1. Development and validation of a Slinky™ ground heat exchanger
model q
Zeyu Xiong ⇑
, Daniel E. Fisher, Jeffrey D. Spitler
School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078, USA
h i g h l i g h t s
A Slinky™ GHX model intended for use in building simulation programs is developed.
A response function (g-function) method is used to model Slinky™ GHX.
A full surface heat balance is adopted to model the effect of ground surface.
The model has been validated against experimental data.
a r t i c l e i n f o
Article history:
Received 14 May 2014
Received in revised form 23 October 2014
Accepted 25 November 2014
Keywords:
Ground source heat pump
Horizontal ground heat exchanger
Slinky™ ground heat exchanger
Analytical solution
Simulation
a b s t r a c t
Ground source heat pump systems are an energy efficient heating and cooling technology for residential
and commercial buildings. The main barrier to adoption is the higher investment cost compared to con-
ventional systems. Where available land area permits, horizontal ground heat exchangers are generally
less expensive than vertical borehole-type ground heat exchangers (GHXs). A further cost reduction
can be made by using Slinky™ GHXs, which require less trench space and hence reduce the installation
cost, in many cases. It is desirable to formulate an accurate model for simulation purposes; such simula-
tions can be used in both design tools and in energy analysis programs.
The model formulated in this paper relies on analytical ring source solutions to compute temperature
response functions for both horizontal and vertical Slinky™ heat exchangers. The algorithms used to cal-
culate the response factors have several features that significantly increase computation speed. The ther-
mal effect of the ground surface temperature variation on the GHXs is considered by superimposing the
undisturbed ground temperature, which is calculated using the numerical approach. For use in system
simulations where the GHX may be connected to other components, the model is formulated to calculate
both heat transfer rate and exiting fluid temperature, given entering fluid temperature. The model has
been validated against the previously published field test data.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Horizontal ground heat exchangers (HGHXs) are most com-
monly laid in trenches at a depth of 0.9–1.8 m [1]. Alternatively,
they are sometimes laid out during site excavation, and then bur-
ied; in this case, they may be deeper than typical trench depths.
When compared to vertical loop ground heat exchangers (GHXs),
HGHXs usually have significantly lower installation costs, since
trench excavation is considerably less expensive than drilling ver-
tical boreholes. However, conventional straight tube HGHXs
require large amounts of land surface for installation, which limits
the application of such GHXs. As an alternative, Slinky™ GHXs, also
known as slinky-coil [2] or slinky-loop [3] GHXs, use coiled tubing,
with the individual rings spread out along the direction of the
trench, as Figs. 1 and 2 show. Slinky™ GHXs are characterized by
the pitch p (the distance between rings) and the diameter of rings
D. Compared to conventional straight tube HGHXs, Slinky™ GHXs
have higher tube density, allowing GHX to extract/inject more heat
per land area. Hence, with the same cooling/heating loads, Slinky™
GHXs require less land area and excavation work than straight tube
HGHXs. The compact Slinky™ GHX needs about one-third trench
length compared to a two-pipe straight tube HGHX [4].
Slinky™ GHXs can be placed either horizontally or vertically, as
shown in Fig. 2. If the excavation is made with a trenching
machine, the Slinky™ is usually placed vertically or near-vertically
http://dx.doi.org/10.1016/j.apenergy.2014.11.058
0306-2619/Ó 2014 Elsevier Ltd. All rights reserved.
q
Slinky is registered trademark of Poof-Slinky, Inc, 45400 Helm Street, Plymouth,
MI 48170, USA.
⇑ Corresponding author.
E-mail address: zeyuxiong@gmail.com (Z. Xiong).
Applied Energy 141 (2015) 57–69
Contents lists available at ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

2. in the narrow trench. However, backfilling the narrow trenches of a
vertical Slinky™ GHX can be difficult and may resulting in air pock-
ets around the tubing if not backfilled correctly, which will affect
its thermal performance significantly. A high-pressure water jet
is recommended [4] when backfilling to prevent bridging of soil
and air pockets. Conversely, when Slinky™ GHXs are placed in a
large single excavation or in a wider trench dug with a backhoe,
they are generally installed horizontally.
Though Slinky™ GHXs have existed for more than two decades,
a suitable design tool is still absent. Due to their compact configu-
ration, there is considerable thermal interaction between individ-
ual rings of the Slinky™ GHX. In addition, because of its
comparatively shallow buried depth, the above ground environ-
ment will affect the overall performance of a Slinky™ GHX signif-
icantly. Previous studies on the thermal performance of Slinky™
GHXs [5,3] are based on numerical simulations of only a few rings
of a Slinky™ GHX using commercial CFD software with simplified
boundary conditions. The thermal interactions and the ground sur-
face effects are not carefully considered in these studies. Therefore,
it is highly desirable to develop a simulation model that can sup-
port both design tools and energy analysis of the GSHP systems uti-
lizing Slinky™ GHXs.
In the last several decades, many numerical and analytical mod-
els of different types of GHXs were developed. Analytical models
based on the point source solution and Green’s function theory are
one type of model. The point source solution may be interpreted
as the temperature variation in an infinite solid due to a finite quan-
tity of heat continuously generated from t = 0 onwards at a point. By
introducing the Green’s function theory, the new solution of a point
source in the semi-infinite solid with isothermal boundary condi-
tion is given. This solution can be derived by applying the method
of images [6], which introduces a virtual point source with same
heat rate yet opposite in sign on symmetrical to the boundary. By
integrating the solutions for point sources with regard to appropri-
ate variables, we may obtain solutions for line, cylindrical, and even
spiral sources [7–9]. These solutions are widely adopted in GHX
models. The theory of applying infinite line source and finite line
Nomenclature
Variables
cp specific heat, J/kg K
d distance between two points, m
di inner diameter of tube, m
do outer diameter of tube, m
D diameter of ring or loop, m
erfc complementary error function
gs temperature response function
h buried depth, m
k thermal conductivity of soil, W/m K
L trench length, m
_m mass flow rate, kg/s
N number of GHXs or rings
p pitch, m
P point
PA phase angle of annual soil temperature cycle, rad
q heat rate, W
Q heat impulse, J
ql heat rate per tube length, W/m
R radius of ring, m
Rpw thermal resistance of tube wall, m K/W
r radius of tube, m
SA annual temperature amplitude at ground surface, °C
T temperature, °C
TAE annual average soil temperature, °C
Tavg mean fluid temperature, °C
TP period of soil temperature cycle (8766 h)
t time, s
x; y; z Cartesian coordinate, m
Greek letters
a thermal diffusivity of soil, m2
/s
u angular parameter, rad
x angular parameter, rad
Subscripts
avg average
ground property of the ground
i; j arbitrary index
in inlet
n time step number
out outlet
tube Slinky™ tube
tw tube wall
ring ring source
ug property of undisturbed ground
Fig. 1. Schematic of a typical Slinky™ tube. Fig. 2. Schematics of horizontal and vertical Slinky™ ground heat exchangers.
58 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

3. source solutions in modeling GHX was presented in the work of
Ingersoll et al. [7]. In recent decades, several studies have extended
the theory. Li and Lai [10] developed a composite-medium line-
source model. Instead of approximating the borehole as a line
source, this model treats the U-shaped tube inside the borehole as
two parallel infinite line sources in a composite medium with the
grout region and the soil region. Finite line source models were
developed for GHXs with vertical boreholes [11], inclined boreholes
[12], and horizontal tubes [13]. Cui et al. [14] propose a ring source
model for pile GHXs that have spiral heat exchangers spiraling
downwards along the outside of the pile. The spiral tubing is mod-
eled as a series of infinite or finite ring sources in the longitudinal
direction. In the finite ring source model, the number of rings is
determined by the length of the pile and the coil pitch. Infinite
and finite ring source solutions of the soil temperature variation in
response to the heat transfer rate per pile length are given.
