Entropy generation for hydrodynamically and thermally developing laminar now in a circular duct at constant wall temperature is investigated. Optimum temperature difference between fluid inlet temperature and wall temperature is determined for minimum entropy generation rate. An equation for optimum temperature difference at minimum entropy generation rate is given. It is seen that the optimum dimensionless temperature difference is a finite value for a duct length approaching zero for low temperature differences. It is possible to calculate the optimum pressure loss for given wall and inlet fluid temperature differences and corresponding optimum dimensions of a circular duct can be determined with these results. An example for the dimensioning of an air-conditioning heat exchanger tube for given wall and air inlet temperatures and heat transferred is presented.