New temperature response functions (G functions) for pile and

Energy 38 (2012) 255e263
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Energy
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New temperature response functions (G functions) for pile and borehole ground
heat exchangers based on composite-medium line-source theory
Min Li, Alvin C.K. Lai*
Department of Civil and Architectural Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 19 September 2011
Received in revised form
1 December 2011
Accepted 3 December 2011
Available online 26 January 2012
This paper presents a new approach to modeling of heat transfer by ground heat exchangers, involving
unsteady heat conduction in composite media together with complex geometry. Analytical solutions for
continuous line and cylindrical-surface sources are developed based on Jaeger’s instantaneous linesource solution for composite media. New temperature response functions (G functions) are also presented for pile ground heat exchangers with spiral coils and for borehole ground heat exchangers with
single or double U-shaped tubes. These temperature response functions can be used to analyze the
impact of difference between properties of materials inside and outside boreholes or piles on the
performance of ground heat exchangers. Theoretical results show that the difference in properties is an
important factor affecting the temperature response of ground heat exchangers. Since the heat capacity
of grout, inside boreholes, is fully considered by this line-source theory, the new G functions should be
particularly suitable for predicting small-time dynamic behaviors of ground heat exchangers. Therefore,
these G functions may be significant for energy analysis and in-situ thermal response tests of groundcoupled heat pump systems.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
Ground heat exchanger
Energy pile
Temperature response function
Ground-coupled heat pump system
Composite medium
1. Introduction
Ground-coupled heat pump systems (GCHPs) use the ground as
a heat source and sink as they transfer heat through buried ground
heat exchangers (GHEs). Compared to ambient air, the ground
provides low temperature for cooling and high temperature for
heating, and the temperature fluctuation is low. Therefore, GCHPs
offer an environment-friendly and energy-efficient way of
providing cooling and heating, as well as hot water. Because of its
economic benefits, installation of GCHP systems in both residential
and commercial buildings has become increasingly commonplace
in many countries [1e7]. However, the promotion of GCHPs is
impeded by relatively high initial cost, large land area requirement,
and difficulties and uncertainty of computation and design of GHEs.
Much research interest exists in resolving these practical problems.
To improve the reliability of design and simulation of GCHEs,
many researchers have investigated heat transfer processes associated with GHEs. A borehole GHE consists of one or two U-shaped
tubes inserted into a vertical borehole and connected to a heat
pump to form a closed loop (Fig. 1), and water with or without
antifreeze is circulated in the closed loop. The space between the
* Corresponding author. Tel.: þ852 3442 6299; fax: þ852 2788 7612.
E-mail addresses: [email protected] (M. Li), [email protected] (A.C.K. Lai).
0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.energy.2011.12.004
borehole wall and the U-tubes is filled with grout to enhance the
heat transfer between soil and the circulated fluid. When analyzing
the heat transfer processes, the borehole GHE is theoretically
separated into two parts with the radius of the borehole. Typically,
the radius of the borehole used for GHE is about 0.06 m, which is
very small compared to its length, which is tens of meters. So, heat
conduction outside a borehole may be modeled as a line of heat
sources of infinite or finite length liberating heat to the surrounding
soil [8,9]. This process can also be treated as unsteady heat
conduction in an infinite region bounded internally by a circular
cylinder [10,11]. In contrast, heat transfer inside boreholes is
generally assumed to be in steady-state due to the much smaller
dimensions and heat capacity. Several models with varying
complexity were developed for this purpose; for example, the
simplest one-dimensional model [12], two-dimensional models
[13], and a quasi-three-dimensional model [14]. But the steadystate assumption may cause somewhat errors when modeling
short-time temperature responses of borehole GHEs, in which the
heat capacity of grout affects the heat transfer process.
Modeling short-time responses of GHEs is significant and
important for analysis of energy consumption of GCHPs, design of
hybrid GSHPs, and in-situ thermal response tests of GHEs [15,16].
