Summary: We use the complex variable boundary element method to analyze conjugate heat transfer in solids with cooling passages of general, convex cross-section. The method is well-suited to duct cross-sections with high curvature and high aspect ratios because the whole-domain boundary integrals are path-independent and analytic. The effects of channel boundary curvature on overall heat transfer are quantified for the first time. Then we present shape-constrained optimal solutions involving fixed pressure drop and fixed pump work. Increased channel boundary curvature is shown to decrease the optimal distance between parallel channels by increasing fin efficiency.