Article pubs.acs.org/IECR
Seeding and Optimization of Batch Reactive Crystallization Hsing-Yu Wang and Jeffrey D. Ward* Department of Chemical Engineering, National Taiwan University, Taipei 106-07, Taiwan ABSTRACT: Conceptual methods are illustrated for the development of operating policies for batch reactive crystallization processes. Two processes, production of barium sulfate and production of L-glutamic acid, are studied. For barium sulfate, the critical seed loading ratio is found to be above the practical limit, which suggests that seeding cannot be employed to suppress nucleation. Conversely, for L-glutamic acid, the critical seed loading ratio is below the practical limit for a wide range of seed sizes, suggesting that seeding can be used to suppress nucleation almost entirely. These results are verified by rigorous process simulation. Furthermore, for the barium sulfate process, a plot of nucleation rate B versus growth rate G is concave, indicating that an early growth trajectory (with supersaturation highest at the beginning of the batch) will minimize the nucleated mass, while for the L-glutamic acid process the plot is convex, indicating that a late growth trajectory will minimize the nucleated mass. These results are also supported by rigorous simulation and optimization.
1. INTRODUCTION Batch crystallization is often used in the chemical engineering field for solid−liquid separation and purification to produce chemical and pharmaceutical products. An operating recipe for a seeded batch crystallization process contains two parts: a specification of seed properties and a trajectory for the supersaturation as a function of time. A review by Nagy and Braatz1 discusses recent advancements in the operation of batch crystallization processes including online monitoring and feedback control during the batch. Several researchers have specifically investigated the effect of seed properties on the performance of batch crystallization processes.2,3 Kubota and his co-workers4 defined critical seed loading, which means the amount of seed loading that just makes the nucleated mass fraction negligible, and proposed a correlation between critical seed loading and initial seed size based on experimental studies of two crystallization systems. Later, Tseng and Ward5 proposed a shortcut method to quickly determine the relationship between nucleated mass fraction, seed mass, and seed size. They determined the critical seed loading curve using only the kinetic expression for the nucleation rate as a function of growth rate. In some cases, reactants that are soluble in some solvent combine to form a product that is sparingly soluble. In this case, it is attractive to combine reaction and crystallization in a single step. Batch reactive crystallization processes have been widely studied by simulation and experiment in the literature.6−22 Many studies focused on the determination of crystallization kinetic parameters and simulation of the crystallization process, but only a few have studied optimization of such processes.22 Therefore, in this work, we study conceptual methods for the rapid development of operating recipes for seeded batch reactive crystallization processes using two case studies: production of barium sulfate and production of glutamic acid, which are model systems for reactive crystallization that are widely studied in the literature. For both systems, before the models were applied in this study, the results presented by the original authors of the models were reproduced to confirm that the model is implemented correctly and makes accurate © XXXX American Chemical Society
predictions. The remainder of this Article is organized as follows. In section 2, models for the two case studies are presented, and the numerical optimization procedure used in this work is described. In section 3, results are drawn for both case studies, and finally in section 4 results are presented.
2. MODEL DEVELOPMENT AND OPTIMIZATION 2.1. Model Equations for Batch Reactive Crystallization. For a well-mixed batch crystallization process with sizeindependent growth rate, neglecting agglomeration and breakage, the population balance equation is ∂n(t , x) ∂n(t , x) =0 + G (t ) ∂x ∂t
(1)
subject to the initial condition, n(0,x) = nseed(x), and the left boundary condition, n(t,0) = B(t)/G(t). Here, n(t,x) is crystal size distribution (CSD) function, G(t) is crystal growth rate (m/s), B(t) is nucleation rate (no./m3·s), x is crystal size (m), and t is time (s). The definition of the moments is μi (t ) =
∫0
∞
x in(t , x) dx ,
i = 0, 1, 2, ...
(2)
and the time derivatives for the moments if growth rate is sizeindependent and volume will change with time are d(μ0 V ) dt
d(μi V ) dt
= BV
= iGVμi − 1 ,
(3)
i = 1, 2, ...
