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Thermodynamic Analysis of Temperature Dependence of the Crystal Growth Rate of Potassium Sulfate Fangqin Cheng,† Yang Bai,‡ Chang Liu,‡ Xiaohua Lu,*,‡ and Chuan Dong† School of EnVironment and Resources, Shanxi UniVersity, Taiyuan 030006, People’s Republic of China, and College of Chemistry and Chemical Engineering, Nanjing UniVersity of Technology, Nanjing 210009, People’s Republic of China
In this work, the effect of temperature on the crystal growth rate of potassium sulfate in aqueous K2SO4 and KCl-K2SO4 solutions is analyzed. The crystal growth rate equation of K2SO4 is developed on the basis of the difference of chemical potentials of K2SO4 at the solid-liquid interface. The Lu-Maurer model is used to predict the activity coefficient of K2SO4. The crystal growth rate of K2SO4 in aqueous K2SO4 solution decreases, then increases with increasing temperature, and reaches its lowest value around 303.15 K, which is in accordance with previous experimental results. Meanwhile, the optimal crystallization temperature of K2SO4 in aqueous KCl-K2SO4 solution is discussed. An effective method is found to increase the yield of K2SO4 by increasing the crystallization temperature to 333.15 K. Introduction Crystallization is an important solid-liquid separation method and has been used to separate and purify a variety of substances including inorganic salt and protein.1-5 Because crystallization temperature shows a notable effect on the solubility of solute, product yield, and crystallization rate, it is one of the key parameters for both the design of a crystallizer and the crystallization process.2 Generally, the crystal growth rate increases with increasing temperature. However, the temperature dependence of the crystal growth rate of K2SO4 is abnormal in the low temperature range; that is, the growth rate decreases with the increase of the temperature from 273.15 to 293.15 K. This abnormal phenomenon is probably due to the deposition of fine crystals on the seed crystal surface which may enhance the crystal growth rate.6 Theoretical explanation is still needed to understand it clearly. On the other hand, the solution is complicated for the large scale production of K2SO4. For example, if K2SO4 is produced from sodium sulfate and potassium chloride, the solution contains Na+, K+, Cl-, and SO42- during the crystallization process.7 The complicated solution causes the complexity of the temperature effect on the growth rate of K2SO4 and the difficulty in theoretical study for crystallization. Most of the available studies are based on empirical equations to correlate experimental data, and the driving force for crystallization is simplified to be the concentration difference between the crystal and the bulk solution. However, as Myerson et al. have pointed out, the fundamental driving force for crystallization is the difference in the chemical potential of crystal at the solid-liquid interface8,9 in which a thermodynamic model is needed to calculate thermodynamic properties of activity coefficients and chemical potential. The rigorous theoretical study on crystallization should be carried out based on this point. In our previous work, thermodynamic properties of aqueous electrolyte solutions were studied. In this paper, the temperature * To whom correspondence should be addressed. E-mail: xhlu@ njut.edu.cn. Tel.: 86-25-83587205. † Shanxi University. ‡ Nanjing University of Technology.
dependence of the crystal growth rate of K2SO4 is investigated from a rigorous theoretical model. The proper crystallization temperature of K2SO4 in aqueous KCl-K2SO4 solution is discussed theoretically and experimentally. Theory Generally, the crystal growth rate is expressed as
G ) kσg
(1)
where G is a empirical parameter regressed from experimental data, σ is the driving force, g is the growth order, and k is the crystal growth rate coefficient related with temperature and the hydrodynamic state in solution.
