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

### **5.1 Overview**

130 Modeling and Optimization of Renewable Energy Systems

1 1 ( , , ) exp exp ' exp ' 42 2 2

*ar a r a*

*Pe R Z Fo Pe R f R Z Fo Pe dZ f R Z Fo Pe dZ* (23)

*<sup>q</sup> v x v r v r T xyz dz dz*

1 1 ( , , ) exp cos( ) exp ' ' exp ' ' 2 '2 '2 *<sup>s</sup> Pe Pe Pe R Z Pe R R dZ R dZ*

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

lines: Moving finite line source model; Dashed lines: Moving infinite line source model. (a)

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

( , , , ) 2 exp cos( ) ( , , , ) ' ( , , , ) ' <sup>2</sup>

0

1' ' ( , , , ) exp ' exp ' 4' 2 2 2 <sup>2</sup> *Pe R PeFo Pe R PeFo f R Z Fo Pe R erfc R erfc <sup>R</sup> Fo Fo*

*H <sup>L</sup> TT T <sup>s</sup>*

0

*H*

 1 0

0 1

 1 0

0 1

(22)

= 2.5 Wm-1K-1, *ql*=20 Wm-1, *q*=1.010-7 ms-1, *t*=20 yrs). Solid

*R R* (25)

(24)

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

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

3

3

3

vicinity of the borehole ends.

Fig. 7. Temperature contours (

Plan view (b) Vertical cross section.

discrepancy between the moving finite and infinite line source models.

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

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 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

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)

Heat Transfer Modeling of the Ground

source:

**5.2.1.1 Infinite ring-coil source model** 

**5.2.1.2 Finite ring-coil source model** 

*r f*

activated at the instant

can be expressed as:

0

*n*

*m*

*r i*

 

*n*

for the finite ring-coil source model is expressed as:

*Fo*

**5.2.2 The spiral source model and its analytical solutions** 

For an instantaneous point heat source with intensity of

exp exp 4 4

 

heat source. Both the infinite and finite spiral source models are studied.

, 0 3 2 <sup>0</sup> 3 2 ' '

*B RR R Z Fo <sup>I</sup>*

exp exp 4 4

1 1 , , exp <sup>8</sup> ' 2 4

*Fo*

Heat Exchangers for the Ground-Coupled Heat Pump Systems 133

The infinite ring-coil source model is first studied, which means the ring-coil source is assumed to be infinite in the longitudinal direction. Define that the *z*-coordinates of the ring coils are *zn b* 0.5 , where n=0, 1, 2,…, +∞. As a consequence, the overall temperature response at a random point in the medium to all the ring sources can be determined as the sum of all the individual temperature rises caused by each ring-coil

, 0 3 2 <sup>0</sup> 3 2 ' <sup>0</sup>

*<sup>B</sup> R R R Z Fo <sup>I</sup>*

 

0.5 0.5

*Z nB B Z nB B dFo Fo Fo Fo Fo*

1 1 , , exp <sup>8</sup> ' <sup>2</sup> <sup>4</sup>

' '

The infinite model neglects the effects of heat flow through the top and bottom ends of the heat source; therefore it is inadequate for the long-term operation of the GHE made of buried spiral coil. While keeping the ring-coil source simplification, the spiral coil is taken as a finite ring-coil source buried in a semi-infinite medium, stretching from *h*1 to *h*2 from the boundary of the ground surface. The coil is then approximated as *m* pieces of rings. Again, the images of the ring coils with negative heating rate *-qlb* are set on symmetry to the boundary in order to keep the constant temperature of the ground surface, and the solution

*Fo Fo Fo Fo Fo Fo*

2

(27)

'

(28)

*dFo*

and

'

 

2 2

, its Green's function in the cylindrical coordinates at point *r z* , ,

*c* , located at *r z* ', ', '

2

 

*Fo Fo Fo Fo Fo Fo*

1 0.5 1 0.5

*Fo Fo Fo Fo*

*ZH n B ZH n B*

On the basis of the ring-coil source model, the spiral source model is further developed with increasing sophistication and accuracy to take the 3-D geometrical characteristic of spiral coil into account. In the spiral source model, the buried coil is represented by a spiral line

' '

2 2

and (b). The ring-coil model further takes into account the discontinuity of the heat source and the impact of the coil pitch by simplifying the buried spiral coil as a set of separated rings located on the cylindrical surface. This model has made big progress from the classical models, however, the ring heat sources is discontinuous and separated with each other, which still deviates from the realistic conditions of the buried spiral coil.

Fig. 9. Established heat source models of the pile GHE

Evolved from the ring-coil source model, the "spiral source model" (Man et al., 2011) is further presented for better analyzing and designing the pile GHE with spiral coils, as shown in Fig. 9 (c) and (d). For this model, the buried spiral coil is approximated by a spiral line heat source. The temperature response of the buried spiral coils can be evaluated according to the analytical solutions of this model.

#### **5.2 Ring model and spiral model**

#### **5.2.1 Ring-coil heat source model and solutions**

In the ring-coil heat source model, the buried spiral coil is simplified as a number of separated rings located on the cylindrical surface. In order to analyze the thermal effect of the heat transfer along the axis on the total heat transfer efficiency, both the infinite and finite ring-coil source models are discussed by means of the Green's function method.

To develop the analytical model of the buried spiral coils a basic and simple starting point is to study a single ring-coil heat source. Suppose the ring coil is located on the plane *z=z'*  with its axis being coincident with *z*-axis with a continuous heating rate of *q* since a starting instant, *τ=τ'*. Based on the governing equation of transient heat conduction along with given boundary and initial conditions, the temperature response at location (*r, z*) in the medium to such a single ring source can be obtained according to the Green's function theory:

$$\Theta\_{r,s} = \frac{1}{8\pi^{3/2}} \int\_0^{Fo} \frac{1}{\left(Fo - Fo'\right)^{3/2}} I\_0\left[\frac{R}{2\left(Fo - Fo'\right)}\right] \exp\left[-\frac{R^2 + 1}{4\left(Fo - Fo'\right)}\right] \cdot \exp\left[-\frac{\left(Z - Z'\right)^2}{4\left(Fo - Fo'\right)}\right] dFo' \tag{26}$$

#### **5.2.1.1 Infinite ring-coil source model**

132 Modeling and Optimization of Renewable Energy Systems

and (b). The ring-coil model further takes into account the discontinuity of the heat source and the impact of the coil pitch by simplifying the buried spiral coil as a set of separated rings located on the cylindrical surface. This model has made big progress from the classical models, however, the ring heat sources is discontinuous and separated with each other,

> *b/2 b*

> > *2r0*

(c) Infinit spiral source model

*b*

*h2 z*

 

2 2

*R R Z Z I dFo*

*2r0*

(d) Finit spiral source model

*h1*

*b/2 b*

*z*

which still deviates from the realistic conditions of the buried spiral coil.

*b*

*h1*

*h2*

(b) Finit ring-coil source model

Evolved from the ring-coil source model, the "spiral source model" (Man et al., 2011) is further presented for better analyzing and designing the pile GHE with spiral coils, as shown in Fig. 9 (c) and (d). For this model, the buried spiral coil is approximated by a spiral line heat source. The temperature response of the buried spiral coils can be evaluated

In the ring-coil heat source model, the buried spiral coil is simplified as a number of separated rings located on the cylindrical surface. In order to analyze the thermal effect of the heat transfer along the axis on the total heat transfer efficiency, both the infinite and finite ring-coil source models are discussed by means of the Green's function method.

To develop the analytical model of the buried spiral coils a basic and simple starting point is to study a single ring-coil heat source. Suppose the ring coil is located on the plane *z=z'*  with its axis being coincident with *z*-axis with a continuous heating rate of *q* since a starting instant, *τ=τ'*. Based on the governing equation of transient heat conduction along with given boundary and initial conditions, the temperature response at location (*r, z*) in the medium to such a single ring source can be obtained according to the Green's function

exp exp <sup>8</sup> 24 4

 *Fo Fo Fo Fo Fo Fo Fo Fo* (26)

1 1 1

*<sup>z</sup> 2r0 2r0*

*z*

(a) Infinit ring-coil source model

Fig. 9. Established heat source models of the pile GHE

according to the analytical solutions of this model.

**5.2.1 Ring-coil heat source model and solutions** 

**5.2 Ring model and spiral model** 

, 0 3 2 <sup>0</sup> 3 2

*Fo*

Pile GHE with spiral coil

theory:

*r s*

The infinite ring-coil source model is first studied, which means the ring-coil source is assumed to be infinite in the longitudinal direction. Define that the *z*-coordinates of the ring coils are *zn b* 0.5 , where n=0, 1, 2,…, +∞. As a consequence, the overall temperature response at a random point in the medium to all the ring sources can be determined as the sum of all the individual temperature rises caused by each ring-coil source:

$$\Theta\_{r,i}\left(R,Z,Fo\right) = \frac{B}{8\pi^{3/2}} \sum\_{n=0}^{\infty} \int\_{0}^{Fo} \frac{1}{\left(Fo - Fo'\right)^{3/2}} I\_0\left[\frac{R}{2\left(Fo - Fo'\right)}\right] \exp\left[-\frac{R^2 + 1}{4\left(Fo - Fo'\right)}\right].\tag{27}$$

$$\left\{\exp\left[-\frac{\left(Z - nB - 0.5B\right)^2}{4\left(Fo - Fo'\right)}\right] + \exp\left[-\frac{\left(Z + nB + 0.5B\right)^2}{4\left(Fo - Fo'\right)}\right]\right\} dF\hat{o}^{\uparrow}.$$

#### **5.2.1.2 Finite ring-coil source model**

The infinite model neglects the effects of heat flow through the top and bottom ends of the heat source; therefore it is inadequate for the long-term operation of the GHE made of buried spiral coil. While keeping the ring-coil source simplification, the spiral coil is taken as a finite ring-coil source buried in a semi-infinite medium, stretching from *h*1 to *h*2 from the boundary of the ground surface. The coil is then approximated as *m* pieces of rings. Again, the images of the ring coils with negative heating rate *-qlb* are set on symmetry to the boundary in order to keep the constant temperature of the ground surface, and the solution for the finite ring-coil source model is expressed as:

$$\Theta\_{r,f}\left(R,Z,Fo\right) = \frac{B}{8\pi^{3/2}} \int\_0^{Fo} \frac{1}{\left(Fo - Fo\right)^{3/2}} I\_0\left[\frac{R}{2\left(Fo - Fo\right)}\right] \exp\left[-\frac{R^2 + 1}{4\left(Fo - Fo\right)}\right].$$

$$\sum\_{n=0}^m \left\{ \exp\left[-\frac{\left(Z - H1 - \left(n + 0.5\right)B\right)^2}{4\left(Fo - Fo\right)}\right] - \exp\left[-\frac{\left(Z + H1 + \left(n + 0.5\right)B\right)^2}{4\left(Fo - Fo\right)}\right] \right\} dF\alpha^\dagger.$$

#### **5.2.2 The spiral source model and its analytical solutions**

On the basis of the ring-coil source model, the spiral source model is further developed with increasing sophistication and accuracy to take the 3-D geometrical characteristic of spiral coil into account. In the spiral source model, the buried coil is represented by a spiral line heat source. Both the infinite and finite spiral source models are studied.

For an instantaneous point heat source with intensity of *c* , located at *r z* ', ', ' and activated at the instant , its Green's function in the cylindrical coordinates at point *r z* , , can be expressed as:

Heat Transfer Modeling of the Ground

Dimensionless temperature rise


Infinite spiral source model *R=r/r0*

0.0

0.1

0.2

0.3

Dimensionless temperature of pile GHE

0.4

0.5

0.6

0.7

0.8

*Fo=*1.0 *B*=1.0

Fig. 10. Temperature response of the infinite and finite spiral source models

with infinite and finite spiral source model are further compared in Fig. 11.

The dimensionless temperature response at the midpoint of spiral heat source calculated

 Infinite spiral source model Finite spiral source model

0.01 0.1 1 10 100 1000 10000

*/r*2 0

Dimensionless time Fo=*a*

Fig. 11. Dimensionless temperature vs. time from infinite and finite spiral source models

18

16

14

12

10

*Z=z/r0*

8

6

4

2

0

Dimensionless temperature rise


Finite spiral source model *R=r/r0*

*Fo=*1.0 *B*=1.0 *H*1*=*2.0 *H*2*=*12.0

*r,f*

> 1.000E-4 0.01250 0.02500 0.03750 0.05000 0.06250 0.07500 0.08100 0.08500 0.09000 0.09400 0.1000

18

16

14

12

10

*Z=z/r*

*0*

8

6

4

2

0

Heat Exchangers for the Ground-Coupled Heat Pump Systems 135

*r,f*

> 1.000E-4 0.01250 0.02500 0.03750 0.05000 0.06250 0.07500 0.08090 0.08390 0.08830 0.1020 0.1100

$$\begin{aligned} \text{G}\left(r,\,\rho,z,\tau;r',\,\rho',z',\tau'\right) &= \frac{1}{8\left[\pi a\left(\tau-\tau'\right)\right]^{3/2}}.\\ \text{b}\cdot\exp\left[-\frac{\left(r\cos\rho-r'\cos\rho'\right)^{2}+\left(r\sin\rho-r'\sin\rho'\right)^{2}+\left(z-z'\right)^{2}}{4a\left(\tau-\tau'\right)}\right] \end{aligned} \tag{29}$$

#### **5.2.2.1 Infinite spiral source model**

The infinite spiral source model is discussed with the axial heat flow neglected, as shown in Fig. 9 (c). This spiral heat source can be considered as the sum, or integral, of numerous point heat sources located on the spiral line with the instantaneous intensity of *q bd d <sup>l</sup>* 2 , which cylindrical coordinates keep *r rz b* <sup>0</sup> , 2 . Then the temperature response in the medium resulted from the step heating of the spiral heat source from the starting instant 0 can be deduced according to the Green's function and superimposing theory based on the temperature response to a point heat source:

$$\Theta\_{i,spin} = \frac{B}{16\pi^{5/2}} \int\_0^{Fo} \left(\frac{1}{Fo - Fo'}\right)^{3/2} \cdot \int\_{-\infty}^{v} \exp\left[-\frac{R^2 + 1 - 2R\cos(\rho - \rho') + \left(Z - B\rho'/2\pi\right)^2}{4\left(Fo - Fo'\right)}\right] d\rho' dFo' \tag{30}$$

#### **5.2.2.2 Finite spiral source model**

In order to take the effects of heat flow through the top and bottom ends of pile into account and investigate the long-term operation performance of the pile GHE, the finite spiral source model is proposed and analyzed. In this model, the coil pipe buried in the pile is considered as a finite-length spiral coil in a semi-infinite medium, as shown in Fig. 9 (d).

With the virtual heat source theory, a virtual spiral heat sink with negative heating rate -*ql* and of identical physical dimensions is set on symmetry to the boundary. Then the finite spiral heat source and heat sink can be approximated as the sum of numerous point heat sources and heat sinks. Again, the Green's function theory is employed to obtain the temperature response of the medium. For the finite-length spiral source starts at 1 *z' h* , or <sup>1</sup> 2 *h b* , and ends at <sup>2</sup> 2 *h b* buried in a semi-infinite medium with a step heating rate per length of the pile, *ql*, the solution may be derived:

$$\Theta\_{f, spiral} = \frac{B}{16\pi^{5/2}} \int\_0^{Fo} \left(\frac{1}{Fo - Fo}\right)^{3/2} \cdot \exp\left[-\frac{R^2 + 1}{4\left(Fo - Fo\right)}\right].$$

$$\int\_{2\pi H\_1/\mathcal{B}}^{2\pi H\_2/\mathcal{B}} \exp\left[\frac{2R\cos(\rho - \rho')}{4\left(Fo - Fo\right)}\right] \cdot \left\{\exp\left[-\frac{\left(Z - B\rho'/2\pi\right)^2}{4\left(Fo - Fo\right)}\right] - \exp\left[-\frac{\left(Z + B\rho'/2\pi\right)^2}{4\left(Fo - Fo\right)}\right]\right\} d\rho' dFo'$$

Take an example of buried spiral source with *B*=1, *H1*=2.0, and *H2*=12.0, the temperature distributions in the longitudinal profile of pile as well as the ground at the dimensionless time of *Fo*=1.0 calculated with infinite and finite spiral source model are compared in Fig. 10. In general, the spiral configuration of the heat source gets well representation.

*a*

The infinite spiral source model is discussed with the axial heat flow neglected, as shown in Fig. 9 (c). This spiral heat source can be considered as the sum, or integral, of numerous point heat sources located on the spiral line with the instantaneous intensity of

temperature response in the medium resulted from the step heating of the spiral heat source

In order to take the effects of heat flow through the top and bottom ends of pile into account and investigate the long-term operation performance of the pile GHE, the finite spiral source model is proposed and analyzed. In this model, the coil pipe buried in the pile is considered

With the virtual heat source theory, a virtual spiral heat sink with negative heating rate -*ql* and of identical physical dimensions is set on symmetry to the boundary. Then the finite spiral heat source and heat sink can be approximated as the sum of numerous point heat sources and heat sinks. Again, the Green's function theory is employed to obtain the temperature response of the medium. For the finite-length spiral source starts at 1 *z' h* ,

 

 

*Fo Fo' Fo Fo' Fo Fo'*

*<sup>R</sup> Z B Z B d dFo'*

2 cos( ') 2 2

Take an example of buried spiral source with *B*=1, *H1*=2.0, and *H2*=12.0, the temperature distributions in the longitudinal profile of pile as well as the ground at the dimensionless time of *Fo*=1.0 calculated with infinite and finite spiral source model are compared in Fig.

8

cos cos sin sin

2 , which cylindrical coordinates keep *r rz b* <sup>0</sup> , 2

superimposing theory based on the temperature response to a point heat source:

as a finite-length spiral coil in a semi-infinite medium, as shown in Fig. 9 (d).

3/2 2

1 1

*Fo Fo' Fo Fo'*

2 2 2

exp exp exp <sup>444</sup>

10. In general, the spiral configuration of the heat source gets well representation.

 

<sup>1</sup> 1 2 cos( ) 2 exp <sup>16</sup> <sup>4</sup>

*Fo Fo' Fo Fo'*

*B R R ' Z B'*

1

3 2

2 2 2

can be deduced according to the Green's function and

3/2 2 2

(30)

2 *h b* buried in a semi-infinite medium with a step heating

 

 

(31)

(29)

. Then the

*d 'dFo'*

 

 

 

*r r r r zz a*

 

 

exp <sup>4</sup>

, ,,; , , ,

*Gr z r z*

**5.2.2.1 Infinite spiral source model** 

from the starting instant 0

*Fo*

**5.2.2.2 Finite spiral source model** 

2 *h b* , and ends at <sup>2</sup>

*Fo*

exp <sup>16</sup> <sup>4</sup>

 

rate per length of the pile, *ql*, the solution may be derived:

*B R*

 

, 5/2 <sup>0</sup>

*q bd d <sup>l</sup>* 

*i spiral*

or <sup>1</sup> 

 

2

*H B*

, 5/2 <sup>0</sup>

1

*H B*

2

*f spiral*

Fig. 10. Temperature response of the infinite and finite spiral source models

The dimensionless temperature response at the midpoint of spiral heat source calculated with infinite and finite spiral source model are further compared in Fig. 11.

Fig. 11. Dimensionless temperature vs. time from infinite and finite spiral source models

Heat Transfer Modeling of the Ground

equation (32) turns to be:

Heat Exchangers for the Ground-Coupled Heat Pump Systems 137

On determination of the temperature rise on a certain borehole wall it is important to distinguish the temperature rise caused by the heat source (U-tubes) in the borehole itself, which is usually the most significant, and those caused by thermal interference from other boreholes in the GHE. The spaces between adjacent boreholes are much greater than borehole radius, as a consequence, the minor discrepancy in the temperature rises on the borehole perimeter in the circumferential direction caused by an adjacent borehole can be neglected, and, then, the distance between the two borehole axes is counted. As a result, for calculation of the temperature rise on a borehole wall

<sup>1</sup>

In the GHE with multiple boreholes the boreholes experience diversified temperature responses owing to their specific locations in the GHE configuration and, then, different heat transfer conditions. A representative borehole needs to be selected to determine the temperature rise on the borehole wall to avoid too large a workload of computation. It is usually recommended for engineering design to take the least favorable borehole as the

 

*m b j*

representative, i.e. the one with largest temperature rise in the GHE. It is defined as:

,

 *Max Max* 

*e m i i ij*

It is easy to locate the least favorable borehole in most of the GHE configurations for pure conduction models; and this choice is conservative for GHE design. While ensuring safe operation of the GHE, it leads to over-sizing of the GHE and aggravating its cost. In view of fact that larger and larger GHEs are constructed consisting of hundreds boreholes in a single GHE, this choice of the representative borehole can result in too severe deviations. A desirable alternative for the representative borehole would be the one whose temperature rise follows closely the average temperature rise of the GHE. This task is demanding even for the pure conduction models due to the wide diversity of possible configurations of the GHE. Some studies are under the way on this subject, and the borehole locating at the nearest vicinity of the geometric center of a quarter of a matrix configuration of the GHE is considered to be an appropriate choice for the representative one in pure conduction models (Lin, 2010). For the advection models it is even more intricate to find out a proper representative borehole because another factor, the velocity of the groundwater infiltration is incorporated into the model while its direction has added more numerous variations relative to the orientation of the GHE configuration. This seems a problem which needs to

GCHP systems can provide buildings with heating and cooling in different seasons, so heat can be extracted from or rejected to the ground. As defined, the heat load *q*l means the heat

where *x*j is the distance between the *j* borehole and the borehole concerned.

 be addressed to properly in such an approach.

**7. Temperature response to variable loads** 

1 ,0.5 , ,0.5 , *N*

 

 

(33)

1

(34)

*j j i*

 

 

*N*

*j*

 *rH xH* 

As shown in Fig. 11, the finite model yields a relatively lower temperature response compared with the infinite model. Temperature rises calculated with these two models are in good agreement for a short period after the start of heating. As time goes on, remarkable discrepancy of the finite model from the infinite one appears since the former takes the heat transfer through the top and bottom ends of the pile into account. While the temperature response rises continuously with time for the infinite model, temperature response for the finite model tends to a steady state as time approaches infinity. This feature indicates the importance to take the finite length effect into account in consideration of the long term operation of the buried spiral heat source. The heat transfer features of the pile GHE can be adequately described by the finite spiral source model, and its analytical solutions have provided a desirable tool for simulating the pile GHE and prompting its applications.

