Modeling of Metastable Zone Width Behavior with Dynamic Equation

Identification of the metastable zone width (MZW) is crucial for product quality control in industrial crystallization. Traditionally, the MZW in batc...
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Modeling of Metastable Zone Width Behavior with Dynamic Equation Dae R. Yang,*,† Kwang S. Lee,‡ Ju S. Lee,† Sung G. Kim,† Do H. Kim,† and Yoo K. Bang‡ Department of Chemical and Biological Engineering, Korea UniVersity, 1 Anamdong 5Ga, Seongbukku, Seoul 136-713, Korea, and Department of Chemical and Biomolecular Engineering, Sogang UniVersity, 1 Shinsoodong, Mapoku, Seoul 121-742, Korea

Identification of the metastable zone width (MZW) is crucial for product quality control in industrial crystallization. Traditionally, the MZW in batch cooling crystallization has been defined as a kind of static property of a solution that primarily depends on cooling rate. In industrial batch crystallization operations, however, the cooling rate does not remain constant but changes with time based on the operation recipe. The existing models with static relationship between MZW and cooling rate have a limit to predict the behavior of the MZW under such a dynamic situation. In this study, the MZW is newly elucidated as a dynamic state that predominantly depends on the cooling rate among other operating variables. For this, a first-order dynamic model with input nonlinearity has been proposed between the cooling rate and the MZW. To verify the proposition, a series of experiments have been conducted for crystallization of ammonium sulfate, potassium aluminum sulfate, and potassium chloride under various cooling scenarios. From our experiments, it is observed that (1) holding temperature after cooling affects the nucleation time, (2) undersaturated starting temperature of cooling affects the nucleation temperature, and (3) solution thermal history before saturation temperature also affects the nucleation temperature. 1. Introduction Crystallization is an indispensable separation and purification process in chemical and related industries, where >70% of products are obtained in the crystal form.1 As the biochemical and pharmaceutical industries have grown fast recently, the importance of the crystallization process has been recognized more. Nevertheless, many aspects of crystallization are not still understood properly, and industrial crystallization is sometimes regarded as an art. In industrial crystallization, the uniformity in size and shape of the final product is very often the most important quality requirement. In batch unseeded as well as seeded crystallizations, suppressing the undesired ancillary nucleation after the seed crystals are fed or generated is the key to achieve the uniform product-size distribution. For this, the solution is required to be carefully maintained within the metastable zone, which refers to the region where spontaneous nucleation is believed to be unlikely.1 Since its concept was first proposed by Ostwald,2 the metastable zone has been one of the essential notions in industrial crystallization. Nonetheless, its definitions have still some vagueness, and the models to estimate the metastable limit (MSL) have not been matured enough to be used in practical operation. The phenomena of underlying physicochemical processes such as solution chemistry, nucleation theory, and crystal growth have been investigated in many ways. However, it is not enough yet to reliably predict the MSL, although some plausible theoretical predictions have been achieved under restricted conditions.3-5 Despite such limitations, it is true that the concept of MSL is very useful. If the MSL can be reliably predicted in real time during a course of crystallization, the crystal productivity can be maximized while effectively suppressing the unwanted generation of small particles. * To whom correspondence should be addressed. Tel.: +82-2-32903298. Fax: +82-2-926-6102. E-mail: [email protected]. † Korea University. ‡ Sogang University.

Figure 1. Solubility-supersolubility curves.

Traditionally, the MSL has been experimentally determined by the isothermal method or the polythermal method. In the isothermal method, the solution is instantaneously undercooled or equivalently supersaturated and then fixed at the supersaturation until the detection of crystal particles is made. The induction period, the duration for the first appearance of crystal particles, is a consequence of not only the solution characteristics but also the many external influences. Nonetheless, it is frequently used for calculation of interfacial energy assuming that the nucleation time governs the induction period.1,3-5 This approach has been more elaborated over the past two decades or so, enhancing the reliability of the nucleation-rate model.6 This model can be used to predict the generation of unwanted crystals after incorporation with the growth model.7,8 Despite its advantages, the irrational assumptions depending on situation often render the estimate of interfacial energy and the resulting nucleation model unreliable for practical use. In the polythermal method, the solution is cooled at a constant rate until visible crystal particles appear and the metastable zone width (MZW), the difference between the saturation and crystal detection temperatures, is measured. The polythermal MZW is easy to be determined and, thus, has been widely used for

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Figure 2. Dynamic behavior of MAUC to changes in the cooling rate: (a) step change and (b) ramp change in the solution temperature.

