Solution of the population balance equation using constant-number Monte Carlo

We formulate a Monte Carlo simulation of the mean-field population balance equation by tracking a sample of the population whose size (number of particles in the sample) is kept constant throughout the simulation. This method amounts to expanding or contracting the physical volume represented by the simulation so as to continuously maintain a reaction volume that contains constant number of particles. We call this method constant-number Monte Carlo to distinguish it from the more common constant-volume method. In this work, we expand the formulation to include any mechanism of interest to population balances, whether the total mass of the system is conserved or not. The main problem is to establish connection between the sample of particles in the simulation box and the volume of the physical system it represents. Once this connection is established all concentrations of interest can be determined. We present two methods to accomplish this, one by requiring that the mass concentration remain unaffected by any volume changes, the second by applying the same requirement to the number concentration. We find that the method based on the mass concentration is superior. These ideas are demonstrated with simulations of coagulation in the presence of either breakup or nucleation.