Analytical Solution to Coupled Chemical Reaction and Thermally Diffusing Systems: Applicability to Self-Propagating Thermopower Waves

Problems involving chemical reaction coupled to thermal diffusion are central in the chemistry and physics of propulsion, self-propagating high-temperature synthesis (SHS) of materials, and new approaches to power generation involving thermopower waves. We present an analytical solution to a one-dimensional Fourier description of heat propagation in a solid reactant, with a first-order chemical reaction of Arrhenius form providing a thermal source. A generalized logistic function can completely determine the form of the temperature and concentration profiles within the solid, as well as the velocity of the reaction wave under a wide range of conditions. This alternative to the common asymptotic and three-zone approximate solution methods in the literature requires fewer assumptions and is valid at all time and length scales. The solution is limited to cases where the reaction wave velocity is constant but can be generalized to nth-order kinetics. Applications to the theory of power generation using self-propagating thermopower waves are discussed.