Heat source solutions have been used in modeling Slinky™
GHXs as well. Mukerji et al. [15] developed a line source approxi-
mation (LSA) model for GHXs that use coiled tubes. In this model,
the spiral or Slinky™ coiled tube is treated as spiral or Slinky™
heat source, with heat source coiled in exactly the same pattern
as the tube. As the result of this study, an analytical solution for
a performance factor (PF) was derived. The PF is the ratio of the
heat transfer rate per unit tube length of coiled tubing to that of
straight tubing. The authors introduce the PF as a simple way to
incorporate design of Slinky™ GHX into existing HGHX design
tools. However, the derived solutions are limited to the steady-
state heat transfer case. Li et al. [16] presented a ring source model
for Slinky™ GHXs, called ‘‘spiral heat exchangers’’. Slinky™ GHXs
are modeled as multiple ring sources in a semi-infinite homoge-
neous medium. The ground surface is assumed to be either an iso-
thermal surface for warm regions or an adiabatic surface for cold
regions. The solutions of the ground temperature response and
the mean tube wall temperature response to the heat transfer rate
per tube length are derived in the work. The multiple ring sources
solution was verified by an experiment.
The objective of this study is to develop a simulation model for
Slinky™ GHXs. A ring source model is established to simulate fluid
temperature variations of Slinky™ GHXs. According to the litera-
ture review, the previous ring source models either focus only on
the calculation of soil temperature response [14], or are intended
to be used to determine the size of the GHX [16]. To develop a
model suitable for whole-building energy simulation, several
important issues are addressed in this study. First, analytical solu-
tions of the mean tube wall temperature response have been spe-
cifically derived for both horizontal and vertical Slinky™ GHXs.
Compared to previous analytical solutions [16], our solutions
improved on the accuracy and are suitable for implementation in
whole-building energy simulation programs. Accuracy was
improved by not neglecting the tube diameter in calculating the
thermal interactions between rings.
Second, the ring source solutions are applied in a short variable
time-step simulation environment. For annual simulations with
short time steps (less than an hour), it is impractical to calculate
the temperature variation of the tubing wall at each time step by
applying the solutions directly. Therefore, a g-function (tempera-
ture response function) theory similar to that used in vertical bore-
hole models [17,18] is introduced in this study. The theory is
implemented in three parts: a computer program written in Visual
Basic for Applications (VBA) to generate temperature response fac-
tors; undisturbed ground temperatures are computed with an
existing Fortran 95 code within EnergyPlus; and computer pro-
grams implemented in both VBA and Fortran 95 (within Energy-
Plus) to simulate the Slinky™ GHX. The generated temperature
response factors are used to calculate temperature variation of
the tubing wall for each time step, which isolates the calculations
of the analytical solution from the simulation process. In addition,
the algorithm used to calculate the temperature response factors
have been optimized with a significantly improvement in compu-
tation speed.
The third issue is the model’s ability to couple to other compo-
nents in a whole-system simulation that is part of a building simu-
lation environment. The energy transfer between components is
captured based on the change of fluid state. Therefore, for a compo-
nent model that intends to be embedded in whole-building simula-
tion environment, the model input is normally inlet fluid state; and
the heat transfer rate is calculated as an output. However, the
reviewed ring source models [14,16] rely on the heat transfer rate
of tubing being specified as an input. To solve this, the model pre-
sented in this paper is formulated to calculate exiting fluid temper-
ature and heat transfer rate, with knowledge of the time-varying
entering fluid temperature. Simulations of fluid temperatures and
heat transfer rates of Slinky™ GHXs can be carried out with either
the stand-alone VBA computer program or within EnergyPlus, but
the results presented here were computed with the VBA program.
Fourth, due to Slinky™ GHXs’ comparatively shallow buried
depth, the thermal effect of the ground surface needs to be consid-
ered. According to the principle of superposition [19], the thermal
effect of the ground surface temperature variation can be
accounted for by superimposing the undisturbed ground tempera-
ture. While a one-dimensional numerical model of the ground
would be sufficient for this purpose, we utilized a three-dimen-
sional numerical model with a detailed ground surface heat
balance [20] that was already implemented within the whole-
building energy simulation software EnergyPlus. The component
model for the simulation of straight-pipe HGHXs in EnergyPlus
[20] is revised to calculate the undisturbed ground temperatures
used in our simulations of Slinky™ GHXs.
Last, but not least, while a number of heat source models have
been proposed to predict the thermal performance of the GHXs with
coiled tubing [8,9,14,16], none of them has been validated against
field experimental data. The model is validated against published
field test data [2]. Fluid temperatures predicted by our model are
compared to measurements from three short-term (5 days) thermal
response tests (TRTs) and one long-term (38 days) GSHP system test.
2. Ring source model and its analytical solutions
In the ring source model, the configuration of a typical Slinky™
tube is simplified as a series of rings, as Fig. 3 shows. Several addi-
tional approximations are made:
The GHXs in a Slinky™ GHX field are assumed to have uniform
heat fluxes. This assumption implies that the heat flux along a
ring is uniform; and the heat flux of every ring in the GHX field
Fig. 3. Simplification of a Slinky™ tube.
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 59

4. is the same. In addition, with this assumption, the mean fluid
temperature will be the mean of the inlet and outlet fluid tem-
perature. This assumption facilitates the application of analyti-
cal solutions in a short variable time-step simulation
environment. It is widely adopted in analytical models. The lim-
itations of this assumption as applied to vertical borehole GHX
are discussed in Malayappan and Spitler [21], Marcotte and Pas-
quier [22] and Beier [23].
The ground is treated as semi-infinite homogenous medium;
the surrounding ground is assumed to have thermal properties
that do not change with depth or time.
The influence of the soil moisture transfer on the heat transfer is
neglected. Past comparisons of field data and models suggest
that effect of soil moisture transfer is small [24].
The thermal storage and thickness of the tube wall are
neglected; the analytical solution treats everything inside the
outer wall of the tube as having the same thermal properties
as the soil.
Although not required by the model formulation, for conve-
nience in user input the Slinky™ GHX is assumed to be placed
in trenches of uniform length with a single trench-to-trench
spacing and a single pitch (ring-to-ring spacing).
The model is based on the principle of superposition, which,
strictly speaking, applies only for heat conduction processes with
linear boundary conditions and governing equations [19]. Our
model relies on superposition for computing the temperature
response of the ring sources to heat inputs and for combining the
temperature response of the ring sources with the time-varying
undisturbed ground temperature. It makes the assumption that
several deviations from linear boundary conditions and governing
equations are not significant enough to adversely impact superpo-
sition of the temperature responses from the ring sources and from
the undisturbed ground temperature model. These deviations
include:
The undisturbed ground temperature model has several nonlin-
ear boundary conditions: thermal radiation to/from the night
sky and evapotranspiration. We assume that the GHX has a very
small influence on the surface temperatures, especially when
compared to thermal radiation and evapotranspiration.
