Estimation of heat flow in circular bars of variable diameter

Upper and lower bounds for the heat flow in nonhomogeneous circular bars of variable diameter are presented. The thermal properties may depend on the axial and radial coordinates, and the boundary conditions of the considered heat conduction problem do not depend on the polar angle. The analysed steady-state heat conduction problem is axisymmetric. Equations of Fourier's theory are used to formulate the thermal boundary value problem of heat conduction in nonhomogeneous circular bars with nonuniform cross-section. The computation of the heat flow is based on the concept of overall heat transfer coefficient. The derivation of bounding formulae for the overall heat transfer coefficient is based on a minimum principle and Schwarz's inequality. Six examples illustrate the applications of the derived upper and lower bound formulae how one can use to estimate the heat flow in a nonhomogeneous circular bar with nonuniform cross section.