k ) k0e-E/(RT)
(2)
The driving force σ is described by the chemistry potential difference at the solid-liquid interface (∆µ/RT) as suggested by Myerson3 instead of by the concentration difference,
() ( )
m(νγ(ν ∆µ a ) ln σ) ) ln RT a* KSP
(3)
where γ( is the mean ionic activity coefficient of an electrolyte in supersaturated solution, m( is the mean ionic molality, ν is the stoichiometric coefficient, and a* (KSP) is the solubility product. Substitute eqs 2 and 3, and eq 1 becomes -E/(RT)
G ) k0e
( ( )) m(νγ(ν ln KSP
g
(4)
For crystal K2SO4, according to the Burton-Cabrera-Frank (BCF) surface diffusion theory,9,10 the index g is equal to 2 at low supersaturations. Equation 4 becomes -E/(RT)
G ) k0 e
(( ln
))
mK2mSO4γ(3 KSP
2
(5)
Results and Discussion (1) Parameters. The solubility product of K2SO4 in aqueous solution is calculated by
10.1021/ie0513649 CCC: $33.50 © 2006 American Chemical Society Published on Web 08/04/2006
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KSP ) m′K2m′SO4γ′(3
(6)
m′K ) 2m*
(7)
m′SO4 ) m*
(8)
where m* is the solubility of K2SO4 in aqueous solution and is affected by temperature notably. In the range of 273.15-333.15 K, the solubility data of K2SO4 are taken from literature11 and regressed using a polynomial of six orders by the least-squares method. The fit results are listed in Table 1, and the corresponding correlation coefficient R2 ) 0.999 988.
m* ) [(a + bx + cx2 + dx3 + ex4 + fx5 + gx6) × 10]/174.25 (9) where
Figure 1. Temperature effect on the crystal growth rate in aqueous K2SO4 solution where the activity coefficient is calculated by the Lu-Maurer model (E ) 48.981 kJ/mol, k0 ) 2.903 × 107 kg/(m2‚s)).
x ) T - 273.15 At a certain temperature, the solubility product (KSP) of K2SO4 can be calculated from eqs 6-9 and by using the LuMaurer model to calculate the corresponding mean ionic activity coefficient γ′( in eq 6.12,13 The temperature dependence of KSP is expressed as in eq 10 with the corresponding coefficients listed in Table 2.
T 1 (T1 - 298.15 ) + U ln(298.15 )+
ln KSP ) U1 + U2
3
U4(T - 298.15) (10)
where U1, U2, U3, and U4 are the model parameters. Parameters E and k0 are regressed from the experimental results reported by Garside et al.,14 and the results are listed in Table 3. Equation 5 can be used to analyze the temperature effect on the crystal growth rate of K2SO4 for both aqueous K2SO4 and K2SO4-KCl solutions. The difference is how mK is calculted. For the aqueous K2SO4 solution,
mK ) 2mK2SO4 ) 2mSO4
(11)
while for the aqueous K2SO4-KCl solution
mK ) 2mK2SO4 + mKCl ) 2mSO4 + mCl
(12)
(2) Temperature Effect on the Crystal Growth Rate in the Aqueous K2SO4 Solution. Figure 1 shows the temperature effect on the crystal growth rate at different supersaturation (∆m ) mK2SO4 - m*) in aqueous K2SO4 solution which is calculated with eq 5. The activity coefficient of K2SO4 in the solution is calculated with the Lu-Maurer model. It is clear that the crystal growth rate increases with increasing supersaturation in solution at the same temperature. At a certain supersaturation, the crystal growth rate decreases first and then increases with increasing temperature, and the minimum growth rate is found at around 303.15 K, which is in accordance with the data reported by Taguchi et al.6 This phenomenon can be explained theoretically according to eq 5. The crystal growth rate G can be written as G ) GTGm, where GT ) k0e-E/(RT) and Gm ) {ln[(mK2mSO4γ(3)/KSP]}2. The term GT represents the Arrhenius relationship of the crystal growth
Figure 2. Effects of the terms GT, Gm, and Gm′ on the crystal growth rate in the supersaturation solution (∆m ) 0.08 mol/kg, E ) 48.981 kJ/mol, k0 ) 2.903 × 107 kg/(m2‚s)). Table 2. Parameters in Equation 10 for the Solubility Products of K2SO4 name of salt
