Calculation of Some Integrals Arising in Heat Transfer in Geothermics

We calculate some integrals involved in the temperature field evaluation of the ground, heated by a borehole heat exchanger. This calculation allows a faster computation of that component of the temperature field which involves the periodic oscillation of the ambient temperature or the ambient heat flux. 1. Introduction Ground-coupled heat pumps (GCHPs) are an attractive choice of system for heating and cooling buildings [1]. By comparison with standard technologies, these pumps offer competitive levels of comfort, reduced noise levels, lower greenhouse gas emissions, and reasonable environmental safety. Furthermore, their electrical consumption and maintenance requirements are lower than those required by conventional systems and, consequently, have lower annual operating cost [2]. A GCHP exchanges heat with the ground through a buried U tube loop. Since this exchange strongly depends on the thermal properties of the ground, it is very important to have a knowledge of these properties when designing GCHP air-conditioning systems. Methods to estimate them include literature searches, conducting laboratory experiments on soil/rock samples, and/or performing field tests. The in situ tests are based on studying the thermal response of the borehole heat exchanger to a constant heat injection or extraction. The outputs of the thermal response test are the inlet and outlet temperature of the heat-carrier fluid as a function of time. From these experimental data, and with an appropriate model describing the heat transfer between the fluid and the ground, the thermal conductivity of the surroundings is inferred. The results presented in this study are based on a model of the underground heat transfer due to a borehole heat exchanger assuming that it works as a line heat source of finite depth inside the ground. We assume as well a constant power of length unit for the heat source considered and a periodic oscillation of the ambient temperature or heat flux on the ground surface. This paper is organized as follows. Section 2 describes the equations that model the heat transfer inside the ground due to a borehole heat exchanger. Section 3 solves the integrals given in literature for periodic oscillations of the ambient temperature or heat flux, on the ground surface. Section 4 shows that our results agree with the literature in some particular cases, such as the solution on the ground surface and the solution of the quasistationary regime. 2. The Equations Let us consider the heat equation in cylindrical coordinates, in which is the temperature field, On the one