A second-law comparison of irreversibilities is used to determine the optimum duct geometry which minimizes losses for a range of laminar flows and constant heat flux. The duct geometries used are circular, square, equilaterally triangular, rectangular with an aspect ratio of 1/2, and sinusoidal with an aspect ratio of root 3/2. Hydraulic diameters are used for the different geometries. The circular geometry is best, especially when the frictional contributions of entropy generation become important. Triangular and rectangular duct geometries are inferior choices for both entropy generation and pumping power.