A numerical comparison of the method of moments for the population balance equation. (English) Zbl 1540.82008
Summary: We investigate the application of the method of moments approach for the one-dimensional population balance equation. We consider different types of moment closures, including polynomial \((\text{P}_N)\) closures and minimum entropy \((\text{M}_N)\) closures. Additionally, we transfer the concept of Kershaw closures \((\text{K}_N)\) from radiative head transfer to the population balance equation. Realizability-preserving schemes for the moment closures as well as for a discontinuous Galerkin discretization of the space-dependent population balance equation are discussed. The numerical examples range from spatially homogeneous cases to a population balance equation coupled with the stationary incompressible Navier-Stokes equation. A detailed numerical discussion of accuracy, order of the moment method and computational time is given. We show that the minimum entropy closures are comparable or superior to the Kershaw closures in these test cases. In addition, we demonstrate that higher-order numerical schemes can be applied to the moment equations of the population balance equation.
MSC:
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
Software:
DUNEReferences:
| [1] | Abramov, R., An improved algorithm for the multidimensional moment-constrained maximum entropy problem, J. Comput. Phys., 226, 1, 621-644 (2007) · Zbl 1125.65010 |
| [2] | Ahiezer, N.; Krein, M., Some Questions in the Theory of Moments (1962), American Mathematical Society · Zbl 0117.32702 |
| [3] | Alldredge, G.; Hauck, C.; O’Leary, D.; Tits, A., Adaptive change of basis in entropy-based moment closures for linear kinetic equations, J. Comput. Phys., 258, 489-508 (2014) · Zbl 1349.82066 |
| [4] | Alldredge, G.; Hauck, C.; Tits, A., High-order entropy-based closures for linear transport in slab geometry II: a computational study of the optimization problem, SIAM J. Sci. Comput., 34, 4, B361-B391 (2012) · Zbl 1297.82032 |
| [5] | Alldredge, G.; Schneider, F., A realizability-preserving discontinuous galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, J. Comput. Phys., 295, 665-684 (2015) · Zbl 1349.82067 |
| [6] | Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE, Computing, 82, 2-3, 121-138 (2008) · Zbl 1151.65088 |
| [7] | Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. Part i: abstract framework, Computing, 82, 2-3, 103-119 (2008) · Zbl 1151.65089 |
| [8] | Bourgade, J.; Filbet, F., Convergence of a finite volume scheme for coagulation-fragmentation equations, Math. Comp., 77, 851-882 (2007) · Zbl 1140.82032 |
| [9] | Bresten, C.; Gottlieb, S.; Grant, Z.; Higgs, D.; Ketchseon, D.; Németh, A., Explicit strong stability preserving multistep rungekutta methods, Math. Comp., 86, 304, 747-769 (2017) · Zbl 1355.65120 |
| [10] | Brezzi, F.; Douglas, J.; Marini, L., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 2, 217-235 (1985) · Zbl 0599.65072 |
| [11] | Cameron, I.; Wang, F.; Immanuel, C.; Stepanek, F., Process systems modelling and applications in granulation: a review, Chem. Eng. Sci., 60, 14, 3723-3750 (2005) |
| [12] | Cockburn, B., An introduction to the discontinuous galerkin method for convection-dominated problems, (Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (1997), Springer), 151-268 · Zbl 0927.65120 |
| [13] | Coulaloglou, C.; Tavlarides, L., Description of interaction process in agitated liquidliquid dispersions, Chem. Eng. Sci., 32, 1289-1297 (1977) |
| [14] | Curto, R.; Fialkow, L., Recursiveness, positivity, and truncated moment problems, Houston J. Math., 17, 4, 603-635 (1991) · Zbl 0757.44006 |
| [15] | Dhanasekharan, K.; Sanyal, J.; Jain, A.; Haidari, A., A generalized approach to model oxygen transfer in bioreactors using population balances and computational fluid dynamics, Chem. Eng. Sci., 60, 1, 213-218 (2005) |
| [16] | Dolbeault, J.; Klar, A.; Mouhot, C.; Schmeiser, C., Hypocoercivity and a fokker – planck equation for fiber lay-down, Appl. Math. Res. Exp., 165-175 (2013) · Zbl 1278.82044 |