A great challenge, however, occurs when analyzing this short-time
responses. The challenge of this problem is solving the unsteady
256
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
Fig. 1. Schematic layout of Borehole GHE with a single or double U-shaped tubes.
heat conduction in composite solids with complex geometric
arrangements. One way to solve this problem is numerical
methods [17,18]. But numerical methods are computationally
expensive and lack enough flexibility to model various forms of
heat transfer tubes. Another way is to simplify the geometric
arrangements, for example, by modeling U-shaped tubes as a pipe
with an “equivalent” diameter. This assumption reduces the
original problem to a problem of transient heat conduction in
a hollow cylindrical composite region. This problem has been
solved by several researchers using the generalized orthogonal
expansion technique [19] or the method of Laplace transform
[20,21]. Better ways to model the short-time response of borehole
GHEs are still lacking.
Another practical problem of GCHPs is that large land area is
required for installing GHEs. To reduce this requirement, foundation
piles of buildings may be used for installing heat transfer tubes,
which are called “energy piles” or pile ground heat exchangers
[4,22]. In energy piles, heat transfer tubes are arranged near the steel
frame of foundation piles in various forms (Fig. 2). These forms
usually have large heat transfer areas compared to single or double
U-shaped tubes used in borehole GHEs, though they cause complications in analysis of relevant heat transfer processes. Since energy
Fig. 2. Schematic layout of Energy piles with a spiral coils and a W-shaped tube.
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
piles can reduce the requirement of land area, this technology has
evoked increasing interest in research community [23e28].
A solid cylindrical-source and a ring-coil-source models have
been proposed for pile GHEs with spiral coils [26,27], assuming that
heat is released from the coil pipe wall and ignoring the difference
between properties of soil and foundation piles. The ring-coil-source
model treats the spiral coil as a series of separated rings liberating
heat in an infinite or semi-infinite homogenous medium [26]. The
solid cylindrical-source model models spiral coil as a cylindrical
surface releasing heat in the ground [27]. Similarly, the traditional
line-source theory has also been used to evaluate the performance of
energy piles [28], ignoring the difference between thermal properties of piles and soil.
So far, there are two knowledge gaps that need to be closed in
the field of modeling heat transfer by GHEs. The first is that no
reliable analytical solution is available for modeling the short-term
unsteady behaviors of borehole GHEs with geometric complexity
inside boreholes. The traditional heat source models ignore the
difference between properties of soil and piles. Soil composition
varies widely not only with locations but also from wet clay to
sandy soil. So, heat conduction associated with an energy pile
definitely occurs in composite media. Therefore, the second gap is
that no analytical model can explicitly consider the difference
between properties of soil and piles.
This paper aims to fill these two gaps by introducing the line-source
theory for composite media to GCHP applications. The advantages of
this theory are that it can be used to analyze transient heat conduction
inside and outside boreholes, and it includes the effect of the difference
between properties of soil and piles/grout. Further, it is flexible enough
to model many configurations of heat transfer tubes, including single
and double U-shaped tubes, W-shaped tubes and spiral coils, etc. In
this paper, we first introduce Jaeger’s solution for an instantaneous
line-source in composite solids. Next, based on the Jaeger’s solution we
develop the continuous line and cylindrical-surface source solutions.
Then, these solutions are used to obtain temperature distributions
around two borehole and two pile GHEs. After that, new temperature
response functions for pile and borehole GHEs are presented, which is
applicable not only for short-time scale but also medium time scale.
Finally, a preliminary quantitative analysis of the influence of difference between properties of grout (or pile) and surrounding soil is
presented.
2. The infinite line-source theory in composite solids
Kelvin’s theory of heat sources and sinks has been proved most
useful in solving the problem of heat conduction [10]. Starting from
fundamental solutions of instantaneous point sources, one can
obtain solutions for continuous sources or line, plane, and volume
sources by integrating the fundamental solutions with respect to
time or appropriate space variables. The fundamental solution used
in this work has been developed by Jaeger [29], which is the infinite
instantaneous line-source solution in composite cylindrical media.
