**1. Introduction**

500 Numerical Simulation – From Theory to Industry

*version 2.1*, MathWorks Inc.

Vladimirescu, A. (1999). *SPICE*, Ed.Tehnica, Bucuresti

Vladimirescu, A. (1997) SIMULINK ; *Dynamic System Simulation for MATLAB, User's Guide,* 

The purpose of this book chapter is to analyze from a numerical point of view a reaction diffusion mathematical modelling of titanium carbide combustion synthesis from a mixture of titanium and carbide reactive powders thanks to self-propagating high temperature synthesis process. This modelling results in the coupling between a nonlinear parabolic equation expressing the enthalpy balance of the system with radiative boundary conditions and a nonlinear differential equation describing the exothermic chemical reaction in the system. An another multiphysics coupling was analyzed in [3]. This Self-propagating High temperature Synthesis (SHS) process was discovered in 1965 by Merzhanov [7], [8] and uses the energy released by the exothermic reaction itself to ensure its self-propagation inside the material after a localized heat supply has been performed for several seconds on the surface of the solid mixture. The stoichiometric solid mixture is made of several kinds of reactive powders. We analyze in this book chapter, the influence of radiative boundary conditions *related to the heat supply* over the ignition and eventual propagation of a combustion front inside the material.

Four sections are used to present our numerical simulation work. Section two presents the governing equations of the modelling. Section three outline the main aspects of the numerical scheme. Section four analyzes and discusses the main numerical simulation results of the combustion synthesis process. A conclusion summarizes the results that were obtained.