T, K
U1
U2
U3
U4
K2SO4
273-373
-4.186 29
79 646.1
443.046
0.622 767
Table 3. Correlated Activation Energy E and Pre-Exponential Factor k0 from Experimental Results by Garside et al.14 E, kJ/mol
k0, kg/(m2‚s)
R2
48.981
2.903 × 107
0.9903
rate coefficient which increases with increasing temperature as shown in Figure 2. The term Gm represents the driving force in crystallization by the chemical potential difference of K2SO4 at the solid-liquid interface. The value (mK2mSO4γ(3)/KSP in the expression for Gm is always more than 1 under the conditions of interest. With increasing temperature, the solubility of K2SO4 in the aqueous solution and the solubility product KSP increase, which results in the decrease of (mK2mSO4γ (3)/KSP and approaching to 1. When temperature is lower than 303.15 K, the effect of temperature on Gm is more than that on GT, and the crystal growth rate of K2SO4 decreases with increasing temperature. However, when temperature is higher than 303.15 K, with increasing temperature, the change of Gm is very small while GT still increases, which causes the crystal growth rate of K2SO4 to increase. If the activity coefficient of K2SO4 is not considered, that is, γ( in the supersaturated solution is assumed to be the same as γ(* in the saturated solution, the crystal growth rates at different
Table 1. Coefficients in Equation 9 for the Solubility of K2SO4 a
b
c
d
e
f
g
7.654 38
0.089 344 3
0.013 013 3
-0.000 835 32
2.635 98 × 10-5
-3.9454 × 10-7
2.230 79 × 10-9
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Figure 3. Effect of temperature on the crystal growth rate in aqueous K2SO4 solution where the activity coefficient is neglected (E ) 48.981 kJ/ mol, k0 ) 2.903 × 107 kg/(m2‚s)).
Figure 4. Effect of temperature on the crystal growth rate in aqueous K2SO4 solution in the case of E ) 30 kJ/mol.
temperatures and supersaturations are also calculated and are shown in Figure 3. At a certain supersaturation, the crystal growth rate increases when the temperature increases from 283.15 K to 333.15 K. The tendency of crystal growth rate decrease with increasing temperature does not exist. This result can be explained from the curve Gm′ in Figure 2. Here
( ( )) ( (
Gm′ ) ln
mK2mSO4 4m*3
2
) ln
))
4(m* + ∆m)3 4m*3
2
which is the contribution due to the concentration difference on crystal growth rate without accounting for the difference of activity coefficients of K2SO4 in supersaturated and saturated solutions. In the expression of Gm and Gm′, KSP, γ(, and m* are all sensitive to temperature. Thus, Gm and Gm′ will change with temperature. The notable difference between Gm and Gm′ in Figure 2 results in the wrong result in the low-temperature range (283.15-303.15 K), which implies that the importance of considering the activity coefficient in studying the crystal growth process of K2SO4, especially at low temperature. The activation energy used in the calculations for Figures 1 and 3 is regressed from the experimental data by Garside et al.14 () 48.9 kJ/mol). However, its value is different for different crystallizers and under different crystallization conditions. Therefore, the effect of activation energy on the crystal growth rate is investigated by assuming that the activation energies are 30 and 70 kJ/mol, respectively. The results of the crystal growth rate are shown in Figures 4 and 5. The crystal growth rate with the activation energy of 30 kJ/mol is about 1000 times that shown in Figure 1 while the minimum growth rate is still at
Figure 5. Effect of temperature on the crystal growth rate in aqueous K2SO4 solution in the case of E ) 70 kJ/mol.
Figure 6. Effect of temperature on the crystal growth rate in aqueous KClK2SO4 solution (E ) 48.981 kJ/mol, k0 ) 2.903 × 107 kg/(m2‚s)).
