Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena
Affiliation(s)
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.
ABSTRACT
The population balance modeling is regarded as a universally accepted mathematical framework for dynamic simulation of various particulate processes, such as crystallization, granulation and polymerization. This article is concerned with the application of the method of characteristics (MOC) for solving population balance models describing batch crystallization process. The growth and nucleation are considered as dominant phenomena, while the breakage and aggregation are neglected. The numerical solutions of such PBEs require high order accuracy due to the occurrence of steep moving fronts and narrow peaks in the solutions. The MOC has been found to be a very effective technique for resolving sharp discontinuities. Different case studies are carried out to analyze the accuracy of proposed algorithm. For validation, the results of MOC are compared with the available analytical solutions and the results of finite volume schemes. The results of MOC were found to be in good agreement with analytical solutions and superior than those obtained by finite volume schemes.
KEYWORDS
Population Balance Modeling, Batch Crystallization, Method of Characteristics, Nucleation and Growth
Population Balance Modeling, Batch Crystallization, Method of Characteristics, Nucleation and Growth
Cite this paper
Rehman, S. and Qamar, S. (2014) Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena. Applied Mathematics, 5, 1853-1862. doi: 10.4236/am.2014.513178.
Rehman, S. and Qamar, S. (2014) Application of the Method of Characteristics to Population Balance Models Considering Growth and Nucleation Phenomena. Applied Mathematics, 5, 1853-1862. doi: 10.4236/am.2014.513178.
References
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http://dx.doi.org/10.1016/0009-2509(64)85047-8 [3] Kumar, S. and Ramkrishna, D. (1996) On the Solution of Population Balance Equations by Discretization-I. A Fixed Pivot Technique. Chemical Engineering Science, 51, 1311-1332.
http://dx.doi.org/10.1016/0009-2509(96)88489-2 [4] Kumar, S. and Ramkrishna, D. (1997) On the Solution of Population Balance Equations by Discretization-III. Nucleation, Growth and Aggregation of Particles. Chemical Engineering Science, 52, 4659-4679.
http://dx.doi.org/10.1016/S0009-2509(97)00307-2 [5] Marchisio, D.L., Vigil, R.D. and Fox, R.O. (2003) Quadrature Method of Moments for Aggregation-Breakage Processes. Journal of Colloid and Interface Science, 258, 476-491.
http://dx.doi.org/10.1016/S0021-9797(02)00054-1 [6] Tsuchiya, H.M., Barrett, J.C. and Jheeta, I.S. (1996) Improving the Accuracy of the Moments Method for Solving the Aerosol General Dynamic Equation. Journal of Aerosol Science, 27, 1135-1142.
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http://dx.doi.org/10.1016/0025-5564(71)90050-2 [8] Smith, M. and Matsoukas, T. (1998) Constant-Number Monte Carlo Simulation of Population Balances. Chemical Engineering Science, 53, 1777-1786.
http://dx.doi.org/10.1016/S0009-2509(98)00045-1 [9] Gunawan, R., Fusman, I. and Braatz, R.D. (2004) High Resolution Algorithms for Multidimensional Population Balance Equations. AIChE Journal, 50, 2738-2749.
http://dx.doi.org/10.1002/aic.10228 [10] Hulburt, H.M. and Katz, S. (1988) Some Problems in Particle Technology. Chemical Engineering Science, 19, 555-574.
http://dx.doi.org/10.1016/0009-2509(64)85047-8 [11] Qamar, S., Ashfaq, A., Warnecke, G., Angelov, I., Elsner, M.P. and Morgenstern, A.S. (2006) Adaptive High Resolution Schemes for Multidimensional Population Balances in Crystallization Processes. Computers & Chemical Engineering, 30, 1119-1131.
http://dx.doi.org/10.1016/j.compchemeng.2006.02.012 [12] Qamar, S., Elsner, M.P., Angelov, I., Warnecke, G. and Morgenstern, A.S. (2006) A Comparative Study of High Resolution Schemes for Solving Population Balances in Crystallization. Computers & Chemical Engineering, 30, 1119-1131.
http://dx.doi.org/10.1016/j.compchemeng.2006.02.012 [13] Févotte, F. and Févotte, G. (2010) A Method of Characteristics for Solving Population Balance Equations (PBE) Describing the Adsorption of Impurities during Crystallization Processes. Chemical Engineering Science, 65, 3191-3198.
http://dx.doi.org/10.1016/j.ces.2010.02.009 [14] Qamar, S. and Warnecke, G. (2007) Numerical Solution of Population Balance Equations for Nucleation, Growth and Aggregation Processes. Chemical Engineering Science, 64, 2088-2095.
http://dx.doi.org/10.1016/j.ces.2009.01.040 [15] Randolph, A.D. and Larson, M.A. (1988) Population Balances: Theory of Particulate Processes. 2nd Edition, Academic Press, Waltham. [16] Leonard, B.P. and Niknafs, H.S. (1991) Sharp Monotonic Resolution of Discontinuities without Clipping of Narrow Extreme. Computers & Fluids, 19, 141-154. [17] Lim, Y.I., Lann, J-M.L., Meyer, X.M., Joulia, X., Lee, G. and Yoon, E.S. (2002) On the Solution of Population Balance Equation (PBE) with Accurate Front Tracking Method in Practical Crystallization Processes. Chemical Engineering Science, 57, 3715-3732.
