Thermal conductivity determination of ground by new modified two dimensional analytical models

Thermal conductivity determination of ground by new modified two dimensional analytical models: Study cases
Babak Dehghan B.
Department of Energy, Politecnico di Milano, 20156 Milano, Italy
[email protected], [email protected]

ABSTRACT
Determining thermal conductivity of ground plays an important role in designing procedure of ground source heat pump (GSHP) systems. In this paper new modified 2D analytical models which are depending on thermal conductivity of ground are derived and results are compared with experimental ones. In an experimental study, a single borehole ground heat exchanger (GHE) with polyethylene U-tube pipe is considered for two different regions. Fluid is pumped into the pipes in a specific temperature and inlet and outlet temperatures are measured as well as volumetric flow rate. Analytical results curves are fitted to experimental one and thermal conductivities of ground are calculated for each region. Based on validated analytical models, long term performance of a single borehole GHE is determined. Additionally, temperature distributions around borehole GHE are investigated analytically in region 1 (N.D.B. residence region). Analytical models given in this study can easily help designers to evaluate thermal conductivity of ground and thermal performance of the borehole GHEs.
Keywords: Ground source heat pump applications, Green’s function method, Analytical model, borehole ground heat exchanger, borehole performance prediction
Nomenclature

Specific heat capacity of ground
Average temperature of inlet fluid

Specific heat capacity of fluid
Average temperature of outlet fluid

borehole diameter
Undisturbed ground temperature

Length of borehole
Time

Dimensionless length
Volumetric flow rate

Thermal conductivity of ground Z Dimensionless z coordinate

Thermal conductivity of U-tube pipe
Density of ground

Mass flow rate
Thermal diffusivity

Experimental HTR value
Dimensionless temperature

HTR value per length of borehole
Dimensionless time

Borehole’s radius

Equal pipe’s radius

Inner radius of U-tube pipe

Outer radius of U-tube pipe

Dimensionless radius

Average fluid temperature
1. INTRODUCTION
Using of renewable energy sources becomes popular due to their large heating load and high efficiency 1. Geothermal energy is known as one the most efficient renewable energies in the world. It is clean, sustainable, suitable energy storage and available for whole day. Geothermal energy can be used through ground source heat pump (GSHP) systems. Ground source heat pump (GSHP) systems are becoming widespread due to their high potential to considerably reduce the primary energy use of space heating and cooling compared with the conventional systems. They can be used for different purposes such as, supplying heating and cooling demands of the building, waste heat recovery of gas turbine exhaust gasses 2, snow melting on pavements and bridge decks, etc. 3. For instance, as it is stated in Ref. 2, waste heat of micro gas turbine exhaust gasses energy can be stored in ground through ground heat exchangers (GHEs) and then recovered by GSHP system for supplying heating demands of the building. Balbay and Esen 3 developed new method for snow melting on pavements by using GSHP system. Their GSHP system consists of U-tube GHEs (vertical length: 30, 60 and 90 m) on pavements and bridge decks. The advantage of using GSHP system in their method is that, melting snow with a hydronic heating system can eliminate need for snow removal by chemical or mechanical means. In another study 4 GSHP system is integrated with solar system. The requirement for alternative low-cost energy sources has given rise to the development of new ground source heat pump systems for residential and commercial heating/cooling applications. Esen et al. 4 established slinky GHEs for a solar assisted GSHP. GSHP consists of two main part, heat pump and GHE. Each part highly influences GSHP system performance and are needed to be designed exactly 5-9. GHEs are divided into shallow and deep ones. Shallow GHEs are helix, slinky etc. and deep one is generally U-tube pipes. U-tube GHEs are usually embedded in the borehole which is filled with grout 9, 10. When GSHP system start working, heat transfer process in ground around borehole is began. Therefore, analyzing this process becomes as an important problem for borehole designers 11, 12. There are different methods that can explore heat transfer procedure such as experimental study, analytical modeling and computational methods (Using software like FLUENT, ANSYS, COMSOL). Each method has its own advantages and disadvantages. Although experimental study gives almost exact results, its building’s cost is high, not applicable everywhere and time consuming. Commercial software takes short period to reach results but their license’s cost is too high and cannot be used by everyone. Between these methods, analytical models are almost favorable because of their clearness and reaching expected results in a short period. Analyzing thermal properties of application area (ground) is one the most essential steps in designing procedure of GSHP systems. Therefore, thermal properties of ground such as thermal conductivity of it is needed to be determined. Thermal conductivity of ground can be simply calculated by fitting analytical (1D, 2D or 3D models) results to experimental ones. 1D analytical method is described in details in our previous work given in Ref. 13 and main goal of this study is to present 2D method. Another important part of GSHP designing procedure is to estimate thermal performance or heat transfer rate (HTR) of borehole accurately. HTR of the borehole will be decreased by the GSHP system operation time due to temperature rise or reduction around borehole. HTR is directly related to temperature differences between borehole wall and ground 11-14. Therefore, it is necessary to find reliable model which can predict HTR of the borehole. Many research with different methods have done analysis on borehole modeling in literature 15-28.