303.15 K at the same supersaturated degree. When the temperature is lower than 303.15 K, the decrease of the crystal growth rate with increasing temperature is more notable than that in Figure 1. When the activation energy is 70 kJ/mol, the crystal growth rate is three orders lower than that shown in Figure 1. Although the minimum growth rate is still at 303.15 K, the temperature effect on the crystal growth rate is not notable at low temperature. (3) Temperature Effect on Crystal Growth Rate in Aqueous KCl-K2SO4 Solution. The crystallization of K2SO4 is usually carried out by cooling or vaporizing its aqueous solution. Another effective method is to add some salts such as KCl which has the common-ion effect. In this case, the solution becomes an aqueous multiple-electrolyte solution. The crystal growth rate of K2SO4 is probably affected by various ions in the multiple-electrolyte solution. Because the effect of the activity coefficients cannot be neglected especially at low temperatures as discussed in the former text, it is important to use a reliable model to predict the activity coefficient of K2SO4 in such a complicated solution at different temperatures. The Lu-Maurer model is used in this paper to solve the problem with high accuracy.15 The temperature effect on the crystal growth rate of K2SO4 in saturated K2SO4 solution with additional KCl is investigated. It is assumed that the addition of KCl does not affect the crystallization of K2SO4, and the activation energy E of crystallization and the pre-exponential constant k0 are the same as those for the aqueous K2SO4 solution listed in Table 3. The calculated results are shown in Figure 6. It is clear that the crystal growth rate of K2SO4 increases with increasing amount of KCl. At a certain amount of KCl, the crystal growth rate decreases and then increases with increasing temperature with
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Figure 7. Comparison of the added amount of KCl (mCl) in saturated K2SO4 solution at 283.15 K and the supersaturation (∆m) in single K2SO4 solution at the same crystal growth rate.
the minimum at 303.15 K, which is in accordance with that for the aqueous K2SO4 solution. As shown in Figure 6, the crystal growth rate of K2SO4 can be enhanced by either increasing or decreasing the temperature from room temperature. When the temperature is higher than 323.15 K, the increase of the crystal growth rate is more notable. Considering that it is more convenient and economical to use high-temperature steam to heat the solution than to use other methods to cool the solution in many factories, heating the solution to 333.15 K is an efficient way to increase the K2SO4 yield. The amount of KCl (mKCl ) mCl) that should be added to the saturated K2SO4 solution to reach the same crystal growth rate as that in the single K2SO4 aqueous solution with supersaturation ∆m is interesting. This question can be resolved by combining the growth rate equations both in the aqueous KCl-K2SO4 solution and in the aqueous K2SO4 solution to obtain the following equation.
mCl )
x
4(m* + ∆m)3γ13 m*γ23
- 2m*
(13)
where γ1 and γ2 are the activity coefficients of K2SO4 in aqueous K2SO4 and KCl-K2SO4 solutions, respectively. The added amount of KCl (mKCl ) mCl) added to the saturated aqueous K2SO4 solution at different temperatures can be calculated from eq 13. The results at 283.15 K are shown in Figure 7. It is found that mCl increases linearly with increasing ∆m with a slope of 3.103 at low supersaturation (∆m < 0.1 mol/kg). At other temperatures the increments of mCl with increasing ∆m are still linear but with different slopes. The effect of temperature on the slopes (mCl/∆m) is shown in Figure 8. The ratio of mCl to ∆m is 3.05-3.1 in the temperature range of 283.15-333.15 K, and the ratio decreases with increasing temperature, which means that the amount of KCl added at higher temperature is less than that at lower temperature. It is also implies that it is an effective method to increase the production of K2SO4 by increasing temperature. (4) Temperature Effect on the Equilibrium Precipitation Amount of K2SO4 in KCl-K2SO4 Aqueous Solution. K2SO4 will be precipitated if a certain amount of KCl is added into the saturated K2SO4 aqueous solution.16 The calculated precipitation amount of K2SO4 versus the concentration of KCl is shown in Figure 9 from 283.15 to 333.15 K. The precipitation amount of K2SO4 increases with increasing amount of KCl; however, it decreases first and then increases with increasing
Figure 8. Ratio of added KCl to K2SO4 in saturated solution of K2SO4 versus temperature at the same crystal growth rate.
