# American Institute of Mathematical Sciences

June  2016, 9(2): 373-391. doi: 10.3934/krm.2016.9.373

## An accurate and efficient discrete formulation of aggregation population balance equation

 1 Department of Mathematics, Indian Institute of Technology Kharagpur, 721302 Kharagpur, India, India 2 Chair of Thermal Process Engineering, Otto-von-Guericke University Magdeburg, D-39106 Magdeburg, Germany

Received  January 2015 Revised  September 2015 Published  March 2016

An efficient and accurate discretization method based on a finite volume approach is presented for solving aggregation population balance equation. The principle of the method lies in the introduction of an extra feature that is beyond the essential requirement of mass conservation. The extra feature controls more precisely the behaviour of a chosen integral property of the particle size distribution that does not remain constant like mass, but changes with time. The new method is compared to the finite volume scheme recently proposed by Forestier and Mancini (SIAM J. Sci. Comput., 34, B840 - B860). It retains all the advantages of this scheme, such as simplicity, generality to apply on uniform or nonuniform meshes and computational efficiency, and improves the prediction of the complete particle size distribution as well as of its moments. The numerical results of particle size distribution using the previous finite volume method are consistently overpredicting, which is reflected in the form of the diverging behaviour of second or higher moments for large extent of aggregation. However, the new method controls the growth of higher moments very well and predicts the zeroth moment with high accuracy. Consequently, the new method becomes a powerful tool for the computation of gelling problems. The scheme is validated and compared with the existing finite volume method against several aggregation problems for suitably selected aggregation kernels, including analytically tractable and physically relevant kernels.
Citation: Jitendra Kumar, Gurmeet Kaur, Evangelos Tsotsas. An accurate and efficient discrete formulation of aggregation population balance equation. Kinetic and Related Models, 2016, 9 (2) : 373-391. doi: 10.3934/krm.2016.9.373
##### References:
 [1] M. Attarakih, C. Drumm and H. J. Bart, Solution of the population balance equation using the sectional quadrature method of moments, (SQMOM), Chemical Engineering Science, 64 (2009), 742-752. [2] J. C. Barrett and J. S. Jheeta, Improving the accuracy of the moments method for solving the aerosol general dynamic equation, Journal of Aerosol Science, 27 (1996), 1135-1142. doi: 10.1016/0021-8502(96)00059-6. [3] M. K. Bennett and S. Rohani, Solution of population balance equations with a new combined Lax-Wendroff/Crank-Nicholson method, Chemical Engineering Science, 56 (2001), 6623-6633. doi: 10.1016/S0009-2509(01)00314-1. [4] S. Bove, T. Solberg and B. H. Hjertager, A novel algorithm for solving population balance equations: The parallel parent and daughter classes. Derivation, analysis and testing, Chemical Engineering Science, 60 (2005), 1449-1464. doi: 10.1016/j.ces.2004.10.021. [5] A. Eibeck and W. Wagner, Stochastic particle approximations for Smoluchowski's coagulation equation, The Annals of Applied Probability, 11 (2001), 1137-1165. doi: 10.1214/aoap/1015345398. [6] M. H. Ernst, R. M. Ziff and E. M. Hendriks, Coagulation processes with a phase transition, Journal of Colloid and Interface Science, 97 (1984), 266-277. doi: 10.1016/0021-9797(84)90292-3. [7] F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski equation, SIAM Journal of Scientific Computing, 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. [8] L. Forestier and S. Mancini, A Finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM Journal of Scientific Computing, 34 (2012), B840-B860. doi: 10.1137/110847998. [9] M. J. Hounslow, R. L. Ryall and V. R. Marshall, A discretized population balance for nucleation, growth and aggregation, AIChE Journal, 38 (1988), 1821-1832. doi: 10.1002/aic.690341108. [10] H. M. Hulburt and S. Katz, Some problems in particle technology: A statistical mechanical formulation, Chemical Engineering Science, 19 (1964), 555-574. doi: 10.1016/0009-2509(64)85047-8. [11] Y. P. Kim and J. H. Seinfeld, Simulation of multicomponent aerosol dynamics, Journal of Colloid and Interface Science, 149 (1992), 425-449. doi: 10.1016/0021-9797(92)90432-L. [12] M. Kostoglou, Extended cell average technique for the solution of coagulation equation, Journal of Colloid and Interface Science, 306 (2007), 72-81. doi: 10.1016/j.jcis.2006.10.044. [13] R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085. [14] J. Kumar, M. Peglow, G. Warnecke, S. Heinrich and L. Mörl, A discretized model for tracer population balance equation: Improved accuracy and convergence, Computers and Chemical Engineering, 30 (2006), 1278-1292. doi: 10.1016/j.compchemeng.2006.02.021. [15] J. Kumar, M. Peglow, G. Warnecke, S. Heinrich and L. Mörl, Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique, Chemical Engineering Science, 61 (2006), 3327-3342. doi: 10.1016/j.ces.2005.12.014. [16] J. Kumar, M. Peglow, G. Warnecke and S. Heinrich, An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation, Powder Technology, 182 (2008), 81-104. [17] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I. A fixed pivot technique, Chemical Engineering Science, 51 (1995), 1311-1332. doi: 10.1016/0009-2509(96)88489-2. [18] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization - II. A moving pivot technique, Chemical Engineering Science, 51 (1996), 1333-1342. doi: 10.1016/0009-2509(95)00355-X. [19] J. D. Litster, D. J. Smit and M. J. Hounslow, Adjustable discretized population balance for growth and aggregation, AIChE Journal, 41 (1995), 591-603. [20] G. Madras and B. J. McCoy, Reversible crystal growth-dissolution and aggregation-breakage: Numerical and moment solutions for population balance equations, Powder Technology, 143-144 (2004), 297-307. doi: 10.1016/j.powtec.2004.04.022. [21] A. W. Mahoney and D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chemical Engineering Science, 57 (2002), 1107-1119. doi: 10.1016/S0009-2509(01)00427-4. [22] D. L. Marchisio and R. O. Fox, Solution of population balance equations using the direct quadrature method of moments, Journal of Aerosol Science, 36 (2005), 43-73. doi: 10.1016/j.jaerosci.2004.07.009. [23] M. Nicmanis and M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Computers and Chemical Engineering, 20 (1996), S261-S266. doi: 10.1016/0098-1354(96)00054-3. [24] V. N. Piskunov and A. I. Golubev, The generalized approximation method for modeling coagulation kinetics-Part 1: Justification and implimentation of the method, Journal of Aerosol Science, 33 (2002), 51-63. doi: 10.1016/S0021-8502(01)00073-8. [25] V. N. Piskunov, A. I. Golubev, J. C. Barrett and N. A. Ismailova, The generalized approximation method for modeling coagulation kinetics-Part 2: Comparison with other methods, Journal of Aerosol Science, 33 (2002), 65-75. doi: 10.1016/S0021-8502(01)00072-6. [26] S. Qamar and G. Warnecke, Numerical solution of population balance equations for nucleation, growth and aggregation processes, Computers and Chemical Engineering, 31 (2007), 1576-1589. doi: 10.1016/j.compchemeng.2007.01.006. [27] D. Ramkrishna, Population Balances. Theory and Applications to Particulate Systems in Engineering, $1^{st}$ edition, Academic Press, New York, USA, 2000. [28] S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139. doi: 10.1002/aic.690490507. [29] W. T. Scott, Analytic studies of cloud droplet coalescence, Analytic Studies of Cloud Droplet Coalescence, 25 (1968), 54-65. doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2. [30] D. J. Smit, M. J. Hounslow and W. R. Paterson, Aggregation and gelation-I. Analytical solutions for CST and batch operation, Chemical Engineering Science, 49 (1994), 1025-1035. doi: 10.1016/0009-2509(94)80009-X. [31] D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett, Population balances for particulate processes-a volume approach, Chemical Engineering Science, 57 (2002), 2287-2303. doi: 10.1016/S0009-2509(02)00118-5. [32] E. Wynn, Improved accuracy and convergence of discretized population balance of Lister et al, AIChE Journal, 42 (1996), 2084-2086. doi: 10.1002/aic.690420729.