Beier et al. 15 developed weighting factor for inlet and outlet temperature of vertical borehole GHEs. They derived 1D radial models than can give temperature variations with depth. They also validated their results by experimental ones and concluded that their models give more accurate results than the mean temperature approximation. Balbay and Esen 16 analyzed temperature distributions in pavement and bridge slabs which are heated by vertical GSHP systems. They developed 3D finite element models for modeling temperature distributions. They also validated their computational results by experimental ones. Their results show that thermal properties related to ground structure are very significant parameters in design of vertical GSHP system. Calvo et al. 17 introduced novel numerical method in ground source heat exchangers modeling. They decoupled short and long term analysis of borehole, borehole to ground model (B2G) was used for short term responses and g-function (In mathematics, the Barnes g-function is a function that is an extension of super factorials to the complex numbers) was used for long term ones. They believed that their novel model is faster and exact in both short and long term analysis. Conti et al. 18 gave analytical heat transfer models for a borehole which contains double U-tube. In their model, heat flux at borehole surface and return temperature with respect to various geometrical properties were evaluated. They used ?-NTU (effectiveness Number of Transfer Units) heat exchanger theory for models derivation. To investigate the effectiveness of their models, they compare results of double and single U-tube borehole ground heat exchangers. Hein et al. 19 constructed a comprehensive numerical model of borehole GHE which include flow and heat transport processes. The effects of various parameters on temperature evaluation around borehole was shown in their work. Holmberg et al. 20 studied on coaxial borehole heat exchangers numerically. They validated analytical results with temperature measurements obtained by thermal response test. Their model shows good accuracy in predicting the behavior of borehole heat exchanger. Furthermore, they investigate the effects of mass flow direction on borehole performance. Mingzhi et al. 21 presented a simplified heat transfer method which can examine efficiency of large scale borehole applications. They considered borehole geometric symmetry and assumed no water flow in ground in their modeling. Priarone et al. 22 developed a numerical model for temperature responses around borehole. Fourier equation was solved for different geometries and boundary conditions and the results were compared with available analytical results in literature. Selamat et al. 23 examined the optimization ways of horizontal GHEs by considering different layouts and pipe materials. They did 3D computational fluid dynamic (CFD) analysis during optimization process. Wang et al. 24 investigated the thermal behavior of horizontal geothermal heat exchangers with vertical spiral coils. Mathematical model was developed to analyze performance of these GHEs. Their model was validated by experimental and also simulation results. Furthermore, they showed temperature distribution around GHEs. Zhang et al. 25 gave optimum design of borehole heat exchangers. They simulated performance of the borehole hourly and determined optimum combination of distance between boreholes, borehole depth, diameter and annually heating/cooling load of the building. They used field test study to validate the effectiveness and feasibility of their model. Aydin et al. 26 investigate thermal performance of the borehole with different numbers of U-tube GHEs experimentally and analytically. They derived an analytical model which was fitted to experimental results. Furthermore, they made long term performance prediction of the borehole as well as cost analysis of it. Molina Giraldo et al. 27 developed a new mathematical method to investigate the thermal performance of the GHE by considering the effects of groundwater flow. They claim that fluid flow in ground highly affect the borehole performance. Gultekin et al. 28 determined HTR values of the borehole computationally by using COMSOL software. They also analyzed thermal interaction between boreholes in large area applications and determined optimum distance between borehole as well as temperature variations.
One of the most important parameters which highly affects thermal behavior of borehole GHE in GSHP applications is thermal conductivity of ground. The goal of this study is to find simple, feasible, accurate and reliable analytical model which can determine thermal conductivity of ground and predict thermal performance of the borehole GHE. To determine thermal conductivity of ground/soil, analytical results curve is fitted to experimental ones.
2. EXPERIMENTAL STUDY
The main goal of this study is to analyze thermal behavior of borehole GHEs and calculate thermal conductivity of soil by fitting analytical results to experimental ones. To investigate thermal performance of borehole GHEs experimentally, test system of N.D.B. residence (region 1) which is located in Tabriz, Iran is used. Fig. 1 shows the schematic view of the entire application area at N.D.B. residence with different types of GHEs. Application area contains 2 vertical U-tubes with depth of 50 m and nine vertical spiral GHEs with depth of 4m. To record the temperature variations in ground during GSHP system operation, 21 temperature sensors (sensor type: Ds18b20) are embedded in ground with various locations and depths (18 sensors with 4m depth and 3 sensors with 25m depth). During the test, inlet and outlet water temperatures of each GHE are measured (by Pt100) as well as volumetric flow rate (Z-4004 Flow Meter 2-20 GPM). Uncertainties for temperature and volumetric flowmeter are + 0.1 oC and +3% respectively. Distance between spiral and U-tube GHEs are 6m. As it is shown in Fig. 1b, a heat exchanger is used to keep the inlet temperature of GHEs in desired test or operation value. In this study we use only a single borehole GHE and test its performance for 100 hours non-stop operation. Aydin et al. 26 work show that this amount of test time (between 60 and 120 hours) is enough for estimating thermal conductivity of ground/soil. To test the thermal performance of the borehole GHE, fluid (mixture of water and glycol) is pumped to each GHEs in a specific temperature. Measurements show that in this case study, average inlet and outlet temperature of fluid is 30 oC during the test. Furthermore, in an application area, average temperature of ground along borehole GHE and its surroundings is measured about 16oC by sensors. Different properties of application area and working conditions are given in Table1.

