The helical tube ground heat exchanger (GHE) surpasses all other types of vertical ground heat exchangers, such as the Utube, double Utube, triple Utube, double Wtube, and doubletube type ground heat exchanger in terms of thermal performance. In the present study, the performance of several helix configurations that have an external helix outlet pipe is investigated and compared with a conventional Utube ground heat exchanger. Another motive of this research is to find the best configuration in terms of heat transfer rate and pressure drop. Water is employed as the working fluid with a fixed flow rate of 2 L/min under laminar flow consideration and a constant inlet temperature of 300.15 K. To determine the optimal configuration among the 17 independent models, several performance parameters are considered, including heat exchange rate, outlet temperature, pressure drop, and the criterion of coefficient of performance (COP) improvement factor. Furthermore, the effects of using different materials for the tube and backfill on the performance of the GHE are evaluated. In a comparison study, it is found that a modified model with a 0.2 m pitch distance between the inlet and outlet pipes, but the upper portion of the outlet is kept straight for 8 m, provides the best thermal performance but consumes the highest pumping power. It has the highest average heat transfer among all configurations, which is 42.69% higher compared to Utube GHE. The average heat transfer rate is increased by 6.54% for steel as a tube material compared to polyethylene. By using concrete as the backfill, there is a 5.41% improvement of average heat transfer rate compared to silica sand. The worst thermal performance is found in the Utube ground heat exchanger.
NOMENCLATURE
Abbreviation
Symbolic
 A_{p}

