Preprint 2026-23
Juan Barajas-Calonge, Raimund Bürger, Stefan Diehl, Luis M. Villada:
A network of conservation laws with discontinuous fluxes and source terms modeling countercurrent decantation
Abstract:
Countercurrent decantation (CCD) is an industrial separation process used to wash a slurry (solid-liquid suspension) by allowing the wash liquid and the slurry to flow in opposite directions through a series of clarifier-thickeners (CTs). The solid-liquid two-phase flow dynamics in a CCD circuit can be represented by a system of $N$ CT models, where the effluent and underflow streams are fed into neighboring units in opposite directions. The slurry to be treated and the wash water are fed into the first and the $N$th unit, respectively. The final CCD circuit model is formulated as a system of $N$ weakly coupled, nonlinear first-order conservation laws with discontinuous flux and smooth source terms combined with $N$~ordinary differential equations representing mixing tanks. The unknowns are the solids volume fraction in each CT, in the connecting pipes, and in the mixing tanks. An $L^1$-stability result for weak entropy solutions for the CCD model is established, which in particular implies uniqueness. It is demonstrated that a monotone finite difference scheme, based on the Engquist--Osher numerical flux, converges to the unique weak entropy solution. Finally, optimal steady states of the CCD circuit are characterized, and numerical simulations are presented.