(a)

(b)
Fig. 1: (a) schematic top view of application area configuration of N.D.B. residence (b) Schematic view of experimental setup
Table 1: Parameters of application area
Parameters Values Definition

Thermal Properties of Ground

2100 Density kg m-3

900 Specific heat capacity J kg-1K-1

1.8
Thermal conductivity W m-1K-1

Thermal Properties of Grout

1650 Density kg m-3

950 Specific heat capacity J kg-1K-1

0.9
Thermal conductivity W m-1K-1
Thermal Properties of Water-glycol mixture
kg m-3
1036 Density kg m-3
J kg-1K-1
3870
Specific heat capacity J kg-1K-1
Thermal Properties of PE
kg m-3
950 Density kg m-3
J kg-1K-1
2300 Specific heat capacity J kg-1K-1
W m-1K-1
0.45
Thermal conductivity W m-1K-1
Geometric Properties of borehole GHE (U-tube)

0.014 Internal radius of PE pipe m

0.017 External radius of PE pipe m

50 Vertical length of borehole GHE m

0.2 Major diameter of borehole GHE m

Working Conditions

18 Average volumetric flow rate L min-1

30 Average fluid temperature oC

1.5 Average Temperature difference between inlet and outlet oC

16 Initial temperature of system oC

16 Undisturbed uniform ground temperature oC

100 Testing period hour
As it was stated previously, in an experimental study, inlet and outlet temperatures of U-tube pipes are measured as well as volumetric flow rate. Thermal performance or HTR of the borehole GHE is calculated by following expression;
(1)
(2)
Where is total HTR amount (W), is total HTR amount per length of borehole (W/m), L is length borehole, is volumetric flow rate of Nth borehole GHE, and are inlet and outlet temperature of Nth borehole GHEs respectively. Since in this study only one borehole GHE is considered, j value is equal to 1. Fig. 2 shows thermal performance of a single borehole GHE which contains a single U-tube GHE for region 1 (N.D.B. residence).

