Optimum fin shape

Abstract Using the steady-state nonlinear differential equation which controls the temperature distribution in a fin of variable cross-section that is exchanging heat with the surroundings by convection and radiation, and an assumed form of the temperature distribution, an expression for the half fin height is developed, and the volume of the resulting fin is found. The total volume of the fin is optimized with respect to the temperature distribution parameter, and the half fin height thus found yields a minimum volume fin. The general solution yields the Schmidt solution for a simple convection fin, and the Wilkens solution for a simple radiation fin. An example problem is solved to find the half fin height for a minimum volume fin that is exchanging heat with the surroundings by convection and radiation.