An Inverse Problem in Propellant Combustion

Abstract

If the temperature or heat flux at the surface of a solid are given as a function of time, then the temperature distribution in the solid’s interior can be calculated. This is termed a direct problem. However, there are many practical situations which arise in engineering where the surface environment may be unsuitable for attaching a sensor such as a thermocouple. One example of such a situation is designing an experiment to determine the temperature distribution and surface heat flux history on an ablating heat shield undergoing chemical reactions with the atmosphere. Since the shield’s surface is receding and located in a hostile environment, the sensor cannot monitor the temperature accurately. However, it is easier to measure the transient temperature at a known interior point of the shield. The mathematical problem of determining the temperature distribution and surface boundary conditions from such an interior measurement is known as the inverse heat conduction problem (IHCP) [1]. There are two general classes of such inverse problems: parameter estimation and function estimation. An example of parameter estimation is the determination of a solid’s thermal diffusivity from time-dependent temperature measurements. The estimation of the surface heat-flux as a function of time is an example of a function estimation problem. There are many papers and some books on both classes of problems. Beck et al. [2] have written the only English language book on the IHCP. This book provides an excellent introduction to various aspects of the IHCP and contains numerous references up to 1985.