**2. Heat transfer inside boreholes**

## **2.1 Overview**

118 Modeling and Optimization of Renewable Energy Systems

However, the commercial growth of the GCHP systems has been hindered by its higher capital cost, of which a significant portion is attributed to the GHE. Besides the structural and geometrical configuration of the exchanger a lot of factors influence the exchanger performance, such as the ground temperature distribution, soil moisture content and its thermal properties, groundwater movement and possible freezing and thawing in soil. Thus, heat transfer between a GHE and its surrounding soil/rock is difficult to model for the purpose of sizing the GHE or simulation of the GCHP systems. In order to assess the thermal behaviour and to optimise the technical as well as economical aspects of GCHP systems, it is crucial to work out appropriate and validated heat transfer models of the GHE. To determine the heat transfer in the GHEs with adequate accuracy is a crucial task, and has great impact on sizing and simulating GHE. The design goal is to control the temperature rise of the ground and the circulating fluid within acceptable limits over the lifetime of the system. A fundamental task for application of the GCHP technology is to grasp the heat transfer process of a single borehole in the GHE. Heat transfer in a field with multiple

There are roughly two categories of approaches in dealing with the thermal analysis and design of the GHEs. Empirical or semi-empirical formulations are recommended in textbooks and monographs for GHE design purposes (Bose et al., 1985; Kavanaugh, 1997). These approaches are relatively simple, and may be manipulated easily by design engineers. However, they do not reveal in detail the impacts of complicated factors on the GHE performance. The second kind of approaches involves numerical simulation of the heat transfer in the GHEs (Mei & Baxter, 1986; Yavuzturk & Spitler, 1999). While having provided important understandings on GHE heat transfer, these studies of numerical simulation have not yet been suitable to design and/or energy analysis of full scale

Theoretical study on the GHE with an analytical approach is presented by some Swedish and American scholars (Eskilson, 1987; Spitler, 2005). Involving a time span of several years, the heat transfer process in the ground around the vertical boreholes is rather complicated, and should be treated, on the whole, as a transient one. Because of all the complications of this problem and its long time scale, the heat transfer process may usually be analyzed in two separated regions. One is the solid soil/rock outside the borehole, where the heat conduction must be treated as a transient process. With the knowledge of the temperature response in the ground, the temperature on the borehole wall can then be determined for any instant on specified operational conditions. Another sector often segregated for analysis is the region inside the borehole, including the grout, the U-tube pipes and the circulating fluid inside the pipes. The main objective of this analysis is to determine the inlet and outlet temperatures of the circulating fluid according to the borehole wall temperature, the

In this approach for GHE analysis a single borehole is investigated in detail experiencing a step heating/cooling. Then, the principle of superimposition is used to deal with the more complicated situation of GHEs with multiple boreholes as well as the variable load. It is more adequate and accurate than the empirical approaches and yet much more convenient for computations than the numerical simulations. In this regard, better understanding of every thermal resistances of the GHE is crucial, and their analytical solutions are especially

boreholes may be analyzed on this basis with the superposition principle.

engineering projects because it takes too substantial computing time.

thermal resistance inside the borehole and the heat rate of the GHE.

The main objective to analyze the heat transfer inside boreholes is to determine the entering and leaving temperatures of the circulating fluid in the GHE according to the borehole wall temperature and its heat flow. Compared with the infinite ground outside it, both the dimensional scale and thermal mass of the borehole are much smaller. Moreover, the temperature variation inside the borehole is usually slow and minor. Thus, it is a common practice that the heat transfer in this region is approximated as a steady-state process.

It is obvious that the double U-tube configuration provides more heat transfer area between the circulating fluid and the ground than the single U-tube GHE does, and will reduce thermal resistance inside the borehole. On the other hand, however, it might require more pipes and consume more pumping power on operation for a certain project. Thus, analysis on performance and costs of different configurations of the GHEs has been a task for scholars and engineers to study.

A few models of varying complexity have been established to describe the heat transfer inside the GHE boreholes. Models for practical engineering designs are often oversimplified in dealing with the complicated geometry inside the boreholes. One-dimensional (1-D) model (Bose et al., 1985) has been recommended for engineering design, conceiving the U-

Heat Transfer Modeling of the Ground

Fig. 2. Cross-section of a borehole with double U-tube

*f b f b f b f b*

source assumption has resulted in the following solution.

2

inside the U-tubes to the pipe outside surface. A linear transformation of equation (2) leads to:

1

2

3

4

11 2

2 2 2

2 2

<sup>1</sup> ln ln

13 2

<sup>1</sup> ln ln

*r kk rD <sup>R</sup> k D kk r*

12 4

<sup>1</sup> ln ln

*r kk rD <sup>R</sup> k D r k k*

*b Pb b*

*r kk rD R R k r kk r*

*bb b*

*bb b b b b*

where *k* denotes the conductivity of soil/rock around the borehole, while *k*b the heat conductivity of the grouting material, and *R*p the heat transfer resistance from the fluid

1 12 13 14

 

*f bf f f f f f*

*T TT T T T T T <sup>q</sup> RR R R*

1 12 13 12 21 2 23 24

*f f f bf f f f*

*T T T TT T T T <sup>q</sup> R RR R*

12 1 12 13 31 1 3 2 1 3 4

*f f f f f bf f*

*T T T T T TT T <sup>q</sup> R R RR*

13 12 1 12 41 42 43 4

12 13 12 1

*f f f f f f fb*

*TT TT TT TT <sup>q</sup> R R RR*

Heat Exchangers for the Ground-Coupled Heat Pump Systems 121

 11 1 12 2 13 3 14 4 21 1 22 2 23 3 24 4 3 311 322 333 34 4 41 1 42 2 43 3 44 4

where *R*ii (i=1, 2, 3, 4) is the thermal resistance between the circulating fluid in a certain Utube leg and the borehole wall, and *R*ij (i, j=1, 2, 3, 4) the resistance between two individual pipes. It is most likely in engineering practice the U-tube legs are disposed in the borehole symmetrically as shown in Fig. 2. In this case one gets *R*ij=*R*ji,*R*ii=*R*jj(i, j=1, 2, 3, 4)and *R*14=*R*12 and so on. Hellstrom (1991) analyzed the steady-state conduction problem in the borehole cross-section in detail with the line-source and multiple approximations. The line-

*bb b b b b*

2 2

2 2

4 4

*P*

 

 

(1)

(2)

(3)

*T T Rq Rq Rq Rq T T Rq Rq Rq Rq T T Rq Rq Rq Rq T T Rq Rq Rq Rq*

tube pipes as a single "equivalent" pipe. By a different approach Hellstrom (1991) has derived two-dimensional (2-D) analytical solutions of the borehole thermal resistances in the cross-section perpendicular to the borehole. On assumptions of identical temperatures and heat fluxes of all the pipes in it, the borehole resistance was worked out. Exchanging heat with the surrounding ground, however, the fluid circulating through different legs of the Utubes is of varying temperatures. As a result, thermal interference, or thermal "shortcircuiting", among U-tube legs is inevitable, which degrades the effective heat transfer in the GHEs. With the assumption of identical temperature of all the pipes, it is impossible for all the models mentioned above to reveal impact of this thermal interference on GHE performances.

