This paper presents spatially averaged temperature equations for modelling macroscale heat transfer in periodic solid structures such as fin and tube arrays. The governing equations for the periodically developed heat transfer regime in isothermal solids are derived. It is shown that the appropriate macro-scale temperature in the periodically developed heat transfer regime is obtained by averaging the temperature with a specific weighting function which is adapted to the temperature decay rate. This matched weighting function allows the representation of the macro-scale interfacial heat transfer and thermal dispersion source by means of a spatially constant interfacial heat transfer coefficient and thermal dispersion vector, which both can be calculated from the periodic rescaled temperature on a unit cell of the solid structures. Moreover, it is proved that for small temperature decay rates, the matched weighting function yields the same macro-scale description as repeated volume averaging. The theoretical framework of this paper is applied to a case study, describing the heat transfer between a fluid and an array of solid cylinders at constant temperature.
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.
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