Heat transfer to constant-property, fully developed, laminar flows in circular-segment ducts with uniform wall temperature (T) has been analyzed. Besides representing a compact surface, the segment duct geometry models the flow cross section of a circular tube with a straight-tape insert. Two variations in the T thermal boundary condition are considered: constant axial and circumferential wall temperature, and constant temperature on the curved surface but an adiabatic flat wall. These two conditions model the extremes of the fin effects of a straight-tape insert, i.e., 100% and zero fin efficiencies, respectively. Numerical solutions, obtained by using finite difference techniques, are presented for both the velocity and temperature fields. The isothermal friction factors are in excellent agreement with analytical solutions reported in the literature. The Nusselt number results for the two thermal boundary conditions are presented for different segment shapes, 0° ≤, 6 ≤, 90°, and they represent the lower limits of the heat transfer enhancement due to twisted-tape inserts.