(4)
There are seed-grown crystals (subscript s) and nuclei-grown crystals (subscript n). Note that for each moment, the total crystal value equals seed-grown value plus nuclei-grown value: Received: January 14, 2015 Revised: August 27, 2015 Accepted: August 30, 2015
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μT, i = μn, i + μs, i
Table 1. Parameters in the Barium Sulfate Crystallization Process Model
(5)
2.2. Case Study Systems. In this work, two case study systems are investigated: reaction of barium chloride with sodium sulfate to produce barium sulfate,23 and reaction of monosodium glutamate with hydrochloric acid to produce Lglutamic acid.20,21,24 In this subsection, the process models for each process are briefly summarized. 2.2.1. Barium Sulfate. A model for a reactive crystallization process for the production of barium sulfate is provided by Wei and Garside.23 The reaction is instantaneous, so the reaction kinetics are not considered. The chemical reaction is Ba 2 +(l) + SO24 −(l) → BaSO4(s)↓
description
(6)
and the mass balance equations for the components in this system are d(C Ba V ) = FBa 2+ dt
k vρC d(m3V ) − MBaSO4 dt
d(CSO24−V )
k vρC d(m3V ) − MBaSO4 dt
2+
= FSO24−
dt d(C BaSO4V ) dt
k vρC d(m3V ) = MBaSO4 dt
dV = FBa 2+ + FSO24− dt
(7)
(9)
kAB
A + H+ ← →B
0
kBA
Bp = B exp[−A p /ln Sa]
RH
+ ka3
67.3 1.46 × 1012 4.0 × 10−11 2 1.10 × 10−11 4480 π/6 233.4 2.46 × 10−3 0.1931 1.0 × 10−5 0.1 16.26 1000
X=
(13)
kmol2/m6 kg/m3
m3 kg/m3 m mol/m3 mol/m3
kAB =
[A][H+] [B]
(15)
[A]0 + [H+]0,HCl + kAB 2 {[A]0 + [H+]0,HCl + kAB}2 − 4{[A]0 [H+]0,HCl − [B]0 kAB}
−
2
(16) +
+
+
where [H ]0,HCl = [H ]0 + [H ]HCl. Afterward, the resulting hydrogen ion concentration can be calculated: [H+] = [H+]0(solution) + [H+]HCl(added) − X
(17)
With the resulting hydrogen ion concentration, the concentration for individual species can be expressed in terms of [H+]: [RH+] = C RH
−1
+ H ↔ RH (MSG) ka2
[H+]3 [H+]3 + ka1[H+]2 + ka1ka2[H+] + ka1ka2ka3
RH−1(MSG) + H+ ↔ RH(GA)solution
(18)
ka1
RH(GA)solution + H+ ↔ RH+1
[RH] = C RH
K sp
RH(GA)solution ← → RHsolid
no./(m3 s) m/s
Solving the new equilibrium for X gives
Values of parameters in these equations are given in Table 1, with some values taken from the literature.23 2.2.2. L-Glutamic Acid. The second case study system considered in this work is production of L-glutamic acid. LGA can exist in solution in one of four states, with the dominant state depending on pH. Adding hydrochloric acid (HCl) to a solution of monosodium glutamate (MSG) produces chargeneutral LGA, which precipitates out of solution. The chemical reactions of this system are −2
AN B0 kg g Ksp ρc kv MBaSO4 V0 m0 L0 W CNa2SO4 CBaCl2
no./(m3 s)
If [H ]HCl of HCl is added to the system, the reaction will move rightward by X and reach new equilibrium. B A H+ initial [A]0 [H+]0 [B]0 change −X [H+]HCl − X +X equilibrium [A]0 − X [H+]0 + [H+]HCl − X [B]0 + X
(11)
(12) 2
2686 1036
+
where c+ and c− are the concentrations of cation and anion dissolved in the solution, v+ and v− are number of moles of cations and anions if 1 mol of solute dissolves into the solution, and v is the number of moles of ions if 1 mol of solute dissolves into the solution. For the crystallization kinetics, semiempirical power-law expressions are used for the growth and nucleation rates: G = kg(1 − Sa)g
AN B0
unit
where RH−2 is the twice dissociated form, RH−1 is the once dissociated form from monosodium glutamate, RH is zwitterion with neutral charge, and RH+1 is protonated form. Moreover, ka1, ka2, and ka3 are equilibrium constants for each reaction. As to the concentration calculation for each species in this system, considering one protonation reaction:
For ionic species, thermodynamic supersaturation ratio Sa is used and is defined as Sa =
value
(8)
(10)
[(c+υ +c −υ −)/K sp]1/ υ
symbols
for Sa ≥ 103 nucleation rate parameter nucleation rate coefficient for Sa < 103 nucleation rate parameter nucleation rate coefficient growth rate coefficient growth rate order solubility product density of crystals volumetric shape factor molecular weight initial volume mass of seeds added mean seed size % width of seed CSD initial concentration feed concentration
ka1[H+]2 + 3
[H ] + ka1[H ] + ka1ka2[H+] + ka1ka2ka3
− Δ[RH]
(14) B
+ 2
(19) DOI: 10.1021/acs.iecr.5b00185 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Table 2. Parameters in the L-Glutamic Acid Crystallization Process Model
+
C RH
+ 3
ka1ka2[H ] + 2 ka1[H ] + ka1ka2[H+]
+ ka1ka2ka3
(20)
[H ] + ka1[H ] + ka1ka2[H+] + ka1ka2ka3
(21)
[H ] +
[RH−2] = C RH
ka1ka2ka3 + 3
+ 2