#### **6. Ground heat exchangers with multiple boreholes**

GHEs in practical GCHP projects usually consist of multiple boreholes. The conduction problems under the assumption of constant properties satisfy the condition of superposition; therefore the temperature rise at a certain location in the GHE with multiple boreholes can be obtained by means of summing up all the individual temperature excesses caused by each of the boreholes at the concerned spot, that is:

$$\theta\_m(\boldsymbol{\tau}) = \sum\_{i=1}^N \theta\_i(\boldsymbol{\tau}) \tag{32}$$

According to simulation requirements and conditions the function for the temperature excess,*<sup>i</sup>* , caused by a single borehole may be determined from models discussed in previous sections such as the finite line source model or cylindrical source model.

As mentioned above, while the superposition approach is employed for thermal analysis of GHEs, the domain involved is divided into two separate regions, i.e. the region inside the borehole and that outside it. For the former region the heat transfer is considered as steadystate. For heat transfer outside the borehole, transient heat transfer models should be used. The temperature on the interface of the two regions, i.e. the borehole wall, constitutes a key link of the thermal analyses in the two regions. The mean temperature of the circulating fluid, and, then its inlet and outlet temperatures, varying with time, can be determined with the borehole temperature plus a temperature difference resulted from the borehole resistance. The temperature on borehole wall, however varies along its depth as indicated in previous discussions, and differs from each other among different boreholes. As a consequence, it is desirable to define a representative temperature of the borehole wall for the entire borehole field so as to keep the analysis concise enough for engineering design and thermal analysis purposes.

Normally, the temperature response at the midpoint of borehole in depth-direction is selected to represent the borehole wall temperature response for each individual borehole. Although more sophisticated approaches have been investigated such as taking the integrated average temperature along the borehole depth as the representative one, the study (Zeng et al. 2003) has shown that the simpler choice of the midpoint temperature as the representative one is acceptable for engineering applications.

As shown in Fig. 11, the finite model yields a relatively lower temperature response compared with the infinite model. Temperature rises calculated with these two models are in good agreement for a short period after the start of heating. As time goes on, remarkable discrepancy of the finite model from the infinite one appears since the former takes the heat transfer through the top and bottom ends of the pile into account. While the temperature response rises continuously with time for the infinite model, temperature response for the finite model tends to a steady state as time approaches infinity. This feature indicates the importance to take the finite length effect into account in consideration of the long term operation of the buried spiral heat source. The heat transfer features of the pile GHE can be adequately described by the finite spiral source model, and its analytical solutions have provided a desirable tool for simulating the pile GHE and prompting its applications.

GHEs in practical GCHP projects usually consist of multiple boreholes. The conduction problems under the assumption of constant properties satisfy the condition of superposition; therefore the temperature rise at a certain location in the GHE with multiple boreholes can be obtained by means of summing up all the individual temperature excesses

> 1

According to simulation requirements and conditions the function for the temperature

As mentioned above, while the superposition approach is employed for thermal analysis of GHEs, the domain involved is divided into two separate regions, i.e. the region inside the borehole and that outside it. For the former region the heat transfer is considered as steadystate. For heat transfer outside the borehole, transient heat transfer models should be used. The temperature on the interface of the two regions, i.e. the borehole wall, constitutes a key link of the thermal analyses in the two regions. The mean temperature of the circulating fluid, and, then its inlet and outlet temperatures, varying with time, can be determined with the borehole temperature plus a temperature difference resulted from the borehole resistance. The temperature on borehole wall, however varies along its depth as indicated in previous discussions, and differs from each other among different boreholes. As a consequence, it is desirable to define a representative temperature of the borehole wall for the entire borehole field so as to keep the analysis concise enough for engineering design

Normally, the temperature response at the midpoint of borehole in depth-direction is selected to represent the borehole wall temperature response for each individual borehole. Although more sophisticated approaches have been investigated such as taking the integrated average temperature along the borehole depth as the representative one, the study (Zeng et al. 2003) has shown that the simpler choice of the midpoint temperature as

the representative one is acceptable for engineering applications.

 

, caused by a single borehole may be determined from models discussed in

(32)

 

previous sections such as the finite line source model or cylindrical source model.

*N m i i*

**6. Ground heat exchangers with multiple boreholes** 

caused by each of the boreholes at the concerned spot, that is:

excess,*<sup>i</sup>* 

and thermal analysis purposes.

On determination of the temperature rise on a certain borehole wall it is important to distinguish the temperature rise caused by the heat source (U-tubes) in the borehole itself, which is usually the most significant, and those caused by thermal interference from other boreholes in the GHE. The spaces between adjacent boreholes are much greater than borehole radius, as a consequence, the minor discrepancy in the temperature rises on the borehole perimeter in the circumferential direction caused by an adjacent borehole can be neglected, and, then, the distance between the two borehole axes is counted. As a result, for calculation of the temperature rise on a borehole wall equation (32) turns to be:

$$\theta\_m(\tau) = \theta\left(r\_b, 0.5H, \tau\right) + \sum\_{j=1}^{N-1} \theta\left(\mathbf{x}\_j, 0.5H, \tau\right) \tag{33}$$

where *x*j is the distance between the *j* borehole and the borehole concerned.

