Stochastic simulation of population balance models with disparate time scales: Hybrid strategies

The Monte Carlo methods have been an effective tool for the numerical solution of population balance models (PBMs). They are particularly useful for complex multidimensional problems. Less attention has been paid to solving population balance models where some species are away from the thermodynamic limit (very dilute or finite) and other species can be considered deterministic (high concentration). These types of problem often result in a stochastic system with rates spanning orders of magnitude for different mechanisms. Using the exact Monte Carlo solution to solve these types of problem is very inefficient because of the simulation time spent sampling fast events. These fast events are associated with species with large populations for which a single event does not change the population appreciably. This frequent sampling of fast events becomes a bottleneck during a simulation in which many single MC steps are required to make an appreciable change in the population.In this work, a hybrid solution strategy is developed to effectively solve this type of problem. The method implements the self-consistent fast/slow partitions used to solve stochastic equations in chemical kinetics. One strategy is found on the capacity of a coarse-grained Markov model called particle ensemble random product (PERP) to accelerate the simulation of fast events of PBMs (Chem. Eng. Sci. 63, 7649–7664; Chem. Eng. Sci. 63, 7665–7675). A second strategy approximates the fast events using mass conservation equations. These models are coupled with the exact MC simulation of slow events. Two extreme cases of heterocoagulation are studied to demonstrate these hybrid strategies.

► The hybrid algorithm is statistically identical to exact MC with an accelerated simulation speed.
► The hybrid strategy exploits the dual structure (individual particles) of the particle ensemble random product (PERP) model.
► Two extreme cases of heterocoagulation are presented.