2.1. Instantaneous line-source in composite media
In a composite medium, heat conduction due to an instantaneous line-source of infinite length is solved by the following
idea [29]. Consider an infinite composite solid, expressed in
cylindrical coordinates, region r < rb of which is of one medium
and region r > rb of another. There is an infinite line-source of
strength q0I (J/m) instantaneously releasing heat into the
composite solid at zero time. The line-source is parallel to the
height direction, z-axis, and is located through a point (r0, q0 ). The
key step to resolve this problem is application of the addition
theorem for the modified Bessel function of the second kind of
257
zeros order. This theorem makes this problem solvable in cylindrical coordinates by the Laplace transform method. The final
result for the case of r0 < rb is [29]:
T1 ðt;r; qÞ ¼
þN
X
q0I
0
cos n q q
2r1 c1 p n¼N
ZþN
0
T2 ðt; r; qÞ ¼
J ðurÞJ ður 0 Þ40 g 0 j0 f 0 n
n
exp a1 u2 t
udu (1a)
402 þ j02
þN
q0I a1 X
0
cos n q q
2
p rb n ¼ N
ZþN
0
J ður0 Þj0 J ðaurÞ 40 Y ðaurÞ
n
n
n
exp a1 u2 t
du(1b)
402 þ j02
where
40 ¼ k1 akJn ðrb uÞJn0 ðarb uÞ Jn0 ðrb uÞJn ðarb uÞ
j0 ¼ k1 akJn ðrb uÞYn0 ðarb uÞ Jn0 ðrb uÞYn ðarb uÞ
(2a)
(2b)
f 0 ¼ k1 akYn ðrb uÞJn0 ðarb uÞ Yn0 ðrb uÞJn ðarb uÞ
(2c)
g0 ¼ k1 akYn ðrb uÞYn0 ðarb uÞ Yn0 ðrb uÞYn ðarb uÞ
(2d)
In these expressions, Jn and Yn denote the Bessel functions of the
first kind and the second kind of order n; u is the integral variable
(1/m); subscripts 1 and 2 denote regions r < rb and r > rb; k, r, and c
are thermal conductivity, density and specific heat, respectively, of
the media;
pffiffiffiffiffiffiffiffiffiffiffiffiffiand a and k are dimensionless variables k ¼ k2/k1,
a ¼
a1 =a2 , where a1 and a2 are thermal diffusivities of the
composite solid. This solution provides a good basis for developing
solutions for other heat sources.
2.2. Continuous line-source in composite media
If a line-source in a composite medium liberates heat continuously at a constant rate ql per unit time and per unit length (W/m),
the temperature response can be easily obtained by integrating the
instantaneous line-source with respect to time t0 :
þN
X
ql
0
cos n q q
2r1 c1 p n ¼ N
Z t ZþN
exp a1 u2 ðt t 0 Þ Jn ðurÞJn ður 0 Þ 40 g 0 j0 f 0
Tl;1 ðt; r; qÞ ¼
0
402 þ j02
0
ududt 0 (3a)
þN
ql a1 X
0
cosn q q
2
p rb n¼N
Z t ZþN
exp a1 u2 ðtt 0 Þ Jn ður0 Þ j0 Jn ðaurÞ 40 Yn ðaurÞ
Tl;2 ðt;r; qÞ¼
0
402 þ j02
0
dudt 0 (3b)
Performing the integration with respect to t0 , we reduce Eqs.
(3a) and (3b) to
Ql;1 ðFo; R; qÞ ¼
þN
X
n ¼ N
0
cos n q q
ZþNh
i
1 exp y2 Fo
0
Jn ðyRÞJn ðyR0 Þð4g jf Þ
dy
y 42 þ j2
(4a)
258
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
Ql;2 ðFo; R; qÞ ¼
2
þN
X
p
n ¼ N
0
cos n q q
ZþNh
1 exp y Fo
2
i
Qs;1 ðFo; R; R0 Þ ¼
0
0
Jn ðyR0 Þ½jJn ðayRÞ 4Yn ðayRÞ
dy
y2 42 þ j2
ZþNh
(4b)
0
Qs;2 ðFo; R; R Þ ¼
Here, we use dimensionless variables Q(Fo, R, q) ¼ 2pk1T/ql,
Fo ¼ a1 t=rb2 , R ¼ r/rb, R0 ¼ r0 /rb, y ¼ urb; and definitions (2a)e(2d)
become:
2
p
i J ðyRÞJ ðyR0 Þð4g jf Þ
0
0
1 exp y2 Fo
dy
y 42 þ j2
ZþNh
1 exp y Fo
2
i
(9a)
0
J0 ðyR0 Þ½jJ0 ðayRÞ 4Y0 ðayRÞ
dy
y2 42 þ j2
(9b)
4 ¼ akJn ðyÞJn0 ðayÞ Jn0 ðyÞJn ðayÞ
(5a)
In Eqs. (8,9), values of n in defining Eqs. (5a)e(5d) are equal to 0.
