Analytical versus Numerical Calculations of Physical Problems, The Benefits of its Combination, Example: Application of the Heat Transfer Equation in Electrical Engineering

Abstract

Due to the available powerful calculation tools, in present day engineering physical and technical problems are more and
more solved with numerical methods. Analytical calculations are left for some few basic questions to show only the physical laws behind. The potential of the possibilities of numerical approaches need not to be emphasized.
The disadvantage of the pure application of numerical approaches, however, is the fact, that
the physicals laws behind are not so easy to visualize,
the results art not so easy to generalize, and
the storage of the information requires mostly a lot of data.

This paper would like to show at some examples the advantages the combination of both methods. One example is the application of the heat transfer equation at conductors and cable harness for vehicles, like cars or airplanes. Up to now the selection of cables is based on data, which were not particular optimized for mobile application where the length is typically short and low weight is important. More concern about this matter, new materials and diverting shapes require the optimization of the possible electrical load under thermal considerations. Other examples are the calculation of the melting behavior of electrical fuses or the thermal conditions in automatic electrical heated production processes.

The key part of this approach is the calculation of the heat transfer by the Finite Volume Method (FVM) and the approximation of the calculated data by the so-called "simplified equations". These simplified equations were received by analytical calculations of the basic heat conduction equation. The required adaptation of the numerical and measured results were done with properly adapted fitting algorithms on the basis of the elaborated analytical solutions, a process which was leading to an enormous reduction of data. As a result it became possible to describe the applied tasks by a few characteristic constants.
Another approach for an analytical solution with a numerical calculation process is the determination of the so-called "properties of mixed magnitudes". As an example this principle has been applied for the numerical calculation of electrical multi conductor containing cables. This way allowed predicting the thermal behavior of any cable harness with the required precision.