Joule-heating power dissipation in a Type-II superconductor tube

We have calculated the radial distribution of Joule-heating power dissipation in the wall of a hollow tube of Nb---25-at.% Zr for changing longitudinal applied magnetic field. The super-conducting critical current density in the cylinder was assumed to obey the formula $J_c={\alpha }_c\times \frac{({\mu }_0{H}_{c2}-B)}{(B_0+B)}$, as reported elsewhere. The power-density distribution across the tube wall was calculated as the field increased and decreased from a series of cooled-in back-ground fields representing the idealized physical situation following flux jumps. It was found that at each radial position in the tube the power-density function is far from monotonic. The function has a pronounced discontinuous amplitude increase and may have from one to as many as five changes in slope. For each radial position the power density can be displayed as a three-dimensional surface. Relative temperature changes to be expected in well-cooled and poorly cooled tubes have been estimated from the power-density curves.