Heat transfer of horizontal parallel pipe ground heat exchanger and experimental verification

Applied Thermal Engineering 29 (2009) 224–233

Contents lists available at ScienceDirect

Applied Thermal Engineering

journal homepage: www.elsevier .com/locate /apthermeng

Heat transfer of horizontal parallel pipe ground heat exchanger andexperimental verification

Hakan Demir *, Ahmet Koyun, Galip TemirDepartment of Mechanical Engineering, Yıldız Technical University, Besiktas, 34349 Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 1 November 2007Accepted 21 February 2008Available online 29 February 2008

Keywords:Ground heat exchangerParallel pipe horizontal ground heatexchangerFinite differencesHeat transferAlternating direction implicit

1359-4311/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.applthermaleng.2008.02.027

* Corresponding author. Tel.: +90 212 383 28 20; fE-mail address: [email protected] (H. Demir).

a b s t r a c t

The ground heat exchangers (GHE) consist of pipes buried in the soil and is used for transferring heatbetween the soil and the heat exchanger pipes of the ground source heat pump (GSHP). Because of thecomplexity of the boundary conditions, the heat conduction equation has been solved numerically usingalternating direction implicit finite difference formulation. A software was developed in MATLAB envi-ronment and the effects of solution parameters on the results were investigated. An experimental studywas carried out to test the validity of the model. An experimental GSHP system is installed at Yıldız Tech-nical University Davupasa Campus on 800 m2 surface area with no special surface cover. Temperaturedata were collected using thermocouples buried in soil horizontally and vertically at various distancesfrom the pipe center and at the inlet and the outlet of the ground heat exchanger. Experimental andnumerical simulation results calculated using experimental water inlet temperatures were compared.The maximum difference between the numerical results and the experimental data is 10.03%. The tem-perature distribution in the soil was calculated and compared with experimental data also. Both horizon-tal and vertical temperature profiles matched the experimental data well. Simulation results werecompared with the other studies.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The GHE is an important part of GSHP systems and its dimen-sions and burial depth should be calculated using an effectivemethod. Particularly, the cost of the assembly of GHE affects thechoice of these systems. For efficiency of the GSHP system, the heatextracted from or dissipated to the soil should not be changed bytime for longer period runs of GSHP systems. Therefore, the burialdepth and the distance between the pipes are important for sizingthe GHE. There are plenty of works for this aim.

In the literature, as a basic, there are two kinds of analytical ap-proaches. The first one is the Kelvin line source theory and theother one is the cylindrical source theory. In addition, there aremany studies using two or three-dimensional steady state andtime-dependent numerical techniques [1–7,10–13,16]. Kelvin linesource and cylindrical source theories find only symmetrical soiltemperature distributions around the pipe. Metz [7] has suggestedan analytical model to find temperature distribution in the soil bydividing the ground into blocks around the coil and has done somemodifications to the Line Source theory. Mei [6] has included theeffects of seasonal ground temperature variation, pipe material,

ll rights reserved.

ax: +90 2122616659.

circulating liquid properties and compared his work with modifiedline source and simple line source models.

A simplification of boundary conditions to solve equations ana-lytically causes some error on results especially shorter simulationtimes. Analytical models do not consider the temperature changeof soil by depth and the surface effects such as radiation, convec-tion and also surface cover are not included. The effects of the con-vection on the ground surface were included in some models [1,2].

A more complicated model for heat transfer of buried pipes wasperformed by Negiz, Hastaoglu and Heidemann for petrol transfer-ring pipes [4,11,12]. Piechowsky included mass transfer in hismodel to take into account the effects of the soil moisture[13,17]. There were two questions. One of them was weather con-ditions and their effects on GHE design and calculation, the otherwas to match or not with the experimental data. In order to simu-late whole effects on GHE, a new model and a big scale experimen-tal set area was built.

2. Development of a new model

Aiming to find three-dimensional temperature distributions inthe soil, a new model including all the weather effects in real lifewas suggested. Heat transfer in the soil is a time dependent, threedimensional heat conduction. Temperature gradient along the pipeaxis is so small that it can be neglected and the heat conduction

mailto:[email protected]

http://www.sciencedirect.com/science/journal/13594311

http://www.elsevier.com/locate/apthermeng

Nomenclature

albedo reflectivity of the surfaceD distance between pipes (m)b constant number according to illumination and absorp-

tivity of surface (dimensionless)Cp,f specific heat of fluid (J/kg K)De latent heat exchange coefficient (m/s)Dh sensible heat exchange coefficient (m/s)ea atmospheric vapor pressure (Pa)ey soil surface vapor pressure (Pa)f constant number according to surface cover and humid-