j ¼ akJn ðyÞYn0 ðayÞ Jn0 ðyÞYn ðayÞ
(5b)
3. Application to borehole and pile GHEs
f ¼ akYn ðyÞJn0 ðayÞ Yn0 ðyÞJn ðayÞ
(5c)
g ¼ akYn ðyÞYn0 ðayÞ Yn0 ðyÞYn ðayÞ
(5d)
The operation of GCHPs spans tens of years but no analytical
model is available for all time scales. In this section, we apply the
line-source theory for composite media to analyze heat transfer
processes of GHEs and examine its performance in different time
scales.
Solution (4) gives the temperature distribution in a composite
medium, where region r < rb is of one medium and r > rb of another.
This solution can be applied to GHE situations as the grout inside
borehole GHEs is different from soil or rocks outside boreholes and
the material of foundation piles is also different from the
surrounding soil or rocks. It should be noted that the common heat
source models ignore these differences in properties.
2.3. Continuous cylindrical-surface source in composite media
In energy piles with spiral coils (Fig. 2), one may model the GHE
as a cylindrical-surface heat source [27]. A cylindrical-surface
source of radius r0 (r0 < rb) can be imagined as many line-sources of
strength qldq0 /2p are distributed around the circle of radius r0,
which is also true for the case of composite media. Therefore,
solutions for a continuous cylindrical surface source in composite
media can be obtained by summing or integrating the distributed
line-source solutions with regard to the azimuth q0 :
Ts;i ¼
1
2p
Z2p
0
Tl;i ðt; r; qÞdq ;
i ¼ 1; 2
(6)
0
Here, subscript s denotes surface source. Because we have the
following relation for integer n:
Z2p
0 0
cos n q q dq ¼
2p
0
n ¼ 0
ns0
(7)
0
the infinite series in Eq. (4) only leave the term with n ¼ 0, and Eq.
(6) finally reduces to
q
Ts;1 ðt; rÞ ¼ l
2pk1
ZþNh
i
1 exp a1 u2 t
0
J0 ðurÞJ0 ður0 Þ 40 g0 j0 f 0
du
u 402 þ j02
q
Ts;2 ðt; rÞ ¼ 2 l
p rb
(8a)
ZþNh
i
1 exp a1 u2 t
3.1. Scale analysis
All analytical models for GHEs make some assumptions relevant
to time scales of heat transfer by GHEs and, therefore, it is necessary
to analyze these time scales before evaluating the performance of
the new method. To the best of our knowledge, these time scales
have never been explicitly analyzed before.
Using the heat conduction equation, the time scale, when
unsteady effect is still dominant, can be estimated by:
twðDrÞ2 =a1
(10)
where t denotes the estimated time scale, Dr is the chosen space
range and a1 denotes thermal diffusivity of the medium. Although
the above scale relation produces only the order-of-magnitude
estimates for the time scale of interest, these results are thought
to provide exact results within a factor of order one [30]. Therefore,
the transient effect may be ignored when the time period is equal to
or larger than (5e10) (Dr)2/a1.
Now we examine three time scales relevant to three space
ranges: radius of the heat transfer pipe, radii of boreholes or piles,
and one half of length of GHEs. The order-of-magnitude of these
three space ranges are about 0.013 m for inner radii of pipes, 0.06 m
for borehole radii or 0.3 m for radii of piles, and 20e50 m for one
half of length of boreholes. Based on one half of length of boreholes,
the scale analysis can give time scales used to determine whether
the ground surface affects the heat transfer processes. The order-ofmagnitude of thermal diffusivities is 106 m2/s in GSHP applications. According to these data, the time scale when the effect of heat
capacity inside a borehole is dominant is t w 1 h. This time scale is
equal to that of energy simulation and variation of cooling loads.
For a foundation pile of radius 0.3 m, the corresponding time equals
25 h. These scales imply that the transient responses inside boreholes or piles are important for hourly energy analysis of GSHPs.
