Parametric Study of the Ground-Air Heat Exchanger (GAHE): Effect of Burial Depth and Insulation Length

Abstract

A parametric study of a ground-to-air heat exchanger (GAHE) using numerical models based on computational fluid dynamics with the finite volume method to evaluate the thermal potential of GAHE is presented. After the validation of the numerical code developed with published experimental data, it is proceeded to the study of the geometric parameters to define those that have the greatest impact on the application potential of GAHE. Climatological variables such as relative humidity, air flow velocity, and inlet air temperature are analyzed, as well as the increase in the thermal conductivity of the soil due to its humidity content. In addition, a study of the optimal installation depth as well as the length of the thermal insulation in the outlet pipe of the GAHE is presented. The results reveal that there is a higher heat exchange potential in the GAHE for an optimal burial depth of 4 m and a length of pipe of 15 m, 30% soil moisture content for heating and 32% for cooling, and a pipe diameter of 0.15 m. The use of thermal insulating is recommended only for the last 2 m of length in the outlet pipe of the GAHE.

1. Introduction

Currently, humanity is faced with the dilemma of continuing to produce and consume energy from fossil fuels for electricity generation, fuel production for transport, food cooking, domestic hot water, and the use of conventional air conditioning and heating systems [1] or developing production systems with renewable energies and promoting the use of ecotechnologies to meet its needs [2]. In this sense, sustainability not only depends on changing the way of generating electricity or the type of fuel used for public and private transport; it also depends on adopting and implementing clean technologies that allow the reduction of energy consumption, without depriving goods and/or services considered as human rights, among them thermal comfort in buildings [3], because conventional heating, ventilation, and air conditioning (HVAC) systems are responsible for more than 40% of a building’s total energy consumption [4]. Moreover, there are several problems associated with the use of conventional HVAC systems: its energy consumption is high, its use increases the peak load, it is harmful to the environment, and it reduces indoor air quality. Therefore, all these issues with conventional HVAC systems led to the exploration of energy-efficient and environment friendly systems for heating/cooling of buildings [5].

To contribute to the current energy transition, it is imperative to take advantage of renewable energies; this is directly related to sustainable architecture and the bioclimatic design of buildings, which focuses on the use of natural resources to meet the needs of the inhabitants or users of buildings [6]. This work focuses on the parametric study of a passive technology to improve the thermal comfort of residential buildings, namely ground-to-air heat exchangers (GAHEs), for climatic conditions in Mexico, since it has been shown that the economic good with the highest rate of scarcity in Mexican homes is thermal comfort [7]. It has also been shown that GAHEs can reduce the energy consumption of a building by up to 30%, generating a considerable reduction in the heating/cooling load of buildings and consequently a reduction in greenhouse gas emissions [8,9].

The study and modelling of GAHEs or EAHEs focuses on identifying the relevant parameters to improve the heat transfer between the ground and the air inside the GAHE to improve the thermal performance of GAHEs in building cooling and heating applications. Some studies analyze the length of pipe required to achieve the desired temperature drop/rise depending on the duration of operation [5]. Additionally, Bordoloi, N. et al. in 2018 concluded that it is necessary to consider the properties of the soil, specifically its moisture content [9]. For these reasons, it is important to analyze parameters such as soil moisture content, wind speed, and relative air humidity in the parametric study.

On the other hand, Serageldin, A. A. et al. (2016) demonstrated that the use of different GAHE construction materials on the thermal performance of the system is negligible, since they obtained the same outlet temperature for copper and steel pipes, while in a PVC pipe a difference of only 0.1 °C was obtained with respect to the previous ones [10]. Likewise, other studies evaluate the performance of GAHE as passive cooling systems and reported that the air velocity increases with the diameter and decreases with the length of the pipe [11].

Mostafaeipour et al. (2021) tested nine different configurations of a GAHE at three depths considering three pipe lengths, determining that the longer the pipe length and pipe depth, the higher the efficiency of the GAHE system, reducing the total annual cooling and heating load [12]. However, the authors did not include modifications in pipe diameter. In contrast, other studies have determined that the coefficient of performance is higher for pipes with circular cross-section [13]. In this sense, Rosa et al. in 2020 reported that there is a decreasing relationship for the thermal performance of GAHEs as the speed of the airflow circulating in the pipe increases, an aspect that is made no-tar to a greater extent for cooling purposes. In addition, they numerically determined that for systems with parallel pipes it is possible to reduce the distance of pipes up to 0.5 m without compromising the performance of the GAHE [14].

Lekhal, M. C. et al. (2021) performed a parametric analysis for the sizing of a GAHE in which they reported that the thermal performance of GAHEs is significantly affected from one region to another for the same geometrical and dynamic operating conditions. They found that the temperature differences between the inlet and outlet of the GAHE can vary up to 4 °C, so the need to perform specific studies in locations where it is intended to install a GAHE is evident. As a result, they recommend a depth of 3 m, a length of 25 m, and a diameter of 0.18 m for a temperate climate [15].

In addition to the geometric factors that affect the performance of GAHEs, the study of climatological parameters is another aspect of importance, such as the ideas raised by Chiesa and Zajch (2020), among which they hypothesized that variations in relative humidity values may be important for localities with high relative air humidity, which can generate conditions of low thermal comfort [16]. To solve this problem, several studies point to the implementation of an air humidity control system to reduce the content of moisture in the air, especially for the rainy season or in climates with a high percentage of relative humidity, as well as intermittent use of the system, depending on operating conditions [17].