Fig. 2: Thermal performance of a single borehole GHE (Eq. 2 results).
2. BOREHOLE GROUND HEAT EXCHANGER MODEL DESCRIPTION
Before installing GSHP system, thermal properties of ground in the application region need to be calculated. Thermal conductivity of ground/soil plays an important role in performance of GSHP system. Thermal conductivity (k) of ground can be calculated by fitting k-based analytical model results to experimental ones. In this study new modified analytical models are derived (2D, r and z directions) for calculation of ground’s thermal conductivity based on Green’s Function method (assumptions: Infinite body, cylindrical coordinates and transient 1D).
To investigate thermal behavior of the borehole, a single equal pipe to inlet and outlet pipes is assumed. Diameter and wall temperature of equal pipe are equal to 4rp and . Fig. 3 shows schematic view of a single borehole GHE.

Fig. 3: schematic view of borehole (a) 2D view of Borehole (b) A-A section top view.
Since there is mostly no fluid flow in layers of ground, only conduction heat transfer model is considered and conductive one is neglected. 2D heat conduction equation is as follow 13, 29;
(3)
In the 2D model the cylindrical heat source is considered of a limited length (h), stretching from the boundary with constant temperature (Tw) to a certain depth, h. 33 and ground is assumed as homogenous infinite medium. These assumptions are very close to our experimental working conditions which was discussed in previous section. Fig. 4 indicates schematic side view of equal pipe model which is assumed to be solved in this study (schematic boundary conditions are shown in this figure).

Fig. 4: schematic side view of equal pipe used in solving procedure
The Green’s function method is used to solve Eq. 3. In Cartesian coordinate and in an infinite medium, heat conduction equation of Green’s function is given by 13, 30-33;
(4)
Green’s function in cylindrical coordinates ( and ), 13, 30-33 is;
(5)
By using Green’s function rules, temperature distribution in the ground can be calculated by following equation:
(6)
As it was stated previously only r and z directions are considered and variations in is neglected. Therefore, temperature distribution by heat conduction around borehole is only a function of r, z and t,
(7)
Where B is the boundary conditions-variable which is calculated by using boundary conditions of related problem. To solve Eq. 7, MATHEMATICA software is used and final solution is derived as follows;
(8)
Where re=r’ is equal pipe’s radius, is thermal diffusivity of ground, t is time, h is borehole length, erf is error function and I0 is Bessel’s function .
Using dimensionless forms of radius, length, time and temperature ( , and , , ,) can speed up and also simplify the solving procedure 13, 30. After applying the above given dimensionless forms of parameters, Eq. 8 becomes,
(9)
In Eq. 9, B value is calculated based on boundary conditions of related case study. In the case study of this paper, following boundary conditions are assumed,
(10)
(11)
(12)
(13)
(14)
More details about boundary conditions are also presented in Table 2.
Table 2: Comparison between experimental and analytical boundary conditions (For N.D.B region, region 1)
Boundary
Conditions Experimental
Study Computational
Study Description

Initial Temperature of application area
( )
• Average initial temperature of application area
• Average initial ground temperature along vertical length of borehole GHEs and their surrounding
• Will be changed after starting for operation
• Equal to ground initial temperature (16 oC)

• Initial temperature of whole application area (all of surfaces and boundaries shown in Fig. 4)
• Initial ground temperature along vertical length of spiral GHEs and their surrounding
• Uniform distribution
• Will be changed after starting for simulation
• Equal to ground initial temperature (16 oC)

– This temperature shows the temperature of application area before starting for any tests. Our available application area consists of two borehole GHEs. Initial temperature of application area is measured by using temperature sensors shown in Fig. 1. Records of temperature sensors show that average initial temperature is about 16 oC.
Temperature of ground in far from application area (in r-direction) • Temperature of locations in more than 6 m from application area
• Not affected during the operation of system
• Constant during operation • Constant during simulation

– Distant locations from our application area are not affected during experiment. Therefore in numerical study temperature of distant points is assumed as constant.