Crosssectional area (m^{2})
 C_{p}

Specific heat (J/kg K)
 D

Helical diameter of the model (m)
 d_{b}

Borehole diameter (m)
 d_{h}

Hydraulic diameter (m)
 d_{i}

Inner diameter of the tube (m)
 d_{o}

Outer diameter of the tube (m)
 F

Body (volume) force (N/m^{3})
 f

Friction factor
 H

Height of the borehole (m)
 h

Heat transfer coefficient (W/m^{2 }K)
 h_{int}

Convective heat coefficient of solid (W/m^{2 }K)
 I

Identity matrix
 K

COP improvement factor
 k

Thermal conductivity (W/m K)
 L

Vertical tube height (m)
 l_{p}

Length of the straight pipe (m)
 $m\u2022$

Mass flow rate (kg/m^{3})
 P

Pressure (Pa)
 p

Helix pitch (m)
 Q_{wall}

Wall heat transfer rate (W/m)
 Re

Reynolds number
 r_{pi}

Inner radius of the tube (m)
 r_{po}

Outer radius of the tube (m)
 Q

Heat transfer rate (W)
 T

Temperature (K)
 t

Time (s)
 V

Volumetric flow rate (m^{3}/s)
 $x1$

Inlet upper part (straight) length (m)
 $x2$

Outlet upper part (straight) length (m)
 z

Wetted perimeter (m)
 ΔP

Pressure drop (Pa)
Greek symbols
Subscripts
I. INTRODUCTION
Geothermal energy is an abundantly available renewable energy source that has longterm potential to meet present global energy demand.^{1} Geothermal energy offers distinct advantages over other renewable energy sources in terms of costs, performance,^{2} steady heat supply, environmentally clean, and sustainable.^{3} Harnessing usable energy from this abundant geothermal source, geothermal heat pumps also known as ground source heat pump (GSHP) systems hold great promise for building cooling and heating.^{4} This energy source plays an important role in reducing energy consumption and environmental pollution.^{5}
Earth's surface temperature remains constant from a depth of ten to a hundred meters. Practical technology uses geothermal energy for a variety of applications, such as agricultural areas, space warming and chilling, and more.^{6} The temperature of the earth below a particular depth is generally between 10 and 20 °C, depending on climatic circumstances, and the soil qualities stay reasonably stable throughout the year.^{7} In winter, the ground temperature at a specific depth is always higher than the air temperature outside, and in summer, the ground temperature is lower than the air temperature outside. The temperature gradient between the outside air and the ground can be used for ventilation air heating or cooling in winter and summer, respectively. In a GSHP system, a ground heat exchanger (GHE) is built to exploit heat energy from the ground and coupled to the heat pump. A ground heat exchanger is often installed by digging into the ground in a horizontal or vertical position to extract heat from the earth.^{8} Ground source heat pumps (GSHPs) are efficiently used for building heating and cooling, which reduces conventional energy consumption as well as carbon intensity.^{9} Thus, the popularity of GSHPs is increasing day by day.^{10} However, the performance of a GSHP system mainly depends on the thermal performance of GHE. Therefore, many researchers focused on improving the performance of GHE by considering different configurations that are installed under the ground vertically or horizontally.
Among them, Casasso and Sethi^{11} analyzed the factors that influence the efficiency of doubletube ground heat exchangers. The findings showed that the borehole depth was the most essential factor in the arrangement of these ground heat exchangers, although the impact of the working fluid and the backfill material on heat energy transfer could not be overlooked. Similarly, Iry and Rafee^{12} analyzed the various diameter ratios along with the hydraulic and thermal performance of the ground heat exchanger in a transient state to increase the performance of the GSHP. The results demonstrated that reducing the diameter ratio increased the temperature gradient between the intake and exit temperatures. Recently, Zanchini et al.^{13} analyzed the influence of thermal shortcircuiting and flow rate on the performance of a 100 m depth coaxial ground heat exchanger. They suggested that a lowconductivity material can be used for the inner tube material, which reduces the shortcircuiting impact. Zarrella et al.^{14} found a costeffective method by comparing the helical and triple Utube configurations inside a foundation pile. According to their findings, the helical pipe continuously outperformed the triple Utube design in terms of thermal performance, and the pressure drop in the helical pipe energy pile under investigation was significantly greater—roughly 20 times—than that of the triple Utube connected in parallel. Cao et al.^{15} analyzed the thermal performance of two different Utube GHEs made of steel and polyethylene both experimentally and numerically. Both studies showed that the steel pipe had better thermal performance compared to polyethylene pipe. Saeidi et al.^{16} numerically investigated a spiral GHE for lowdepth and wide boreholes. It was found that the aluminum fins increased the heat transfer rate as well as increased the thermal performance of the spiral GHE. In an analogous study, Serageldin et al.^{17} found that spiral double GHEs had 40.8% better thermal performance than single Utube. Cui et al.^{18} performed a shortterm simulation of GHE's different operation modes. They observed that operating the GHEs in discontinuous operation mode as well as the alternate cooling/heating mode efficiently reduced heat generation in the surrounding soil. Spitler et al.^{19} evaluated the thermal performance of a single wellinstrumented groundwaterfilled borehole with a single Utube at a depth of 80 m. Hasan et al.^{20} numerically investigated a doubletube vertical GHE. They concluded that when the heat transfer rate and pressure drop were balanced, doubletube vertical GHE performed better in laminar flow. They also showed that reducing the inlet and outlet diameters of GHEs did not substantially impact the heat transfer rate in the laminar flow mode. Tarnawski et al.^{21} investigated a 200 m^{2} villa's GSHP system with horizontal GHEs computationally. Their research revealed that horizontal GHE installation was suitable for home and industrial purpose in northern Japan. Mehrizi et al.^{22} analyzed numerically and compared the efficiency of Wtube and Utube geothermal heat exchangers in a cooling operation mode. The results showed that arranging six series Utube pipes of a Wtube ground heat exchanger gave the highest thermal efficiency. Recently, Aresti et al.^{23} determined that the thermal conductivities of borehole backfill materials must be close to the thermal conductivity of the earth to achieve maximum efficiency. Nalla et al.^{24} investigated the impacts of flow rate, borehole diameter, outer and inner pipe diameters, drilling depth, grout, and pipe materials on the performance of the deep borehole heat exchanger. In an analogous study, Aydın et al.^{25} investigated the effects of Utube numbers on both borehole performance and initial cost. They highlighted that double and triple Utube configurations had 14% and 25% better performance, respectively, compared to the single Utube.
Numerous studies were conducted on helical tube ground heat exchangers due to their superior thermal performance. Among them, Luo et al.^{26} compared the performances of various types of geothermal heat exchangers, including Wtube, helical, double, and triple Utube. The results showed that the triple Utube ground heat exchanger had the maximum efficiency and the highest cost reduction. Dehghan^{27} determined the heat transfer rate and thermal interference of various numbers of helical geothermal heat exchangers positioned at various distances. The result suggested that the number of helical ground heat exchangers and the distance between them should be 6–9 m for better thermal performance. Similarly, Yang et al.^{28} investigated the impacts of several parameters including inlet water temperature, intermittent operation mode, spiral pitch, and pile material on the thermal performance of an energy pile having a spiral ground heat exchanger. They found that the heat transfer rate of spiral tube ground heat exchangers was improved by increasing the inlet water flow rate. Additionally, the results of their research showed that the soil temperature growth could be maintained effectively by intermittent operation resulting in an improvement in the thermal performance. However, Congedo et al.^{29} conducted a computational study on a horizontal helical ground heat exchanger. They compared this design to linear and slinky ground heat exchangers in both winter and summer. They concluded that the helical heat exchanger offered the best thermal performance. Etghani and Baboli^{30} investigated the numerical model of a shell and helical tube heat exchanger to estimate the heat transfer coefficient exergy loss. Four design factors, which were more important for heat exchanger performance, were considered: pitch coil, tube diameter, and hot and cold flow rates. According to the findings, tube diameter and cold flow rate were the most important design factors for heat transfer and exergy loss, respectively. Carotenuto et al.^{31} investigated the effects of operational and design parameters on the performance of typical Utube GHEs. They observed that the thermal conductivity of backfill materials and fluid flow rate could increase the performance of a GHE by 42% and 15%, respectively. Interestingly, the finding also showed that by reducing helix pitch by 64%, thermal performance could be increased by 68%. Later, Suzuki et al.^{32} performed numerical and experimental investigations of a spiral tube GHE in the heating mode. Recently, Javadi et al.^{33} compared eight new types of helical ground heat exchangers with a single Utube heat exchanger to find the best one. The results showed that the triple helix heat exchanger had the best thermal performance.
Jalaluddin and Miyara^{34} performed a numerical analysis of Utube and spiral tube GHEs, determining that decreasing pitch led to a notable increase in pressure drop. Zhao et al.^{35,36} thoroughly explored various vertical GHE configurations, including single, double, and triple Utube, W, and spiral shapes and found that spiralshaped GHEs consistently provided the highest heat transfer rates. In addition, Dehghan^{37} numerically assessed the performance of several spiral GHEs and identified that a configuration with nine spiral GHEs gained the best thermal performance. Wang et al.^{38} and Leroy and Bernier^{39} introduced new approaches to evaluate the heat transmission effectiveness of spiral GHEs. Ahmadi^{40} investigated a doubletube heat exchanger with a spiral turbulator, altering both the spiral pitch and radius. The study concluded that improved fluid mixing at lower pitch led to enhanced heat transmission. Additionally, corrugated pipes were found to boost the thermal performance of heat exchangers relative to smooth tubes.^{41} More recently, Hasan et al.^{42} employed a novel method by using variablepitched spiral GHEs and adjusting the placement of the exit straight tube to diminish thermal interference. Table I provides key details about significant previous research on various types of vertical spiral tube GHEs.
Author(s) .  Methods .  Helical pitch dimension (m) .  Spiral pitch distance .  Flow condition .  Remarks .  

Flow Type .  Inlet temperature (K) .  
Jalaluddin and Miyara^{34}  Numerical  0.05, 0.1, 0.2  Uniform  Laminar  300.15  Spiral tube GHEs were analyzed and contrasted to Utube GHEs. The spiral GHEs with lower pitch demonstrated the highest heat transmission and pressure drop. 
Zhao et al.^{35}  Numerical  0.25, 0.5, 1, 2  Uniform  Turbulent  ⋯  The effect of varying spiral pitch on the thermal performance of spiral GHEs was studied, showing improved performance at lower pitch. 
Zhao et al.^{36}  Experimental and numerical  0.25  Uniform  Laminar  301.07  A relative investigation was conducted on the performances of Ushaped, Wshaped, and spiralshaped GHEs, revealing that the spiralshaped GHE provided the best performance under similar conditions. 
Dehghan^{37}  Numerical  0.1  Uniform  Turbulent  270.25  The overall performance of nine spiral tube GHEs was evaluated. Results indicated that thermal performance increased with higher vertical lengths, even though this led to greater construction expenses. 
Wang et al.^{38}  Numerical  1  Uniform  ⋯  ⋯  An analytical method utilizing the Laplacian approach was formed to study spiral GHEs. 
Leroy and Bernier^{39}  Numerical  0.4  Uniform  Turbulent  280.15  A novel helical coil GHE model was proposed to incorporate axial heat transfer effects. 
Luo et al.^{26}  Experimental  0.3  Uniform  Turbulent  278.15  DoubleU, tripleU, doubleW, and spiral tube GHEs were examined. The spiral configuration showed a 15% improvement in performance compared to the W configuration. 
Carotenuto et al.^{31}  Numerical  0.1, 0.25, 0.5, 0.7  Uniform  Turbulent  308.28  The performance of single, double, and triple Utube GHEs as well as spiral GHEs was analyzed. It was found that spiral GHEs with very small pitch led to an greater pressure drop. 
Dehghan^{27}  Experimental and numerical  0.1  Uniform  Turbulent  323.22  Different types of uniformly pitched spiral GHEs were analyzed. The arrangement with nine spiral GHEs achieved the maximum heat transmission rate. 
Ahmadi^{40}  Numerical  0.05, 0.1, 0.25, 0.5  Uniform  Turbulent  333, 293  Various spiral GHE configurations were examined by altering the spiral pitch and radius. A reduction in pitch led to improved heat transfer, attributed to better fluid mixing. 
Hasan et al.^{42}  Numerical  0.10, 0.11, 0.17, 0.19, 0.20, 0.25, 0.29, 0.45  Variable  Laminar  300.15  Variablepitched spiral GHEs were employed to enhance performance, and the position of the outlet straight pipe was changed to reduce thermal interference with the inlet spiral pipe. Optimization efforts also focused on GHE pipe material, backfill, and ground soil material. Furthermore, the impacts of different flow rates on thermal performance and pressure drop were analyzed. 
Serageldin et al.^{17}  Numerical  0.05,0.1,0.15,0.2  Variable  Laminar  300  This study focuses on parametric analysis, compares thermohydraulic performance through simulations, and optimizes design to increase efficiency and installation. 
Present study  Numerical  0.2, 0.3, 0.4  Variable  Laminar  300.15  The present study simultaneously employs both helical and straight pipes as the inlet and outlet of the GHEs and optimizes key design parameters, including pitch and straight pipe length, to obtain optimal thermal performance. Moreover, optimal material selection is also performed. Additionally, the selection of optimal GHE materials is carried out. 
Author(s) .  Methods .  Helical pitch dimension (m) .  Spiral pitch distance .  Flow condition .  Remarks .  