On the other hand, Mei & Baxter (1986) considered the 2-D model of the radial and longitudinal heat transfer, which was solved with a finite difference scheme. Yavuzturk et al. (1999) employed the 2-D finite element method to analyze the heat conduction in the plane perpendicular to the borehole for short time step responses. Requiring numerical solutions, these models are of limited practical value for use by designers although they may result in more exact solutions for research and parametric analysis of GHEs.

Taking the fluid axial convective heat transfer and thermal "short-circuiting" among U-tube legs into account, a quasi three-dimensional (3-D) model for boreholes in GHEs has been established as an extension of the work of Eskilson (1987) and Hellstrom (1991) to reveal the thermal interference between the U-tube legs. Analytical solutions of the fluid temperature profiles along the borehole depth have been obtained (Zeng et al., 2003). This model takes into account more factors than previous models ever did before, including the geometrical parameters (borehole and pipe sizes and pipe disposal in the borehole) and physical parameters (thermal conductivity of the materials, flow rate and fluid properties) as well as the flow circuit configuration. Its solutions have provided a reliable tool for GHE sizing and performance simulation and a solid basis for technical and economic assessment of different borehole configurations.

#### **2.2 Quasi 3-D model on heat transfer inside the borehole**

This section focuses on the model of heat transfer inside the borehole taking into account the 2- D heat conduction in the transverse cross-section as well as the convective heat transfer in the axial direction by the fluid inside the U-tubes, which is referred to as the quasi 3-D model. To keep the problem analytically manageable some simplifications are assumed. They are:


Number the pipes in the borehole clockwise as shown in Fig. 2. If the temperature on the borehole wall is taken as the reference of the temperature excess, the fluid temperature excess in the pipes may be expressed as the sum of four separate temperature excesses caused by the heat fluxes per unit length, *q*1, *q*2, *q*3 and *q*4 from the four legs of the U-tubes. Thus, the following expressions may be obtained

tube pipes as a single "equivalent" pipe. By a different approach Hellstrom (1991) has derived two-dimensional (2-D) analytical solutions of the borehole thermal resistances in the cross-section perpendicular to the borehole. On assumptions of identical temperatures and heat fluxes of all the pipes in it, the borehole resistance was worked out. Exchanging heat with the surrounding ground, however, the fluid circulating through different legs of the Utubes is of varying temperatures. As a result, thermal interference, or thermal "shortcircuiting", among U-tube legs is inevitable, which degrades the effective heat transfer in the GHEs. With the assumption of identical temperature of all the pipes, it is impossible for all the models mentioned above to reveal impact of this thermal interference on GHE

On the other hand, Mei & Baxter (1986) considered the 2-D model of the radial and longitudinal heat transfer, which was solved with a finite difference scheme. Yavuzturk et al. (1999) employed the 2-D finite element method to analyze the heat conduction in the plane perpendicular to the borehole for short time step responses. Requiring numerical solutions, these models are of limited practical value for use by designers although they

Taking the fluid axial convective heat transfer and thermal "short-circuiting" among U-tube legs into account, a quasi three-dimensional (3-D) model for boreholes in GHEs has been established as an extension of the work of Eskilson (1987) and Hellstrom (1991) to reveal the thermal interference between the U-tube legs. Analytical solutions of the fluid temperature profiles along the borehole depth have been obtained (Zeng et al., 2003). This model takes into account more factors than previous models ever did before, including the geometrical parameters (borehole and pipe sizes and pipe disposal in the borehole) and physical parameters (thermal conductivity of the materials, flow rate and fluid properties) as well as the flow circuit configuration. Its solutions have provided a reliable tool for GHE sizing and performance simulation and a solid basis for technical and economic assessment of different

This section focuses on the model of heat transfer inside the borehole taking into account the 2- D heat conduction in the transverse cross-section as well as the convective heat transfer in the axial direction by the fluid inside the U-tubes, which is referred to as the quasi 3-D model. To

2. The heat conduction in the axial direction is negligible, and only the conductive heat flow among the borehole wall and the pipes in the transverse cross-section is counted. 3. The borehole wall temperature, *T*b, is constant along its depth, but may vary with time. 4. The ground outside the borehole and grout in it are homogeneous, and all the thermal

Number the pipes in the borehole clockwise as shown in Fig. 2. If the temperature on the borehole wall is taken as the reference of the temperature excess, the fluid temperature excess in the pipes may be expressed as the sum of four separate temperature excesses caused by the heat fluxes per unit length, *q*1, *q*2, *q*3 and *q*4 from the four legs of the U-tubes.

keep the problem analytically manageable some simplifications are assumed. They are:

1. The heat capacity of the materials inside the borehole is neglected.

may result in more exact solutions for research and parametric analysis of GHEs.

**2.2 Quasi 3-D model on heat transfer inside the borehole** 

properties involved are independent of temperature.

Thus, the following expressions may be obtained

performances.

borehole configurations.

$$\begin{aligned} T\_{f1} - T\_b &= R\_{11}q\_1 + R\_{12}q\_2 + R\_{13}q\_3 + R\_{14}q\_4\\ T\_{f2} - T\_b &= R\_{21}q\_1 + R\_{22}q\_2 + R\_{23}q\_3 + R\_{24}q\_4\\ T\_{f3} - T\_b &= R\_{31}q\_1 + R\_{32}q\_2 + R\_{33}q\_3 + R\_{34}q\_4\\ T\_{f4} - T\_b &= R\_{41}q\_1 + R\_{42}q\_2 + R\_{43}q\_3 + R\_{44}q\_4 \end{aligned} \tag{1}$$

where *R*ii (i=1, 2, 3, 4) is the thermal resistance between the circulating fluid in a certain Utube leg and the borehole wall, and *R*ij (i, j=1, 2, 3, 4) the resistance between two individual pipes. It is most likely in engineering practice the U-tube legs are disposed in the borehole symmetrically as shown in Fig. 2. In this case one gets *R*ij=*R*ji,*R*ii=*R*jj(i, j=1, 2, 3, 4)and *R*14=*R*12 and so on. Hellstrom (1991) analyzed the steady-state conduction problem in the borehole cross-section in detail with the line-source and multiple approximations. The linesource assumption has resulted in the following solution.

$$\begin{aligned} R\_{11} &= \frac{1}{2\pi k\_b} \left[ \ln \left( \frac{r\_b}{r\_P} \right) - \frac{k\_b - k}{k\_b + k} \ln \left( \frac{r\_b^2 - D^2}{r\_b^2} \right) \right] + R\_P\\ R\_{12} &= \frac{1}{2\pi k\_b} \left[ \ln \left( \frac{r\_b}{\sqrt{2}D} \right) - \frac{k\_b - k}{2(k\_b + k)} \ln \left( \frac{r\_b^4 + D^4}{r\_b^4} \right) \right] \\ R\_{13} &= \frac{1}{2\pi k\_b} \left[ \ln \left( \frac{r\_b}{2D} \right) - \frac{k\_b - k}{k\_b + k} \ln \left( \frac{r\_b^2 + D^2}{r\_b^2} \right) \right] \end{aligned} \tag{2}$$

where *k* denotes the conductivity of soil/rock around the borehole, while *k*b the heat conductivity of the grouting material, and *R*p the heat transfer resistance from the fluid inside the U-tubes to the pipe outside surface.

