio for such nucleation dominant systems. Another popularly studied process is batch cooling crystallization of potassium nitrate from aqueous solution.4−6,15−21 Simulation and optimization studies reported have shown that seeding characteristics affect the achievable final CSD along with the optimal temperature profile.4 It has also been demonstrated that the shape of the temperature profile varies according to the choice of the Received: Revised: Accepted: Published: A

February 10, 2017 April 27, 2017 May 4, 2017 May 4, 2017 DOI: 10.1021/acs.iecr.7b00586 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research objective studied.6,19 A process control strategy has been evaluated to obtain large product crystals where dissolution of crystals by temperature raise has also been considered to seek optimal cooling profiles for various seeding temperatures.18 Recently, the effect of seed size and seed loading on final CSD has been evaluated, and seed loading charts have been proposed using nucleation kinetics.20 For unseeded paracetamol crystallization,11 although multiobjective optimization has been reported, the focus was only the size and shape of the crystals, and the uniformity of crystal size has not been considered. For seeded potassium nitrate crystallization, only single objective optimization has been attempted. Therefore, in this work, multiobjective optimization of unseeded paracetamol crystallization and seeded potassium nitrate crystallization are attempted using maximization of mean size and minimization of CV as the objectives in the unseeded case and, additionally, minimization of nucleated mass in the seeded case. An attempt has also been made to study and compare multiobjective optimization using different objectives. A procedure has been presented where apart from investigating the effect of seed loading, MOO is used to find feasible limits for various combinations of seed properties. The remainder of this article is organized as follows. In section 2, dynamic models for the two case studies are presented. In section 3, the multiobjective optimization procedure used in this work is illustrated. In this section, along with the solution methodology, a detailed description of choice of performance objectives for the present work is given in the context of recent literature. In section 4, the results obtained for each case study are discussed, and in the final section, concluding remarks are given.

dt dμi dt

=B

= iGμi − 1 + Br0i

G = kg ΔC g

(5)

(6)

ΔC = C − C*

C* is solubility in g solute/g solvent and is a function of temperature (T in K) as shown in Table 1. Table 1. Crystallization Model Parameters and Variables Used in Simulation for the Systems Studied model parameters and variables for paracetamol system (Nagy et al.14) nucleation parameter (kb) nucleation parameter (b) growth parameter (kg) growth parameter (g) density (ρc) volumetric shape factor (kv) nucleated crystal size (r0) maximum temperature (Tmax) minimum temperature (Tmin) batch time (tf) solubility (C*)

7.775 × 1019 6.2 0.0166 1.5 1.296 1

#/g min m/min g/cm3

0 318

m K

293

K

300 min 1.3066 − 0.0090567T + g/g water 0.000015846T2 model parameters and variables for potassium nitrate system (Chung et al.4)

nucleation parameter (kb) nucleation parameter (b) growth parameter (kg) growth parameter (g) density (ρc) volumetric shape factor (kv) nucleated crystal size (r0) mass of the solvent (Msolv) maximum temperature (Tmax) minimum temperature (Tmin) maximum seed mass (Mmax s ) minimum seed mass (Mmin s ) maximum initial mean seed size (Lmax s ) minimum initial mean seed size (Mmin s ) batch time (tf) solubility (C*)

(1)

The standard method of moments is usually used to reduce the population balance equation into ordinary differential equations which are easy to solve for optimization. A wellmixed crystallizer is assumed with the size-independent growth process and with negligible crystal breakage and agglomeration and no growth rate dispersion. Thus, the properties of CSD can be described by the following moment equations

dμ0

(4)

where kg, kb, g, and b are kinetic parameters. To obtain larger and fewer crystals, it is desirable to suppress nucleation and enhance growth. This can be achieved by controlling the supersaturation, which is the actual driving potential for both the nucleation and growth rates and is represented as a concentration difference ΔC.