Flow Type .  Inlet temperature (K) .  
Jalaluddin and Miyara^{34}  Numerical  0.05, 0.1, 0.2  Uniform  Laminar  300.15  Spiral tube GHEs were analyzed and contrasted to Utube GHEs. The spiral GHEs with lower pitch demonstrated the highest heat transmission and pressure drop. 
Zhao et al.^{35}  Numerical  0.25, 0.5, 1, 2  Uniform  Turbulent  ⋯  The effect of varying spiral pitch on the thermal performance of spiral GHEs was studied, showing improved performance at lower pitch. 
Zhao et al.^{36}  Experimental and numerical  0.25  Uniform  Laminar  301.07  A relative investigation was conducted on the performances of Ushaped, Wshaped, and spiralshaped GHEs, revealing that the spiralshaped GHE provided the best performance under similar conditions. 
Dehghan^{37}  Numerical  0.1  Uniform  Turbulent  270.25  The overall performance of nine spiral tube GHEs was evaluated. Results indicated that thermal performance increased with higher vertical lengths, even though this led to greater construction expenses. 
Wang et al.^{38}  Numerical  1  Uniform  ⋯  ⋯  An analytical method utilizing the Laplacian approach was formed to study spiral GHEs. 
Leroy and Bernier^{39}  Numerical  0.4  Uniform  Turbulent  280.15  A novel helical coil GHE model was proposed to incorporate axial heat transfer effects. 
Luo et al.^{26}  Experimental  0.3  Uniform  Turbulent  278.15  DoubleU, tripleU, doubleW, and spiral tube GHEs were examined. The spiral configuration showed a 15% improvement in performance compared to the W configuration. 
Carotenuto et al.^{31}  Numerical  0.1, 0.25, 0.5, 0.7  Uniform  Turbulent  308.28  The performance of single, double, and triple Utube GHEs as well as spiral GHEs was analyzed. It was found that spiral GHEs with very small pitch led to an greater pressure drop. 
Dehghan^{27}  Experimental and numerical  0.1  Uniform  Turbulent  323.22  Different types of uniformly pitched spiral GHEs were analyzed. The arrangement with nine spiral GHEs achieved the maximum heat transmission rate. 
Ahmadi^{40}  Numerical  0.05, 0.1, 0.25, 0.5  Uniform  Turbulent  333, 293  Various spiral GHE configurations were examined by altering the spiral pitch and radius. A reduction in pitch led to improved heat transfer, attributed to better fluid mixing. 
Hasan et al.^{42}  Numerical  0.10, 0.11, 0.17, 0.19, 0.20, 0.25, 0.29, 0.45  Variable  Laminar  300.15  Variablepitched spiral GHEs were employed to enhance performance, and the position of the outlet straight pipe was changed to reduce thermal interference with the inlet spiral pipe. Optimization efforts also focused on GHE pipe material, backfill, and ground soil material. Furthermore, the impacts of different flow rates on thermal performance and pressure drop were analyzed. 
Serageldin et al.^{17}  Numerical  0.05,0.1,0.15,0.2  Variable  Laminar  300  This study focuses on parametric analysis, compares thermohydraulic performance through simulations, and optimizes design to increase efficiency and installation. 
Present study  Numerical  0.2, 0.3, 0.4  Variable  Laminar  300.15  The present study simultaneously employs both helical and straight pipes as the inlet and outlet of the GHEs and optimizes key design parameters, including pitch and straight pipe length, to obtain optimal thermal performance. Moreover, optimal material selection is also performed. Additionally, the selection of optimal GHE materials is carried out. 
From the aforementioned literature studies, it can be concluded that helical tube ground heat exchangers can achieve higher energy efficiency due to their improved heat transfer capabilities. Previous studies on the vertical helical tube GHE primarily focused on configurations where helical coils function as the inlet pipe, and the straight pipe serves as the outlet, either positioned inside or outside the helical coil. It is important to note that thermal interference between the inlet and exit tubes can significantly influence the overall system efficiency. However, among existing literature, no studies exclusively investigated how to reduce thermal interference. Moreover, prior studies primarily focused on enhancing helical coil configurations without considering how varying the length of straight pipes at both the inlet and outlet positions impacts thermal performance and interference. To address these research gaps, this study provides an indepth analysis of helical tube GHEs, simultaneously employing both helical and straight pipes as the inlet and outlet of the GHEs and optimizing key design parameters, including pitch and straight pipe length, to obtain optimal thermal performance. Hence, this study optimizes helical tube GHE performance, resulting in more efficient and sustainable geothermal energy use.
II. PHYSICAL MODEL
The physical model along with different geometric parameters is illustrated in Fig. 1. The performance of a GHE depends on several parameters, such as GHE configurations, pipe materials and dimension, grout material, length and depth of GHE installation, soil conditions, working fluids, etc.^{7,23} Therefore, it is a biggest challenge to select appropriate parameters for the GHE design. Additionally, many previous studies focused mainly on improving helical coil configurations, but there is a lack of analysis of how varying the length of straight pipes at both the inlet and outlet positions affects thermal performance and reduces thermal interference. This research aims to fill these gaps by offering an indepth study of helical tube GHEs, focusing on employing both helical and straight pipes simultaneously as the inlet and outlet of the GHEs and optimizing key design parameters, including pitch and straight pipe length, to optimize thermal performance while reducing thermal interference. From the several prior literature studies as summarized in Table I, the technical parameters for present simulation study on helical GHE are assumed based on typical values from study conducted by Jalaluddin and Miyara.^{34} After that, the traditional helical GHE is modified as listed in Table II.
GHE models .  Inlet upper part (straight) length, x_{1} (m) .  Outlet upper part (straight) length, x_{2} (m) .  Inlet pipe helix pitch, p_{1}(m) .  Outlet pipe helix pitch, p_{2} (m) . 

