Modeling, analysis, and numerical solution of stirred liquid–liquid dispersions

In this paper, we derive a mathematical model describing the different processes in stirred liquid–liquid systems. The turbulent flow field in the stirred tank is modeled by the Reynolds-averaged Navier–Stokes equations together with a k  –εε turbulence model, whereas the population dynamics of the dispersed phase are described by an averaged population balance equation. This system of equations is then semi-discretized in space so that the dynamic behavior of the resulting differential-algebraic system can be analyzed. The strangeness-index of the system is determined, which tells us something about the problems that may arise when solving the system numerically. It is shown that the index of the semi-discretized Navier–Stokes equations is not increased by the coupling with the population balance equation and by turbulence modeling. Thus, all implicit time discretization schemes that are applicable to the solution of the Navier–Stokes equations can also be adapted for solving the discussed coupled system. Suggestions for the practical realization are given.