A linear transformation of equation (2) leads to:

$$\begin{aligned} q\_1 &= \frac{T\_{f1} - T\_b}{R\_1^\Lambda} + \frac{T\_{f1} - T\_{f2}}{R\_{12}^\Lambda} + \frac{T\_{f1} - T\_{f3}}{R\_{13}^\Lambda} + \frac{T\_{f1} - T\_{f4}}{R\_{12}^\Lambda} \\ q\_2 &= \frac{T\_{f2} - T\_{f1}}{R\_{12}^\Lambda} + \frac{T\_{f2} - T\_b}{R\_1^\Lambda} + \frac{T\_{f2} - T\_{f3}}{R\_{12}^\Lambda} + \frac{T\_{f2} - T\_{f4}}{R\_{13}^\Lambda} \\ q\_3 &= \frac{T\_{f31} - T\_{f1}}{R\_{13}^\Lambda} + \frac{T\_{f3} - T\_{f2}}{R\_{12}^\Lambda} + \frac{T\_{f1} - T\_b}{R\_1^\Lambda} + \frac{T\_{f3} - T\_{f4}}{R\_{12}^\Lambda} \\ q\_4 &= \frac{T\_{f4} - T\_{f1}}{R\_{12}^\Lambda} + \frac{T\_{f4} - T\_{f2}}{R\_{13}^\Lambda} + \frac{T\_{f4} - T\_{f3}}{R\_{12}^\Lambda} + \frac{T\_{f4} - T\_b}{R\_1^\Lambda} \end{aligned} \tag{3}$$

Heat Transfer Modeling of the Ground

<sup>1</sup> <sup>2</sup>

configurations

Heat Exchangers for the Ground-Coupled Heat Pump Systems 123

All these options have been analyzed separately. Analytical expressions of the fluid temperature profiles along the channel have been obtained for all these option. The dimensionless solution of the temperature profile along the borehole depth takes the following form for the single U-tube and the double U-tube in parallel configurations.

*P*

1

*P P P P Z ZZ P P P P P P*

 

 

> 

  (5)

 

 

 

*<sup>P</sup> <sup>Z</sup> <sup>Z</sup> <sup>P</sup> ZZ P*

<sup>1</sup> cosh sinh sinh <sup>1</sup> cosh <sup>1</sup>

<sup>1</sup> <sup>1</sup> cosh sinh

Fig. 3. Temperature profiles along the borehole depth with different U-tube

The effective borehole thermal resistance defines the proportional relationship between the heat flow rate transferred by the borehole and the temperature difference between the

*b*

*R*

*f b*

(6)

*l T T*

*q*

*P P*

1 1 cosh sinh cosh sinh 1 1 sinh cosh 1 1 <sup>1</sup> cosh sinh cosh sinh 1 1

More intricate but less-frequently-used solutions for the double U-tube in series can be found elsewhere (Zeng et al., 2003) together with the definitions of the dimensionless parameters in above equations. Typical temperature profiles along the single and double U-

tubes in different flow patterns are plotted in Fig. 3.

circulating fluid and the borehole wall, that is:

<sup>2</sup> <sup>2</sup>

where

$$\begin{aligned} R\_1^{\Lambda} &= R\_{11} + R\_{13} + 2R\_{12}, \; R\_{12}^{\Lambda} = \frac{R\_{11}^2 + R\_{13}^2 + 2R\_{11}R\_{13} - 4R\_{12}^2}{R\_{12}}, \\\\ R\_{13}^{\Lambda} &= \frac{\left(R\_{11} - R\_{13}\right)\left(R\_{11}^2 + R\_{13}^2 + 2R\_{11}R\_{13} - 4R\_{12}^2\right)}{R\_{13}^2 + R\_{11}R\_{13} - 2R\_{12}^2}. \end{aligned}$$

In this model the convective heat flow along the fluid channel is balanced by the conductive heat flows among the fluid channels and borehole wall. According to equation (3) the heat equilibrium of the fluid in individual pipes can be formulated as:

$$\begin{aligned} \pm Mc \frac{dT\_{f1}(z)}{dz} &= \frac{T\_{f1}(z) - T\_b}{R\_1^{\Lambda}} + \frac{T\_{f1}(z) - T\_{f2}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f1}(z) - T\_{f3}(z)}{R\_{13}^{\Lambda}} + \frac{T\_{f1}(z) - T\_{f4}(z)}{R\_{12}^{\Lambda}} \\ \pm Mc \frac{dT\_{f2}(z)}{dz} &= \frac{T\_{f2}(z) - T\_{f1}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f2}(z) - T\_b}{R\_{1}^{\Lambda}} + \frac{T\_{f2}(z) - T\_{f3}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f2}(z) - T\_{f4}(z)}{R\_{13}^{\Lambda}} \\ \pm Mc \frac{dT\_{f3}(z)}{dz} &= \frac{T\_{f3}(z) - T\_{f1}(z)}{R\_{13}^{\Lambda}} + \frac{T\_{f3}(z) - T\_{f2}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f3}(z) - T\_b}{R\_{1}^{\Lambda}} + \frac{T\_{f3}(z) - T\_{f4}(z)}{R\_{12}^{\Lambda}} \\ \pm Mc \frac{dT\_{f4}(z)}{dz} &= \frac{T\_{f4}(z) - T\_{f1}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f4}(z) - T\_{f2}(z)}{R\_{13}^{\Lambda}} + \frac{T\_{f4}(z) - T\_{f3}(z)}{R\_{12}^{\Lambda}} + \frac{T\_{f4}(z) - T\_b}{R\_{1}^{\Lambda}} \end{aligned} \tag{4}$$

Here the signal ± on the left side of the equations depends on the condition whether the fluid flows in the same direction as the *z*-coordinate, which is designated to be downwards. When the fluid moves downwards, the signal is positive, and vice versa. Combined with certain connecting conditions from the flow circuit arrangement, the energy equilibrium equation can be solved by means of Laplace transformation. Then, the temperature distribution of circulating fluid along the channels can be analytically worked out, and the thermal resistance inside the borehole can be determined more adequately.

#### **2.3 Fluid temperature profiles along the depth and borehole resistance**

The fluid temperature profiles in the flow channels and, then, the borehole resistance are affected by borehole configuration. As mentioned above, there are single and double U-tube boreholes. The latter can be arranged in series or parallel flow circuits, and each of them includes a few connecting patterns. For the two U-tubes in the borehole connected in parallel circuit, different combinations of circuit arrangement come down to two options that make difference to its heat transfer. They may be represented by notations of (1-3, 2-4) and (1-2, 3-4). Here 1-3 denotes that the fluid flows through pipes 1 and 3 as indicated in Fig. 2, and also through pipes 2 and 4 in parallel. When the fluid circulates through the four legs of the U-tubes in a series circuit, there are quite a few possible layouts. Only three of them, however, bear different impact on the performance of GHE on the assumption of symmetrical disposal of the pipes. The three representative layouts in series are marked as 1-3-2-4, 1-2-3-4 and 1-2-4-3, where the sequence indicates flow succession of the pipes as shown in Fig. 2.

 2 2 <sup>2</sup> 11 13 11 13 11 13 12

.

*R R R R RR R*

*R RR R*

In this model the convective heat flow along the fluid channel is balanced by the conductive heat flows among the fluid channels and borehole wall. According to equation (3) the heat

13 11 13 12

1 1 12 13 14

*f f bf f f f f f*

*dz RR R R dT z T z T z T z T T z T z T z T z*

41 42 43 4

thermal resistance inside the borehole can be determined more adequately.

