The distribution of temperature along a thin rod electrically heated in vacuo. V. Time lag

Abstract

On the basis of the expressions obtained in parts I and II of this series for the distribution of temperature in the steady state along a filament electrically heated in vacuo, the growth of temperature accompanying a small increase in the heating current is investigated in the present part. Over a considerable region about the centre of the filament, which is the region of practical interest, it is found that to a close approximation the growth of temperature can be completely represented by a simple exponential law involving a single relaxation time, whose magnitude is readily calculated. This method of investigating the time lag, which is general and applicable to any filament, is compared with the well-known method of Fourier expansion developed by Straneo for the special case where the temperature everywhere in the filament is only slightly higher than the room temperature, and hence the loss by radiation conforms to Newton's law of cooling. Each of the Fourier terms is assigned in his method a separate relaxation time that will make the term separately satisfy the differential equation and the boundary conditions. In principle the Fourier method also should be applicable to any filament. But the actual temperature distribution is in general too complicated for an analytical Fourier expansion. In the special case treated by Straneo the temperature distribution over practically the whole length of the filament is parabolic. The actual distribution near the centre of any filament is also known to be parabolic. Hence a comparison of the results obtained by the two methods in the above special case suggests a convenient adaptation of the Fourier method also to the calculation of the time lag near the centre of any filament. The adaptation lies essentially in the use of a certain effective length to determine the period of the Fourier expansion, instead of the actual length generally used. The magnitude of this length is obtained from the results of the present investigation. The distinction between the two lengths is not significant in the special case treated by Straneo, but it is in other cases. Though the occurrence of a single effective relaxation time is not directly obvious from the Fourier expansion, it is shown to follow from it as a close approximation. This result is convenient for practical application. For a given central temperature the relaxation time is found to vary inversely as the ratio of the surface to the volume, and is therefore smaller for a ribbon filament than for one of circular cross-section, as observed by Prescott & Morrison. For a given central temperature and length, the ribbon filament is found to approximate closer than one of circular section, to an infinitely long one. The variation of the relaxation time near the centre with the length of the filament is investigated in some detail.