Model 1 (Traditional Utube)  19.5  19.5  0  0 
Model 2  0  0  0.2  0.2 
Model 3  0  0  0.2  0.3 
Model 4  0  0  0.3  0.3 
Model 5  0  0  0.3  0.4 
Model 6  0  0  0.4  0.4 
Model 7  5  5  0.2  0.2 
Model 8  5  5  0.2  0.3 
Model 9  5  5  0.2  0.4 
Model 10  5  5  0.3  0.3 
Model 11  5  5  0.3  0.4 
Model 12  0  8  0.2  0.2 
Model 13  0  10  0.2  0.2 
Model 14  0  12  0.2  0.2 
Model 15  5  10  0.2  0.2 
Model 16  5  19.5  0.2  0 
Model 17  10  10  0.2  0.2 
GHE models .  Inlet upper part (straight) length, x_{1} (m) .  Outlet upper part (straight) length, x_{2} (m) .  Inlet pipe helix pitch, p_{1}(m) .  Outlet pipe helix pitch, p_{2} (m) . 

Model 1 (Traditional Utube)  19.5  19.5  0  0 
Model 2  0  0  0.2  0.2 
Model 3  0  0  0.2  0.3 
Model 4  0  0  0.3  0.3 
Model 5  0  0  0.3  0.4 
Model 6  0  0  0.4  0.4 
Model 7  5  5  0.2  0.2 
Model 8  5  5  0.2  0.3 
Model 9  5  5  0.2  0.4 
Model 10  5  5  0.3  0.3 
Model 11  5  5  0.3  0.4 
Model 12  0  8  0.2  0.2 
Model 13  0  10  0.2  0.2 
Model 14  0  12  0.2  0.2 
Model 15  5  10  0.2  0.2 
Model 16  5  19.5  0.2  0 
Model 17  10  10  0.2  0.2 
The physical problem of typical helical GHE consists of helical and straight pipe sections inserted in the grout, all of which are surrounded by soil. The helical diameter (D) and height (H) of the model are 0.1398 and 19.5 m, respectively. The inner and outer diameters of the tube are kept at 33 and 26 mm, respectively. The centertocenter distance between the inlet and outlet tubes is maintained at 200 mm. The borehole diameter is kept to be 400 mm, with a surrounding soil diameter of 5 m. Polyethylene is chosen as the pipe material, silica sand as the backfill material, and sandy clay as the soil material.
In this research work, a helical ground heat exchanger is modified in different configurations, and a comparison study is to be done to choose the best one. A traditional Utube GHE is modified by using a helical pipe as portrayed in Fig. 1. The different geometric parameters of the modified models and thermophysical properties of different materials are listed in Tables II and III, respectively.
Parameters .  Value .  Unit . 

General dimensions of helical GHE  
Inner diameter  26  mm 
Outer diameter  33  mm 
Pitch  0.2  m 
Helix diameter  0.1398  m 
Borehole/GHE length  20  m 
Polyethylene (tube materials)  
Thermal conductivity  0.35  Wm^{−1 }K^{−1} 
Specific heat  2300  Jkg^{−1 }K^{−1} 
Density  920  kg m^{−3} 
Highdensity polyethylene (tube materials)  
Thermal conductivity  0.45  Wm^{−1 }K^{−1} 
Specific heat  2300  Jkg^{−1 }K^{−1} 
Density  950  kg m^{−3} 
Steel (tube materials)  
Thermal conductivity  16.27  Wm^{−1 }K^{−1} 
Specific heat  502.48  Jkg^{−1 }K^{−1} 
Density  8030  kg m^{−3} 
Silica sand (backfill materials)  
Thermal conductivity  1.4  Wm^{−1 }K^{−1} 
Specific heat  750  Jkg^{−1 }K^{−1} 
Density  2210  kg m^{−3} 
Concrete pile (backfill materials)  
Thermal conductivity  1.65  Wm^{−1 }K^{−1} 
Specific heat  1000  Jkg^{−1 }K^{−1} 
Density  2200  kg m^{−3} 
Gravel (backfill materials)  
Thermal conductivity  0.5  Wm^{−1 }K^{−1} 
Specific heat  1200  Jkg^{−1 }K^{−1} 
Density  1680  kg m^{−3} 
Sandyclay (ground materials)  
Thermal conductivity  2.1  Wm^{−1 }K^{−1} 
Specific heat  1200  Jkg^{−1 }K^{−1} 
Density  1960  kg m^{−3} 
Parameters .  Value .  Unit . 

