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December  2014, 7(4): 713-737. doi: 10.3934/krm.2014.7.713

## Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations

 1 RICAM, Austrian Academy of Sciences, Altenberger Strasse 69, Linz, 4040, Austria 2 Department of Mathematics, IIT Kharagpur, Kharagpur, 721302, India 3 Institute for Analysis and Numerics, Otto-von-Guericke University Magdeburg, Universitatsplatz 2, Magdeburg, D-39106, Germany

Received  March 2014 Revised  August 2014 Published  November 2014

This paper presents stability and convergence analysis of a finite volume scheme for solving aggregation, breakage and the combined processes by showing consistency of the method and Lipschitz continuity of numerical fluxes. It is investigated that the finite volume scheme is second order convergent independently of the meshes for pure breakage problem while for pure aggregation and coupled problems, it indicates second order convergence on uniform and non-uniform smooth meshes. Furthermore, it gives only first order accuracy on non-uniform meshes. The mathematical results of convergence analysis are also demonstrated numerically for several test problems.
Citation: Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic and Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
##### References:
 [1] J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Mathematics of Computation, 77 (2008), 851-882. doi: 10.1090/S0025-5718-07-02054-6. [2] R. B. Diemer and J. H. Olson, A moment methodology for coagulation and breakage problems: Part 1-analytical solution of the steady-state population balance, Chem. Eng. Sci., 57 (2002), 2193-2209. [3] P. B. Dubovskii, V. A. Galkin and I. W. Stewart, Exact solutions for the coagulation-fragmentation equations, J. of Physics A: Mathematical and General, 25 (1992), 4737-4744. doi: 10.1088/0305-4470/25/18/009. [4] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, Journal of Differential Equations, 195 (2003), 143-174. doi: 10.1016/S0022-0396(03)00134-7. [5] F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM Journal on Scientific Computing, 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. [6] F. Filbet and P. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation, Archiv der Mathematik, 83 (2004), 558-567. doi: 10.1007/s00013-004-1060-9. [7] Y. P. Gokhale, R. Kumar, J. Kumar, G. Warnecke, J. Tomas and W. Hintz, Disintegration process of surface stabilized sol-gel $\text{TiO}_2$ nanoparticles by population balances, Chem. Eng. Sci., 64 (2009), 5302-5307. [8] W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, $1^{st}$ edition, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-662-09017-6. [9] J. Kumar, Numerical Approximations of Population Balance Equations in Particulate Systems, Ph.D thesis, Otto-von-Guericke-University Magdeburg in Germany, 2006. [10] J. Kumar and G. Warnecke, Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique, Numer. Math., 111 (2008), 81-108. doi: 10.1007/s00211-008-0174-6. [11] R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Math. Models and Methods in App. Sc., 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085. [12] R. Kumar, Numerical Analysis of Finite Volume Schemes for Solving Population Balance Equations, Ph.D thesis, Otto-von-Guericke-University Magdeburg in Germany, 2011. [13] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I: a fixed pivot technique, Chem. Eng. Sci., 51 (1996), 1311-1332. doi: 10.1016/0009-2509(96)88489-2. [14] P. L. C. Lage, Comments on the "an analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles" by D.P. Patil and J.R.G. Andrews, Chem. Eng. Sci., 57 (2002), 4253-4254. [15] W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation, Mathematical Methods in the Applied Sciences, 27 (2004), 703-721. doi: 10.1002/mma.496. [16] K. Lee and T. Matsoukas, Simultaneous coagulation and breakage using constant-N Monte Carlo, Powder Technology, 110 (2000), 82-89. [17] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, $1^{st}$ edition, Cambridge University Press, Cambridge, UK, 2002. doi: 10.1017/CBO9780511791253. [18] P. Linz, Convergence of a discretization method for integro-differential equations, Numer. Math., 25 (1975), 103-107. doi: 10.1007/BF01419532. [19] J. Makino, T. Fukushige, Y. Funato and E. Kokubo, On the mass distribution of planetesimals in the early runaway stage, New Astronomy, 3 (1998), 411-417. doi: 10.1016/S1384-1076(98)00021-9. [20] 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. [21] D. J. McLaughlin, W. Lamb and A. C. McBride, Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation, Mathematical Methods in the Applied Sciences, 21 (1998), 1067-1084. doi: 10.1002/(SICI)1099-1476(19980725)21:11<1067::AID-MMA985>3.0.CO;2-X. [22] Z. A. Melzak, A scalar transport equation, Trans. of the American Math. Society, 85 (1957), 547-560. doi: 10.1090/S0002-9947-1957-0087880-6. [23] S. Motz, A. Mitrovic and E. D. Gilles, Comparison of numerical methods for the simulation of dispersed phase systems, Chem. Eng. Sci., 57 (2002), 4329-4344. doi: 10.1016/S0009-2509(02)00349-4. [24] S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme, Chem. Eng. Sci., 62 (2007), 679-693.\vspace*{2pt} doi: 10.1016/j.ces.2006.10.001. [25] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, $1^{st}$ edition, Academic Press, New York, USA, 2000. [26] W. T. Scott, Analytic studies of cloud droplet coalescence, J. of the Atmospheric Sci., 25 (1968), 54-65. doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2. [27] M. Sommer, F. Stenger, W. Peukert and N. J. Wagner, Agglomeration and breakage of nanoparticles in stirred media mills-a comparison of different methods and models, Chem. Eng. Sci., 61 (2006), 135-148. doi: 10.1016/j.ces.2004.12.057. [28] M. Vanni, Approximate population balance equations for aggregation-breakage processes, J. of Colloid and Interface Science, 221 (2002), 143-160. doi: 10.1006/jcis.1999.6571. [29] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization, J. of Physics A: Mathematical and General, 18 (1985), 3027-3037. doi: 10.1088/0305-4470/18/15/026. [30] R. M. Ziff, New solution to the fragmentation equation, J. of Physics A: Mathematical and General, 24 (1991), 2821-2828. doi: 10.1088/0305-4470/24/12/020.