2. SYSTEMS STUDIED In the present study, two case studies are considered for simulation and multiobjective optimization, namely, unseeded and seeded batch cooling crystallization processes. The dynamic models for the two case studies are discussed below. 2.1. Unseeded Batch Cooling Crystallization of Paracetamol. In this case study, unseeded cooling crystallization of paracetamol from aqueous solution as studied by Nagy and coworkers13 is considered for MOO. The general mathematical framework for modeling crystallization processes is the population balance equation which is expressed as ∂n ∂n + G (t ) =0 ∂t ∂L

B = k bΔC b

4.6401 × 1011 1.78 1.1612 × 10−4 1.32 2.11 × 103 1 0 7.57 × 103 32 22 0.0145 1.0 × 10−6 600

#/m3 s

m kg °C °C kg/kg solv kg/kg solv μm

5

μm

160 0.1286 + 0.00588T + 0.0001721T2

min g/gwater

m/s kg/m3

The initial condition for eq 2 and eq 3 is given as μi(0) = 0 (i = 0,1,2,3) and C(0) = 0.0256 g/gwater for eq 7. Assuming constant volume, the amount of solute leaving the solution must be accounted for by crystal growth and nucleation, and the mass balance of solute concentration is given as

(2)

(3)

dC = −3ρc kvGμ2 − ρc kvBr03 dt

The expressions for crystallization kinetics (i.e., nucleation rate, B, and growth rate, G) are given as B

(7) DOI: 10.1021/acs.iecr.7b00586 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research where ρC is the density of the crystal and C is the solute concentration in g solute/g solvent. 2.2. Seeded batch cooling crystallization of potassium nitrate. In this case study, an industrial scale batch cooling crystallizer producing potassium nitrate reported by Chung and co-workers4 is considered as a benchmark problem for determination of optimal temperature profiles along with optimal seed properties. In addition to the overall moment equations, eqs 2 and 3, the final crystals can be characterized in terms of the final amount of nucleated crystal mass or seed crystal mass in seeded crystallization. Quantification of the crystals grown from seeds is done by writing a mass balance over the seed crystals only using the method of moments. dμ0s dt dμjs dt

=0 = jGμjs− 1

3.1. Performance Objectives. The optimal profiles are determined based on surrogate objective functions attributed to some property of CSD, formulated in terms of moments of CSD at final batch time, when the population balance model is represented by moment differential equations. Usually, in most of the optimization studies, mean size and CV are considered as objectives.3−5 Review of different objective functions used by various authors has been presented recently with the focus on seeded crystallization processes.6 In a study by Ward and coworkers,6 it has been concluded that the choice of objectives determines the shape of the temperature profile and the objectives have been categorized as those leading to either early growth or late growth profiles. In a similar way, Hsu and Ward19 have classified the objectives into three categories: those aiming to maximize a product average size (either the number-average size or weight-average size), those aiming to maximize a moment of the seed crystals or minimize a moment of the nucleated crystals, and those attempting to minimize the CV (either the number CV or the mass CV). They have also stated that the most suitable objective function for seeded crystallization processes in several instances is minimizing the nucleated mass, which is the third moment of nucleation. Yet, in another recent study,21 nine different objectives which include four moments of nucleation and different definitions of mean size and CV have been studied and compared in a single objective optimization context. All these studies19,21 consistently stated that using number mean size and number coefficient of variation for optimization of seeded crystallization may lead to large nucleated mass, which is undesirable. Hence, for the present MOO study of seeded crystallization, maximizing weight mean size and minimizing weight coefficient of variation based on higher-order moments are considered as objectives in addition to minimizing the nucleated mass. Suitable seed properties like seed mass and seed size are known to suppress the nucleation of crystals and in turn the nucleated mass. Thus, in our work, the effect of seeding on the performance of MOO is also evaluated. Comparison of various objectives has not been reported in the context of unseeded cooling crystallization. Thus, for our study of unseeded crystallization, two MOO cases are considered where maximizing number mean size (NMS) and minimizing number CV are chosen as the objectives in the first case and maximizing surface weighted mean size, which is another representation of WMS, along with minimizing number CV are considered as objectives in the second case. 3.2. Problem Formulation for Systems Studied. 3.2.1. Unseeded Batch Cooling Crystallization of Paracetamol. Usually in batch crystallization processes, the desirable crystal characteristics are large mean size and low coefficient of variation. The two objective functions, mean size and CV are considered for optimization. There are different representations for mean size in the literature.6−8 In accordance with the moments considered for mean size calculation, the number mean size6 is represented as μ1/μ0, which is denoted as NMS, and surface-weighted mean size8 is represented as μ3/μ2, which is denoted as WMS throughout this section. Optimization studies are carried out for this system in two ways, referred to as two cases. The problem formulation is done with the chosen representations of objective functions to be optimized simultaneously with respect to the temperature profile and time of operation and appropriate constraints are incorporated. In the first case (case 1), NMS and number CV are considered as objective functions. These individual

