Optimal shapes and geometries are determined for systems involving liquid and gas coolants. The shape of the channel boundary, channel width, and wall thickness are varied to minimize overall thermal resistance under flow constraints involving pressure drop and pump work. The effect of boundary curvature is studied systematically by employing a parameterized boundary shape that spans from rounded rectangles to ellipses to rounded diamonds. The results indicate that increased channel boundary curvature can decrease the optimal distance between channels, and that the optimal boundary shapes of fully embedded channels can be non-rectangular. In particular, elliptic and nearly elliptic shapes are found to produce equivalent optimal thermal performance as rounded rectangular shapes under practical conditions.

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