Influence of Nonuniform Development of Heat Upon the Temperature Distribution in Electrical Coils and Similar Heat Sources of Simple Form

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Abstract
The maximum temperature in an electrical coil of simple shape can be found by measuring the surface temperature and the increase of the electrical resistance under load which yields the mean temperature. This well-known method is based upon the assumption that a coil is a body in which heat is developed almost uniformly all over the volume. But, owing to the increase of the electrical resistance with temperature, more heat per unit volume is developed in the warmer sections of the coil than in the colder ones. The purpose of the present paper is to find the influence of this lack of uniformity in some simple cases. It is further shown how for some usual shapes of coils the maximum temperature can be found approximately from the equations derived for certain simple shapes, and the results for such a case are compared with experimental results taken from literature. The results of the paper refer, but are not restricted to electrical coils. They can be used also for exothermal chemical reactions in cases where steady-state and linear increase of the heat of reaction with temperature can be assumed, and the method can be extended to endothermal reactions. The application to the cases of plate, hollow cylinder, and hollow sphere with different temperature at the two free surfaces will form the subject of another paper.

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