(8)

(9)

The nucleation rate and growth rate is represented in standard power-law form4 B = kbS bμ3

(10)

G = kg S g

(11)

where kg and g are the growth parameters and S is the relative supersaturation expressed as S=

C − C* C*

(12)

where C* is the solubility given in Table 1. For a seeded cooling crystallization process, the initial moments of seeds can be expressed as μjs = NsLsj

(13)

where Ls is the size of the seeds introduced at the initial temperature and Ns is the number of crystals with uniform distribution, which is calculated from the mass of seeds Ms as Ns =

Ms ρc kvLs3

(14)

and initial values for eqs 2 and 3 are μi(0) = μsj where i,j = 0,1,2,3,4,5 and C(0) = 0.493g/gwater for eq 7.

3. MULTIOBJECTIVE OPTIMIZATION For crystallization processes, multiple performance objectives should be considered to seek open loop optimal temperature trajectories.3−5 Increase in performance of one of the objectives hinders the performance of other objectives and vice versa. Thus, a trade-off between these objectives exists which ultimately results in multiple solutions for an optimization problem. Before formulation of a MOO problem, a better understanding regarding the suitable choice of objectives is necessary. In this section, discussion regarding selection of objectives is presented first. MOO problem formulation of unseeded and seeded batch cooling crystallization processes is discussed later. The optimization technique or algorithm used for the present study is discussed in detail, where the representation of input variables and solution methodology considered for optimization are presented. C

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Industrial & Engineering Chemistry Research objectives are optimized simultaneously (J1=[J11,J12]), and the optimization problem is represented as follows with the objective function J11, representing maximization of NMS; the number coefficient of variation6 is represented by the objective function J12. max J11 = T (t )

c1: Tmin ≤ T (t ) ≤ Tmax dT ≤0 dt c3: t ≤ tfinal c 2:

μ1

c4: M smin ≤ Ms ≤ M smax

μ0 ⎛μ μ ⎞ ⎜⎜ 2 20 − 1⎟⎟ ⎝ μ1 ⎠

min J12 = T (t )

In the second case (case 2), initial seed mass is fixed at a particular value and initial seed mean size is considered as a decision variable along with the temperature variables. The multiobjective optimization for this case is represented similar to the first case as (J4 = [J41, J42, J43]), and the optimization problem is formulated as follows

(15)

subject to c1: Tmin ≤ T (t ) ≤ Tmax

max J41 =

T (t ), Ls

dT ≤0 dt c3: t ≤ tfinal c 2:

min J42 =

(16)

T (t ), Ls

In the same way, for the second case (case 2), WMS and number CV are considered as objective functions represented by J2 = [J21, J22] as follows max J21 = T (t )

max J31 =

min J32 =

T (t ), Ms

(20)

c1: Tmin ≤ T (t ) ≤ Tmax

⎛μ μ ⎞ ⎜⎜ 2 20 − 1⎟⎟ ⎝ μ1 ⎠

dT ≤0 dt c3: t ≤ tfinal c 2:

(17)

c4: Lsmin ≤ Ls ≤ Lsmax

(21)