Figure 3. Experimental apparatus.

identifying the metastable zone. However, the MZW is not only a property of the solution itself but also varies with internal as well as external influences on the solution. It was Ny´vlt who first expressed the MZW as a nonlinear function of cooling rate.9 Since then, his model and its variants have been employed predominantly. The polythermal MZW is a complicated consequence of both nucleation and growth and has scarcely been used for estimation of fundamental solution properties, such as interfacial energy and growth kinetics, while the isothermal method has been related to the estimation of these fundamental properties. Among those attempts, So¨hnel and Mullin10 proposed a method to determine growth parameters in a special polythermal method composed of cooling followed by holding under the assumption that the nucleation time is negligible. Mersmann and Bartosch11 have predicted the MZW using the nucleation and growth models and could have obtained reasonable results. Despite those attempts, sound physical-chemical interpretation of MZW seems to be premature yet, and the worth of the polythermal MZW still lies in its empirical model that can be directly used for practical purpose. The existing MZW models have limitations in that they are valid only for constant rate cooling under constant solution concentration and, thus, cannot provide a safeguard in practical crystallization operation where the solution concentration, as well as solution temperature, varies dynamically. For example, if the cooling is ceased at some temperature before nucleation occurs, the present MZW model predicts the appearance of visible crystals at the moment the cooling stops, which is not true in reality, as the crystals appear after some period.10,12 The existing MZW model also gives a contradictory prediction to the induction-period model. In the case of instantaneous subcooling, the polythermal MZW model predicts instantaneous outcome of crystal particles, whereas the induction-time model

does not. To correctly predict the MZW behavior of underdynamically changing solution state, therefore, it is necessary to improve the existing MZW model. If a reliable prediction of MZW is possible in real time, one may push the solution temperature to the point where the driving force for crystal growth is maximized while not violating the MSL, which results in maximum crystal production. The objective of the present research has been placed in elucidating the MZW as a dynamic state in a macroscopic sense and developing a novel empirical model that can appropriately represent the behavior of MZW under dynamically changing solution state. Through reasoning with physically more probable response of MZW to various changes in cooling rate, it has been proposed that the MZW is described as an output of a first-order linear dynamic model plus input nonlinearity with cooling rate as an input. To investigate the performance of the proposed model, careful experiments have been conducted with (NH4)2SO4, KAl(SO4)2, and KCl solutions under various cooling scenarios, including changes in the cooling rate, holding temperature after cooling, undersaturated starting temperature, and thermal history in the undersaturated state. Some of the experimental results were used for parameter estimation of the proposed model, and the others were used for model validation. 2. Modeling of Metastable Zone Width Behavior 2.1. Metastable Limit. The basic notions related to the cooling crystallization can be described in the solubilitysupersolubility curves shown in Figure 1. The solubility curve is well-defined and determined uniquely through experiments.13 On the other hand, the supersolubility curve, or, equivalently, the metastable limit, is not. The metastable limit is defined as the boundary of the region within which spontaneous crystallization up to a visible size under supersaturation is unlikely. The nucleation is brought about through different mechanisms by different causes, and hence, the metastable limit needs to be defined differently according to the nucleation mechanism. For example, the metastable limit by the primary homogeneous nucleation is quite different from that by the surface nucleation.14 Therefore, the governing nucleation mechanism should be defined first when we determine the metastable limit in industrial crystallization. In Figure 1, at a certain solution state (T, C) in the metastable zone, C - C*(T) represents the driving force for crystal growth where C*(T) denotes the saturation concentration at T. ∆Cmax(T) is called the maximum allowable supersaturation (MASS) at T. Likewise, ∆Tmax(C) is called the maximum allowable

