Analytical time-stepping solution of the discretized population balance equation

The prediction of the particle-size distribution (PSD) of the particulate systems in chemical engineering is very important in a variety of different contexts, such as parameter identification, troubleshooting, process control, design, product quality, production economics etc. The time evolution of the PSD can be evaluated by means of the population balance equation (PBE), which is a complex integro-differential equation, whose solution in practical cases always requires sophisticated numerical methods that may be computationally tedious. In this work, we propose a novel technique that tackles this issue by using an analytical time-stepping procedure (ATS) to resolve the PSD time dependency. The ATS is an explicit time integrator, taking advantage of the linear or almost linear time dependency of the discretized population balance equation. Thus, linear approximation of the source term is a precondition for the ATS simulations. The presented technique is compared with a standard variable step time integrator (MATLAB ODE15s stiff solver), for practical examples e.g. emulsion, aging cellulose process, cooling crystallization, reactive dissolution, and liquid-liquid extraction. The results show that this advancing in time procedure is accurate for all tested practical examples, allowing reproducing the same results given by standard time integrators in a fraction of the computational time.