The constraints c4 in eqs 19 and 21 ensure that the initial seed distribution is practical. In this study, mass of the seeds is constrained to less than 10% of the final crystal mass4 and initial seed mean size ranges from 5 to 600 μm. 3.3. Solution Methodology. The problems defined in equations starting from eq 15 to eq 21 are dynamic optimization problems since the independent variable T(t) is a continuous function of time. Solution methodology for dynamic optimization is complex since it involves application of Pontryagin’s maximum principle and solution of a two-point boundary value problem. Therefore, these MOO problems are converted into algebraic optimization problems by approximating the independent variable profile using control vector parametrization as a piecewise linear profile with a finite number of decision variables. There are different methods available for MOO, among them, the well-known and widely used evolutionary algorithm is the real-coded elitist nondominated sorting genetic algorithm popularly known as NSGA-II.22 It has been successfully applied to various optimization problems related to chemical engineering.23 In the present study, the NSGA-II algorithm is employed to solve the MOO problems through control vector parametrization. The control variable (i.e., coolant temperature) is discretized by piecewise linear function which has been proposed and used earlier by various researchers.7,8,11 The method used by Sarkar et al.7 is briefly reviewed in this section. In order to parametrize the input temperature trajectories T(t), the time interval (0, tfinal) is divided into P time stages of equal length (tfinal/P) and piecewise linear control policy is sought in each time interval (tk,tk+1).

μ4 μ3 ⎛μ μ ⎞ ⎜⎜ 5 2 3 − 1⎟⎟ ⎝ μ4 ⎠

min J33 = μ3n = μ3 − μ3s

T (t ), Ms

⎛μ μ ⎞ ⎜⎜ 5 2 3 − 1⎟⎟ ⎝ μ4 ⎠

subject to

subject to eq 16 The temperature should always be a decreasing function of time for cooling crystallization throughout the batch. The first two inequality constraints in eq 16 ensure that the temperature profile can be implemented. 3.2.2. Seeded Batch Cooling Crystallization of Potassium Nitrate. In seeded crystallization processes, the goal is to increase the size of seed grown crystals. This may be achieved by suppressing the nucleation of new crystals or total mass of nucleated crystals. To investigate the effect of suitable seeding on final product characteristics, two different cases are considered in the study of this system. In the first case (case 1), initial seed mean size is fixed at a particular value and initial seed mass is also considered as a decision variable in addition to the nine temperature variables. In the multiobjective optimization of this case (J3 = [J31, J32, J33]), three properties representing CSD are considered as objective functions expressed in terms of moments of CSD [i.e., volume weighted mean size (J31), mass coefficient of variation (J32), and third moment of nucleation (J33)] and the optimization problem is represented as follows T (t ), Ms

μ3

min J43 = μ3n = μ3 − μ3s

μ2

T (t )

μ4

T (t ), Ls

μ3

min J22 =

(19)

(18)

subject to D

DOI: 10.1021/acs.iecr.7b00586 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research T (t ) = T (k ) +

(T (k + 1) − T (k)) × (t − t(k)) tfinal /P

WMS) are shown in Figure1. The trade-off between the two objectives chosen is observed clearly for both cases. It is

(22)

where T(k) is the temperature at the beginning of kth interval. The optimal control problem then is to find T(k), where k = 1,2, ···, P. This temperature profile is used as the input for simulation and optimization through NSGA-II. In the present study, P is considered as 10. In this study, nine (P-1) temperature variables (excluding the initial and final temperature values) are represented by a real valued vector known as a chromosome that forms input to compute objective functions during the NSGA-II simulation procedure. These values are generated randomly in the initial population between the upper and lower bounds of the admissible cooling range of temperature [T ∈ (Tmin, Tmax)], for the given time of operation, tfinal, thus satisfying the first constraint c1 in all the cases. In order to satisfy the second constraint c2, the randomly generated temperature values are sorted in descending order. The third constraint is automatically satisfied since the time of simulation is fixed as tfinal. Additional variables like mass of the seed or size of the seed are considered as additional decision variables, and the additional real-values are appropriately added to the chromosome to account for and these are also generated randomly in the initial population between their lower and upper bounds, thus satisfying the additional constraint c4 in the seeded case. The objective functions are evaluated for each chromosome with known values of these quantities through the simulation of the model equations.