**2.3 Fluid temperature profiles along the depth and borehole resistance** 

*dz R RR R dT z T z T z T z T z T z T T z T z*

*dz R R RR*

*dT z T z T T z T z T z T z T z T z*

2 21 2 23 24

*f f f f bf f f f*

3 31 32 3 34

Here the signal ± on the left side of the equations depends on the condition whether the fluid flows in the same direction as the *z*-coordinate, which is designated to be downwards. When the fluid moves downwards, the signal is positive, and vice versa. Combined with certain connecting conditions from the flow circuit arrangement, the energy equilibrium equation can be solved by means of Laplace transformation. Then, the temperature distribution of circulating fluid along the channels can be analytically worked out, and the

The fluid temperature profiles in the flow channels and, then, the borehole resistance are affected by borehole configuration. As mentioned above, there are single and double U-tube boreholes. The latter can be arranged in series or parallel flow circuits, and each of them includes a few connecting patterns. For the two U-tubes in the borehole connected in parallel circuit, different combinations of circuit arrangement come down to two options that make difference to its heat transfer. They may be represented by notations of (1-3, 2-4) and (1-2, 3-4). Here 1-3 denotes that the fluid flows through pipes 1 and 3 as indicated in Fig. 2, and also through pipes 2 and 4 in parallel. When the fluid circulates through the four legs of the U-tubes in a series circuit, there are quite a few possible layouts. Only three of them, however, bear different impact on the performance of GHE on the assumption of symmetrical disposal of the pipes. The three representative layouts in series are marked as 1-3-2-4, 1-2-3-4 and 1-2-4-3, where the sequence indicates flow succession of the pipes as

*f f f f f f bf f*

1 12 13 12

12 1 12 13

13 12 1 12

12 13 12 1 *T zTz T zT z T zTz T zT f f f f f f fb R R RR*

 

12

13 2 2

2 2 2 11 13 11 13 12

*R R RR R* 2 4 *<sup>R</sup> R* ,

2 4

2

12

 

(4)

1 11 13 12 *RRR R*<sup>2</sup> ,

equilibrium of the fluid in individual pipes can be formulated as:

*R*

4

*dz*

*f*

*dT z*

*Mc*

*Mc*

*Mc*

*Mc*

shown in Fig. 2.

where

All these options have been analyzed separately. Analytical expressions of the fluid temperature profiles along the channel have been obtained for all these option. The dimensionless solution of the temperature profile along the borehole depth takes the following form for the single U-tube and the double U-tube in parallel configurations.

$$\begin{aligned} \Theta\_1(Z) &= \cosh(\beta Z) - \frac{\sinh(\beta Z)}{\sqrt{1-P^2}} \left[ 1 - P \frac{\cosh(\beta) - \sqrt{\frac{1-P}{1+P}} \sinh(\beta Z)}{\cosh(\beta) + \sqrt{\frac{1-P}{1+P}} \sinh(\beta)} \right] \\ \Theta\_2(Z) &= \frac{\cosh(\beta) - \sqrt{\frac{1-P}{1+P}} \sinh(\beta)}{\cosh(\beta) + \sqrt{\frac{1-P}{1+P}} \sinh(\beta)} \cosh(\beta Z) + \frac{\sinh(\beta Z)}{\sqrt{1-P^2}} \left[ \frac{\cosh(\beta) - \sqrt{\frac{1-P}{1+P}} \sinh(\beta)}{\cosh(\beta) + \sqrt{\frac{1-P}{1+P}} \sinh(\beta)} - P \right] \end{aligned} (5)$$

More intricate but less-frequently-used solutions for the double U-tube in series can be found elsewhere (Zeng et al., 2003) together with the definitions of the dimensionless parameters in above equations. Typical temperature profiles along the single and double Utubes in different flow patterns are plotted in Fig. 3.

Fig. 3. Temperature profiles along the borehole depth with different U-tube configurations

The effective borehole thermal resistance defines the proportional relationship between the heat flow rate transferred by the borehole and the temperature difference between the circulating fluid and the borehole wall, that is:

$$R\_b = \frac{T\_f - T\_b}{q\_l} \tag{6}$$

Heat Transfer Modeling of the Ground

medium.

model with =0.

where

axis.

numerical calculation is sophisticated and time-consuming.

**3.2 Vertical and inclined finite line source model** 

0

4

0 0 *r xx s*sin cos 

0 0 *r xx s*sin cos 

**3.3 Infinite and finite solid cylindrical source model** 

0

*H l*

Heat Exchangers for the Ground-Coupled Heat Pump Systems 125

A progress in the short time-step simulations of the GHEs is the short time-step response factor model developed by Yavuzturk & Spitler (1999) based on numerical solution by taking the heat capacity of grout and pipe into account. This model is validated to be accurate and has been implemented as part of a component model of the TRNSYS, but its

Considering the axial heat flow and taking the long-term effect of the limited borehole depth into account, a 2-D finite line source model was established by Zeng et al. (2002) to analyze the heat transfer outside vertical borehole GHE. This 2-D model assumes the ground as a homogeneous semi-infinite medium with a uniform initial temperature, and assumes the borehole as a line source with finite length releasing heat at a constant rate per length. Evolved from the vertical borehole systems, inclined boreholes are considered as a favorable alternative to further reduce the land areas required for the GHEs. Then a 3-D finite line source model for inclined borehole GHE was also proposed by Cui et al. (2006). In order to take characteristics of the pile GHEs into proper consideration and to deal with the short term temperature response, Man et al. (2010) proposed a "solid" cylindrical source model, which suppose that the cylinder is no longer a cavity, but filled with the medium identical to that out of the cylinder, so that the whole infinite domain is composed of a homogeneous

As mentioned, the 2-D vertical finite line source model and the 3-D inclined finite line source model are established based on the Green's function method to analyze the heat conduction outside the vertical and inclined boreholes, as shown in Fig. 4. An analytical solution has been derived for the 3-D inclined finite line source model as follows (Cui et al., 2006). Then, the 2-D model of vertical finite line source becomes a special case of the 3-D

2 2

<sup>2</sup> <sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup> <sup>2</sup>

*z s* .

For the solid cylindrical source model, a heat source shaped in a cylindrical surface of a radius *r0* is supposed to be buried in the medium with its axis being coincident with *z*-

 *yy s*sin sin cos 

 *yy s*sin sin cos 

 

(10)

 

 

*r r erfc erfc q a a t t dS k r r* 

*z s*

where 2 *T TT f ff* denotes the arithmetic mean of the inlet and outlet fluid temperatures. In view of heat balance for the single U-tube and double U-tube in series one also has:

$$
\eta\_l \mathcal{H} = \mathcal{M} \mathcal{c} \left( T\_f' - T\_f'' \right) \tag{7}
$$

As a result the borehole resistance can be determined according to the analytical solutions of fluid temperature profile in the borehole presented in previous discussion.

