This paper is a proposal to design flow structures with maximal heat transfer rate per unit volume, by shaping each duct so that it fits optimally on the body of the convective flow. Optimally shaped ducts can be assembled into larger constructs. Two examples are given. In the first, a heat-generating strip is cooled inside a duct of rectangular cross-section. The duct geometry has two degrees of freedom, which can be selected so that the fixed duct volume packs a maximum of heat transfer rate. In the second example, the duct is a tube with isothermal internal surface, and the flow is sufficiently slow so that boundary layers do not form inside the duct. Once again, the duct aspect ratio can be optimized for maximal heat transfer rate density. Further improvements can be sought by endowing the duct geometry with more degrees of freedom. (C) 2003 Elsevier Science Ltd. All rights reserved.