Aging of Reaction-Crystallized Benzoic Acid - American Chemical

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Ind. Eng. Chem. Res. 2004, 43, 6694-6702

Aging of Reaction-Crystallized Benzoic Acid Marie Ståhl, Bengt Åslund, and Åke C. Rasmuson* Department of Chemical Engineering and Technology, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Benzoic acid has been crystallized by mixing sodium benzoate and hydrochloric acid in a T-mixer. The suspension is collected in an agitated crystallizer, and aging of the solid product is evaluated by microscopy. In addition, the kinetics of aging is examined by population balance modeling and nonlinear parameter estimation. Microscopic observations reveal that the crystal shape is quite irregular immediately after the T-mixer crystallizer, but during the aging the crystals turn into nice well-shaped platelets. A model on Ostwald-type ripening gives a fairly good description of the evolution of the crystal size distribution during the aging, provided that the observed changes in crystal shape are accounted for in the model. Introduction Reaction crystallization is extensively used in production of inorganic and organic compounds, e.g., calcium carbonate, photographic emulsions, and organic fine chemicals. In crystallization, the primary processes are nucleation and crystal growth. However, secondary processes such as aging and agglomeration can have a significant influence on the final properties of the product, for example, by changing the crystal habit and the size distribution. Industrially, it is quite common to allow a crystallization batch to stand for some time before it is drained. Sometimes it is more for practical reasons (e.g., time of the day), and sometimes it is a deliberate aging of the product. Aging is a rather broad term that includes all changes in the solid product that take place after the actual crystallization has been terminated but with the particles still in the suspension. How we define when crystallization ends and aging begins is a matter of terminology. In a batch cooling or an evaporation crystallization it seems reasonable to define the starting point of the aging when the generation of supersaturation has finished. This definition could also be reasonable for a fed-batch reaction crystallizer, but not in a continuous plug flow reactive crystallizer. Still aging denotes changes that takes place over longer time scales and at low or very low driving forces. Important aging phenomena are changes in the size distribution, changes in crystal shape, and polymorphic transformations. When a precipitate size distribution is observed to broaden with time accompanied by a decrease in number, this is generally interpreted as an effect of Ostwald ripening, especially when the crystals are smaller than 10 µm. A number of theoretical and experimental studies have been published concerning Ostwald ripening in suspension (e.g., refs 4, 5, 8-10, and 13). In experimental studies of Ostwald ripening, it has been found that small crystals dissolve preferentially when exposed to undersaturated conditions.5 Generally, Ostwald ripening has been reported to give an increased average size, a decreased number of particles, and a broader size distribution.12 The theoretical work on Ostwald ripening is mostly based on the fundamental * To whom correspondence should be addressed. Tel.: +468-790 8227. Fax: +46-8-105228. E-mail: [email protected].

work done by Lifshitz and Slyozov10 and independently by Wagner.17 Using several assumptions, such as negligible volume fraction and diffusion-controlled growth, they studied the long-time regime of Ostwald ripening and drew conclusions concerning the average growth rate and the shape of the size distribution. Kahlweit9 and Chan2 took a similar approach but included a dispersion in growth rate in their model of Ostwald ripening. Dadyburjor and Ruckenstein3 modeled a similar dispersion of ripening rates using a microscopic approach. They modeled Ostwald ripening by modeling the absorption and emission of individual atoms at the crystal surface. Despite its potential importance, Ostwald ripening is seldom included in models of reaction crystallization. The studies by Tavare14 and Tavare and Garside15 are exceptions. They found that Ostwald ripening had a significant influence on the simulated weight mean size and coefficient of variation. In this study, we focus on the aging of benzoic acid precipitates. A combination of experiments and modeling with parameter estimation is used to investigate the kinetics and governing processes. A model of Ostwald ripening with growth and dissolution rates controlled by the rate of mass transfer gives a fairly good description of the experimental results, provided that the gradual change of shape during aging is accounted for in the model. Theory The basis for Ostwald ripening is that solubility depends on crystal size, in the same way as vapor pressure above a nonflat liquid surface depends on the curvature. The Gibbs-Thomson equation describes the dependency of crystal solubility on particle size:12

ln

2kaMcγs c*(L) ) csol 3kvRTFcL

(1)

where ka and kv are the area and volume shape factors, Mc is the molar mass, L is the crystal length, Fc is the density of crystals, R is the general gas constant, and csol is the solubility of large crystals. The equation shows that small crystals are more soluble than large ones, and will dissolve preferentially. Normally, Ostwald ripening is insignificant for crystals larger than a few

