Vertical-Borehole Ground-coupled Heat Pumps

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Vertical-borehole ground-coupled heat pumps: A review of models and systems

H. Yang a,*, P. Cui a, Z. Fang b

a Renewable Energy Research Group, Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, Chinab Ground Source Heat Pump Research Center, Shandong Jianzhu University, Jinan, China

a r t i c l e i n f o

Article history:

Received 13 March 2009

Received in revised form 28 April 2009Accepted 28 April 2009

Available online 29 May 2009

Keywords:

Ground-coupled heat pump

Ground heat exchanger

Hybrid ground-coupled heat pump

Heat transfer

a b s t r a c t

A large number of ground-coupled heat pump (GCHP) systems have been used in residential and com-

mercial buildings throughout the world due to the attractive advantages of high efficiency and environ-

mental friendliness. This paper gives a detailed literature review of the research and developments of the

vertical-borehole GCHP technology for applications in air-conditioning. A general introduction on the

ground source heat pump system and its development is briefly presented first. Then, the most typical

simulation models of the vertical ground heat exchangers currently available are summarized in detail

including the heat transfer processes outside and inside the boreholes. The various design/simulation

programs for vertical GCHP systems primarily based on the typical simulation models are also reviewed

in this paper. Finally, the various hybrid GCHP systems for cooling or heating-dominated buildings are

well described. It is found that the GCHP technology can be used both in cold and hot weather areas

and the energy saving potential is significant.

2009 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172. Simulation models of vertical GHEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1. Heat conduction outside borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1. Kelvins line source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2. Cylindrical source model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.3. Eskilsons model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.4. Finite line-source solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.5. Short time-step model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.6. Other typical numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2. Heat transfer inside borehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1. One-dimensional model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2. Two-dimensional model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3. Quasi-three-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3. Comparisons of the analytical and numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3. Computer programs for GCHP design/simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1. IGSHPA approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2. Design tools based on the line-source model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1. The Lund programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2. The GLHEPRO program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3. The GeoStar program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.4. The building simulation programs integrated with GHE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3. The GchpCalc program based on cylindrical source model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4. Numerical simulation programs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. Hybrid GCHP systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1. HGCHP systems with supplemental heat rejecters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

0306-2619/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.apenergy.2009.04.038

* Corresponding author. Tel.: +852 2766 7801; fax: +852 2774 6146.

E-mail addresses: [email protected] (H. Yang), [email protected] (P. Cui), [email protected] (Z. Fang).

Applied Energy 87 (2010) 1627

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4.2. HGCHP systems with hot water supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3. HGCHP systems with solar collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5. Conclusions and recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1. Introduction

Ground source heat pump (GSHP) systems use the ground as a

heat source/sink to provide space heating and cooling as well as

domestic hot water. The GSHP technology can offer higher energy

efficiency for air-conditioning compared to conventional air condi-

tioning (A/C) systems because the underground environment pro-

vides lower temperature for cooling and higher temperature for

heating and experiences less temperature fluctuation than ambient

air temperature change.

The first known record of the concept of using the ground as

heat source for a heat pump was found in a Swiss patent issued

in 1912 [1]. Thus, the research associated with the GSHP systems

has been undertaken for nearly a century. The first surge of interestin the GSHP technology began in both North America and Europe

after World War Two and lasted until the early 1950s when gas

and oil became widely used as heating fuels. At that time, the basic

analytical theory for the heat conduction of the GSHP system was

proposed by Ingersoll and Plass [2], which served as a basis for

development of some of the later design programs.

The next period of intense activity on the GSHPs started in

North America and Europe in 1970s after the first oil crisis, with

an emphasis on experimental investigation. During this time peri-

od, the research was focused on the development of the vertical-

borehole system due to the advantage of less land area require-

ment for borehole installation. In the ensuing two decades, consid-

erable efforts were made to establish the installation standard and

develop design methods [36].To date, the GSHP systems have been widely used in both resi-

dential and commercial buildings. It is estimated that the GSHP

system installations have grown continuously on a global basis

with the range from 10% to 30% annually in recent years [7].

