Accepted Manuscript

Heat transfer in horizontal ground heat exchangers

Krzysztof Kupiec, Barbara Larwa, Monika Gwadera

PII: S1359-4311(14)00874-6

DOI: 10.1016/j.applthermaleng.2014.10.003

Reference: ATE 6017

To appear in: Applied Thermal Engineering

Received Date: 1 July 2014

Revised Date: 27 August 2014

Accepted Date: 3 October 2014

Please cite this article as: K. Kupiec, B. Larwa, M. Gwadera, Heat transfer in horizontal ground heatexchangers, Applied Thermal Engineering (2014), doi: 10.1016/j.applthermaleng.2014.10.003.

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HEAT TRANSFER IN HORIZONTAL GROUND HEAT EXCHANGERS

Krzysztof Kupiec*, Barbara Larwa, Monika Gwadera Cracow University of Technology, Faculty of Chemical Engineering and Technology

ul.Warszawska 24, Cracow, Poland Tel/Fax: +48 12 6282731, e-mail: kkupiec@chemia.pk.edu.pl (K.Kupiec, corresponding

author)

Abstract: The mathematical model of a horizontal ground heat exchanger was presented. The model was based on a one-dimensional equation of the transient heat conduction with an internal source of heat. The model was correctly verified by comparison of computational results and experimental measurements presented in literature. Temporal changes of the ground surface temperature and rates of heat transferred from the ground to the working fluid and between the ground surface and the environment were determined. Moreover, temporal variations of the average temperature of the subsurface layer of the ground were determined. It was found that after about 10 years of operation, the ground temperature reaches a cyclic steady state. Thermal calculations concerning the ground under natural conditions were also presented.

1. Introduction The heat contained in the ground can be utilized by using a working fluid circulation

between the lower heat exchanger arranged in the ground and the upper exchanger, which is a part of a heat pump. During the heating season in the lower heat exchanger working fluid takes heat from the ground and transfers it to the boiling thermodynamic medium in the heat pump. The exchanger is turned off when the fluid temperature decreases excessively, and there is a risk that further extraction of heat will cause frost penetration near the exchanger pipes.

In parallel with transfer of heat to/from the working fluid, the ground receives/gives heat from/to the environment. The direction of heat transfer between the ground and the environment depends on the relationship between ambient temperature and the temperature of the surface of the ground. In order to characterize these relationships, semi-annual periods: warmer and colder were considered. In natural conditions (no heat exchanger installed in the ground), the amount of heat received by the ground from the environment during the warmer half of the year is equal to the amount of heat lost during the colder half of the year. This causes that the average temperature of the ground in a concerned area does not change over the years. When a ground heat exchanger is installed, the temperature averaged over the volume of the sub-surface layer of the ground changes in subsequent years because the amount of heat received by the heat pump from the ground in the heating season is not generally compensated by the amount of heat supplied to the ground during the next half of the year.

Vertical and horizontal ground heat exchangers are used. Vertical exchangers require to make deep holes in the ground while horizontal exchangers are installed in shallow subsurface layers but they need large surface of the ground. Modeling and design of both types of ground exchangers are based on different calculation relationships. Numerical simulation of a vertical ground heat exchanger is presented e.g. in the paper [13]. Benli [2] presented a comparison of both types of ground exchangers.

Horizontal ground heat exchangers have been widely used in many countries as a heat source for ground-source heat pump systems. Therefore, ground heat exchangers are the subject of many studies that are both experimental as well as numerical. An overview of the

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ACCEPTED MANUSCRIPTapplications of ground heat exchangers and numerical models related to them was presented by Florides and Kalogirou [7]. Tarnawski and Leong [12] described a two-dimensional mathematical model of a horizontal ground heat exchanger based on the heat transfer equation. Their model takes into consideration phenomena referring to the ground humidity: freezing/thawing and drying/rewetting. Piechowski [11] included mass transfer in his model to take into account the effects of soil moisture.

