In this paper we address the optimization of a heat sink formed by parallel circular or non-circular ducts in a finite volume. The flow is considered to be laminar and steady and the fluid properties are assumed to be constant. Results for optimum dimensionless thermal length, optimum hydraulic diameter and maximum heat transfer rate density are presented for five different duct shapes subjected to a fixed pumping power constraint. Simple equations for the calculation of these optimum values are presented, and the influence of the local pressure drops at the inlet and outlet plenums, and of Prandtl number is discussed. The optimization results are then extended for the case of pumping power minimization at fixed heat transfer density. (C) 2010 Elsevier Ltd. All rights reserved.