Mathematical apparatus for determination of homogenous scalar medium thermal resistance

Abstract

Introduction: the article suggests a method for determining a thermal resistance of small and large-sized areas (one-dimensional and multidimensional problems) of wall enclosure. The subject of the study is the thermal resistance of homogeneous scalar medium (homogeneous wall enclosure). The aim is the determination of thermal resistance of a wall structure for areas of arbitrary dimension (by the coordinates xi, where 1 ≤ i ≤ d and d is the area dimension) filled with a scalar (homogeneous and isotropic) heat-conducting medium. Materials and methods: the article used the following physical laws: Fourier law (the value of the heat flow when transferring heat through thermal conductivity) and continuity condition for the heat flow rate leading to the thermal conductivity equation. Results: this method extends the standard definition of thermal resistance. The research proved that the active thermal resistance does not increase with increasing of the area dimension (for example, when switching from a thin shell or plate to a rectangle with length and width of the same order of magnitude). That is the sense of geometric inclusion, i.e., increase of the dimension of an area filled with a homogeneous isotropic medium. Evident expressions are obtained for the determination of active, reactive, and total thermal resistance. It is proved that the total resistance is higher than the active resistance since the reactive resistance is positive, and the wall possesses an ability to suppress the temperature fluctuations and accumulate/give up the heat. Conclusions: the appearance of an additional wall dimension (comparable length-to-thickness ratio) does not increase its active resistance. In the general case, the total thermal resistance exceeds the active thermal resistance no more than four times. Geometric inclusions must be considered in the calculation of wall enclosures that are variant from one-dimensional bodies.