General dimensions of helical GHE  
Inner diameter  26  mm 
Outer diameter  33  mm 
Pitch  0.2  m 
Helix diameter  0.1398  m 
Borehole/GHE length  20  m 
Polyethylene (tube materials)  
Thermal conductivity  0.35  Wm^{−1 }K^{−1} 
Specific heat  2300  Jkg^{−1 }K^{−1} 
Density  920  kg m^{−3} 
Highdensity polyethylene (tube materials)  
Thermal conductivity  0.45  Wm^{−1 }K^{−1} 
Specific heat  2300  Jkg^{−1 }K^{−1} 
Density  950  kg m^{−3} 
Steel (tube materials)  
Thermal conductivity  16.27  Wm^{−1 }K^{−1} 
Specific heat  502.48  Jkg^{−1 }K^{−1} 
Density  8030  kg m^{−3} 
Silica sand (backfill materials)  
Thermal conductivity  1.4  Wm^{−1 }K^{−1} 
Specific heat  750  Jkg^{−1 }K^{−1} 
Density  2210  kg m^{−3} 
Concrete pile (backfill materials)  
Thermal conductivity  1.65  Wm^{−1 }K^{−1} 
Specific heat  1000  Jkg^{−1 }K^{−1} 
Density  2200  kg m^{−3} 
Gravel (backfill materials)  
Thermal conductivity  0.5  Wm^{−1 }K^{−1} 
Specific heat  1200  Jkg^{−1 }K^{−1} 
Density  1680  kg m^{−3} 
Sandyclay (ground materials)  
Thermal conductivity  2.1  Wm^{−1 }K^{−1} 
Specific heat  1200  Jkg^{−1 }K^{−1} 
Density  1960  kg m^{−3} 
III. NUMERICAL PROCEDURE AND BOUNDARY CONDITIONS
A. Numerical procedure and boundary conditions
Based on the flow rate of 2 L/min, tube inner diameter, and other parameters, the corresponding Reynolds number is 1829 at 25 °C, which indicates the flow is laminar. From Table I, although several many researchers^{27,31,35,37,39,40} conducted numerical studies on helical/spiral GHE based on turbulent flow assumption, there are numerous existing studies,^{17,34,36,42} where laminar flow assumption was employed for the simplicity and to reduce computational time. Furthermore, the present simulation model has been validated by comparing the results of numerical study^{34} and experimental study.^{46} Therefore, as the Reynolds number (Re = 1900) is low, simulations have been conducted using laminar assumption to investigate helical GHEs with different geometric configurations, aiming to save computational time. The temperature of the inlet water is maintained at 300.15 K. The initial ground temperature is considered to be 290.85 K. The inlet velocity is kept at 0.0625 m/s corresponding to a mass flow rate of 2 L/min, and at the outlet, the outflow condition is considered. To generalize these findings for different flow rates and temperatures, empirical correlations can be applied, which relate performance to varying conditions. The temperature at the top of the borehole and the ground is assumed to be the same as the ambient temperature. The ground surrounding temperature and the bottom temperature are set at 290.85 K. In the simulation setup, the pressurebased transient option is applied for all cases. Gravity acts in the downward (−y) direction. Inlet and outlet are considered as mass flow inlet and outlet, respectively. For the initialization of the solution, hybrid initialization is applied.
Due to the asymmetric nature of all geometric configurations in this study, it is not feasible to conduct numerical simulations in two dimensions. Therefore, the numerical problems in this study are solved threedimensionally. Specifically, threedimensional Navier–Stokes and thermal energy equations are solved to obtain numerical solutions. As the model involves both the solid and liquid interfaces for transferring heat, the solver is set to simulate it as a problem of conjugate heat transfer. The geothermal energy transfer process involves both convectiondominant and conductiondominant.^{43} Details of heat transfer mechanisms between the working fluid, GHE pipe, and the grout in a GHE system are shown in Fig. 2. The heat transfer in the fluid flow region is primarily influenced by convection, while in the solid regions of grout and ground soil, the heat transfer is governed by low thermal conduction coefficient of grout and ground soil.^{44}
Since the fluid to the pipe wall forms the interface between the two regions, it is known as the “twosided region.” The interface is coupled where the thermal conditions are applied, the wall thickness is given in boundary conditions, and the solver calculates the heat transmission. All of the GHE models listed in Table II have been simulated in a cooling mode of continuous operation for 72 h.
B. Governing equations
The numerical simulations of different configurations of helical GHE were conducted by using the CFD software ANSYS FLUENT. The computational domain of helical GHE consists of the fluid flow inside the pipe, materials of the pipe, backfill, and ground soil. FLUENT solves the problem by considering the following conservation of mass, momentum, and energy equations.^{20,42,45} These equations describe water flow inside the tube, the materials of the tube, the borehole, and the ground soil. Since water is used as a working fluid, the fluid flow is incompressible. ANSYS FLUENT solves these equations in the following form.
C. Mesh element refinement assessment
The mesh has been generated using the ANSYS Meshing tool. Fine meshing is utilized in proximity to the boundary regions between fluid and solid contact surfaces to avoid the creation of singular points and better grasp the faster alteration of unknown variables. Body and edge sizing has been done on the fluid and solid regions in the computational domain. The mesh size of the grout is set to be larger compared to the mesh size of the fluid region to reduce the number of mesh elements as well as the computation time.
A grid refinement test is performed to avoid the longer computation time and to ensure that the simulation is independent of mesh size, which will provide accurate and stable results. For a helical GHE, four Grid systems of meshes with different element sizes are generated, ranging from coarse to fine. The Grid1 system consists largest element sizes, including the lowest element number of 9 527 822, and then, the element sizes gradually decreased to generate a finer mesh with the smallest element sizes in the Grid4 system consisting of 13 274 691 element number as shown in Table IV. Then, simulations are conducted for these four types of meshes. The inlet temperature for the model is assumed to be 300.15 K, the flow rate is assumed 2 L/min, and the GHE is operated for 72 h. From Table IV, it is seen that the obtained outlet temperature differences between subsequent Grid systems decrease with the increases in element number, and there is no significant difference in temperature between 11 380 456 and 13 274 691 elements systems. Therefore, the 11 380 456 number of elements system is considered in this study, which provides a reasonable balance between computational time and accuracy. Finally, Fig. 3 shows the optimal mesh pattern, and for all proposed models, a similar technique of meshing is employed during simulation.
Grid configuration .  Fluid zone element size (mm) .  Grout element size (mm) .  Number of elements .  Outlet temperature (K) . 

Grid1  8  250  9 527 822  297.79 
Grid2  6  200  10 235 784  297.49 
Grid3 (optimum)  4  150  11 380 456  297.31 
Grid4  2  100  13 274 691  297.26 
Grid configuration .  Fluid zone element size (mm) .  Grout element size (mm) .  Number of elements .  Outlet temperature (K) . 