Figure 1. Pareto fronts of two cases of optimization with (CV, WMS) and (CV, NMS) as competing objectives.

observed that the solutions obtained with CV and NMS combination has a range of mean size from 89 to 120 μm and CV varying from 16% to 30%, whereas solutions obtained with CV and WMS combinations has a wider range of mean size from 95 to 150 μm and CV from 16% to 90%. It is intended to compare the performance of WMS and NMS as objective functions from the two MOO cases, and the procedure adopted is explained as follows: Step 1: optimal operating policies are determined by carrying out MOO with two objective functions NMS and CV (case 1). The values of objective function-NMS for all the Pareto solutions are denoted as NMS_optimized. Step 2: optimal operating policies are implemented through simulation for all the Pareto solutions obtained in step 1, and the corresponding WMS values are calculated. These sets of WMS values are called WMS_calculated. Step 3: optimal operating policies are determined by carrying out MOO with two objectives WMS and CV (case 2). The values of objective function-WMS for all the Pareto solutions are denoted as WMS_optimized. Step 4: optimal operating policies are implemented through simulation for all the Pareto solutions obtained in step 3, and the corresponding NMS values are calculated. These sets of NMS values are called NMS_calculated. Step 5: Figure 2a represents comparison of values obtained in step 3 and step 2 (i.e., WMS_optimized and WMS_calculated). Step 6: Figure 2b represents comparison of values obtained in step 1 and step 4 (i.e., NMS_optimized and NMS_calculated). It is observed from Figure 2a that the solutions of the WMS_calculated from NMS optimization overlap and form a part of Pareto solutions of WMS optimization. It is evident that when WMS is chosen for optimization, the solutions cover a wider optimal region (wider ranges of WMS and CV) and when NMS is chosen, the solutions correspond to a narrower range of WMS and CV where the upper and right-most part of the Pareto front with WMS optimization is not included. In the same way, in Figure 2b, the values of NMS_calculated from solutions of WMS optimization overlaps with the solutions of NMS optimization up to 120 μm. The upper Pareto region of NMS_calculated for WMS optimization has high CV and low

4. RESULTS AND DISCUSSION Unlike the SOO problem, the solution of the MOO problem corresponds to a set of trade off solutions, each expressing a particular compromise between different objectives. These set of solutions are nondominated or noninferior solutions and are called “Pareto optimal solutions”. The image of these solutions formed by the objective components in the objective space is known as a “Pareto front”, where each objective component of any solution along the Pareto front can only be improved by degrading one or more of its other objective components. Since none of the solutions in the nondominated set is exactly better than any other, any one of these is an acceptable solution. The NSGA parameters in this study for the two case studies are predefined and fixed to a particular value. The population size is considered as 100 and the number of generations is predetermined to 100 with the crossover probability of 0.9 and mutation probability of 0.05. Though the population size and number of generations are increased beyond these values, there is not much improvement in the final results. Therefore, for interpretation of results, the above values are chosen. The crossover distribution index is chosen as 10 and the mutation index as 20 for all the cases. 4.1. Unseeded Batch Cooling Crystallization of Paracetamol. The simulation of the paracetamol crystallization is carried out using the parameters listed in Table 1. The temperature is decreased from 45 to 20 °C for the unseeded batch cooling crystallization process. For each of the cooling profiles generated randomly through NSGA-II, the simulation is carried out and the objectives are evaluated using these profiles. MOO is carried out for the two cases explained in section 3.2 with the chosen objectives, and optimization is carried out until the predetermined number of generations is reached. The Pareto solution sets obtained for the two MOO cases where the objective functions are (CV, NMS) and (CV, E

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Figure 2. Comparison of Pareto fronts for (a) optimized and calculated WMS (b) optimized and calculated NMS.

