Electrical Analogy for Two-Dimensional Conduction

Steady-state electric conduction in a homogeneous material of constant resistivity is analogous to steady-state heat conduction in a body of similar geometric shape. For two-dimensional electric conduction the Laplace equation applies  

Once the electric potential is impressed on the paper, an ordinary voltmeter may be used to plot lines of constant electric potential. With these constant-potential lines available, the flux lines may be easily constructed since they are orthogonal to the potential lines. These equipotential and flux lines have precisely the same arrangement as the isotherms and heat-flux lines in the corresponding heat-conduction problem. The shape factor is calculated immediately using the method which was applied to the curvilinear squares. 

There is a myriad of analytical solutions for steady-state conduction heat-transfer problems available in the literature. In this day of computers most of these solutions are of small utility, despite their exercise in mathematical facilities. This is not to say that we cannot use the results of past experience to anticipate answers to new problems. But, most of the time, the problem a person wants to solve can be attacked directly by numerical techniques, except when there is an easier way to do the job. As a summary, the following suggestions are offered  

When tackling a two- or three-dimensional heat-transfer problem first try to reduce it to a one-dimensional problem. An example is a cylinder with length much larger than its diameter. 

If possible, select a simple shape factor model which may either exactly or approximately represent the physical situation. See comments under items 4 and 5. 

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