ity (dimensionless)g gravity (m/s2)hy convection heat transfer coefficient (W/m2 K)H soil depth used in computer simulation as lower bound-

ary (m)I total irradiance (W/m2)IR rain intensity (kg/m2 s)kf fluid thermal conductivity (W/m K)ks soil thermal conductivity (W/m K)kt,u thermal conductivity of upper soil layer (W/m K)ksnow thermal conductivity of snow layer (W/m K)L pipe length (m)Ls latent heat of snow layer (J/kg)_mf mass flow rate of the fluid (kg/s)

P period (s)Pa atmospheric pressure (Pa)QC conduction heat flux through snow layer (W/m2)QE heat flux due to evaporation (W/m2)QH emitted long wave radiation heat flux (W/m2)QLI incident long wave radiation heat flux (W/m2)QP heat flux due to precipitation (W/m2)QSI surface solar incident radiation heat flux (W/m2)Qt total heat flux at soil surface (W/m2)Ri Richardson number (dimensionless)S soil surface incident solar radiation (W/m2)Sa amplitude of solar radiation (W/m2)

Sm mean annual solar radiation (W/m2)T temperature (�C)t time (h)Tf fluid temperature (K)Ti initial temperature (�C)Ts,a amplitude of soil temperature (�C)Ta air temperature (K)Ts,a soil temperature (K)Tdp daily dew point temperature (K)Ts,m mean soil surface temperature (�C)Ty soil surface temperature (K)Tf,i fluid inlet temperature (�C)Tf,o fluid outlet temperature (�C)Ts soil temperature (�C)Uz wind speed at a height of z (m/s)Y burial depth (m)x soil depth (m)z reference height (m)z0 roughness (m)zt,u thickness of upper soil level (m)zsnow snow layer thickness (m)a soil absorptivity (dimensionless)at soil thermal diffusivity (m2/h)e soil emissivity (dimensionless)f stability function (dimensionless)j Von Karman constant (dimensionless)q soil density (kg/m3)qa air density (kg/m3)r Stefan-Boltzmann constant (5.67 � 10�8 W/m2 K4)/1 phase angle (rad)x angular velocity (rad)qf fluid density (kg/m3)Dx mesh size in the X direction (m)Dy mesh size in the Y direction (m)Dt time step (s)

H. Demir et al. / Applied Thermal Engineering 29 (2009) 224–233 225

equation can be solved using dynamical boundary conditions intwo-dimensional geometry. By means of conservation of energy,the temperature distribution of the fluid along the pipe was calcu-lated and used for linking two-dimensional solution domains. Be-cause of the complexity of the boundary conditions, the heatconduction equation has been solved numerically using alternatingdirection implicit (ADI) finite difference formulation. ADI method isstable for every time step and grid size and the resulting matrix sys-tem is tri-diagonal. Tri-diagonal matrix systems can be solved eas-ily using the Thomas algorithm. For this purpose, software wasdeveloped in MATLAB environment and the effects of solutionparameters on the results were investigated. The simulation resultswere acceptable when a mesh size of 0.1 m in x and y directions,1 m in z direction and 1800 s as time step were used.

The solution domain and boundary conditions were prepared asdepicted in Fig. 1. In order to simplify the problem some assump-tions were made as below:

� All of the parallel pipes are buried at the same depth and everyduct includes only one pipe.

� The fluid temperature inside the pipe is constant and does notchange in a section perpendicular to the pipe axis.

� The effects of the collectors are neglected.� Volumetric flow rates in every pipe were the same.� The soil thermal properties (soil thermal conductivity and diffu-

sivity) are constant and the same at every point of domain.

� Mass transfer and its effects to total heat transfer rate areneglected.

� Thermal properties of the pipe are constant.

The model consists of parallel pipes buried at the depth ofY. Distance between the pipes is D. Region shown in Fig. 2 is takenas the domain for the solution. This is a two-dimensional space andis presented in Cartesian coordinate system. Two-dimensionaltime-dependent heat conduction and boundary conditions of theproblem are of the form:

o2Tox2 þ

o2Toy2 ¼

1a

oTot

ð1Þ

T i ¼ Tðx; tÞ; t ¼ 0 ð2ÞoTox

��x¼D=2

¼ 0 ð3Þ

oTox

��x¼0¼ 0 ð4Þ

Q ðW=m2Þ; y ¼ L ð5ÞQ t ðW=m2Þ; y ¼ 0 ð6ÞT f ;i ¼ C ð7Þ

For the initial boundary condition, the distribution of temperature inthe soil at the beginning of the simulation time should be known. Eq.(8) can be used for calculating the soil temperature as a function ofdepth

Fig. 1. Parallel pipe horizontal GHE and solution area in the soil.

