Calculation of non-stationary heat exchange in multi-layer media by means of the theory of Markov chains

Abstract

Calculation of non-stationary heat exchange in multi-layer dissimilar media is the problem, which is often found in many industries including high temperature metallurgic technologies of production and treatment of metals, alloys and their products. The similar problems arise during the design of buildings and structures. The models used for calculation are mainly based on approximate solutions of classical heat conduction equations. The drawback of such models is a complex computational procedure, associated with the need to solve a large number of equations and to define a large number of identification parameters. At present, the calculation of one mode of heat transfer between the sections of a composite structure requires 6–8 hours of operation of a computer of average power. In this regard, the development of discrete models that require less computer time is of current importance. The proposed model of the process is based on the theory of Markov chains. A multi-layer medium is presented as a chain of small but finite cells. Each of them contains a certain amount of heat that can be transferred to the neighboring cells. The part of the transferred heat is directly proportional to the heat conduction coefficient and in inverse proportion to the material heat capacity, material density and the cell length. The matrix model to describe heat transfer between cell in a multi-layer media based on the theory of Markov chains is developed. Construction of the matrix of transition probabilities is described, evolution of the state vectors i.e. distribution of heat and temperature is carried out, and non-uniformity of the heater temperature is taken into account. Comparison of calculated and experimental data has showed the adequate description of the real process using the model. Analysis of identification parameters has given a satisfactory result.