where CRH is the summation of the concentrations of four species. Note that Δ[RH] in eq 19 means the consumption of [RH] due to crystallization. Because reactant is fed to the system, the volume of the system changes with time, and the resulting total concentration is C RH(after adding HCl) = ([RH−2] + [RH−1] + [RH] + [RH+1])
V0 V0 + Ft
(22)
Additionally, the decrease of [RH] due to crystal nucleation and growth is KAρcry K vρcry L0 3 Δ[RH] − = −B −G Mcry 2Mcry Δt
∫0
Lmax
L2n(L) dL
As to the crystallization kinetics, the definition of supersaturation S and growth rate kinetics are taken from the literature:20 ⎛ −A ⎞ ⎛ −A 2 ⎞ 2/3 ⎟(S − 1) G B + S = A1T exp⎜ (ln S)1/6 exp⎜ 2 3 ⎟ ⎝ T ⎠ ⎝ T ln S ⎠
where c and c* are the concentration and solubility, respectively. The growth rate is the birth and spread (B+S) mechanism. For nucleation rate kinetics, expression from the literature21 is used, and both homogeneous and heterogeneous primary nucleations are considered: ⎛ −A ⎞ BHom = A4 exp⎜ 2 5 ⎟ ⎝ ln S ⎠
(27)
⎛ −A ⎞ BHet = A 6 exp⎜ 2 7 ⎟ ⎝ ln S ⎠
(28)
6.31 × 10−3 5.62 × 10−5 2.14 × 10−10 3.63 × 10−4 3720 54 200 1.3 × 1027 163 1.4 × 108 10 1460 1 147 6.61 × 10−3 1.0 × 10−4 0.5 1.183 6
unit
m/(s K) K K2 no./(m3 s) no./(m3 s) kg/m3
kg m M M
(29)
To implement the cubic trajectory, the feed rate is given by the following expression:
(25)
(26)
value
ka1 ka2 ka3 A1 A2 A3 A4 A5 A6 A7 ρc kv Mcry mseed Ls W CMSG,0 CHCl
3 ⎛⎛ ⎞ Gt̃ ⎞ F = ms⎜⎜⎜1 + ⎟ − 1⎟⎟ x0 ⎠ ⎝⎝ ⎠
(24)
B = BHom + BHet
symbols
implement the linear trajectory, reactant is added at a constant rate during the batch. To implement the constant supersaturation trajectory, reactant is added so that the supersaturation remains constant during the batch. To implement the Mullin−Nyvlt trajectory, the feed rate is specified by the following equation:25
(23)
S = c / c*
description equilibrium constant 1 equilibrium constant 2 equilibrium constant 3 growth rate coefficient growth rate parameter growth rate parameter nucleation rate coefficient nucleation rate parameter nucleation rate coefficient nucleation rate parameter density of crystals volumetric shape factor molecular weight mass of seeds added mean seed size % width of seed CSD initial concentration feed concentration
F = (t /tf )3 Ff
(30)
Finally, two optimal trajectories are considered for two different objective functions, minimizing the nucleated mass at the end of the batch and minimizing the number of nuclei at the end of the batch. These optimal trajectories are determined using the procedure described in the following subsection. All of the trajectories considered in this work are practical and are reasonable candidates for implementation. We compare many different trajectories because it is useful to better understand the relationship between the batch trajectory and the seed loading and the resulting product crystal size distribution. Because the definitions of supersaturation are different for the two compounds, it is not meaningful to directly compare numerical values of the supersaturation between the two systems. However, qualitative shapes of supersaturation trajectories (early growth, late growth, constant growth) can be compared. 2.4. Optimization. Several optimization methods for batch crystallization have been discussed in the literature,8 including simulated annealing. In this work, the method of iteration3 is used for optimization. For each optimization, several initial guesses are tested, and the algorithm always converged to the same reasonable answer, suggesting that the algorithm is not getting stuck in a local minimum. Two objective functions were considered in this work. The first is to minimize the ratio of the nucleated mass to the total product mass:
Parameters for these equations are given in Table 2, with some values taken from the literature.20,21,24 2.3. Supersaturation and Feeding Trajectories. For both processes, one reactant is charged initially in the reactive crystallizer, and the other reactant is fed continuously during the batch. For the L-glutamic acid process, sodium glutamate is charged initially and hydrochloric acid is fed continuously, while for the barium sulfate process, sodium sulfate is charged initially and barium chloride is fed continuously. The reactant feed rate can be changed with time, and the feed rate trajectory determines the supersaturation in the batch as a function of time. In this work, several supersaturation or feeding trajectories are compared. In this subsection, the meaning of trajectories is explained. To implement the batch trajectory, all of the reactant is added at the beginning of the batch. To C