Fig. 2. Computational solution domain.

226 H. Demir et al. / Applied Thermal Engineering 29 (2009) 224–233

Tðx; tÞ ¼ Ts;m þ Ts;ae�xffiffiffiffip

asP

pcos 2p

tP� x

ffiffiffiffiffiffiffip

asP

r� �ð8Þ

When annual mean temperature and the amplitude of the annualtemperature change are known, the distribution of temperature inthe soil can be easily calculated.

The surface heat flux was taken into account by means of en-ergy balance equations. The effects of solar radiation, long waveradiation, latent and sensible heat transfer, convection, surfacecover and precipitation are included. Eq. (9) can be used for thesurface energy balance

_Q t ¼ _QC þ _QE þ _QH þ _Q LE þ _Q LI þ _Q SI þ _QP ð9Þ

The sensible heat flux can be written using the following equations[9].

_QH ¼ qaCp;aDhfðTa � TyÞ ð10Þ

Dh ¼j2Uz

½lnðz=z0Þ�2ð11Þ

f ¼ 1ð1þ 10RiÞ ð12Þ

Ri ¼ gzðTa � TyÞTaU2

z

ð13Þ

The latent heat flux on the soil surface can be investigated in twoparts. The first one is transportation from soil surface and the sec-ond one is sublimation of snow. Sublimation of snow is given by [9]

_QE ¼ qaLsDef 0:622ea � ey

Pa

� �ð14Þ

De ¼j2Uz

½lnðz=z0Þ�2ð15Þ

log10ea ¼ 11:40� 2353Tdp

ð16Þ

and the transportation from the soil surface [8]

_QE ¼ 0:0168 fha½ðaTy þ bÞ � raðaTa þ bÞ� ð17Þ

can be used to calculate latent heat flux. Hourly solar radiation dataand the daylight time should be known in order to calculate solarradiation heat flux. Solar radiation absorbed from soil can be easilycalculated using the following equations [8]:

_QSI ¼ bS ð18Þb ¼ 1� albedo ð19Þ

By using Fourier analysis and first harmonic function, Eq. (18) maybe expressed in the form

_QSI ¼ b½Sm þ SaReðexpðixt þ u1Þ� ð20Þ

Daylight time is calculated using astronomical solar positionequations according to longitude and altitude of the location whereGSHP system is installed. Long wave radiation diffused from soilsurface and incident long wave radiation absorbed from soil sur-face can be expressed with the following equations, respectively[9]:

Table 1The effects of the solution parameters on horizontal temperature distributions

Distance frompipe axis (m)

Temperature (K)(Dx = Dy = 0.5 mDt = 1800 s)

Temperature (K)(Dx = Dy = 0.3 m Dt = 1800 s)





0 303.15 303.15 303.15 303.15 303.15 303.150.1 289.7889 289.7149 289.5488 289.92140.2 287.8433 287.3988 287.9659 287.43670.3 287.045 286.6479 286.4232 286.7145 286.45270.4 286.1047 286.0019 286.1402 286.02140.5 286.2003 285.8801 285.8365 285.8965 285.8460.6 285.6489 285.7964 285.7795 285.8025 285.78260.7 285.7683 285.7623 285.7696 285.76240.8 285.7597 285.7579 285.7591 285.75670.9 285.4668 285.7574 285.7568 285.756 285.75531 285.4873 285.7568 285.7566 285.7552 285.7551.1 285.7566 285.7566 285.755 285.75491.2 285.4504 285.7566 285.7566 285.7549 285.75491.3 285.7566 285.7566 285.7549 285.75491.4 285.7566 285.7566 285.7549 285.75491.5 285.452 285.4494 285.7566 285.7566 285.7549 285.7549

250

255

260

265

270

275

280

285

290

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Time (Day)

Tem

per

atu

re (

K)

Fig. 3. Fluid inlet temperatures to the GHE for winter operation [6].

250

255

260

265

270

275

280

285

290

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

Time (Day)

Tem

per

atu

re (

K)

New Model Experimental Mei Model Modified Line Source

Fig. 4. Comparison of fluid outlet temperatures of three models for winter operation.


