Application of preferential crystallization for racemic compound integrating thermodynamics, kinetics and optimization 4

Chapter Crystallization kinetics of mandelic acid and ketoprofen 93 5.1 Introduction In order to obtain a solid product with high purity and desired crystal size distribution, which determines the final product quality, design and optimization of the crystallizer, crystallization kinetics are essential. It often affects downstream processing such as filtration, centrifugation, and milling. In recent years there has been an increasing recognition of the importance of crystallization kinetics in assessing the design and performance of crystallizers. The characterization of crystallization kinetics, such as nucleation and crystal growth, in an environment is therefore of utmost importance. A number of methods of determining crystallization kinetics have been published in literature. The classical approach to establish crystallization kinetics includes the determination of the kinetics of growth and nucleation process separately and estimation their kinetics simultaneously. The method of initial derivatives was used to determine the crystal growth kinetic. Garside et al (1982) suggested a simple technique, known as initial derivatives, to evaluate crystal growth kinetics from integral mode batch experiments operated in the integral mode. Tavare (1985) extended this analysis to a batch cooling crystallizer and showed that the initial derivatives of the supersaturation and temperature profiles obtained in a series of integral batch experiments can be used to evaluate directly the kinetic parameters in crystal growth relation. Pincombe (1982) has also proposed a theoretical formula describing the dependence of crystal growth rate on temperature, concentration and supersaturation, based on the interaction between crystals and solution. Some researchers (Bijvoet et al., 1983; Blomen et al., 1983; Will et al., 1983) developed a method to measure the growth rate kinetics of calcium oxalate monohydrate based on 94 the direct uptake of 45Ca tracer into the crystals. Fractional uptake was defined as the fraction of calcium taken up into the crystals at any time t and represented a measure of the overall growth of the crystals. Palwe et al. (1985) reported three different methods to obtain the growth rate kinetics of ammonium nitrate, including polynomial fitting, initial derivatives, and optimization procedure. The growth rate also can be obtained using the theory of molecular diffusion (Mersmann et al., 1992). For the nucleation kinetics, Nyvlt (1968) showed a rapid, simple and indirect method to determine the nucleation kinetics in a simple power law nucleation rate expression in supersaturation by measuring the maximum allowable undercooling at a constant cooling rate. Because crystal growth and nucleation are almost simultaneous processes, it is difficult to determine their rate kinetics separately. Therefore, scientists proposed many methods to estimate the kinetics of growth and nucleation process simultaneously. These methods have been used even more greatly. Randolph and Larson (1971) structured a model combining differential equations for mass and energy balances, partial differential equations (PDE) for population balance, and algebraic equations for nucleation and growth kinetics. Among them the population balance equation (PBE) is the most classical equation, which much evaluation of crystallization kinetics relies on the use of techniques to solve. PBE is a nonlinear first order partial differential equation, which can be transformed into a set of differential equation by some transformation and then the problem may be converted into an ordinary differential equation parameter estimation problem. Tavare (1987; 1891; 1993) reported that there were several transforming methods such as moment analysis, Laplace transform analysis, Fourier