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Table 1. Experimental and Predicted Values of the Metastable Zone Width for Ammonium Sulfatea T(0) (°C)

u (°C/h)

Th (°C)

measured tn (s)

predicted tn (s)

60 60 60 60 60 60 60 60 60 60 60 60

30.0 30.0 30.0 30.0 27.5 25.0 22.5 20.0 17.5 15.0 12.5 10.0

44.0 45.0 47.0

2390 2543 3314 2278 2373 2543 2574 2799 2968 3276 3692 4259

2448 2619 3231 2213 2303 2415 2555 2734 2968 3283 3730 4406

measured ∆Tmax(tn) (°C)

8.98 8.13 7.66 6.09 5.55 4.43 3.65 2.82 1.83

predicted ∆Tmax(tˆn) (°C)

8.44 7.59 6.77 5.97 5.19 4.43 3.68 2.95 2.24

a Solution conc. ) 0.8425 ((NH ) SO kg/water kg) and saturation temp. 4 2 4 ) 50 (°C). Estimated model parameters: k ) 0.0858, p ) 1.4701, and τ ) 0.5651. Root-mean-squared prediction error of the induction time ) 2.6214%.

Table 2. Experimental and Predicted Values of the Metastable Zone Width for Potassium Aluminum Sulfatea T(0) (°C)

u (°C/h)

Th (°C)

measured tn (s)

predicted ˆtn (s)

measured ∆Tmax(tn) (°C)

predicted ∆Tmax(tˆn) (°C)

60 60 60 60 60 60

30.0 30.0 30.0 25.0 20.0 10.0

44.0 45.0

1894 1896 2111 2435 2812 5332

1881 1904 2153 2460 2722 5404

7.59 6.91 5.62 4.81

7.94 7.08 5.12 5.01

a Solution conc. ) 0.34(KAl(SO ) kg/water kg) and saturation temp. 4 2 ) 50 (°C). Estimated model parameters: k ) 1.862, p ) 0.4096, and τ ) 0.0959. Root-mean-squared prediction error of the induction time ) 0.7008%.

Table 3. Experimental and Predicted Values of the Metastable Zone Width for Potassium Chloridea T(0) (°C)

u (°C/h)

Th (°C)

measured tn (s)

predicted ˆtn (s)

measured ∆Tmax(tn) (°C)

predicted ∆Tmax(tˆn) (°C)

60 60 60 60 60 60

20.0 20.0 25.0 20.0 15.0 10.0

44.0 45.0

2601 2696 2335 2650 3287 4359

2584 2721 2388 2687 3403 4375

6.22 4.72 3.69 2.11

6.58 4.93 4.18 2.15

a Solution conc. ) 0.425 (KCl kg/water kg) and saturation temp. ) 50 (°C). Estimated model parameters: k ) 0.1070, p ) 1.2649, and τ ) 0.2361. Root-mean-squared prediction error of the induction time ) 0.7661%.

undercooling (MAUC) at C. Either MASS or MAUC can be used to define the MZW. In this research, the MZW is defined as MAUC. 2.2. Existing MZW Models. In Figure 1, if a solution in the stable zone is cooled at a constant rate, tiny crystals begin to appear at a certain point beyond the saturation temperature. It is usually observed that the higher cooling rate results in a lower temperature for crystals appearance. On the basis of these observations, Ny´vlt et al.9 proposed the following model that relates the cooling rate to the MZW,

∆Tmax ) kup

(1)

where u ≡ -dT/dt represents the cooling rate of the solution. In the above, the parameters p and k usually appear to be strongly dependent on the solution concentration. The MZW model in eq 1 has been widely accepted and used for the design

of cooling curves in industrial crystallization. In terms of ∆Cmax, Choi and Kim have proposed the model15

∆Cmax ) k0uk1 + k2(T0 - T)