| [17] | Dubroca, B.; Feugeas, J.-L.; Frank, M., Angular moment model for the fokker-planck equation, Eur. Phys. J. D, 60, 2, 301-307 (2010) |
| [18] | Duclous, R.; Dubroca, B.; Frank, M., A deterministic partial differential equation model for dose calculation in electron radiotherap, Phys. Med. Biol., 55, 13, 3843 (2010) |
| [19] | Filbet, F.; Laurençot, P., Numerical simulation of the smoluchowski coagulation equation, SIAM J. Sci. Comput., 25, 2004-2028 (2004) · Zbl 1086.65123 |
| [20] | Gerhard, N.; Müller, S., Adaptive multiresolution discontinuous galerkin schemes for conservation laws: multi-dimensional case, Comput. Appl. Math., 35, 2, 321-349 (2016) · Zbl 1372.65271 |
| [21] | Gokhale, Y.; Kumar, R.; Kumar, J.; Hintz, W.; Warnecke, G.; Tomas, J., Disintegration process of surface stabilized sol-gel tio2 nanoparticles by population balances, Chem. Eng. Sci., 64, 24, 5302-5307 (2009) |
| [22] | Golub, G., Some modified matrix eigenvalue problems, SIAM Rev., 15, 2, 318-334 (1973) · Zbl 0254.65027 |
| [23] | Gottlieb, S., On high order strong stability preserving runge-kutta and multi step time discretizations, J. Sci. Comput., 25, 112, 105-128 (2005) · Zbl 1203.65166 |
| [24] | Hauck, C., High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9, 1, 187-205 (2011) · Zbl 1284.82050 |
| [25] | Hauck, C.; Garett, C., A comparison of moment closures for linear kinetic transport equations: the line source benchmark, Transport Theor. Stat. Phys. (2013) · Zbl 1302.82088 |
| [26] | Hill, P.; Ng, K., Statistics of multiple particle breakage, AIChE, 42, 6, 1600-1608 (1966) |
| [27] | Kershaw, D., Flux limiting, nature’s own way-a new method for numerical solution of the transport equation (1976), Lawrence Livermore Laboratory |
| [28] | Klar, A.; Schneider, F.; Tse, O., Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker – planck equations, Kinet. Relat. Mod., 7, 3, 509-529 (2014) · Zbl 1306.82012 |
| [29] | Kolb, O., A third order hierarchical basis WENO interpolation for sparse grids with application to conservation laws with uncertain data, J. Sci. Comput., 74, 3, 1480-1503 (2018) · Zbl 1397.65204 |
| [30] | Kumar, J., Numerical approximations of population balance equations in particulate systems (2007), Otto-von-Guericke-Universität Magdeburg, Ph.D. thesis |
| [31] | Kumar, R.; Kumar, J.; Warnecke, G., Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Math. Models Methods Appl. Sci., 23, 1235-1273 (2012) · Zbl 1269.65137 |
| [32] | Kumar, R.; Kumar, J.; Warnecke, G., Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations, Kinet. Relat. Mod., 7, 713-737 (2014) · Zbl 1316.65116 |
| [33] | Kumar, J.; Peglow, M.; Warnecke, G.; Heinrich, S.; Mörl, L., Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique, Chem. Eng. Sci., 61, 10, 3327-3342 (2006) |
| [34] | Levermore, C., Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83, 5-6, 1021-1065 (1996) · Zbl 1081.82619 |
| [35] | Levermore, C., Moment closure hierarchies for the boltzmann-poisson equation, VLSI Design, 6, C, 97-101 (1998) |
| [36] | Marchisio, D.; Vigil, D.; Fox, R., Quadrature method of moments for aggregation breakage processes, J. Colloid Interface Sci., 258, 322-334 (2003) |
| [37] | McGraw, R., Description of aerosol dynamics by the quadrature method of moments, Aerosol Sci. Tech., 27, 2, 255-265 (1997) |
| [38] | T. Müller, R. Dürr, B. Isken, J. Schulze-Horsel, U. Reichl, A. Kienle, Distributed modeling of human influenza a virus host cell interactions during vaccine production, Biotechnol. Bioeng. 110 (8) 2252-2266.; T. Müller, R. Dürr, B. Isken, J. Schulze-Horsel, U. Reichl, A. Kienle, Distributed modeling of human influenza a virus host cell interactions during vaccine production, Biotechnol. Bioeng. 110 (8) 2252-2266. |
| [39] | Müller, L.; Meurer, A.; Schneider, F.; Klar, A., A numerical investigation of flux-limited approximations for pedestrian dynamics, Math. Models Methods Appl. Sci., 27, 06, 1177-1197 (2017) · Zbl 1365.90096 |
