Temperature distribution along a fiber embedded in a matrix under steady state conditions

The temperature distribution along a high thermal conductivity bar (fiber) embedded in a low thermal conductivity half-space (matrix) subjected to an axial differential of temperature is studied. It is assumed that the fiber and matrix are perfectly bonded along the entire interface between them. The system is assumed to be in steady state condition and no heat is generated internally. An approximated analytical solution to the problem based on the heat conduction equation, the principle of conservation of energy and the idea of boundary layer is presented. The problem is also solved numerically by means of the finite element method using commercial software. The results obtained by both approaches, analytical and numerical, are compared. The discrepancy between the two approaches appears very small in most of the cases although substantial relative error can be found at specific points in specific cases.