transform analysis, frequency analysis and methods based on empirical fittings to solve the PDE to obtain the crystallization 95 kinetics. In addition, Nývlt (1982) suggested a new method for dealing with PDE to study crystallization kinetic parameters from a batch crystallizer, assuming that the behavior of the crystal size distribution in a continuous MSMPR crystallizer may be applied to a batch crystallizer under certain circumstances. Several researchers have reported using the discretization methods to solve the PDE to derive the crystallization kinetics (Marchal et al., 1988; Yokota and Kubota, 1996; Wojcik and Jones, 1998). Qiu and Rasmuson (1991) used concentration and final crystal size distribution data to determine kinetic parameters for an aqueous succinic acid system. They simplified the population balance equation to a first order quasi-linear equation by assuming that the crystal growth rate can be expressed as a product of a function of supersaturation and a function of crystal size. Crystallization kinetics of potassium chloride was studied by Dash and Rohani (1993). They included crystal mass distribution from a sieve analysis in the objective function. Miller (1993) suggested and solved a model for cooling batch crystallization using the numeric method of lines. Several researchers (David et al., 1991; Livk et al., 1995; Tadayon et al., 2002) have used weighted least squares to determine kinetics in batch crystallization. Among so many methods of determining the crystal growth and nucleation, the most classical methodologies are using method of moment analysis and Laplace transform analysis to reduce the population balance equation (PBE) to a set of ordinary differential equations (ODE) or change PBE to simple forms under some assumptions. In this Chapter, the linear crystal growth rate and nucleation rate of S enantiomer and racemate were calculated for mandelic acid and ketoprofen. The Laplace transform analysis (s plane method) was used for the (S)- and (RS)-mandelic acid kinetics calculation. The methods of moments analysis and Laplace transform analysis are both used to obtain crystal growth rate and nucleation rate for (S)- 96 ketoprofen and (RS)-ketoprofen separately. The transient crystal size distribution data, the concentration curve and suspension density measured from experiments in a batch crystallizer were used in these calculations. 5.2 Mathematical model As mentioned above, the classical equation for determining all kinetics of crystallization process is population balance equation: Q ∂n ∂(Gn) d (ln V ) Q + +n + n = B'− D'− i ni ∂t ∂L dt V V (5-1) For a well-mixed batch crystallization process, Q = Qi = , in which crystal breakage and agglomeration are negligible, the population balance equation for sizeindependent of growth rate is given by (Randolph and Larson, 1971) ∂n ∂n +G =0 ∂t ∂L (5-2) The method of moment analysis and Laplace transform analysis were used to reduce the population balance equation to a set of ordinary differential equations and obtain the crystallization kinetics. 5.2.1 Method of moment analysis 97 The moment transformation is a kind of transformation of the population balance equation that will average the distribution with regard to the internal coordinate properties and that will reduce the dimensionality to that of the transport equations. This closed set can be thought of as a complete mathematical representation of particulate systems (Randolph and Larson, 1988). The kth moment for the population density distribution about the origin, obtained by moment transformation with size, is defined as (5-3) ∞ µ k = ∫ nLk dL The moment equations obtained by moment transformation of PBE with respect to size are: dµ = B0 dt dµ1 = µ 0G dt (5-4) If the moments of the experimental population density function are available at two times, and the time interval ∆t is so small that the linearity of the model may be assumed between them, the kinetic parameters can be expressed in terms of moments with respect to size at an average time as ∆µ0 =B ∆t ∆µ1 ∆t µ =G (5-5) Then, the average nucleation rate B and average overall linear growth rate G may be determined from two experimental population densities, if the crystal size distribution of different time can be obtain. In addition, when the suspension density 98 and supersaturation concentration can be provided, all the kinetics parameter can be derived. 