(2)

where T0 is the initial temperature of the solution. The expression for ∆Cmax is convertible to an expression for ∆Tmax and vice versa under the assumption that the slope of the supersolubility curve is the same as that of the solubility curve. 2.3. Drawbacks in Existing MZW Models. The existing MZW models provide correct prediction under constant rate cooling with constant solution concentration. In industrial batchcooling crystallization operation, however, the cooling rate varies with time while the solution concentration continuously decreases. The existing models cannot properly predict the behavior of MZW under such situations. For example, the model in eq 1 yields ∆Tmax ) 0 when u ) 0. This implies that visible crystals appear at the moment when the cooling rate is turned to zero if the solution temperature is lower than the saturation temperature, as shown in Figure 2a. In reality, crystal particles would be observed after some period of time from the instant when the cooling is stopped. So¨hnel and Mullin10 proposed a model for induction time of crystal appearance under constantrate cooling followed by holding and showed that a certain induction period exists before the crystal is visible after the cooling is stopped. As a next case, if the solution temperature drops stepwise, ∆Tmax(t) is predicted as an impulse function by eq 1. This implies that the MZW is pushed back to infinity momentarily and returned to zero instantaneously, which is impossible to occur. A more probable response of ∆Tmax(t) to this change would be a sudden increase followed by a slow decay, as shown in the Figure 2b. Indeed, such cooling is done for induction-time measurement under constant supersaturation. This tells that the model in eq 1 is in conflict with the MZW model by the isothermal method. As an additional case, consider the situation where the solution is cooled at a constant rate from an initial saturation temperature. Since the cooling rate increases stepwise, the MZW by eq 1 increases stepwise where the step size depends on the cooling rate. If this truly happens, the faster initial cooling provides the larger MZW by eq 1 and always becomes advantageous over the slower initial cooling. In real operations, however, such a cooling strategy is seldom employed. Perhaps a more likely response of ∆Tmax(t) would be a smooth lagged increase over a period, as shown in Figure 2c. From the system theoretic point of view, the responses of ∆Tmax(t) proposed above can be interpreted as a consequence of dynamic behavior of the MZW, and a new model needs to be introduced to appropriately describe the dynamic responses. 2.4. Dynamic Model for MZW. The existing MZW models consider that u(t) determines ∆Tmax(t), i.e., the present cooling rate determines the present MZW. However, the appearance of visible crystals is a consequence of continuous nucleation and growth over a period. In other words, ∆Tmax(t) is an integral effect of past u(τ), 0 e τ e t, instead of an instantaneous effect of u(t). This can be written as

∆Tmax(t) )

∫0t F(t, τ, u(τ)) dτ

(3)

for some nonlinear function F. When F is linear in u, i.e.,

∆Tmax(t) )

∫0t g(t, τ)u(τ) dτ

(4)

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Figure 4. Prediction of nucleation time based on simulation for different linear cooling rates for ammonium sulfate.

an equivalent state-space equation of the following form exists for some matrices A(t), B(t), C(t):

dx(t) ) A(t)x(t) + B(t)u(t), ∆Tmax(t) ) C(t)x(t) dt

(5)

In the above, x(t) is an n-dimensional vector called state. To represent the nonlinear relationship between u(t) and ∆Tmax(t), one may introduce a nonlinear state-space equation. In this research, we propose the following one-dimensional state-space model with input nonlinearity for the MZW model:

dx(t) + x(t) ) ku(t)p, ∆Tmax(t) ) x(t); x(0) ) x0 τ dt

(6)

When the solution remains in a saturated state for a sufficiently long period, it obviously holds that ∆Tmax(t) ) x0 ) 0. By extending the idea, we define x0 ) 0 as the state when the solution rests in an undersaturated state for a sufficiently long period as well. Upon integration with x(0) ) 0, ∆Tmax(t) is represented as

∆Tmax(t) )

∫0t e-(t-s)/τku(s)p ds

1 τ

(7)

It can easily be seen that the above model represents eq 1 at the steady state under constant u since ∆Tmax(t) f kup as t f ∞. During the transient period, x(t) is subject to the first-order dynamics instead. The supposed responses of ∆Tmax(t) in Figure

