Smooth and non-smooth optimal pin fin profiles beyond the Schmidt optimality assumption and “length-of-arc” approximation

A parametric model of the pin fin shape has been developed. The pin fin profile optimization is performed without adopting the “length-of-arc” approximation. The model is based on the accurate treatment of the arc length. The heat transfer coefficient h is not constant but variable along the length of the pin fin. The objective is to minimize the pin fin volume for given value of the transferred heat flux. Direct optimal control methods are used. The model is validated for a specific case compatible with the application of the Schmidt optimality assumption. Results are shown for pin fins with circular cross section. Both forced convection and natural convection have been considered. Generally, the optimal profile of the pin fin consists of two regions; in the region near the pin fin base, the thickness is constant or decreases almost linearly while in the region near the pin fin tip the thickness is constant or decreases, depending on the thermal load. Including (instead of omitting) in calculation the accurate description of the arc length and using variable (instead of constant) h may yield thinner and shorter pin fins. The main conclusion is that, by including in calculation the accurate treatment of the arc length, non-smooth pin fin profiles may be obtained. Such profiles consist of a series of steps and are more effective than the smooth profiles.