Mathematical model for acid water neutralization with anomalous and fast diffusion

TL;DR

This paper develops a hyperbolic diffusion model using Cattaneo's equation to describe acid water neutralization, revealing multi-scale dynamics and solving a free boundary problem with numerical validation.

Contribution

It introduces a novel hyperbolic diffusion model for acid neutralization, extending classical models with Cattaneo's equation and analyzing multi-scale and free boundary aspects.

Findings

The model captures fast diffusion effects in acid neutralization.

The problem reduces to a wave equation in certain time scales.

Numerical simulations validate the theoretical results.

Abstract

In this paper we model the neutralization of an acid solution in which the hydrogen ions are transported according to Cattaneo's diffusion. The latter is a modification of classical Fickian diffusion in which the flux adjusts to the gradient with a positive relaxation time. Accordingly the evolution of the ions concentration is governed by the hyperbolic telegraph equation instead of the classical heat equation. We focus on the specific case of a marble slab reacting with a sulphuric acid solution and we consider a one-dimensional geometry. We show that the problem is multi-scale in time, with a reaction time scale that is larger than the diffusive time scale, so that the governing equation is reduced to the one-dimensional wave equation. The mathematical problem turns out to be a hyperbolic free boundary problem where the consumption of the slab is described by a nonlinear differential equation. Global well posedness is proved and some numerical simulations are provided.