Figure 9. Effect of the added amount of KCl on the equilibrium precipitation amount of K2SO4 crystal in aqueous KCl-K2SO4 solution.
temperature to reach the minimum at 313.15 K, which is in accordance with the results shown in Figure 6. Throughout the whole comparison concentration ranges (mKCl ) 0.04-0.10 mol/kg), the effect of mKCl on the precipitation amount of K2SO4 at equilibrium is linear with the slope of 0.35. It means that the amount of KCl increases by 1 mol, and the precipitation amount of K2SO4 will always increase by 0.35 mol, which is independent of the crystallization temperature. The results shown in Figure 9 also imply that for the KCl-K2SO4 aqueous solution, the amount of K2SO4 crystal by crystallization at 333.15 K is more than that at room temperature. Experimental Verification The crystal growth rate of K2SO4 in aqueous K2SO4 solutions has been measured experimentally; however, to the best of our knowledge, there is no experimental data about the crystal growth rate of K2SO4 in aqueous K2SO4-KCl solutions. The calculation in aqueous K2SO4-KCl solution is based on the assumption that the activation energy E of crystallization and the pre-exponential constant k0 in the aqueous K2SO4-KCl solution are the same as those in the aqueous K2SO4 solution. New experimental data are measured to verify the reasonableness of this assumption and the reliability of the calculated result of the crystal growth rate of K2SO4 in the aqueous K2SO4KCl solution. The experiments are carried out in a batch crystallizer. A total of 1.192 g of crystals of KCl is added to the saturated solution of K2SO4 (with 200 g of water included, and the concentration of KCl is 0.08 mol/kg) at a given temperature. At the same time 1 g of sieved seed crystals of K2SO4 (0.3-0.4 mm) is added in the solution with a magnetic force stirring speed of 500 rpm
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Table 4. Effect of Temperature on the Crystal Growth Rate of K2SO4 in Aqueous KCl-K2SO4 Solution temperature (K)
precipitation amount of K2SO4 (g)
growth rate of K2SO4 (kg/m2‚s)
283.15 293.15 298.15 303.15 313.15 323.15 333.15
0.3023 0.1383 0.5459 0.5935 1.0380 1.0322 1.3918
2.7921 × 10-5 1.2774 × 10-5 5.0420 × 10-5 5.4817 × 10-5 9.5871 × 10-5 9.5336 × 10-5 1.2855 × 10-4
to avoid the primary and second nucleation. After the temperature is kept constant for about 0.5 h, crystals are taken out, dried for 0.5 h, and then weighed. The composition of the crystals has been measured by the X-ray diffraction XRD method. The experimental data of the crystal growth rate of K2SO4 in aqueous K2SO4-KCl solution at different temperatures are listed in Table 4 in which the surface area (S) of the seed crystals is calculated from the following equation with the mean diameter of 0.375 mm on the assumption that the shape of the seed crystals is a ball.
S)
Figure 10. XRD of K2SO4 with 1.192 g of KCl added into 200 mL of saturated K2SO4 aqueous solution at 333.15 K and after being kept isothermal for 0.5 h. b, K2SO4.