Combined with the definition of dimensionless temperature, equations (6) and (7) result in the following expression of borehole resistance for boreholes with the single U-tube and double U-tube in series.

$$R\_b = \frac{H}{2Mc} \cdot \frac{1 + \Theta\_f'}{1 - \Theta\_f''} \tag{8}$$

In the case of double U-tubes in parallel the fluid mass rate in the borehole is doubled, i.e. 2*M*, thus the borehole thermal resistance can be expressed as:

$$R\_b = \frac{H}{4Mc} \cdot \frac{1 + \Theta\_f'}{1 - \Theta\_f''} \tag{9}$$

The borehole thermal resistance defined above takes into accounts both the geometrical parameters (borehole and pipe sizes and pipe disposal in the borehole) and physical parameters (thermal conductivity of materials, flow rate and fluid properties). Therefore the concept of effective thermal resistance facilitates heat transfer analysis. Analyses have shown that the single U-tube boreholes yield considerably higher borehole resistance than the double U-tube boreholes do while the other conditions are identical. Practical choice of the U-tube configuration should be taken in accordance with economic consideration in practical engineering.

#### **3. Heat conduction outside boreholes**

#### **3.1 Overview**

There have been some classical models for GHE thermal analysis based on analytical 1-D solutions. A most widely used 1-D model for this purpose is Kelven's line source model (Carslaw & Jeager, 1947). In this model the borehole is replaced by a line heat source with its radial dimension neglected. Another best known 1-D model, referred as the cylindrical heat source model (Carslaw & Jeager, 1947; Ingersoll & Zobel, 1954) is an alternative approach to sizing GHEs. Although the radial dimension of the borehole is taken into consideration in this model, the heat capacity of the cylinder is ignored, so the geometrical domain of the model can be regarded as an infinite medium with a cylindrical cavity in it. We refer it to as the "hollow" cylindrical heat source model to distinguish it from the "solid" cylindrical heat source model we are to propose below. Significant simplifications are made in these two classical models, which result in substantial deviations of the temperature response from the actual situation especially in the initial period of the heating pulse. Therefore, neither of these two models is suitable for short time-step analysis of GHEs.

A progress in the short time-step simulations of the GHEs is the short time-step response factor model developed by Yavuzturk & Spitler (1999) based on numerical solution by taking the heat capacity of grout and pipe into account. This model is validated to be accurate and has been implemented as part of a component model of the TRNSYS, but its numerical calculation is sophisticated and time-consuming.

Considering the axial heat flow and taking the long-term effect of the limited borehole depth into account, a 2-D finite line source model was established by Zeng et al. (2002) to analyze the heat transfer outside vertical borehole GHE. This 2-D model assumes the ground as a homogeneous semi-infinite medium with a uniform initial temperature, and assumes the borehole as a line source with finite length releasing heat at a constant rate per length. Evolved from the vertical borehole systems, inclined boreholes are considered as a favorable alternative to further reduce the land areas required for the GHEs. Then a 3-D finite line source model for inclined borehole GHE was also proposed by Cui et al. (2006). In order to take characteristics of the pile GHEs into proper consideration and to deal with the short term temperature response, Man et al. (2010) proposed a "solid" cylindrical source model, which suppose that the cylinder is no longer a cavity, but filled with the medium identical to that out of the cylinder, so that the whole infinite domain is composed of a homogeneous medium.

#### **3.2 Vertical and inclined finite line source model**

As mentioned, the 2-D vertical finite line source model and the 3-D inclined finite line source model are established based on the Green's function method to analyze the heat conduction outside the vertical and inclined boreholes, as shown in Fig. 4. An analytical solution has been derived for the 3-D inclined finite line source model as follows (Cui et al., 2006). Then, the 2-D model of vertical finite line source becomes a special case of the 3-D model with =0.

$$t - t\_0 = \frac{q\_l}{4\pi k} \int\_0^H \left| \frac{\operatorname{erfc}\left(\frac{r\_+}{2\sqrt{a\pi}}\right)}{r\_+} - \frac{\operatorname{erfc}\left(\frac{r\_-}{2\sqrt{a\pi}}\right)}{r\_-} \right| dS \tag{10}$$

where

124 Modeling and Optimization of Renewable Energy Systems

where 2 *T TT f ff* denotes the arithmetic mean of the inlet and outlet fluid temperatures.

As a result the borehole resistance can be determined according to the analytical solutions of

Combined with the definition of dimensionless temperature, equations (6) and (7) result in the following expression of borehole resistance for boreholes with the single U-tube and

2 1

In the case of double U-tubes in parallel the fluid mass rate in the borehole is doubled, i.e.

4 1

The borehole thermal resistance defined above takes into accounts both the geometrical parameters (borehole and pipe sizes and pipe disposal in the borehole) and physical parameters (thermal conductivity of materials, flow rate and fluid properties). Therefore the concept of effective thermal resistance facilitates heat transfer analysis. Analyses have shown that the single U-tube boreholes yield considerably higher borehole resistance than the double U-tube boreholes do while the other conditions are identical. Practical choice of the U-tube configuration should be taken in accordance with economic consideration in

There have been some classical models for GHE thermal analysis based on analytical 1-D solutions. A most widely used 1-D model for this purpose is Kelven's line source model (Carslaw & Jeager, 1947). In this model the borehole is replaced by a line heat source with its radial dimension neglected. Another best known 1-D model, referred as the cylindrical heat source model (Carslaw & Jeager, 1947; Ingersoll & Zobel, 1954) is an alternative approach to sizing GHEs. Although the radial dimension of the borehole is taken into consideration in this model, the heat capacity of the cylinder is ignored, so the geometrical domain of the model can be regarded as an infinite medium with a cylindrical cavity in it. We refer it to as the "hollow" cylindrical heat source model to distinguish it from the "solid" cylindrical heat source model we are to propose below. Significant simplifications are made in these two classical models, which result in substantial deviations of the temperature response from the actual situation especially in the initial period of the heating pulse. Therefore, neither of

1

1

*f*

*f*

*f*

*f*

*H Mc T T f f* (7)

(8)

(9)

In view of heat balance for the single U-tube and double U-tube in series one also has:

*ql*

fluid temperature profile in the borehole presented in previous discussion.

*b*

*b*

*<sup>H</sup> <sup>R</sup> Mc*

2*M*, thus the borehole thermal resistance can be expressed as:

*<sup>H</sup> <sup>R</sup> Mc*

double U-tube in series.

practical engineering.

**3.1 Overview** 

**3. Heat conduction outside boreholes** 

these two models is suitable for short time-step analysis of GHEs.

$$r\_+ = \sqrt{\left(x - x\_0 - s\sin\alpha\cos\beta\right)^2 + \left(y - y\_0 - s\sin\alpha\sin\beta\right)^2 + \left(z - s\cos\alpha\right)^2}$$

$$r\_- = \sqrt{\left(x - x\_0 - s\sin\alpha\cos\beta\right)^2 + \left(y - y\_0 - s\sin\alpha\sin\beta\right)^2 + \left(z + s\cos\alpha\right)^2} \dots$$

#### **3.3 Infinite and finite solid cylindrical source model**

For the solid cylindrical source model, a heat source shaped in a cylindrical surface of a radius *r0* is supposed to be buried in the medium with its axis being coincident with *z*axis.

Heat Transfer Modeling of the Ground

0.0

0.1

0.2

0.3

0.4

Dimensionless temperature response

0.5

0.6

0.7

0.8

**3.3.2 The finite solid cylindrical source model** 

 

boundary. Then, the solution of this problem can be obtained as:

*h*

 

source model in simulating the long-term operation of the GHE.

 relevant finite problem.

