Thermal networks considering graph theory and thermodynamics

Abstract

Heat transfer in solids may be dealt with the heat equation, which is a partial differential equation, from which different analytical solutions for the study of heat transfer throughout solids and at their surfaces may be found. This implies the resolution of a distributed parameter model. On the other hand, the possibility of considering the thermal-electrical analogy is usually assumed, this being based mainly on the similarity between Ohm’s and Fourier’s laws under the assumption that the different variables used in electrical networks may be regarded as analogues to the thermal network variables. This implies the use of a lumped parameter model, which may be represented as a system of differential and algebraic equations (DAE) linked to the graphical representation of the thermal network. In this latter case the limitations of such analogy for describing heat flow should be taken into account. Therefore, it would be important to consider thermal networks independently of the thermal-electrical analogy. For this, thermal networks may be built as particular cases of directed graphs, within graph theory, since thermal networks may have physical meaning without the electrical analogy. The interpretation of a graph as a thermal network may directly use physical principles of heat and thermodynamics. This enables us to propose an alternative to the use of the electrical analogy, since electrical networks are only a particular application of graph theory consistent with electromagnetic laws which are not analogous to thermodynamic laws. Furthermore, the construction and the use of thermal networks for analysing heat transfer problems may be simplified from this perspective.