250

260

270

280

290

300

310

320

330

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Time (Day)

Tem

per

atu

re (

K)

Fig. 5. Fluid inlet temperatures to the GHE for summer operation [6].

250

260

270

280

290

300

310

320

330

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Time (Day)

Tem

per

atu

re (

K)

New Model Experimental Mei Model Modified Line Source

Fig. 6. Comparison of fluid outlet temperatures of three models for summer operation.

Fig. 7. Experimental setup (heat pump and measuring system).


Fig. 8. Experimental setup (ground heat exchanger and thermocouple locations).


_Q LE ¼ �erT4y ð21Þ

_Q LI ¼ 1:08ð1� exp �ð0:01eaÞTa

2016

� �rT4

a ð22Þ

The heat conduction occurs through the snow layer if the snow ex-ists on the soil surface. The heat conduction is expressed as [9]

270

272

274

276

278

280

282

284

286

288

290

1 3 5 7 9 11 13 15 17

Tim

Tem

per

atu

re (

K)

Fig. 9. Experimental fluid inlet and experimen

_QC ¼ �ðTy � Tt;aÞzsnow

ksnowþ zt;u

kt;u

� ��1

ð23Þ

The heat flux due to precipitation can be taken as (24)

_QP ¼ IRCp;wðTa � TyÞ ð24Þ

Using all the equations above, the distribution of temperature intwo dimensions can be calculated. The soil temperature profiles inthree dimensions might be calculated by dividing the whole pipeinto small parts in order to calculate fluid inlet and outlet temper-atures by using the fluid outlet temperatures as fluid inlet temper-ature of the next part. The temperature profile along with the pipemay be expressed as below [18]

T f ;o ¼ Ts � ðTs � T f ;iÞe�ksL_mf Cp;f ð25Þ

3. Numerical solution

Eq. (1) is solved using alternating direction implicit (ADI) finitedifference scheme [14,15]. The finite difference form of Eq. (1) at(n + 1)th time is

Tnþ1i;j � Tn

i;j

a � Dt¼

Tnþ1i�1;j � 2 � Tnþ1

i;j þ Tnþ1iþ1;j

ðDxÞ2þ

Tni;j�1 � 2 � Tn

i;j þ Tni;jþ1

ðDyÞ2ð26Þ

and at (n + 2)th time in the same manner

Tnþ2i;j � Tnþ1

i;j

a � Dt¼

Tnþ1i�1;j � 2 � Tnþ1

i;j þ Tnþ1iþ1;j

ðDxÞ2þ

Tnþ2i;j�1 � 2 � Tnþ2

i;j þ Tnþ2i;jþ1

ðDyÞ2ð27Þ

for equal spacing in x and y directions

Dx ¼ Dy ð28Þa:Dt

ðDxÞ2¼ a:Dt

ðDyÞ2¼ r ð29Þ

we may write for (n + 1) and (n + 2)

� r �Tnþ1i�1;jþð1þ2rÞ � Tnþ1

i;j � r � Tnþ1iþ1;j ¼ r �Tn

i;j�1þð1�2rÞ � Tni;jþ r � Tn

i;jþ1

ð30Þ� r �Tnþ2

i;j�1þð1þ2rÞ � Tnþ2i;j � r � Tnþ2

i;jþ1 ¼ r �Tnþ1i�1;jþð1�2rÞ � Tnþ1

i;j þ r �Tnþ1iþ1;j

ð31Þ

19 21 23 25 27 29 31 33 35 37

e (Day)

Fluid inlet

Experimental

Theoretical

tal/theoretical fluid outlet temperatures.

250

255

260

265

270

275

280

285

290

295

300

0 0.5 1 1.5 2

Distance from center of the pipe (m)

Tem

per

atu

re (

K)

Theoretical

Experimental

Fig. 10. Vertical temperature distributions after 1 h.

250

255

260

265

270

275

280

285

290

295

300


Tem

per

atu

re (

K)

Theoretical

Experimental

0 0.5 1 1.5 2



Finite difference form of the boundary conditions can be expressedas

dTdx¼ 0) Ti�1;j ¼ Tiþ1;j ð32Þ

q ¼ �kdTdx) Tiþ1;j ¼ Ti�1;j �

2qDxk

ð33Þ

� kdTdx¼ hðT1 � TÞ ) Ti�1;j ¼ Tiþ1;j þ

2DxhkðT1 � Ti;jÞ ð34Þ

The above equations are solved successively to obtain tempera-ture distribution at a given time. Since the resulting matrixes aretri-diagonal, Thomas Algorithm can be used to solve the equations.The effects of the solution parameters have been investigated onthe results. Table 1 shows the effects of time step and mesh sizeon horizontal temperature distribution from the pipe axis. Temper-atures for smaller time step and mesh size became more stable butthe simulation time increases. Since the error arising from mea-surement of the temperatures is ±0.3 �C, the optimum time stepand mesh size in x and y directions have been chosen as 1800 s,0.1 m and 0.1 m, respectively.