10.1021/ie049828a CCC: $27.50 © 2004 American Chemical Society Published on Web 09/18/2004

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micrometers,12 because its only for quite small particles that there is an appreciable increase in the solubility. The driving force for different crystallization phenomena such as crystal growth and dissolution is the difference in chemical potential between the solute in the solid phase and the solute dissolved in the solution. At low supersaturation the driving force can be approximated by the concentration difference as is done in this work. Crystal growth is generally described as a two-step process, in which mass transport of solute molecules by volume diffusion is followed by a surface integration where solute molecules are incorporated into the crystal lattice (or detached from it in the case of dissolution). It is usually assumed that the dissolution rate is determined by the rate of mass transfer only, since the process of surface detachment is faster. In the present study preliminary work showed that the same was true for the growth rate under the used conditions. The mass-transfer step can be described by ordinary mass-transport equations. Assuming that the solubility of benzoic acid is sufficiently low to discard the bulk flow term, the rate of mass transfer through the boundary layer to the interface can be described by

nA )

1 Fc dV ) kd(c - ci) A Mc dt

(2)

where nA is the molar flux, A the crystal surface area, V the crystal volume, kd the mass-transfer coefficient, Mc the molar mass, Fc the density of the solid phase, c the bulk concentration, and ci the concentration at the interface.7 For diffusion-controlled growth and dissolution, the change in a crystal’s characteristic size L is then described by

ka Mc dL ) k (c - c*) dt 3kv Fc d

(3) Model

Kirwan1

examined the contributions Armenante and of convection and molecular diffusion to the mass transfer to microparticles and proposed the following correlation for the mass-transfer coefficient:

Sh )

kdL ) 2 + 0.52Rep0.52Sc1/3 Dv

(4)

where

1/3L4/3 ν Rep ) and Sc ) ν Dv

glass, has an internal diameter of 2 mm and a length of 560 mm. The outlet tube is mounted in the T-block up to the mixing point. The superficial velocity in the outlet tube is normally 5 m/s (flow rate of 15.7 mL/s). The experimental procedure and conditions are described in detail by Ståhl et al.11 In the experiments, the suspension is led directly from the T-mixer by a plastic tube into a stirred vessel temperature-controlled at 30 °C. The size of the suspension sample is approximately 100 mL. The agitation is gentle but sufficient to keep the crystals suspended (approximately 200 rpm). When the agitation of the suspension sample is stopped, large flocks appear rapidly in the suspension. To obtain a representative sample, it is necessary to add a dispersant to the vessel. With the dispersant, the suspension is homogeneous and reproducible data are obtained. Since the addition of the dispersant makes further aging of the sample not representative, the entire experiment is repeated to obtain data corresponding to different times of aging. After the desired aging time, a sample is taken and the size distribution is determined by electrosensing zone measurements (Elzone 180 XY). As was reported in previous work,11 an undercooling of 0.5-0.7 °C is needed during the size determination to keep the size distribution of the original crystals (before aging) stable. The crystals are examined under a microscope to ensure that they are not agglomerated and to determine the crystal shape. In addition, both primary and aged crystals have been analyzed by X-ray powder diffraction. The particles are fully crystalline and of equal structure. Thus, we may conclude that the observed changes are not the result of a polymorphic transformation. The experimental data consist of size distributions from 10 experiments, performed at three different initial supersaturations in the T-mixer.

The stirred vessel is modeled as an isothermal batch reactor, in which crystal growth and dissolution occur due to Ostwald ripening. All crystals are assumed to have the same shape, and nucleation, agglomeration, and breakage are assumed to be negligible during aging. The population balance for the batch reactor is written as

∂n ∂(Gn) + )0 ∂t ∂L

(6)

