The drop population balance model describes the drop size distribution and the mass distribution of the solute in a liquid-liquid extraction column. It accounts for drop interactions (like breakage and coalescence) and the axial transport in the column, considering the phase inlets as point sources. Both integro-differential equations are discretized by utilizing Galerkin’s method. This leads to a coupled, nonlinear, dominantly hyperbolic system of conservation laws with additional drop interaction terms. We apply a diagonally implicit space-time discretization in conservative form to this system of PDEs.