Non-dimensional finite element formulation for thermal problems

The use of dimensional analysis and dimensionless parameters is very common in the field of heat transfer; nevertheless the concept of non-dimensional finite element formulation has been applied to a limited type of thermo-fluid problems. The non-dimensional finite element method should provide the dimensionless solution for a given problem. The aim of present work is to develop a non-dimensional thermal finite element for getting dimensionless solution of the problems that do not have a closed form solution. An example is a fin (or extended surface) design. Fin efficiency is a performance characteristic that can be used as design criterion; thus closed form dimensionless solutions for fin efficiency are available in the literature. The results are for different geometry, single material fins. In case, if the fin problem has some geometric and/or material complexities then closed form solutions are not available and finite element approach can be used. However, the obtained finite element solution would not be in dimensionless form. For example, no closed form solutions are available for variable thickness composite fins (i.e. a fin having a base material with a coating over its surface), and the literature shows that finite element solution has been used to study thermal performance of the variable thickness composite fins. Therefore, non-dimensional finite element approach can be applied to directly obtain the dimensionless solution for the problem. The current work consists of presenting a non-dimensional finite element formulation for thermal problems. The element formulation is first validated by solving a test case study that has known closed form solution. The objective is to demonstrate the usefulness of the non-dimensional finite element approach by obtaining dimensionless finite element solutions for some applied problems that do not have a closed form solution.