In the works of Wu et. al. [14, 15] the thermal performance of slinky heat exchangers for Ground Source Heat Pump systems for a UK climate was investigated. The authors presented results of experimental measurements as well as of numerical simulation using a 3D model. The results of the further experiments for the same ground heat exchanger were presented by Gonzalez et. al. [9].

The results of experimental work were presented also by Neuberger et. al. [10]; the measurements were related to the temperature of ground at two different dephts: 1.5 m (where the exchanger had been installed) and 1.3 m.

Computational fluid dynamics CFD is frequently used for digital simulations. For cooling purposes in a continuous and cyclic operating mode three-dimensional numerical simulation was carried out by Benazza et al [1]. The influence of thermal conductivities and geometrical parameters on the heat exchanger efficiency has been studied. Condego et. al. [5] performed calculations using Fluent and the simulations covered one year of system operation both in summer and winter for typical climate conditions of the South of Italy. In particular, horizontal type heat exchangers have been investigated for different configurations, in order to evaluate the characteristics of these systems in the most common layouts and under different working conditions. A three-dimensional numerical model was developed by Chong et. al. [4] and the results of the thermal performance for various heat exchanger configurations were presented. Comparison was made for the heat transfer rate, the amount of pipe material needed, as well as excavation work required for the horizontal slinky-loop heat exchanger.

Demir et. al. [6] developed software in MATLAB environment and the effects of different parameters on the results were investigated. An experimental study (temperature data were collected using thermocouples buried in soil at various distances from the pipe center and at the inlet and the outlet of the horizontal ground heat exchanger) was conducted to test the validity of the model. Experimental and numerical simulation results for experimental water inlet temperatures were compared.

This study presents a mathematical model of a horizontal ground heat exchanger and the results of simulations concerning heat transfer in the ground with the exchanger. Particular attention is focused on the ground temperature changes occurring in the long term in a few (or more) years.

Numerical calculations were preceded by a theoretical analysis concerning ground without an exchanger. Under these conditions of heat transfer, the analytical solutions exist.

Temperature conditions of the northern hemisphere were considered in this work.

2. Calculations for the ground under natural conditions The ground was treated as a semi-infinite body. It was assumed that conduction is the only

mechanism of heat transfer in the ground. The heat conduction equation has the form:

ρc

q

x

Ta

t

T v+∂∂=

∂∂

2

2

(1)

Under natural conditions (no ground heat exchanger installed): qv = 0. The initial condition has the general form:

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( )xfTt == :0 (2)

It was assumed that heat between the ground and the environment is transferred by convection; the boundary condition for the surface of the ground is:

( )00:0 TThdx

Tkx a −=∂−= (3)

Ambient temperature Ta changes periodically according to the relationship:

( )[ ]maxcos ttBTT aa −⋅+= ω (4)

where frequency ω is equal to:

ct

πω 2= (5)

where the cycle time tc is equal to 365 days. Hence the value of ω = 0.199·10-6 s-1 is obtained. The second boundary condition relates to the ground at great depth where the temperature

is constant and it has the form:

bTTx =∞→ : (6)

wherein the values aT and Tb are equal.

The solution of equation (1) with boundary conditions (3-4) and (6) leads to the following relationship for cyclic steady state (Carslaw, Jaeger [3]):

( )

+−−−⋅

−⋅

+

+

+= −

1

1tancosexp

110

1max2

0

0

k

LhL

xtt

L

x

k

Lh

k

Lh

BTT b ω (7)

where the quantity L (with length dimension) is defined as follows:

ωa

L2= (8)

Equation (7) contains two dimensionless quantities. The dimensionless constant:

k

LhBi 0= (9)

is the Biot number characterizing the ratio between internal and external resistance to heat transfer. The variable X defined as follows:

L

xX = (10)

is a dimensionless position coordinate. When the dimensionless quantities are used, boundary condition (3) takes the form:

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( )0:0 TTBiX

TX a −−=

∂∂= (3a)

while formula (7) can be written in the form:

( ) ( )[ ]2max1 cosexp CXttXBCTT b −−−⋅−⋅+= ω (7a)

where C1 and C2 are dependent on the Biot number (Fig.1):

( ) 11 21

++=

Bi

BiC (11)

+= −

1

1tan 1

2 BiC (12)

If the external resistance between the ground and the environment can be neglected (Bi→∞), instead of condition (3a) one gets:

aTTX == 0:0 (13)

Since in this case it is C1 = 1 and C2 = 0 (Fig.1), thus formula (7a) simplifies to the form (Carslaw, Jaeger [3]):

( ) ( )[ ]XttXBTT b −−⋅−⋅+= maxcosexp ω (14)

In general, the temperature gradient at the ground surface follows from differentiation of formula (7a):

( )

+−−⋅−=

∂∂=

∂∂

== 4cos2 2max1

00

πω CttBCx

TL

X

T

xX

(15)

The heat flux on the surface of the ground, in accordance with the Fourier equation, is:

0=

∂∂−=

xx

TkF (16)

In Fig.2 reaching a cyclic steady state of the ground was shown. The ground temperature profiles for the day corresponding to tmax at yearly intervals were presented. For numerical calculations it was assumed that initial condition (2) has the form f(x) = Tb. The calculations were performed for Bi→∞ and thermal diffusivity of the ground a = 0.5·10-6 m2/s. The curve concerning the cyclic steady state was determined analytically by formula (14); the other curves – numerically. As one can see the system has not reached the cyclic steady state after 5 years, so it might be concluded that the initial temperature profile was far from the target profile.

Effect of external resistance to heat transfer on the shape of temperature profiles in the ground was shown in Fig. 3. It can be seen from the figure that the external heat transfer resistance has a significant impact on the temperature profile in the subsurface layer of the ground. The curves relate to the day corresponding to tmax and the cyclic steady state. To generate the data formulas (7a) and (14) were used. For Bi = 0 the external thermal resistance

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Temporal changes in the heat flux between the ground and the environment were determined from relationships (15) and (16) and were shown in Fig.4. The influence of the Biot number, inversely proportional to the external thermal resistance was analyzed. The greater the external thermal resistance, the smaller the heat flux between the ground and the environment for certain values of B, k and L is transferred. The value of the external thermal resistance affects not only the heat flux, but also the date of occurrence of the maximum flux. For B = 11 K, k = 1.5 W/(mK), L = 2.24 m and tmax = 182 (the warmest day in the year − 1st of July) exemplary values were read from the graph. If the external thermal resistance can be neglected, the maximum heat flux Fmax = 9.5 W/m2 (FmaxL/(kB) = 1.41) is reached on the 14th of May. This date is 46 days before warmest day of the year. While for Bi = 5 the maximum value of heat flux is Fmax = 7.8 W/m2 (FmaxL/(kB) = 1.16) and it is reached 10 days later.

3. Mathematical model of a horizontal ground heat exchanger When a heat exchanger is installed in the ground, then qv ≠ 0 and heat conduction equation

(1) with conditions (3-4) and (6) should be solved numerically. A simplified, one-dimensional model of a horizontal ground heat exchanger is presented below.

3.1. System description In the considered system a lower heat exchanger consists of pipes located under the surface

of the ground. In general the upper exchanger is an evaporator or condenser (depending on the season) which is a part of a heat pump or air conditioner. When the ambient air temperature is lower than the ground temperature (colder season) the heat is transferred (in general) from the lower exchanger to the upper one. Otherwise the direction of heat transfer is reversed (if the system is used for air-conditioning). In Fig.4 the location and orientation of a horizontal ground heat exchanger is presented. It was assumed that heat is transported in the ground in the direction of the x-axis by conduction.

In Fig.5 the ground exchanger was shown as a fictitious cuboid wherein heat is generated. Inside a cuboid of dimensions Lg×b×δ temperature is equalized along the axis x and z at any time. Temperature variation along the y-axis was taken into account by dividing the heat exchanger into stages. A working fluid flows in series through the neighboring stages, which are treated in the model as perfect-mixing tanks. One of these tanks is shown demonstratively in Fig.5.