| [40] | Nguyen, T.; Fox, R.; Massot, M.; Laurent, F., Solution of population balance equations in applications with fine particles: mathematical modeling and numerical schemes, J. Comput. Phys., 325, 129-156 (2016) · Zbl 1375.76204 |
| [41] | Nguyen, N.; Peraire, J.; Cockburn, B., A hybridizable discontinuous galerkin method for stokes flow, Comput. Methods Appl. Mech. Eng., 199, 9-12, 582-597 (2010) · Zbl 1227.76036 |
| [42] | Nguyen, N.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous galerkin method for the incompressible navier-stokes equations, J. Comput. Phys., 230, 4, 1147-1170 (2011) · Zbl 1391.76353 |
| [43] | Olbrant, E.; Frank, M., Generalized fokker-planck theory for electron and photon transport in biological tissues: application to radiotherapy., Comput. Math. Methods Med., 11, 4, 313-339 (2010) · Zbl 1202.92045 |
| [44] | Olbrant, E.; Hauck, C.; Frank, M., A realizability-preserving discontinuous galerkin method for the m1 model of radiative transfer, J. Comput. Phys., 231, 5612-5639 (2012) · Zbl 1277.65083 |
| [45] | Pigou, M.; Morchain, J.; Fede, P.; Penet, M.-I.; Laronze, G., An assessment of methods of moments for the simulation of population dynamics in large-scale bioreactors, Chem. Eng. Sci., 171, 218-232 (2017) |
| [46] | Ramkrishna, D., Population Balances. Theory and Applications to particulate Systems in Engineering (2000), Academic Press |
| [47] | Randolph, A.; Larson, M., Theory of Particulate Processes (1971), Elsevier |
| [48] | J. Ritter, A. Klar, F. Schneider, Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions, 2016.; J. Ritter, A. Klar, F. Schneider, Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions, 2016. · Zbl 1382.65257 |
| [49] | Rosner, D.; McGraw, R.; Tandon, P., Multivariate population balances via moment and Monte Carlo simulation methods: an important sol reaction engineering bivariate example and mixed moments for the estimation of deposition, scavenging, and optical properties for populations of nonspherical suspended particles, Ind. Eng. Chem. Res., 42, 12, 2699-2711 (2003) |
| [50] | Ruuth, S.; Spiteri, R., Two barriers on strong-stability-preserving time discretization methods, J. Sci. Comput., 17, 211-220 (2002) · Zbl 1003.65107 |
| [51] | Schneider, F., Moment models in radiation transport equations, 260 (2016), Dr. Hut Verlag |
| [52] | Schneider, F., Kershaw closures for linear transport equations in slab geometry I: model derivation, J. Comput. Phys., 322, 920-935 (2016) · Zbl 1351.82087 |
| [53] | Schneider, F., Kershaw closures for linear transport equations in slab geometry II: high-order realizability-preserving discontinuous-galerkin schemes, J. Comput. Phys., 322, 920-935 (2016) · Zbl 1351.82087 |
| [54] | Schneider, F.; Alldredge, G.; Frank, M.; Klar, A., Higher order mixed moment approximations for the Fokker-Planck equation in one space dimension, SIAM J. Appl. Math., 74, 4, 1087-1114 (2014) · Zbl 1307.35301 |
| [55] | Schneider, F.; Kall, J.; Alldredge, G., A realizability-preserving high-order kinetic scheme using weno reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinet. Relat. Mod., 9, 1, 193-215 (2015) · Zbl 1326.65115 |
| [56] | Schneider, F.; Kall, J.; Roth, A., First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions, Kinet. Relat. Mod. (2016), (in press) |
| [57] | Schütz, J.; May, G., A hybrid mixed method for the compressible navier-stokes equations, J. Comput. Phys., 240, 58-75 (2013) · Zbl 1426.76566 |
| [58] | Smoluchowski, M., Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Phys. Z., 14, 557-571,587-599 (1926) |
| [59] | Vikas, V.; Hauck, C.; Wang, Z.; Fox, R., Radiation transport modeling using extended quadrature method of moments, J. Comput. Phys., 246, 221-241 (2013) · Zbl 1349.78080 |
| [60] | Wheeler, J., Modified moments and gaussian quadrature, Rocky Mountain J. Math., 4, 2, 287-296 (1974) · Zbl 0309.65009 |
| [61] | Yuan, C.; Fox, R., Conditional quadrature method of moments for kinetic equations, J. Comput. Phys., 230, 8216-8246 (2011) · Zbl 1231.82007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
