Approximation of an optimal convective heat transfer coefficient

The problem studied in this paper is that of a heat source (electronic chip) placed on the top surface of a flat thermal spreader which is cooled by convection on the opposite surface. An optimal convection heat transfer coefficient yielding maximal heat removal from the chip is found using an optimal control technique. We control the solution of the heat equation with the convective boundary condition, taking the heat transfer coefficient as the control. We show the existence and uniqueness of an optimal control. A conjugate gradient method is used to solve the optimal control problem. The results show that the temperature distributions corresponding to the controlled solution are lower and display a flatter profile than those corresponding to the uncontrolled solution. This study can provide guidance in designing micro heat pipe heat sinks, which have emerged as an effective technique for cooling electronic components.