Analytical solutions of the Arrhenius-Semenov problem for constant volume burn

TL;DR

This paper develops approximate and exact analytical solutions for the Arrhenius-Semenov thermal ignition problem under constant volume conditions, providing a simple method for solving the differential equation across various reaction orders.

Contribution

It introduces a novel approximation technique for the Arrhenius-Semenov equation applicable to any positive reaction order and constructs exact solutions for orders up to three.

Findings

Approximate solutions closely match numerical results.

The method is simple, robust, and applicable to various reaction orders.

Exact solutions are derived for reaction orders n ≤ 3.

Abstract

Analytical solutions to the Semenov thermal ignition problem for constant volume burn governed by Arrhenius reaction kinetics are derived. Specifically, an approximate analytical solution technique for the Arrhenius-Semenov differential equation is derived for reaction orders $n \in \mathbb{R}_{>0}$ and exact solutions are also constructed for reaction orders $n \in \mathbb{N}: n \leq 3$. The approximation technique relies on expansion of the respective nondominant terms in the differential equation at the lower and upper bounds of the reaction progress variable in order to create a pair of integrable series. The two integrated series are then connected to create a single continuous analytical solution. Excellent agreement is observed between the analytical approximation and solutions obtained numerically. The presented approximation constitutes a simple and robust strategy for solving the Arrhenius-Semenov problem analytically.