A general algorithm for exact simulation of multicomponent aggregation processes. (English) Zbl 1139.82331
J. Comput. Phys. 177, No. 2, 418-449 (2002); erratum 217, No. 2, 866-867 (2006).
Summary: A Monte Carlo (MC) algorithm is presented for the simulation of the time evolution of aggregation processes featuring multiple components, properties, or conservation laws. Instead of using deterministic differential population balance equations, the MC algorithm utilizes a stochastic approach to aggregation kinetics. As a result, exact simulation of spatially independent aggregation processes is possible without the need for numerical approximations. Furthermore, simulations exactly predict all moments of the size and composition distributions of aggregating particles for both nongelling and gelling kernels and extend these results to the postgelation period. The algorithm is shown to require at most \(O((\Omega_1\Omega_2\dots\Omega_\kappa)^{1/(\kappa+1)})\) rate-limiting operations per time step for a \(\kappa\)-component aggregation process featuring \(\Omega_i\) monomers of each component \(i\) – a substantial performance improvement over the potential of previous methods. Simulation results are presented for bivariate sum, product, and constant kernels, and for the perikinetic (Brownian) kernel.
MSC:
| 82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 76T30 | Three or more component flows |
References:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001 |
| [2] | Bayewitz, M. H.; Yerushalmi, J.; Katz, S.; Shinnar, R., The extent of correlations in a stochastic coalescence process, J. Atmos. Phys., 31, 1604 (1974) |
| [3] | Friedlander, S. K., Smoke, Dust and Haze (2000), Oxford University Press: Oxford University Press New York |
| [4] | Gelbard, F. M.; Seinfeld, J. H., Coagulation and growth of a multicomponent aerosol, J. Colloid Interface Sci., 63, 472 (1978) |
| [5] | Gillespie, D. T., The stochastic coalescence model for cloud droplet growth, J. Atmos. Sci., 29, 1496 (1972) |
| [6] | Gillespie, D. T., An exact method for numerically simulating the stochastic coalescence process in a cloud, J. Atmos. Sci., 32, 1977 (1975) |
| [7] | Gillespie, D. T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22, 304 (1976) |
| [8] | Golovin, A. M., The solution of the coagulation equation for cloud droplets in a rising air current, Izv. Zkad. Nauk. SSSR Ser. Geofiz., 5, 482 (1963) |
| [9] | Hendriks, E. M.; Spouge, J. L.; Eibl, M.; Schreckenberg, M., Exact solutions for random coagulation processes, Z. Phys. B Condens. Matter, 58, 219 (1985) |
| [10] | Hounslow, M. J.; Ryall, R. L.; Marshall, V. R., A discretized population balance for nucleation, growth, and aggregation, AIChE J., 34, 1821 (1988) |
| [11] | Kourti, N.; Schatz, A., Solution of the general dynamic equation (GDE) for multicomponent aerosols, J. Aerosol Sci., 21, 41 (1998) |
| [12] | Krapivsky, P. L.; Ben-Naim, E., Aggregation with multiple conservation laws, Phys. Rev. E., 53, 291 (1996) |
| [13] | Kumar, S.; Ramkrishna, D., On the solution of population balance equations by discretization. I. A fixed pivot technique, Chem. Eng. Sci., 51, 1311 (1996) |
| [14] | Kumar, S.; Ramkrishna, D., On the solution of population balance equations by discretization. II. A moving pivot technique, Chem. Eng. Sci., 51, 1333 (1996) |
| [15] | Laurenzi, I. J.; Diamond, S. L., Monte Carlo simulation of the heterotypic aggregation kinetics of platelets and neutrophils, Biophys. J., 77, 1733 (1999) |
| [16] | Lushnikov, A. A., Evolution of coagulating systems. III. Coagulating mixtures, J. Colloid Interface Sci., 54, 94 (1976) |
| [17] | Lushnikov, A. A., Coagulation in finite systems, J. Colloid Interface Sci., 65, 276 (1978) |
| [18] | McLeod, J. B., On an infinite set of non-linear differential equations, Q. J. Math. Oxford, 2, 119 (1962) · Zbl 0109.31501 |
| [19] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in C (1992) · Zbl 0778.65003 |
| [20] | Ramkrishna, D.; Borwanker, J. D., A puristic analysis of population balance. I, Chem. Eng. Sci., 28, 1423 (1973) |
| [21] | Ramkrishna, D.; Shah, B. H.; Borwanker, J. D., Analysis of population balance. III. Agglomerating populations, Chem. Eng. Sci., 31, 435 (1976) · Zbl 0333.92016 |
| [22] | Rényi, A., A discussion of chemical reactions using the theory of stochastic processes, MTA Alk. Mat. Int. Közl., 2, 83 (1953) |
| [23] | Shah, B. H.; Ramkrishna, D.; Borwanker, J. D., Simulation of particulate systems using the concept of the interval of quiescence, AIChE J., 23, 897 (1977) |
| [24] | Smith, M.; Matsoukas, T., Constant-number Monte Carlo simulation of population balances, Chem. Eng. Sci., 53, 1777 (1998) |
| [25] | Smoluchowski, M.v., Vensuch einer mathematischen theorie der koagulationkinetic kolloider lösungen, Z. Phys. Chem., 92, 129 (1917) |
| [26] | Spouge, J. L., Monte-Carlo results for random coagulation, J. Colloid Interface Sci., 107, 38 (1985) |
| [27] | Tanaka, H.; Nakazawa, K., Stochastic coagulation equation and validity of the statistical coagulation equation, J. Geomeg. Geoelectr., 45, 361 (1993) |
| [28] | Tanaka, H.; Nakazawa, K., Validity of the statistical coagulation equation and runaway growth of protoplanets, Icarus, 107, 404 (1994) |
| [29] | Tandon, P.; Diamond, S. L., Hydrodynamic effects and receptor interactions of platelets and their aggregates in linear shear flow, Biophys. J., 73, 2819 (1997) |
| [30] | Thorn, M.; Breuer, H. P.; Petruccione, F.; Honerkamp, J., A master equation investigation of coagulation reactions—Sol-gel transition, Makromol. Chem. Theory. Simul., 3, 585 (1994) |
| [31] | Vigil, R. D.; Ziff, R. M., On the scaling theory of two-component aggregation, Chem. Eng. Sci., 53, 1725 (1998) |
| [32] | Wright, D. L.; McGraw, R.; Rosner, D. E., Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations, J. Colloid Interface Sci., 236, 242 (2001) |
| [33] | Ziff, R. M., Kinetics of polymerization, J. Stat. Phys., 23, 241 (1980) |
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.
