Analysis of Transient Heat Conduction in a Hollow Sphere Using Duhamel Theorem

Abstract

In this article, analytical modeling of two-dimensional heat conduction in a hollow sphere is presented. The hollow sphere is subjected to time-dependent periodic boundary conditions at the inner and outer surfaces. The Duhamel theorem is employed to solve the problem where the periodic and time-dependent terms in the boundary conditions are considered. In the analysis, the thermophysical properties of the material are assumed to be isotropic and homogenous. Moreover, the effects of the temperature oscillation frequency, the thickness variation of the hollow sphere, and thermophysical properties of the sphere are studied. The temperature distribution obtained here contains two characteristics, the dimensionless amplitude (A) and the dimensionless phase difference (ϕ). Moreover, the obtained results are shown with respect to Biot and Fourier numbers. Comparison between the present results and the findings from a previous study for a hollow sphere subjected to the reference harmonic state show good agreement.

Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

“…The traditional Fourier law implies an unbounded pace for thermal disturbance propagation [1] , [2] , which disagrees with some experimental observations [3] , [4] , [5] . In this model, any thermal disturbance imposed on a medium is instantly perceived through the entire body [6] , [7] .…”

Numerical Simulation Analysis of Unsteady Temperature in Thermal Insulation Supporting Roadway

“…Based on the basic assumption, the heat dissipation problem of surrounding rock along the axis is simplified [8]. According to the law of conservation of energy and Fourier law, the integral equation of unsteady heat conduction in the temperature field of roadway surrounding rock is established as [13]…”

“…In recent years, finite element method and finite volume method are also used to simulate the temperature field of roadway surrounding rock [8,9]. With the wide application of numerical calculation methods, the distribution of temperature field [10,11], the analysis of influencing factors [12,13], and the characteristics of heat and moisture exchange [14][15][16] are analyzed. In the site investigation and experimental testing, similar experimental platform for temperature field of roadway surrounding rock was established, and the unsteady state process of temperature field of roadway surrounding rock was analyzed [4,17].…”

An experimental investigation of transient heat transfer in surrounding rock mass of high geothermal roadway

“…Compared with site investigation method, finite element method and finite difference method are more commonly used to study the temperature field in the roadway rock mass. More research results were found by the numerical calculation method concerning the rock mass temperature field prediction [3,11], influence factors of temperature distribution in the rock mass [12][13][14][15], and the disturbed domain of the temperature in the surrounding rock mass [16]. Most of the existing researches on the rock mass temperature distribution of the roadway, however, were not validated in laboratory.…”