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##### References:
 [1] M. Attarakih, C. Drumm and H. J. Bart, Solution of the population balance equation using the sectional quadrature method of moments, (SQMOM), Chemical Engineering Science, 64 (2009), 742-752. [2] J. C. Barrett and J. S. Jheeta, Improving the accuracy of the moments method for solving the aerosol general dynamic equation, Journal of Aerosol Science, 27 (1996), 1135-1142. doi: 10.1016/0021-8502(96)00059-6. [3] M. K. Bennett and S. Rohani, Solution of population balance equations with a new combined Lax-Wendroff/Crank-Nicholson method, Chemical Engineering Science, 56 (2001), 6623-6633. doi: 10.1016/S0009-2509(01)00314-1. [4] S. Bove, T. Solberg and B. H. Hjertager, A novel algorithm for solving population balance equations: The parallel parent and daughter classes. Derivation, analysis and testing, Chemical Engineering Science, 60 (2005), 1449-1464. doi: 10.1016/j.ces.2004.10.021. [5] A. Eibeck and W. Wagner, Stochastic particle approximations for Smoluchowski's coagulation equation, The Annals of Applied Probability, 11 (2001), 1137-1165. doi: 10.1214/aoap/1015345398. [6] M. H. Ernst, R. M. Ziff and E. M. Hendriks, Coagulation processes with a phase transition, Journal of Colloid and Interface Science, 97 (1984), 266-277. doi: 10.1016/0021-9797(84)90292-3. [7] F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski equation, SIAM Journal of Scientific Computing, 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. [8] L. Forestier and S. Mancini, A Finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM Journal of Scientific Computing, 34 (2012), B840-B860. doi: 10.1137/110847998. [9] M. J. Hounslow, R. L. Ryall and V. R. Marshall, A discretized population balance for nucleation, growth and aggregation, AIChE Journal, 38 (1988), 1821-1832. doi: 10.1002/aic.690341108. [10] H. M. Hulburt and S. Katz, Some problems in particle technology: A statistical mechanical formulation, Chemical Engineering Science, 19 (1964), 555-574. doi: 10.1016/0009-2509(64)85047-8. [11] Y. P. Kim and J. H. Seinfeld, Simulation of multicomponent aerosol dynamics, Journal of Colloid and Interface Science, 149 (1992), 425-449. doi: 10.1016/0021-9797(92)90432-L. [12] M. Kostoglou, Extended cell average technique for the solution of coagulation equation, Journal of Colloid and Interface Science, 306 (2007), 72-81. doi: 10.1016/j.jcis.2006.10.044. [13] R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085. [14] J. Kumar, M. Peglow, G. Warnecke, S. Heinrich and L. Mörl, A discretized model for tracer population balance equation: Improved accuracy and convergence, Computers and Chemical Engineering, 30 (2006), 1278-1292. doi: 10.1016/j.compchemeng.2006.02.021. [15] J. Kumar, M. Peglow, G. Warnecke, S. Heinrich and L. Mörl, Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique, Chemical Engineering Science, 61 (2006), 3327-3342. doi: 10.1016/j.ces.2005.12.014. [16] J. Kumar, M. Peglow, G. Warnecke and S. Heinrich, An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation, Powder Technology, 182 (2008), 81-104. [17] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I. A fixed pivot technique, Chemical Engineering Science, 51 (1995), 1311-1332. doi: 10.1016/0009-2509(96)88489-2. [18] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization - II. A moving pivot technique, Chemical Engineering Science, 51 (1996), 1333-1342. doi: 10.1016/0009-2509(95)00355-X. [19] J. D. Litster, D. J. Smit and M. J. Hounslow, Adjustable discretized population balance for growth and aggregation, AIChE Journal, 41 (1995), 591-603. [20] G. Madras and B. J. McCoy, Reversible crystal growth-dissolution and aggregation-breakage: Numerical and moment solutions for population balance equations, Powder Technology, 143-144 (2004), 297-307. doi: 10.1016/j.powtec.2004.04.022. [21] A. W. Mahoney and D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chemical Engineering Science, 57 (2002), 1107-1119. doi: 10.1016/S0009-2509(01)00427-4. [22] D. L. Marchisio and R. O. Fox, Solution of population balance equations using the direct quadrature method of moments, Journal of Aerosol Science, 36 (2005), 43-73. doi: 10.1016/j.jaerosci.2004.07.009. [23] M. Nicmanis and M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Computers and Chemical Engineering, 20 (1996), S261-S266. doi: 10.1016/0098-1354(96)00054-3. [24] V. N. Piskunov and A. I. Golubev, The generalized approximation method for modeling coagulation kinetics-Part 1: Justification and implimentation of the method, Journal of Aerosol Science, 33 (2002), 51-63. doi: 10.1016/S0021-8502(01)00073-8. [25] V. N. Piskunov, A. I. Golubev, J. C. Barrett and N. A. Ismailova, The generalized approximation method for modeling coagulation kinetics-Part 2: Comparison with other methods, Journal of Aerosol Science, 33 (2002), 65-75. doi: 10.1016/S0021-8502(01)00072-6. [26] S. Qamar and G. Warnecke, Numerical solution of population balance equations for nucleation, growth and aggregation processes, Computers and Chemical Engineering, 31 (2007), 1576-1589. doi: 10.1016/j.compchemeng.2007.01.006. [27] D. Ramkrishna, Population Balances. Theory and Applications to Particulate Systems in Engineering, $1^{st}$ edition, Academic Press, New York, USA, 2000. [28] S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139. doi: 10.1002/aic.690490507. [29] W. T. Scott, Analytic studies of cloud droplet coalescence, Analytic Studies of Cloud Droplet Coalescence, 25 (1968), 54-65. doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2. [30] D. J. Smit, M. J. Hounslow and W. R. Paterson, Aggregation and gelation-I. Analytical solutions for CST and batch operation, Chemical Engineering Science, 49 (1994), 1025-1035. doi: 10.1016/0009-2509(94)80009-X. [31] D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett, Population balances for particulate processes-a volume approach, Chemical Engineering Science, 57 (2002), 2287-2303. doi: 10.1016/S0009-2509(02)00118-5. [32] E. Wynn, Improved accuracy and convergence of discretized population balance of Lister et al, AIChE Journal, 42 (1996), 2084-2086. doi: 10.1002/aic.690420729.
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