http://dx.doi.org/10.1016/S0009-2509(02)00236-1 [18] Majumder, A., Kariwala, V., Ansumali, S. and Rajendran, A. (2010) Fast High-Resolution Method for Solving Multidimensional Population Balances in Crystallization. Industrial & Engineering Chemistry Research, 349, 3862-3872.
http://dx.doi.org/10.1021/ie9016946
[1] Mesbah, A., Kramer, H.J.M., Huesman, A.E.M. and Vanden Ho, P.M.J. (2009) A Control Oriented Study on the Numerical Solution of the Population Balance Equation for Crystallization Processes. Chemical Engineering Science, 64, 4262-4277.
http://dx.doi.org/10.1016/j.ces.2009.06.060 [2] Hulburt, H.M. and Katz, S. (1988) Some Problems in Particle Technology. Chemical Engineering Science, 19, 555-574.
http://dx.doi.org/10.1016/0009-2509(64)85047-8 [3] Kumar, S. and Ramkrishna, D. (1996) On the Solution of Population Balance Equations by Discretization-I. A Fixed Pivot Technique. Chemical Engineering Science, 51, 1311-1332.
http://dx.doi.org/10.1016/0009-2509(96)88489-2 [4] Kumar, S. and Ramkrishna, D. (1997) On the Solution of Population Balance Equations by Discretization-III. Nucleation, Growth and Aggregation of Particles. Chemical Engineering Science, 52, 4659-4679.
http://dx.doi.org/10.1016/S0009-2509(97)00307-2 [5] Marchisio, D.L., Vigil, R.D. and Fox, R.O. (2003) Quadrature Method of Moments for Aggregation-Breakage Processes. Journal of Colloid and Interface Science, 258, 476-491.
http://dx.doi.org/10.1016/S0021-9797(02)00054-1 [6] Tsuchiya, H.M., Barrett, J.C. and Jheeta, I.S. (1996) Improving the Accuracy of the Moments Method for Solving the Aerosol General Dynamic Equation. Journal of Aerosol Science, 27, 1135-1142.
http://dx.doi.org/10.1016/0021-8502(96)00059-6 [7] Subramanian, G. and Ramkrishna, D. (1971) On the Solution of Statistical Models of Cell Populations. Mathematical Biosciences, 10, 1-23.
http://dx.doi.org/10.1016/0025-5564(71)90050-2 [8] Smith, M. and Matsoukas, T. (1998) Constant-Number Monte Carlo Simulation of Population Balances. Chemical Engineering Science, 53, 1777-1786.
http://dx.doi.org/10.1016/S0009-2509(98)00045-1 [9] Gunawan, R., Fusman, I. and Braatz, R.D. (2004) High Resolution Algorithms for Multidimensional Population Balance Equations. AIChE Journal, 50, 2738-2749.
http://dx.doi.org/10.1002/aic.10228 [10] Hulburt, H.M. and Katz, S. (1988) Some Problems in Particle Technology. Chemical Engineering Science, 19, 555-574.
http://dx.doi.org/10.1016/0009-2509(64)85047-8 [11] Qamar, S., Ashfaq, A., Warnecke, G., Angelov, I., Elsner, M.P. and Morgenstern, A.S. (2006) Adaptive High Resolution Schemes for Multidimensional Population Balances in Crystallization Processes. Computers & Chemical Engineering, 30, 1119-1131.
http://dx.doi.org/10.1016/j.compchemeng.2006.02.012 [12] Qamar, S., Elsner, M.P., Angelov, I., Warnecke, G. and Morgenstern, A.S. (2006) A Comparative Study of High Resolution Schemes for Solving Population Balances in Crystallization. Computers & Chemical Engineering, 30, 1119-1131.
http://dx.doi.org/10.1016/j.compchemeng.2006.02.012 [13] Févotte, F. and Févotte, G. (2010) A Method of Characteristics for Solving Population Balance Equations (PBE) Describing the Adsorption of Impurities during Crystallization Processes. Chemical Engineering Science, 65, 3191-3198.
http://dx.doi.org/10.1016/j.ces.2010.02.009 [14] Qamar, S. and Warnecke, G. (2007) Numerical Solution of Population Balance Equations for Nucleation, Growth and Aggregation Processes. Chemical Engineering Science, 64, 2088-2095.
http://dx.doi.org/10.1016/j.ces.2009.01.040 [15] Randolph, A.D. and Larson, M.A. (1988) Population Balances: Theory of Particulate Processes. 2nd Edition, Academic Press, Waltham. [16] Leonard, B.P. and Niknafs, H.S. (1991) Sharp Monotonic Resolution of Discontinuities without Clipping of Narrow Extreme. Computers & Fluids, 19, 141-154. [17] Lim, Y.I., Lann, J-M.L., Meyer, X.M., Joulia, X., Lee, G. and Yoon, E.S. (2002) On the Solution of Population Balance Equation (PBE) with Accurate Front Tracking Method in Practical Crystallization Processes. Chemical Engineering Science, 57, 3715-3732.
http://dx.doi.org/10.1016/S0009-2509(02)00236-1 [18] Majumder, A., Kariwala, V., Ansumali, S. and Rajendran, A. (2010) Fast High-Resolution Method for Solving Multidimensional Population Balances in Crystallization. Industrial & Engineering Chemistry Research, 349, 3862-3872.
http://dx.doi.org/10.1021/ie9016946