Monte Carlo (MC) methodologies have become mainstream numerical tools for the simulation of particulate processes that require the solution of the population balance equation. Such methodologies have been shown to be very efficient in simulations of well-mixed systems. Implementing population balance strategies within a larger 2D or 3D simulator remains a challenge, however, because the computational cost of solving the population balance equation in multiple dimensions is high. Standard simulators advance time and then compute changes within that interval and for this reason, time-driven MC has been proposed as the most appropriate MC variant for simulations that combine population balances with standard numerical integrators. However, time-driven MC as a stand-alone algorithm is slower than standard event-driven MC with the same number of particles; when interfaced with 2- or 3D effects (e.g. fluid dynamics) it is rather slow for practical implementations. Event-driven MC, while generally faster, operates in the opposite way of standard integrators: it effects a change in the population and then advances time by a corresponding amount. Thus the time step is not directly controlled.

Here we present a new hybrid algorithm, which (i) utilizes a fixed (predefined) number of simulation particles, N, and (ii) which is capable of transforming a sample of N simulation particles at t to a corresponding sample of N particles at time t+Δt, where Δt is a predefined time increment. This is internally accomplished by temporarily inflating the number of simulation particles at time t to a value N' that matches the chosen time step. The inflated population is subjected to a standard Monte Carlo operation that simulates the particulate processes of interest and produces a sample of the population at the end of the time step. Finally, the number of particles is deflated to N and the process is repeated. We formulate the mathematical basis of the algorithm and demonstrate the method in simulations of particle aggregation and breakup. We examine the accuracy as a function of the number of simulation particles and length of time step, and discuss the potential extension to 2- and 3-dimensional problems.