Temperature of borehole GHE surface (TW) • Wall temperature of borehole GHE between z=0 m and z=50 m
• Will be changed during experiment
• Average inlet fluid temperature of each GHE, ( ) : 30.75 oC
• Average outlet fluid temperature of each GHE, ( ): 29.25oC
• Volumetric flow rate: 18 L/min • Is assumed as constant and equal to average temperature of inlet and outlet fluid temperature




– Experimental results show that difference between inlet and outlet fluid temperature is less than 2 oC and the velocity of the fluid is fast enough. Therefore, in numerical study it can be reasonable to assume wall temperature of borehole GHEs as constant and equal to average temperature of inlet and outlet fluid temperature and temperature variations are negligible. This assumption is the nearest one to our experimental conditions. Long term validation of experimental and numerical study results approves the accuracy of assumptions.

By applying above given boundary conditions to Eq. 9, final dimensionless temperature distribution around case study of this paper is determined as 30:
(15)
In order to analyze HTR values per length of borehole, below simple equations are used;
(16)
By substituting dimensionless forms of radius and temperature, Eq. 16 becomes:
(17)
Final solution of HTR values per length of borehole for the case study of this paper is derived as;
(18)
Where,
(19)
(20)
Above given analytical model (Eq. 18) for HTR values of a single borehole GHE is a modified form of given model in Ref. 30.
3. RESULTS
One of the most common ways in examining and checking the accuracy of analytical models is to compare their results with experimental ones. In this study a new modified 2D analytical equation is derived for thermal performance (HTR) of a borehole by using Green’s function method (Eq.18). HTR values (Eq. 18 results) are calculated in mid-point of borehole (Z/H=0.5, Z=z/re=735.3, H=h/re=1470.6 and R=1). Fig. 5 shows experimental and analytical results of a single borehole GHE for region 1. By fitting experimental and analytical results, thermal conductivity of ground/soil can be calculated. Results show that average thermal conductivity of ground/soil around N.D.B. residence (region 1) is approximately 1.8 W m-1K-1. Both fitted curves shown in Fig. 5 confirm the accuracy of our analytical models. It is important to note that second test was done to determine the precision of analytical models (Results are obtained from two boreholes shown in green in Fig. 1a). Determining thermal conductivity of ground is very important during GSHP system’s designing process.

Fig. 5: Experimental results and fitted curve for N.D.B. residence (Eqs. 2 and 18 results)
To better show the applicability of using Eq.18 in calculating thermal conductivity of ground, thermal behavior of another region (region 2) which is located approximately 2 km far from river is analyzed under the same experimental conditions given in experimental part. Fig. 6 indicates experimental HTR values per length of borehole and fitted curve in region 2. By using analytical model (equation 18) given in this study, average thermal conductivity of ground in this region is approximately calculated as 2.32 W m-1K-1. Since this region is located near river, thermal conductivity of ground there is higher than that of N.D.B residence (Region1).

Fig. 6: Experimental results and fitted curve for region 2 (Eqs.2 and 18 results)
After validating analytical models and determining average thermal conductivity of ground long term thermal performances of a single borehole GHE are calculated for region 1(N.D.B residence region) and region 2. During a year, GSHP systems are used to provide heating and cooling demands of buildings for different periods. The system work for several days continuously and this issue highly affect borehole thermal performance. Therefore, long term performance analysis of heating and cooling source is considered as an important problem. The verification of analytical and experimental results proves the accuracy and reliability of given models. Fig. 7 gives the long term HTR values of a borehole under critical working conditions. The performance of a borehole is predicted at the end of 5-month non-stop operation. Under actual working conditions GSHP system operates intermittently Therefore, actual results are better than the results given in Fig. 7.