(5)

is the specific mean power input, Dv the diffusivity, and ν the kinematic viscosity. Experimental Work Benzoic acid is crystallized by mixing equimolar hydrochloric acid and sodium benzoate solutions in a T-mixer. Reactant concentrations (cr) range from 0.179 to 0.340 M, and concentrations and volume flow rates of reactant solutions are equal in each experiment. The reactants are contacted by impingement from opposite directions in a T-coupling, and the mixture is forced into an outlet tube and finally into a sampling beaker. The inlet tubes for the reactant solutions have bore diameters of 1 mm, and the outlet tube, which is made of

where the growth rate is positive or negative (dissolution) depending on the size of the crystals compared to the critical size given by the Gibbs-Thomson equation (eq 7) at the prevailing supersaturation. The growth process of benzoic acid is assumed to involve molecular entities, not ions, and the driving force is described by the concentration difference. The method of characteristics is used to transform the population balance into a system of ordinary differential equations. The experimental size distribution is given as the number of crystals in each given size interval. In the model, the initial size distribution is divided into monosized subpopulations at the mean size of each interval. The Gibbs-Thomson equation is used to calculate the solubility of crystals of a certain size. It can be rear-

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ranged to express the critical size, Lcrit, the size of a crystal in equilibrium with the solution at a certain supersaturation:

Lcrit )

2kaMcγs 3kvRTFc ln S

(7)

Crystals smaller than the critical size dissolve, while crystals larger than the critical size grow. When the crystals become smaller than a minimum size, Lmin, they are assumed to be completely dissolved and the corresponding subpopulation is removed. The growth and dissolution processes are coupled to the supersaturation through the mass balance. The mass balance yields the change in concentration as

dMT dc )dt dt

(8)

where MT is the molar magma concentration. It is calculated from the third moment of the subpopulations as

MT )

Fckvimax Mc

NiLi3 ∑ i)1

(9)

The population density is determined by sorting the subpopulations into the corresponding size intervals, and dividing the total number of particles in each interval by the interval size:

ni(Li) )

Ni ∆Li

(10)

where Ni is the number of crystals, Li is the mean size, and ∆Li is the size range of size interval i. In the objective function the size intervals are the same as those used in the characterization of the experimental distributions; however, in the simulations the size intervals are smaller. Kinetic expressions describing dissolution and growth must be introduced before the model can be used in simulations. Preliminary optimizations showed that both dissolution and growth are controlled by the rate of mass transfer. Equation 4 is used to calculate the mass-transfer coefficient. A power number of 1.5 is used, which gives an estimated power input of 0.0054 W/kg in the stirred beaker. The dissolution and growth rates are then given by eq 3. In the experimental determination of size distributions, the volume of each crystal is measured with an electrosensing zone instrument, and the crystal size is reported as the equivalent spherical diameter. Consequently, the volume shape factor of a sphere is used to obtain a correct mass balance. The area shape factor relates the linear growth rate to the mass consumption. The crystals are shaped like thin platelets, and the exact area shape factor is unknown. Therefore, it is determined in the parameter estimation. The experiments show that the shape of the crystals changes with time. This change is included in the simulations by using a time-dependent area shape factor:

ka(t) ) (kamax - kamin)e-kt + kamin

(11)

Figure 1. Influence of shape parameter k on the evolution of the area shape factor, eq 11, for kamax ) 7 and kamin ) π.

This function is plotted in Figure 1 for different values of the constant k. The function is bound and decreases gradually with time. The constant k determines the slope of the curve, and at infinite time the area shape factor approaches kamin. The system of equations is solved numerically using a fourth-order Runge-Kutta method. In the simulations a time step of between 10 and 100 ms is used. The minimum size Lmin is set to 10-8 m. Parameter Estimation. The kinetic parameters are determined in a nonlinear optimization, minimizing the difference between simulated and experimental population densities. Optimizations have been performed for each experimental series separately and with two or three experimental series combined in one optimization. One experimental series consists of all experiments with the same initial supersaturation in the T-mixer. The objective function, F, is the sum of squares of all residuals. The estimated parameters are the constants k, kamin, and kamax in eq 11. In previous work,11 the interfacial energy was determined to 0.015 J/m2. This value has been used in the optimizations. An initial supersaturation of 1.0015 is assumed. We have used a logarithmic residual with absolute differences:

Ri )

{

log nisim - log niexp nisim > 1 nisim < 1 1 F)

∑i Ri2

}

(12) (13)

We choose to focus on the model’s ability to capture the general shape of the experimental size distributions rather than the correct prediction of the population densities in every measured interval. Since we found early on that the simulated distribution always was more narrow than the experimental one, it was necessary to include the second condition in eq 12 to ensure that the empty intervals did not completely dominate the objective function. The residual is set to a chosen value of 1 for intervals where the simulated population density is very low. The polytope method (also called the simplex method) has been used in the optimizations. Since the discretization of the initial distribution introduces discontinuities in the objective function, more sophisticated opti-


