American Institute of Mathematical Sciences

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

An accurate and efficient discrete formulation of aggregation population balance equation

Jitendra Kumar 1, , Gurmeet Kaur 1, and  Evangelos Tsotsas 2,

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.
Keywords: nonuniform meshes, preservation of moments., aggregation, Population balances, finite volume method.
Mathematics Subject Classification: Primary: 45K05, 65R20; Secondary: 45L0.
Citation: Jitendra Kumar, Gurmeet Kaur, Evangelos Tsotsas. An accurate and efficient discrete formulation of aggregation population balance equation. Kinetic & 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), 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski equation,, SIAM Journal of Scientific Computing, 25 (2004), 2004.  doi: 10.1137/S1064827503429132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski equation,, SIAM Journal of Scientific Computing, 25 (2004), 2004.  doi: 10.1137/S1064827503429132.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[2]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[3]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[4]

Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
[5]

Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
[6]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[7]

Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020400
[8]

Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163
[9]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[10]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[11]

Jianguo Huang, Sen Lin. A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048
[12]

Qilong Zhai, Ran Zhang. Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 403-413. doi: 10.3934/dcdsb.2018091
[13]

Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic & Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237
[14]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004
[15]

Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913
[16]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
[17]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427
[18]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020226
[19]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[20]

Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences