Optimum shape and dimensions of ducts for convective heat transfer in laminar flow at constant wall temperature

One knows that a certain geometry exists for a given pressure loss in a duct, where maximum heat transfer occurs. In this work, the maximum heat transfer and the optimum geometry for a given pressure loss have been calculated for forced convective heat transfer in different duct shapes for laminar flow conditions. Simple equations which enable calculation of these optimum values for all Pr numbers and for all shapes of duct cross-sectional areas have been derived.