Figure 3. (a) Pareto front with NMS and CV as competing objectives, (b) temperature profiles, (c) mean size profiles, and (d) coefficient of variation profiles, corresponding to the solutions chosen from the Pareto front in (a).

chosen from the Pareto set to represent three points in different regions (i.e., upper end of the Pareto, lower end of the Pareto, and a point in the middle region which is chosen as the Pareto optimal point). The temperature profiles for these different points in the Pareto front are plotted in Figure 3. In Figure 3a, the upper end of the Pareto represents a maximum mean size of 120.02 μm and 30.15% of CV, whereas the lower end of the Pareto front represents a low mean size of 88.66 μm but a low CV of 15.97%. A point in the middle region of the Pareto set has been chosen as the best solution for the unseeded crystallization, which has a mean size of 112.27 μm and CV of 19.73%. The temperature profiles corresponding to the three points are plotted in Figure 3b, and it can be observed that the profiles are different for different objective function values but exhibit a similar behavior, where the temperature trajectory takes a deep dip initially showing nucleation of the crystals followed by a slow decrease in temperature where the growth of the crystals also takes place; subsequently, fast cooling is

NMS, showing that there is no trade-off between the objectives in that region, and it can also be observed that for the same NMS values, two solutions with different CV are obtained, of which only the lower Pareto region is optimal. For the same reason, the solutions corresponding to the upper Pareto front in Figure 2b do not appear in Figure 2a as part of the Pareto front for NMS optimization. It can be concluded that, though a part of the Pareto front of two cases are overlapping, the results of two cases are different. Hence, it is necessary to decide on the correct choice of objective function (i.e., either NMS or WMS prior to the optimization study based on the problem requirement). The optimization of CV and WMS (case 2) has been studied, and optimal temperature profiles are reported and discussed for Paracetamol crystallization elsewhere by Hemalatha and Rani.24 Therefore, in the present work, the MOO of CV and NMS (case 1) is chosen for the interpretation of results. Any solution in the Pareto set is an optimal solution. Three solutions are F

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dimension of the three-dimensional figure is represented by one of the objectives. In the provided objective space, NSGA-II randomly chooses seed mass within the limits for a fixed seed size and generates optimal solutions. Different Pareto fronts obtained for five different seed sizes are plotted in Figure 4 for comparison. From Figure 4, the trade-off between the objectives (WMS, CV) and (WMS, μn3) can be seen very clearly for different initial masses of the seeds with various seed sizes. Each point on the Pareto set represents an optimal temperature profile corresponding to a given seed mass and seed size. In Figure 5, the values of three objectives are plotted individually as a function of the mass of seeds that are obtained

observed at the end of the batch. The corresponding NMS and CV profiles throughout the batch for different temperature profiles chosen in three different regions of the Pareto front are shown in Figure 3c and d. From Figure 3c, it is observed that the mean size is zero initially until 30 min of operation following a steep increase corresponding to the nucleation of crystals and further increase in mean size throughout the batch due to growth phenomena. In the same way from Figure 3d, a sudden increase in CV is observed due to nucleation, followed by a steep decrease in CV and then stabilizing toward the end of the operation showing the attainment of uniform distribution of the crystals. The results of MOO of case 1 in this study, when compared with the results of MOO of case 2 carried out by Hemalatha and Rani,24 reveal that the shape of the temperature trajectories obtained in both the cases of optimization is the same, although the Pareto front obtained is different. 4.2. Seeded Batch Cooling Crystallization of Potassium Nitrate. The system of equations, eqs 2 and 3 and eqs 7 to 11 are simulated with the parameters given in Table 1 for seeded batch cooling crystallization of potassium nitrate. The temperature inside the crystallizer is decreased from 32 to 22 °C. Two different cases are considered in the optimization as discussed in section 3.2. In case 1, along with the nine temperature variables, the initial mass of the seeds is considered as a tenth decision variable, while fixing the initial seed mean sizes. Five independent optimization runs are conducted with 5 different initial seed mean sizes (i.e., 50, 150, 300, 450, and 600 μm, respectively). In case 2, the optimization is studied, considering initial mean size of seeds as a decision variable keeping the initial seed mass fixed. Seeding is assumed to be done at the initial temperature and at the beginning of the batch operation. Six independent optimization runs are conducted with 6 different initial seed mass values chosen between the minimum and maximum limits (i.e., 1.32 × 10−5, 1.45 × 10−4, 0.001, 0.0044, 0.009, and 0.0145 kg/kg solvent, respectively). In case 1, the resulting three-dimensional Pareto set obtained for MOO through NSGA-II for three objectives [i.e., maximization of WMS, minimization of CV and minimization of nucleated mass (μn3)] is as shown in Figure 4. Each