Though soil freezing and thawing around the tubes is neglected,
near-surface soil freezing and thawing near the surface are
modeled as part of the computation of undisturbed ground tem-
perature. As shown by Xu and Spitler [24] including soil freez-
ing and thawing significantly reduced the RMSE when
predicting temperatures at depths of 0.5 m and 1.0 m. In our
judgment, this improvement in prediction of the undisturbed
ground temperature outweighs any loss in accuracy due to
deviating from the conditions required for superposition to be
mathematically valid.
2.1. Description of the model
The principle of superposition serves as the basis for this ring
source model. It is valid for the heat conduction process with linear
boundary conditions and governing equation [19]. For the ring
source model, the complicated heat transfer process of the buried
coiled tubing is considered as the superposition of two indepen-
dent elementary heat transfer processes, as Fig. 4 shows. One is
the heat transfer process of multiple ring sources in a semi-infinite
medium with isothermal boundary. The other is the heat transfer
process of a semi-infinite medium with changing boundary tem-
perature. Therefore, to calculate the temperature variation of the
ground, we need to consider the ground temperature response to
multiple ring sources (the first heat transfer process) as well as
the undisturbed ground temperature (the second heat transfer
process).
2.2. Solutions for Horizontal Slinky™ GHXs
The solution of the continuous point source gives the tempera-
ture variation at distance d after time t, due to a point source emit-
ting continuously q units of heat per unit of time from t = 0
onwards. The continuous point source solution is given in Marcotte
and Pasquier [25] in the following form:
DTðd; tÞ ¼
q
4pkd
erfc
d
ffiffiffiffiffiffiffiffi
4at
p
ð1Þ
Marcotte and Pasquier [25] gave the derivation of the finite line
source solution from the point source solution. Applying a similar
method, ring source solutions for Slinky™ GHXs are derived. The
solutions cover both horizontal and vertical Slinky™ GHXs, with
one or multiple Slinky™ heat exchangers (e.g. interference
between two nearby heat exchangers in separate trenches or in
the same excavation can be analyzed). In the subsequent discus-
sion, an individual ring will be labeled as ring i or ring j.
As Fig. 5 shows, point Pi is a fictitious representative point of a
cross-section of ring tube i at angle u. The distance between ficti-
tious point Pi and point Pj is the average value of the distance
between the outer point Pio and point Pj and the distance between
the inner point Pii and point Pj. The average temperature perturba-
tion of the cross-section is assumed as the temperature perturba-
tion of the fictitious representative point Pi. Specially, when i is
equal to j, the dashed line in Fig. 5 shows the ring source of ring
i itself. x0 and y0 are Cartesian coordinates of a ring’s center. x0
and y0 are calculated based on the parameters of the Slinky™
GHX such as the pitch, the distance between Slinky™ tubes, and
the number of rings.
Assuming ql is the heat input rate per tube length, then for a
point source Pj on ring j, the intensity of heat input for the point
source is ql Á R Á dx. A large number of such point sources located
along ring j constitute a ring source. The effect of this ring source
can be viewed as the summation of the effect of these point
sources. The temperature perturbation at point Pi caused by ring
source j is then calculated by integrating the results of the point
sources on ring j.
DTðPi; tÞ ¼
qlR
4pk
Z 2p
0
erfc½dðPj; PiÞ=2
ffiffiffiffiffi
at
p
Š
dðPj; PiÞ
dx ð2Þ
where
dðPj; PiÞ ¼
dðPii;PjÞþdðPio;PjÞ
2
dðPii; PjÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR À rÞ cos u À x0j À R cos xŠ2
þ ½y0i þ ðR À rÞ sin u À y0j À R sin xŠ
2
q
dðPio; PjÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR þ rÞ cos u À x0j À R cos xŠ2
þ ½y0i þ ðR þ rÞ sin u À y0j À R sin xŠ
2
q
60 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

5. As discussed above, for Slinky™ tubing buried underground,
elementary heat transfer process (1) in Fig. 4 can be deemed as
happening in a semi-infinite medium with isothermal boundary
condition. The solution for this case is obtained by applying the
method of images. A fictitious ring source j0
is created for ring j,
as Fig. 6 shows. For horizontal Slinky™ GHXs, the fictitious ring
source j0
is located a distance 2h above the ring source j and has
the same heat input rate and the opposite sign. Eq. (2) is revised
to include the thermal effect of the fictitious ring source on the
point. The distance between Pj0 and Pi can be expressed as
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
. Then the revised solution is given as:
DTðPi; tÞ ¼
qlR
4pk
Z 2p
0
erfc dðPj; PiÞ=2
ffiffiffiffiffi
at
pÀ Á
dðPj; PiÞ
À
erfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
=ð2
ffiffiffiffiffi
at
p
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
3
7
7
5dx ð3Þ
While Eq. (3) gives the temperature perturbation of the ficti-
tious point Pi (represent a cross-section of ring i), the average tem-
perature perturbation of ring i caused by ring source j can be
obtained by integration:
DTjÀiðtÞ ¼
qlR
8p2k
Z 2p
0
Z 2p
0
erfc dðPj; PiÞ=2
ffiffiffiffiffi
at
pÀ Á
dðPj; PiÞ
À
erfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
=ð2
ffiffiffiffiffi
at
p
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
3
7
7
5dxdu ð4Þ
When i is equal to j, DTjÀi gives the temperature perturbation of
the ring tube wall caused by the ring itself as a ring source. When i
does not equal j, DTjÀi counts the thermal influence of ring source j
on ring i. When summing up all the DTjÀi for any two rings, all the
Fig. 4. Schematic of the principle of superposition.
Fig. 5. Distance between point Pi and point Pj on ring source j.
Fig. 6. Three-dimensional view of fictitious ring source of ring j.
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 61

6. thermal interactions between ring sources are considered. The
mean tube wall temperature variation of a Slinky™ GHX can then
be computed based on Eq. (5):
DTðtÞ ¼
1
Nring
XNring
i¼1
XNring
j¼1
DTjÀiðtÞ ð5Þ
The g-function (temperature response function) methodology is
an approach proposed by Eskilson [17] to calculate the vertical
borehole temperature response. A g-function can be represented
by a set of non-dimensional temperature response factors. The g-
function method is widely adopted in solving vertical borehole
related problem. Here, we have applied it to the solution of the ele-
mental heat transfer process (1) in Fig. 4. Following the definition
of the g-function, the temperature response function for Slinky™
GHXs is defined as:
gsðtÞ ¼
2pk
ql
DTðtÞ ð6Þ
The reason that we use 2pk instead of k in Eq. (6) is to keep con-
sistent with the definition of g-function as originally defined by
Claesson and Eskilson [26]. Therefore, the temperature response
function for different type of GHXs can be compared in the same
scale. By combining Eqs. (4)–(6), the analytical solution of the tem-
perature response function for horizontal Slinky™ GHXs is given
as:
gsðtÞ ¼
XNring
i¼1
XNring
j¼1
R
4pNring
Z 2p
0
Z 2p
0
erfc dðPj; PiÞ=2
ffiffiffiffiffi
at
pÀ Á
dðPj; PiÞ
À
erfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
=ð2
ffiffiffiffiffi
at
p
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðPj; PiÞ2
þ 4h
2
q
3
7
7
5dxdu ð7Þ
2.3. Solutions for Vertical Slinky™ GHXs
For vertical Slinky™ GHXs, the temperature variation at a point
Pi caused by a ring source j can still be calculated by using Eq. (2),
while the expressions for d(Pii, Pj) and d(Pio, Pj) need to be revised.
