The distribution of temperature along electrically heated tubes and coils. I. Theoretical

Abstract

The distribution of temperature along a filament electrically heated in vacuo has been studied in detail in previous papers, both theoretically and experimentally. The investigations are extended in the present paper to the case of a thin-walled tube. The major new factor that appears here is the radiational transfer of energy in the core of the tube, and if one can evaluate the rate of gain in energy by a given annular ring on this account one can readily formulate the differential equation governing the distribution of temperature along the tube. Taking ε to be the emissivity, and hence also the absorptivity, of the surface, and taking the fraction (1 - ε) of the radiation incident on the surface that is not absorbed by it to be specularly reflected, we have calculated the radiational gain by the annular ring per second; the expression consists of two terms, proportional to (dT/dx)² and to d²T/dx² respectively, and their coefficients point to a temperature-dependent thermal conductivity of the core equal to 16/3σDT³(2 - ε )/ε . It is as though the conduction is due to the thermal diffusion of the photons, and they had a mean free path equal to the diameter D of the tube, enhanced by a factor (2 - ε )/ε as a result of the specular reflexions, in the same manner in which the `coefficient of slip' of the molecules of a rarefied gas in its passage through a narrow tube is enhanced by the specular reflexions of the molecules from the walls of the tube. The expression for the conductivity of the core bears a close analogy to the corresponding expression for other transport phenomena in which the mean free path of the diffusing particle is limited by the dimensions of the medium or of the enclosure, e.g. the thermal conductivity of a hot gas in a narrow tube due to the diffusion of the photons emitted by the molecules, or the thermal conductivity of a dielectric cylinder at low temperatures due to the diffusion of thermal phonons. Though the differential equation determining the temperature distribution along a tube is more complicated than that for a filament, a practically general solution can be obtained; it is found to be similar to that for the filament, except that the natural length is now considerably greater, and the longitudinal variation of the temperature considerably flatter, than in the filament. The case of a closely wound coil is very similar to that of the tube, except that the conductivity through the material of the walls is now through the wire and hence much smaller than in the tube.