Heat Exchangers for the Ground-Coupled Heat Pump Systems 127

This solution is independent of the axial coordinate *z*, and it provides a means to tackle the

A 2-D finite solid cylindrical source model has also been presented (Man et al., 2010) in order to consider the influences of the finite length of the cylindrical heat source and the boundary. Similar to the approach for the finite line source problem, a virtual cylindrical sink with the same length but a negative heating rate may be set on symmetry to the

> 2 0 3 ' '0 0

*<sup>l</sup> <sup>q</sup> rr r z dz <sup>I</sup> c a a*

<sup>1</sup> (,,) 2( ) 8()

 

exp exp 4( ) 4( )

*a a*

 

Temperature response at the half depth of the cylinder surface calculated with the infinite and finite solid cylindrical source models are compared in Fig. 5. Clear distinction can be seen between the responses of these two models. While the temperature response rises continuously with time for the infinite model, the temperatures of those GHEs simulated by the finite model with different finite lengths tend to produce different steady-state temperatures as time approaches infinity. This feature indicates that, it is important to take the finite length effect of heat source into account and to utilize the finite solid cylindrical

' '

0.01 0.1 1 10 100 1000 10000

*Fo=a/r*2 0

Fig. 5. Temperature response vs. time from infinite and finite solid cylindrical source models

*r r zz r r zz <sup>d</sup>*

2 2 2 2 2 2 0 0

0

*Z/H*=0.5 *R*=1

*H*=100

*H*=50

*H*=20

(14)

 

> 

The infinite solid-cylindrical

source model

Fig. 4. Schematic diagram of an inclined finite line source in a semi-infinite medium

#### **3.3.1 The Infinite solid cylindrical source model**

First, the 1-D infinite solid cylindrical source model is studied with the axial heat flow neglected (Man et al., 2010). The analytical solution of this 1-D problem can be obtained directly with the Green's function method. The cylindrical heat source can be regarded as a collection of numerous line sources disposed along a circle of the radius *r0*. The temperature rise at any location with the radial coordinate *r* should be the sum, or integral, of all the individual temperature rises caused by the corresponding line sources. Then the analytical solution of the infinite solid cylindrical source model can be written as:

$$\theta\_1(r,\tau) = -\frac{q\_l}{4\pi\mathbf{k}} \Big|\_{0}^{\pi} \frac{1}{\pi} \text{Ei}\left(-\frac{r^2 + r\_0^2 - 2rr\_0\cos\varphi}{4a\tau}\right) d\varphi \tag{11}$$

Alternatively, the cylindrical heat source in this model can be reckoned as a collection of numerous ring line heat sources piled along the axial direction. Induced by a single instant ring line source with radius *r=r0* which lies on the plane *z=z'* and releases heat at the instant *τ'*, the temperature rise at any location of the coordinate (*r, z*) and at the instant *τ* can be obtained according to the Green's function theory as:

$$\theta^{\*} = \frac{Q}{\rho^{\circ}} \frac{1}{8\left[\sqrt{\pi a(\tau - \tau^{\prime})}\right]^3} \exp\left[-\frac{r^2 + r\_0^{\prime} + \left(z - z^{\prime}\right)^2}{4a(\tau - \tau^{\prime})}\right] I\_0\left[\frac{r r\_0}{2a(\tau - \tau^{\prime})}\right] \tag{12}$$

Thus, the overall temperature rise caused by the infinite cylindrical heat source at any location should be the sum, or integral, of all the individual temperature rises caused by the corresponding ring sources over the duration from 0 and *τ*. Then, the solution of the infinite solid cylindrical source model can be written, alternatively, as:

$$\partial\_{1}\left(r,\tau\right) = \frac{q\_{l}}{\rho c} \int\_{-\infty}^{\tau} \int\_{-\infty}^{a} \frac{1}{8\left[\sqrt{\pi a(\tau-\tau')}\right]^3} \exp\left[-\frac{r^2 + r\_0^2 + z'^2}{4a(\tau-\tau')}\right] I\_0\left[\frac{r r\_0}{2a(\tau-\tau')}\right] dz' d\tau' \tag{13}$$

This solution is independent of the axial coordinate *z*, and it provides a means to tackle the relevant finite problem.

#### **3.3.2 The finite solid cylindrical source model**

126 Modeling and Optimization of Renewable Energy Systems

Fig. 4. Schematic diagram of an inclined finite line source in a semi-infinite medium

solution of the infinite solid cylindrical source model can be written as:

 

0

First, the 1-D infinite solid cylindrical source model is studied with the axial heat flow neglected (Man et al., 2010). The analytical solution of this 1-D problem can be obtained directly with the Green's function method. The cylindrical heat source can be regarded as a collection of numerous line sources disposed along a circle of the radius *r0*. The temperature rise at any location with the radial coordinate *r* should be the sum, or integral, of all the individual temperature rises caused by the corresponding line sources. Then the analytical

> <sup>1</sup> 2 cos , Ei 4 k 4 *<sup>l</sup> <sup>q</sup> r r rr r d*

Alternatively, the cylindrical heat source in this model can be reckoned as a collection of numerous ring line heat sources piled along the axial direction. Induced by a single instant ring line source with radius *r=r0* which lies on the plane *z=z'* and releases heat at the instant *τ'*, the temperature rise at any location of the coordinate (*r, z*) and at the instant *τ* can be

> <sup>1</sup> \* exp 4 ( ') 2 ( ') 8 ( ') *<sup>Q</sup> r r zz rr <sup>I</sup> c a a a*

Thus, the overall temperature rise caused by the infinite cylindrical heat source at any location should be the sum, or integral, of all the individual temperature rises caused by the corresponding ring sources over the duration from 0 and *τ*. Then, the solution of the infinite

> <sup>1</sup> , exp 4 ( ') 2 ( ') 8 ( ') *<sup>l</sup> q r r z rr r I dz d c a a a*

2 2

0 0

 

> 

  (12)

 

(11)

*a*

2 2 <sup>2</sup>

222

(13)

3 0

0 0

0 0

**3.3.1 The Infinite solid cylindrical source model** 

1

obtained according to the Green's function theory as:

 

solid cylindrical source model can be written, alternatively, as:

 

1 0 3

 0

  A 2-D finite solid cylindrical source model has also been presented (Man et al., 2010) in order to consider the influences of the finite length of the cylindrical heat source and the boundary. Similar to the approach for the finite line source problem, a virtual cylindrical sink with the same length but a negative heating rate may be set on symmetry to the boundary. Then, the solution of this problem can be obtained as:

$$\begin{split} \theta\_{2}(r,z,\tau) &= \frac{q\_{l}}{\rho\varepsilon} \Big|\_{0}^{h} dz' \Big| \frac{1}{\theta \, \bigg|} \frac{1}{8 \left[\sqrt{\pi a(\tau-\tau')}\right]^3} I\_{0} \Big[ \frac{r r\_{0}}{2a(\tau-\tau')} \Big]. \\ &\cdot \Big| \exp\Big[ -\frac{r^{2} + r\_{0}^{2} + \left(z - z'\right)^{2}}{4a(\tau-\tau')} \Big] - \exp\Big[ -\frac{r^{2} + r\_{0}^{2} + \left(z + z'\right)^{2}}{4a(\tau-\tau')} \Big] \Big| \, d\tau' \end{split} \tag{14}$$

Temperature response at the half depth of the cylinder surface calculated with the infinite and finite solid cylindrical source models are compared in Fig. 5. Clear distinction can be seen between the responses of these two models. While the temperature response rises continuously with time for the infinite model, the temperatures of those GHEs simulated by the finite model with different finite lengths tend to produce different steady-state temperatures as time approaches infinity. This feature indicates that, it is important to take the finite length effect of heat source into account and to utilize the finite solid cylindrical source model in simulating the long-term operation of the GHE.

