Search for: All records

Eshelby’s equivalent inclusion method (EIM) has been formulated to solve harmonic heat transfer problems of an infinite or semi-infinite domain containing an inclusion or inhomogeneity. For the inclusion problem, the heat equation is reduced to a modified Helmholtz’s equation in the frequency domain through the Fourier transform, and the harmonic Eshelby’s tensor is derived from the domain integrals of the corresponding Green’s function in the form of Helmholtz’s potential. Using the convolution property of the Fourier space, Helmholtz’s potential with polynomial-form source densities is integrated over an ellipsoidal inclusion, which is reduced to a one-dimensional integral for spheroids and an explicit, exact expression for spheres. The material mismatch in the inhomogeneity problem is simulated by continuously distributed eigen-fields, namely, the eigen-temperature-gradient (ETG) and eigen-heat-source (EHS) for thermal conductivity and specific heat, respectively. The proposed EIM formulation is verified by the conventional boundary integral method with the harmonic Green’s function and multi-domain interfacial continuity, and the accuracy and efficacy of the solution are discussed under different material and load settings. 

Abstract The Green’s function of a bimaterial infinite domain with a plane interface is applied to thermal analysis of a spherical underground heat storage tank. The heat transfer from a spherical source is derived from the integral of the Green’s function over the spherical domain. Because the thermal conductivity of the tank is generally different from soil, the Eshelby’s equivalent inclusion method (EIM) is used to simulate the thermal conductivity mismatch of the tank from the soil. For simplicity, the ground with an approximately uniform temperature on the surface is simulated by a bimaterial infinite domain, which is perfectly conductive above the ground. The heat conduction in the ground is investigated for two scenarios: First, a steady-state uniform heat flux from surface into the ground is considered, and the heat flux is disturbed by the existence of the tank due to the conductivity mismatch. A prescribed temperature gradient, or an eigen-temperature gradient, is introduced to investigate the local temperature field in the neighborhood of the tank. Second, when a temperature difference exists between the water in the tank and soil, the heat transfer between the tank and soil depends on the tank size, conductivity, and temperature difference, which provide a guideline for heat exchange design for the tank size. The modeling framework can be extended to two-dimensional cases, periodic, or transient heat transfer problems for geothermal well operations. The corresponding Green’s functions are provided for those applications.