Applying the same method as deriving the solutions for horizontal
Slinky™ GHXs, the analytical solution of the temperature response
function for vertical Slinky™ GHXs is obtained:
gsðtÞ ¼
XNring
i¼1
XNring
j¼1
R
4pNring
Z 2p
0
Â
Z 2p
0
erfc dðPj; PiÞ=2
ffiffiffiffiffi
at
pÀ Á
dðPj; PiÞ
À
erfc dðPj0 ; PiÞ=2
ffiffiffiffiffi
at
pÀ Á
dðPj0 ; PiÞ
#
dxdu
ð8Þ
where
3. Calculation of temperature response factors
A computer program is created to calculate temperature
response factors for Slinky™ GHXs. The computer program can
generate a set of temperature response factors for a specified
Slinky™ GHX field. The generated factors cover the time range
from 1 min to ten years, which would satisfy most situations for
whole-building energy simulations. While vertical borehole simu-
lations are often done for longer periods, horizontal ground heat
exchangers, being comparatively close to the surface do not have
such long time constants. In simulations, the temperature response
function value at any desired time is calculated by interpolating
the values of the g-function, which are computed for discrete
times.
As Eqs. (7) and (8) indicate, in our original algorithm, the calcu-
lation of one temperature response factor requires ðNringÞ2
calcula-
tions of the double integral. These double integrals are calculated
with a numerical method. In real applications, the configuration
of a Slinky™ GHX may consist of more than a hundred rings. There-
fore, computing one temperature response factor for this Slinky™
GHX requires more than ten thousand calculations of the double
integral. Such calculation could be so time consuming that it hin-
ders application of the model in simulation software.
To shorten the calculation time of temperature response factors,
three improvements to the algorithm are applied. First, by taking
the advantage of symmetry of the configurations of Slinky™ GHXs,
as Fig. 7a demonstrates, we need only calculate around a quarter of
rings’ tube wall temperature perturbation to get the mean wall
temperature perturbation of the whole Slinky™ GHX field.
Second, according to Eq. (4), the interaction effect between two
rings decreases exponentially, as the distance increases. Therefore,
if the distance between two rings is considered large enough, neglect
of the interaction or calculating it with less accuracy will have little
effect on the overall temperature perturbation of the ring tube wall.
A study was conducted to investigate the affected region of a ring
source. The ground temperature response to a ring source with a unit
heat rate (1 W) after ten years was calculated and plotted against the
distance to the ring source, as Fig. 8 shows. The distance to the ring
source is defined as d À R (d minus R), where d is the distance to the
center of the ring; R is the radius of the ring source. The range of the
diameter of ring source is from 0.2 to 1.8 m. The soil thermal diffu-
sivity and conductivity are set as 1.11 W/m K and 5.4 Â 10À7
m2
/s,
following Salomone and Marlowe [27].
The temperature perturbation at d À R = 0.0134 m is considered
as the temperature perturbation of the tube surface, as a common
outside tube diameter in the USA is 26.7 mm. According to Fig. 8,
the temperature perturbations at d À R = 2.5 m are less than 5%
of those at d À R = 0.01 m, while the temperature perturbations
at d À R = 10 m are less than 0.1% of those at d À R = 0.01 m. The
thermal influence was investigated for other soil thermal proper-
ties, with thermal conductivity ranging from 0.77 to 2.5 W/m K,
dðPj; PiÞ ¼
dðPii;PjÞþdðPio;PjÞ
2
dðPj0 ; PiÞ ¼
dðPii;Pj0 ÞþdðPio;Pj0 Þ
2
dðPii; PjÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR À rÞ cos u À x0j À R cos xŠ2
þ ½y0i À y0jŠ2
þ ½z0i þ ðR À rÞ sin u À z0j À R sin xŠ
2
q
dðPio; PjÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR þ rÞ cos u À x0j À R cos xŠ2
þ ½y0i À y0jŠ2
þ ½z0i þ ðR þ rÞ sin u À z0j À R sin xŠ
2
q
dðPii; Pj0 Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR À rÞ cos u À x0j À R cos xŠ2
þ ½y0i À y0jŠ2
þ ½z0i þ ðR À rÞ sin u À z0j À 2h À R sin xŠ
2
q
dðPio; Pj0 Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½x0i þ ðR þ rÞ cos u À x0j À R cos xŠ2
þ ½y0i À y0jŠ2
þ ½z0i þ ðR þ rÞ sin u À z0j À 2h À R sin xŠ
2
q
62 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

7. thermal diffusivity ranging from 4.5 Â 10À7
to 9.3 Â 10À7
m2
/s. The
results of these studies have the same characteristics as discussed
above. Therefore, by considering the distance to a ring source i, the
surrounding rings are divided into three categories: near-field,
middle-field and far-field rings. Near-field rings are defined as
the rings whose centers are within 2.5 m + D distance of the center
of ring i, where D is diameter of ring (m). Following the same rule,
the middle-field rings are defined as the rings whose centers are
located between 2.5 m + D and 10 m + D distance of the center of
ring i. Far-field rings are those rings beyond 10 m + D distance. In
the calculation of tube wall temperature perturbation of ring i,
the near-field rings are treated as ring sources, as described above.
However, the interaction between the ring i and the middle-field
rings is considerably simplified – the interaction is approximated
as that of two point sources at their centers with the same heat
input strength. The thermal effect of the far-field rings is ignored.
The temperature influence of any middle-field ring on ring i is cal-
culated by using Eq. (9):
DTjÀiðtÞ ¼
qlR
2k
erfcðdjÀi=2
ffiffiffiffiffi
at
p
Þ
djÀi
À
erfc
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d
2
jÀi þ 4h
2
q
=ð2
ffiffiffiffiffi
at
p
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d
2
jÀi þ 4h
2
q
2
6
6
4
3
7
7
5 ð9Þ
As Eq. (9) shows, the double integral is eliminated from the solu-
tion. djÀi is the distance between the centers of two rings.
Third, in the calculation of the thermal interaction between any
two rings in a Slinky™ GHX field, since the other parameters are
set, the variables are time and the distance between the centers
of the two rings. Due to the uniform spacing of the rings, many
of the distances occur over and over again. (For example, the 1st
and the 5th rings in one trench are the same distance apart as
the 2nd and 6th rings, the 3rd and the 7th rings, etc.) Therefore,
the calculated values for pairs of rings at specific distances are
saved and used when the same distance comes up again. In addi-
tion, since the diameters of the ring sources and the tubes are set
for a Slinky™ GHX field, the temperature response of a ring tube
wall to itself as ring source is the same for all the rings in this
Slinky™ field. Reuse of the calculated temperature perturbations
eliminates many of the calculation and significantly reduces com-
putation time, especially for Slinky™ fields that have a large num-
ber of rings.
To examine the improvement in computation time and the
impact on accuracy by applying these improvements, comparisons
were made between the original algorithm and the fast algorithm
with the three enhancements just described. Several calculations
were performed by using the original algorithm and the new algo-
rithm. In each of these calculations, temperature response factors
were generated for 7 discrete times, ranging from 6 min to
11 years. The calculation time and the calculated temperature
response factors were compared. Following Jones [4], three typical
Slinky™ configurations in a single trench that are used in the U.S.
were selected for these calculations. The parameters of these three
Slinky™ configurations are listed in Table 1, labeled as Slinky™ 1–
3, respectively.