DOI: 10.1021/acs.iecr.5b00185 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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the supersaturation−time trajectory have only a minor impact on the product crystal size distribution when the critical seed loading is employed. Therefore, it is reasonable to consider the case where the batch crystallizer operates with constant supersaturation. The method is useful because it allows the engineer to rapidly determine for what seed sizes (if any) it will be possible to suppress nucleation by seeding. Integrating the method of moments assuming constant growth rate gives
(31)
The trajectory S1(t) is the trajectory that minimizes the ratio of the final nucleated mass to the final total crystal mass. The second objective is to minimize the number of nuclei: S2(t ) = arg min μ0n (tf ) S(t )
(32)
The trajectory S2(t) is the trajectory that minimizes the total number of nuclei at the end of the batch. In all cases, the value of the independent variable S(t) was represented by a continuous, piece-wise linear spline with 10 points. When the optimal supersaturation trajectory is determined, the required feeding trajectory can be determined by back-calculation. The optimization problem is solved by adjusting the value of each point in the spline iteratively to minimize the objective function, until the change in position of all points is less than a certain tolerance. 2.5. Critical Seed Loading. Critical seed loading analysis is applied to study the effect of seeding on the operation of the batch reactive crystallization processes. This method was first developed by Kubota and co-workers.4 To apply the method, the normalized product mean crystal size is plotted as a function of seed loading ratio for several seed sizes. Applying a material balance in the case where all of the solid precipitate is consumed in the growth of seeds gives Ws k vρc Ls
3
=
μ3,n =
3
=
∫0
t3
G
∫0
t2
B(t1) dt1 dt 2 dt3 dt4
tf
t4
t3
∫0 ∫0 ∫0 ∫0
1 ̃3 G B(G̃ )tf 4 4
t2
dt1 dt 2 dt3 dt4 (39)
where G̃ is the constant growth rate. This equation gives the relationship between nucleated mass, seed size, and seed mass. Therefore, the critical seed loading curve can be determined by application of this equation.
3. RESULTS AND DISCUSSION 3.1. Seeding Policy. To investigate the effect of seed loading on the operation of the batch crystallization, seed loading diagrams for both systems were prepared as shown in Figure 1. For each seed size considered, the critical seed loading ratio was determined, and the values for each seed size for both systems are plotted in Figure 2 with circular and square markers for barium sulfate and LGA, respectively. The seed mean size in Figures 1 and 2 and the correlation of Kubota and co-workers is the number mean size μ1/μ0. Also plotted in the figure are the results of the shortcut method (eq 39). Also plotted in the figure are the results for many nonreactive crystallization systems first reported by Tseng and Ward, and the correlation (eq 36) recommended by Kubota. For both systems, it is observed that the shortcut method gives a very accurate prediction of the actual critical seed loading ratio. For the LGA process, the critical seed loading ratio curve lies within the range of values observed for nonreactive crystallization systems and below the suggested maximum practical value of the seed loading ratio, suggesting that it will be practical to suppress nucleation by seeding for a wide range of seed sizes. By contrast, for the barium sulfate process, the critical seed loading ratio curve lies slightly above the highest curve for nonreactive crystallization, and above the suggested maximum practical seed loading ratio. This suggests that it will not be practical to suppress nucleation entirely using seeds. Figures 3 and 4 show the final product volume size distribution for different seed properties and different feed policies for the barium sulfate and LGA processes, respectively. For the barium sulfate process, it is observed that even for a very small seed size (5 μm) and a very large seed mass (0.290 kg/m3 corresponding to a seed loading ratio of 0.0162), it is not possible to suppress nucleation completely. This can be seen from the fact that for all seeding scenarios, the product crystal
(33)
(34)
(35)
When the normalized product mean size curve for a given seed size just approaches the ideal growth line, this value of the seed loading ratio is called the critical seed loading ratio, Cs*. In this work, we set the critical seed loading ratio to be where the ratio of nucleated mass to the total net crystal yield is 0.01. The critical seed loading ratio then is plotted versus seed size. Kubota and co-workers4 suggested the following empirical relationship between critical seed loading ratio and seed mean size based on their experimental work: Cs* = 2.17 × 10−6Ls 2
