The distribution of temperature along a thin rod electrically heated in vacuo. I. Theoretical

Abstract

The paper describes a detailed investigation of the distribution of temperature along a thin rod or wire electrically heated in vacuo. The distribution is determined uniquely by three parameters, namely, (1) the temperature at the centre of the rod, (2) that at the centre of a similar infinitely long rod heated by the same current, and (3) a simple constant that involves directly the cross-section of the rod, its thermal conductivity and the emissivity of its surface. Though the integral that determines the temperature distribution cannot be evaluated directly, it is shown that it can be expanded as a convergent power series. The rod may be divided for this purpose into two well-defined regions, region A comprising a certain length in the middle of the rod, and region B consisting of the portions outside A, the appropriate series for the evaluation of the integral being different in the two regions. The temperature coefficients of the physical constants involved can also be taken into account in the evaluation. The temperature distribution in the A-region is found to be parabolic. Hence when the rod is too short to have any B-region at all, the theoretical relations become particularly simple, and are also practically independent of the finite temperature coefficients of the physical constants involved. The distribution in the B-region is found to be practically the same as in the corresponding end-portions of an infinitely long rod, except near the upper limit of the B-region when this region is very short. An analytical expression is also obtained for the temperature at the centre of the rod as a function of its length. The criterion for the validity of the well-known logarithmic formula for temperature distribution is shown to be much less stringent than is generally taken to be; the range of its applicability is thus extended far into the B-region even when the rod is not long. The additive constant in the formula, which previously had to be left undetermined, also gets evaluated now.