5.2.2 Method of Laplace transformation Tavare (1986) firstly reported that an experimentally determined population density can be converted into the Laplace transformed response with respect to size L as ∞ n(t , s ) = ∫ n(t , L) exp(− sL)dL (5-6) Then transformation of the population balance equation using Eq.5-6 as [ ] d n( s , t ) + G s n( s, t ) − n(0, t ) = dt (5-7) In Eq. 5-7, n and G both are function of time t, when the time interval is small enough, their value can be expressed by the average value on this time interval, n and G. G n(0, t ) = B (5-8) Therefore, ∆ n( s, t ) = −Gs n( s, t ) + B ∆t (5-9) Over an optimal range of the Laplace transform variable, a plot of ∆ n(t , s ) / ∆t versus s n(t , s ) for Eq. 5-9 should yield a straight line with slope equal to –G and an intercept B. 99 The estimated nucleation B and growth rate G can then be correlated with the degree of supersaturation ∆C and slurry density MT by regression of the data with linear least squares. G = K G ∆C g (5-10) B = K B M Tj G i (5-11) 5.3 Experimental Procedure The batch crystallization experiments were carried out to measure the crystallization kinetics for (S)-, (RS)-MA and (S)-, (RS)-Kp. The set-up was as same as that described in Chapter 4. The loads of crystallizer were 600 ml 35 oC saturated solutions for (S)-MA and (RS)-MA respectively. In both experiments, the operation temperature ranges was 2035 oC. For ketoprofen, the loads were 600ml saturated solution for (S)-Kp at 25 oC and 600 ml (RS)-Kp under 30 oC in the mixture solvent of ethanol and water with volume ratio 0.9:1.0, respectively. The temperature range was 3.5 oC- 30 oC for (S)-Kp and 0.2 oC - 25 oC for (RS)-Kp. During each run of experiment, after slightly supersaturated, pure seeds were added into the solutions and cooled down the specified temperature. 3-5ml samples were withdraw and filtered every certain time. Based on these samples, the crystal size distribution was measured using a Malvern Mastersizer 2000 with a Hydro 2000µP dispersion cell unit with the analytical hexane as the liquid dispersion medium. The sample filtrate was used for solute concentration analysis. The 100 concentration was measured using Shimadzu 2450 UV-visible spectrophotometer. The crystal slurry concentration was calculated using mass balance. 5.4 Results and discussion 5.4.1 Crystal nucleation and growth kinetics for the (S)-MA and (RS)-MA 5.4.1.1 Crystal suspension density and supersaturation The method of Laplace transform analysis was used to obtain the crystallization kinetics for (S)- and (RS)-MA. During each experiment process, the crystal suspension density MT and the solute concentration C were analyzed. According to the solubility data (35oC, 30oC, 25oC, 20oC) in Chapter 4, the regression equations of saturated concentration of (S)- and (RS)-MA can be derived as Eq.5-12 and 5-13 over the studied temperature range. C * = 0.00001466667 × T − 0.00092 × T + 0.02243 × T − 0.111 (5-12) C * = 0.000117333 × T − 0.00766 × T + 0.17757 × T − 1.283 (5-13) Then the ∆C is calculated. All results are listed in the Table 5.1 for the (S)-MA and Table 5.2 for the (RS)- MA. The corresponding concentration profiles of crystal suspension density MT, solute concentration C and supersaturation ∆C are presented in Fig. 5.1 for the (S)-MA and Fig. 5.2 for the (RS)-MA. 101 Table 5.1 Crystallization kinetics measurement of (S)-MA ∆t (min) T (oC) C(g/ml) C*(g/ml) MT(g/ml) ∆C(g/ml) 