Figure 2. Evolution of the experimental total number concentration and weight mean size during aging. Table 1. Experimental Results from the Aging Study initial supersaturation ratio in the T-mixer 3.7 4.6

5.5

aging time (min)

total number concn (number/m3)

number median (µm)

weight mean size (µm)

0 20 150 0 3 17 145 0 3 21 145

1.2 × 1014 3.2 × 1013 1.2 × 1013 4.7 × 1014 2.4 × 1014 1.4 × 1014 5.6 × 1013 8.6 × 1014 3.8 × 1014 2.1 × 1014 9.4 × 1013

3.9 6.5 9.1 3.0 3.8 4.9 6.3 2.4 3.5 4.3 5.6

6.6 9.9 14.5 4.6 5.4 6.9 8.9 3.5 4.8 6.0 7.9

mization methods are not successful since they require a continuous objective function. The method is implemented in the Matlab Optimization Toolbox (The Mathworks). The simulation program is written in C++ for speed and is coupled to Matlab as a mex file to use their optimization routine. For each model, several optimizations are carried out starting from different initial parameter values. A disadvantage of the polytope method is that no information about the confidence intervals of the parameters is obtained. Results and Evaluation The experimental conditions and results of the aging study are summarized in Table 1 and Figure 2. The weight mean size increases with time, while the number of crystals decreases. Figure 3 shows the changes of the crystal size distribution for crystals formed at an initial supersaturation ratio of 4.6. A significant broadening of the distribution with time can be seen. During aging the crystal shape changes. In the photographs in Figure 4 these shape changes are captured. There is not a dramatic change in the overall shape of the crystals. However, initially the crystals are irregular thin plates, and during aging they gradually change into nicely shaped platelets. For the experiments with precipitates formed at a supersaturation ratio lower than 4.0 there are also dendritic crystals present. The shape changes are initially rapid enough to be observed visually, as was done in a small microscopic study carried out by Marika Torbacke (personal communication, 1999). Mainly irregularities in the edges fill out, and large protruding pieces may break off. There is no evidence of polymorphic transformation. The experiments with precipitates formed at supersaturation ratios of 4.6 and 5.5 are optimized both

Figure 3. Evolution of the experimental size distribution during aging for initial supersaturation ratio S0 ) 4.6. Table 2. Optimal Parameters from Optimizations with Each Experimental Series Separately and Combined S0 ) 4.6 S0 ) 5.5 combined

k

kamin

kamax

5.69 × 10-3 2.81 × 10-3 4.3 × 10-3

2.69 2.16 2.31

7.74 6.75 7.77

separately and together. The obtained kinetic parameters are given in Table 2. Simulated and experimental size distributions for each experiment are compared in Figures 5 and 6. The parameters are obtained in an optimization including both experiments. For improved clarity in the figures, the size distribution from the second aging time is not shown (17 and 21 min, respectively). The experiments with precipitates formed at a supersaturation ratio of 3.7 are not included in the optimization since the crystal shapes are quite different. This is commented on further in the discussion. The fit of the simulated distribution is tolerably good even though there are systematic deviations, especially for larger sizes. The simulated distributions are generally less broad than the corresponding experimental ones, and the peak of the distribution is shifted to the right. These differences are also reflected in the obtained weight mean sizes, which are compared in Figure 7. Generally, the simulated weight mean size is lower than the experimental value. The evolution of the supersaturation is seen in Figure 8, and the total number concentrations in experiments and simulations are compared in Figure 9. The supersaturation ratio decreases steadily as expected, even

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Figure 4. Photographs of crystals after different times of aging. From left to right, t ) 0 min for S0 ) 4.6, t ) 145 min for S0 ) 4.6, and t ) 18 h for S0 ) 4.5.

agreement between the total number concentrations from experiments and simulations. Discussion

Figure 5. Comparison between simulated and experimental size distributions for initial supersaturation ratio S0 ) 4.6. Parameters in the simulation are from the combined optimization, Table 2.