In the GHE with multiple boreholes the boreholes experience diversified temperature responses owing to their specific locations in the GHE configuration and, then, different heat transfer conditions. A representative borehole needs to be selected to determine the temperature rise on the borehole wall to avoid too large a workload of computation. It is usually recommended for engineering design to take the least favorable borehole as the representative, i.e. the one with largest temperature rise in the GHE. It is defined as:

$$\Theta\_{\varepsilon} = \text{Max}\left(\Theta\_{m,i}\right) = \text{Max}\left(\Theta\_{i} + \sum\_{\substack{j=1\\j\neq i}}^{N} \Theta\_{ij}\right) \tag{34}$$

It is easy to locate the least favorable borehole in most of the GHE configurations for pure conduction models; and this choice is conservative for GHE design. While ensuring safe operation of the GHE, it leads to over-sizing of the GHE and aggravating its cost. In view of fact that larger and larger GHEs are constructed consisting of hundreds boreholes in a single GHE, this choice of the representative borehole can result in too severe deviations. A desirable alternative for the representative borehole would be the one whose temperature rise follows closely the average temperature rise of the GHE. This task is demanding even for the pure conduction models due to the wide diversity of possible configurations of the GHE. Some studies are under the way on this subject, and the borehole locating at the nearest vicinity of the geometric center of a quarter of a matrix configuration of the GHE is considered to be an appropriate choice for the representative one in pure conduction models (Lin, 2010). For the advection models it is even more intricate to find out a proper representative borehole because another factor, the velocity of the groundwater infiltration is incorporated into the model while its direction has added more numerous variations relative to the orientation of the GHE configuration. This seems a problem which needs to be addressed to properly in such an approach.

#### **7. Temperature response to variable loads**

GCHP systems can provide buildings with heating and cooling in different seasons, so heat can be extracted from or rejected to the ground. As defined, the heat load *q*l means the heat

Heat Transfer Modeling of the Ground

can be obtained by superposition as:

intensity is *ql* /*C* , as shown in Fig. 14 (a).

Fig. 14. A cyclic pulse load and its simplification

span of a single pulse.

Heat Exchangers for the Ground-Coupled Heat Pump Systems 139

On the assumption of a uniform heating rate *q*l in all the boreholes of the GHE, the *g*function is independent of *q*l. The temperature response to the step heating on the representative borehole wall can be determined on basis of the models presented in previous sections together with the superposition procedures such as that of equation (34). For sequential heating pulses shown in Fig. 12, the borehole wall temperature rise at time

1 0

(36)

 

<sup>1</sup> . , ( 0) <sup>2</sup> *i i ll i l*

*qq g q <sup>k</sup>*

Let's consider a simple case of periodic on-off operation of the GCHP system to demonstrate the impact of discontinuous heating. Assume that the discontinuous GCHP operation is cyclic with a period T, of which the on-time is T1. Then, C=T1/T denotes the on-time ratio. If the average heating intensity over the operating period is denoted by *<sup>l</sup> q* , the pulse heating

The borehole wall temperature response in a single borehole GHE to such cyclic pulse heating is calculated based on the line source model. Fig. 15 shows the borehole wall temperature response with the same average heating intensity but different operating time ratios. Simulations indicate that on such conditions the borehole wall temperature oscillates significantly while rising gradually over cycles. Furthermore, the smaller the operating time ratio is, which corresponds to stronger pulse intensities, the larger temperature swings are resulted in. The maximum temperature rise of the fluid after an operating period is an important criterion in design and thermal analysis of the GHEs. Study has shown that the maximum borehole wall temperature rise due to the periodic on-off heating load can be approximated by superposition of temperature rises caused by a continuous mean load and a single heating pulse as shown in Fig. 14 (b). Fig. 16 shows that the maximum temperature rise obtained from the simplified model is equivalent to that from the exact periodical pulse load model. Thus, this simplification provides an approach to analyze discontinuous loads over long durations. The results also indicate that the maximum temperature rise depends not only on the mean load over the whole duration, but also on the intensity and operating

1

*i*

 rejected to the ground, so the temperature rise in ground is positive; they both turn to negatives if heat is extracted from the ground.

In order to analyze the intricate heat transfer process in the GHEs effectively and efficiently, a basic and simple model must be established and solved first, and then, more complicated factors are added gradually. The basic problem is the heat transfer of a single borehole under a step heating, which means a constant heating rate starting from a certain instant. All the models for outside boreholes heat transfer discussed in previous sections deal with the primary problem of step heating.

The heat extracted from or rejected to the ground varies with time because the GCHP load usually varies with time. The variable heat flow can be approximated by a pulse train of heating load as shown in Fig. 12. A heating pulse imposed in the time interval *i i* <sup>1</sup> can be considered as superposition of two step heating fluxes, as shown in Fig. 13.

Fig. 12. Continuous heating approximatedby a rectangular pulse train

Fig. 13. A pulse heating equals two step heating fluxes

In order to facilitate computation of the temperature response of the GHEs to such sequential heating pulses the concept of so-called *g*-function is usually introduced. The *g*function represents the non-dimensional temperature response on the representative borehole wall to the step heating for a specific configuration of the GHE, which is defined as

$$\log\left(\tau\right) = \frac{2\pi k \theta\_e}{q\_l} \tag{35}$$

rejected to the ground, so the temperature rise in ground is positive; they both turn to

In order to analyze the intricate heat transfer process in the GHEs effectively and efficiently, a basic and simple model must be established and solved first, and then, more complicated factors are added gradually. The basic problem is the heat transfer of a single borehole under a step heating, which means a constant heating rate starting from a certain instant. All the models for outside boreholes heat transfer discussed in previous sections deal with the

The heat extracted from or rejected to the ground varies with time because the GCHP load usually varies with time. The variable heat flow can be approximated by a pulse train of heating load as shown in Fig. 12. A heating pulse imposed in the time interval *i i* <sup>1</sup>

In order to facilitate computation of the temperature response of the GHEs to such sequential heating pulses the concept of so-called *g*-function is usually introduced. The *g*function represents the non-dimensional temperature response on the representative borehole wall to the step heating for a specific configuration of the GHE, which is defined as

> <sup>2</sup> *<sup>e</sup> l k*

*q* 

*g*

can be considered as superposition of two step heating fluxes, as shown in Fig. 13.

