Analytical solutions of the steady or unsteady heat conduction equation in industrial devices: A comparison with FEM results

Heat transfer is one of the most salient and fundamental research areas for any engineer, due to its ubiquity. Today the energy efficiency requirements are becoming more and more demanding. This motivates engineers to continuously improve the efficiency of heat transfer processes. For such analysis nowadays, the common and the popular practice to infer the temperature field is now commercially available in computer codes. Analytical solutions for the temperature field are also available under the assigned conditions such as Dirichlet, Neumann or Robin, if the thermal conductivity is constant and isotropic. In the cases where conductivity is anisotropic and strongly dependent by temperature or the material is not homogenous, the exact solution of the energy conservation in the body is not possible due to high non linearity in the equations. Despite the complexity of many engineering structures, the present work is undertaken to demonstrate that the reduction to a simpler version of more complex heat conduction equations is possible and the exact analytical solution is comparable with the approximate finite element solution. The topic of the present research study is the resolution of the problems in various engineering fields through the analysis of conduction heat transfer in rigid bodies, transition bodies, steady and transient bodies and the provision of analytical solutions with graphical representation of the results. Namely; in an electrolytic capacitor, a food container and a gas turbine blade. The modelling of them plays an important role because excessive temperatures drastically reduce the lifetime of capacitors, turbine vanes and blades. In the food sterilization it is necessary to know the thermal wave behaviour in order to reduce its associated costs or to guarantee the sterilization time. The last part of this study is the evaluation of the numerical simulation results in order to make a comparison with the analytical results and, when possible, with experimental data. For such analysis a finite element method has been utilized by both commercial and free software; namely, ANSYS™ and FreeFem++. Very good agreements are obtained between both of them.