6 6 ) ) DF 0.375 × 10-3 × 2.66 × 106
6.015 × 10-3 m2 (14)
The experimental results listed in Table 4 show the minimum crystal growth rate of K2SO4 at 293.15 K at a certain amount of KCl (0.08 mol/kg). In addition, the growth rate of K2SO4 at 333.15 K is obviously faster than that at any other temperature, which means that the crystal growth rate of K2SO4 in aqueous KCl-K2SO4 solution can be increased by increasing the crystallization temperature. All these results are in accordance with the results in aqueous K2SO4 solution, which proves the reasonable assumption and the reliability of the calculated results for aqueous K2SO4-KCl solution. Figure 10 showed the XRD of crystals. It is shown that only K2SO4 diffractions appear in the solid phase, which implies that KCl has been dissolved completely under the experiment conditions. In addition, the experimental result also implies that high quality of the K2SO4 crystal can be produced effectively in aqueous KCl-K2SO4 solution by the described experimental method. Discussion It is notable that the calculated minimum crystal growth rate appears at 303.15 K for both aqueous K2SO4 and KCl-K2SO4 solutions. However, the experimental results by Taguchi et al.6 and those in this work show the minimum at about 293.15 K. It may be due to the fact that the activation energy comes from the experimental data of Garside et al.14 However, if the activation energy E and the pre-exponential constant k0 is correlated from the experimental data by Taguchi et al.,6 the correlation coefficient R2 of the regression is only 0.70, which implies the uncertainty of the experimental data. According to the diffusion-reaction mechanism for crystal growth,14,17 there is a thin stagnant film near the surface of the crystal, and the surface concentration (mi) is different from the bulk concentration of the solution (m) as shown in Figure 11. The stirring speed used in the experiment of Taguchi et al. is 350 rpm, while it is 500 rpm in the experiment of Garside et al. Therefore, in the work of Taguchi et al., the diffusion of the solute in the solution may show a high effect on the crystal growth rate, and it is more accurate to regress by eq 5 using
Figure 11. Schematic diagram for the crystal growth in aqueous solution.
the data reported by Garside et al. However, during the crystallization, it is difficult to eliminate the contribution of the diffusion of the solute in the solution on the crystal growth rate, and the difference between the concentration of the bulk solution m and the surface concentration mi still exists. Therefore, the accurate equation of the crystal growth rate should be written as
( )
( ( ))
ai 2 4mi3γ(3 G ) k0e-E/(RT) ln ) k0e-E/(RT) ln a* KSP
2
(15)
It is possible that the difference between m and mi results in the calculation deviation for the minimum crystal growth rate. Furthermore, eq 15 is only for aqueous K2SO4 solution. For a multiple-electrolyte solution such as the KCl-K2SO4 solution, the accurate equation of the crystal growth rate will become more complicated. Compared the experimental data of G (from Table 4) with calculated results by eq 5 (from Figure 6), it can be found that G(exptl) values are smaller than the G(calcd). This should be in connection with the contribution of the concentration difference in the diffusion layer by use of mi instead of m in eq 15. Therefore, further thermodynamic studies on the temperature dependence of crystal growth are needed, and the work is underway. Conclusion In this paper, a crystal growth rate equation of K2SO4 is developed on the basis of the basic driving force for crystallization, that is, the difference of the chemical potential of K2SO4 at the solid-liquid interface. A Lu-Maurer model is used to predict the activity coefficient of K2SO4 in aqueous K2SO4 and KCl-K2SO4 solutions. The effect of temperature on the crystal growth rate of K2SO4 in aqueous K2SO4 and KCl-K2-
