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Ind. Eng. Chem. Res. 2006, 45, 70-75
Application of a Neural Network for the Prediction of Crystallization Kinetics Meng Yang and Hongyuan Wei* School of Chemical Engineering and Technology Tianjin UniVersity, Tianjin, 300072, People’s Republic of China
A method for more-accurate prediction of crystallization kinetics is greatly needed in the field of industrial crystallization. Traditional empirical correlations cannot give reliable predictions, because of the highly nonlinear behavior of crystallization kinetics, although they have been used for a long time. In this paper, the development of a neural network model is presented. The model was trained with limited data obtained from an antisolvent crystallization system (ciprofloxacin hydrochloride, H2O, and ethanol). The predictions from the network then were validated against newly measured data. The results confirm that this approach gives much moreaccurate predictions of the kinetics, in terms of crystal growth and agglomeration as examples. The mean relative error of the predicted growth rates from this model, versus the measured data, is generally 40%). To summarize the aforementioned comparisons, the traditional approach for kinetic predications is not satisfactory and can lead to large errors.
6. The Neural Network Approach for Kinetics Predictions The ANN model is sensitive to the number of neurons in its hidden layers. Too few neurons may lead to underfitting. However, too many would contribute to overfitting, in which the fitting curve exhibits wild oscillations between well-fitted training points. Hence, it is very important to select a suitable number of neurons. After many trials, the model can give highly satisfactory results with 10 neurons in the hidden layer and 1 neuron in the output layer. The weightings and biases in the model are as follows: W1 ) [wij], where i is the number of input parameters and j is the number of neurons in the hidden layer W2 ) [wij], where i is the number of output parameters and j is the number of neurons in the hidden layer b1 ) [bj], where j is the number of neurons in the hidden layer b2 ) [bj], where j is the number of output parameters The network was first trained with limited data obtained from experiments, and predictions from the trained network were then validated against newly measured data. Network for Linear Growth Rate Prediction under the Number-Length Coordinate. The model constructed in this case was trained with 31 sets of experimental data. Figure 8 shows that the resulting correlation between the predicted values
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Figure 8. Training the multilayer ANN network: correlation between the experimental and predicted values of G.
Figure 10. Training the multilayer ANN network: correlation between the experimental and predicted values of B0.
Figure 9. Multilayer ANN network prediction: comparison between the experimental and predicted values of G. Table 1. Validation: Experimental and Predicted Values of the Growth Rate, G Growth Rate, G (m/s) experiment value
predicted value
3.07 × 10-10 4.64 × 10-10 4.97 × 10-10 5.20 × 10-10 7.47 × 10-10 9.54 × 10-10
3.04 × 10-10 4.52 × 10-10 5.64 × 10-10 5.82 × 10-10 6.39 × 10-10 1.00 × 10-9
Figure 11. Training the multilayer ANN network: correlation between the experimental and predicted values of Ka. Table 2. Validation: Experimental and Predicted Values of the Nucleation Rate, B0 Nucleation Rate, B0 (number/(m3 s))
(the solid line) and measured data (the open circles) is extremely good. The mean relative error Q is only ∼1%. Another six sets of experimental data were selected to validate the model predictions away from training points. The experimental value and the predicted values are shown as Table 1 and Figure 9. All predictions lie within 15% of the measured values, with a mean relative error of Q ) 8.01%. Network for Simultaneous Prediction of Nucleation Rate and Agglomeration Coefficient under the Number-Volume Coordinate. The model constructed in this case was trained with 34 sets of experimental data. Figures 10 and 11 show the correlation between the predicted and measured values of B0 and Ka, respectively. The corresponding mean relative errors are 9.42% and 5.8%. Another six sets of experimental data were selected to validate the model predictions between training points. The experimental
experimental value
predicted value
2.17 × 1011 2.16 × 1011 2.30 × 1011 3.27 × 1011 3.00 × 1011 2.789 × 1011
2.06 × 1011 2.16 × 1011 2.38 × 1011 2.84 × 1011 2.57 × 1011 2.63 × 1011
values and the predicted values for B0 and Ka are shown in Tables 2 and 3, respectively, and, correspondingly, Figures 12 and 13. The predicted values of B0 and Ka were within 15% and 6%, respectively, of measured data, with corresponding mean relative errors of 7.05% and 3.66%. Network for Growth Rate Prediction with Different Crystallizers, under the Number-Length Coordinate. Two sizes of crystallizers were used for the experiments; these had working volumes of 0.5 and 1.5 L.
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Figure 12. Multilayer ANN network prediction: comparison between the experimental and predicted values of B0.
Figure 14. Training the multilayer ANN network: correlation between the experimental and predicted values of growth rate.
