Analytical approximations of the shape factors for conductive heat flow in circular and regular polygonal cross-sections

The heat flux in steady heat conduction through cylinders whose cross-section has an inner or an outer contour in the form of a regular polygon or a circle is considered. To calculate the shape factor the temperature field is determined. Three cases are considered: (a) temperature field for hollow prismatic cylinders bounded by isothermal inner circles and outer regular polygons, (b) temperature field for hollow prismatic cylinders bounded by isothermal inner regular polygons and outer circles, (c) temperature field for hollow prismatic cylinders bounded by isothermal inner and outer regular polygons. The boundary collocation method in the least squares sense for solving appropriate boundary value problems is used. By means of nonlinear approximation (Marquardt method), for the three considered geometry cases, the simple analytical formulas for the shape factors are proposed.