In the case of ground heat exchangers thermal resistance occurs in working fluid and walls of pipes but mainly in the ground. Individual resistances are related to areas through which heat is transported. The presented model includes only the resistance to heat transfer in the ground. The consequence of this is that the surface of the pipes Ap is not used in the model. For the modeled exchanger shown in Fig.5 it can be assumed that the heat transfer surface is the surface of the ground Ag wherein exchanger pipes are installed. Therefore:

gg bLA = (15)

However, this relationship is valid only if pipes are densely packed (i.e. for high value of ratio Ap/Ag). For large spaces between the exchanger pipes temperature of the ground between pipes is not equalized in a z-axis direction. This case refers to Ag < bLg. The quantity Ag should be treated as a adjustable parameter of the model that depends on the way of pipes arrangement and depth of their location under the surface of the ground. Quantitative assessment of Ag can be performed for example by using a CFD simulation for various geometric configurations of heat exchanger pipes and their installation at various depths.

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treated in the model as a m-stage cascade of perfect-mixing tanks, is coupled with the upper exchanger, wherein heat is transported to (or received from) the thermodynamic medium undergoing a phase transition (condensation or evaporation). The more stages in the cascade, the model more precisely reflects the change in temperature of the working fluid flowing through the heat exchanger pipes. When m→∞, temperature profile of the fluid in exchanger becomes continuous.

3.2. Computational relationships

In the presented model the heat conduction equation for an infinite slab with an internal heat source was used (1). The upper side of the slab is the ground surface, while the lower one is located at great depth, which ensures constancy of temperature.

A fluid from the upper exchanger flows to the tank 1 at a temperature of Tin = TL0. Fluid leaving the tank 1 flows into the tank 2, then to the tank 3, etc. Fluid from the last mth tank is transported at a temperature of Tout = TLm to the upper exchanger (Fig.6). Fluid temperature TLj, ground temperature Tj and the rate of heat transfer jQ& are different in the various stages

of the cascade j = 1, 2,…, m. Heat transfer area for each stages is equal to Agj = Ag/m, while volume: Vj = V/m. The rate of heat transfer between the ground and fluid in the jth stage is equal to:

( )1, −−= jLLjLLj TTcmQ && (16)

and the total rate of heat transfer in the exchanger equals:

∑=

=m

jjQQ

1

&& (17)

The quantity qv in equation (1) is the thermal power generated per unit volume. This value is related to the transport of heat between the working fluid flowing through the ground heat exchanger and the ground. Since the heat source is placed at a distance h from the ground surface (Fig.4), thus:

≠

=−=

hx

hxV

Q

q j

j

v

for0

for&

(18) Because the thermal resistances to heat transfer in a fluid and in pipe’s wall were

neglected, hence the temperature of the ground in contact with the outer surface of the heat exchanger pipes is equal to the temperature of the working fluid.

3.3. Initial and boundary conditions As an initial condition the temperature profile for cyclic steady state under natural conditions was assumed:

( )

−−⋅

−⋅+==L

xtt

L

xBTTt b maxcosexp:0 ω (19)

The boundary condition for the surface of the ground takes the form (3). In numerical calculation it must be assumed that the temperature will remain constant for some distance from the ground surface hinf. (theoretically for the semi-infinite body x → ∞, Eq.(6)). Therefore, the second boundary condition has the form:

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bTThx == :inf (20)

It was assumed that the environment temperature periodically varies in time according to the relation:

( )[ ] ( )

−+−⋅+== ddyba ttBttBTTx max,max, 24

2coscos:0

πω (21)

where Bd is half of the maximum daily temperature range. In the above formula the second term applies to temperature variability in the annual cycle (y) and the third term – in the diurnal cycle (d).

Model equations were solved by the finite difference method using the Crank-Nicolson scheme.