Figure 13. Multilayer ANN network prediction: comparison between the experimental and predicted values of Ka. Table 3. Validation: Experimental and Predicted Values of Agglomeration Coefficient, Ka Agglomeration Coefficient, Ka (µm3/s) experimental value
predicted value
3.38 × 10-20 3.55 × 10-20 4.50 × 10-20 3.27 × 10-20 7.09 × 10-20 7.37 × 10-20
3.22 × 10-20 3.43 × 10-20 4.24 × 10-20 3.26 × 10-20 6.84 × 10-20 7.08 × 10-20
The model constructed for this study was trained with 34 sets of experimental data. Figure 14 shows the correlation between the predicted and measured values of G. The mean relative error is Q ) 2%. Another six sets of experimental data were selected to validate the model predictions between training points. The experimental and predicted values are shown in Table 4 and Figure 15. Predicted values are within 7.5% of the experimental data, with a mean relative error of Q ) 1.73%. 7. Conclusions Crystallization kinetics (such as crystal nucleation, growth, and agglomeration rates) is essential for the analysis, design, and operation of industrial crystallization processes. The traditional correlation approach for predicting kinetics is arbitrary and subjective in its selection of assumed functional
Figure 15. Multilayer ANN network prediction: comparison between the experimental and predicted values of G. Table 4. Validation: Experimental and Predicted Values of the Growth Rate, G Growth Rate, G (µm/s) experimental value 10-5
2.91 × 2.59 × 10-5 4.32 × 10-5 4.62 × 10-5 5.49 × 10-5 5.15 × 10-5
predicted value 2.93 × 10-5 2.61 × 10-5 4.00 × 10-5 4.60 × 10-5 5.46 × 10-5 5.18 × 10-5
forms.3 Therefore, making use of these empirical correlations can lead to large errors. The mechanism of crystallization kinetics is very complex with strong nonlinearity, which cannot be reflected accurately by a simple regression function. However, the neural network approach is better able to reproduce overall system behaviors, with its nonlinearities and parameter interaction effects. It can not only capture the interactions between each influential element but also provide the mapping procedure from input to output. Its ability to predict the complex interplays of influential factors in crystallization processes result in more-accurate predictions of the crystallization kinetics, as demonstrated in this paper.
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In summary, the application of an artificial neutral network (ANN) overcomes, to a significant degree, the problems that result from the traditional use of a regressional analysis of kinetic data to an assumed functional form. It provides us with an effective and applicable way to obtain dependable kinetics parameters. Future work will examine the applicability of this method to other systems and situations. Nomenclature B0 ) nucleation rate (number/(m3 s)) b1 ) hidden-layer bias b2 ) outer-layer bias f1(x) ) tansig function f2(x) ) purelin function G ) growth rate (m/s) Ka ) agglomeration coefficient (µm3/s) mi ) ith moment Mt ) suspension density (g/L) Ni ) number concentration at size i (number/m3) Nstr ) stirring rate (rpm) P ) input vector S ) supersaturation t ) temperature (°C) W1 ) inner-layer weighting W2 ) outer-layer weighting Acknowledgment The authors wish to acknowledge the National Natural Science Foundation of China for financial support, and they also thank Dr. Liu Yong (Tianjin University) and Dr. Simon Leefe (Chembridge Technologies, Limited, U.K.) for their valuable assistance with this work.
Literature Cited (1) Hosking, J. C.; Himmelblau, B. M. Artificial neural network models of knowledge representation in Chemical Engineering. Comput. Chem. Eng. 1988, 12, 881-891. (2) Bhagat, P. An introduction to neural nets. Chem. Eng. Prog. 1990, 8, 55-65. (3) Mullin, J. W. Crystallization; 3rd Edition; Butterworth-Heinemann: Oxford, U.K., 1993. (4) Larachi, F.; Iliuta, I.; Chen, M.; Grandjean, B. P. A. Onset of Pulsing in Trickle Beds: Evaluation of Current Tools and State-of-art Correlation. Can. J. Chem. Eng. 1999, 77, 751-757. (5) Larachi, F.; Bensetiti, Z.; Grandjean, B. P. A.; Wild, G. Two-Phase Frictional Pressure Drop in Flooded-Bed Reactors: A State-of-the-art Correlation. Chem. Eng. Technol. 1998, 21 (11), 887-893. (6) Bensetiti, Z.; Larachi, F.; Grandjean, B. P. A.; Wild, G. Liquid saturation in cocurrent upflow fixed-bed reactors: a state-of-the-art correlation. Chem. Eng. Sci. 1997, 52 (21-22), 4239-4247. (7) Iliuta, I.; Larachi, F.; Grandjean, B. P. A. Pressure Drop and Liquid Holdup in Trickle Flow Reactors: Improved Ergun Constants and Slip Correlations for the Slit Model. Ind. Chem. Eng. Res. 1998, 37, 45424550. (8) Iliuta, I.; Larachi, F.; Grandjean, B. P. A.; Wild, G. Gas-liquid interfacial mass transfer in trickle-bed reactors: state-of-the-art correlations. Chem. Eng. Sci. 1999, 54, 5633-5645. (9) Liu, Y.; Wang, J.; Yin, Q. Study on the thermodynamics of crystallization of ciprofloxacin hydrochloride monohydrate. Tianjin Daxue Xuebao, Ziran Kexue Yu Gongcheng Jishuban 2002, 35 (3), 313-316. (10) Liu, Y.; Wang, J. K.; Wei, H. Y. Determination of crystallization kinetics in solution. J. Cryst. Growth 2004, 271, 238-244. (11) Liu, Y. Study on the synthesis and crystallization process of ciprofloxacin hydrochloride monohydrate, Ph.D. Thesis, Tianjin University, Tianjin, PRC, 2003.
ReceiVed for reView December 14, 2004 ReVised manuscript receiVed October 25, 2005 Accepted November 3, 2005 IE0487944