To approve the validity of the new model, a simulation has beencarried out using simplified boundary conditions and compared toMei’s work. Experimental results from Mei’s study and results frommodified line source model were compared. Figs. 3 and 5 show thefluid inlet temperatures to the ground heat exchanger. Figs. 4–6show a comparison of the fluid outlet temperatures from theground heat exchanger between the models for winter and sum-mer, respectively. The new model shows a good agreement withthe experimental results and other models. The temperatures ob-tained from the simulation are slightly higher than the experimen-tal results for winter operation of GHE. The effect of backfillmaterial is neglected for summer operation and the model outputtemperatures are higher than the experimental results.

4. Experimental study

A GSHP having 4 kW heating and 2.7 kW cooling capacity is usedfor experimental study. The ground heat exchanger consists of threeparallel pipes which have 40 m length and 1/200 diameter buried in soil

250

255

260

265

270

275

280

285

290

295

300


Tem

per

atu

re (

K)

Theoretical

Experimental

0 0.5 1 1.5 2


250

255

260

265

270

275

280

285

290

295

300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


Tem

per

atu

re (

K)

Theoretical

Experimental

Fig. 13. Horizontal temperature distributions after 1 h.


at 1.8 m depth. The distance between the parallel pipes is 3 m. Exper-imental GSHP system is installed at Yıldız Technical University Dav-upas�a Campus on an 800 m2 surface area with no special surfacecover. Temperature data were collected using T-type thermocouplesburied in soil horizontally and vertically at various distances from thepipe center and at the inlet and outlet of the ground heat exchanger.All the thermocouples are connected to a 64 channel PLC system capa-ble of saving data of hourly temperature measurements for 8 days.Figs. 7 and 8 represent the experimental setup.

In the beginning of the experimental study for determining thesoil thermal conductivity fluid inlet and outlet temperatures havebeen measured and used by means of conservation of energy[18]. Collected fluid inlet temperatures were used in computersimulation as fluid inlet temperature to the GHE. Parameters usedin computer simulation are as below:

Experimental study started on 13th December, 2005.Soil thermal conductivity, ks = 2.18 W/m K.Soil thermal diffusivity, as = 0.00000068 m2/s.Length of parallel pipes, L = 40 m.Number of parallel pipes, n = 3.Distance between pipes, D = 3 m.Burial depth, Y = 1.8 m.Volumetric flow rate of working fluid, Vf = 0.42768 m3/h.Working fluid = water.Pipe material = PPRC.Pipe thermal conductivity, kp = 0.8999 W/m K.Pipe outer/inner diameter, do/di = 20/14.6 mm.Mesh step in x and y direction, dx = dy = 0.1 m.Mesh step along pipe axis, dz = 1 m.Time step, dt = 1800 s.

250

255

260

265

270

275

280

285

290

295

300


Tem

per

atu

re (

K)

Theoretical

Experimental

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6



250

255

260

265

270

275

280

285

290

295

300


Tem

per

atu

re (

K)

Theoretical

Experimental

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


The experimental fluid inlet/outlet and theoretical fluid outlettemperatures are shown in Fig. 9. Difference between experimentaland theoretical results is maximum 10.03% and simulation resultsshow good agreement with experimental data. Figs. 10–12 repre-sent vertical temperature distribution from the pipe axis after 1,50 and 910 h, respectively. Figs. 13–15 represent horizontal tem-perature distribution from the pipe axis after 1, 50 and 910 h,respectively.

5. Results and discussion

A heat transfer model including real life conditions for horizon-tal pipe GHE has been validated by means of experimental study.The ground heat exchanger outlet temperatures can be predictedaccurately using the new model developed. It can be easily used

for simulation and design of parallel pipe horizontal GHE. Meteoro-logical data and soil thermal properties are crucial for an accuratedesign and simulation of parallel pipe horizontal GHE. The soilthermal conductivity is highly dependent on soil moisture content.It is observed that soil thermal conductivity is the most influentialsoil property on the results. The burial depth and distance betweenpipes are very important design parameters as far as the thermalperformance of a parallel pipe horizontal GHE is concerned.