Figure 6. Comparison between simulated and experimental size distributions for initial supersaturation ratio S0 ) 5.5. Parameters are from the combined optimization, Table 2.

though very small fluctuations can be seen if the data are magnified. These fluctuations are a consequence of the discretization in the model and are not large enough to influence the result significantly. There is a fair

Model. A model based on Ostwald ripening, supplemented by crystal shape change during the processs, gives a decent description of the experimental data. If a constant area shape factor is used, the fit is inadequate. An example from such a simulation (with optimized parameters) is shown in Figure 10. The fit of the simulated distribution after 3 min is rather good, but beyond that major discrepancies develop. The evolution of the size distribution is in principle well predicted, but in the model the changes are more rapid. Either the changes are generally slower than is predicted by the model, or the changes in the size distribution in reality slow during the course of the process in a way that is not captured by the model. We have examined whether this discrepancy may be related to the weights different portions of the size distribution have in the objective function. No such relation has been found since there is no major change in the total number of particles over time and because of the logarithmic number densities that are used in the residuals. Please note that the model does account for decay in supersaturation. Thus, additional features need to be considered. Experimentally, it is clearly documented that the particle shape changes with time. The photographs in Figure 4 also show that the crystals are relatively imperfect when formed, probably as a consequence of rapid formation at high supersaturation in the T-mixer. As aging proceeds, the crystals grow more perfect with regular edges, and the surface area decreases. In the experiments, the particle volume is measured with an electrosensing zone instrument. How fast the volume of a crystal changes depends on both the linear growth rate and the surface area of the crystal. In our case, accounting for the decreasing area by including a timedependent area shape factor improves the ability of the model to describe the experimental data considerably, as is seen in the Results and Evaluation. However, we still recognize that the model does not provide an optimal fit of the experimental data. There are still deviations that seem to be systematic. The experimental distribution is broader than the simulated


Figure 7. Comparison of the weight mean size obtained in experiments and in simulations. Initial supersaturation ratio: (left) S0 ) 4.6, (right) S0 ) 5.5. Parameters are from the combined optimization, Table 2.

Figure 8. Supersaturation ratio during the simulation for initial supersaturation ratio (left) S0 ) 4.6 and (right) S0 ) 5.5. Parameters are from the combined optimization, Table 2.

Figure 9. Comparison of the total number concentration from the experiments and from the simulations. Initial supersaturation ratio: (left) S0 ) 4.6, (right) S0 ) 5.5. Parameters are from the combined optimization, Table 2.

one, and the mode of the model distribution is shifted toward larger sizes. The deviations become more pronounced for longer aging times, indicating that the model cannot fully describe the experiments and, thus, introduces deviations that propagate as the simulation proceeds. The photographs show not only that the shape changes with time but also that there is a distribution of shapes with varying degrees of perfection at each aging time. This variation may lead to a broadening of the size distribution similar to the effect of growth rate dispersion. Although it might lead to a better description of experimental data, at present, it is not possible to include this variation in the model, since it introduces a second dimension in the population balance and significantly increases the time to run an optimization. The imperfect shape of the crystals could also be an

indication of imperfect crystal lattices, with higher chemical potential and thus a higher solubility than those of ordinary benzoic acid crystals. The fact that the size distribution must be undercooled to give stable size measurements seems to indicate that this is the case. If there are significant solubility variations in the crystal population, a development similar to the observed aging could result. However, introductory simulations have shown that the solubility variation must be rather large to explain the observed experimental behavior. Solubility differences of this magnitude between crystals would make it very difficult to obtain reproducible size distribution measurements in the experiments, a difficulty which was not observed as long as a small undercooling of 0.5-0.7 °C was used. We thus conclude that the crystals formed in the T-mixer are

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Figure 10. Comparison between simulated and experimental size distributions when the area shape factor is constant, S0 ) 4.6.