4. Experimental verification of the model

In order to verify the presented mathematical model of the horizontal ground heat exchanger, temperature profiles generated computationally were compared with the measurements results presented by Wu et. al. [14, 15]. The cited authors did the research on ground heat exchanger used as a lower heat source of a heat pump used for space heating. Horizontal coils of exchanger pipes were arranged in four rows with a width of 1 m and a length of 80 m each, at a depth of 1.14 m. The flow rate of an aqueous solution of glycol was 0.57 kg/s. The exchanger was working continuously at almost constant increment of temperature of the fluid c.a. 1.9 K. Thermal diffusivity of the ground (averaged over the depth) was 0.533·10-6 m2/s. Ground temperature was measured every 4 hours at seven depths in the range of 0 to 1.14 m. The measurements were conducted in close proximity to the heat exchanger pipes and, separately, away from any of the heat exchanger pipes (reference hole).

Characteristic values for the climate in which the research was conducted (around London) were assumed as follows: Tb = 9.5°C, B = 9.0 K, Bd = 5.0 K. In the calculations according to the model the following values of parameters were used: hinf = 20 m, m = 4, n = 300 (number of calculation nodes), time step ∆t = 1 h. The value of the averaged external thermal resistance should be also assumed, which depends on, i.a., heat transfer coefficient (including radiation) between the air and the ground. It was assumed that h0 = 20 W/(m2K).

The measurement results together with the results of calculations according to the presented model are shown in Figs. 7 and 8. The symbols represent the experimental values read out from the drawings shown in [15]. These values were obtained on the 7th of November. Fig.7 presents temperature profiles for the ground without a heat exchanger, while Fig.8 refers to temperature profiles around the working heat exchanger in which the working fluid receives the heat from the ground. As one can see, the predictions based on the model are correct, although there is a small difference in temperature determined experimentally and computationally. It should be noted that the compatibility of temperature profiles in Fig.7 depends largely on whether the actual air temperature in the period in which the research was conducted (and the preceding period) was consistent with the values which were used in the calculations. Averaging the physical properties of the ground (which determine the thermal diffusivity) and roughly assumed value of the heat transfer coefficient h0 negatively affect the accuracy of the results obtained from the digital simulation. To generate data for Fig.8 Ag = 160 m2 was assumed. This is an area twice smaller than the value derived directly from the calculation of the area of trenches dug out for the installation of the heat exchanger pipes: 4·1·80 = 320 m2.

5. The results of simulations

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measurement results described in [14, 15] provides a basis for utilization of the model for the simulation. The calculations were carried out to investigate the relationship between the environmental temperature and the temperature of the ground surface where a ground heat exchanger is installed. In addition, the temporal variations of rate of heat transfer received from the ground by the working fluid was determined; the variation of this quantity is caused by switching off the exchanger due to temperature limitation of the working fluid. The temporal variations of an average ground temperature during many years of ground heat exchanger work were also determined.

In the calculations it was assumed that the ground heat exchanger is turned on when the following conditions are met:

CTCT La °>°< 014 0 (22) It was assumed that the exchanger is used for heating purposes only. When the average daily temperature exceeds 14°C the exchanger is turned off. Furthermore the exchanger does not work when the temperature of the working fluid at the inlet of the exchanger TL0 drops below 0°C. Thus, during the heating season the exchanger works periodically. The numerical values of the process parameters used in the calculations are shown in Table 1. In the presented results diurnal temperature changes were not taken into account.

In Fig.9 the temporal dependence of the ambient temperature Ta, the ground surface temperature T0 and the difference between these temperatures was presented. Noteworthy is fourfold change in the sign of Ta−T0 throughout the year. Under natural conditions, the sign of this difference changes twice a year. In contrast, while the ground heat exchanger is turned on the relationship between Ta and T0 is much more complicated. Transfer of heat from the ground causes a reduction of the ground surface temperature, which leads to changes in the rate of heat transferred between the air and the ground (in relation to natural conditions). The chilled ground takes over more heat from the environment than the ground under natural conditions.

