Moment Method for the Boltzmann Equation of Reactive Quaternary Gaseous Mixture

TL;DR

This paper develops a novel moment method for the Boltzmann equation in reactive gases, introducing a general collision moment algorithm, deriving non-equilibrium reaction rates, and demonstrating the importance of the 14-moment system through numerical analysis.

Contribution

It presents a new collision model and a general algorithm for computing moments of the Boltzmann collision operator, extending the Grad's-14 moment method to reactive gases.

Findings

Reaction rates depend explicitly on the 14th moment.

The derived reaction rates recover Arrhenius law at equilibrium.

Numerical experiments show relaxation to equilibrium.

Abstract

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad's-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision operator. Our algorithm is general in the sense that it can be used to compute arbitrary order moments of the collision operator and not just the moments included in the Grad's-14 moment system. For a first-order chemical kinetics, we derive reaction rates for a chemical reaction outside of equilibrium thereby, extending the Arrhenius law that is valid only in equilibrium. We show that the derived reaction rates (i) are consistent in the sense that at equilibrium, we recover the Arrhenius law and (ii) have an explicit dependence on the scalar fourteenth moment, highlighting the importance of considering a fourteen moment system rather than a thirteen one. Through numerical experiments we study the relaxation of the Grad's-14 moment system to the equilibrium state.