A transient natural convection heat transfer modelfor geothermal borehole heat exchangers

S. A. Ghoreishi-Madiseh,1,a) F. P. Hassani,2 A. Mohammadian,3 andP. H. Radziszewski11Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 2A7,Canada2Department of Mining and Materials Engineering, McGill University, Montreal,Quebec H3A 2A7, Canada3Department of Civil Engineering, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada

(Received 18 September 2012; accepted 14 June 2013; published online 2 July 2013)

The effect of buoyancy-driven natural convection on the performance of ground-

coupled heat exchangers of closed loop geothermal systems is investigated. The

governing equations of continuity, momentum, and energy balance are derived, taking

into account a porous ground medium fully saturated with liquid water. Boussinesq

approximation is used to model the effect of buoyancy forces in water. A three-

dimensional finite-volume discretization method over a structured mesh is used to

solve the governing equations numerically. The performance of the ground-coupled

heat exchanger system is assessed based on the rate of energy extraction and the outlet

fluid temperature. The effects of hydraulic conductivity of the heat exchange medium

and seasonal variations of heat load on the heat transfer phenomenon are studied. The

results are evaluated by comparing them against the results of existing conduction-

based heat transfer models. The influence of natural convection on the sustainable

rate of heat extraction from a geothermal resource is underlined and interpreted.VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4812647]

NOMENCLATURE

Abbreviations

GCHE Ground Coupled Heat Exchanger

Symbols

b Coefficient of thermal expansion of water (1/ �C)

c Ratio of natural convection over conduction

u Porosity

l Dynamic viscosity (N s/m2)

K Permeability (m2)

~g Gravity acceleration vector (m/s2)

h~ui Darcy velocity (m/s)

V Representative elementary volume (m3)

DV Volume of the finite volume cell

hPif Intrinsic average pressure of fluid (N/m2)

DF Dimensionless form-drag constant

q Density (kg/m3)

C Specific heat capacity (J/kg/ �C)

k Thermal conductivity of porous medium (W/m/ �C)

a)Author to whom correspondence should be addressed. Electronic mail: seyed.ghoreishimadiseh@mail.mcgill.ca

1941-7012/2013/5(4)/043104/15/$30.00 VC 2013 AIP Publishing LLC5, 043104-1

JOURNAL OF RENEWABLE AND SUSTAINABLE ENERGY 5, 043104 (2013)

_q Rate of heat generation (or extraction) per unit volume (W/m3)

T Temperature (�C)

_m Mass flow rate of water through the borehole (kg/s)

U0 Overall heat transfer coefficient (W/m2/ �C)

h Convection coefficient of water flowing through the tube (W/m2/ �C)

NuD Nusselt number

ReD Reynolds number

Pr Prandtl number

DL Length of the tube cell (m)

D Inner diameter of the tube cell (m)

r Radius (m)

Subscripts

f Water

m Porous medium

wall U-tube wall

tube U-tube

1 Inner surface of tube

2 Outer surface of tube

I. INTRODUCTION

The escalating price of fossil fuels, their pending scarcity, and the greenhouse gas byprod-

ucts of their combustion have motivated researchers to look for alternative, renewable, and

emission-free sources of energy. Geothermal energy is one of the most promising types of

such energy sources, suitable for heating (or cooling) as well as electricity generation pur-

poses. As a result, the application of geothermal systems is growing significantly around the

world. There are two different techniques for extraction of geothermal heat from underground

reservoirs. The first type is called open loop, in which water (from underground aquifers) is

delivered to the surface through the production well(s), and its heat content removed from it.

The water is usually returned underground through injection well(s) to recapture geothermal

heat. The second type is the closed loop, in which geothermal heat extraction is made possible

by circulating a working fluid in a closed-loop network of tubes embedded into the ground.

Although both types are quite popular, the application of open loop systems is constrained by

the availability of excessive amounts of water and requires considerable electric pumping

power. In some occasions, open loop systems are even associated with environmental issues

such as underground water displacement to the surface and its exposure to air.1 Since closed

loop geothermal systems consume less electricity and do not raise those environmental issues,

they have been integrated into the heating systems of various buildings over the last three

decades.

