This paper presents the development of the three-dimensional flow architecture of conjugate cooling channels in forced convection with internal heat generation within a solid. Two types of cross-section channel geometries were used. The first involved equilateral triangles with three equal legs in length and all three internal angles of 60°. The second was isosceles right triangles with two legs of equal length and internal angles of 90°, 45° and 45°. Both the equilateral triangle and isosceles right triangle are special case of triangle that can easily and uniformly be packed and arranged to form a larger construct. The configurations were optimised in such a way that the peak temperature of the heat generating solid was minimised subject to the constraint of a fixed global volume of the solid material. The cooling fluid was driven through the channels by the pressure difference across the channel. The degrees of freedom of the channels were aspect ratio, hydraulic diameter and channel to channel spacing ratio. The shape of the channel was allowed to morph to determine the best configuration that gives the lowest thermal resistance. A gradient-based optimisation algorithm was applied in order to search for the best optimal geometric configurations that improve thermal performance by minimising thermal resistance for a wide range of dimensionless pressure difference. The effects of porosities, applied pressure and heat generation rate on the optimal aspect ratio and channel to channel spacing are reported. It was found that there are unique optimal design variables for a given pressure difference. The numerical results that were obtained were in agreement with the theoretical formulation using scale analysis and method of intersection of asymptotes. Results obtained show that the effects of applied dimensionless pressure drop on minimum thermal resistance were consistent with those obtained in the open literature.