25.23 0.130 0.105 0.014 0.025 30 23.64 0.115 0.099 0.019 0.016 15 22.79 0.109 0.096 0.025 0.013 15 22.00 0.104 0.093 0.041 0.011 30 21.40 0.099 0.091 0.044 0.008 Table 5.2 Crystallization kinetics measurement of (RS)-MA ∆t (min) T (oC) C(g/ml) C*(g/ml) MT(g/ml) ∆C(g/ml) 30.40 0.371 0.332 0.174 0.039 72 26.18 0.255 0.221 0.311 0.034 30 24.69 0.226 0.198 0.333 0.028 22 23.43 0.208 0.182 0.369 0.026 20 22.30 0.193 0.169 0.376 0.024 102 and (RS)-MA. They are actually keeping a good mass balance for the batch crystallization system. In addition, the change of supersaturation ∆C is moderate for both (S)-MA and (RS)-MA as well. It may supports the assumption that ∆C is constant over a small time interval ∆t. 5.4.1.2 Crystal size distribution (CSD) As mentioned above, the population density, which is necessary for the crystallization kinetics, can be calculated from crystal size distribution. CSD is important to obtain product with good quality and purity and ensure the successful operation of the crystallizer. Therefore, CSD of every (RS)-MA and (S)-MA sample was measured from the batch cooling process. The typical results are shown in Fig 5.3 and Fig. 5.4. It was found that the volume percentage distribution shifted to the right side of large size, which suggests the crystal was growing. 104 12 Volume percentage (%) Time 10 Time 10 100 1000 10000 Crystal size (µm) Fig 5.3 Typical crystal size distribution in kinetic measurement of (S)-MA. Volume percentage (%) Time 12 Time 10 10 100 1000 10000 Crystal size (µm) Fig 5.4 Typical crystal size distribution in kinetic measurement of (RS)-MA. 105 The population density n is computed as ni = M T (∆ w i / ∆ Li ) ρ s kv ( L i )3 (5-14) Where the L i is average crystal size in the given size range. 5.4.1.3 Kinetic evaluation on the measured data The estimated linear growth rate and nucleation rate results are reported at Tables 5.3 and 5.4. Table 5.3 Estimated (S)-MA crystal nucleation rate B and growth rate G with s plane analysis from the experiment B(#/min.m3) G(m/min) ∆C (g/m3) MT (g/m3) T (oC) 1.94E+08 1.88E-07 2.06 E+04 1.65E+04 24.43 9.66E+08 8.94E-08 1.43 E+04 2.21E+04 23.21 9.61E+08 8.61E-08 1.20E+04 3.30E+04 22.40 4.98E+08 3.25E-08 9.18E+03 4.24E+04 21.70 106 Table 5.4 Estimated (RS)-MA crystal nucleation rate B and growth rate G with s plane analysis from the experiment B(#/min.m3) G(m/min) ∆C (g/m3) MT (g/m3) T (oC) 3.48E+09 1.43E-07 3.58E+04 2.43E+05 28.29 4.40E+09 1.24E-07 3.08E+04 3.22E+05 25.75 2.23E+09 7.24E-08 2.71E+04 3.51E+05 24.06 1.18E+09 4.85E-08 2.53E+04 3.72E+05 22.86 One of the obvious advantages of using the Laplace domain is that the sensitivity to experimental errors in the determination of the experimental response is greatly reduced provided that a suitable value of the Laplace transform variable s is used. In formulating a linear regression to estimate G and B from Eq 5-9, the limits on the optimal values of s must be known firstly. The selection of the optimum Laplace transform parameter in the analysis of crystal growth dispersion in a batch crystallizer has been reported by Tavare and Garside (1982). They indicated that the limits on the values of s used should be constrained by s f L ~ 1-2 where L is the population average size at time t2. However, in our analyses, the s f L ~0.1-0.5 was most suitable for (S)-MA and (RS)-MA, which gave better linear regression. The results of analysis of two typical sets of data for (S)-MA and (RS)-MA are shown in Figs. 5.5 to 5.8. 107 Fig 5.5 Typical s plane analysis to estimate crystal nucleation and growth rates for (S)-MA. s f L =0.1, G=3.25×10-8m/min, B= 4.98×108#./min.m3. Fig 5.6 Typical s plane analysis to estimate crystal nucleation and growth rates for (S)-MA. s f L =1.0, G=5.16×10-8m/min, B= 5.15×108#./min.m3. 