Another important aspect that influences the performance of GAHEs is the type of soil and its physical and chemical properties, such as moisture content, density, and porosity. These properties modify the thermal conductivity and specific heat of the soil and consequently affect heat transfer. Therefore, quantifying or measuring the thermal conductivity and specific heat of the soil remains a problem due to the scarce information reported in the literature [18].

On the other hand, GAHEs have the necessary characteristics to be considered as a sustainable ecotechnology [19], because their principle of operation is based on the use of low-enthalpy geothermal energy [20] to promote an environment of thermal comfort, either for heating or cooling the interior space of a building, through the exchange of heat between the ambient air and the ground at a depth between 2 to 5 m (Figure 1), taking advantage of the thermal inertia of the soil, since the temperature distribution in the soil is much more stable than the temperature of the air of the troposphere [21]. In addition, the soil temperature at depths of 2 to 5 m tends to be equal to the average annual temperature recorded in the geographic location of study.

Most studies have analyzed the thermal performance of GAHE by means of the air temperature difference at the inlet and outlet, such as the work reported by Díaz H. et al. (2017), in which, using the commercial software FLUENT, they demonstrated that the thermal performance of the GAHE improves by reducing the air inlet velocity [22].

Kaushal, M. (2017) conducted a comprehensive review of several theoretical and experimental studies of GAHEs, concluding that a well-designed GAHE is capable of reducing the electricity consumption of a building by 25–30%; however, to achieve this level of electricity consumption reduction it is necessary to perform modelling and/or simulations before deciding whether or not the installation of a GAHE is feasible [23]. Additionally, Rodrigues, M. K. et al. (2022) used CFD numerical modelling to model a GAHE in a “Y” configuration and found that the implementation of GAHE in a building can reduce monthly electricity consumption by up to 117.72 kWh for heating purposes and 74.13 kWh for cooling [24].

In the numerical modelling of the GAHE, it is very important to establish the appropriate boundary condition at the soil surface to represent the physical phenomenon in a realistic way, since more physical interactions with the environment occur in this zone. In this sense, Badescu, V. (2007) studied parameters such as solar radiation absorbed by the soil surface and the heat flux lost by the system due to evaporation and soil moisture, and this allows to more accurately calculate the soil temperature profile at the surface and at different depths [25].

Xamán, J. et al. (2015) conducted a numerical study of a GAHE to evaluate the cooling and heating potential considering the coldest day and the warmest day of the year in a specific location. The results show whether the system can operate year-round or whether it is only recommended for some periods in which the GAHE performs the function of cooling, heating, or both [26]. Most theoretical studies usually perform their investigations by applying global energy balances due to the simplicity of the mathematical models. However, the length of thermal insulation at the outlet as well as the optimal burial depth of the GAHE are parameters that have not been reported in previous work until this paper, for GAHE focused on residential building air conditioning.

This parametric study considers the GAHE burial depth variation of 2–4 m, lengths of 5–25 m, soil moisture content of 0–50%, pipe diameters of 0.1–0.25 m, and relative air humidity of 0–100%. The length of thermal insulation in the GAHE outlet pipe is also analyzed. Furthermore, it is known that CFD modelling is considered as a robust tool that generates more detailed and accurate information about the behavior, conditions, and operating parameters of the GAHE [27]. Thus, in this work, numerical modelling based on the Finite Volume Method (FVM) is used. The nomenclature used in this paper is presented in

Table 1. Nomenclature and units.

Variable	Symbol (Unit)	Variable	Symbol (Unit)
Thermal conductivity constant	$\lambda$ (W/m°C)	Density	$\rho$ (kg/m3)
Specific heat at constant pressure	$Cp$ (J/kg°C)	Conductive heat flux	Qcond (W/m2)
Temperature	$T$ (°C)	Convective heat flux	Qconv (W/m2)
Overall horizontal length	Hx (m)	Evaporative heat flux	Qevap (W/m2)
Overall vertical length	Hy (m)	Solar irradiance	Gb (W/m2)
Number of horizontal nodes	Nx	Vertical velocity component	$v$ (m/s)
Number of vertical nodes	Ny	Horizontal velocity component	$u$ (m/s)
Horizontal element size	$\Delta x$ (m)	Pressure	P (Pa)
Vertical element size	$\Delta y$ (m)	Soil-to-air convection coefficient	$h_{Sup}$ (W/m2°C)
Horizontal nodal distance	$\delta x$ (m)	Emissivity coefficient	ɛ
Vertical nodal distance	$\delta x$ (m)	Absorptivity coefficient	$\alpha$
Source term	g (W)	Dynamic viscosity	$\mu$ (Pa·s)

.

2. Materials and Methods

In this work, the Finite Volume Numerical Method is used by means of software development (structured programming in Matlab) for the modelling of the GAHE.

2.1. Physical Model

Figure 2 shows the geometric model of the GAHE with its boundary conditions, which is a graphical representation of the real system [28], in which the sections that will be modelled with CFD can be observed. The direct radiation incident on the ground is shown in yellow, and the ground is represented with brown color. The blue arrows indicate the air inlet and outlet to the inside of the buried duct, respectively, and the purple shows the thickness of a thermal insulation around the air outlet duct to conserve the thermal energy obtained by the air during its travel inside the GAHE.

Table 2. Dimensions of the GAHE.

Dimensions	Value
Total depth (Hy)	3.15 m.
Total length (Hx)	6.3 m.
Hy1	2 m.
Hx3	5 m.
Hy3	1 m.
Hx1 = Hx5	0.5 m.
Insulation thickness (e)	0.05 m.
Diameter = Hy2 = Hx2 = Hx4 = D	0.15 m.

shows the dimensions used as a reference for the parametric study of the GAHE, these dimensions were proposed by Rodríguez-Vázquez et al. (2020) due to the characteristics and limitations of land available for social housing in Mexico [29].