Grid1  8  250  9 527 822  297.79 
Grid2  6  200  10 235 784  297.49 
Grid3 (optimum)  4  150  11 380 456  297.31 
Grid4  2  100  13 274 691  297.26 
D. Model validation
To ensure the accuracy of the computational approach used, it is essential to validate the findings against previously published studies under similar conditions. The present simulation model of spiral GHE with a 0.2 m pitch interval is validated by comparing it with the findings of Miyara and Jalaluddin^{34} before performing the parametric simulations of other models. For validating the model, the geometric parameters are kept the same, and the properties of pipe materials, borehole materials, ground materials, and boundary conditions are taken according to the paper.^{34} The pipe material is polyethylene, the backfill material is silica sand, and the ground material is sandy clay; the pitch is kept at 200 mm, and the borehole depth is kept at 20 m. The ground temperature and the inlet temperature of the fluid are maintained at 290.85 and 300.15 K, respectively. Figures 4 and 5 illustrate the variations in outlet water temperature and heat transfer rate, respectively, after 24, 48, and 72 h of continuous operation at a flow rate of 2 L/min. From Fig. 4, it is seen that the current simulated results closely match the published data.^{34}
On the other hand, there is a slight deviation in the heat transfer rate of the present study and of Ref. 34. The error on heat transfer between the current model and the published study^{34} can be calculated using the formula: Error (%) = Q_{c} − Q_{p}×100/Q_{p}, where Q_{c} and Q_{p} denote the current model and the published results, respectively. The results of Fig. 5 indicate that the % error on heat transfer rates of the current simulation model and those reported by Jalaluddin and Miyara^{34} is 0.59%, 1.92%, and 2.03% for operation time after 24, 48, and 72 h, respectively. The reasons for the variations in the results of heat transfer are mesh sensitivity, errors from different computers used in the present study, and may be for different versions of ANSYS FLUENT software. Therefore, from the nearly identical values of exit temperature and heat transmission rate, the results confirm the validity of the present numerical method.
Additionally, Jalaluddin et al.^{46} conducted an experimental investigation on a spiral pipe GHE having a flow rate of 3.6 L/min and an inlet water temperature of 35 °C. Figure 6 compares the outlet water temperature variations from the present approach with the experimental results obtained by Jalaluddin et al.^{46} The observed patterns in these graphs show good agreement with the findings from the prior study, further supporting the validity of the current numerical approach. Nevertheless, the differences observed in Fig. 6 are due to inherent uncertainties and changes in experimental conditions.
IV. RESULTS AND DISCUSSION
All GHEs are operated continuously in cooling mode for 72 h. The performance parameters considered in this study are outlet temperature, heat exchange rate, and pressure drop of water flow in the GHEs. The thermophysical properties of fluid and other materials are assumed to be the same for all models.
A. Variation of the outlet temperature in terms of time
The performance of the GHEs depends upon the difference between the inlet and outlet circulating water temperatures. A large temperature difference denotes higher thermal performance. After continuous operation of 72 h, the outlet temperature of fluid for all models is listed in Table IV.
Due to the interaction between the working fluid (water) and the ground, heat transfer occurs. In the cooling mode, the water inside the ground heat exchanger has a higher inlet temperature than the ground temperature. Hence, the water releases heat to the ground and reduces its temperature gradually. On the other hand, due to heat transfer, the ground temperature gets higher. After a certain period, the rate of heat transfer starts to reduce, which increases the outlet temperature. The outlet water temperature is substantially increased at first because the backfill materials and ground surface are unaffected at that time as shown in Fig. 7. After 72 h of continuous operation, model 12 has the lowest outlet temperature among all the models. It is due to the large heat transfer area; therefore, the ground does not become saturated rapidly. As a result, greater heat exchange occurs between the ground and the working fluid for both the inlet and outlet tubes. The temperature variation is minimal for model 1 because it occupies the smallest area on the ground since its inlet and outlet pipe are straight, whereas the other models have a considerable impact on the ground. Because of their helical design that enhances the heat transmission area, all of the modified models have a lower output temperature than the Utube GHEs.
Table V shows that after 24 h, the outlet temperature is lower for all models, indicating that rapid heat transfer occurs between the water inside the ground heat exchanger and the ground. Since the soil temperature increases with operational time, the heat transfer rate decreases gradually, resulting in higher outlet temperature after 72 h compared to 24 and 48 h of operation. Model 12 outperforms all other models in terms of outlet temperature providing a value of 295.92 K after 72 h of continuous operation.
GHE models .  Outlet temperature (K) .  

After 24 h .  After 48 h .  After 72 h .  
Model 1  296.78  297.09  297.26 
Model 2  295.44  295.84  296.07 
Model 3  295.6  295.99  296.2 
Model 4  295.59  295.97  296.16 
Model 5  295.70  296.06  296.25 
Model 6  295.88  296.21  296.39 
Model 7  295.80  296.14  296.32 
Model 8  295.53  295.90  296.10 
Model 9  295.58  295.94  296.14 
Model 10  295.75  296.10  296.29 
Model 11  295.70  296.06  296.25 
Model 12  295.29  295.70  295.92 
Model 13  295.36  295.77  295.99 
Model 14  295.54  295.91  296.10 
Model 15  295.54  295.92  296.12 
Model 16  295.55  295.96  296.17 
Model 17  296.04  296.37  296.55 
GHE models .  Outlet temperature (K) .  

