Dual variational model of the temperature state of the disk of a unipolar generator

A dual variational form of a mathematical model of the steady process of heat conduction in a rotating disk of a unipolar direct-current generator is constructed. The model contains two alternative functionals that have coinciding stationary points at which these functionals reach the same extreme values (minimum and maximum if the desired temperature distribution in the disk is unique). This property of the functionals makes it possible to estimate the error of the approximate solution of the considered nonlinear heat conduction problem and control its convergence. The features of the radial temperature distribution in the disk are revealed, and the influence of thermal conductivity and electrical resistivity of the disk material (both dependent on temperature) on this distribution is established. The limiting value of the temperature coefficient of electrical resistivity is determined, at which a steady temperature distribution in a disk of a hyperbolic profile is impossible.