In Fig.10 the temporal variability of heat transfer rates associated with receiving and releasing heat by/from the ground is presented. The 1Q& concerns the heat received from the ground by the working fluid and it was determined by the formula:

QQ && χ=1 (23) where χ is the fraction of time in which the ground heat exchanger is turned on in a certain period (χ = 0 for turned off exchanger in this period, χ = 1 for still working exchanger). The

2Q& refers to the rate of heat transfer between air and ground. It was determined from the relationship:

( ) ga ATThQ 002 −=& (24)

The difference between the fluxes 1Q& and 2Q& results in the average ground temperature (heat accumulation). The similarity of temporal variations of temperature difference in Fig.9 and rate of heat transfer 2Q& in Fig.10 follows from the relation between these quantities (formula (24)).

The results presented in Fig.9 and 10 refer to the first season of a ground heat exchanger operation. In subsequent years the graphs of temperature and heat transfer rate change.

A multiannual course of the average ground temperature is shown in Fig.11. The average temperature of the ground has an apparent sense because its value depends on the maximum depth of the ground (hinf), which is taken into account when calculating this average. The greater the depth, the smaller the temporal changes in the average temperature of the ground. On the other hand, the maximum considered depth should be greater than the depth at which

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∫=inf

0inf

1h

dxTh

T (25)

As one can see from the Fig.11, the average ground temperature reaches a cyclic steady state after a certain time (about 10 years). After several years of decreasing, the ground temperature stabilizes in the annual cycle. Under cyclic steady state conditions the amount of heat which has been taken by a heat pump is completely compensated by the net heat delivered from the air (it refers to the annual cycle).

Gan [8] drew attention to the fact that horizontal ground heat exchangers have a significant advantage over vertical ones. In the former ones, shallow layers of the ground are being cooled as a result of extraction of heat from the ground; these layers are in a direct contact with the environment. This results in the fact that the cold ground takes more heat from the environment during the warm period of the year and loses less heat during the cold period. Vertical exchangers do not have this beneficial feature as the cold ground at great depth is not able to compensate the heat loss by receiving heat from the environment.

6. Conslusions • Heat transfer in a horizontal ground heat exchanger can be described by a model based on

the one-dimensional transient heat conduction equation with the internal source of heat. • The variability of the temperature of a fluid flowing through the heat exchanger can be

taken into account by treating the ground heat exchanger as a cascade of perfect-mixing tanks.

• The temperature profiles in the ground determined with the use of the presented mathematical model are consistent with the results of the measurements shown in the literature.

• The applied mathematical model correctly describes the heat transfer in ground heat exchangers.

• The presented model can be used to analyze the influence of various process parameters on the efficiency of a ground heat exchanger.

• The performed calculations show that under the typical operating conditions of a horizontal ground heat exchanger a cyclic steady state is reached in the ground after about 10 years of operation.

LITERATURE 1. Benazza A., Blanco E., Aichouba M., Rio J.L., Laouedj S., 2011, Numerical investigation

of horizontal ground coupled heat exchanger, Energy Procedia, 6, 29-35. 2. Benli H., 2013, A performace comparison between a horizontal source and vertical source

heat pump system for a greenhouse heating in the mild climate Elazig, Turkey, Applied Thermal Engineering 50, 197-206.

3. Carslaw H.S., Jaeger J.C., 1959. Conduction of Heat in Solids, second ed., Clarendon Press, Oxford, 65-74.

4. Chong C.S.A., Gan G., Verhoef A., Garcia R.G., Vidale P.L., 2013, Simulation of thermal performance of horizontal slinky-loop heat exchangers for ground source heat pumps, Applied Energy, 104, 603-610.

5. Congedo P.M., Colangelo G., Starace G., 2012, CFD simulations of horizontal ground heat exchangers: A comparison among different configurations, Applied Thermal Engineering 33-34, 24-32.