Every closed loop system has a specially designed network of tubes installed into the

ground, called a Ground Coupled Heat Exchanger (GCHE) or borehole heat exchanger. The

overall performance of a closed loop geothermal system is directly dependent on the efficiency

of heat transfer in its borehole heat exchanger units.2,3 Accordingly, a number of research

works have been dedicated to understanding the heat transfer in GCHEs. Early studies of heat

transfer in GCHE proposed empirical conduction-based models in which U-tube heat exchang-

ers are considered as line heat sink/source elements and conduction is assumed to be the domi-

nant heat transfer mechanism.1,4,5 However, these models do not have the capability of simulat-

ing complicated GCHE tube network geometries or intermittent heat loads. To address such

deficiencies, transient conduction-based numerical models have been used to allow the simula-

tion of complex GCHE geometries (e.g., U-tubes serried sequentially) and intermittent heat

loads.6–8 It is important to note that conduction-based models are valid for GCHEs with little

or no underground water movement where the advection mechanism is negligible compared to

conduction.

043104-2 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

For convection to happen in the ground, there is a need for pressure gradient. There are

two distinctive mechanisms which can stimulate such pressure gradients. The first mechanism

is the hydrologic head exerted to the ground located below the water table. In this case, the

GCHE will experience a pressure-driven convection. The second mechanism is the buoyancy

force which will result in buoyancy-driven natural convection.

In order to investigate the effect of pressure-driven underground water movement on heat

transfer in GCHEs, some researchers have suggested a conductive-advective model in which

the movement of underground water is simulated assuming a steady uniform underground fluid

flow field.9–12 This conductive-advective approach proved that underground water flow can

affect the heat transfer of a GCHE and cannot be neglected in all cases.

With a growing number of closed loop geothermal cycles installed every year, engineers

are proposing the implementation of GCHEs in fields where natural convection heat transfer

mechanism can be significant. An example of such applications includes embedding GCHEs in

highly permeable geo-materials.13 In such new applications, dealing with a porous ground me-

dium with relatively high permeability can bring in the possibility of having fluid movement

due to buoyancy forces. This buoyancy driven movement of ground water through the pores

enhances the heat transfer in the ground due to better fluid mixing. In other words, the naturally

convected heat component helps improve heat exchange between the tube network of a GCHE

and its geothermal reservoir. Since natural convection cannot be captured using the existing

conductive-advective models, it is necessary to develop a new heat transfer model capable of

simulating natural convection heat transfer in borehole heat exchangers installed in porous geo-

thermal resources. To the best of the authors’ knowledge, the study of buoyancy-driven natural

convection in GCHEs has not so far been carried out.

It is important to note that the significance of pressure-driven convection is dependent on

the downstream hydrologic conditions as well as the pressure gradient. This means that if an

impermeable rock formation lies below a permeable geothermal reservoir, or if the hydrologic

pressure gradient is deemed little, the effect of pressure-driven convection will be insignificant.

Contrary to pressure-driven water movement, buoyancy-driven ground flow is not dependent on

downstream ground permeability or hydrologic pressure-gradient. In this case, the buoyancy

force may be capable of stimulating water circulation inside the porous medium. To distinguish

the effect of buoyancy-driven natural convection from pressure-driven convection, it is required

to study the significance of the former in the absence of the latter.

The main objective of this paper is to assess the effect of natural convection in borehole

heat exchangers installed in porous geothermal resources. To achieve this goal, the authors

have referred to the existing literature dedicated to fluid flow and heat transfer in porous media.

An extensive study of fluid flow and heat transfer in porous media was undertaken by

Whitaker14 and Vafai and Tien15 who have also analyzed fluid flow and heat transfer in satu-

rated porous media profoundly. The derivation of the equations of mass, momentum, and

energy conservation for porous media evolved through the supplementary discussions between

researchers in the last two decades.16–19 The essence of these discussions was later collected

and published by Nield and Bejan20 and Kaviany.21 The study of natural convection in porous

media was carried out by Beckermann et al.,22 Lage,23 Hossain et al.,24 and Saada et al.25

Based on the state of the art knowledge of fluid flow and heat transfer in porous media,

a heat transfer model is developed and numerically solved. The model is then used to exam-

ine possibility of improving the performance of borehole heat exchangers by means of natural

convection. The analyses conducted in this paper help with determining the significance of

natural convection and its contribution to the performance of borehole heat exchangers with

various permeability values. Finally, the current study aims at investigating the effects of

geometric and thermo-physical properties of a GCHE on the performance of its energy

production.

The paper is organized as follows. In Sec. II, the mathematical model and the associated

GCHE geometry are explained and the related physical domain and boundary conditions are

introduced. The governing equations and their physical meaning are discussed in this section as

well. Section III includes the numerical solution procedure and discretization methods. Model

043104-3 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

validation, results, and discussions are described in Sec. IV. Concluding remarks in Sec. V

complete the study.

