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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 |
References:
[1] |
M. Attarakih, C. Drumm and H. J. Bart, Solution of the population balance equation using the sectional quadrature method of moments,, (SQMOM), 64 (2009), 742. Google Scholar |
[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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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. Google Scholar |
[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.
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.
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. Google Scholar |
[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), 143.
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.
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.
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).
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.
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.
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.
doi: 10.1016/j.compchemeng.2007.01.006. |
[27] |
D. Ramkrishna, Population Balances. Theory and Applications to Particulate Systems in Engineering,, $1^{st}$ edition, (2000). Google Scholar |
[28] |
S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation,, AIChE Journal, 49 (2003), 1127.
doi: 10.1002/aic.690490507. |
[29] |
W. T. Scott, Analytic studies of cloud droplet coalescence,, Analytic Studies of Cloud Droplet Coalescence, 25 (1968), 54.
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.
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.
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.
doi: 10.1002/aic.690420729. |
show all references
References:
[1] |
M. Attarakih, C. Drumm and H. J. Bart, Solution of the population balance equation using the sectional quadrature method of moments,, (SQMOM), 64 (2009), 742. Google Scholar |
[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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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. Google Scholar |
[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.
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.
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. Google Scholar |
[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), 143.
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.
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.
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).
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.
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.
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.
doi: 10.1016/j.compchemeng.2007.01.006. |
[27] |
D. Ramkrishna, Population Balances. Theory and Applications to Particulate Systems in Engineering,, $1^{st}$ edition, (2000). Google Scholar |
[28] |
S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation,, AIChE Journal, 49 (2003), 1127.
doi: 10.1002/aic.690490507. |
[29] |
W. T. Scott, Analytic studies of cloud droplet coalescence,, Analytic Studies of Cloud Droplet Coalescence, 25 (1968), 54.
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.
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.
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.
doi: 10.1002/aic.690420729. |
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