108 Fig 5.7 Typical s plane analysis to estimate crystal nucleation and growth rates for (RS)-MA. s f L =0.1, G=1.43×10-7m/min, B= 3.48×109#./min.m3. Fig 5.8 Typical s plane analysis to estimate crystal nucleation and growth rates for (RS)-MA. s f L =1.0, G=2.19×10-7m/min, B= 3.58×109#./min.m3. 109 The kinetic expressions for the crystal growth and nucleation of (S)- and (RS)MA were obtained by regression of the data in Table 5.3 and 5.4 with linear least squares. G = 3.21 × 10 −16 ∆C 2.04 (S)-MA: 5-15 B = 6.46 × 10 M T1.15 G 0.17 5-16 G = 1.24 × 10 −21 ∆C 3.10 5-17 B = 3.52 × 1011 M T1.76 G 1.68 5-18 (RS)-MA: The nucleation power dependency on the suspended density is 1.76 for (RS)MA, which is higher than 1.15 for (S)-MA. It could because that (S)-MA has a lower crystal mass concentration than that of (RS)-MA. 5.4.2 Crystal nucleation and growth kinetics for the (S)-Kp and (RS)- Kp 5.4.2.1 Solubility Besides the solubility data obtained in chapter 4, the saturation concentration of (S)-Kp and (RS)-Kp in the mixed solvent was measured by the polythermal method with temperatures range from 0.2 oC to 30 oC. The solubilities of (RS)-Kp and (S)-Kp are listed in Table 5.5 and shown in Fig. 5.9, respectively. 110 Table 5.5 Solubility of (RS)-Kp and (S)-Kp (RS)-Kp (S)-Kp T(oC) Solubility(mg/ml) T(oC) Solubility(mg/ml) 30.0 49.40 30.0 80.40 25.0 31.05 25.0 69.00 20.0 19.91 20.0 57.84 18.0 18.00 15.0 45.24 15.0 15.21 11.3 40.12 13.6 13.36 10.7 37.56 11.5 11.53 9.2 34.16 7.8 9.01 5.0 22.86 5.0 7.16 3.5 20.19 3.5 6.30 2.6 18.31 2.3 5.84 1.0 16.56 1.3 5.46 0.2 15.52 0.3 5.00 111 90 RS 80 S Solubility (mg/ml) 70 y = 0.0001x4 - 0.0088x3 + 0.1717x2 + 1.0629x + 15.013 R2 = 0.9984 60 50 40 30 20 y = 0.0001x4 - 0.0038x3 + 0.0622x2 + 0.2102x + 5.0014 R2 = 0.9996 10 0 10 15 20 25 30 35 Temperature (oC) Fig 5.9 Solubility of (RS)-Kp and (S)-Kp with different temperature. From these data, the regression equations of concentration of (S)-Kp and (RS)Kp over the studied temperature range can be derived as Eq. 5-19 and 5-20. C* = 0.0001T − 0.0088T + 0.1717T + 1.0629T + 15.013 (5-19) C* = 0.0001T − 0.0038T + 0.0622T + 0.2102T + 5.0014 (5-20) 5.4.2.2 Crystal size distribution The CSD results of (RS)-Kp and (S)-Kp are shown in Fig. 5.10 and Fig. 5.11. From these results, it was easy to find that the crystals were growing. Then the population density would be calculated accordingly. 112 Volume (%) T5 T4 13 12 11 10 T3 T2 T1 10 100 1000 10000 Partical Size (µm) Fig 5.10 Crystal size distribution of (RS)-Kp in kinetic measurement of (RS)-Kp. 20 T1 18 T2 16 T3 Volume (%) 14 12 T4 10 T5 10 100 1000 10000 Patical size (µm) Fig 5.11 Crystal size distribution of (S)-Kp in kinetic measurement of (S)-Kp. 113 5.4.2.3 Crystal growth and nucleation kinetics evaluation The method of moment analysis and Laplace transform analysis were used to obtain the crystallization kinetics for (RS)-Kp and (S)-Kp. For the (RS)-Kp, the estimated linear growth rate and nucleation rate results are reported at Table 5.6. From these results, it is clear to found that the values of growth and nucleation rates obtained from these two methods are agreement with each other in most case. For the selection of the optimum Laplace transform parameter in this analysis, s f L ~0.1-0.5 was most suitable, which gave better linear regression with growth rates. The results of analysis of two typical sets of data are shown in Figs. 5.12 and 5.13. Table 5.6 The estimated linear growth rate and nucleation rate for (RS)-Kp o T( C) ∆C (g/m ) G (m/min) B (#/min.m3) MT(g/m ) Moment Laplace Moment Laplace 15.0 15648.000 1.98E+04 3.24E-8 3.23E-8 1.25E+10 1.25E+10 14.0 12538.200 2.47E+04 2.24E-8 2.11E-8 5.23E+09 5.22E+09 10.4 10543.090 2.94E+04 8.22E-9 7.32E-9 1.99E+09 1.99E+09 5.6 8789.862 6.06E-9 5.61E-9 4.73E+09 4.73E+09 3.42E+04 114 Fig 5.12 Typical s-plane analysis to estimate crystal nucleation and growth rate for (RS)-Kp. s f L =0.1, G=2.105×10-8m/min, B= 5.216×109#/min.m3. Fig 5.13 Typical s-plane analysis to estimate crystal nucleation and growth rate for (RS)-Kp. s f L =1.5, G=1.959×10-8m/min, B= 5.2379×109#/min.m3. 