This two-part paper addresses the design of a U-bend for serpentine internal cooling channels optimized for minimal pressure loss. The total pressure loss for the flow in a Ubend is a critical design parameter, as it augments the pressure required at the inlet of the cooling system, resulting in a lower global efficiency. In this first part of the paper, the design methodology of the cooling channel is presented. The minimization of the total pressure loss is achieved by means of a numerical optimization method that uses a metamodel-assisted differential evolution algorithm in combination with an incompressible Navier-Stokes solver. The profiles of the internal and external side of the bend are parameterized using piece-wise Bezier curves. This allows for a wide variety of shapes, respecting the manufacturability constraints of the design. The pressure loss is computed by the Navier-Stokes solver, which is based on a two-equation turbulence model and is available from the open source software OpenFOAM. The numerical method predicts an improvement of 36% in total pressure drop with respect to a circular U-bend, mainly due to the reduction of the separated flow region along the internal side of the bend. The resulting design is subjected to experimental validation, presented in Part II of the paper.