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.


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.

The Equations
Let us consider the heat equation in cylindrical coordinates, in which T r, z, t is the temperature field, On the one hand, the thermal parameters of 2.1 are the ground thermal diffusivity k m 2 s −1 , the heat flow per length unit Q z Wm −1 , and the volumetric heat capacity of ground C Jm −3 K −1 . On the other hand, the geometry of the heat source, depicted in Figure 1, has been taken into account introducing the Heaviside function H for its finite length L and the Dirac delta distribution δ for its infinitely thin width. Equation 2.1 is subject to to the initial condition where T 0 is the undisturbed ground temperature and k geo Km −1 is the geothermal gradient. We may consider that 2.1 is subject to a Dirichlet boundary condition, which represents a time-dependent temperature on the surface z 0, or to a Neumann boundary condition, which represents a time-dependent heat flux on the surface z 0, Figure 1: Borehole heat exchanger model.
where λ Wm −1 K −1 is the ground thermal conductivity. We split the Cauchy problem posed in 2.1 , 2.2 , and 2.3 or 2.4 in three terms 3 , so that and where the functions v 0 , v s and v d satisfy the Cauchy problems presented below.

The Problem for v d
The partial differential equation that v d r, z, t satisfies is subject to a homogeneous initial condition, and to a homogeneous Dirichlet boundary condition, or to a Neumann boundary condition, homogenous as well, Mathematical Problems in Engineering The solution to 2.6 , 2.7 , and 2.8 is 4 2.10 and the solution to 2.6 , 2.7 , and 2.9 is that we will present in a future work.

The Problem for v 0
The partial differential equation that v 0 z, t satisfies is and to a Dirichlet boundary condition, v 0 0, t 0, t ≥ 0, 2.14 or to a Neumann homogeneous boundary condition The solution to 2.12 , 2.13 , and 2.14 is 5,

Harmonic Analysis of the Boundary Condition
We may consider that on the ground surface we have a periodic temperature or heat flux oscillation, due to daily or annual cycles. Therefore, ψ t is a periodic function of a certain period τ, continuous and bounded, that we may expand in Fourier series ψ t a 0 2 ∞ n 1 a n cos nωt b n sin nωt , 3.1 where the frequency is ω 2π/τ, and the coefficients a n and b n are given by where n ∈ AE.

Dirichlet Boundary Condition
where erfc z 1 − erf z is the complementary error function. The integrals given in 3.3 for t ≥ 0 may be rewritten as where we have defined In order to calculate 3.5 , let us apply the integral 6, Equation 7.4.33

3.7
Finally, substituting 3.7 in 3.4 , we obtain for z, t ≥ 0, 3.8 where we have defined As far as we know, the integrals given in 3.8 are not reported in the literature.

Neumann Boundary Condition
Integrating by parts and performing the substitution ζ z/2 √ ku, considering z, t ≥ 0, we may calculate the integral given in 3.10

Mathematical Problems in Engineering
In order to calculate the integrals given in 3.11 and 3.12 , let us perform the substitution u v 2 , considering t ≥ 0, so that 2 Re e inωt I N,n z, t , 3.14 and similarly t 0 sin nω t − u √ u e −z 2 /4ku du 2 Im e inωt I N,n z, t , 3.15 where we have defined 3.16 Taking a inω, b z 2 /4k, and z ≥ 0 in 3.6 , we may rewrite 3.16 as

3.18
As far as we know, the integrals given in 3.18 are not reported in the literature.

Particular Cases
As a consistency check, let us verify that in certain particular cases the new integrals given in 3.8 and 3.18 are reduced to integrals reported in the literature. These particular cases have to do with the quasistationary regime and the solution on the ground surface.

Quasistationary Regime
Since there is a periodic oscillation on the ground surface, the surroundings of the borehole heat exchanger never reach a stationary regime. However, according to 5, Section 2.6 , we may define a quasistationary regime in which the ground temperature field is stabilized periodically. In this quasistationary regime, 3