II. MODEL DESCRIPTION

A. Model geometry

A ground-coupled heat exchanger embedded in a porous ground medium, as shown in

Fig. 1, forms the physical domain of the borehole heat exchanger. The porous ground structure

is comprised of solid particles and its pores are assumed to be fully saturated by water (i.e., po-

rous ground is fully saturated). The tube network of the GCHE is assumed to be installed inside

this control volume of porous material. In order to focus on canonical cases, GHCE tubes are

placed vertically in an organized matrix form. The boundaries of the medium are considered to

be sufficiently far from the tubes so that extending these boundary walls does not change the

interior flow and temperature fields. Water is assumed to be the working fluid of all the bore-

hole heat exchangers. The U-tube units can be connected in series to increase the outlet water

temperature. For instance, Fig. 1 shows a tube network consisting of eighteen single tubes

forming nine U-tube units.

B. Two-dimensional vs. three-dimensional models

In some numerical simulations of heat transfer in GCHEs, such as the models developed

by Diao et al.,10 Fan et al.,11 and Chiasson et al.,9 it is considered that the temperature gradient

along the length of a borehole heat exchanger is negligible, and therefore a two-dimensional

(2D) Cartesian heat transfer model would suffice. However, since the bulk temperature of the

working fluid in a borehole heat exchanger approaches that of the ground while it travels along

the tube length, it is probable to observe a temperature gradient along the bore length.12 This

temperature gradient, if significant, disproves the validity of the 2D model assumption. Also,

capturing the natural convection is not possible using a 2D heat transfer model. The above-

mentioned potential downsides of 2D heat transfer models necessitate full-scale three-dimen-

sional (3D) modeling of the heat transfer, as presented in this paper.

C. Governing equations

It is assumed that the hydrological pressure gradient does not introduce any underground

water movement. Thus, the main moving force responsible for fluid movement through the

pores of the permeable ground medium is the buoyancy mechanism. The fully saturated porous

FIG. 1. (a) 3D representation of the model, and (b) mid-plane cross section.

043104-4 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

ground material is assumed to be homogeneous (i.e., porosity is constant). Also, the density of

pore water is assumed to be related to its temperature by Boussinesq’s approximation:

q ¼ qf

�1� bðT � T0Þ

�, where, q, qf , and b are density of water at temperature T, density of

water at temperature T0, and coefficient of thermal expansion of water, respectively. Using the

notion of volume-averaged variables, in this paper the derivation of equations of mass and mo-

mentum is based on,16–20,23 which have given extensive discussion about the momentum equa-

tion in a porous medium. The governing equations of conservation of mass and momentum are,

respectively, expressed by

~r:h~ui ¼ 0; (1)

qf

1

u@h~ui@tþ 1

uðh~ui:~rÞ 1

uh~ui

� �( )¼ �~rhPif þ l

ur2h~ui þ l

Kh~ui þ

qf DFffiffiffiffiKp jh~uijh~ui

þ qf ð1� bðT � T0ÞÞ~g; (2)

where u, l, K, b, qm, qf , and ~g are, respectively, porosity, dynamic viscosity of water, perme-

ability, coefficient of thermal expansion of water, porous medium density, water density, and

gravity acceleration vector. Based on the volume averaging technique, h~ui ¼ 1V

ÐV~u dV is the

Darcy velocity, where V is a representative elementary volume of a porous medium. Similarly,

hPif ¼ 1Vf

ÐVf

P dV is the intrinsic average pressure of fluid taken over Vf (fluid representative

volume). Also, DF is the dimensionless form-drag constant for which the results of Beavers

et al.26 are used in this paper. Based on the discussions of Hsu and Cheng16 and Lage,23 the

momentum equation (2) includes Brinkman, Darcy, and Forchheimer terms. Therefore, it can

be used for the Darcy flow regime (ReK ¼ qf jh~uijffiffiffiffiKp

=l�1) as well as the Forchheimer flow

regime (ReK ¼ qf jh~uijffiffiffiffiKp

=l�1), and it also satisfies the no-slip boundary conditions.20

As for the equation of energy conservation, it is reasonable to assume that the fluid and

solid phases of the porous medium are in local thermal equilibrium:23 T ¼ hTif ¼ hTis, where

hTif , and hTis are, respectively, intrinsic averaged temperature of fluid and intrinsic averaged

temperature of solid. So, the resulting energy balance equation will be expressed by

qmCm@T

@tþ qf Cf ðh~ui:~rTÞ ¼ ~r:ðkm

~rTÞ þ _q; (3)

where Cm, Cf , km, and _q are, respectively, specific heat capacity of porous medium, specific

heat capacity of fluid, thermal conductivity of porous medium, and rate of heat generation (or

extraction) per unit volume of the porous medium. Depending on the position, two different

sets of material properties are substituted into Eqs. (2) and (3); as shown in Fig. 2, the proper-

ties of the bore fill material are used for the points located inside the borehole while the proper-

ties of porous ground material are used elsewhere.