As may be determined from Table 1, the computational speed
the first three configurations (Slinky™ 1–3) are increased by fac-
tors of 48–60. Another computation was performed on a case with
700 rings, labeled Slinky™ 4. The computation time is 23 s, which
is an acceptable calculation time for such a large number of rings.
With the original algorithm, the computation time for this config-
uration exceeded a day. The reader should keep in mind that while
these computations were done on a modern desktop computer,
they were performed using an interpreted language (VBA). They
Fig. 7. Schematics of first and second improvements to the algorithm.
Fig. 8. The ground temperature response to a ring source.
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 63

8. should be much faster in a compiled language. The calculated tem-
perature response factors by using the original and fast algorithm
were also compared. Among these comparisons, the relative errors
at the early time can reach 4%, due to the very small base values.
After one hour, the biggest difference is below 0.5%. According to
our following case studies, ±1% error in temperature response fac-
tors resulted in maximum ±0.2 °C difference in the predicted heat
pump entering fluid temperature in annual simulations. In conclu-
sion, the fast algorithm shows good accuracy and reduces the com-
putation time significantly, making the computation time of
temperature response factors within a reasonable range for use
in whole building energy simulation programs.
4. Calculation of undisturbed ground temperature
For Slinky™ GHXs, the ground surface temperature variation
will have a significant impact on the tube wall temperature. As
described above, in our Slinky™ GHX model, this impact can be
accounted for by superimposing the undisturbed ground tempera-
ture. A one-dimensional numerical model would be sufficient to
calculate the undisturbed ground temperature. But, because it
was already available in EnergyPlus, a coarse-grid three-dimen-
sional numerical model [20] is utilized. Lee et al.’s [20] model
was intended for use with a buried horizontal tubing system and
can account for heat transfer to/from basement walls. Here, the
ground heat exchanger geometry is too complex to model within
the Lee et al. [20] model, so we do not utilize either of these fea-
tures and only utilize the capability to estimate the undisturbed
ground temperature. A uniform, rectangular grid is applied, as
Fig. 9 shows. According to Lee et al.’s [20] sensitivity study, for a
1989 m3
domain, with a non-uniform grid due to the presence of
buried tubes and basement walls, 4600 cells balances the compu-
tation time for use in building simulation with acceptable accu-
racy. We have a similar criteria as the computation is also used
within a building simulation program. Therefore, based on Lee
et al.’s [20] sensitivity study, we chose a domain of size
20 m  20 m  5 m with 15  15  25 cells. The undisturbed
ground temperature is taken from the cells that are in the middle
of the domain (horizontally) and that cover the vertical range of
the Slinky™ GHX.
The ground domain is governed by the following equation:
@T
@t
¼ a
@2
T
@x2
þ
@2
T
@y2
þ
@2
T
@z2
!
ð10Þ
Since a three-dimensional model is used, boundary conditions are
required at the top, bottom, and sides. The initial temperatures
throughout the ground domain and the temperatures at the side-
walls and lower boundary are set using the Kusuda and Achenbach
[28] model. A detailed heat balance is applied at the ground surface,
with the considerations of the convection heat transfer, the incident
short wave radiation, the net long wave radiation and the evapo-
transpiration and the heat conducted to the interior of the ground
domain:
qcond ¼ qconv þ qrad s þ qrad L þ qevap ð11Þ
The calculations of the convection, radiation and evapotranspiration
depend on the ground cover; e.g. bare ground is different from short
grass cover which is different from tall grass cover. As our model is
implemented, we have assumed the ground cover is short grass
(12 cm high), with evapotranspiration calculated as explained by
Lee et al. [20] and Xing [29]. The soil freezing and melting process
due to heat transfer to and from the surface is considered in this
numerical model because past work [24] has shown that it is impor-
tant in predicting the undisturbed ground temperature in colder cli-
mates. The effective heat capacity method [30] is used to model the
soil freezing and melting process. However, one limitation of our
model is that it cannot separately simulate the soil freezing/melting
around the Slinky™ heat exchanger buried tubes, since the tubes
are modeled with g-functions. Incidentally, the Lee et al. [20] model
can model the soil freezing/melting around the tubes, but it cannot
handle the complex Slinky™ heat exchanger geometry.
Table 1
Comparison between fast and original algorithms.
Name Slinky™
1
Slinky™
2
Slinky™
3
Slinky™
4
Pitch (m) 0.91 0.46 0.25 0.25
Number of rings 35 51 70 700
Single trench length (m) 31.9 23.9 18.2 18.2
Number of trenches 1 1 1 10
Total tube length (m) 100 145.8 200.1 2001
Distance between trenches (m) – – – 2
Diameter of rings (m) 0.91 0.91 0.91 0.91
Computation time,
original algorithm (s)
240 480 900 1 day
Computation time,
fast algorithm (s)
5 9 15 23
Fig. 9. Schematic of the numerical ground heat transfer model.
64 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

9. 5. Calculation of tube wall and fluid temperature
According to the principle of superposition, the mean tube wall
temperature of a Slinky™ GHX can be calculated as the superposi-
tion of two terms: the mean temperature response of the tube wall
and the undisturbed ground temperature at the GHX’s average
buried depth. By knowing the heat input and the temperature
response factors, we can obtain the mean temperature response
of the tube wall at a given time. For simulations of GHXs, the heat
transfer rates are functions of time, instead of constant values. In
this study, the heat rate function is decomposed into piecewise
constant step heat inputs. Similar to the superposition of piecewise
linear step heat pulses in the vertical borehole model [18,31], the
step heat inputs are imposed at each time step with the corre-
sponding temperature response function value. The superposition
of the response of each step heat input gives the temperature
response at the end of the last time step. Therefore, the mean tube
wall temperature can be calculated using the following equation
[18]:
TtwðtnÞ ¼ Tugðh; tnÞ þ
Xn
i¼1
ðqli À qliÀ1Þ
2pk
gsðtn À tiÀ1Þ ð12Þ
While the above equation allows the calculation of the mean tube
wall temperature, the heat transfer rate at the current time step
requires simultaneous simulation of the building, heat pump, and
ground heat exchanger. In practice, this requires an initial guess
and iteration at each time step. The system simulation in the
whole-building simulation software EnergyPlus is configured such
that the entering fluid temperature and mass flow rate is an input
to the ground heat exchanger model and the heat transfer rate
and exiting fluid temperature are outputs. To utilize the tempera-
ture response factors in calculating the exiting fluid temperature
as well as the heat transfer rate, a simple tube model, as shown
in Fig. 10, is established with the following assumptions regarding
the buried tubing:
The heat transfer between tube wall and fluid is at quasi-steady
state.
The mean fluid temperature is the average value of the inlet
fluid temperature and outlet fluid temperature.
According to these assumptions, we may have the following
equations at time step n. Eq. (13) is the expression of the second
assumption we made above. Eq. (14) gives the quasi-steady state
heat transfer rate in terms of the tube wall resistance Rtw, which
can be calculated as the sum of the conduction resistance of the
wall and the convection resistance between the wall and the fluid.
Eq. (15) gives the quasi-steady state heat transfer rate in terms of
fluid transport.