more soluble than ordinary crystals, but the solubility variation is not likely to be of a magnitude sufficient to explain the broadening of the size distributions. Kinetics. The experience in the optimizations is that the process is well described by growth and dissolution processes governed by the rate of mass transfer through the boundary layer. The mass-transfer coefficient is calculated using eq 4, and the only parameters that are determined in the optimizations are those related to the shape change. kamax receives a value on the order of 8. The value on k reveals that the shape change with time is quite nonlinear. Initially, the changes are rapid, but quite soon the rate of change becomes much slower. The parameter kamin, which represents the lower limit of the area shape factor, receives a value of about 2.3, which is below the expected range. Since we measure the equivalent spherical diameter, a natural lower limit for the area shape factor would be that of a sphere. Thus, kamin should not be less than π. The low value may be explained by the fact that the used mass-transfer correlation was obtained for spherical particles. In eq 4, the first term on the right-hand side is the limiting Sherwood number, which describes the contribution by pure diffusion. This limiting number is 2 for spherical particles. The second term is the contribution by convection, which is determined by the mixing conditions and the size of the particles. At the mixing conditions in the aging experiments, the second term is between 2 and 10 times smaller than the first term, depending on the crystal size. For nonspherical particles, the limiting Sherwood number is not 2. Tsubouchi and Sato16 obtained a limiting Sherwood number of 0 for cylinders and wires. Fitchett and Tarbell’s interpretation of their results was that particles with large aspect ratios should have a limiting Sherwood number of 0.6 A smaller limiting Sherwood number leads to a smaller mass-transfer coefficient, which in turn influences the value of the area shape factor at the optimum. In Table 3 we present the shape factor parameters determined by optimization on the experiments of both supersaturation levels: 4.6 and 5.5, at lower limiting Sherwood numbers. At limiting values of Sh ) 1 and Sh ) 0, kamin becomes 3.14 and 3.4, respectively. This is still somewhat lower than expected, but the values are within the range that is reasonable from a physical standpoint. It turns out that neither the objective function value at the optimum nor the cor-

Figure 11. Comparison between simulated and experimental size distributions for initial supersaturation ratio S0 ) 3.7. Table 3. Optimal Parameters from a Combined Optimization (S0 ) 4.6 and S0 ) 5.5) When the Limiting Sherwood Number Is Set to 1 or 0 kmin a

kmax a

3.14 3.39

8.21 11.6

k 10-3

combined, Shlimit ) 1 combined, Shlimit ) 0

4.9 × 3.0 × 10-3

Table 4. Optimal Parameters for the Experiments with Precipitates Formed at Initial supersaturation Ratio S0 ) 3.7 S0 ) 3.7

k

kmin a

kmax a

1.0 × 10-3

4.35

6.41

relation of the size distributions versus time is sensitive to the value of the limiting Sherwood number, but there is some influence on the shape factor values obtained. The kinetics of nucleation and growth of benzoic acid have been determined in a previous study by Ståhl et al.11 The growth rate was then well correlated with an expression of the form G ) kg∆cg. The constant kg received a value of 2.4 × 10-9 and the exponent g a value of 2, indicating surface integration control. There was also evidence of a significant growth rate dispersion. In that study the nucleation and growth kinetics were determined from the size distributions immediately after the T-mixer, and hence relate to the conditions inside the T-mixer. In the present work, we evaluate the growth and dissolution kinetics in the agitated vessel after the T-mixer, where mixing is much more gentle, and the supersaturation is quite low. In addition, the crystals are of product crystal size. The growth rates in the previous study are mainly governed by the conditions where the growth process consumes the supersaturation, leading to a slowing and termination of nucleation, and hence concern much smaller particles. The governing surface integration mechanism depends on the supersaturation level. While a surface nucleation mechanism may prevail at the high supersaturation when the crystals are formed in the T-mixer, it is not likely to be governing at the very low supersaturation during the aging. However, more important is the fact that, in our previous work, surface integration controls because mixing is much more intense and because the particles are much smaller. As is clear from eq 4, the mass-transfer rate constant increases with decreasing size and increasing mixing intensity. At such widely different conditions as in these two studies, it is not to be expected that the same growth kinetics should apply.


Figure 12. Evolution of the supersaturation ratio and comparison of the total number concentration from the experiments and from the simulations for initial supersaturation ratio S0 ) 3.7.