2.2. Mathematical Model

The assumptions of the mathematical model are as follows:

-

Incompressible flow in laminar regime.

-

The soil is considered as a solid and isotropic medium.

-

Boussinesq approximation: density is assumed constant except in the gravity term in the momentum equations where it is varied linearly [30].

-

Radiatively non-participating fluid.

-

For the entire non-fluid domain (soil and thermal insulator), the blocking technique is applied, which implies defining with a value of 0 permanently all flow variables at all nodes of the solid domain (u = v = P = 0) so that for the solid domain only conduction heat transfer is evaluated [31].

The steady state governing equations inside the physical model of the GAHE (Figure 2) are the mass, momentum, and energy equations for natural convection described as follows:

$$\frac{\partial \left(\rho u\right)}{\partial x}+\frac{\partial \left(\rho v\right)}{\partial y}=0$$

(1)

$$\frac{\partial \left(\rho u u\right)}{\partial x}+\frac{\partial \left(\rho v v\right)}{\partial y}=\frac{\partial }{\partial x}\left(\mu \frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mu \frac{\partial u}{\partial y}\right)-\frac{\partial P}{\partial x}$$

(2)

$$\frac{\partial \left(\rho u u\right)}{\partial x}+\frac{\partial \left(\rho v v\right)}{\partial y}=\frac{\partial }{\partial x}\left(\mu \frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial y}\left(\mu \frac{\partial v}{\partial y}\right)-\frac{\partial P}{\partial y}$$

(3)

$$\frac{\partial \left(\rho u T\right)}{\partial x}+\frac{\partial \left(\rho v T\right)}{\partial y}=\frac{\partial }{\partial x}\left(\frac{\lambda }{Cp}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{\lambda }{Cp}\frac{\partial T}{\partial y}\right)$$

(4)

2.3. Boundary Conditions

For the temperature, the East, West, and South boundaries are considered adiabatic:

$${\frac{\partial T}{\partial x}|}_{x=0}=0 \mathrm{for} 0\le y\le Hy$$

(5)

$${\frac{\partial T}{\partial x}|}_{x=Hx}=0 \mathrm{for} 0\le y\le Hy$$

(6)

$${\frac{\partial T}{\partial y}|}_{y=0}=0 \mathrm{for} 0\le x\le Hx$$

(7)

On the other hand, the energy balance proposed by G. Mihalakakou et al. [32] was implemented for the Northern boundary:

$$-\lambda \frac{\partial T}{\partial y}{|}_{y=Hy}=CE+-\left(LR\right)+\left(SR\right)-LE \mathrm{for} 0\le x\le Hx$$

(8)

CE represents the convective energy exchanged by the air with the soil surface, calculated as:

$$CE=h_S\left(T_{amb}\right)$$

(9)

SR represents the solar radiation absorbed from the surface of the ground.

LR represents the solar radiation emitted by the soil.

According to Badescu et al. (2007), the convective heat transfer coefficient ($h_S$) at the ground surface is a function of wind speed [25]:

$$h_S=5.678\left[0.775+0.35\left(\frac{V_{wind}}{0.304}\right)\right] \mathrm{for}{V}_{wind}<4.88$$

(10)

$$h_S=5.678\left[0.775+0.35{\left(\frac{V_{viento}}{0.304}\right)}^{0.78}\right] \mathrm{for}{V}_{wind}\ge 4.88$$

(11)

$$-\left(LR\right)+\left(SR\right)=-\epsilon \Delta R+\alpha G_b$$

(12)

$$SR=\alpha G_b$$

(13)

In Equation (12), ΔR represents the term that depends on the relative humidity of the soil surface, the effective atmospheric temperature, and the radiative properties of the soil.

LE represents the latent heat flux from the ground surface due to evaporation:

$$\mathrm{LE}=0.0168f{\mathrm{h}}_{\mathrm{Sup}}\left(\right(\mathrm{a}{\mathrm{T}}_{\sup }+\mathrm{b})-HR(\mathrm{a}{\mathrm{T}}_{\mathrm{amb}}+\mathrm{b}\left)\right)$$

(14)

In Equation (14), HR represents the relative humidity of the air and f is a fraction which depends mainly on the soil surface.

The factors a, b, and f have a constant value for a mean soil moisture of [32]:

a = 103 (Pa/K)    b = 609 (Pa)    f = 0.7

The boundary condition for the air inlet flow velocity (y = Hy) is calculated as a function of the Reynolds number:

$$v=f\left(Re\right), u=0 \mathrm{for}Hx1\le x\le Hx3$$

(15)

The boundary condition for the outlet air flow (y = Hy) and the pressure are:

$${ \frac{\partial u}{\partial y}|}_{y=Hy}=0, {\frac{\partial v}{\partial y}|}_{y=Hy}=0, {\frac{\partial P}{\partial y}|}_{y=Hy}=0 \mathrm{for} Hx3\le x\le Hx5$$

(16)

2.4. Methodology

The numerical models for the GAHE and boundary conditions, represented by the equations group (1)–(16), were solved by the FVM [33], and the coupling of the continuity and momentum equations were performed with the SIMPLEC algorithm. The convective terms were approximated by the upwind scheme, and the diffusive terms were approximated by the central differential scheme [31]. The technique used to solve the set of discretized linear equations was the Line Gauss Seidel-Alternating Direction Implicit (LGS-ADI), which is highly efficient for the solution of this type of system [31], and a non-uniform mesh was also used. The numerical methodology is shown in the flow chart in Figure 3.