2 are derived, though not perfectly, from the first-order dynamics. It is true that a higher-order dynamic model may provide more reasonable responses. In this case, however, the number of parameters to estimate is increased. In this model, the state x is an internal state that relates the ∆Tmax to the time-integrated effects of the cooling rate. Then, by solving the above differential equation with the time trajectory of the cooling rate changes as input, the time trajectory of ∆Tmax can be calculated. 3. Experimental Part 3.1. Apparatus and Procedure. The experimental apparatus is shown in Figure 3. Solutions with saturation concentration at 50 °C were prepared by dissolving EP-grade (NH4)2SO4 in distilled water. The solution was transferred to a 100 mL conical glass flask whose top has a long and narrow glass tube on it. The solution was filled to the top of the tube for suppressing the undesired nucleation by the contact with air to be localized at the top of the small tube. The solution temperature was measured by thermocouples calibrated within (0.01 °C accuracy and controlled using a water bath. The solution was heated to 5-10 °C above the saturation temperature and maintained at least for 180 min to eliminate the effect of solution thermal history. Then, the solution was cooled at a specified rate and cooling scenario using a proportional-integral-derivative (PID) controller until the crystals appeared. The crystals were detected visually with the aid of a stroboscope against a black

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Figure 5. Prediction of the nucleation time based on simulation for cooling rate of 30 °C/h and holding temperature at (a) 44 °C, (b) 45 °C, and (c) 47 °C for ammonium sulfate.

background. The solutions were prepared newly for each experiment to exclude the effect of recrystallization. Throughout the experiments, each experiment was repeated several times and the deviations were th (10) where Th and th denote the holding temperature and the corresponding time, respectively, and S(t) represents the Heaviside function, which is zero for t < 0 and 1 otherwise. Th and th are related by Th ) T0 - uth. If the cooling rate changes in an arbitrary manner, the ∆Tmax(t) has to be obtained by solving the ordinary differential equation in eq 6. 4. Results and Discussion 4.1. Comparisons of Induction Time between Prediction and Experimental Values. After stabilization of the prepared solution, the solution was cooled according to various cooling scenarios until the crystals were observed. The data consist of two different cooling strategies. One is the constant-cooling strategy in which the cooling rate is between 10 and 30 °C/h, and the other is the cooling-holding strategy, which starts at 60 °C with a cooling rate of 30 °C/h and holds the temperature at several different values. The holding temperatures should be in the supersaturation states. The scenario of cooling and holding is crucial to find the model parameters, especially the time constant. In Tables 1-3, the parameter estimates for the proposed MZW model are given together with the experimental data set used for the estimation and the polythermal induction time predicted by the model for the cases of ammonium sulfate, potassium aluminum sulfate, and potassium chloride, respectively. Using these data, the parameters of the proposed model were determined in the least-squares sense to the first-order dynamic model. The second-order model was tested. However, the improvement over the first-order model was insignificant. The rms errors of the induction-time prediction were found to be around 0.70-2.62%. The case for ammonium sulfate had

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Figure 6. Prediction of the nucleation temperature using the proposed model for different initial cooling temperatures. The initial cooling temperatures are (a) 60 °C, (b) 55 °C, and (c) 52 °C at a cooling rate of 30 °C/h for ammonium sulfate.

Figure 7. Prediction of the nucleation temperature using the proposed model for different solution thermal histories. The initial resting temperature is 60 °C, and the cooling rates from 60 to 55 °C are (a) 30 °C/h, (b) 5 °C/h, and (c) 2.5 °C/h. The cooling rate of 30 °C/h is applied after 55 °C for ammonium sulfate.

the largest error, but the predictions of induction time were quite accurate in a reasonable range for crystallization experiments. In Figures 4 and 5, the model quality is shown by comparing the measured MZWs with the predicted values graphically for the ammonium sulfate system. Figure 4 compares the predictions of induction time for different constant-cooling-rate policies. Figure 5 shows the model accuracy for cooling-holding strategies. For different cooling policies, the trajectories of solution temperature and metastable limit estimated by the MZW

model are shown together with the observed induction times. The figures depict how the metastable limit moves while the solution is cooled following a certain pattern. It can be seen that the predicted induction times agree quite accurately with the experimentally observed values. 4.2. Verification of the Proposed Model. If the solution is held at a certain undersaturated state for a long period, the solution reaches equilibrium and the effects of the thermal history of the solution fade out. Otherwise, the thermal history