2G
(38)
μ3,n = 6G̃ B(G̃ )
When the seed loading is low, the equation can be simplified
= Cs−1/3
t4
Then
to
Ls
∫0
B = B (G )
Lp
Lp
3G
If the nucleation rate B is only a function of growth rate G:
where Ls and Lp are the seed and product mean sizes, respectively, and ρc and kv are the crystal density and volumetric shape factor, respectively. Rearranging the equation gives
⎛ 1 + Cs ⎞1/3 =⎜ ⎟ Ls ⎝ Cs ⎠
tf
(37)
Ws + Wth k vρc Lp3
∫0
(36)
where Ls is in micrometers. 2.6. Shortcut Method. Tseng and Ward proposed a shortcut method for determining the critical seed loading ratio as a function of seed size requiring only an expression for the nucleation rate as a function of growth rate and solid crystal mass such as an empirical correlation of the form B = kBGiMTj. In particular, the nucleation and growth rates do not have to be related to the supersaturation. The method relies on the fact that the shape (i.e., breadth) of the crystal size distribution and D
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By contrast, in Figure 4 it is observed that even for a larger seed size (50 μm) and a lower seed mass (9.9 g/m 3 corresponding to a critical seed loading ratio of 0.011), nucleation is essentially entirely suppressed. Comparing the trajectories for the LGA process, it is observed that the optimal trajectory again performs best as expected and that the linear trajectory performs worst. The batch trajectory is not considered for the LGA process because if all of the acid is added immediately, the protonated LGA ion is formed and no crystals are formed. 3.2. Determination of Optimal Trajectories and Comparison of Trajectories. In most crystallization processes, nucleation and growth are the phenomena that have the greatest effect on the change in the crystal size distribution during the batch. Typically both of these phenomena are related to the supersaturation in the batch, and so therefore determination of the optimal batch trajectory can be understood as balancing a trade-off between nucleation and growth. For both of the processes studied in this work, both nucleation and growth depend only on supersaturation, and so there is a bijective relationship between them. Figures 5 and 6 show the relationship between supersaturation and growth rate, supersaturation and nucleation rate, and growth rate and nucleation rate for the LGA and barium sulfate processes, respectively. The relationship between the nucleation rate and growth rate for the LGA process is typical of most crystallization systems (both reactive and nonreactive) in that it is convex; that is, the second derivative (d2B/dG2) of the function B(G) is positive. Ward et al. (2006)26 argued that for systems of this type, the trajectory that minimizes the nucleated mass will be a lategrowth trajectory (with supersaturation higher at the end of the batch). If the rate of secondary nucleation depends on the mass of crystals in the system (secondary nucleation), then an early growth trajectory will minimize the number of nuclei. If the rate of nucleation is independent of the crystal mass (as is the case for both examples considered in this work), then the trajectory that minimizes the number of nuclei has constant supersaturation. For the barium sulfate process, however, the situation is different. As shown in Figure 6, the function B(G) is concave. Chiang (2012) observed that for such processes, the trajectory that minimizes the nucleated mass may be early growth. Figure 7 shows the results of several different trajectories for the L-glutamic acid process. The results are similar to those observed previously for nonreactive crystallization processes. The linear trajectory results in an early growth profile, with the supersaturation reaching the maximum value early in the batch. The trajectory that minimizes the nucleated mass is late-growth trajectory. The trajectory that minimizes the number of nuclei is a constant supersaturation trajectory, which is expected because the nucleation rate does not depend on the solid crystal mass. The cubic and Mullin−Nyvlt trajectories have peaks in the supersaturation in the middle of the batch and perform somewhat worse than the optimal trajectory but much better than the linear trajectory. As stated previously, the batch trajectory is not considered for the LGA process because if all of the acid is added immediately, the protonated LGA ion is formed and no crystals are formed. Table 3 shows the values of the objectives μ3n(tf)/μ3T(tf) and μ0n(tf) for all trajectories. The tabulated results also show that the linear trajectory performs much worse than the others.
