Optimal profile of heat transfer pin fins under technological constraints

Abstract

Direct optimal control methods are used. The pin fin's volume is minimized for given heat flux value. Schmidt's criterion is not adopted. The optimal pin fin profile is neither parabolic nor circular. It consists of two regions. In the first region, close to the basis, the pin thickness decreases linearly. In the second region the pin thickness is constant or may decrease, depending on thermal loads and operation. The optimal control solution is usually singular but may be approximated by a bang–bang solution. The Schmidt criterion works better at larger heat flux values. Results obtained for specific assumptions adopted in the paper (fluid temperature 300 K, transverse Biot number ranging between 0.00041 and 0.041) are summarized next. For very small values of the heat flux (=0.1 W) the reduced minimum pin fin volume (i.e. the ratio between pin fin volume and the volume of a cylinder of similar length) is about one tenth. The technology and design constraints have important effects on the optimal pin fin's profile. The reduced minimum pin fin volume decreases from 0.30 to 0.20 when the maximum slope of the pin fin profile increases from 1 to 100. The reduced pin fin volume is a minimum minimorum for a maximum allowable pin fin temperature ranging between two and three times the fluid temperature.