For the solution of the SIMPLEC algorithm, it is necessary to establish an initial distribution of initial speed, pressure, and temperature. For temperature, an initial distribution model as a function of soil depth proposed by Baruch G. [34] is used.

2.4.1. Mesh Independence Analysis

A mesh independence analysis was performed to determine the number of control volumes suitable for numerical modeling [33], considering a critical case; hence, this analysis was performed for a Reynolds of 1500, an ambient temperature of 38 °C, a completely dry silty type of soil (zero moisture content), and an average annual temperature at the soil surface of 26 °C. The computing time to achieve the convergence of the numerical solution is also analyzed.

The mesh independence analysis is divided into two stages; the first stage consists of determining the number of nodes required for the pipe diameter, and the second stage consists of determining the number of nodes for the depth and horizontal length of the buried GAHE pipe. The number of nodes for the soil section (Hy3, Hx1, Hx5) is kept fixed with a finer mesh than the other sections. The average temperature values at the outlet of the GAHE and their respective computation time for five different meshes is presented in

Table 3. Number of nodes and computation time.

Number of Nodes Diameter	Average Outlet Temperature (°C)	Computing Time (Hours)
57	21.53	1.47
71	21.94	1.34
87	22.31	1.27
101	22.63	1.48
117	22.73	2.08

.

The number of nodes for the pipe diameter will be 101 nodes, as this is the solution that presents the smallest difference with respect to the result with 117 nodes, in addition to the fact that the additional computation time for the solution with 117 nodes is higher.

Table 4. Average outlet temperature and computing time for each modeling.

Number of Nodes (H × V)	Average Outlet Temperature (°C)	Computing Time (Hours)
57 × 57	22.69	1.20
71 × 71	22.65	1.37
87 × 87	22.60	1.42
101 × 101	22.63	1.49
117 × 117	22.59	1.67

shows the variation of the outlet temperature with its respective computation time based on the number of nodes used for the air flow in the vertical and horizontal pipe, while Figure 4 shows the temperature distribution obtained at the outlet of the GAHE according to the number of nodes used for the pipe diameter.

The outlet temperature difference between the 101 × 101 and 117 × 117 meshes is 0.04 °C, which is considered negligible. Moreover, the computational time is shorter. Consequently, the mesh used for the numerical results is 101 × 101 nodes and has a total of 83,835 computational nodes (Figure 5).

2.4.2. Numerical Model Validation

The validation of the numerical model developed with CFD consists of comparing the results obtained by the numerical solution with the experimental results reported for real GAHE operating conditions. In this work, the comparison was made with experimental data from two GAHEs installed in Chetumal Quintana Roo, Mexico; one was installed at 1 m and the other at 2 m depth and 6 m horizontal length.

Table 5. Operating conditions of the GAHE, Chetumal Quintana Roo (30 July 2016 and 9 August 2016).

Date	Hour	Solar Irradiance (W/m2)	Air Relative Humidity (%)	Wind Speed (m/s)	Environment Temperature (°C)
30/07/2016	1	0	84.17	4.02	28.72
30/07/2016	2	0	83.83	3.65	28.63
30/07/2016	3	0	82.83	3.27	28.73
30/07/2016	4	0	82.5	3.52	28.68
30/07/2016	5	0	84	3.27	28.6
30/07/2016	6	0	83.5	3.6	28.58
30/07/2016	7	4.67	83.83	3.35	28.43
30/07/2016	8	105.83	83.33	3.27	28.73
30/07/2016	9	319.17	81.5	3.27	29.12
30/07/2016	10	544.67	80	2.53	29.63
30/07/2016	11	790.5	75.83	1.83	30.48
30/07/2016	12	910.33	72.67	2.53	31.03
30/07/2016	13	952.33	69.33	2.3	31.52
30/07/2016	14	972	65.17	2.83	31.85
30/07/2016	15	913.67	65.83	2.75	31.97
30/07/2016	16	797	65.83	2.68	32.07
30/07/2016	17	626.5	64.33	2.7	32.08
30/07/2016	18	427.17	67.33	3.27	31.72
30/07/2016	19	195.17	73.67	4.1	30.52
30/07/2016	20	13.83	78	2.97	29.77
30/07/2016	21	0	91.25	2.2	28.74
30/07/2016	22	0	93.59	2.72	29.3
30/07/2016	23	0	93.56	3.1	28.94
30/07/2016	24	0	93.5	2.69	28.24
09/08/2016	1	0	87.17	2.62	28.72
09/08/2016	2	0	87.83	2.77	28.67
09/08/2016	3	0	88.33	3.1	28.58
09/08/2016	4	0	88.5	2.62	28.47
09/08/2016	5	0	88.5	1.7	28.43
09/08/2016	6	0	89	1.03	28.5
09/08/2016	7	3.67	88.83	1.92	28.53
09/08/2016	8	74.33	88.17	2.77	28.67
09/08/2016	9	345.33	84.67	2.53	29.6
09/08/2016	10	570.17	82.33	2.75	30.22
09/08/2016	11	747.33	76.67	3.15	31.22
09/08/2016	12	886.67	75.83	2.83	31.62
09/08/2016	13	960.5	78	2.23	31.75
09/08/2016	14	970.83	77.67	3.2	31.52
09/08/2016	15	915.17	78.83	4.43	31.33
09/08/2016	16	793.17	81.33	3.18	31.13
09/08/2016	17	617.33	80.67	3.43	31.22
09/08/2016	18	406.33	80.5	3.03	31.1
09/08/2016	19	149.33	81.83	2.53	30.68
09/08/2016	20	5.33	83.17	3.2	30.08
09/08/2016	21	0	85.67	3.58	29.57
09/08/2016	22	0	87	2.27	29.23
09/08/2016	23	0	87.33	4.17	29.05
09/08/2016	24	0	88.17	3.87	28.85

presents the operating weather conditions reported by the authors for 30 July and 9 August 2016, which were used for the numerical modelling. The validation was performed by comparing the average temperatures at the outlet of the GAHE with hourly intervals, values that were reported by the authors [35,36].