Fig. 7: GHE performance prediction at the end of 5 months non-stop operation (Eq.18 results)
Furthermore, in this paper temperature distribution around a borehole GHE of region 1 (N.D.B. residence) is investigated analytically. Fig. 8 shows the dimensionless temperature variations in ground around borehole for five different operation times. By increasing the value of (operation time) the temperature of ground is increased. These temperature distributions are the results of Eq. 15. Since wall temperature of equal pipe is assumed as constant ( ), the beginning point (R=0) of all curves are . In distant positions from borehole ( ), the temperature is equal to ground temperature ( ), which means that . Fig. 9 also shows the temperature variations in ground around borehole for five different R values. From this figure, temperature of ground around borehole is decreased by increasing R values. Furthermore, just like Fig. 7, in distant positions, . Temperature distribution figures can give good idea to borehole designers especially in designing procedure of more than one borehole. The approximate position of additional borehole (second, third…) can easily be estimated by using these figures and Eq. 15. Furthermore, to better show the temperature variations around single borehole in region 1, Fig. 10 is given. This figure shows dimensional temperature distribution in mid-point of borehole through A-A direction at the end of one, two and three months. Temperature is started from ground initial temperature in distant positions of borehole and then increased up to borehole wall temperature. This procedure is same for all three curves. By the operation time of GSHP system, temperature of ground around borehole is increased as shown in this figure too. This figure also shows that; additional boreholes can be drilled in more than five meters’ distance. In this case performance losses will be reduced.

Fig. 8: Temperature distribution around mid-point of borehole with respect to R (Eq.15 results)

Fig. 9: Temperature distribution around mid-point of borehole with respect to (Eq.15 results).

Fig. 10: Dimensional temperature distribution (Eq.15 results). around mid-point of borehole along A-A direction for region 1(N.D.B. residence).
T be sure that analytical procedure given in this study is reliable and to determine its accuracy, thermal conductivity of one more region is determined by fitting analytical results to the 100 hours experimental ones. Fig. 11 illustrates experimental and analytical results for region 3. Fitting of the results shows that thermal conductivity of region 3 is approximately 1.7 W m-1K-1.

Fig. 11: Experimental results and fitted curve for region 3 (Eqs.2 and 18 results)

