Influence of Initial Temperature and Convective Heat Loss on the Self-Propagating Reaction in Al/Ni Multilayer Foils

A two-dimensional numerical model for self-propagating reactions in Al/Ni multilayer foils was developed. It was used to study thermal properties, convective heat loss, and the effect of initial temperature on the self-propagating reaction in Al/Ni multilayer foils. For model adjustments by experimental results, these Al/Ni multilayer foils were fabricated by the magnetron sputtering technique with a 1:1 atomic ratio. Heat of reaction of the fabricated foils was determined employing Differential Scanning Calorimetry (DSC). Self-propagating reaction was initiated by an electrical spark on the surface of the foils. The movement of the reaction front was recorded with a high-speed camera. Activation energy is fitted with these velocity data from the high-speed camera to adjust the numerical model. Calculated reaction front temperature of the self-propagating reaction was compared with the temperature obtained by time-resolved pyrometer measurements. X-ray diffraction results confirmed that all reactants reacted and formed a B2 NiAl phase. Finally, it is predicted that (1) increasing thermal conductivity of the final product increases the reaction front velocity; (2) effect of heat convection losses on reaction characteristics is insignificant, e.g., the foils can maintain their characteristics in water; and (3) with increasing initial temperature of the foils, the reaction front velocity and the reaction temperature increased.

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction–Diffusion–Advection-Type with Data on the Position of a Reaction Front

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction–diffusion–advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.