The interfacial energy and the initial supersaturation have been set to constant values in the optimizations. The interfacial energy was calculated from nucleation data in a previous study.11 The value of the interfacial energy affects the kinetic parameters, but the general ability of the model to describe the experimental data does not change significantly. The initial supersaturation has no large impact on the result either. No general improvement of the fit between simulated and experimental data is obtained for optimizations with a higher initial supersaturation. Therefore, the initial supersaturation ratio is set to a value of 1.0015, a value that allows the Ostwald ripening process to start immediately in the simulation. The experimental program also includes experiments with precipitates formed in the T-mixer at an initial supersaturation ratio of 3.7. The obtained kinetic parameters differ from those from the other experiments, and the fit between experimental and simulated data is not as good. The optimal parameters are given in Table 4, and the obtained size distributions are compared in Figure 11. The evolution of the supersaturation ratio during the simulation and a comparison between the total number concentration in experiments and simulations are given in Figure 12. The difference in the kinetic parameters between this experimental series and the ones presented previously makes a combined optimization with all experiments unsuccessful. The broadening of the experimental distribution is more marked in experiments at S0 ) 3.7, and the model fails to predict it. This is probably a consequence of the presence of dendrites, which make the variation in shape more pronounced than in the other experiments. Photographs of this have been presented previously.11 Objective Function and Optimization. The location of the optimum depends to some extent on the choice of the objective function, especially when there are systematic deviations. However, the choice of objective function will not affect the general impression of the model’s applicability, but rather move the deviations from one part of the size distribution to another. In general, size distributions corresponding to shorter aging times are better correlated than those corresponding to longer times, and this is not influenced by the choice of the objective function. In the optimizations, the residuals are defined using the logarithm of the population density instead of using the population density directly. The advantage is that negative deviations are not favored over positive deviations, and relative deviations are given equal weight, even though

the number of crystals may differ significantly for different size intervals as well as for different aging times. The discretization of the initial size distribution is important for the quality of the simulation result. The experimental data are given in 32 size intervals. If this distribution is taken as input to the simulation, it is impossible to simulate the experimental changes in the number distribution. In the simulations a finer discretization has been obtained by dividing each size interval into several parts and distributing the crystals in the original interval evenly over these refined intervals. Conclusions We have experimentally investigated the aging at low supersaturation of benzoic acid crystals formed by reaction crystallization in a T-mixer at high supersaturation. The experimental results have then been evaluated by population balance modeling and direct nonlinear parameter estimation. During the aging of the precipitate in an agitated crystallizer, the weight mean size increases, and the total number of crystals decreases. This is in accordance with the expected Ostwald ripening. The results indicate that both growth and dissolution are governed by the rate of mass transfer through the particle boundary layer. However, a simple Ostwald ripening model is not capable of describing the decreasing rate of change of the size distribution. Microscopic analysis reveals that the crystals originally are quite irregular but gradually become more well shaped during the course of the aging. A fairly good description of the changes of the crystal size distribution due to aging is obtained, provided that also the shape changes of the crystals are taken into account in the model. Acknowledgment The financial support of the Swedish Council for Planning and Coordination of Research, the Swedish Research Council for Engineering Sciences, now The Swedish Research Council, and the Industrial Association for Crystallization Research and Development is gratefully acknowledged. Notation A ) crystal surface area, m2 c ) solute concentration, mol/m3 csol ) solubility of large crystals, mol/m3

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c* ) solubility, mol/m3 Dv ) diffusivity, m2/s F ) objective function G ) crystal growth rate, m/s g ) exponent in growth rate expressions k ) constant, l/s ka, kv ) area and volume shape factors kd ) mass-transfer coefficient, m/s kg ) growth rate constant in power laws, (m3/mol)g L ) crystal size, m Lcrit ) critical size, m Mc ) molar mass, kg/mol MT ) molar magma density, mol/m3 N ) number of crystals, number/m3 n ) population density, number/m, m3 nA ) molar flux, mol/(m2 s) R ) gas constant () 8.314), J/(mol K) Rj ) residual for interval j Rep ) Reynolds number for a particle ) (1/3L4/3)/ν S ) supersaturation ratio, c/c* Sc ) Schmidt number ) ν/Dv Sh ) Sherwood number ) kdL/Dv T ) temperature, K t ) time, s V ) volume, m3 Greek Letters ) power input, W/kg γs ) interfacial surface energy, J/m2 ν ) kinematic viscosity, m2/s F ) density, kg/m3 Subscripts c ) crystal 0 ) initial

Literature Cited (1) Armenante, P. M.; Kirwan, D. J. Mass Transfer to Microparticles in Agitated Systems. Chem. Eng. Sci. 1989, 44 (12), 2781-2796. (2) Chan, S. K.; Ostwald Ripening of Precipitates I: The Path Integral Solution. II: The Irreversible Thermodynamics. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 745-751.

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Received for review March 2, 2004 Revised manuscript received June 18, 2004 Accepted June 30, 2004 IE049828A

Aging of Reaction-Crystallized Benzoic Acid - American Chemical

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