115 The following expressions for (RS)-Kp were obtained by regression of the data in Table 5.6 with linear least squares. G = 8.91 × 10 −24 ∆C 3.29 (5-21) B = 8.36 × 1011 M T1.44 G 1.07 (5-22) For (S)-Kp, the estimated linear growth rate and nucleation rate are shown in Table 5.7. The (S)-Kp kinetic results of moments analysis are also comparable with those of Laplace transform analysis. The most suitable s value for (S)-Kp was as same as (RS)-Kp, s f L ~0.1-0.5. The results of Laplace analysis of two typical sets of data are shown in Figs. 5.14 and 5.15. Table 5.7 The estimated linear growth rate and nucleation rate for (S)-Kp o T( C) G (m/min) B (#/min.m3) ∆C (g/m ) MT(g/m ) Moment Laplace Moment Laplace 4.75 37930.05 8.73E+03 3.34E-06 3.56E-06 8.16E+08 8.22E+08 4.00 20757.5 2.83E+04 1.39E-06 1.34E-06 5.12E+08 5.12E+08 2.75 8560.6 4.30E+04 8.04E-07 8.39E-07 7.38E+08 7.36E+08 1.10 4749.1 4.90E+04 3.86E-08 3.10E-08 1.91E+08 1.90E+08 116 Fig 5.14 Typical s-plane analysis to estimate crystal nucleation and growth rate for (S)-Kp. s f L =1.0, G=11.22×10-7m/min, B= 7.5679×108#/min.m3. Fig 5.15 Typical s-plane analysis to estimate crystal nucleation and growth rate for (S)-Kp. s f L =0.1, G=8.39×10-7m/min, B=7.3593×108#/min.m3. 117 The growth rate and nucleation rate expressions for (S)-Kp were expressed as following equations with linear least squares. G = 3.018 × 10 −15 ∆C 2.01 5-23 B = 5.862 × 1010 M T0.17 G 0.35 5-24 It was found that the nucleation power dependency on the suspended crystal density was just 0.17 for (S)-Kp, which is lower than that for (RS)-Kp and lower than those found in industrial inorganic crystallization process. It could because of the relatively low crystal mass concentration used here. Another possible reason is that the primary nucleation occurred especially in the case that the supersaturation was under somewhat arbitrary control in the kinetic measurement experiments (Wang and Ching, 2006). 5.5 Conclusion The classical methods of moment analysis and Laplace transform analysis for derive the crystal growth rate and nucleation rate in the batch crystallization process were introduced. The crystallization kinetics of (S) and (RS)-MA were obtained by moment analysis and the crystallization kinetics of (S) and (RS)-Kp were derived by using both methods. The kinetic results of moment analysis are comparable with those of Laplace transform analysis for ketoprofen. The enantiomer and racemate of mandelic acid and ketoprofen show different characteristics in crystal nucleation and growth. 118 Among these measurements, the transient crystal size distribution data, the concentration curve and suspension density measured from experiments in a batch crystallizer were used. A more suitable the Laplace transform variable s range was determined for both current mandelic acid and ketoprofen crystallization system. The results of these kinetics data are useful to control the critical supersaturation and optimize the crystallizer in batch operation. 119 [...]... (S)-MA crystal nucleation rate B and growth rate G with s plane analysis from the experiment B(#/min.m3) G(m/min) ∆C (g/m3) MT (g/m3) T (oC) 1.94E+08 1.88E-07 2.06 E+ 04 1.65E+ 04 24. 43 9.66E+08 8.94E-08 1 .43 E+ 04 2.21E+ 04 23.21 9.61E+08 8.61E-08 1.20E+ 04 3.30E+ 04 22 .40 4. 98E+08 3.25E-08 9.18E+03 4. 24E+ 04 21.70 106 Table 5 .4 Estimated (RS)-MA crystal nucleation rate B and growth rate G with s plane analysis... (oC) 3 .48 E+09 1 .43 E-07 3.58E+ 04 2 .43 E+05 28.29 4. 40E+09 1.24E-07 3.08E+ 04 3.22E+05 25.75 2.23E+09 7.24E-08 2.71E+ 04 3.51E+05 24. 06 1.18E+09 4. 85E-08 2.53E+ 04 3.72E+05 22.86 One of the obvious advantages of using the Laplace domain is that the sensitivity to experimental errors in the determination of the experimental response is greatly reduced provided that a suitable value of the Laplace transform... measurement of (RS)-Kp 20 T1 18 T2 16 T3 Volume (%) 14 12 T4 10 T5 8 6 4 2 0 1 10 100 1000 10000 Patical size (µm) Fig 5.11 Crystal size distribution of (S)-Kp in kinetic measurement of (S)-Kp 113 5 .4. 