An exact analytic solution to the problem of steady heat conduction with convective and radiative
heat transfer normal to the conduction heat flow is presented. The solution is unique as it does not
impose any assumptions on the surroundings and fluid temperature values and addresses all possible
tip boundary conditions. The temperature profiles in the direction of heat conduction are produced for
constant temperature boundary condition at the base and three different boundary conditions at the
tip: adiabatic, constant temperature and radiative/convective heat transfer. Approximate solutions to
the implicit exact solution are also developed. The analytic solutions, exact and approximate for the
adiabatic tip boundary condition, compare very well to experimental data.