The results of the numerical modelling and the relative percentage error with the outlet temperature of the GAHEs installed at 2 and 1 m depth, respectively, are shown in

Table 6. Comparison of experimental vs modeled outlet temperature for 30 July 2016 (2 m depth).

Time	Experimental Outlet Temperature (°C)	Outlet Temperature (°C) for Re = 500	Relative Error (%)
1	28.95	27.3	5.7
2	28.95	27.2	6.04
3	28.99	27.2	6.17
4	28.95	27.16	6.18
5	28.92	27.16	6.09
6	28.91	27.14	6.12
7	28.86	27.05	6.27
8	29.04	27.6	4.96
9	29.47	28.54	3.16
10	30.03	29.53	1.67
11	30.91	32.68	−5.73
12	30.94	32.22	−4.14
13	31.76	32.87	−3.49
14	31.5	32.35	−2.7
15	31.41	32.32	−2.9
16	31.37	31.97	−1.91
17	31.03	31.18	−0.48
18	30.74	30.06	2.21
19	30.27	28.65	5.35
20	29.79	27.74	6.88
21	25.39	27.51	−8.35
22	25.53	28.09	−10.03
23	26.48	27.85	−5.17
24	27.93	27.3	2.26
	Average Error.		4.75

and

Table 7. Comparison of experimental vs. modeled outlet temperature for 30 July 2016 (1 m depth).

Time	Experimental Outlet Temperature (°C)	Outlet Temperature (°C) for Re = 500	Relative Error (%)
1	28.66	26.62	7.12
2	28.68	26.48	7.67
3	28.72	26.43	7.97
4	28.68	26.39	7.98
5	28.65	26.43	7.75
6	28.64	26.44	7.68
7	28.58	26.32	7.91
8	28.78	27.15	5.66
9	29.36	28.65	2.42
10	29.88	31.03	−3.85
11	30.74	34.74	−13.01
12	31.25	34.26	−9.63
13	31.51	35.16	−11.58
14	31.55	34.05	−7.92
15	31.73	33.92	−6.9
16	31.67	33.25	−4.99
17	31.18	31.82	−2.05
18	30.74	29.99	2.44
19	30.25	28.08	7.17
20	29.7	26.86	9.56
21	26.86	26.98	−0.45
22	27.28	27.73	−1.65
23	27.44	27.5	−0.22
24	27.07	26.87	0.74
	Average Error		6.01

.

The relative percent error and the average error are calculated as:

$$\begin{gathered} Relative error \%=\frac{\left(Experimental temperature-Modeled temperature\right)}{Experimental temperature}\ast 100\% \\ Average relative error \%=\frac{{{\displaystyle \sum }}_1^{24}\left|Relative error \%\right|}{24} \end{gathered}$$

(17)

As a better approximation is observed in the results obtained by the numerical modelling for the GAHE installed at 2 m depth with respect to the GAHE installed at 1 m, then validation results with experimental data reported for 9 August 2016 are presented in

Table 8. Comparison of experimental vs. modeled outlet temperature for 9 August 2016 (2 m depth).

Time	Experimental outlet Temperature (°C)	Outlet Temperature (°C) for Re = 500	Relative Error (%)
1	28.43	27.34	3.83
2	28.4	27.35	3.7
3	28.38	27.33	3.7
4	28.34	27.22	3.95
5	28.31	27.06	4.42
6	28.22	26.96	4.46
7	28.28	27.21	3.78
8	28.44	27.66	2.74
9	29.18	29.33	−0.51
10	29.75	30.5	−2.52
11	30.38	31.36	−3.23
12	30.66	32.37	−5.58
13	30.65	33.58	−9.56
14	30.79	32.4	−5.23
15	30.75	31.41	−2.15
16	30.61	31.68	−3.5
17	30.11	30.97	−2.86
18	29.8	30.28	−1.61
19	29.53	29.12	1.39
20	29.3	28.19	3.79
21	29	27.95	3.62
22	28.84	27.66	4.09
23	28.72	27.69	3.59
24	28.61	27.57	3.64
		Average Error.	3.64

. This is realized because from 2 m depth the soil temperature distribution starts to stabilize, as the influence of the boundary conditions at the soil surface and of the climatological variables decreases with increasing soil depth.

The results for the GAHE installed at 2 m depth for 30 July 2016 are shown in Figure 6.

Now that the numerical model has been validated with experimental results and because the average percentage error was less than 10%, it is considered that the developed computer program is capable of adequately predicting the thermal behavior of the GAHE. Therefore, the operating conditions can now be modified, and the parametric study can be carried out.

3. Results and Discussion

The results of this study are presented in two stages; the first stage presents the study of geometric and climatological parameters, and the second stage presents the results for the optimum burial depth and the length of insulation of the GAHE outlet pipe.