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exchanger and experimental verification, Applied Thermal Engineering, 29, 224-233. 7. Florides G., Kalogirou S., 2007, Ground heat exchangers – A review of systems, models and

applications, Renewable Energy, 32, 2461-2478. 8. Gan G., 2013, Dynamic thermal modelling of horizontal ground-source heat pumps,

International Journal of Low-Carbon Technologies 8, 95–105. 9. Gonzalez R.G., Verhoef A., Vidale P.L., Main B., Gan G., Wu Y., 2012, Interactions between

the physical soil environment and a horizontal ground coupled heat pump for a domestic site in the UK, Renewable Energy 44, 141-153.

10. Neuberger P., Adamovsky R., Sed’ova M., 2014, Temperatures and heat flows in a soil enclosing a slinky horizontal heat exchanger, Energies 7, 972-987.

11. Piechowski M., 1999, Heat and mass transfer model of a ground heat exchanger: Theoretical development, International Journal of Energy Research 23, 571-588.

12. Tarnawski V.R., Leong W.H., 1993, Computer analysis, design and simulation of horizontal ground heat exchangers, International Journal of Energy Research 17, 467-477.

13. Wang J., Long E., Qin W., 2013, Numerical simulation of ground heat exchangers based on dynamic thermal boundary conditions in solid zone, Applied Thermal Engineering 59, 106-115.

14. Wu Y., Gan G., Gonzalez R.G., Verhoef A., Vidale P.L., 2011, Prediction of the thermal performance of horizontal-coupled ground-source heat exchangers, International Journal of Low-Carbon Technologies 0, 1-9.

15. Wu Y., Gan G., Verhoef A., Vidale P.L., Gonzalez R.G., 2010, Experimental measurement and numerical simulation of horizontal-coupled slinky ground source heat exchangers, Applied Thermal Engineering, 30, 2574-2583.

Nomenclature a − thermal diffusivity of the ground, A g − surface area of the ground, B − half of the annual maximum temperature range, Bi − Biot number, C1, C2 − constants dependent on the Biot number, c − heat capacity of the ground, F − heat flux, h – distance between the heat exchanger and the surface of the ground, h0 – heat transfer coefficient, k – thermal conductivity of the ground, m − number of stages, m& − mass flow rate, qv – rate of heat generation per unit of volume, Q& - rate of heat transfer, power of a heat pump, t − time, tc − cycle time, tmax − time from the beginning of the year until the maximum ambient temperature is reached, T – temperature, Tb − temperature of the ground at great depth, x − position coordinate,

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X − dimensionless position coordinate, ρ − density of the ground, ω – frequency.

Indexes: 0 − surface of the ground, a – ambient (environment), g − ground, j = 1, 2, …, m, L − working fluid, − − average value. Captions Fig.1. C1 and C2 dependence on the Biot number Fig.2. Ground temperature profiles in the subsequent years Fig.3. The influence of the external heat transfer resistance on temperature profiles in the

ground Fig.4. Temporal variation of heat flux density to/from the ground surface Fig.4. The location and orientation of the ground heat exchanger Fig.5. The ground heat exchanger as a cascade of a perfect-mixing tanks Fig.6. A working fluid circulation between the lower and upper heat exchanger Fig.7. Comparison of computational and experimental values – ground under natural

conditions Fig.8. Comparison of computational and experimental values – working ground heat

exchanger Fig.9. Time courses of the ambient temperature and the temperature of the ground surface Fig.10. Time courses of heat transfer rates 1Q& and 2Q&

Fig.11. The multi-annual course of the average temperature of the subsurface layer of the ground

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Symbol Value

Q 5000 W k 1.24 W/(m·K) c 1670 J/(kg·K) ρ 1800 kg/m3

Lm& 1.5 kg/s

cL 3700 J/(kg·K) A 250 m2 h 1 m h0 10 W/(m2·K) B 11 K

tmax 200 days Tb 10°C

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ACCEPTED MANUSCRIPTThe mathematical model of horizontal ground heat exchanger was presented The model was correctly verified by comparison experimental and computational results Simulated calculations based on the presented model were performed Thermal calculations were carried out for ground under natural conditions