2.3 Crystal growth and nucleation kinetics evaluation The method of moment analysis and Laplace transform analysis were used to obtain the crystallization kinetics for (RS)-Kp and (S)-Kp For the (RS)-Kp,... and Ching, 2006) 5.5 Conclusion The classical methods of moment analysis and Laplace transform analysis for derive the crystal growth rate and nucleation rate in the batch crystallization process were introduced The crystallization kinetics of (S) and (RS)-MA were obtained by moment analysis and the crystallization kinetics of (S) and (RS)-Kp were derived by using both methods The kinetic results of. .. are shown in Figs 5. 14 and 5.15 Table 5.7 The estimated linear growth rate and nucleation rate for (S)-Kp o T( C) 3 3 G (m/min) B (#/min.m3) ∆C (g/m ) MT(g/m ) Moment Laplace Moment Laplace 4. 75 37930.05 8.73E+03 3.34E-06 3.56E-06 8.16E+08 8.22E+08 4. 00 20757.5 2.83E+ 04 1.39E-06 1.34E-06 5.12E+08 5.12E+08 2.75 8560.6 4. 30E+ 04 8.04E-07 8.39E-07 7.38E+08 7.36E+08 1.10 47 49.1 4. 90E+ 04 3.86E-08 3.10E-08... The results of analysis of two typical sets of data are shown in Figs 5.12 and 5.13 Table 5.6 The estimated linear growth rate and nucleation rate for (RS)-Kp o T( C) 3 ∆C (g/m ) 3 G (m/min) B (#/min.m3) MT(g/m ) Moment Laplace Moment Laplace 15.0 15 648 .000 1.98E+ 04 3.24E-8 3.23E-8 1.25E+10 1.25E+10 14. 0 12538.200 2 .47 E+ 04 2.24E-8 2.11E-8 5.23E+09 5.22E+09 10 .4 10 543 .090 2.94E+ 04 8.22E-9 7.32E-9 1.99E+09... Solubility(mg/ml) 30.0 49 .40 30.0 80 .40 25.0 31.05 25.0 69.00 20.0 19.91 20.0 57. 84 18.0 18.00 15.0 45 . 24 15.0 15.21 11.3 40 .12 13.6 13.36 10.7 37.56 11.5 11.53 9.2 34. 16 7.8 9.01 5.0 22.86 5.0 7.16 3.5 20.19 3.5 6.30 2.6 18.31 2.3 5. 84 1.0 16.56 1.3 5 .46 0.2 15.52 0.3 5.00 111 90 RS 80 S Solubility (mg/ml) 70 y = 0.0001x4 - 0.0088x3 + 0.1717x2 + 1.0629x + 15.013 R2 = 0.99 84 60 50 40 30 20 y = 0.0001x4 - 0.0038x3... crystal nucleation and growth rates for (RS)-MA s f L 2 =0.1, G=1 .43 ×10-7m/min, B= 3 .48 ×109#./min.m3 Fig 5.8 Typical s plane analysis to estimate crystal nucleation and growth rates for (RS)-MA s f L 2 =1.0, G=2.19×10-7m/min, B= 3.58×109#./min.m3 109 The kinetic expressions for the crystal growth and nucleation of (S)- and (RS)MA were obtained by regression of the data in Table 5.3 and 5 .4 with linear least... ∆C 2. 04 (S)-MA: 1 B = 6 .46 × 10 4 M T.15 G 0.17 (RS)-MA: 5-15 5-16 G = 1. 24 × 10 −21 ∆C 3.10 5-17 1 B = 3.52 × 1011 M T.76 G 1.68 5-18 The nucleation power dependency on the suspended density is 1.76 for (RS)MA, which is higher than 1.15 for (S)-MA It could because that (S)-MA has a lower crystal mass concentration than that of (RS)-MA 5 .4. 2 Crystal nucleation and growth kinetics for the (S)-Kp and (RS)-... 8.91 × 10 − 24 ∆C 3.29 (5-21) 1 B = 8.36 × 1011 M T .44 G 1.07 (5-22) For (S)-Kp, the estimated linear growth rate and nucleation rate are shown in Table 5.7 The (S)-Kp kinetic results of moments analysis are also comparable with those of Laplace transform analysis The most suitable s value for (S)-Kp was as same as (RS)-Kp, s f L 2 ~0.1-0.5 The results of Laplace analysis of two typical sets of data are . (g/m 3 ) T ( o C) 1.94E+08 1.88E-07 2.06 E+ 04 1.65E+ 04 24. 43 9.66E+08 8.94E-08 1 .43 E+ 04 2.21E+ 04 23.21 9.61E+08 8.61E-08 1.20E+ 04 3.30E+ 04 22 .40 4. 98E+08 3.25E-08 9.18E+03 4. 24E+ 04 21.70 . 3.58E+ 04 2 .43 E+05 28.29 4. 40E+09 1.24E-07 3.08E+ 04 3.22E+05 25.75 2.23E+09 7.24E-08 2.71E+ 04 3.51E+05 24. 06 1.18E+09 4. 85E-08 2.53E+ 04 3.72E+05 22.86 One of the obvious advantages of using. 5 .4 Results and discussion 5 .4. 1 Crystal nucleation and growth kinetics for the (S)-MA and (RS)-MA 5 .4. 1.1 Crystal suspension density and supersaturation The method of Laplace transform