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Gelfand-type problem for two-phase porous media.

Peter V Gordon1, Vitaly Moroz2

  • 1Department of Mathematics , University of Akron , Akron, OH 44325, USA.

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|March 11, 2014
PubMed
Summary

Thermal explosion in two-phase materials occurs without stable temperature distribution. Interphase heat exchange delays this explosion, simplifying to the classical Gelfand problem with adjusted constants under high heat exchange conditions.

Keywords:
Gelfand problemblow-upporous media combustionreaction–diffusion systemssingular limitsthermal explosion

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Area of Science:

  • Chemical Engineering
  • Materials Science
  • Thermodynamics

Background:

  • The Gelfand problem is a key model in Frank-Kamenetskii theory for thermal explosion.
  • Classical models often use a single temperature, which is insufficient for multi-phase systems.
  • Two-phase materials require distinct temperature descriptions for accurate modeling.

Purpose of the Study:

  • To generalize the Gelfand problem for two-phase materials.
  • To analyze the conditions leading to thermal explosion in these systems.
  • To investigate the effect of interphase heat exchange on explosion dynamics.

Main Methods:

  • Mathematical modeling of thermal processes in two-phase media.
  • Analysis of stationary temperature distributions.
  • Asymptotic analysis in the limit of infinite interphase heat exchange.

Main Results:

  • Thermal explosion in two-phase systems is driven by the absence of stationary temperature states.
  • Interphase heat exchange acts as a delaying factor for thermal explosion.
  • Under conditions of infinite interphase heat exchange, the problem simplifies to the classical Gelfand problem with modified constants.

Conclusions:

  • The study extends thermal explosion theory to two-phase materials.
  • Interphase heat exchange significantly influences the onset and dynamics of thermal explosion.
  • The findings provide a theoretical basis for understanding thermal safety in composite materials.