3.1. Results of the Parametric Study

The parametric study of the GAHE consists of modifying the different operating parameters and design configurations that allow the best use of the thermal inertia of the soil, as well as obtaining the greatest temperature difference between the inlet and outlet of the GAHE, depending on the locality and climatic zone under study [16]. For the first stage of the parametric study, it is proposed to model different pipe lengths, burial depths, and different pipe diameters.

The modeling as a cooling system of the GAHE corresponds to the hour of highest temperature for the hottest day of the year 2020 in the municipality of Jicarero in Jojutla Morelos, Mexico, while the modeling as a heating system corresponds to the hour of lowest temperature reported for the coldest day of the year 2020 in the municipality of La Rosilla in Durango, Mexico. The climatological conditions for modelling of the GAHE as a passive cooling and heating system are presented in

Table 9. Climatological conditions for modelling of the GAHE.

Condition	Solar Irradiance (W/m2)	Air Relative Humidity (%)	Humidity of Soil (%)	Wind Speed (m/s)	Environment Temperature (°C)
Cooling Jojutla Morelos	849.46	19.81	30	2.85	37.08
Heating La Rosilla Durango	0	100	30	1.32	−6.96

; these parameters remain constant for the numerical modelling in this first stage of the parametric study. Therefore, the starting point for this study will be the reference dimensions used for the physical model presented in Figure 2 and Table 2.

The first stage of the study starts by modelling the cooling operation conditions, it is expected to reduce the thermal input level, according to the climatological conditions for the hottest hour reported in 2020 in the municipality of Jojutla Morelos, Mexico, the geometric parameters studied in this first stage and the result obtained for the cooling potential are listed in

Table 10. Numerical evaluation of the cooling potential according to the dimensions of the GAHE.

Pipe Diameter (m)	Pipe Depth (m)	Horizontal Length (m)	ΔT (°C) Outet—Inlet for Re = 500	ΔT (°C) Outet—Inlet for Re = 250	ΔT (°C) Outet—Inlet for Re = 100
Horizontal length evaluation
0.15 (6 in)	2	5	−3.03	−9.24	−10.69
0.15 (6 in)	2	10	−4.46	−10.18	−10.97
0.15 (6 in)	2	15	−4.7	−10.68	−10.87
0.15 (6 in)	2	25	−5.77	−11.04	−10.71
Vertical length evaluation
0.15 (6 in)	2	5	−3.03	−9.24	−10.69
0.15 (6 in)	3	5	−7.47	−10.23	−11.73
0.15 (6 in)	4	5	−7.36	−10.34	−11.86
Diameter Evaluation
0.1 (4 in)	2	5	−4.3	−10.15	−11.75
0.15 (6 in)	2	5	−3.03	−9.24	−10.69
0.2 (8 in)	2	5	−2.76	−8.18	−9.79
0.25 (10 in)	2	5	−2.77	−6.41	−9.25

. Based on the temperature difference between the inlet and outlet of the GAHE, it is observed that the greater the temperature difference, the greater the application potential according to each modified parameter with respect to the reference parameter. Modelling was performed for laminar flow considering Reynolds numbers of 100, 250 and 500.

Table 10 and

Table 11. Numerical evaluation of the heating potential according to the dimensions of the GAHE.

Pipe Diameter (m)	Pipe Depth (m)	Horizontal Length (m)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Inlet for Re = 250	ΔT (°C) Outlet—Intlet for Re = 100
Horizontal length evaluation
0.15 (6 in)	2	5	0.41	5.32	6.55
0.15 (6 in)	2	10	1.74	5.54	6.33
0.15 (6 in)	2	15	1.3	5.61	5.86
0.15 (6 in)	2	25	1.79	5.46	3.97
Vertical length evaluation
0.15 (6 in)	2	5	0.41	5.32	6.55
0.15 (6 in)	3	5	5.04	6.63	8.29
0.15 (6 in)	4	5	6.19	8.1	10.11
Diameter Evaluation
0.1 (4 in)	2	5	1.55	6.1	7.34
0.15 (6 in)	2	5	0.41	5.32	6.55
0.2 (8 in)	2	5	−0.19 *	4.75	6.06
0.25 (10 in)	2	5	−0.3 *	4.32	5.23

presents the results obtained for the geometrical parameters of a GAHE for both climatic conditions, Jojutla Morelos, with a climate considered warm and Rosilla Durango, Mexico; this locality is considered the coldest in the country.

The values marked with (*) indicate unsatisfactory results, as the system does not fulfill its purpose; for cooling, a lower outlet temperature than the inlet temperature and for heating, a higher temperature than the inlet of the GAHE is obtained.

Based on the most relevant geometrical parameters and due to the results presented in Table 10 and Table 11, the GAHE is now evaluated for both applications (cooling and heating). The additional parameters and configurations with their respective results are shown in

Table 12. Additional GAHE settings.

Pipe Diameter (m)	Pipe Depth (m)	Horizontal Length (m)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Intlet for Re = 250	ΔT (°C) Outlet—Intlet for Re = 100
Cooling
0.15 (6 in)	3	15	−8.95	−11.72	−11.74
0.15 (6 in)	4	15	−9.82	−12.58	−11.78
Heating
0.15 (6 in)	3	15	5.32	7.68	6.86
0.15 (6 in)	4	15	6.71	9.61	8.65

.

The next stage of the parametric study consists of evaluating the variation of the thermal behavior by modifying the climatological parameters of relative air humidity, air flow velocity, as well as the variation in soil moisture, because some studies have reported an increase in the GAHE potential caused by the increase in soil moisture content [5,37]. The geometrical parameters to be used will be the dimensions of Table 2, and the solar irradiance remains constant because the study focuses on evaluating the change of a single variable considering real operating conditions.