The GSHPs comprise a wide variety of systems that may use

ground water, ground, or surface water as heat sources or sinks.

These systems have been basically grouped into three categories

by ASHRAE [8], i.e. (1) ground water heat pump (GWHP) systems,

(2) surface water heat pump (SWHP) systems and (3) ground-cou-

pled heat pump (GCHP) systems. The schematics of these different

systems are shown in Fig. 1. The GWHP system, which utilizes

ground water as heat source or heat sink, has some marked advan-

tages including lowinitial costand minimal requirementfor ground

surfacearea over other GSHP systems[9]. However, a numberof fac-

tors seriously restrict the wide application of the GWHP systems,

such as the limited availability of ground water and the high main-

tenance cost dueto foulingcorrosion in pipelines and equipment. In

addition, many legal issues have arisen over ground water with-

drawal and re-injection in some regions, which also restrict the

GWHP applications to a large extent. In a SWHP system, heat rejec-

tion/extraction is accomplished by the circulating working fluid

through high-density polyethylene (HDPE) pipes positioned at an

adequate depth withina lake, pond, reservoir, or other suitableopen

channels. Natural convection becomes the primary role in the heatexchangers of the SWHP systemrather than heat conduction in the

heat transfer process in a GCHP system, which tends to have higher

heat exchange capability than a GCHP system. The major disadvan-

tage of the systemis that the surface water temperature is more af-

fected by weather condition, especially in winter.

In a GCHP system, heat is extracted from or rejected to the

ground via a closed loop, i.e. ground heat exchanger (GHE), through

which pure water or antifreeze fluid circulates. The GHEs com-

monly used in the GCHP systems typically consist of HDPE pipes

which are installed in either vertical boreholes (called vertical

GHE) or horizontal trenches (horizontal GHE). In the horizontal

GCHP systems, the GHEs typically consist of a series of parallel pipe

arrangements laid out in dug trenches approximately 12 m below

the ground surface. A major disadvantage is that the horizontalsystems are more affected by ambient air temperature fluctuations

because of their proximity to the ground surface. Another disad-

vantage is that the installation of the horizontal systems needs

much more ground area than vertical systems.

In the vertical GCHP systems, the GHE configurations may in-

clude one, tens, or even hundreds of boreholes, each containing

one or double U-tubes through which heat exchange fluid is circu-

lated. Typical U-tubes have a diameter in the range of 1938 mm

and each borehole is normally 20200 m deep with a diameter

ranging from 100 mm to 200 mm. The borehole annulus is

Fig. 1. Schematics of different ground source heat pumps.

H. Yang et al./ Applied Energy 87 (2010) 1627 17


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generally backfilled with some special material (named as grout)

that can prevent contamination of ground water. A typical bore-

hole with a single U-tube is illustrated in Fig. 2.

The worldwide growing energy shortage and increasing energydemand have recently driven a great incentive of the GSHP appli-

cations in air conditioning field. Among the various GSHP systems,

the vertical GCHP system has attracted the greatest interest in re-

search field and practical engineering as well, owing to its advan-

tages of less land area requirement and wide range of

applicability. During the past few decades, a considerable number

of studies have been carried out to investigate the development

and applications of the GCHP systems with various types of GHEs

and addressed their individual advantages and disadvantages in

detail. Furthermore, various hybrid GCHP systems which couple

the conventional GCHP equipment with a supplemental heat rejec-

tion/generation device have been recently developed in order to

improve the economics of the GCHP systems for unbalanced cli-

mates. Several literature reviews on the GCHP technology have

been reported [7,1012]. This paper mainly presents a detailed lit-

erature review of the vertical-borehole GCHP systems, primarily

related to the typical heat transfer models of the GHEs and the rep-resentative design/simulation programs as well as advanced engi-

neering applications of hybrid GCHP systems.

2. Simulation models of vertical GHEs

The major difference between the GCHP system and a conven-

tional A/C system is the use of a special heat exchanger (i.e. GHE)

instead of a cooling tower. The construction costs of the GHEs

are critical for the economical competitiveness of a GCHP system

for a heating or an A/C system. On the other hand, the GHE size also

plays a decisive role on the operation performance of the GCHP

system. Thus, it is of great importance to work out sophisticated

and validated tools by which the thermal behavior of any GCHP

system can be assessed and then, optimized in technical and eco-nomical aspects.