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SO4 solutions is analyzed. The crystal growth rate of K2SO4 decreases and then increases with increasing temperature, which is in accordance with the experimental results in the literature. The optimal crystallization operating condition of K2SO4 in KCl-K2SO4 aqueous solution is obtained by increasing the crystallization temperature to 333.15 K to increase the yield of K2SO4. Acknowledgment The authors would like to thank Dr. Xiaoyan Ji (Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, Wyoming) for a critical review of the manuscript and helpful discussions. This work was supported by the National Basic Research Program of China (2003CB615700), Chinese National Science Foundation for Outstanding Young Scholars (No. 20428606), Chinese National Natural Science Foundation (No. 20376032, No. 20236010) and the Natural Science of Foundation of Jiangsu Province (BK2004215, BK2002016). Notation a ) activity on the basis of molality E ) activation energy of crystallization (kJ/mol) G ) growth rate (kg/(m2‚s)) g ) growth order KSP ) solubility product k ) overall growth rate coefficient (kg/(m2‚s)) k0 ) pre-exponential constant (kg/(m2‚s)) m ) concentration on the basis of molality (mol/kg) R ) universal gas constant (J/(mol‚K)) D ) mean diameter of seed crystal (m) S ) surface area of seed crystal (m2) T ) temperature (K) Greek Letters γ ) activity coefficient ∆µ ) the difference of chemical potentials (J/mol) F ) density (g/m3) σ ) relative supersaturation Superscripts * ) saturated condition ′ ) equilibrium condition
Literature Cited (1) Bergfors, T. M. Protein Crystallization; International University Line: La Jolla, CA, 1999. (2) Mullin, J. W. Crystallization, 3rd ed.; Butterworth-Heinemann: Woburn, MA, 1997. (3) Myerson, A. S. Handbook of Industrial Crystallization, 2nd ed.; Butterworth-Heinemann: Woburn, MA, 2001. (4) Pan, X. J.; Glatz, C. E. Solvent Role in Protein Crystallization as Determined by Pressure Dependence of Nucleation Rate and Solubility. Cryst. Growth Des. 2002, 2 (1), 45. (5) Taboada, M. E.; Cisternas, L. A.; Cheng, Y. S.; Ng, K. M. Design of Alternative Purification Processes for Potassium Sulfate. Ind. Eng. Chem. Res. 2005, 44 (15), 5845. (6) Taguchi, Y.; Yoshida, M.; Kobayashi, H. Temperature Dependence of the Crystal Growth Rate of Potassium Sulfate at Low Temperature. J. Chem. Eng. Jpn. 2002, 35 (11), 1038. (7) Feng, X.; Liu, C.; Ji, X. Y.; Chen, D. L.; Lu, X. H. The Simulation and Analysis for the Process of Potassium Sulfate Production by Glauber Salt Method. J. Chem. Eng. Chin. UniV. 2000, 14 (6), 583. (8) Kim, S.; Myerson, A. S. Metastable Solution Thermodynamic Properties and Crystal Growth Kinetics. Ind. Eng. Chem. Res. 1996, 35 (4), 1078. (9) Mohan, R.; Myerson, A. S. Growth Kinetics: a Thermodynamic Approach. Chem. Eng. Sci. 2002, 57 (20), 4277. (10) Burton, W. K.; Cabrera, N.; Frank, F. C. The Growth of Crystals and the Equilibrium Structure of Their Surfaces. Philos. Trans. R. Soc. London, Ser. A 1951, 243 (866), 299. (11) So¨hnel, O.; Novotny, P. Densities of Aqueous Solutions of Inorganic Substances; Elsevier: Amsterdam, 1985. (12) Ji, X. Y.; Zhang, L. Z.; Lu, X. H.; Wang, Y. R.; Shi, J. Calculation of Solid-Liquid Equilibria for Mixed Aqueous Electrolyte Solutions. J. Chem. Ind. Eng. (China) 1997, 48 (5), 532. (13) Lu, X. H.; Zhang, L. Z.; Wang, Y. R.; Shi, J.; Maurer, G. Prediction of Activity Coefficients of Electrolytes in Aqueous Solutions at High Temperatures. Ind. Eng. Chem. Res. 1996, 35 (5), 1777. (14) Garside, J.; Mullin, J. W.; Das, S. N. Growth and Dissolution Kinetics of Potassium Sulfate Crystals in an Agitated Vessel. Ind. Eng. Chem. Fundam. 1974, 13 (4), 299. (15) Lu, X.; Maurer, G. Model for Describing Activity Coefficients in Mixed Electrolyte Aqueous Solutions. AIChE J. 1993, 39 (9), 1527. (16) Ji, X. Y.; Feng, X.; Lu, X. H.; Zhang, L. Z.; Wang, Y. R.; Shi, J.; Liu, Y. D. A Generalized Method for the Solid-liquid Equilibrium Stage and Its Application in Process Simulation. Ind. Eng. Chem. Res. 2002, 41 (8), 2040. (17) Ji, X. Y.; Chen, D. L.; Wei, T.; Lu, X. H.; Wang, Y. R.; Shi, J. Determination of Dissolution Kinetics of K2SO4 Crystal with Ion Selective Electrode. Chem. Eng. Sci. 2001, 56 (24), 7017.
ReceiVed for reView December 7, 2005 ReVised manuscript receiVed June 1, 2006 Accepted June 6, 2006 IE0513649