TavgðtnÞ ¼ ½TinðtnÞ þ ToutðtnÞŠ=2 ð13Þ
TavgðtnÞ À TtwðtnÞ ¼ qln
Rtw ð14Þ
2pRNringqln
¼ _mcp½TinðtnÞ À ToutðtnÞŠ ð15Þ
For the first time step (n = 1), the heat transfer rate per trench
length at the last time step ðql0
Þ and the time at the beginning of this
time step (t0) are equal to zero. The interpolation of the obtained
temperature response factors would generate the temperature
response function value at time t1 (g(t1)). Then the above four equa-
tions have four unknowns: the heat transfer rate ðql1
Þ, the mean
fluid temperature (Tavg(t1)), the outlet (exiting) fluid temperature
(Tout(t1)), and the mean tube wall temperature (Ttw(t1)) at the first
time step. By solving the four equations above simultaneously, we
have the outlet (exiting) fluid temperature (Tout(t1)) and the heat
transfer rate at the first time step ðql1
Þ calculated. For the next time
step, by using the heat transfer rate at the last step ðql1
Þ, the outlet
(exiting) fluid temperature (Tout(t2)) and the heat transfer rate at the
second time step ðql2
Þ could be calculated by solving the same set of
equations. Following this algorithm and accounting for each past
time step, the outlet (exiting) fluid temperature and heat transfer
rate at any desired time step can be calculated. For long-term sim-
ulations (e.g. annual simulations), a load aggregation algorithm
similar to that presented in Yavuzturk and Spitler [18] is applied.
The algorithm aggregates past loads into two groups: the most
recent history consists of a minimum of one week of hourly loads;
more distant history is grouped into blocks of 730 h each.
6. Validation against experimental data
The model is validated against published experimental data [2]
for three horizontal Slinky™ ground heat exchangers. The GHXs
were installed in 1.5 m deep horizontal trenches, each of which
was U-shaped as illustrated in the top portion of Fig. 11. The three
GHXs had different tubing lengths and pitches, but all were buried
in identically dimensioned trenches. The GHX dimensions and
other experimental parameters are summarized in Tables 1 and 2.
The authors conducted three short-term thermal response tests
(TRTs) and one long-term system test in 2008 and 2009.
The thermal conductivity of the unsaturated soil was measured
at 1.5 m depth at the test site. The authors mention that multiple
soil samples were collected, though no distribution of thermal con-
ductivity values is given. Flow rates of the working fluid, water,
Fig. 10. Schematic of simple tube model. Fig. 11. Simplification of the Slinky™ GHXs used in the field tests.
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 65

10. were controlled and measured using valves equipped with flow
meters. Resistance temperature detectors (Pt 100) were used to
measure the inlet and outlet fluid temperatures of the Slinky™
GHXs, the far field ground temperature at 1.5 m depth (the buried
depth of the GHXs), etc. Experimental uncertainties for tempera-
ture measurements and flow rates are not given. In each TRT, the
GHX was connected to an electrical heater and a pump; the system
ran continuously in heat rejection mode for 5 days. The long-term
system test was performed for 38 days from January 19, 2009 to
February 27, 2009. In the long-term system test, the GHX was con-
nected to a water-source heat pump and a pump. The house condi-
tioned by the heat pump utilized an air-source heat pump to
maintain a constant room temperature and, hence, the ability to
run the water-source heat pump in heating mode continuously
for 20 h each day. Measured inlet (entering) and outlet (exiting)
water temperature of GHX in each TRT and system test were plot-
ted in Fujii et al.’s paper [2].
To model the horizontal Slinky™ GHX described in Fujii et al.
[2], the U-shaped Slinky™ tubing is simplified as two parallel
Slinky™ heat exchangers with the same total tube length. The
two parallel GHXs connected in series, as Fig. 11 shows. As dis-
cussed earlier, the implementation has purposely (to minimize
user-required input) been limited to parallel linear trenches with
uniform Slinky™ pitch and trench-to-trench spacing. Using the
parameters in Table 2 and 3, the temperature response factors
for the Slinky™ GHXs were generated, shown in Fig. 12. The time
ranges corresponding to the 5-day TRT and the 38-day system test
are marked. The temperature response factors of these GHXs have
similar values in the first few hours. After that, the curve of the
temperature response factors of GHX1 starts to deviate from the
other two. This indicates the onset of the interaction between
loops, which is affected by the pitch of the rings. The pitch of
GHX1 is 0.4 m, which is smallest among the three, and so the inter-
action between the rings shows up sooner. The deviation between
the curves of GHX2 (pitch = 0.6 m) and GHX3 (pitch = 0.8 m)
begins at around the 30th hour. All three curves tend to constant
values after about 5 years. To check the effect of treating the U-
shaped trench as two parallel trenches, we computed the temper-
ature response factors for the U-shaped GHX and compared those
to the values of the simplified parallel configuration. The maxi-
mum difference is only 0.8%. As discussed before, ±1% error in tem-
perature response factors resulted in maximum ±0.2 °C difference
in the predicted heat pump entering fluid temperature in annual
simulations. The inaccuracy brought by the simplification of U-
shaped configuration can thus be ignored.
The undisturbed (far-field) ground temperatures at the buried
depth (1.5 m) during short-term TRTs are considered to be con-
stant over the 5-day tests and are given by Fujii et al. [2] as
25.6 °C (TRT1), 24.4 °C (TRT2) and 24.0 °C (TRT3). For the long-term
system test, the measured undisturbed (far-field) ground tempera-
ture is plotted against time by Fujii et al. [2]. The hourly measured
undisturbed ground temperature is then calculated by interpolat-
ing the digitized data points from the figure shown in Fujii et al.
[2]. To simulate the undisturbed ground temperature at the field
experiment location (Fukuoka City, Japan) during the experiment
period (January 19–February 27, 2009), historical measured
weather data (June 2008–May 2009) from the location is used.
The hourly calculated and measured undisturbed ground tempera-
tures during the system test period (38 days) are compared, with a
root mean square error (RMSE) of 0.3 °C.
With the three sets of temperature response factors, the mea-
sured undisturbed ground temperature, and the parameters given
in Tables 2 and 3, the hourly outlet fluid temperature of the GHXs
in three TRTs can be simulated. The measured inlet and outlet fluid
temperatures in TRTs are plotted in Fujii et al. [2]. Using the digi-
tized data points from the figure, the hourly measured inlet and
outlet fluid temperatures are determined by interpolation. TheTable 2
Common parameters in the field tests.
Parameter Value Parameter Value
Diameter of loops 0.8 m Soil thermal conductivity 1.09 W/m K
Inner diameter of tube 24 mm Tube thermal conductivity 0.35 W/m K
Buried depth 1.5 m Distance between trenches 2 m
Trench length 72 m Thickness of tube 5 mm
Table 3
Changing parameters in the field tests.
Test Pitch
(m)
Tube
length
(m)
Avg. mass
flow rate
(kg/s)
Avg.
electrical
heater load
(W)
Running
period
TRT1 0.4 500 0.17 4400 5 Days
(continuous)
TRT2 0.6 380 0.19 4540 5 Days
(continuous)
TRT3 0.8 308 0.20 4560 5 Days
(continuous)
Long-term
System
0.4 500 0.23 – 38 Days (cyclic)
Fig. 12. Calculated temperature response factors for GHXs Used in the field tests.
Fig. 13. Comparisons of the hourly outlet fluid temperatures and heat transfer rates
in TRT1.
66 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

11. validation of the model is then based on the comparison of the
hourly measured and simulated outlet fluid temperature. In this
study, both component-level and system-level validations are
conducted.
For the component-level validation, the hourly measured inlet
fluid temperature is input to calculate the outlet fluid temperature.