The main objective of the GHE thermal analysis is to determine

the temperature of the heat carrier fluid, which is circulated in the

U-tubes and the heat pump, under certain operating conditions. A

design goal is then to control the temperature rise of the ground

and the circulating fluid within acceptable limits over the system

lifespan.

The rule of thumb approximation method was in fashion for a

long time, which was discussed by Ball et al. [1]. Rules of thumb

can serve well for specific localities where soil and weather condi-

tions are fairly uniform because design specifications are primarily

based on the experience with related installations. However, some

systems have suffered from the inability of the rule of thumb

designers to properly assess the effect of varied design parameters,such as shallower burial depth, lower shank spacing between U-

tube legs and larger borehole space in ground surface.

In addition to the rule of thumb method, several models with

different complexity have been developed for the design and per-

formance prediction of the GHEs in engineering applications. Actu-

ally, the heat transfer process in a GHE involves a number of

uncertain factors, such as the ground thermal properties, the

ground water flow and building loads over a long lifespan of sev-

eral or even tens of years. In this case, the heat transfer process

is rather complicated and must be treated, on the whole, as a tran-

sient one. In view of the complication of this problem and its long

time scale, the heat transfer process may usually be analyzed in

two separated regions. One is the solid soil/rock outside the bore-

hole, where the heat conduction must be treated as a transient pro-cess. With the knowledge of the temperature response in theFig. 2. Schematic of a vertical grouted borehole.

Nomenclatures

a ground thermal diffusivity (m2/s)c water heat capacity (kJ/(kgK))D uppermost part of the borehole (m)F run fractionH borehole length (m)

k ground thermal conductivity (W/m K)M fluid mass flow rate per borehole (kg/s)ql heating rate per borehole length (W/m)rb borehole radius (m)R thermal resistance (m2 K/W)Tor t temperature (C)t0 ground initial temperature (C)z variable of borehole depth (m)

e heat transfer efficiency of a boreholes time (s)

Subscriptsb borehole wall

C cooling modef fluidH heating modep pipe of the U-tubes soil or ground

AbbreviationGLHEPRO ground loop heat exchange program

18 H. Yang et al./ Applied Energy 87 (2010) 1627


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ground, the temperature on the borehole wall can then be deter-

mined for any instant on specified operational conditions. Another

sector often segregated for analysis is the region inside the bore-

hole, including the grout, the U-tube pipes and the circulating fluid

inside the pipes. This region is sometimes analyzed as being stea-

dy-state or quasi-steady-state and sometimes analyzed as being

transient. The analyzes on the two spatial regions are interlinked

on the borehole wall. The heat transfer models for the two separate

regions are as follows.

2.1. Heat conduction outside borehole

A number of simulation models for the heat transfer outside the

borehole have been recently reported, most of which were based

on either analytical methodologies or numerical methods. A few

models were developed based on the incorporation of the analyti-

cal and numerical solutions, such as Eskilsons model [6].

2.1.1. Kelvins line source

The earliest approach to calculating the thermal transport

around a heat exchange pipe in the ground is the Kelvinline-source

theory, i.e. the infinite line source [2,13]. In the Kelvins line-source

theory, the ground is regarded as an infinite medium with an initial

uniform temperature, in which the borehole is assumed as an infi-

nite line source. The heat transfer in the direction of the borehole

axis, including the heat flux across the ground surface and down

the bottom of the borehole, is neglected. The heat conduction pro-

cess in the ground is, therefore, simplified as one-dimensional one.

According to the Kelvins line-source theory, the temperature re-

sponse in the ground due to a constant heat rate is given by:

tr; s t0 ql4pk

Z1r2

4as

eu

udu 1

where r is the distance from the line-source and s the time since

start of the operation; t the temperature of the ground at distance

rand time s; t0 the initial temperature of the ground; ql the heating

rate per length of the line source; and k and a are the thermal con-

ductivity and diffusivity of the ground.