In Eq. (3), _q represents the rate of heat exchanged between the ground and borehole heat

exchanger tube(s). Fig. 3 illustrates a tube cell and its surrounding control volume. Obviously,

FIG. 2. Illustration of borehole heat exchanger tubes, borehole fill material, and porous ground medium.

043104-5 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

_q is non-zero only in locations where the tubes rest, and zero elsewhere. Since the bulk temper-

ature of the water changes along the borehole length, assuming a constant _q may lead to unreal-

istic results. Accordingly, in this paper, _q is calculated using the local rate of heat exchanged

between the ground and the borehole tube,

_q ¼ � _mCf ðTf þ DTf � Tf Þ=DV ¼ � _mCf DTf=DV; (4)

where Tf , _m, and DV are, respectively, the bulk temperature of water, mass flow rate of water

through the borehole, and volume of the finite volume cell surrounding the tube cell. To guar-

antee the satisfaction of energy balance, the rate of enthalpy change in water must be equal to

the rate of heat transferred through the tube cell,

_mCf DTf ¼ UOðpDDLÞðTwall � Tf Þ; (5)

where Twall, U0, DL, and D are, respectively, wall temperature, overall heat transfer coefficient,

length of the tube cell, and inner diameter of the tube cell. Integrating the differential form of

Eq. (5) over the tube cell length leads to

_mCf

ðTfþDTf

Tf

dTf

Twall � Tf¼ pDUO

ðDL

0

dL; (6)

and, therefore,

DTf ¼ ðTwall � Tf Þð1� e�bÞ; (7)

where

b ¼ ðpDDLUOÞ=ð _mCf Þ: (8)

It is important to note that in Eq. (7), Twall is the temperature in the center of the control vol-

ume that resides inside the tube cell of Fig. 3. In other words, Eq. (7) relates the temperature of

the water flowing inside the heat exchange tube to local bore temperature. The overall heat

transfer coefficient is formulated as27

FIG. 3. Tube cell and its surrounding control volume.

043104-6 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

1

UO¼ r2

r1

� 1

hþ r2Lnðr2=r1Þ

ktube; (9)

where h, r1, r2, and ktube are, respectively, the convection coefficient of water flowing through

the tube, inner radius of the tube, outer radius of the tube, and thermal conductivity of the tube.

The convection coefficient, h, is obtained using the relation developed by Dittus and Boelter,28

NuD ¼ 3:66; Laminar U-tube flow ðReD � 2500Þ ; (10a)

NuD ¼ hD=kf ¼ 0:023ReD0:8Pr0:3; Turbulent U-tube flow ðReD > 2500Þ ; (10b)

where NuD, ReD ¼ 4 _m=ðpDlf Þ, Pr, kf , and lf are, respectively, the Nusselt number, Reynolds

number, Prandtl number, thermal conductivity of water, and dynamic viscosity of water. The phys-

ical interpretation of Eqs. (7)–(10) is better understood by examining the following extreme cases:

Case (1) UO � 1 or _m � 1 (associated to b� 1) leads to DTf � 0, meaning that if the

overall heat transfer coefficient is extremely low or the fluid flow rate is extremely high, the

fluid temperature at the outlet of the tube cell will be the same as at its inlet.

Case (2) UO � 1 or _m � 1 (associated to b� 1) leads to DTf � Twall � Tf , meaning that

if the overall heat transfer coefficient is extremely high or the fluid flow rate is extremely low,

the temperature of the fluid at the outlet of the tube cell will approach the ground temperature

at the location of the tube cell.

D. Initial and boundary conditions

The following initial temperatures and flow conditions are assumed:

Tjt¼0 ¼ T0; (11a)

h~uijt¼0 ¼~0; (11b)

where T0 is the initial ground temperature. Also, zero-velocity and isothermal boundary condi-

tions are assumed for all the boundaries except for the adiabatic z ¼ H2 boundary,

Tjx¼0 ¼ Tjx¼L2¼ Tjy¼0 ¼ Tjy¼W2

¼ Tjz¼0 ¼ T0; (12a)

@T

@z

����z¼H2

¼ 0; (12b)

h~uijx¼0 ¼ h~uijx¼L2¼ h~uiy¼0 ¼ h~uijy¼W2

¼ h~uijz¼0 ¼ h~uijz¼H2¼~0: (12c)

Equations (1)–(3) include four unknowns (h~ui, hPif , T, and _q). Also, _q is calculated using Eqs.

(4) and (7)–(10). Together with the initial and boundary conditions, Eqs. (11) and (12), this sys-

tem of equations must be solved numerically and the unknowns calculated at each time step.