Figure 5. Pareto solution sets obtained for seed mass as a decision variable with respect to a given initial seed mean size for the three objectives individually: (a) WMS, (b) CV, and (c) μn3.

through optimization for five different optimization runs with fixed seed sizes. From Figure 5, it can be seen that a very small amount of seed mass favors large weight mean size, resulting in large CV and μn3 which is undesirable. It can also be observed that for a small initial seed mean size of 50 μm, the Pareto set consists of solutions constituting a wide range of mass of the seeds, where an increase in the initial seed mass resulted in the decrease of WMS favoring a decrease in CV and nucleated mass. But, for an initial seed mass over 0.002 kg/kg solvent, this

Figure 4. Pareto fronts with initial seed mass as a decision variable for various given initial seed mean sizes. G

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Industrial & Engineering Chemistry Research effect is not seen and the values of objective functions has reached a constant value. With 150 μm initial seed mean size, the same trend is observed with the Pareto solutions where increase in seed mass over 0.009 kg/kg solvent is exerting a little effect over the three objectives considered. In a similar way, when larger seed crystals of 300 μm mean size are considered, WMS is insensitive to the increase in seed mass. A final mean size of 620 μm is obtained for all the solutions with a seed mass ranging from 0.0044 to 0.0145 kg/kg solvent, whereas the values of CV and μn3 showed a decreasing trend for higher values of mass of the seeds. The solutions for seed size of 450 μm are found to concentrate at a single point, corresponding to high seed mass showing maximum WMS but with high values of CV and moderate values of nucleated mass, which is actually undesirable. For the case of 600 μm, some solutions are found to concentrate near a high initial seed mass region with high WMS of 718 μm, corresponding to a high CV of 51% and a nucleated mass of 3.8 × 10−5. Although many optimal solutions for 600 μm seed mean size concentrated around very low seed mass values representing maximum final WMS in the range from 570 to 610 μm, they show maximum μn3 and maximum CV pertaining to the dominant growth of nucleated crystals as opposed to the growth of seed crystals. In case 2, where initial seed mean size is considered as a decision variable with a fixed seed mass for various optimization runs, the Pareto front obtained is plotted as shown in Figure 6.

Figure 7. Pareto solution sets obtained for initial seed mean size as a decision variable with respect to a given seed mass for the three objectives: (a) WMS, (b) CV, and (c) μn3.

that when 3% of seed loading is considered, the limit on the choice of seed size is up to 400 μm beyond which no solutions are generated in the Pareto front. With the maximum seed loading of about 10% (0.01453 kg/kg solvent), larger size product crystals can be achieved with a compromise on high CV and nucleated mass. An intermediate seed mass of 0.009 kg/kg solvent (7% seed loading) is also considered, where selection of seed size >400 μm results in a reduced WMS when compared to a maximum seed loading case and corresponding values of CV and nucleated mass are also higher. From these results, it is evident that considering seed properties ranging from low initial seed loading and smaller initial seed mean sizes to high seed loading and larger seed sizes, it is possible to derive optimal temperature trajectories and it is left to the decision maker to choose a solution. For very low initial mean sizes and seed mass combinations, the maximum weight mean size is obtained due to primary nucleation of crystals, which can be seen from the high values of μn3 with high CV. However, the focus in seeded crystallization is more on crystal growth rather than nucleation, and therefore, such solutions cannot be considered as optimal in a true sense.

Figure 6. Pareto fronts with initial seed size as a decision variable for various given initial seed mass values.

The trade-off between the objectives (WMS, CV) and (WMS, μn3) can be seen clearly for the solutions obtained with initial seed size as a decision variable and for six optimization runs with different fixed values of initial seed mass. The Pareto solutions are plotted in Figure 7 with all three objectives individually against the initial mean seed sizes for various fixed initial mass of seeds. When a low seed mass value of 1.32 × 10−5 is considered for optimization, the Pareto solution set contains solutions that require very small seed initial mean sizes of