Simulation shows good agreement with the experimental re-sults for vertical and horizontal temperature distributions. The ini-tial soil temperature was calculated using meteorological soiltemperature data of the previous years. Air temperature and solarradiation are modeled using the mean of the previous year’s mete-orological data. The surface effects on vertical temperature distri-butions can be seen in Figs. 11 and 12.


Finally, heat transfer of horizontal parallel pipe GHE is modeledand simulated including all meteorological and surface conditions,and temperature distribution in the soil is obtained. Validity of thenew model is approved by the experimental study.

6. Conclusion

The surface effects are successfully included in the new modelby means of energy balance equation and validated by an experi-mental study, and the following conclusions are achieved:

1. Meteorological soil, weather temperatures and solar radiationdata have been used in computer simulation and the resultswere satisfactory. As seen in Figs. 10 and 13, the soil tempera-tures after 1 h of the simulation are very close to the tempera-tures measured from the experimental work.

2. The developed software capable of estimating the fluid outlettemperatures with a small error can be used for calculatingoptimum ground heat exchanger dimensions and burial depthfor a given location if meteorological data are available.

3. Simulation time can be shortened by means of reducing thenode number in the y direction if the heat flux is known at adepth of H.

4. Neural network approach may be used for modeling soil, airtemperature and solar radiation for further studies.

References

[1] B. Bohm, On transient heat losses from buried district heating pipes, Int. J.Energy Res. 24 (2000) 1311–1334.

[2] M. Chung, P.S. Jung, R.H. Rangel, Semi-analytical solution for heat transfer froma buried pipe with convection on the exposed surface, Int. J. Heat Mass Transf.42 (1999) 3771–3786.

[3] Y. Gu, D.L. O’Neal, An analytical solution to transient heat conduction in acomposite region with a cylindrical heat source, Trans. ASME 117 (1995) 242–248.

[4] M.A. Hastaoglu, A. Negiz, R.A. Heidemann, Three-dimensional transient heattransfer from a buried pipe – part III comprehensive model, Chem. Eng. Sci. 50(1995) 2545–2555.

[5] T.K. Lei, Development of a computational model for a ground-coupled heatexchanger, ASHRAE Trans. Res. 99 (1993) 149–159.

[6] V.C. Mei, Heat transfer of buried pipe for heat pump application, J. Solar EnergyEng. 113 (1991) 51–55.

[7] P.D. Metz, A Simple computer program to model three-dimensionalunderground heat flow with realistic boundary conditions, Trans. ASME 105(1983) 42–49.

[8] G. Mihalakakou, On estimating soil surface temperature profiles, Energy Build.34 (2002) 251–259.

[9] F. Ling, T. Zhang, A numerical model for surface energy balance and thermalregime of the active layer and permafrost containing unfrozen water, Cold Reg.Sci. Technol. 38 (2004) 1–15.

[10] S. Mukerji, K.A. Tagavi, W.E. Murphy, Steady-state heat transfer analysis ofarbitrary coiled buried pipes, J. Thermophys. Heat Transf. 11 (1997) 182–188.

[11] A. Negiz, M.A. Hastaoglu, R.A. Heidemann, Three-dimensional heattransfer from a buried pipe – I. laminar flow, Chem. Eng. Sci. 48 (1993) 3507–3517.

[12] A. Negiz, M.A. Hastaoglu, R.A. Heidemann, Three-dimensional transient heattransfer from a buried pipe: solidification of a stationary fluid, Numer. HeatTransf. 28 (1995) 175–193.

[13] M. Piechowsky, Heat and mass transfer of a ground heat exchanger:theoretical development, Int. J. Energy Res. 23 (1999) 571–588.

[14] M.N. Ozisik, Heat Conduction, A Wiley Interscience Publication, 1980.[15] D.U. Von Rosenberg, Methods for the Numerical Solution of Partial Differential

Equations, Publishing Division Gerald L. Farrar & Associates Inc., Tulsa, 1969.[16] A.D. Chiasson, Advances in Modeling of Ground Source Heat Pump Systems,

MSc thesis, Oklahoma State University, 1999.[17] M. Piechowski, A Ground Coupled Heat Pump System with Energy Storage,

PhD thesis, The University of Melbourne, 1996.[18] H. Demir, Ground Source Heat Pumps and Optimization and Development of

Ground Heat Exchangers, PhD thesis, Yıldız Technical University, 2006.

Documents

Heat transfer of horizontal parallel pipe ground heat exchanger and experimental verification