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Table 4. Comparison of Observed and Predicted Nucleation Temperature for Different Initial Cooling Temperatures cooling condition initial T (°C)

cooling rate (°C/h)

observed nucleation T (°C)

predicted nucleation T (°C)

prediction error (°C)

60 55 52

30 30 30

41.02 43.65 46.80

41.56 43.87 46.44

-0.44 0.22 0.36

Table 5. Comparison of Observed and Predicted Nucleation Temperatures for Four Cases of Thermal History of the Solution with Saturation Temperature of 50 °C; The Second Cooling Rate Is Applied When the Solution Temperature Reaches 55 °C experiment condition initial T (°C)

first cooling rate (°C/h)

second cooling rate (°C/h)

observed nucleation T (°C)

predicted nucleation T (°C)

prediction error (°C)

60 60 60

30 5 2.5

30 30 30

41.02 43.98 44.50

41.56 43.26 43.60

-0.54 0.72 0.90

of the solution has significant effects on the solution state.1 Indeed, the effect of overheating treatment on nucleation is one of the recent issues in crystallization.16,17 In this section, it was investigated if the proposed dynamic model can appropriately represent the effect of the thermal history of the solution in the undersaturated state on the metastable limit. For this, a series of experiments has been carried out with an ammonium sulfate solution saturated at 50 °C. First, the effects of initial resting temperature were investigated. After at least 3 h of resting at 52, 55, and 60 °C, respectively, the solution underwent constant-rate cooling at 30 °C/h. Second, the effects of the thermal history of the solution during the undersaturated state are investigated. The initial resting temperature was fixed at 60 °C. From this initial condition, the solution temperature is cooled down to 55 °C with different cooling rates of 30, 5, and 2.5 °C/h, and then the cooling rate of 30 °C/h is applied from 55 °C, which is still at the undersaturated temperature. From the existing models, these variations in operating condition cannot be explained. The experimental and simulation results of the nucleation temperature at which the nucleation is detected are summarized in Tables 4 and 5. Figures 6 and 7 show how the nucleation time is predicted using the proposed dynamic model for different cooling scenarios. As can be seen, the maximum discrepancy between the observed and predicted nucleation temperatures for all cases was 0.9 °C, and the performance of the proposed dynamic model is outstanding. It is interesting to note that our model could represent the effect of thermal history of solutions although the model is a purely empirical one. 5. Conclusions In this paper, mathematical representation of the relationship between the MZW and the cooling rate on cooling crystallization has been addressed. Previous studies have confined the relationship to static ones, i.e., the present cooling rate determines the present MZW value. In this study, it was first noted that the metastable limit varies as an integral effect of the past history of cooling rate, and based on this consideration, a first-order dynamic model with input nonlinearity has been proposed. To investigate the validity of the proposed model, a series of experiments have been conducted for crystallization of am-