4. CONCLUSIONS
In this paper, new modified 2D (r and z directions) analytical models are presented for thermal behavior investigation of a single borehole GHE. The models, which are coupling short and long term responses of a borehole, are derived based on Green’s function method. Since there is no fluid flow in ground, Green’s function of conduction heat transfer is used. The results of analytical model are validated by available experimental HTR values. Comparison between test and analytical results shows that, they are in good agreement. Based on validated results, long term performance of a borehole is predicted. By using analytical model given in this study, thermal conductivity of ground can be calculated easily. Steps are given below;
• GSHP application area is chosen.
• A single borehole (with single U-tube pipe) is drilled with depth h.
• Fluid is pumped into the pipes in a specific temperature.
• Inlet and outlet fluid temperatures are measured as well as volumetric flow rate for t hours.
• HTR values per length of borehole are calculated (Eq. 18).
• Results of equation 18 in this study are fitted to test ones.
• Average amount of thermal conductivity (effective value) for chosen application area is determined.
• Long term performance of a single borehole is predicted.
Another useful part of this study is temperature distribution analysis around borehole. The results show that, given analytical model for temperature variation entirely satisfies boundary conditions. In equal pipe wall element and distant positions from borehole, dimensionless temperatures are and respectively. In large application area more than one borehole is needed, therefore, temperature distribution results and function around borehole can give good idea to borehole designers in finding more appropriate drilling positions.
Newly modified presented 2D analytical models (Eqs.15 and 18) can be integrated with GSHP monitoring system and they can provide good real time performance predictor tool for GSHP applications. Sudden changes in different parameters such as environmental (weather and ground) temperature changes, extra heating or cooling demand of building and etc. highly affect performance of the borehole. Integrated performance predictor model can predict borehole performance under these sudden changes and it manages existing conditions. This study can also be expanded to 3D analysis of borehole in future.
ACKNOWLEDGEMENT
Author also would like to appreciate Mr. Nader Dehghan, Mrs. Shahnaz Asadnasab and Mr. Ata Dehghan who provided financial supports to this study.
REFERENCES
1 Esen, M., Yuksel, T. Experimental evaluation of using various renewable energy sources for heating a greenhouse, Energy and Buildings, 2013; 65, 340–351.
2 Dehghan, B. Performance assessment of ground source heat pump system integrated with micro gas turbine: Waste heat recovery, Energy Conversion and Management 2017; 152, 328-341.
3 Balbay, A., Esen, M. Experimental investigation of using ground source heat pump system for snow melting on pavements and bridge decks, Scientific Research and Essays, 2010; 24, 3955–3966.
4 Esen, H., Esen, M., Ozsolak, O. Modelling and experimental performance analysis of solar-assisted ground source heat pump system, Journal of Experimental and Theoretical Artificial Intelligence, 29; 24, 1–17.
5 Banks D. An introduction to thermogeology: ground source heating and cooling. Blackwell Publishing; 2008.
6 Hagihara , K., Iba, C., Hokoi, S., Effective use of a ground-source heat-pump system in traditional Japanese “Kyo-machiya” residences during winter, Energy and Buildings, 2016; 128, 262–269.
7 Atam, E., Helsen, L. Ground-coupled heat pumps: Part 1 – Literature review and research challenges in modeling and optimal control, Renewable and Sustainable Energy Reviews, 2016; 54, 1653–1667
8 Atam, E., Helsen, L. Ground-coupled heat pumps: Part 2 – Literature review and research challenges in optimal design, Renewable and Sustainable Energy Reviews, 2016; 54, 1668–1684
9 Kharseh, M., Al-Khawaja, M., Suleiman, M. Potential of ground source heat pump systems in cooling-dominated environments: Residential buildings, Geothermics, 2015; 57, 104–110.
10 Molavi, J., McDaniel, J. A Review of the Benefits of Geothermal Heat Pump Systems in Retail Buildings, Procedia Engineering, 2016; 145, 1135–1143.
11 Florides GA, Christodoulides P, Pouloupatis P. Single and double U-tube ground heat exchangers in multiple-layer substrates, Applied Energy, 2013; 102, 364–73.
12 Capozza, A., De Carli, M., Zarrella, A. Design of borehole heat exchangers for ground-source heat pumps: A literature review, methodology comparison and analysis on the penalty temperature, Energy and Buildings, 2012; 55, 369–379.