Fig. 5. Temperature response vs. time from infinite and finite solid cylindrical source models

Heat Transfer Modeling of the Ground

can be expressed in dimensionless forms:

2

**4.2 Moving finite line-source model** 

3

transient solution reads as:

Heat Exchangers for the Ground-Coupled Heat Pump Systems 129

Introducing the dimensionless variable *Pe v H a <sup>T</sup>* (Peclet number), equations (16) and (17)

<sup>1</sup> ( , , ) exp cos( ) exp <sup>2</sup> <sup>16</sup> *Pe Fo Pe Pe R R Fo Pe R* 

> 2 0 ( , ) 2exp cos( ) 2 2 *<sup>s</sup> Pe Pe R Pe R KR*

advection on GHE performance may be found in details elsewhere (Diao et al., 2004).

Fig. 6. Isotherms of a GHE field of 18 boreholes with groundwater advection

*L T*

For long-term period simulations of the GHEs axial effects become more evident. Therefore, the moving finite line source model has been further established based on the Green's function by applying the method of images (Eskilson, 1987) and the moving source theory (Carslaw & Jaeger, 1959). The detailed derivation is presented by Nelson et al. (2011). The

> 0 ( , , , ) exp ( , , , ) ' ( , , , ) ' 2 2 *H*

<sup>1</sup> ( , , , ) exp exp 4 2 2 2 <sup>2</sup> *T TT T vr r vt vr r vt f xyzt erfc erfc*

*a*

(20)

*r a at a at* 

*<sup>q</sup> v x T xyzt f x y z t dz f x y z t dz*

0

*H*

(21)

The isotherms of a GHE field with groundwater advection simulated with the moving infinite line source model are shown in Fig. 6. Analysis on the influence of the groundwater

0

(18)

<sup>2</sup> <sup>4</sup> 2 2

(19)

 *d*

 

#### **4. Heat transfer with groundwater infiltration**

In practice, the boreholes of GHEs may penetrate several geologic strata. Below the water table, water is held and moves between the grains of geologic formations in response to hydraulic gradients. In general, a moderate groundwater advection is expected to make notable difference in alleviating the possible heat buildup around the borehole over time. As a result, it is desirable to account for the groundwater infiltration in the heat transfer model to avoid over-sizing of the GHEs. However, all of the GHE design tools available at present are based simply on principles of heat conduction, and do not consider the implications of groundwater flow in carrying away heat due partly to lack of appropriate analytical tools.

Taking the groundwater advection into account, the combined heat transfer models of conduction and advection in the GHE have been solved with analytical approach. Explicit expressions about temperature response of the moving infinite/finite line source models have been derived (Diao et al., 2004; Nelson et al., 2011). In these studies the ground around the boreholes is assumed to be a homogeneous porous medium saturated by groundwater. The groundwater velocity is uniform in the whole domain concerned and parallel to the ground surface. Heat is transported through the saturated porous medium in a combined mechanism: by conduction through its solid matrix and liquid in its pores as well as by convection of the moving liquid.

#### **4.1 Moving infinite line-source model**

The partial differential equation for advective and conductive heat transport in porous media can be expressed in a 2-D form (x-y plane) as follows (Domenico & Schwartz, 1998):

$$
\rho c \frac{\partial T}{\partial t} + \mu\_x \rho\_w c\_w \frac{\partial T}{\partial x} - \lambda \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = 0 \tag{15}
$$

The solution of equation (15) for an infinite porous medium with a uniform initial temperature was given by Sutton et al. (2003) and Diao et al. (2004):

$$\Delta T\_2(\mathbf{x}, y, t) = \frac{q\_L}{4\lambda\pi} \exp\left[\frac{v\_T \mathbf{x}}{2a}\right]^{v\_T^2 t} \int\_0^{v\_T^2 t} \frac{1}{\nu} \exp\left[-\nu - \frac{v\_T^2 (\mathbf{x}^2 + y^2)}{16a^2 \nu}\right] d\nu \tag{16}$$

This analytical solution applies for the response of a constant line source with infinite length along the *z*-direction with a continuous heat flow rate per unit length of the borehole, *qL*. Although a GHE is composed of a buried pipe that commonly is surrounded by grouting material, approximation by a line source is commonly accepted in heat transport models of GCHP systems (Diao et al., 2004; Eskilson, 1987; Sutton et al., 2003). The underground is assumed to be homogeneous with respect to the thermal and hydraulic parameters. For steady state conditions equation (16) becomes:

$$
\Delta T\_{2s}(\mathbf{x}, \mathbf{y}) = \frac{q\_L}{2\pi\mathcal{U}} \exp\left[\frac{v\_T \mathbf{x}}{2a}\right] K\_0 \left[\frac{v\_T \sqrt{\mathbf{x}^2 + \mathbf{y}^2}}{2a}\right] \tag{17}
$$

In practice, the boreholes of GHEs may penetrate several geologic strata. Below the water table, water is held and moves between the grains of geologic formations in response to hydraulic gradients. In general, a moderate groundwater advection is expected to make notable difference in alleviating the possible heat buildup around the borehole over time. As a result, it is desirable to account for the groundwater infiltration in the heat transfer model to avoid over-sizing of the GHEs. However, all of the GHE design tools available at present are based simply on principles of heat conduction, and do not consider the implications of groundwater flow in carrying away heat due partly to lack of appropriate

Taking the groundwater advection into account, the combined heat transfer models of conduction and advection in the GHE have been solved with analytical approach. Explicit expressions about temperature response of the moving infinite/finite line source models have been derived (Diao et al., 2004; Nelson et al., 2011). In these studies the ground around the boreholes is assumed to be a homogeneous porous medium saturated by groundwater. The groundwater velocity is uniform in the whole domain concerned and parallel to the ground surface. Heat is transported through the saturated porous medium in a combined mechanism: by conduction through its solid matrix and liquid in its pores as well as by

The partial differential equation for advective and conductive heat transport in porous media can be expressed in a 2-D form (x-y plane) as follows (Domenico & Schwartz, 1998):

*T T TT*

 

*t x x y*

2 2 0 <sup>1</sup> ( ) ( , , ) exp exp 4 2 <sup>16</sup> *Tvt a L T <sup>T</sup> <sup>q</sup> v x vx y T x <sup>y</sup> t d*

> 2 0 ( , ) exp 22 2 *L T T*

*<sup>q</sup> v x vx y T xy <sup>K</sup>*

*a a*

This analytical solution applies for the response of a constant line source with infinite length along the *z*-direction with a continuous heat flow rate per unit length of the borehole, *qL*. Although a GHE is composed of a buried pipe that commonly is surrounded by grouting material, approximation by a line source is commonly accepted in heat transport models of GCHP systems (Diao et al., 2004; Eskilson, 1987; Sutton et al., 2003). The underground is assumed to be homogeneous with respect to the thermal and hydraulic parameters. For

The solution of equation (15) for an infinite porous medium with a uniform initial

*c uc*

temperature was given by Sutton et al. (2003) and Diao et al. (2004):

steady state conditions equation (16) becomes:

*s*

2 2

<sup>2</sup> <sup>4</sup> 22 2

*a a* 

 (15)

(17)

(16)

2 2

2 2 0 *x ww*

**4. Heat transfer with groundwater infiltration** 

analytical tools.

convection of the moving liquid.