Figure 1. Seed loading diagrams: (a) barium sulfate system, and (b) Lglutamic acid system.
Figure 2. Critical seed loading chart.
size distribution is bimodal, with the peak at smaller product crystal sizes corresponding to nucleated crystals and the peak at larger crystal size corresponding to seed-grown crystals. The situation is worse for larger seed size or smaller seed mass. Comparing the trajectories, the optimal trajectory (minimizing the nucleated mass) performs best as expected, while the batch trajectory performs worst. E
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Figure 3. Product volume size distribution of barium sulfate system for different seed loading ratio and mean seed size.
Figure 4. Product volume size distribution of LGA system for different seed loading ratio and mean seed size. F
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Figure 5. Supersaturation, growth rate, and nucleation rate relation for L-glutamic acid system.
Figure 6. Supersaturation, growth rate, and nucleation rate relation for barium sulfate system.
Figure 7. (a) Feeding policy, (b) supersaturation trajectory, (c) crystal mass trajectory, and (d) product volume size distribution for six different feeding policies for L-glutamic acid system.
of nuclei also performs well but has a noticeably larger nucleated mass. The linear, cubic, constant supersaturation, and Mullin−Nyvlt trajectories all perform about the same, although the linear trajectory has a larger supersaturation at the beginning of the batch. The batch trajectory, in which all reactant is added at the beginning of the batch, performs the
Figure 8 shows the results for the barium sulfate process. The results are quite different from those of the LGA process and from most other nonreactive crystallization processes. For the optimal trajectory (the trajectory that minimizes the nucleated mass), the greatest supersaturation is at the beginning of the batch, not at the end. The trajectory that minimizes the number G
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that nucleation can be effectively suppressed by seeding for the LGA process, whereas it will not be possible to suppress nucleation by seeding for the barium sulfate process. These results are supported by rigorous numerical simulation. The methods illustrated would be useful for understanding experimental results and guiding experimental work and also for development of practical seeding policies for industrial implementation. Different feed policies are studied, including batch, optimal, linear, constant supersaturation, cubic, Mullin−Nyvlt, and the trajectory that minimizes the number of nuclei. Late-growth trajectories generate smaller final nucleated mass fraction for LGA system, and early growth trajectories generate smaller final nucleated mass fraction for BaSO4 system. These results can be understood on the basis of the relationship between the nucleation rate and growth rate. They are also useful for guiding and understanding experimental work and for practical implementation.
Table 3. Objective Function Value for All Case Studies case study
trajectory
μ3n(tf)/μ3T(tf)
barium sulfate
optimal min μ0 const S cubic linear MN batch optimal const S min μ0 MN cubic linear
0.1255 0.2849 0.4842 0.4875 0.4982 0.5043 0.6690 0.0011 0.0046 0.0047 0.1459 0.2782 0.8128
L-glutamic
acid
μ0n(tf) (105 no./m3) 8.91 5.63 20.67 20.99 19.12 20.07 48.43 77.81 2.39 2.35 61.18 208.15 2190.4
× × × × × × ×
107 107 107 107 107 107 107
worst because the large degree of supersaturation at the beginning of the batch causes excessive nucleation. These unexpected results are related to the shape of the nucleation rate curve in Figure 6. Because the nucleation rate is convex when plotted versus the growth rate, the ratio of nucleation rate to growth rate actually decreases with increasing growth rate. In this case, it is favorable to operate with a very large supersaturation (growth rate) at the beginning of the batch.
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AUTHOR INFORMATION
Corresponding Author
*Tel.: +886-2-3366-3037. Fax: +886-2-2369-1314. E-mail: jeff
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This research was funded by the National Science Council of Taiwan under grants 102-2218-E-002-006 and 103-2218-E-002010.
4. CONCLUSIONS In this work, conceptual methods for the rapid determination of seed loading and operating policy are applied to two batch reactive crystallization processes. Seed loading analysis predicts
Figure 8. (a) Feeding policy, (b) supersaturation trajectory, (c) crystal mass trajectory, and (d) product volume size distribution for different feeding policies for barium sulfate system. H
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DOI: 10.1021/acs.iecr.5b00185 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