Fig. 12. Continuous heating approximatedby a rectangular pulse train

Fig. 13. A pulse heating equals two step heating fluxes

 

(35)

negatives if heat is extracted from the ground.

primary problem of step heating.

On the assumption of a uniform heating rate *q*l in all the boreholes of the GHE, the *g*function is independent of *q*l. The temperature response to the step heating on the representative borehole wall can be determined on basis of the models presented in previous sections together with the superposition procedures such as that of equation (34).

For sequential heating pulses shown in Fig. 12, the borehole wall temperature rise at time can be obtained by superposition as:

$$\theta = \frac{1}{2\pi k} \sum\_{i=1}^{n} \left( q\_{l\_i} - q\_{l\_{i-1}} \right) \cdot \mathbf{g} \left( \tau - \tau\_i \right), \qquad \left( q\_{l\_0} = 0 \right) \tag{36}$$

Let's consider a simple case of periodic on-off operation of the GCHP system to demonstrate the impact of discontinuous heating. Assume that the discontinuous GCHP operation is cyclic with a period T, of which the on-time is T1. Then, C=T1/T denotes the on-time ratio. If the average heating intensity over the operating period is denoted by *<sup>l</sup> q* , the pulse heating intensity is *ql* /*C* , as shown in Fig. 14 (a).

Fig. 14. A cyclic pulse load and its simplification

The borehole wall temperature response in a single borehole GHE to such cyclic pulse heating is calculated based on the line source model. Fig. 15 shows the borehole wall temperature response with the same average heating intensity but different operating time ratios. Simulations indicate that on such conditions the borehole wall temperature oscillates significantly while rising gradually over cycles. Furthermore, the smaller the operating time ratio is, which corresponds to stronger pulse intensities, the larger temperature swings are resulted in. The maximum temperature rise of the fluid after an operating period is an important criterion in design and thermal analysis of the GHEs. Study has shown that the maximum borehole wall temperature rise due to the periodic on-off heating load can be approximated by superposition of temperature rises caused by a continuous mean load and a single heating pulse as shown in Fig. 14 (b). Fig. 16 shows that the maximum temperature rise obtained from the simplified model is equivalent to that from the exact periodical pulse load model. Thus, this simplification provides an approach to analyze discontinuous loads over long durations. The results also indicate that the maximum temperature rise depends not only on the mean load over the whole duration, but also on the intensity and operating span of a single pulse.

Heat Transfer Modeling of the Ground

method.

Heat Exchangers for the Ground-Coupled Heat Pump Systems 141

to a heat pump for a given set of design conditions, such as building load, ground thermal properties, borehole configuration, and heat pump operating characteristics. The heat transfer models mentioned above are employed in the software, including the analytical solution of the finite line source model for the thermal resistance outside boreholes and the quasi 3-D model for the thermal resistance inside boreholes. In addition, the modeling procedure uses spatial superimposition for multiple boreholes and sequential temporal superimposition to dealing with the dynamic heating and cooling loads of the systems. The flow chart of the computing procedure for the model implementation is described in Fig. 17. The design process is actually a simulation-based process by means of the trial-and-error

Begin

Set the max and min Temp of EFT Input the GHE size

Assume the GHE size Cal resistance of borehole

Cal the resistance of borehole Cal the predicted EFT, ExFT

output

Cal the heat transfer rate of GHE and power consumption of HP

end

The program with a friendly interface and visual graph has been developed under the Delphi Environment. In the visual interface, all the geometry parameters and inlet conditions can be set up in dialog boxes which can be popped up by clicking the different pages, as shown in Fig. 18. When all the required parameters are set up, the pre-compiled

Cal the predicted EFT, ExFT

Design

*EFTset EFTCal*

Fig. 17. The flowchart of the GeoStar program

program will begin to simulate or design under the specific conditions.

Yes

No

Adjust GHE size

Input heat pump data and building loads

Simulate

Fig. 15. Borehole wall temperature response to cyclic pulse loads

Fig. 16. Temperature response to a cyclic load and its simplification

#### **8. Design and simulation software for ground heat exchangers**

As mentioned above, the heat transfer process in a GHE involves quite a number of factors. It is necessary to further develop an accurate, reliable and convenient program for GHE design and simulation. In the last decade, a number of GHE models have been developed and they have been combined, directly or indirectly, with models of the building, heat pumps, and other components in various modeling environments such as TRNSYS, EnergyPlus, eQuest, and HVACSIM+. The GHE model used in TRNSYS (Hellström, 1989) is called the Duct Ground Heat Storage model, originally intended for underground thermal storage systems. The model uses numerical solutions for the global heat transfer between the storage volume and the far-field, and for the local problem of the heat transfer around the boreholes. An analytical method is employed to solve the steady-flux problem around the nearest pipe. The three models implemented in HVACSIM+ (Xu & Spitler, 2006), EnergyPlus (Fisher, 2006) and eQuest (Liu, 2008) have a common heritage, which are based on extensions of Eskilson's model (1987). The programs are based on pre-computed response functions for specific GHE geometries.

On basis of the study on the heat transfer modeling of the GHEs, a software package in Chinese interface named GeoStar has been developed and spread for the design and simulation of the GHEs mainly in China (Fang et al., 2002). This software package is able to size GHEs to meet the user-specified minimum and maximum entering fluid temperatures

Fig. 15. Borehole wall temperature response to cyclic pulse loads

Fig. 16. Temperature response to a cyclic load and its simplification

response functions for specific GHE geometries.