One half of length of the borehole gives a time scale of at least
10 years, implying that the impact of ground surface is negligible
during the first several years. In the space of heat transfer tubes (Ushaped or spiral tubes), expression (10) gives an estimation of time,
3 min. These time periods are useful when evaluating the performance of the following new models.
0
J0 ður 0 Þ j0 J0 ðaurÞ 40 Y0 ðaurÞ
du
u2 402 þ j02
or in dimensionless form
3.2. Temperature distributions of borehole and pile GHEs
(8b)
Fig. 3 shows the layout of line and surface sources for modeling
heat transfer by borehole and pile GHEs, where line and surface
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
sources are there, in place of tubes. Points A and B in Fig. 3
correspond to those in Figs. 1 and 2. These two points are used
for developing temperature response functions.
It is important to analyze the temperature distribution around
a GHE. Fig. 4 shows some contours of temperatures near energy
piles and boreholes, corresponding to the arrangements of Fig. 3.
These temperature fields are obtained by Eq. (4) or (9) with
prescribed positions of heat sources. Each line-source is assumed
to be of the same strength. Obviously, temperature fields around
boreholes (Fig. 4a and b) are symmetric because of symmetrical
arrangements of U-tubes or line-sources. The temperature around
the energy pile with a spirally bound tube is a one-dimensional
distribution in a cylindrical coordinate system (Fig. 4c) because
we adopted the solid cylindrical surface source model. Fig. 4d
shows an example of an energy pile with a W-tube, showing the
line-source theory has the flexibility for dealing with any arbitrary
arrangement of heat transfer pipes. As shown below, when
developing temperature response functions of GHEs we are only
interested in average temperature at walls of U-shaped pipes or
heat transfer coils. These temperature contours clearly indicate
that temperatures at the pipe walls vary, and temperatures at
positions A and B (as labeled in Fig. 3) should be the maximum and
minimum of temperatures at pipe walls for these symmetrical
arrangements. Therefore, arithmetic mean values of temperatures
at A and B can approximate average temperature of pipe walls
with good accuracy. Based upon this observation, we derive
response functions of GHEs as follows.
3.3. Temperature response functions
When deriving the line-source solutions we assumed the
strength of line or surface sources to be constant. In fact, loads
of GCHPs vary continuously due to varied cooling or heating
loads of buildings. In such situations, solutions can be obtained
by the Duhamel’s theorem [10,31], or the principle of superposition, a method that uses solutions to cases of constant load
ql. Now, we assume that the load of GHEs is a function of time
ql(t). According to Duhamel’s theorem, the temperature, T1, say,
reads
T1 ðr; q; tÞ ¼ T1;0 þ
259
Zt
ql ðsÞ
0
vGðr; q; t sÞ
ds
vt
(11)
where T1,0 is the initial temperature of the ground, and G, called G
function in GSHP literature, is the temperature response function.
In this context, physical significance of the G function represents
temperature response in the composite media due to a unit-step
change in GHEs loads ql. Since heating and cooling loads of buildings are commonly expressed in step-wise constant values (on
hourly, daily or other bases), Eq. (11) further reduces to [31]
T1 ðr; q; tÞ ¼ T1;0 þ
N1
X
Dql;j Gðr; q; t jDtÞ
(12)
j¼0
The G function is generally evaluated at borehole walls r ¼ rb,
representing the average temperature response at a bore wall. Thus,
the temperature of circulating fluid Tf of GHE loops is obtained by
Tf ¼ Tb þ ql Re;b
(13)
where Re,b is the effective fluid-to-ground thermal resistance, and
Tb is the average temperature at a borehole wall obtained by
substituting coordinates of the borehole wall into Eq. (12). Thermal
resistance Re,b is derived traditionally by assuming that heat
conduction inside a borehole is steady-state.
3.3.1. New response functions for GHEs with a U-tube
Line-source models for composite solids can do without the
steady-state assumption for Re,b and model the U-shaped pipe
inside a borehole as two line-sources in a composite medium. In
terms of definition of the G function, Eq. (11), the G function for
single U-tube should be the temperature response of two line heat
sources of strength 1/2 at (R0 , 0) and (R0 , p). More importantly, by
using the new solutions, the G function can be evaluated at the Upipe wall, and not at the borehole wall, representing the average
temperature response at the pipe wall. For simplicity, average value
of temperature responses at points A and B at the wall of U-shaped
tube is used to evaluate the G function (based on the observation in
Fig. 4a):
Fig. 3. Schematic layout of heat sources for pile and borehole GHEs.
