Simulation of processes in heat sensors based on solution of inverse problems in heat conduction

Abstract

The paper deals with nonstationary problems in heat conduction, which arise in connection with the determination of the heat flux density and temperature on the surface of a model in intermittent high-enthalpy wind-tunnel facilities by the results of temperature measurements using intramodel heat sensors. The solution of inverse problem in heat conduction in a one-dimensional formulation with an arbitrary time dependence of the heat flux density is obtained by two methods, namely, by iterations and by integral transformations with finite limits. In the former method, the inverse problem is reduced to a system of two coupled integral and integro-differential equations of the Volterra type relative to the temperature and heat flux density on the external boundary. Calculations demonstrate that the numerical solution asymptotically approaches the exact solution, and the iteration method exhibits smoothing properties and is stable with respect to random errors of measurement. In the integral method, an inverse problem for the class of boundary functions satisfying the Dirichlet conditions and represented by a partial sum of the Fourier series reduces to a set of algebraic equations which has a unique solution. In the absence of measurement errors, the solution of inverse problem is exact. Examples are given of constructing solutions in the presence of random noise; it is demonstrated that, in the case of reasonable restriction of the range of frequencies to be analyzed, the errors in the solution do not exceed the mean-square level of noise.