monium sulfate, potassium aluminum sulfate, and potassium chloride in an aqueous solution. From the experiments, it was found that the proposed model quite accurately predicts the polythermal induction time at which the nucleation is detected under various cooling profiles and different thermal histories. Even though the proposed model cannot explain the behavior of the metastable limit in physicochemical terms, it successfully expresses the phenomenological solution behavior for cooling crystallization, at least for the concerned inorganic materials. It is worth mentioning that this model can be applied to other crystallization processes such as evaporative crystallization, drowning out, and so on, since other processes are very similar except they have different driving forces. More investigations for different solution systems are underway. Also, it is believed that the proposed MZW model can be utilized as a real-time operation tool that provides a cooling strategy by which the undesired nucleation is suppressed while optimizing the crystal production and other objectives. Acknowledgment This work was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by the Korea government (MOST) (No. R01-2006-000-11377-0) and also by the Korea Energy Management Cooperation. Nomenclature C ) concentration of solution (solute kg/solvent kg) C* ) saturation concentration of solution (solute kg/solvent kg) ∆C ) supersaturation (C - C*) ∆Cmax ) maximum allowable supersaturation (MASS) k ) constant for metastable limit dynamics p ) kinetic order of cooling rate for metastable limit dynamics S(t) ) Heaviside function T ) temperature of solution (°C) T* ) saturation temperature of solution (°C) ∆Tmax ) maximum allowable undercooling (MAUC) (°C) t ) time (h) tn ) nucleation time at which visible crystal is detected (h) th ) time at which the cooling stops (h) τ ) time constant for metastable limit dynamics (h) u ) cooling rate (°C/h) Literature Cited (1) Mullin, J. W. Crystallization, 4th ed.; Butterworth-Heinemann: Oxford, U.K., 2001. (2) Ostwald, W. Studien uber die Umwandlung und Bildung fester Korper. Eitschrift Phys. Chem. 1897, 22, 289. (3) Boistelle, R.; Astier, J. P. Crystallization Mechanism in Solution. J. Cryst. Growth 1988, 90, 14. (4) Selvaraju, K.; Valluvan, R.; Kumararaman, S. Experimental Determination of Metastable Zone Width, Induction Period, Interfacial Energy and Growth of Non-linear Optical L-Glutamic Acid Hydrochloride Single Crystals. Mater. Lett. 2006, 60, 1565. (5) Ny´vlt, J. Induction Period of Nucleation and Metastable Zone Width. Collect. Czech. Chem. Commun. 1983, 48, 1977. (6) Kashchiev, D.; van Rosmalen, G. M. Review: Nucleation in Solutions Revisited. Cryst. Res. Technol. 2003, 38, 555. (7) Gerstlauer, A.; Motz, S.; Mitrovic, A.; Gilles, E.-D. Development, Analysis and Validation of Population Models for Continuous and Batch Crystallizers. Chem. Eng. Sci. 2002, 57, 4311. (8) Hu, Q.; Rohani, S.; Jutan, A. Modeling and Optimization of Seeded Batch Crystallizers. Comput. Chem. Eng. 2005, 29, 911. (9) Ny´vlt, J.; Rychly, R.; Gottfried, J.; Wurzelova, J. Metastable ZoneWidth of Some Aqueous Solutions. J. Cryst. Growth 1970, 6, 151. (10) So¨hnel, O.; Mullin, J. W. The Role of Time in Metastable Zone Width Determination. Chem. Eng. Res. Des. 1988, 66, 537.

Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8165 (11) Mersmann, A.; Bartosch, K. How to Predict the Metastable Zone Width. J. Cryst. Growth 1998, 183, 240. (12) So¨hnel, O.; Mullin, J. W. Interpretation of Crystallization Induction Periods. J. Colloid Interface Sci. 1988, 123, 43. (13) Lide, D. R. Handbook of Chemistry and Physics, 81st ed.; Taylor and Francis: Boca Raton, FL, 2000. (14) Mersmann, A. Super-saturation and Nucleation. Trans. Inst. Chem. Eng. 1996, 74 (Part A), 812. (15) Choi, C. S.; Kim, I. S. Growth Kinetics of (NH4)2SO4 in the Ternary System (NH4)2SO4-NH4SO4-H2O. Ind. Eng. Chem. Res. 1990, 29, 1562.

(16) Sugimoto, T.; Mori, A.; Inoue, T. Effect of “Overheating Treatment” on the Stability of KCl Aqueous Solutions. J. Cryst. Growth 2006, 292, 108. (17) Hussain, K.; Thorsen, G.; Malthe-Sorenssen, D. Nucleation and Metastability in Crystallization of Vanillin and Ethyl Vanillin. Chem. Eng. Sci. 2006, 56, 2295.

ReceiVed for reView February 5, 2007 ReVised manuscript receiVed August 12, 2007 Accepted August 13, 2007 IE070209M