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

Source: [Analytical solutions of the Arrhenius-Semenov problem for constant volume burn](<https://www.sota2.com/research/paper/68fa44aa1e209131ce912bc9>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T04:22:11.760Z

Authors: Galen T. Craven

Published: 2023-07-07T23:31:21.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2307.03862>)

[PDF](<https://arxiv.org/pdf/2307.03862.pdf>)