III. NUMERICAL METHOD

The finite volume method was employed to solve the governing equations (1)–(3).29 Based

on the Marker and Cell (MAC) scheme developed by Harlow and Welch,30 Eqs. (1) and (2)

were discretized over a staggered structured Cartesian mesh while fully explicit discretization

was used for discretization of Eq. (3). Patankar’s Harmonic Mean Method29 was employed to

calculate the thermal conductivity values for the links between each grid cell and its neighbor-

ing grid cells. In each time step of the numerical procedure, DTf of each tube cell, calculated

through Eq. (7), was substituted into Eq. (4) to calculate _q associated with each tube cell. The

temperature of water at the inlet of each U-tube heat exchanger was assumed to be Tin.

However, if two (or more) U-tubes were connected in series, the temperature of water at the

043104-7 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

outlet of the preceding U-tube was set as the inlet temperature of the sequentially following

one. The discretized set of equations was then solved using a FORTRAN computer code named

Convective Geothermal Solver (CGS), self-devised to carry out the numerical calculations.

Fig. 4 shows the block diagram of this computer program. As indicated in Fig. 4, in order to

simulate the seasonal variations of heat load demand, an Inlet Temperature Adjuster Subroutine

was devised in the computational code which takes advantage of the Newton-Raphson method

to adjust the inlet temperature of the U-tube heat exchanger(s) in a way that the extracted heat

power would match the heat load demand power (set by the Demanded Power Calculator

Subroutine). In other words, in each time step of the numerical procedure, the resulting temper-

ature difference between the outlet and inlet water ðTout � TinÞ was used to calculate the value

of generated heat power using Pn ¼ _mt Cp ðTout � TinÞ, where Tout, Tin, and _mt are, respectively,

temperatures of water at the outlet and inlet of the ground-coupled heat exchanger, and the total

water mass flow through the heat exchanger. The dynamic adjustment of the inlet temperature

of borehole U-tube(s), Tin, provides the opportunity to match the rate of heat extraction with

the seasonal heat load variation. This method also provides a more realistic approach for the

simulation of heat load in borehole heat exchangers when compared to the existing heat transfer

models, which assume a uniform heat flux distribution over the borehole length.

The step by step procedure of the numerical simulation, shown in Fig. 4, is presented in

the following:

Step (1) At start time (t¼ 0), the initial values are used to define the velocity and tempera-

ture fields, and an initial guess is made for the borehole inlet temperature.

Step (2) The numerical solution for Eqs. (1)–(3) (discretized in accordance to Marker and

Cell method30) is calculated, and the values of velocity and temperature at each node as well as

the inlet and outlet temperature of each tube cell are obtained.

Step (3) Based on the values of the inlet and outlet temperature of each U-tube calculated

in Step 2, the value of Extracted Heat Power is calculated using Pn ¼ _m Cp ðTout � TinÞ.Step (4) Using the Newton-Raphson method, the Inlet Temperature Adjuster Subroutine

adjusts the value of borehole inlet temperature, so that the Extracted Heat Power would match

the value of Heat Load Demand (set by Demanded Power Calculator Subroutine).

Step (5) The temperature and velocity fields are updated, the time step is moved forward,

and the loop restarts from Step 2 until the final time step is reached and the numerical calcula-

tions are stopped.

FIG. 4. Flow chart of the flow control scheme applied in the numerical simulation model.

043104-8 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

IV. RESULTS AND DISCUSSION

To make sure of the independency of the results from the size of the computational region

and also to evaluate the mesh-independency of the results, a series of tests are carried out. The

computational region is chosen such that extending its size does not have any significant effect

on the results of temperature and flow fields (i.e., relative difference less than 10�5). Thus, the

opted size of the computational region is the size for which the infinite boundary requirement is

satisfied. Similarly, the number of grid points of the structured mesh is chosen such that

increasing the number of nodes does not change the resulting temperature field (i.e., the mesh

size is reduced until the relative difference between the results of two consecutive grid size val-

ues becomes less than 10�5).

A. Model validation

To validate the results of the proposed model, a comparison is made between its numerical

results and the results of Lee and Lam.12 Fig. 5 illustrates the bore temperature rise for a 110 m

long U-tube heat exchanger with a bore diameter of 0.11 m and a continuous heat injection of

3300 W into the ground. Since the focus of the present paper is on heat extraction, the study of

heat injection is done for comparison and validation purposes only. The thermal conductivity

and thermal diffusivity of the impervious ground material are assumed to be, respectively,

3.5 W/m �C and 1.62� 10�6 m2/s. According to Fig. 5, the numerical results of the proposed

method agree with the results of Lee and Lam.12 However, it is important to note that Lee and

Lam’s model is not capable of capturing the effect of buoyancy-driven natural convection. To

show the importance of natural convection, Fig. 5 shows the results associated with a ground

hydraulic conductivity of kh¼Kl=ðqgÞ¼ 10�5 m/s. According to Fig. 5, the bore temperature

rise would be up to 8.4% lower compared to the impermeable case of kh¼ 0 m/s. This differ-

ence between the case of kh¼ 10�5 m/s and the case of kh¼ 0 m/s, shows the possibility of

improving the borehole performance by taking advantage of the natural convection

phenomenon.