After 24 h .  After 48 h .  After 72 h .  
Model 1  296.78  297.09  297.26 
Model 2  295.44  295.84  296.07 
Model 3  295.6  295.99  296.2 
Model 4  295.59  295.97  296.16 
Model 5  295.70  296.06  296.25 
Model 6  295.88  296.21  296.39 
Model 7  295.80  296.14  296.32 
Model 8  295.53  295.90  296.10 
Model 9  295.58  295.94  296.14 
Model 10  295.75  296.10  296.29 
Model 11  295.70  296.06  296.25 
Model 12  295.29  295.70  295.92 
Model 13  295.36  295.77  295.99 
Model 14  295.54  295.91  296.10 
Model 15  295.54  295.92  296.12 
Model 16  295.55  295.96  296.17 
Model 17  296.04  296.37  296.55 
B. Analysis of heat exchange rate
Using the helical pipe in the borehole increases the heat transfer rate per meter of borehole depth compared with that of a straight pipe. The heat transfer rate of the helical pipe increases with a reduced helix pitch.^{34} Figure 8 demonstrates that higher heat transfer between the water and soil occurs at the very beginning since the temperature difference is high between the ground and the working fluid. Therefore, the water releases more heat to the ground soil at the beginning of operation in the cooling mode, and the heat transfer rate rapidly decreases until about 10 h of operation. After that, the heat transfer rate decreases slowly and tends to be almost steady with the increases in operation time. This happens because with the operation time, the water continuously releases heat to the ground soil, and hence, the surrounding ground soil thermal energy degraded (heat is developed in the soil) the ground soil temperature around the GHE, which increases gradually. As a result, the temperature difference between water inside the GHE and ground soil around the GHE decreases and reduces the heat transfer rate.
For model 1, the heat transfer rate is the lowest among all the models. This happens because of lesser circulating residence time and a smaller heat exchange area than the other models. It has the highest average heat transfer among all configurations, which is 42.69% higher compared to Utube GHE. On the other hand, model 12 exchanges more heat than the other models, and within the operation time average, it exchanges 201.93 W higher heat compared to the Utube GHE (model 1). This is because model 12 has the highest heat exchange area. Moreover, the inlet and outlet have a 200 mm helical pitch spacing, and the upper portion of the outlet is kept 8 m straight to avoid the atmospheric effects on the outlet water temperature. After a prolonged operational time, the surrounding soil gets saturated due to the helix conformation as well as the reduced helix pitch distance. To avoid this, the upper half of the outlet is kept straight, resulting in improved performance. The heat transfer rate after 24, 48, and 72 h of continuous operation for all the models is shown in Fig. 9, which shows the operational time of after every 24 h. Models 12, 11, and 2 have significant heat transfer rates compared to the others.
It is worth mentioning that at the beginning of the operation, a higher heat transfer occurs, and after the first 24 h of operation, the heat transfer rate is almost at a constant level, as shown in Fig. 9. For model 12, after 48 and 72 h of operation, the difference in the heat transfer rate with the 24 h of operation is 57 and 87 W, respectively. From Fig. 10, it can be seen that the modified helical GHE models have a greater average heat transfer rate than the Utube ground heat exchanger. It indicates that the working fluid gets adequate time to interact with the surrounding soil to exchange heat, and due to the helical configuration, the heat exchange area is large enough, which results in the longer time required to get the surrounding soil saturated. The highest average heat exchange rate is 674.98 W for model 12, whereas the lowest heat exchange rate is 473.05 W for model 1, which is 42.69% (1.43 times) higher as compared to model 1 (Utube).
C. Analysis of pressure drop
From Fig. 11, it is seen that the highest pressure drops occur for model 2 since it has a low helix pitch indicating the highest number of turns. As helix pitch is inversely proportional to the number of turns, model 2 has the highest number of turns among all models, which results in a higher pressure drop. The increased number of turns creates a high resistance that restricts the smooth flow of the working fluid, necessitating the use of more pumping power to provide it. Due to its straight pipe, model 1 has the lowest pressure drop, which is 119.5 Pa, whereas model 2 has the highest pressure drop, which is 775.28 Pa, which is nearly 6.5 times greater than model 1. Model 2 has the highest pressure drop since the helix pitch (200 mm) is the same for both the inlet and outlet. Model 12 has the secondhighest pressure drop, which is 730.3 Pa. It has the secondhighest number of turns; therefore, a substantial amount of pumping power is required to overcome the restriction.
Models 3–11 and 13–17 have lower turns than the models 2 and 12 resulting in a comparatively lesser pressure drop. The number of turns is kept almost the same for models 8, 9, and 11, resulting in a pressure drop of equal magnitude. The geometric parameters for models 2 and 12 are nearly the same; however, in model 12, the upper portion of the outlet tube is retained 8 m straight, which reduces pressure drop, and the pressure drop difference between these two is nearly 45 Pa. Model 16 has the lowest pressure drop among the modified helical configurations. For model 16, the outlet pipe is straight, whereas the inlet pipe has a helical pitch of 200 mm. The difference between the two is that model 1 has the entire pipe helical, whereas model 12 has the bottom 14.5 m designed as helical, while the remainder of the upper portion is straight. It is investigated that by increasing the helix pitch as well as reducing the turns, the pressure drops can be decreased, which indicates that less pumping power is required to push the working fluid through the helical tube.
This study focuses only on the laminar flow regime, following insights from Jalaluddin and Miyara.^{34} They examined how varying flow rates affect the heat exchange rate and the pressure drop. Their observations revealed that in a spiral tube GHE with a 0.05 m pitch, increasing the flow rate from the laminar regime (2 L/min) to the turbulent regime (8 L/min) resulted in only a 1.5fold increase in the heat exchange rate. However, the pressure drop throughout the GHE increased by six times. Since pumping power is proportional to pressure drop, a significant amount of pumping power would be required to operate the GHE in a turbulent regime. However, the heat exchange rate did not increase significantly in this regime. Therefore, based on this specific finding, this study aims to maximize thermal performance by modifying geometric configurations while keeping the flow rate at the laminar regime (2 L/min). Based on the prior studies,^{34,48} it is evident that the heat transfer rate and the pressure drop for a specific GHE configuration remain consistent across different flow rates. For example, lowerpitched GHEs consistently exhibited higher heat exchange rates and pressure drops regardless of flow rate variation. For other flow rate, tube diameter, and inlet temperature, empirical correlation could be developed to generalize the present results.
D. Effect of using different tube materials
Although models 2 and 12 offer a higher heat transfer rate of 655.5 and 675.0 W (see Fig. 10), respectively, followed by model 8 (642.2 W) and other models, the pressure drops for models 2 and 12 are 775.3 and 730.3 Pa (see Fig. 11), respectively, also much higher than model 8 (535.5 Pa). The model 2 is fully helical in shape both in inlet and outlet pipes. Furthermore, helical loop distribution according to model 12 (see Table II) needs precise adjustment for any other vertical depth of GHE. On the other hand, helical loop arrangement according to model 8 is easy to select upper straight and lower helical parts by subdividing the total number of loops into two segments. Therefore, the analysis of the impact of different materials on thermal performance has been conducted using the simpler geometry of model 8, despite its slightly lower heat exchange rate compared to model 12. This analysis is done using polyethylene, steel, and HDPE as the pipe materials. Polyethylene is widely used in GHEs because of its flexibility, therefore easy to make spiral parts and resistance to chemical corrosion.^{47} Steel is chosen due to its high thermal conductivity and mechanical strength.^{20} Additionally, HDPE is chosen due to its higher strength, more resistance to variations in temperature, and longer durability.^{44,47} These materials are chosen for the pipe of GHE models to conduct a comparative analysis of feasibility and obtain the best performance. Properties of these materials are listed in Table III.
Figure 12 shows that the average heat transfer rate increases by 6.54%, while steel is used as the pipe material. Because of steel's high thermal conductivity, heat is transferred quickly. Polyethylene has lower thermal conductivity than steel, which results in a slower heat transfer. It is worth mentioning that steel is 46 times more thermally conductive compared to polyethylene. By using HDPE, the heat transfer rate is increased by about 1.5% compared to polyethylene, which is not significant. As the surrounding soil temperature (290.85 K) remains constant, there is an ultimate threshold of heat transfer rate that cannot be exceeded. That is why, even when employing a highly thermally conductive material like steel, there is no substantial change in the heat transfer rate compared to polyethylene.
E. Effect of using different backfill materials
As backfill materials, silica sand and concrete piles are used for analysis. Figure 13 demonstrates the average heat transfer rate of model 8 for silica sand and concrete piles as the backfill materials.
Since the concrete piles have 1.33 times more specific heat than silica sand, it can hold more heat. So, comparatively more heat transfer occurs. Gravel has lesser specific heat and thermal conductivity; hence, the average heat transfer rate is comparatively less than silica sand and concrete piles. During this analysis, the pipe material and other parameters are kept unchanged.
F. Criterion of COP improvement factor
GHE models .  $QC\u2212c$ (W/m) .  $QC$ (W/m) .  $Q\u2032C$ (W/m) .  V $(m3/s)$ .  $\Delta pc$ (Pa/m) .  $\Delta p$ (Pa/m) .  $\Delta p\u2032$ (Pa/m) .  COP improvement factor, K . 

