Analytical solution of heat transfer in a cone made of functionally graded materials

In the current study, the problem of two-dimensional heat conduction in a truncated hollow cone made of functionally graded materials is referred and an exact analytical solution is presented. Functionally graded materials are materials with special production processes in which different thermophysical properties can be gradually changed. In the present study, the properties of a material are modified in accordance with a power function. The thermal boundary conditions are also assumed to be non-homogeneous. The separation of variable (SOV) method is implemented to acquire the exact steady-state temperature distribution. The obtained solution is adequately verified using numerical data. To further demonstrate the ability of the solution, an illustrative case which is exposed to a combination of boundary conditions is studied. In particular, the influences of effective parameters on the temperature distribution are investigated for the current geometry. It is shown that by using functionally graded material more flexible attitudes in terms of temperature distribution are seen. The outcome of this study would be helpful to shed light on the process of designing and optimizing relatively complex geometries. Also, considering the analyticity of the present solution, the results of this study can be useful for a better understanding of the heat transfer mechanisms of functionally graded materials. In the present case, increasing the amount of m and κ, the thermal conductivity increased by about 8 and 2 percent respectively, which would increase the distribution of cone temperature.