Another interesting result of the above-mentioned test case is the demonstration of non-

uniformity of heat load distribution along the U-tube length. Fig. 6 shows the rate of heat flux

into the bore along the bore length over 10 years of borehole heat exchanger operation.

According to Fig. 6, the rate of heat exchanged along the bore length is not constant,

FIG. 5. Comparison of the results of the proposed numerical model with the results of Lee and Lam.12

043104-9 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

suggesting the validity of the method proposed in this paper (which calculates the heat

exchange based on the bulk temperature of the water). In other words, the approach of assum-

ing a uniform heat flux along the tube length is not realistic, and therefore not suitable for cap-

turing temperature gradients exhibited along the length of U-tube heat exchanger. It is impor-

tant to note that Fig. 6 indicates that the difference between the rate of heat flux into the bore

in the 4th year and the 10th year of operation is very small; meaning that after 10 years of

operation the local heat flux into the borehole will not change.

B. Single borehole

To investigate the impact of natural convection in borehole heat exchangers, the perform-

ance of heat extraction from a typical single borehole heat exchanger is studied. The geometric

and thermo-physical properties of the test case are given in Tables I and II, respectively. A

wide range of ground hydraulic conductivity values have been chosen to represent various pos-

sible ground permeability cases. Also, in order to simulate seasonal variations in the demanded

heat load, the system is assumed to operate under a time-dependent heat extraction schedule,

shown in Fig. 7, which resembles the heat load demand in northern Canadian territories. The

seasonal variation of the rate of heat extraction provides the opportunity of covering the peak

heat demand seasons. In other words, more heat can be produced in winter, while, by producing

less heat during summer, the resource can recover part of its heat content. The resulting outlet

temperatures of the single borehole for various hydraulic conductivity values are shown in

Fig. 8. According to Fig. 8, the results associated with kh � 10�5 m/s are the same. In other

words, the effect of natural convection for kh � 10�5 m/s is negligible. However, as hydraulic

FIG. 6. Heat exchange in borehole along the bore depth at different times.

TABLE I. Thermal properties of tailings, tube, and water.

Ground Bore fill material Tube Water

Density (kg/m3) 2160 2300 … 998

Heat capacity (J/kg �C) 1000 1100 … 4180

Thermal conductivity (W/m �C) 1.5 1.3 0.4 0.58

Hydraulic conductivity (m/s) 10�5 to10�2 10�10 … …

043104-10 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

conductivity increases, natural convection leads to a higher outlet temperature; for example, af-

ter10 years of operation, the outlet temperature for kh¼ 10�3 m/s is 13% higher than the outlet

temperature for kh¼ 10�5 m/s, while the resulting outlet temperature for kh¼ 10�4 m/s is only

1% higher than the outlet temperature associated with kh¼ 10�5 m/s.

Fig. 9 shows the resulting underground velocity field for the case of a single borehole with

hydraulic conductivity of kh¼ 10�4 m/s after 10 years of operation. As can be seen in Fig. 9,

the underground water movement is observed only in a small zone around the single borehole.

C. Multiple boreholes

Natural convection in a borehole comprised of 9 U-tube boreholes distanced 6 m apart (as

shown in Fig. 1) with properties given in Tables I and II is studied here, assuming the rate of

heat extraction shown in Fig. 7. Considering various ground hydraulic conductivity values

(10�8 to 10�3m/s), the resulting outlet bore temperature trends are shown in Fig. 10. According

to Fig. 10, the results of kh¼ 10�8 m/s and kh¼ 10�5 m/s are identical. However, a higher outlet

temperature is observed for kh¼ 10�4m/s and kh¼ 10�3 m/s; after 10 years of operation, the

outlet temperature for kh¼ 10�3 m/s is found to be 23.8 percent higher when compared to the

outlet temperature associated with kh¼ 10�3 m/s. Fig. 11 shows the resulting underground water

TABLE II. Properties of the borehole heat exchanger.