Model 2  33.61  24.26  9.35  $3.33\xd710\u22125$  39.76  6.13  33.63  0.39 
Model 3  32.41  24.26  8.15  $3.33\xd710\u22125$  35.48  6.13  29.35  0.34 
Model 4  32.44  24.26  8.18  $3.33\xd710\u22125$  29.63  6.13  23.50  0.34 
Model 5  31.63  24.26  7.37  $3.33\xd710\u22125$  26.83  6.13  20.71  0.30 
Model 6  30.37  24.26  6.10  $3.33\xd710\u22125$  24.00  6.13  17.87  0.25 
Model 7  31.00  24.26  6.74  $3.33\xd710\u22125$  33.31  6.13  27.18  0.28 
Model 8  32.93  24.26  8.67  $3.33\xd710\u22125$  27.46  6.13  21.33  0.36 
Model 9  32.55  24.26  8.29  $3.33\xd710\u22125$  26.96  6.13  20.83  0.34 
Model 10  31.26  24.26  7.00  $3.33\xd710\u22125$  24.62  6.13  18.48  0.29 
Model 11  31.63  24.26  7.37  $3.33\xd710\u22125$  26.77  6.13  20.65  0.30 
Model 12  34.61  24.26  10.35  $3.33\xd710\u22125$  37.45  6.13  31.32  0.43 
Model 13  34.10  24.26  9.84  $3.33\xd710\u22125$  35.08  6.13  28.95  0.41 
Model 14  32.87  24.26  8.61  $3.33\xd710\u22125$  32.67  6.13  26.54  0.35 
Model 15  32.83  24.26  8.57  $3.33\xd710\u22125$  31.5  6.13  25.37  0.35 
Model 16  32.60  24.26  8.34  $3.33\xd710\u22125$  17.02  6.13  10.89  0.34 
Model 17  29.35  24.26  5.09  $3.33\xd710\u22125$  25.93  6.13  19.8  0.20 
GHE models .  $QC\u2212c$ (W/m) .  $QC$ (W/m) .  $Q\u2032C$ (W/m) .  V $(m3/s)$ .  $\Delta pc$ (Pa/m) .  $\Delta p$ (Pa/m) .  $\Delta p\u2032$ (Pa/m) .  COP improvement factor, K . 

Model 2  33.61  24.26  9.35  $3.33\xd710\u22125$  39.76  6.13  33.63  0.39 
Model 3  32.41  24.26  8.15  $3.33\xd710\u22125$  35.48  6.13  29.35  0.34 
Model 4  32.44  24.26  8.18  $3.33\xd710\u22125$  29.63  6.13  23.50  0.34 
Model 5  31.63  24.26  7.37  $3.33\xd710\u22125$  26.83  6.13  20.71  0.30 
Model 6  30.37  24.26  6.10  $3.33\xd710\u22125$  24.00  6.13  17.87  0.25 
Model 7  31.00  24.26  6.74  $3.33\xd710\u22125$  33.31  6.13  27.18  0.28 
Model 8  32.93  24.26  8.67  $3.33\xd710\u22125$  27.46  6.13  21.33  0.36 
Model 9  32.55  24.26  8.29  $3.33\xd710\u22125$  26.96  6.13  20.83  0.34 
Model 10  31.26  24.26  7.00  $3.33\xd710\u22125$  24.62  6.13  18.48  0.29 
Model 11  31.63  24.26  7.37  $3.33\xd710\u22125$  26.77  6.13  20.65  0.30 
Model 12  34.61  24.26  10.35  $3.33\xd710\u22125$  37.45  6.13  31.32  0.43 
Model 13  34.10  24.26  9.84  $3.33\xd710\u22125$  35.08  6.13  28.95  0.41 
Model 14  32.87  24.26  8.61  $3.33\xd710\u22125$  32.67  6.13  26.54  0.35 
Model 15  32.83  24.26  8.57  $3.33\xd710\u22125$  31.5  6.13  25.37  0.35 
Model 16  32.60  24.26  8.34  $3.33\xd710\u22125$  17.02  6.13  10.89  0.34 
Model 17  29.35  24.26  5.09  $3.33\xd710\u22125$  25.93  6.13  19.8  0.20 
It can be seen that the values of the COP improvement factor for all the GHE models are positive. It indicates that all of the models are effective and feasible for the ground source heat pump (GSHP) system. In this study, the findings are presented in dimensional forms, including heat transfer rate and pressure drop, which depend on various geometric parameters (pipe diameter, thickness, length, mas flow rate, inlet temperature, etc.) and operational time. Consequently, the results are specific to the models examined. However, by expressing the results as dimensionless performance parameters, such as the Nusselt number, the findings could be extended to models with different dimensions.
G. Comparative analysis of the helical model
Figure 14 depicts the contours of the temperature distribution of GHE at the XY plane. As the upper 4 m of the two pipes is kept straight, the temperature drop in this region is less, as shown in Fig. 14(d). However, at the helix portion, the thermal gradient is higher. Due to the helical path, the heat transfer rate increases, and the temperature of the fluid drops subsequently. It is worth mentioning that helical models affect the surrounding soil significantly, as shown in Fig. 14.
V. CONCLUSION
This study investigates several modified configurations of helical ground heat exchangers numerically to assess the thermal performance. In this research work, 16 different helical configurations are compared with a Utube ground heat exchanger, and their outlet temperature, heat exchange rate, and pressure drop are investigated. The main outcomes are summarized as follows:

Utube GHE has the minimum variation of outlet water temperature compared to the rest of the helical GHE models.

Model 12 has the highest average heat transfer among all configurations, which is 42.69% higher compared to Utube GHE.

Model 8 has nearly 1.5 times more heat exchange rate compared to Utube GHE.

The highest pressure drop (775.28 Pa) is observed for model 2 due to its lowest pitch and highest turn numbers, and the lowest pressure drop (119.5 Pa) is observed for Utube GHE.

The average heat transfer rate is increased by 6.54% for steel as a tube material compared to polyethylene.

The average heat transfer rate is increased by 5.41% for concrete as the backfill material compared to silica sand.

The criterion of COP improvement factor for all the modified models is positive.

It is worth mentioning that the COP improvement factor for model 12 is nearly 43%, indicating its feasibility to integrate with the GSHP system.
Furthermore, based on the prior studies,^{34,48} it is evident that the heat transfer rate and the pressure drop for a specific GHE configuration remain consistent across different flow rates. For example, lowerpitched GHEs consistently exhibited higher heat exchange rates and pressure drops regardless of flow rate variation. Therefore, it is unnecessary to reassess the performance of selected geometric configurations for different flow regimes.
ACKNOWLEDGMENTS
The authors would like to acknowledge the contribution of Khulna University of Engineering & Technology (KUET), Bangladesh, for the logistic support to carry out this work.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Pijus Roy: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nahid Hasan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nushrat Jahan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Md. Sohag Hossain: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Md. Asaduzzaman: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Md. Hasan Ali: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Akio Miyara: Conceptualization (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.