The predicted hourly outlet temperatures and heat transfer rates in
three TRTs are plotted in Figs. 13–15. According to these figures,
the simulated outlet fluid temperatures show a good agreement
with the measured values. The RMSEs of the simulated outlet tem-
peratures in TRT1, TRT2 and TRT3 are 0.2 °C, 0.2 °C, and 0.1 °C,
respectively. The maximum errors are À0.3 °C, À0.7 °C, and
À0.2 °C, which appear at the 3rd, 1st and 99th hour of the opera-
tion time, respectively. The RMSEs of the predicted heat rejection
rates are 2%, 3%, and 2%, respectively.
For the system-level validation, instead of inputting the mea-
sured inlet fluid temperature, the hourly load on the GHXs is used
as the boundary condition. With the use of heat transfer rates as
inputs, both the inlet and outlet fluid temperatures of the heat
exchanger are determined based on current and past values of the
heat inputs. Any inaccuracy in the temperatures can accumulate
with time. As the system-level validation allows error to accumulate
at every time step and may be considered a more rigorous validation,
with the results more directly applicable to using the model in a
building simulation. The calculated results are plotted in Figs. 13–
15 as well. Under this condition, the RMSEs of the predicted outlet
fluid temperature in three TRTs become 0.2 °C, 0.3 °C, and 0.3 °C,
respectively. The maximum errors increase to 0.4 °C, À0.7 °C, and
0.5 °C, happening at the 120th, 3rd, and 120th hour, respectively.
In the long-term system test, the inlet and outlet fluid temper-
atures were recorded every 10 min during the running period. The
author of the paper [32] provided the original recorded data of the
inlet and outlet fluid temperature. During the system off time,
there is no heat transfer to the Slinky™ GHX. For the compo-
nent-level validation, the measured inlet temperatures at 10 min
intervals are input; for the system-level validation calculated heat
transfer rates are input at the same intervals. Both the component-
level and system-level validations are shown in Fig. 16. In the com-
ponent-level validation, the RMSE is 0.2 °C and the maximum error
is À0.6 °C. The RMSE for the predicted heat transfer rate is 9%. The
maximum error appears at the beginning of the running period at
the 20th day. As can be seen, the largest errors always appear at
the beginning of the running periods and is likely due to the fluid
temperature approaching room temperature during the off-cycle.
It is also possible that natural circulation may lead to some heat
transfer occurring during the off-cycle.
Fig. 14. Comparisons of the hourly outlet fluid temperatures and heat transfer rates
in TRT2.
Fig. 15. Comparisons of the hourly outlet fluid temperatures and heat transfer rates
in TRT3.
Fig. 16. Comparison of 10 min-interval outlet fluid temperatures and heat transfer
rates in long-term system test.
Fig. 17. Daily average error in predicting outlet fluid temperature in long-term
system test.
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 67

12. With heat transfer rates as inputs in the system-level validation,
the RMSE increases to 0.5 °C, with a maximum error of À1 °C
appearing at the beginning of the 33rd day. The difference between
the predicted and measured outlet temperature increases after the
15th day and reaches its peak on the 31st day for the system-level
validation. Fig. 17 compares the error between measured and mod-
eled daily-average outlet fluid temperature from the ground heat
exchanger. By comparing the component-level and system-level
validation errors, it may be inferred that model does accumulate
some error when heat transfer rates are input. Keeping in mind
that the test involved continuous heat extraction for 20 h each
day yet the error does not grow monotonically but rather rises
and falls, it may also be inferred that the relatively close proximity
of the ground heat exchanger to the surface mitigates some of the
potential for accumulation of error.
In summary, the model predicted outlet temperature shows a
good agreement with the measured data of three TRTs and one
long-term system test. In all the simulations, the errors are
within ±1 °C. The RMSEs range from 0.1 to 0.5 °C in the compo-
nent-level validations. Since this model is designed for simulation
purposes, the time scale may vary from minutes to years. The
ranges of the temperature response function applied in the vali-
dations are indicated in Fig. 12. Though the time range (10 min
to 38 days) is not as wide as may be desired, these tests actually
cover quite a wide range of the temperature response function.
Future research with longer tests and additional geometries is
recommended.
7. Conclusions and recommendations
This paper presents the development and validation of a
Slinky™ GHX model intended for use in building simulation pro-
grams. The intended usage requires that the model be both suffi-
ciently accurate and fast. Detailed numerical solutions which
may be accurate will not be acceptably fast. Therefore, we have
adopted a response function method similar to that used for simu-
lation of vertical borehole heat exchangers. As far as we know, this
is the first time such an approach has been applied to horizontal
ground heat exchangers. There are two notable differences
between this response function method and those used for simula-
tion of vertical borehole heat exchangers. First, the methodology
for calculating the response function is quite different and, second,
since Slinky™ GHX are often buried at shallow depths, the annu-
ally-varying ground temperature can be significant and it is
included in the superposition done in the simulation. In order for
this approach to be sufficiently accurate and fast, it is necessary
to have relatively fast and accurate methods for computing both
the response functions and the undisturbed ground temperatures.
Analytical solutions have been used for computing response
functions for Slinky™ ground heat exchangers. We calculate the
response functions by superimposing ring sources. While this
method gives good accuracy, it can be quite slow for larger Slinky™
GHX with many rings. As an example, computation of the response
function for a Slinky™ GHX with 700 rings took over a day to cal-
culate using VBA (an interpreted language.) In order to improve the
speed, we refined the superposition approach so that rings that are
farther than 10 m away from a given ring are ignored; rings that
are between 10 m and 2.5 m from a ring are approximated as being
point sources; and rings within 2.5 m of a ring are treated in a fully
detailed manner. By also taking advantage of symmetry and mak-
ing use of the fact that many pairs of rings are the same distance
apart, we were able to develop a model that is both sufficiently
accurate and acceptably fast. With the refinements to the algo-
rithm, computation of the response function for the 700 ring exam-
ple took 23 s in VBA, and should be considerably faster when
implemented in a compiled language. These refinements have rel-
atively little impact on the response functions – after one hour the
difference is less than 0.5% – and this should have less than a 0.1 °C
error on the simulated GHX exiting fluid temperature.
The undisturbed ground temperatures are computed with a
numerical model that performs a full surface heat balance, including
the effects of solar radiation, longwave radiation, convection, evapo-
transpiration and conduction. We took advantage of the existing
three-dimensional coarse-grid numerical model already imple-
mented in EnergyPlus to calculate the undisturbed ground temper-
atures. However, a one-dimensional model is all that is needed and
we recommend that a one-dimensional model be implemented in
order to increase the overall computational speed of the model.
The procedure for calculating the temperature response func-
tions has been formulated to minimize user input for standard con-
figurations of horizontal or vertical Slinky™ GHX fields in parallel
rows. This is not an inherent limitation, but allowing specification
of GHX in, for example, U-shaped configurations would require
additional user inputs.
We validated our model against the experimental data pub-
lished by Fujii et al. [2]. These data were collected for a U-shaped
configuration, which we approximated as being in two parallel
trenches. The simulated hourly and sub-hourly exiting fluid tem-
peratures were compared with the measured data from three
short-term (5 days) TRTs and a long-term (38 days) system test,
with RMSEs of 0.2 °C, 0.3 °C, 0.3 °C, and 0.5 °C, respectively. The
maximum error observed is À1 °C. To check the effect of approxi-
mating the GHX as being in two parallel trenches, we also calcu-
lated response functions for the U-shaped configuration, but
found it would only make a very small difference. Calculation of
response functions for other shapes could be done in the future.
