Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

Highlights

Analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying cylindrical heat sources.

The model is exact at all radial points, highly elegant, robust and easy to implement in computer codes.

The proposed analytical solution is computationally very efficient.

The model can be utilized for forward heat flow analysis and design of geothermal borehole heat exchangers.

The model can be utilized for inverse calculation of material parameters.

Abstract

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.

Keywords

  • Heat equation;
  • Cylindrical heat source;
  • Geothermal system;
  • Modified bessel series;
  • FFT;
  • Borehole heat exchanger BHE

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