Mathematical Determination of Transient Temperatures in Solids (Classic Reprint)

Excerpt from Mathematical Determination of Transient Temperatures in Solids

In Sections 1-7 we formulate the mathematical prob lems and the numerical procedures which can be used to evaluate the solutions of these problems.

In Section 10 we present a description of the tech niques that were employed to make the large-scale auto matic machine computation feasible on the Mark I. In ad dition an analysis of the pertinent operational and nu merical procedures is given which would be of use in the future should other problems of this type be tackled.

In order to measure the temperature in the steel wall of a rocket, a thin cylindrical thermocouple is inserted into a hole that was drilled through the outer surface of the wall. The conductivity and the heat capacitance of the various parts of the thermocouple are different from those of steel and consequently the temperatures which are measured are not those which would obtain in the wall when the thermocouple is absent. The purpose of this report is to mathematically determine the difference between the measured temperature and the actual temperature.

We consider a simplified physical model. A steel cylinder of radius 5l and height 5l has a concentric hole of radius L and depth 4l. Into this hole a thermocouple of radius L and height 4l is inserted. The thermocouple is composed of a nonconducting cylindrical outer shell of thickness and a conducting cylindrical core of radius (figure We assume that, except for the hot wall of the rocket, all of the outer surface is nonconducting.

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