We explore the following optimal design problem in 3-D heat transfer: given fixed amounts of two isotropic conducting materials, decide how we are to mix them in a three-dimensional domain so as to optimize a certain cost functional depending on the underlying temperature gradient. By relying on a suitable reformulation of the problem, and examining its relaxation ([1, 2]), easier relaxations for the design problem are obtained in most cases. We provide numerical evidence, based on optimality conditions for the new relaxations, that Tartar's result ([3]) is verified when the target field is zero (also for divergence-free fields) and that optimal solutions can be interpreted as graded materials. In such cases, they are given by first-order laminates somewhat special. This same evidence also holds for a general quadratic functional in the field and as a particular case, for the compliance one.