Another reason why solar irradiance remains constant is that the cooling condition would lose its purpose if the solar irradiance were modified, and the same happens with the operation of the GAHE as a heating system. The results of the GAHE as a cooling system are presented in

Table 13. Numerical evaluation of the cooling potential according to the climatological conditions of the GAHE.

Air Relative Humidity (%)	Humidity of Soil (%)	Wind Speed (m/s)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Intlet for Re = 250	ΔT (°C) Outlet—Intlet for Re = 100
Air relative humidity evaluation
0	32	2.85	−4.59	−10.73	−12.69
19.81	32	2.85	−3.03	−9.24	−10.69
40	32	2.85	−1.6	−7.01	−8.66
60	32	2.85	−0.11	−4.73	−5.59
80	32	2.85	1.39 *	−2.43	−2.33
100	32	2.85	2.88 *	−0.11	1.03 *
Humidity of soil evaluation
19.81	0	2.85	−1.58	−6.84	−9.84
19.81	10	2.85	−1.95	−7.81	−10.64
19.81	20	2.85	−1.51	−8.63	−10.86
19.81	32	2.85	−3.03	−9.24	−10.69
19.81	40	2.85	−3.31	−9.51	−10.71
19.81	50	2.85	−3.38	−9.44	−10.53
Wind speed evaluation
19.81	32	0.5	4.48 *	2.41 *	4.8 *
19.81	32	1	1.11 *	−2.86	−2.95
19.81	32	2	−1.89	−7.44	−9.15
19.81	32	2.85	−3.03	−9.24	−10.69
19.81	32	4	−4.06	−10.29	−12.02

and the results as a heating system are presented in

Table 14. Numerical evaluation of the heating potential according to the climatological conditions of the GAHE.

Air Relative Humidity (%)	Humidity of Soil (%)	Wind Speed (m/s)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Intlet for Re = 250	ΔT (°C) Outlet—Intlet for Re = 100
Air relative humidity evaluation
0	30	1.32	0.71	5.61	6.97
20	30	1.32	0.67	5.55	6.88
40	30	1.32	0.64	5.49	6.8
60	30	1.32	0.6	5.44	6.72
80	30	1.32	0.56	5.38	6.63
100	30	1.32	0.41	5.32	6.55
Humidity of soil evaluation
100	0	1.32	0.19	4.96	6.78
100	10	1.32	0.45	5.59	7.6
100	20	1.32	0.26	4.9	6.28
100	30	1.32	0.41	5.32	6.55
100	40	1.32	0.76	5.66	6.78
100	50	1.32	0.65	5.42	6.38
Wind speed evaluation
100	30	0.5	0.05	4.59	5.5
100	30	1.32	0.41	5.32	6.55
100	30	2	0.7	5.59	6.93
100	30	3	0.84	5.8	7.24
100	30	4	0.92	5.93	7.41

.

3.2. Optimum Burial Depth and Insulation Length

As a result of the parametric study, specifically, at depths greater than 2 m, an interesting thermal behavior was observed as shown in Figure 7, in which the temperatures profile of the GAHE is observed.

Marked with a red box in Figure 7, a stratified zone of temperatures is observed at a depth of 4 m and in the vertical riser pipe inside a building. The red box in Figure 7a,b represents the zone of greatest heat exchange, because this zone also represents the maximum and minimum temperature reached by the system, so that, in this zone, it is possible to increase the thermal potential of the GAHE for either heating or cooling applications, respectively. Accordingly, the optimum installation depth of the GAHE is 4 m.

On the other hand, J. Xamán et al. (2014) determined numerically that the use of a thermal insulator in the outlet pipe increases both the cooling and heating potential of the GAHE. In addition, the authors determined that the appropriate thickness for the thermal insulator in the outlet pipe is 0.05 m (2 in) [38]; however, and due to what is observed in Figure 7, it is possible that this benefit is only present in the last 2 m of the upward air flow inside the GAHE. Therefore, modelling of the GAHE with different lengths of thermal insulation in the outlet pipe was carried out, always keeping the insulation in a length of 1.5 m deep up to 4 m completely thermally insulated for a GAHE with a pipe depth of 4 m and a horizontal length of 5 m, according to the real operating conditions established in Table 8.

Four laminar flows corresponding to Reynolds numbers of 150, 220, 290, and 500 were modelled, considering the operating potential of the GAHE as a function of the temperature difference between the outlet and inlet air temperature (ambient temperature).

Table 15. Results of the cooling potential according to the isolated length of the GAHE outlet pipe.

Pipe Depth (m)	Insulated Length (m)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Intlet for Re = 290	ΔT (°C) Outlet—Intlet for Re = 220	ΔT (°C) Outlet—Intlet for Re = 150
4	4	−7.36	−8.73	−10.4	−11.62
4	3.5	−7.42	−8.78	−10.45	−11.65
4	3	−7.46	−8.82	−10.49	−11.68
4	2.5	−7.49	−8.85	−10.53	−11.69
4	2	−7.53	−8.89	−10.56	−11.71
4	1.5	−7.57	−8.92	−10.59	−11.7

presents the results for the numerical modelling according to the climatic conditions of the lowest temperature hour for the year 2020 in the municipality of Jojutla Morelos, Mexico.

Table 16. Results of the heating potential according to the isolated length of the GAHE outlet pipe.