**4.1 Moving infinite line-source model** 

Introducing the dimensionless variable *Pe v H a <sup>T</sup>* (Peclet number), equations (16) and (17) can be expressed in dimensionless forms:

$$\Theta\_2(R, Fo, Pe) = \exp\left[\frac{Pe}{2}R\cos(\varphi)\right] \int\_0^{Pe^2} \frac{1}{\nu} \exp\left[-\nu - \frac{Pe^2R^2}{16\nu}\right] d\nu \tag{18}$$

$$\Theta\_{2s}(R, Pe) = 2 \exp\left[\frac{Pe}{2}R\cos(\phi)\right] K\_0 \left[\frac{Pe}{2}R\right] \tag{19}$$

The isotherms of a GHE field with groundwater advection simulated with the moving infinite line source model are shown in Fig. 6. Analysis on the influence of the groundwater advection on GHE performance may be found in details elsewhere (Diao et al., 2004).

Fig. 6. Isotherms of a GHE field of 18 boreholes with groundwater advection

#### **4.2 Moving finite line-source model**

For long-term period simulations of the GHEs axial effects become more evident. Therefore, the moving finite line source model has been further established based on the Green's function by applying the method of images (Eskilson, 1987) and the moving source theory (Carslaw & Jaeger, 1959). The detailed derivation is presented by Nelson et al. (2011). The transient solution reads as:

$$
\Delta T\_3(\mathbf{x}, y, z, t) = \frac{q\_L}{2\lambda\pi} \exp\left[\frac{v\_T \chi}{2a}\right] \left[\int\_0^H f(\mathbf{x}, y, z, t) dz' - \int\_{-H}^0 f(\mathbf{x}, y, z, t) dz'\right] \tag{20}
$$

$$f(\mathbf{x}, y, z, t) = \frac{1}{4r} \left[ \exp\left(-\frac{\upsilon\_T r}{2a}\right) \text{erfc}\left(\frac{r - \upsilon\_T t}{2\sqrt{at}}\right) + \exp\left(\frac{\upsilon\_T r}{2a}\right) \text{erfc}\left(\frac{r + \upsilon\_T t}{2\sqrt{at}}\right) \right] \tag{21}$$

Heat Transfer Modeling of the Ground

**5.1 Overview** 

**5. Heat conduction around a buried spiral coil** 

the initial cost as well as ground requirement for the borehole field.

and the pile GHE with spiral coil are compared in Fig. 8.

Fig. 8. Schematic diagram of a vertical borehole and a pile with a coil

grout

U-tube

Piles are much thicker in diameter but shorter in depth than boreholes. Obviously, either the line source models or the "hollow" cylindrical model mentioned in previous sections is no longer valid in this case. Due to its limited application history few analytical models on the buried spiral coils have been seen in literature. In order to better understand and simulate the heat transfer of buried spiral pipes, the authors have proposed two new kinds of models. The first model is referred as the "ring-coil source model" (Cui et al., 2011), which is developed on the basis of cylindrical source model (Man et al., 2010), as shown in Fig. 9 (a)

concrete pile

spiral coil

Heat Exchangers for the Ground-Coupled Heat Pump Systems 131

The GHE with vertical boreholes (Bose et al., 1985) has been the mainstream for the GCHP systems, which is also a major obstacle to apply the GCHP technology because its installation needs a substantial initial cost and requires additional ground area. In recent years foundation piles of buildings start to be utilized as part of the GHEs. These so-called "energy piles" combining the heat exchanger with building foundation piles are a notable progress in the GCHP applications, and its most competitive advantage is that it can reduce

Literature review has shown that most of existing studies of pile GHE were based on either experiments or numerical simulations (Morino & Oka, 1994; Pahud et al., 1996; Pahud et al., 1999; Laloui et al., 2006; Hamada et al., 2007; Sekine et al., 2007). Besides, pipes are buried in concrete piles in configurations of U-tubes in most of such applications. The effective heat transfer area in a certain pile is limited, and air choking may occur in the turning tips of the tubes connected in series. In order to overcome these drawbacks, a novel configuration of the foundation pile GHE with a spiral coil has been proposed (Man et al., 2010). The distinct advantage of this novel GHE is that it can offer higher heat transfer efficiency, reduce pipe connection complexity, and decrease the thermal "short-circuiting" among the feed and return pipes. The schematic diagrams of a conventional single U-tube vertical borehole GHE

As time approaches infinity, the steady state solution is derived as follows:

$$\Delta T\_{3s}(\mathbf{x}, y, z) = \frac{q\_L}{4\lambda\pi} \exp\left[\frac{v\_T \mathbf{x}}{2a}\right] \left| \int\_0^H \frac{1}{r} \exp\left[-\frac{v\_T r}{2a}\right] dz' - \int\_{-H}^0 \frac{1}{r} \exp\left[-\frac{v\_T r}{2a}\right] dz'\right| \tag{22}$$

Equations (20), (21) and (22) can be expressed in dimensionless forms:

$$\Theta\_3(R, Z, Fo, Pe) = 2 \exp\left[\frac{Pe}{2}R\cos(\rho)\right] \left[\int\_0^1 f(R, Z, Fo, Pe)dZ' - \int\_{-1}^0 f(R, Z, Fo, Pe)dZ'\right] \tag{23}$$

$$f(R, Z, Fo, Pe) = \frac{1}{4R'} \left[ \exp\left(-\frac{Pe}{2}R'\right) \text{erfc}\left(\frac{R' - PeFo}{2\sqrt{Fo}}\right) + \exp\left(\frac{Pe}{2}R'\right) \text{erfc}\left(\frac{R' + PeFo}{2\sqrt{Fo}}\right) \right] \tag{24}$$

$$\Theta\_{3s}(R, Z, Pe) = \exp\left[\frac{Pe}{2}R\cos(\varphi)\right] \left[\frac{1}{R}\frac{1}{\Gamma}\exp\left[-\frac{Pe}{2}R^{\dagger}\right]dZ' - \int\_{-1}^{0} \frac{1}{R^{\dagger}}\exp\left[-\frac{Pe}{2}R^{\dagger}\right]dZ'\right] \tag{25}$$

Temperature contours obtained according to the moving finite and infinite line source models with groundwater advection considered are compared in Fig. 7. Note that temperature plumes are shorter for the finite model (Fig. 7a). Axial effects yields lower temperature changes at any given distance from the source due to the vertically dissipated heat. Temperature anomaly created in the vertical direction due to the axial effects can be observed in Fig. 7b. Obviously the differences between the models are most evident in the vicinity of the borehole ends.

Fig. 7. Temperature contours ( = 2.5 Wm-1K-1, *ql*=20 Wm-1, *q*=1.010-7 ms-1, *t*=20 yrs). Solid lines: Moving finite line source model; Dashed lines: Moving infinite line source model. (a) Plan view (b) Vertical cross section.

By comparisons based on the simulation results, the shorter the borehole length is, the larger the discrepancy is between the moving finite and infinite line source models, and the shorter is the time when resulted temperature responses of the moving finite and infinite line source models start to differ. Besides, it is noticeable that the larger the Peclet number the lesser the discrepancy between the moving finite and infinite line source models.