**8. Design and simulation software for ground heat exchangers** 

As mentioned above, the heat transfer process in a GHE involves quite a number of factors. It is necessary to further develop an accurate, reliable and convenient program for GHE design and simulation. In the last decade, a number of GHE models have been developed and they have been combined, directly or indirectly, with models of the building, heat pumps, and other components in various modeling environments such as TRNSYS, EnergyPlus, eQuest, and HVACSIM+. The GHE model used in TRNSYS (Hellström, 1989) is called the Duct Ground Heat Storage model, originally intended for underground thermal storage systems. The model uses numerical solutions for the global heat transfer between the storage volume and the far-field, and for the local problem of the heat transfer around the boreholes. An analytical method is employed to solve the steady-flux problem around the nearest pipe. The three models implemented in HVACSIM+ (Xu & Spitler, 2006), EnergyPlus (Fisher, 2006) and eQuest (Liu, 2008) have a common heritage, which are based on extensions of Eskilson's model (1987). The programs are based on pre-computed

On basis of the study on the heat transfer modeling of the GHEs, a software package in Chinese interface named GeoStar has been developed and spread for the design and simulation of the GHEs mainly in China (Fang et al., 2002). This software package is able to size GHEs to meet the user-specified minimum and maximum entering fluid temperatures to a heat pump for a given set of design conditions, such as building load, ground thermal properties, borehole configuration, and heat pump operating characteristics. The heat transfer models mentioned above are employed in the software, including the analytical solution of the finite line source model for the thermal resistance outside boreholes and the quasi 3-D model for the thermal resistance inside boreholes. In addition, the modeling procedure uses spatial superimposition for multiple boreholes and sequential temporal superimposition to dealing with the dynamic heating and cooling loads of the systems. The flow chart of the computing procedure for the model implementation is described in Fig. 17. The design process is actually a simulation-based process by means of the trial-and-error method.

Fig. 17. The flowchart of the GeoStar program

The program with a friendly interface and visual graph has been developed under the Delphi Environment. In the visual interface, all the geometry parameters and inlet conditions can be set up in dialog boxes which can be popped up by clicking the different pages, as shown in Fig. 18. When all the required parameters are set up, the pre-compiled program will begin to simulate or design under the specific conditions.

Heat Transfer Modeling of the Ground

expounded in particular, which include mainly

interference among the legs of the U-tube.

**9. Conclusion** 

engineering.

account.

applications of the GCHP systems.

ASHRAE.

pp. 1169-1175.

University Press, New York.

**10. References** 

Heat Exchangers for the Ground-Coupled Heat Pump Systems 143

This chapter presents a comprehensive review of the study on heat transfer modeling of the ground heat exchangers in ground-coupled heat pump systems by means of superposition principle and analytical solutions. An entire set of techniques are provided to develop computer software for thermal analysis and design of GHEs in practical

Contributions of our research group to improvement of the GHE heat transfer modeling are

1. A quasi 3-D model for heat transfer inside borehole has been proposed, and its analytical solution derived to account for the borehole geometry and thermal

2. The explicit analytical solutions of 2-D and 3-D models for vertical and inclined finite line source model for heat transfer outside borehole have been obtained to consider the axial heat flow and take the long-term effect of the limited borehole depth into

3. A solid cylindrical source model has been developed which gives a better description of the short-term temperature response for boreholes than the traditional 1-D models do,

4. The advection models of infinite and finite line source have been proposed, and their analytical solutions derived to deal with the combined conductive and convective heat

5. The ring-coil source model and the spiral source model established for the heat transfer of a buried spiral coil to simulate heat transfer in the foundation pile GHE, which

The entire modeling uses the techniques of spatial superimposition for multiple boreholes and sequential temporal superimposition for arbitrary heating/cooling loads of the systems. The heat transfer models for the borehole GHE have been incorporated into a computer program, developed by our research group for providing a reliable and useful tool to design and simulate the GHE of GCHP systems. These studies on GHE heat transfer modeling in this chapter is expected to provide supports for developing the technique and promoting

Bose, J. E.; Parker, J. D. & McQuiston, F. C. (1985). Design/data manual for

Cui, P; Yang, H. X.; & Fang, Z. H. (2006). Heat transfer analysis of ground heat

Carslaw, H. S. & Jeager, J. C. (1947). Conduction of heat in solids, Oxford Press, Oxford. Carslaw, H. S. & Jaeger, J. C. (1959). Conduction of Heat in Solids, second ed. Oxford

closed-loop ground coupled heat pump systems, Oklahoma State University for

exchangers with inclined boreholes. Applied Thermal Engineering, 2006, 26:

and may also serve as a tool for the pile GHE thermal analysis.

having created a new frontier of the GCHP applications.

transfer in GHEs with groundwater infiltration taken into account.


Fig. 18. Wizard user interface

In the past ten years a number of GHE models have been developed for use with various building simulation programs. In view of this, one of the authors together with other researchers conducted a comparison of GHE models developed for use with programs, including GeoStar, TRNSYS, HVACSIM+, GEOEASEⅡ and eQuest (Spitler et al., 2009). The experimental validation was also carried out between the models and the experimental results. One of the research results is described in Fig. 19, which illustrated the predicted monthly average borehole ExFT and the measured data, when the hourly heat transfer rate was specified. It can be seen that all of the models (including GeoStar) predicted the ExFT within 1°C, except HVACSIM+ which overpredicted a maximum of 2°C, for the summer cooling months during the first year. For the shoulder seasons and heating months, the errors decreased slightly.

Fig. 19. Comparisons of experimental and predicted monthly average borehole ExFTs
