Blundell Thermal Physics · Problem 10.8
The thermal diffusion equation
Physics — Statistical Mechanics · heat-flow-concentric-cylinders
Problem
The space between two concentric cylinders is filled with material of thermal conductivity . The inner (outer) cylinder has radius () and is maintained at temperature (). Derive an expression for the heat flow per unit length between the cylinders.Intuitive explanation
Heat flow in cylindrical geometry is fundamentally different from the flat-plate case because the conducting area changes with radius—the farther you go from the axis, the more surface there is to conduct heat through. The key insight is to use Fourier's law in cylindrical coordinates and recognize that in steady state, the heat flow rate must be constant at every radius. This constraint lets you separate variables and integrate to find how temperature varies with , ultimately revealing that thermal resistance in cylindrical geometry depends logarithmically on the radius ratio.
Intuitive explanation
Heat flow in cylindrical geometry is fundamentally different from the flat-plate case because the conducting area changes with radius—the farther you go from the axis, the more surface there is to conduct heat through. The key insight is to use Fourier's law in cylindrical coordinates and recognize that in steady state, the heat flow rate must be constant at every radius. This constraint lets you separate variables and integrate to find how temperature varies with , ultimately revealing that thermal resistance in cylindrical geometry depends logarithmically on the radius ratio.