Tube length 100 m

Bore diameter 0.125 m

Outer diameter of the tube, Dtube 0.016 m

Tube thickness 1.5 mm

Number of U-tubes 1

Center-to-center distance of the tubes, d1 0.1 m

Porosity of ground, u 0.4

Dimensionless form-drag of ground, DF 0.55

FIG. 7. Seasonally variable rate of heat load.

043104-11 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

velocity field for kh¼ 10�4 m/s. According to Fig. 11, the maximum value of underground

water flow is 13.6% lower for a multiple borehole heat exchanger when compared to a single

borehole heat exchanger. However, the presence of multiple boreholes has extended the size of

the zone affected by the buoyancy-driven water movement by almost one order of magnitude;

from barely 1 m around the single U-tube to almost 12 m for a multiple borehole heat

exchanger. In other words, the effect of natural convection in a multiple borehole heat exchang-

ers is more significant when compared to a similar single borehole heat exchanger. Also, Fig.

10 indicates that a GCHE exhibiting natural convection is capable of providing heat at a higher

outlet temperature as compared to an identical, impervious GCHE. This proves the possibility

of enhancing the rate of heat extraction in long-term operation of a GCHE by taking advantage

of natural convection.

To compare the effect of natural convection on the performance of GCHEs, a scale analysis is

developed. In this analysis, a network of heat exchange tubes embedded in a porous medium, as

shown in Fig. 3, is considered. In Eq. (3), natural convection component of heat transfer is repre-

sented by qf Cf ðh~ui:~rTÞ¼qf Cf hui@T@xþhvi@T

@yþhwi@T@z

� while ~r:ðkm

~rTÞ¼km@2T@x2þ@2T

@y2þ@2T@z2

� represents conduction.

Since vertical gravity is responsible for buoyancy-driven fluid movement, it is reasonable

to assume that w� u v. Therefore, it can be assumed that natural convection is represented

by qf Cf hwi @T@z . Also, it is reasonable to assume that conduction can be represented by its hori-

zontal components (i.e., km@2T@x2 km

@2T@y2 � km

@2T@z2 ). Defining c as the ratio of natural convection

over conduction, it is found that

c ¼ qf Cf w@T

@z

km@2T

@x2

� �

qf Cf

kf

kf

kmw

@T

@z

� �@2T

@x2

� �¼ 1

af

kf

kmw

@T

@z

� �@2T

@x2

� �: (13)

Using the notion of hydraulic conductivity and considering Darcy flow inside the porous me-

dium, it can be assumed that w ¼ khDH=Dz, where DH and kh are the difference in hydraulic

head and the hydraulic conductivity of the medium, respectively. In order to find the scale of

the hydraulic head, Eq. (2) should be analyzed. The last terms on the right hand side of Eq. (2)

are responsible for creating hydraulic head difference. Therefore, w can be estimated as

FIG. 8. Effect of hydraulic conductivity of ground on the outlet temperature of a single borehole heat exchanger.

043104-12 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

FIG. 9. Underground water velocity contour in the mid plane (y¼ 0.5W2) of a single borehole with kh¼ 10�4m/s after 10

years of operation.

FIG. 10. Effect of hydraulic conductivity of ground on the outlet temperature of a ground heat exchanger comprised of

multiple boreholes.

043104-13 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

w ¼ khDH=Dz kh

qf bðT0 � TinÞ gH1

qf g

!H1 khbðT0 � TinÞ: (14)

Equation (14) shows that if the hydrologic-pressure gradient is capable of stimulating under-

ground water movement, the effect of this factor will be more significant than buoyancy-driven

ground water movement. Substituting Eq. (14) into Eq. (13), c will be as follows:

c 1

af

kf

km

� �kh bðT0 � TinÞ

T0 � Tf

H1

� �T0 � Tf

ðL1Þ2

! kh bðT0 � TinÞH1

af

kf

km

� �L1

H1

� �2

; (15)

where Tf is the average of fluid bulk temperature at the inlet and the outlet of the GCHE. For

the multiple borehole heat exchanger assumed in section (C), it is found that c 105kh. This

means that if kh � 10�5 m/s, conduction is more important than natural convection. However,

if kh � 10�5 m/s, natural convection will considerably influence the performance of the GCHE.

The analogy presented here is in agreement with the numerical results of this paper.

V. SUMMARY AND CONCLUSION

A heat transfer model for the simulation of natural convection in borehole heat exchangers

of closed-loop geothermal cycles was developed. Using a finite volume discretization method,

FIG. 11. Underground water velocity contour in the mid-plane (y¼ 0.5W2) of a multiple borehole heat exchanger with

kh ¼ 10�4 m/s after 10 years of operation.

