## Simulation of coalescence and breakage: an assessment of two stochastic methods suitable for simulating liquid-liquid extraction

Industrial problems such as liquid-liquid extraction often require the use of population balances, so a thorough understanding of the advantages and disadvantages of all the available solution techniques is essential. The mathematical literature on stochastic solution methods for the population balance equation is often quite fragmentary and tends to focus mainly on existence proofs rather than examining the applicability of a particular algorithm to a practical problem of interest to researchers or industrialists. 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 & Wagner 2000, Stoch. Anal. App., 18(6), 921-948). The second is an extension of the Mass Flow Algorithm (MFA), which was proposed in (Eibeck & Wagner 2001, Ann. Appl. Probab., 11(4), 1137-1165) for coagulation only and in (Jourdain 2003, Markov Processes and Related Fields, 9(1), 103-130) for discrete coagulation-fragmentation. MFA is extended to include breakage in the continuous case, 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 Ni1=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. Using models for coalescence and breakage from literature, DSA is successfully applied to the case of a laboratory scale rotating disc contactor.

- This paper draws from preprint 9: Stochastic simulation of coalescence and breakage processes: a practical study