13 Dehghan, B., Kukrer, E. A new 1D analytical model for investigating the long term heat transfer rate of a borehole ground heat exchanger by Green’s function method, Renewable Energy 2017; 108, 615-621
14 Luo, J., Rohn, J, Xiang, W., Bertermann, D., Blum, P. A review of ground investigations for ground source heat pump(GSHP) systems, Energy and Buildings, 2016; 117, 160–175.
15 Beier, R., Spitler, J. Weighted average of inlet and outlet temperatures in borehole heat exchangers, Applied Energy, 2016; 174, 118–124.
16 Balbay, A., Esen, M. Temperature distributions in pavement and bridge slabs heated by using vertical ground-source heat pump systems, Acta Scientiarum-Technology, 2013; 35, 677–685.
17 Ruiz-Calvo, F., De Rosa, M, Monzó, P., Montagud, C., Corberán, J. Coupling short-term (B2G model) and long-term (g-function) models for ground source heat exchanger simulation in TRNSYS. Application in a real installation, Applied Thermal Engineering, 2016; 102, 720–732.
18 Conti, P., Testi, D., Grassi, W. Revised heat transfer modeling of double-U vertical ground-coupled heat exchangers, Applied Thermal Engineering, 2016; 106, 1257–1267
19 Hein, P., Kolditz, O, Gorke, U., Bucher, A., Shao, H. A numerical study on the sustainability and efficiency of borehole heat exchanger coupled ground source heat pump systems, Applied Thermal Engineering, 2016; 100, 421–433.
20 Holmberg, H., Acuna, J, Naes, E., Sønju, O. Thermal evaluation of coaxial deep borehole heat exchangers, Renewable Energy, 2016; 97, 65–76.
21 Mingzhi, Y., Tengteng, M., Kai, Z., Ping, C., Aijuan, H., Zhaohong, F. Simplified heat transfer analysis method for large-scale boreholes ground heat exchangers, Energy and Buildings, 2016; 116, 593–601.
22 Priarone, A., Fossa, M. Modelling the ground volume for numerically generating single borehole heat exchanger response factors according to the cylindrical source approach, Geothermics, 2015; 58, 32–38.
23 Selamat, S., Miyara, A, Kariya, K. Numerical study of horizontal ground heat exchangers for design optimization, Renewable Energy, 2016; 95, 561–573.
24 Wang, D., Lu, L, Cui, P. A new analytical solution for horizontal geothermal heat exchangers with vertical spiral coils, International Journal of Heat and Mass Transfer, 2016; 100, 111–120.
25 Zhang, C., Hu, S, Liu, Y., Wang, Q. Optimal design of borehole heat exchangers based on hourly load simulation, Energy, 2016; 116, 1180–1190.
26 Aydin, M., Sisman, A. Experimental and computational investigation of multi U-tube boreholes, Applied Energy, 2015; 145, 163–171.
27 Molina-Giraldo, N., Blum, P., Zhu, K., Bayer, P., Fang, Z. A moving finite line source model to simulate borehole heat exchangers with groundwater advection, International Journal of Thermal Sciences, 2011; 50, 2506–2513.
28 Gultekin, A., Aydin, M., Sisman, A. Thermal performance analysis of multiple borehole heat exchangers, Energy Conversion and Management, 2016; 122, 544–551.
29 Dehghan, B., Sisman, A., Aydin, M., Parametric investigation of helical ground heat exchangers for heat pump applications, Energy and Buildings, 2016; 127, 999–1007.
30 Dehghan, B., Aydin, A., Sisman, A., Long Term Performance Prediction of a Borehole Ground Heat Exchanger by Green’s Function Method, World Geothermal Congress April 2015, Australia-New Zealand.
31 Carslaw H.S., Jaeger J.C., Heat Conduction in Solids, Claremore Press, Oxford, 1947.
32 Ozisik N. Heat conduction second edition. USA: A Wiley Inter Science Publication John Wiley ; Sons, Inc.; 1993. Chapter III.
33 Man, Y., Yang, H., Diao, N., Liu, J., Fang Z. A new model and analytical solutions for borehole and pile ground heat exchangers, International Journal of Heat and Mass Transfer, 2010; 53, 2593–2601.
?
“Figures captions”
Fig. 1: (a) schematic top view of application area configuration of N.D.B. residence (b) Schematic view of experimental setup
Fig. 2: Thermal performance of a single borehole GHE (Eq. 2 results).
Fig. 3: schematic view of borehole (a) 2D view of Borehole (b) A-A section top view.
Fig. 4: schematic side view of equal pipe used in solving procedure
Fig. 5: Experimental results and fitted curve for N.D.B. residence (Eqs.2 and 18 results)
Fig. 6: Experimental results and fitted curve for region 2 (Eqs.2 and 18 results)
Fig. 7: GHE performance prediction at the end of 5months non-stop operation (Eq.18 results)
Fig. 8: Temperature distribution around mid-point of borehole with respect to R (Eq.15 results)
Fig. 9: Temperature distribution around mid-point of borehole with respect to (Eq.15 results).
Fig. 10: Dimensional temperature distribution (Eq.15 results). around mid-point of borehole along A-A direction for region 1(N.D.B. residence).
Fig. 11: Experimental results and fitted curve for region 3 (Eqs.2 and 18 results) ?
“Table caption”
Table 1: Parameters of application area
Table 2: Comparison between experimental and analytical boundary conditions (For N.D.B region, region 1)