Macro-scale heat transfer in periodically developed flow through isothermal solids

Abstract

This paper presents spatially averaged temperature equations for modelling macro-scale 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.