show all references

##### References:
 [1] J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Mathematics of Computation, 77 (2008), 851-882. doi: 10.1090/S0025-5718-07-02054-6. [2] R. B. Diemer and J. H. Olson, A moment methodology for coagulation and breakage problems: Part 1-analytical solution of the steady-state population balance, Chem. Eng. Sci., 57 (2002), 2193-2209. [3] P. B. Dubovskii, V. A. Galkin and I. W. Stewart, Exact solutions for the coagulation-fragmentation equations, J. of Physics A: Mathematical and General, 25 (1992), 4737-4744. doi: 10.1088/0305-4470/25/18/009. [4] M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, Journal of Differential Equations, 195 (2003), 143-174. doi: 10.1016/S0022-0396(03)00134-7. [5] F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM Journal on Scientific Computing, 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132. [6] F. Filbet and P. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation, Archiv der Mathematik, 83 (2004), 558-567. doi: 10.1007/s00013-004-1060-9. [7] Y. P. Gokhale, R. Kumar, J. Kumar, G. Warnecke, J. Tomas and W. Hintz, Disintegration process of surface stabilized sol-gel $\text{TiO}_2$ nanoparticles by population balances, Chem. Eng. Sci., 64 (2009), 5302-5307. [8] W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, $1^{st}$ edition, Springer-Verlag, New York, 2003. doi: 10.1007/978-3-662-09017-6. [9] J. Kumar, Numerical Approximations of Population Balance Equations in Particulate Systems, Ph.D thesis, Otto-von-Guericke-University Magdeburg in Germany, 2006. [10] J. Kumar and G. Warnecke, Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique, Numer. Math., 111 (2008), 81-108. doi: 10.1007/s00211-008-0174-6. [11] R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Math. Models and Methods in App. Sc., 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085. [12] R. Kumar, Numerical Analysis of Finite Volume Schemes for Solving Population Balance Equations, Ph.D thesis, Otto-von-Guericke-University Magdeburg in Germany, 2011. [13] S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I: a fixed pivot technique, Chem. Eng. Sci., 51 (1996), 1311-1332. doi: 10.1016/0009-2509(96)88489-2. [14] P. L. C. Lage, Comments on the "an analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles" by D.P. Patil and J.R.G. Andrews, Chem. Eng. Sci., 57 (2002), 4253-4254. [15] W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation, Mathematical Methods in the Applied Sciences, 27 (2004), 703-721. doi: 10.1002/mma.496. [16] K. Lee and T. Matsoukas, Simultaneous coagulation and breakage using constant-N Monte Carlo, Powder Technology, 110 (2000), 82-89. [17] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, $1^{st}$ edition, Cambridge University Press, Cambridge, UK, 2002. doi: 10.1017/CBO9780511791253. [18] P. Linz, Convergence of a discretization method for integro-differential equations, Numer. Math., 25 (1975), 103-107. doi: 10.1007/BF01419532. [19] J. Makino, T. Fukushige, Y. Funato and E. Kokubo, On the mass distribution of planetesimals in the early runaway stage, New Astronomy, 3 (1998), 411-417. doi: 10.1016/S1384-1076(98)00021-9. [20] 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. [21] D. J. McLaughlin, W. Lamb and A. C. McBride, Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation, Mathematical Methods in the Applied Sciences, 21 (1998), 1067-1084. doi: 10.1002/(SICI)1099-1476(19980725)21:11<1067::AID-MMA985>3.0.CO;2-X. [22] Z. A. Melzak, A scalar transport equation, Trans. of the American Math. Society, 85 (1957), 547-560. doi: 10.1090/S0002-9947-1957-0087880-6. [23] S. Motz, A. Mitrovic and E. D. Gilles, Comparison of numerical methods for the simulation of dispersed phase systems, Chem. Eng. Sci., 57 (2002), 4329-4344. doi: 10.1016/S0009-2509(02)00349-4. [24] S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme, Chem. Eng. Sci., 62 (2007), 679-693.\vspace*{2pt} doi: 10.1016/j.ces.2006.10.001. [25] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, $1^{st}$ edition, Academic Press, New York, USA, 2000. [26] W. T. Scott, Analytic studies of cloud droplet coalescence, J. of the Atmospheric Sci., 25 (1968), 54-65. doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2. [27] M. Sommer, F. Stenger, W. Peukert and N. J. Wagner, Agglomeration and breakage of nanoparticles in stirred media mills-a comparison of different methods and models, Chem. Eng. Sci., 61 (2006), 135-148. doi: 10.1016/j.ces.2004.12.057. [28] M. Vanni, Approximate population balance equations for aggregation-breakage processes, J. of Colloid and Interface Science, 221 (2002), 143-160. doi: 10.1006/jcis.1999.6571. [29] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization, J. of Physics A: Mathematical and General, 18 (1985), 3027-3037. doi: 10.1088/0305-4470/18/15/026. [30] R. M. Ziff, New solution to the fragmentation equation, J. of Physics A: Mathematical and General, 24 (1991), 2821-2828. doi: 10.1088/0305-4470/24/12/020.
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