The model is based on two approximations that likely lead to
some error: (1) the heat transfer rates of the tubes in a Slinky™
GHX field are uniform; (2) the mean fluid temperature is the aver-
age value of the entering and exiting fluid temperature, with the
second assumption relying on the first assumption. Theoretically,
these two assumptions will result in inaccuracy in calculating the
temperature response factors [21] as well as the exiting fluid tem-
perature [22,23]. However, for most practical installations of hori-
zontal Slinky™ GHX when used in conjunction with heat pumps,
including our validation cases, the error caused by these two
assumptions is insignificant. Past research on vertical boreholes
[22,23,21] suggests that the error caused by these approximations
should be low as long as the flow rates are typical for systems with
heat pumps and the ground heat exchangers are shallow, where
interaction with the surface will reduce any long-term effects.
Nevertheless, future research would be useful to help identify the
limits on use of these approximations.
Finally, the response function computation also assumes that
the ground is homogeneous, so freezing around the tube cannot
be accounted for. Experimental measurements of GHX in cold cli-
mates would be useful in informing future model development.
Likewise, the homogeneous ground has uniform properties and
the effects of moisture transport are not accounted for. Further
field measurements could be helpful in identifying needs for more
sophisticated models of undisturbed ground temperature and/or
heat transfer to/from the GHX tubing.
Acknowledgement
The authors gratefully acknowledge Dr. Hikari Fujii, professor at
Akita University, for providing the experimental data.
References
[1] Chiasson A.D. Advances in modeling of ground-source heat pump systems.
Master thesis. School of Aerospace and Mechanical Engineering, Oklahoma
State University, USA; 1999.
68 Z. Xiong et al. / Applied Energy 141 (2015) 57–69

13. [2] Fujii H, Nishi K, Komaniwa Y, Chou N. Numerical modeling of slinky-coil
horizontal ground heat exchangers. Geothermics 2012;41:55–62.
[3] Chong CSA, Gan G, Verhoef A, Garcia RG, Vidale PL. Simulation of thermal
performance of horizontal slinky-loop heat exchangers for ground source heat
pumps. Appl Energy 2013;104:603–10.
[4] Jones F. Closed loop geothermal systems – Slinky™ installation guide. Rural
Electric Research Project 86-1, National Rural Electric Cooperative Association,
Electric Power Research Institute, and International Ground Source Heat Pump
Association; 1995.
[5] Wu Y, Gan G, Verhoef A, Vidale PL, Gonzalez RG. Experimental measurement
and numerical simulation of horizontal-coupled slinky ground source heat
exchangers. Appl Therm Eng 2010;30:2574–83.
[6] Carslaw H, Jaeger J. Conduction of heat in solids. 2nd ed. Oxford, UK: Oxford
University Press; 1959.
[7] Ingersoll LR, Zobel OJ, Ingersoll AC. Heat conduction with engineering
geological and other applications. Madison, WI: The University of Wisconsin
Press; 1954.
[8] Man Y, Yang H, Diao N, Liu J, Fang Z. A new model and analytical solutions for
borehole and pile ground heat exchangers. Int J Heat Mass Transf
2010;53:2593–601.
[9] Man Y, Yang H, Diao N, Cui P, Lu L, Fang Z. Development of spiral heat source
model for novel pile ground heat exchangers. HVACR Res
2011;17(6):1075–88.
[10] Li M, Lai ACK. Analytical model for short-time responses of ground heat
exchangers with U-shaped tubes: model development and validation. Appl
Energy 2013;104:510–6.
[11] Zeng HY, Diao NR, Fang ZH. A finite line-source model for boreholes in
geothermal heat exchangers. Heat Transf-Asian Res 2002;31(7):558–67.
[12] Cui P, Yang H, Fang Z. Heat transfer analysis of ground heat exchangers with
inclined boreholes. Appl Therm Eng 2006;26:1169–75.
[13] Fontaine P-O, Marcotte D, Pasquier P, Thibodeau D. Modeling of horizontal
geoexchange systems for building heating and permafrost stabilization.
Geothermics 2011;40:211–20.
[14] Cui P, Li X, Man Y, Fang Z. Heat transfer analysis of pile geothermal heat
exchangers with spiral coils. Appl Energy 2011;88:4113–9.
[15] Mukerji S, Tagavi KA, Murphy WE. Steady-state heat transfer analysis of
arbitrary coiled buried pipes. J Thermophys Heat Transfer 1997;11(2):182–8.
[16] Li H, Nagano K, Lai Y. A new model and solutions for a spiral heat exchanger
and its experimental validation. Int J Heat Mass Transf 2012;55:4404–14.
[17] Eskilson P. Thermal analysis of heat extraction boreholes. Doctoral thesis. Lund
University, Department of Mathematical Physics, Lund, Sweden; 1987.
[18] Yavuzturk C, Spitler JD. A short time step response factor model for vertical
ground loop heat exchangers. ASHRAE Trans 1999;105(2):475–85.
[19] Claesson J, Dunand A. Heat extraction from the ground by horizontal pipes—a
mathematical analysis. Sweden: Department of Mathematical Physics, Lund
University; 1983.
[20] Lee ES, Fisher DE, Spitler JD. Efficient horizontal ground heat exchanger
simulation with zone heat balance integration. HVACR Res
2013;19(3):307–23.
[21] Malayappan V, Spitler JD. Limitations of using uniform heat flux assumptions
in sizing vertical borehole heat exchanger fields. In: Proceedings of Clima
2013, June 16–19, Prague; 2013.
[22] Marcotte D, Pasquier P. On the estimation of thermal resistance in borehole
thermal conductivity test. Renewable Energy 2008;33:2407–15.
[23] Beier RA. Vertical temperature profile in ground heat exchanger during in-situ
test. Renewable Energy 2011;36:1578–87.
[24] Xu H, Spitler JD. The relative importance of moisture transfer, soil freezing and
snow cover on ground temperature predictions. Renewable Energy
2014;72:1–11.
[25] Marcotte D, Pasquier P. The effect of borehole inclination on fluid and ground
temperature for GLHE systems. Geothermics 2009;38(4):392–8.
[26] Claesson J, Eskilson P. Conductive heat extraction to a deep borehole: thermal
analyses and dimensioning rules. Energy 1988;13(6):509–27.
[27] Salomone LA, Marlow JI. Soil and rock classification for the design of ground
coupled heat pump systems: field manual. EPRI CU-6600. Palo Alto,
CA: Electric Power Research Institute; 1989.
[28] Kusuda T, Achenbach PR. Earth temperature and thermal diffusivity at selected
stations in the United States. Technical Report. National Bureau of Standards,
Washington, D.C.; 1965.
[29] Xing L. Analytical and numerical modeling of foundation heat exchangers.
Master thesis. School of aerospace and mechanical engineering, Oklahoma
State University, USA; 2010.
[30] Lamberg P, Lehtiniemi R, Henell AM. Numerical and experimental
investigation of melting a freezing processes in phase change material
storage. Int J Therm Sci 2004;43(3):277–87.
[31] Michopoulos A, Kyriakis N. Predicting the fluid temperature at the exit of the
vertical ground heat exchangers. Appl Energy 2009;86:2065–70.
[32] Fujii H. Professor at Akita University, Japan; 2013 [private communication].
Z. Xiong et al. / Applied Energy 141 (2015) 57–69 69