260
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
Fig. 4. (a) Borehole with single U-pipe. (b) Borehole with double U-pipes. (c) Pile with a spiral pipe. (d). Pile with a W-tube.
G Fo; Rp ; R0 ¼
1
2pk1
þN
X
m ¼ N
ZþNh
i
1 exp y2 Fo
0
ðyR0 Þð4g
J2m ðyRA Þ þ J2m ðyRB Þ J2m
jf Þ
dy (14)
2
y 4 2 þ j2
In Eq. (14), RA and RB are dimensionless radial coordinates of
points A and B; values of n in definitions Eqs. (5a)e(5d) equal 2m.
function is also evaluated at points A and B, because of the
symmetry:
G Fo; Rp ; R0 ¼
þN
þN Z h
i
X
1
1 exp y2 Fo
2pk1
l ¼ N
0
J ðyRA Þ þ J4l ðyRB Þ J4l ðyR0 Þð4g jf Þ
dy
4l
2
y 42 þ j2
(15)
In Eq. (15), values of n in definitions Eqs. (5a)e(5d) equal 4l.
3.3.2. New response functions for GHEs with double U-tubes
Similarly, the response function G for a double U-tube can be
obtained by assuming four line-sources of strength 1/4, located
at positions (R0 , 0), (R0 , p), (R0 , p/2) and (R0 , 3p/2), and the G
3.3.3. New response functions for energy piles with spiral tubes
In this case, the temperature field is one-dimensional, as shown
in Fig. 4c. The average value of temperatures at points A and B can
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
also approximate the average temperature response at tube walls
with enough accuracy. Therefore, the G function for energy piles
with spirally arranged tubes is
G Fo; Rp ; R0 ¼
1
2pk1
ZþNh
i
1 exp y2 Fo
0
J ðyRA Þ þ J0 ðyRB Þ J0 ðyR0 Þð4g jf Þ
dy
0
2
y 42 þ j2
(16)
In Eq. (16), values of n in definitions Eqs. (5a)e(5d) equal 0.
The three new G functions provide a new way of modeling the
heat transfer by pile or borehole GHEs. The G functions can be used
to determine the temperature of the circulating fluid by the
following expression:
Tf Tp ¼ ql Rp
(17)
Here, Tp denotes the average temperature at pipe walls obtained by
substituting the new G functions into Eq. (12). Thermal resistance of
pipe Rp is evaluated by
Rp ¼
1
ro
ln
2pkp ri
261
used for predicting the behaviors of GHEs during short-time
periods, except the very short periods (15e30 min) and that the
new method should be more reasonable for short-time responses
analysis than traditional ones that ignore all heat capacities inside
a borehole (including those of U-shaped pipes and grout).
The composite-medium G functions are derived from the theory
of line-sources of infinite length, so they suffer from the common
limit of infinite line-sources, i.e. they ignore the influence of the
ground surface and are unable to predict long-term behaviors of
GHEs in some situations. If cooling and heating loads balance each
other, the ground surface has no influence on long-term behaviors
of GHEs, and our method can model both short-term and long-term
heat transfer by GHEs. Contrarily, if there is an imbalance between
cooling and heating loads, the redundant heat due to unequal loads
is accumulated in the ground, and the influence of the ground
surface becomes significant after several years’ operations. The
scale analysis suggests that the span of this time is ten years. As the
time approaches theoretical infinity, the temperature field reaches
a steady-state, in which redundant heat is transferred to ambient
air through the ground surface. The models of infinite line-sources,
however, ignore the ground surface, resulting in the calculated
(18)
where kp, ro, ri denote thermal conductivity, outer radius and inner
radius of pipes, respectively. This modeling approach is different
from traditional ones in several aspects, and its performance is
further explored in the following subsection.
3.3.4. Analysis of G functions
Before analyzing performance of the new G functions, the new
line-source model was validated against the conventional infinite
line-source solution for a homogenous medium. Table 1 indicates
that when a medium is homogenous, the composite-medium linesource solution indeed approaches to that predicted by the
conventional model, which has been widely used in design and
simulation of GCHPs and has been partly validated by real engineering practice.
