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TRANSCRIPT
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: [email protected]
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.
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