Optimal placement depth for air–ground heat transfer systems

Abstract

Heat transfer between the air and the ground is important for a variety of applications including buried pipes and cables, ground source heat pumps, and building insulation. A recent application considered a ground-source heat engine using a thermoelectric module sandwiched between two heat exchangers to produce a very small amount of electricity from the daily temperature difference between the air and the ground. For a given device, the power produced is proportional to the square of the temperature difference across the module and it is desirable to maximize the power. For a sinusoidal (in time) temperature boundary condition, the solution to the heat equation for a semi-infinite body yields an attenuated sinusoid shifted in phase from the boundary condition. The phase shift and attenuation both increase with depth, so while the phase shift may be exploited to increase the temperature difference, the attenuation tends to reduce this effect. In this paper, an optimal depth for the ground side heat exchanger is derived which maximizes the square of the difference between the surface temperature and the ground temperature. The maximum temperature difference in the optimized case is 7% higher than what would be obtainable from an infinitely deep heat exchanger. The range of depths in which the temperature difference is an improvement over the infinitely deep case is derived and presented. The improvement is asymmetric so that away from the optimum depth, it declines rapidly with decreasing depth while it declines slowly with increasing depth. An alternative optimization that reflects the temperature drop through the heat engine device is derived and presented. The optimal depth decreases by slightly more than half in this case. The effect of a convection boundary condition at the free surface is explored in terms of a Biot number for the ground. The Biot number affects both the attenuation and the phase shift. The attenuation is approximately doubled for Biot of order unity, and becomes inversely proportional to Biot number if it is much less than unity. The phase shift varies from an additional shift of 45° for Biot much less than unity to an additional shift of 27° for Biot of order unity. For Biot much greater than unity, the temperature boundary condition applies.