043104-14 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)

the model was numerically solved to simulate the performance of typical single and multiple

borehole heat exchanger units. The results of the proposed model showed good agreement with

the results of the conduction-based model when the hydraulic conductivity of the heat exchange

medium was lower than 10�5 m/s. However, it was found that as the ground hydraulic conduc-

tivity increases, kh 10�4m/s for the typical cases studied, natural convection cannot be

neglected. The findings of this study suggest that for a typical GCHE, if the hydraulic conduc-

tivity of the heat exchange medium is smaller than 10�5m/s, natural convection can be

neglected and conduction-based heat transfer models would suffice. However, as the hydraulic

conductivity increases from 10�5 to 10�3 m/s, the role of natural convection grows from me-

dium to considerably effective. The study shows that as the number of boreholes of a heat

exchanger increases, the extent of the zone exhibiting a naturally driven flow of underground

water becomes greater. Thus, compared to a single borehole heat exchanger, the significance of

buoyancy-driven natural convection is more considerable in a similar multiple borehole heat

exchanger. Also, it was found that natural convection becomes more important in the long-term

performance of a ground-coupled heat exchanger.

1S. P. Kavanaugh and K. Rafferty, Ground-source Heat Pumps, Design of Geothermal Systems for Commercial andInstitutional Buildings (American Society of Heating, Refrigerating and Air-conditioning Engineers, Inc., Atlanta, 1997).

2L. Xinguo, Energy Convers. Manage. 39, 295 (1998).3N. R. Diao, H. Y. Zeng, and Z. H. Fang, HVAC&R Res. 10, 459 (2004).4H. Zeng, N. Diao, and Z. Fang, Int. J. Heat Mass Transfer 46, 4467 (2003).5A. M. Omer, Renewable Sustainable Energy Rev. 12, 344 (2008).6A. J. Philippacopoulos and M. L. Berndt, Geothermics 30, 527 (2001).7R. Al-Khoury and P. G. Bonnier, Int. J. Numer. Methods Eng. 67, 725 (2006).8Z. Li and M. Zheng, Appl. Thermal Eng. 29, 920 (2009).9A. D. Chiasson, S. J. Rees, and J. D. Spitler, ASHRAE Trans. 106, 380 (2000).

10N. Diao, Q. Li, and Z. Fang, Int. J. Thermal Sci. 43, 1203 (2004).11R. Fan, Y. Jiang, Y. Yao, S. M. Deng, and Z. L. Ma, Energy 32, 2199 (2007).12C. K. Lee and H. N. Lam, Renewable Energy 33, 1286 (2008).13S. A. Ghoreishi-Madiseh, F. P. Hassani, A. Mohammadian, and P. H. Radziszewski, “A study into extraction of geother-

mal energy from tailings ponds,” Int. J. Min. Reclam. Environ. 2012, 1.14S. Whitaker, Adv. Heat Transfer 13, 119 (1977).15K. Vafai and L. Tien, Int. J. Heat Mass Transfer 24, 195 (1981).16C. T. Hsu and P. Cheng, Int. J. Heat Mass Transfer 33, 1587 (1990).17D. A. Nield, Int. J. Heat Fluid Flow 12, 269 (1991).18K. Vafai and S. J. Kim, Int. J. Heat Fluid Flow 16, 11 (1995).19B. Alazmi and K. Vafai, J. Heat Transfer 122, 303 (2000).20D. A. Nield and A. Bejan, Convection in Porous Media (Springer-Verlag Inc, New York, 1992).21M. Kaviany, Principles of Heat Transfer in Porous Media (Springer, Berlin, 1995).22C. Beckermann, R. Viskanta, and S. Ramadhyani, J. Fluid Mech. 186, 257 (1988).23J. L. Lage, Int. Commun. Heat Mass Transfer 20, 501 (1993).24M. A. Hossain, K. Vafai, and K. M. N. Khanafer, Int. J. Therm. Sci. 38, 854 (1999).25M. A. Saada, S. Chikh, and A. Campo, Int. J. Heat Fluid Flow 28, 483 (2007).26G. S. Beavers, E. M. Sparrow, and D. E. Rondez, J. Appl. Mech. 40, 655 (1973).27J. P. Holman, Heat transfer (McGraw-Hill Science/Engineering/Mat, New York, 1997).28F. W. Dittus and L. M. K. Boelter, Univ. Calif. Publ. Eng. 2, 443 (1930).29S. V. Patankar, Numerical Heat Transfer and Fluid Flow (McGraw-Hill Book Co., New York, 1980).30F. H. Harlow and J. E. Welch, Phys. Fluids 8, 2182 (1965).

043104-15 Ghoreishi-Madiseh et al. J. Renewable Sustainable Energy 5, 043104 (2013)