An important aspect of heat transfer models for GHEs is applicability in different time scales. First of all, heat capacity of grout or
concrete piles can be fully considered by the new models, replacing
heat-transfer tubes by composite-medium heat sources. This
replacement assumes that the convective heat transfer inside the
tubes is replaced by heat conduction. So, the new line-source
models is unsuitable for very small times when transient effect is
dominant within the region of heat-transfer tubes. Heat transfer
tubes have much small geometric dimensions and heat capacity
compared to grout and boreholes. The scale analysis indicates that
the unsteady effect within tubes is significant only for a few
minutes and can be ignored when time scale is equal to or larger
than 15e30 min. Therefore, we infer that the new method can be
Table 1
Comparison of infinite line-source solutions with a ¼ 1 and k ¼ 1. An infinite linesource is parallel to the z-axis, through original point, and liberates heat at the
rate of ql per unit time per unit length.
Fourier number, Fo
1
10
100
1000
10,000
Dimensionless temperature 4pk1T/ql, at r ¼ rb
Common line-source
solution
Line-source
solution Eq. (4a)
1.04428
3.13651
5.41675
7.71708
10.0194
1.04422
3.13644
5.41668
7.71702
10.0194
Fig. 5. The variation of the dimensionless G functions of borehole GHEs with Fourier
number and a (a) or k (b).
262
M. Li, A.C.K. Lai / Energy 38 (2012) 255e263
temperatures increasing continuously and never reaching steadystates, as shown in Figs. 5 and 6. Although infinite line-source
models cannot predict the long-term performance of GHEs, in the
case of loads imbalance, the scale analysis demonstrates that these
models still can predict the performance of GHEs in medium times,
for example, over several years, depending on the length of GHEs.
Another outstanding characteristic of the new models is that
they can explicitly analyze the influence of the difference between
thermal properties of soil and grout (or foundation piles). The
influences of the differences in heat capacity on G functions are
summarized in Fig. 5 for boreholes with U-pipes and in Fig. 6 for
energy piles with spiral coils. In these figures, the differences
between G functions along horizontal axes denote phase differences in temperature responses, and those along vertical axes are
amplitude differences. According to the definition of the variables,
k ¼ k2/k1 and a ¼ ða1 =a2 Þ1=2 , fixing k at 1 while increasing a can be
understood as increasing heat capacity of soil, and fixing a at 1
while increasing k can be understood as increasing thermal
conductivity or heat capacity of soil. Variations of G function with
these two situations are shown in Figs. 5 and 6. These diagrams
illustrate that the temperature magnitude decreases with
increasing heat capacity of soil. Inferring from the figures, it can
also be seen that the temperature response delays with heat
capability of soil. The decrease and time-delay become significant
for large values of time as the influence of soil outside boreholes or
piles becomes more involved as time increases. In our cases, the
impact of difference in properties is insignificant and may be
ignored when Fo < 0.3, but when Fo > 0.3, the differences become
significant and influence the temperature responses greatly. This
observation suggests that ignoring differences in thermal properties, associated with GHEs, may induce considerable errors,
including phase errors and amplitude errors.
4. Conclusions
This paper presents a novel approach to modeling heat
conduction by pile and borehole GHEs which uses the line-source
solution to heat conduction in composite media. New solutions
for continuous line and cylindrical surface sources are derived
based on Jaeger’s instantaneous line-source solution. Several
important conclusions are drawn, as follows:
1) Based on the new line-source solutions, this work develops
new temperature response functions (G functions) for borehole
GHEs with U-tubes and for energy piles with spiral coils, which
provide a new way of designing GHEs and analyzing energy
consumption of GCHPs.
2) The new G functions can be used to model the performance of
GHEs in different time scales from an hour to several years. This
characteristic is remarkable for annual energy analysis of
GCHPs.
3) The property difference between the media inside and outside
boreholes or piles is an important factor that should be
considered when analyzing heat transfer by GHEs. Ignoring this
property difference may induce considerable phase and
amplitude errors.
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Fig. 6. The variation of the dimensionless G functions of pile GHEs with Fourier
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