The solution to the integral term in Eq. (1) can be found from

the related references [5,14,15]. Although it is characterized by

the simplicity and less computation time, this model can only be

applied to small pipes within a narrow range of a few hours to

months because of the assumption of the infinite line source

[6,16]. It was estimated that using the Kelvins line source may

cause a noticeable error when asr2

b

< 20 [14].

This approach has been widely utilized in some analytical de-

sign methods that are currently used to analyze the heat transfer

of GHEs [5,15,17]. A number of improvements for this approach

have been proposed to account for some complicated factors so

that the accuracy can be comparable to that of the numerical

methods. Of all these methods employing Kelvins line-source the-ory, the Hart and Couvillion method may be more accurate than

others [15].

2.1.2. Cylindrical source model

The cylindrical source solution for a constant heat transfer rate

was first developed by Carslaw and Jaeger [18], then refined by

Ingersoll et al. [14], and later employed in a number of research

studies [1921]. It is actually an exact solution for a buried cylin-

drical pipe with infinite length under the boundary condition of

either a constant pipe surface temperature or a constant heat

transfer rate between the buried pipe and the soil. In the cylindri-

cal source model, the borehole is assumed as an infinite cylinder

surrounded by homogeneous medium with constant properties,

i.e. the ground. It also assumes that the heat transfer between

the borehole and soil with perfect contact is of pure heat

conduction.

Based on the governing equation of the transient heat conduc-

tion along with the given boundary and initial conditions, the tem-

perature distribution of the ground can be easily given in the

cylindrical coordinate:

@2t@r2

1r@t@r

1a

@t@s

rb < r 0t t0 0 s 0; r> rb

8>: 2

where rb is the borehole radius.

The cylindrical source solution is given as follows:

t t0 qlkGz; p 3

where z asrb; p r

rb.

As defined by Carslaw and Jaeger [18], the expression G(z, p) is

only a function of time and distance from the borehole center.

The temperature on the borehole wall, where r= rb, i.e. p = 1, is

of interest as it is the representative temperature in the design of

GHEs. However, the expression G(z, p) is relatively complex and in-

volves integration from zero to infinity of a complicated function,which includes some Bessel functions. Fortunately, some graphical

results and tabulated values for the G(z, p) function at p = 1 are

available in some related references [14,19]. An approximate

method for G was proposed by Hellstrom [22] and presented by

Liu et al. [23].

2.1.3. Eskilsons model

Both the one-dimensional model of the Kelvins theory and the

cylindrical source model neglect the axial heat flow along the bore-

hole depth; therefore they are inadequate for the long-term oper-

ation of the GCHP systems. A major progress was made by Eskilson

(1987) to account for the finite length of the borehole [6]. In Eskil-

sons model, the ground is assumed to be homogeneous with con-

stant initial and boundary temperatures, and the thermalcapacitance of the borehole elements such as the pipe wall and

the grout are neglected. The basic formulation of the ground tem-

perature is governed by the heat conduction equation in cylindrical

coordinates:

@2t@r2

1r@t@r

@2t@z2

1a

@t@s

tr; 0; s t0tr;z; 0 t0qls 1H

RDHD

2prk @t@r

jrrbdz

8>>>>>>>:

4

where His the borehole length; D means the uppermost part of the

borehole, which can be thermally neglected in engineering practice.

In Eskilsons model, the numerical finite-difference method is

used on a radialaxial coordinate system to obtain the temperaturedistribution of a single borehole with finite length. The final

expression of the temperature response at the borehole wall to a

unit step heat pulse is a function ofs=ss and rb=H only:

tb t0 ql2pk

gs=ss; rb=H 5

where ss H2=9a means the steady-state time. The g-function is

essentially the dimensionless temperature response at the borehole

wall, which was computed numerically.

Another important achievement of Eskilsons model is that the

special superimposition was employed to account for the temper-

ature responses for multiple boreholes. In addition, the sequential

temporal superimposition was used to calculate the temperature

response (i.e. g-functions) to any arbitrary heat rejection/extrac-tion which can be decomposed into a set of single pulses. In other

H. Yang et al./ Applied Energy 87 (2010) 1627 19


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words, the overall temperature response of the GHE to any heat

rejection/extraction at any time can be determined by the special

and temporal superimpositions.

