Transient heat transfer analysis of laminated materials based on extended separation of variables

Abstract

In general, when the one-dimensional heat conduction equation is solved by the method of separation of variables, we need to know the governing equations, two boundary conditions and initial condition. Because the thermophysical parameters in different layers of laminated materials are different, the heat conduction model cannot be expressed by the same governing equation. For each layer of laminated material, the boundary condition is unknown. That equation can-not be solved directly by the general separation variable method. In this work the separation of variable method is extended. The temperature field of laminated material's heat transfer is divided into many minute time intervals on the time axis. Based on differential conception, in a minimum time interval, the temperature at the junction of laminated materials can be considered to be proportional to time. Assume that the slope coefficient makes the boundary condition known, then for each layer of laminated materials, the general separation of variables method will be used to solve the temperature field. According to the same temperature and the energy continuity at the junction of laminated materials, one can solve the slope coefficient. The temperature field in the whole time domain can be obtained through cycling. Then the three-layer insulation materials are analyzed by the extended separation variable method. The correctness of the method is verified by comparing the calculated results with those from the finite element method. The influences of the type and thickness of heat insulation layer, heat transfer coefficient, air temperature on the heat insulation are studied. It is found that the thermal conductivity of the thermal insulation layer has a great influence on the insulation. The material with low heat conduction coefficient can enhance the heat insulation effect. The thicker the thickness of the insulation layer, the more slowly the surface temperature of the heat insulation material rises, and the lower the final temperature, the better the insulation effect is. The thicker the thickness of the insulation layer, the smaller the heat flux density of the heat insulation material shell is, and the better the heat insulation effect when the heat transfer reaches a stable state. All calculation results are consistent with physical phenomena. In this work, the analytical method is used to solve the heat transfer problem of laminated materials. Compared with the general numerical methods, the analytical method presents clear physical meaning and high efficiency of operation as well.