Technical Report 9, c4e-Preprint Series, Cambridge

Authors: Mike J. Goodson and Markus Kraft*

Reference: Technical Report 9, c4e-Preprint Series, Cambridge, 2002

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In this paper, a practical study is made of two stochastic solution methods for the population balance equation, simulating coalescence and binary breakage. The first algorithm studied is the existing Direct Simulation Algorithm (DSA), proposed in (Eibeck and Wagner, Stoch. Anal. App., 18(6):921-948,2000). The second is an extension of the Mass Flow Algorithm (MFA), which was proposed in (Eibeck and Wagner, Ann.Appl.Probab.,11(4):1137-1165,2001) for coagulation only. MFA is extended to include breakage and a binary search method of distribution generation is introduced, leading to improved efficiency. Numerical investigation of the performance of the two algorithms is carried out by applying them both to a test case, for which an analytical solution is calculated. For both algorithms, convergence of the predicted moments to the analytical solution goes as the inverse of the number of stochastic particles, N, except for the zeroth moment predicted by MFA. This exhibits large fluctuations, due to the presence of very small particles, and converges approximately as N-1/3. The new algorithm, MFA, exhibits significant variance reduction - and therefore improved simulation efficiency - for the prediction of higher moments, but for our test case the zeroth moment (the total number of particles) is predicted with better efficiency by DSA. In many breakage models for liquid-liquid systems however, the introduction of a minimum particle size reduces the advantage held by DSA for predicting the zeroth moment. Depending on the minimum particle size, MFA can perform comparably with DSA for predicting the zeroth moment.

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