Solution of Inverse Problems in Thermal Systems

Abstract

Abstract A common occurrence in many practical systems is that the desired result is known or given, but the conditions needed for achieving this result are not known. This situation leads to inverse problems, which are of particular interest in thermal processes. For instance, the temperature cycle to which a component must be subjected in order to obtain desired characteristics in a manufacturing system, such as heat treatment or plastic thermoforming, is prescribed. However, the necessary boundary and initial conditions are not known and must be determined by solving the inverse problem. Similarly, an inverse solution may be needed to complete a given physical problem by determining the unknown boundary conditions. Solutions thus obtained are not unique and optimization is generally needed to obtain results within a small region of uncertainty. This review paper discusses several inverse problems that arise in a variety of practical processes and presents some of the approaches that may be used to solve them and obtain acceptable and realistic results. Optimization methods that may be used for reducing the error are presented. A few examples are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. These examples include the heat treatment process, unknown wall temperature distribution in a furnace, and transport in a plume or jet involving the determination of the strength and location of the heat source by employing a few selected data points downstream. Optimization of the positioning of the data points is used to minimize the number of samples needed for accurate predictions.