Discrete solution of the breakage equation using markov chains

Analytical solution of population balance equations (PBEs) may be impossible except for some simple cases. In the literature there are a number of methods to solve PBEs including discrete methods, Monte Carlo simulation, and method of moments. In this paper, the Markov chain is presented as a discrete solution for a population balance equation of a breakage process for determining the particle size distribution (PSD) over time. The transition matrix P, which is the key operator of a Markov chain, is built using breakage equations. Thereafter, from calculating transition matrix, P, the particle size distribution of the system is easily evaluated using the Markov chain. According to simulation results, if the size range of the system is divided into a sufficient number of states and an appropriate transition time step was chosen, then results from the Markov chain are in agreement with the analytical solution of PBEs governed by the same breakage functions. In addition to theoretical illustration, the Markov theory was employed to model the breakage process of aggregated food products passing through a pneumatic conveying pipeline rig.