Simulations of population balance systems with one internal coordinate using finite element methods

The paper presents an approach for simulating a precipitation process which is described by a population balance system consisting of the incompressible Navier–Stokes equations, nonlinear convection–diffusion–reaction equations and a transport equation for the particle size distribution (PSD). The Navier–Stokes equations and the convection–diffusion–reaction equations are discretized implicitly in time and with finite element methods in space. Two stabilization techniques for the convection–diffusion–reaction equations are investigated. An explicit temporal discretization and an upwind finite difference method are used for discretizing the equation of the PSD. Simulations of the calcium carbonate precipitation in a cavity are presented which study the influence of the flow field on the PSD at the outflow. It is shown that variations of the positions of the inlets change the volume fraction of the PSD at the center of the outlet. The corresponding medians of the volume fraction differ up to a factor of about three. In addition, it is demonstrated that the use of the two different stabilized finite element methods for the convection–diffusion–reaction equations leads to completely different numerical results.