This paper treats the macro-scale description of the periodically developed conjugate heat transfer regime, in which heat transfer takes place between an incompressible viscous flow and spatially periodic solid structures through a spatially periodic interfacial heat flux. The macro-scale temperature of the fluid and the solid structures are defined through a spatial averaging operator with a specific weighting function. It is shown that a double volume average is necessary in order to have a linearly changing macro-scale temperature in response to a constant macro-scale heat flux. Furthermore, with the aid of a double volume average, the thermal dispersion source, the thermal tortuosity and the interfacial heat transfer coefficient all become spatially constant in the developed regime. That way, these closure terms of the macro-scale temperature equations can be exactly determined from the periodic temperature part on a unit cell of the solid structures without taking the spatial moments of the solid into account. The theoretical derivations of this paper are illustrated for a case study describing the heat transfer between a fluid flow and an array of solid squares with a uniform volumetric heat source.