Critical conditions in the Frank–Kamenetsky problem with equal opposite pinpoint impacts

Igor G. Donskoy

Melentiev Energy Systems Institute, Irkutsk, Russian Federation

Abstract: A model of thermal explosion in a one-dimensional flat sample is considered in the paper. It is related to the following two problems. The first one concerns transition from the Frank–Kamenetsky thermal explosion model to the Semenov model. The second one is connected with the control of exothermic reactions using point effects. The model is presented in the form of a non-linear differential equation with an exponential term, and includes a source and a sink in the form of sharp peaks with equal integrals over the region. The equation is solved numerically. It is shown that averaging leads to the Semenov model for the original and modified formulations. However, the critical conditions depend significantly on the intensities and positions of the source and sink.

Keywords: differential equation, thermal explosion, numerical solution, pinch control.

UDC: 544.45, 536.2, 519.62

Received: 27.06.2025
Received in revised form: 02.07.2025
Accepted: 24.08.2025

Language: English