Radiative transfer of energy in the core of a heated tube

Abstract

While investigating the temperature distribution along a thin-walled tube heated by passing an electric current through it in vacuowe had naturally to calculate the radiative transfer of energy occurring in the hollow core of the tube. Taking the tube to be a circular cylinder of diameter D, and considering an annular ring of width dx, the net gain in energy by the ring due to radiative transfer can be readily calculated. If the ring is not too near the ends, this is found to be equal to : 4πσD³T² (dT/dx)² dx + 4/3 πσD³T³ (d²T/dx²)dx which may be expressed in the form: d/dx (4/3 (πσD³T² dT/dx)dx where σ is Stefan's constant of radiation, and the emissivity of the walls is taken to be unity. It is as though the hollow core had a thermal conductivity of magnitude : X_r = 16/3 σDT³.