The disadvantage of this approach, however, is time-consum-

ing, and it can hardly be incorporated directly into a design and en-

ergy analysis program for practical applications, since the g-

functions of the GHEs with different configurations have to be

pre-computed and stored in the program as a massive database.

The interpolation function is also needed in using the database

which may lead to some computing errors.

2.1.4. Finite line-source solution

Based on the Eskilsons model, an analytical solution to the fi-

nite line source has been developed by a research group which

considers the influences of the finite length of the borehole and

the ground surface as a boundary [24]. Some necessary assump-

tions are taken in the analytical model in order to derive an analyt-

ical solution:

The ground is regarded as a homogeneous semi-infinite medium

with constant thermophysical properties.

The boundary of the medium, i.e. the ground surface, keeps aconstant temperature (t0), same as its initial one throughout

the time period concerned.

The radial dimension of the borehole is neglected so that it may

be approximated as a line-source stretching from the boundary

to a certain depth, H.

As a basic case of study, the heating rate per length of the source,

ql, is constant since the starting instant, s = 0.

The computation results from the analytical solution were com-

pared with the data from numerical solutions in references [6,24],

and they agreed with each other perfectly when as=r2b P 5: The

solution of the temperature excess was given by Zeng et al. [24]:

t r;z; s t0 ql4kp

ZH

0

erfcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2zh

2p

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6It can be seen from Eq. (6) that the temperature on the bore-

hole wall, where r= rb, varies with time and borehole depth. The

temperature at the middle of the borehole depth (z= 0.5H) is

usually chosen as its representative temperature. An alternative

is the integral mean temperature along the borehole depth,

which may be determined by numerical integration of Eq. (6).

For the convenience of applications, the former is usually ac-

cepted as the representative temperature in the design and

analysis program. It is obvious that the integral of Eq. (6) can

be computed much faster than the numerical solution of thesame heat conduction problem in the semi-infinite domain with

long duration. The methodology has been complied in the later

design and simulation software developed by other researchers

[25,26].

As discussed in the aforementioned section, when time tends to

infinity, the temperature rise of the Kelvins theory tends to infin-

ity, whereas the temperature from the finite line-source model ap-

proaches steady state, which corresponds to the actual heat

transfer mechanism. With respect to long duration, substantial dis-

crepancy between the Kelvins model and the finite line source

may yield [27].

2.1.5. Short time-step model

Since both Eskilsons model and the finite line-source model ne-glect the effect of the thermal capacity of the borehole including

the U-tubes, circulating fluid and the grout, the dimensionless tem-

perature responses on the borehole wall are only valid approxi-

mately for the time greater than5r2

b

a, estimated by Eskilson [6].

For a typical borehole with a radius of 55 mm, the required time

may be any time period between 2 and 6 h.

Yavuzturk and Spitler presented a short time-step model for the

simulation of the transient heat transfer in vertical GHEs, which

can be accurate down to an hour and below [28,29]. This short

time-step model is based on the two-dimensional, fully implicit fi-

nite volume formulation and utilizes an automated parametric grid

generation algorithm for different pipe sizes, shank spacing and

borehole geometry. The numerical results are expressed in terms

of a short time-step response factor (i.e. g-function), which is a

very useful extension of the long time-step response factor devel-

oped by Eskilson [6]. The numerical model was cast as a compo-

nent model for TRNSYS developed by Klein et al. [30]. The

authors stated that it can evaluate the energy consumption and

electrical demand of the GCHP system in hourly or shorter time

intervals.

2.1.6. Other typical numerical models

Hellstrom (1989, 1991) and Thornton et al. (1997) proposed a

simulation model for ground heat stores, which are densely packed

ground loop heat exchangers used for seasonal thermal energy

storage [22,31,32]. This type of systemmay be directly used to heat

buildings with or without a heat pump. The duct storage model

(named as DST) divides the ground storage volume with multiple

boreholes into two regions: one is the volume that surrounds a sin-

gle borehole, described as the local region; the other is called

global region, which denotes the ground volume between the

bulk of the heat store volume and the far field. A two-dimensional

finite difference scheme is used to solve the ground temperature in

the global region while the one-dimensional numerical method

is employed to calculate the temperature in the local region.

