On the Effectiveness of Wastewater Cylindrical Reactors: an Analysis Through Steiner Symmetrization

Abstract

The mathematical analysis of the shape of chemical reactors is studied in this paper through the research of the optimization of its effectiveness g such as introduced by R. Aris around 1960. Although our main motivation is the consideration of reactors specially designed for the treatment of wastewaters our results are relevant also in more general frameworks. We simplify the modeling by assuming a single chemical reaction with a monotone kinetics leading to a parabolic equation with a non-necessarily differentiable function. In fact we consider here the case of a single, non-reversible catalysis reaction of chemical order q; 0\q\1 (i.e.,the kinetics is given by bðwÞ ¼ kwq for some k [0). We assume the chemical reactor of cylindrical shape X ¼ G ð0; HÞ with G and open regular set of R2 not necessarily symmetric. We show that among all the sections G with prescribed area the ball is the set of lowest effectiveness gðt; GÞ. The proof uses the notions of Steiner rearrangement. Finally, we show that if the height H is small enough then the effectiveness can be made as close to 1 as desired.