Since the ground heat storage is mainly used to provide a heating

function, the boreholes are generally spaced in a quite dense field,

which may be not suitable for some buildings with a considerableamount of cooling loads [33].

Muraya, et al. (1996) developed a transient finite-element mod-

el of the heat transfer around a vertical U-tube heat exchanger for a

GCHP system to study the thermal interference that occurred be-

tween the adjacent legs of the U-tube [34]. The thermal interfer-

ence was quantified by defining a heat exchanger effectiveness.

The impacts of the separation distance, leg temperatures, different

ambient soil temperatures and backfills were all investigated.

Rottmayer, et al. (1997) presented a finite difference model that

simulated the heat transfer process of a U-tube heat exchanger

[35]. A geometric factor was introduced to account for the noncir-

cular geometry used to represent the pipes in the borehole. The

model was validated for simple conditions and compared with an

existing model, resulting in good agreement.A three-dimensional unstructured finite volume model for the

vertical GHEs was developed by Li and Zheng [36]. The model uses

Delaunay triangulation method to mesh the cross-section domain

of the borehole field, and consequently retains the geometric struc-

ture in the borehole. The surrounding soil is divided into many lay-

ers in the vertical direction in order to account for the effect of

changing fluid temperature with depth on the thermal process in

the bore field. Finally, a comparison of the model predictions and

experimental data shows that the model has good prediction

accuracy.

2.2. Heat transfer inside borehole

The thermal resistance inside the borehole, which is primarilydetermined by the thermal properties of the grouting materials

20 H. Yang et al./ Applied Energy 87 (2010) 1627


6/12

and the arrangement of flow channels of the borehole, has a signif-

icant impact on the GHE performance. The main objective of this

analysis is to determine the entering and leaving temperatures of

the circulating fluid in the borehole according to the borehole wall

temperature, its heat flow and the thermal resistance. A few mod-

els with varying degrees of complexity have been established to

describe the heat transfer inside the GHE boreholes.

2.2.1. One-dimensional model

A simplified one-dimensional model has been recommended for

GHE design, which considers the U-tube as a single equivalent

pipe [5,37]. In this model, both the thermal capacitance of the

borehole and the axial heat flow in the grout and pipe walls are

negligible as the borehole dimensional scale is much smaller com-

pared with the infinite ground outside the borehole. Thus, the heat

transfer in this region is approximated as a steady-state one-

dimensional process. The authors stated that the simplified one-

dimensional model was appropriate and convenient for most engi-

neering practices except for the analyzes to deal with dynamic re-

sponses within a few hours. However, this oversimplified model

seems inadequate and unsatisfactory because it is incapable of

evaluating the impact of the thermal short circuiting between

the U-tube legs on the performance of the GHEs.

2.2.2. Two-dimensional model

Hellstrom (1991) derived the analytical two-dimensional solu-

tions of the thermal resistances among pipes in the cross-section

perpendicular to the borehole axis, which is superior to empirical

expressions and one-dimensional model [22]. In the two-dimen-

sional, the temperature of the fluid in the U-tubes is expressed as

a superposition of the two separate temperature responses caused

by the heat fluxes per unit length, q1 and q2, from the two pipes of

the U-tube, as shown in Fig. 3. If the temperature on the borehole

wall, tb, which is also considered as uniform along the borehole

depth, is taken as a reference of the temperature excess, the fluid

temperatures in the U-tubes can be obtained from the following

equations:

tf1 tb R11q1 R12q2tf2 tb R12q1 R22q2

7

where R11 and R22 are the thermal resistances between the circulat-

ing fluid in each pipe and the borehole wall, and R12 is the resistance

between the two pipes. A linear transformation of Eq. (7) leads to:

q1 tf1tbRD1

tf1tf2RD12

q2 tf2tbRD2

tf2tf1RD12

8

Vertical-Borehole Ground-coupled Heat Pumps

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