Pipe Depth (m)	Insulated Length (m)	ΔT (°C) Outlet—Intlet for Re = 500	ΔT (°C) Outlet—Intlet for Re = 290	ΔT (°C) Outlet—Intlet for Re = 220	ΔT (°C) Outlet—Intlet for Re = 150
4	4	6.19	7.69	8.5	9.57
4	3.5	6.2	7.69	8.52	9.61
4	3	6.23	7.71	8.54	9.64
4	2.5	6.25	7.72	8.56	9.67
4	2	6.28	7.74	8.59	9.69
4	1.5	6.3	7.75	8.6	9.69

presents the results for the operation of the GAHE as a heating system.

4. Discussion

The CFD-based numerical code was used to perform a parametric study of the GAHE considering cooling and heating applications.

The recommended dimensions for a GAHE according to the results obtained in the first stage of the parametric study are presented in Table 12, and these results coincide with those obtained by Lekhal, M. C. et al. (2021) in which the recommended burial depth is 3 m with a pipe diameter of 0.18 m for a temperate climate, while the recommended length is 25 m [15], although 25 m length for the GAHE describes a higher thermal potential than the 15 m recommended in Table 11 for flows with Reynolds numbers of 500 and 250. Other studies also recommend a burial depth of 2.5 to 3 m for GAHEs [9].

For a Reynolds number of 250 the increase from 5 to 25 m in horizontal pipe length generates an increase in the cooling and heating potential of −1.8 °C and 0.14 °C, respectively, while for a Reynolds number of 500 the increases are −2.74 °C and 1.38 °C, the increase from 2 to 4 m in pipe depth generates an increase in the cooling and heating potential for a Reynolds of 250 of −1.1 °C and 2.78 °C, respectively.

For a Reynolds of 500, the increases are −4.34 °C and 5.78 °C, with respect to the reference dimensions of 2x5 m. The increase of the burial depth of the GAHE also describes the advantage of its possible implementation in confined installation spaces.

Regarding the diameter of the GAHE, in this work it was numerically demonstrated that the smaller the diameter, the greater the thermal potential of operation. Table 10 and Table 11 show that for the three laminar flows evaluated, with the smallest diameter the greatest difference in temperatures between the outlet and inlet of the GAHE is obtained, it can be said that the diameter of the pipe has an inversely proportional effect on the application potential of the GAHE, which coincides with other studies such as the work carried out by Benhammou, M. et al. (2015) [11].

Agrawal, K. K. et al. (2020) in their published work concerning GAHEs reported that the outlet temperature difference with respect to the inlet temperature increases with increasing soil moisture content, which was confirmed in this work for laminar flows corresponding to Reynolds numbers of 250 and 500. In addition, the authors recommend reducing the length of the GAHEs by using wet soils so that it is possible to install them in less space. They also recommend reducing the air flow velocities and the diameter of the pipe. This is also corroborated by the results obtained in this work, since in all cases the maximum thermal potential obtained of the GAHEs is for air flows a lower velocity (low numbers of Reynolds) and smaller diameters.

Serageldin, A. A. et al. (2016) found similar characteristics to those reported in this work—that the outlet air temperature of a GAHE is reduced from 20.4 °C to 18.7 °C when the pipe diameter is increased from 2 to 3 in, the increase in the horizontal length of the pipe from 5.45 m to 7 m only produces an increase of 0.2 °C in the inlet temperature difference with respect to the outlet [10].

Xamán, J. et al. (2014) studied the effect of the thermal insulation thickness at the GAHE outlet and determined that the optimum thickness is 0.05 m, because increasing that thickness does not improve the thermal heating and cooling potential of GAHEs in temperate climates [38]. Therefore, in this work, it was determined that it is not necessary to cover the optimal length of the outlet pipe (4 m) with thermal insulation; it is only necessary to use insulation in the last 2 m length at the outlet of the GAHE to ensure the thermal potential for heating and cooling.

The use of GAHE as a heating system is not recommended for the conditions evaluated in Table 9 with diameters greater than 8 inches and a flow with a Reynolds number greater than 500. For cooling, the use of GAHE is not recommended in places with air speeds less than or equal to 0.5 m/s and relative humidity of 100%.

5. Conclusions and Future Work

1. A higher cooling and heating potential was observed when increasing the pipe depth with respect to increasing the horizontal length of the air flow path, except for a Reynolds number of 250 as a cooling system, even so, taking into account that the difference between 5 and 25 m of horizontal length is five times greater than doubling the depth, so that, equally, a greater benefit is obtained by increasing the burial depth of the pipe with respect to the result obtained by increasing the horizontal length.

2. The thermal potential of the GAHE evaluated as the temperature difference between outlet and inlet decreases with increasing pipe diameter for both applications (cooling and heating).

3. A higher thermal potential was obtained in the operation of the GAHE for cooling applications with respect to heating; however, in both cases it was observed that, as the relative humidity of the air decreases, the thermal potential of the GAHE increases.

4. The thermal potential of the GAHE increases with increasing soil moisture for laminar flows with Reynolds numbers of 250 and 500; the laminar flow with a Reynolds of 100 describes little variation and does not describe a behavioral trend with soil moisture content.

5. The air flow velocity generates an increase in the operating potential of the GAHEs as the air flow velocity increases up to 4 m/s for both cooling and heating applications.

6. It was also determined that the thermal insulation in the outlet pipe of the GAHE should be 1.5 to 2 m in the last pipe section for systems with burial depth of 3 m or more, because the reduction of the thermal insulation in the outlet pipe presents an average negligible variation of −0.17 °C for cooling applications and 0.098 °C for heating applications, reducing the installation cost of the GAHEs.

The limitations of this study are as follows: The numerical modelling is valid and applicable only to laminar flow; mass transfer between the soil and the GAHE system is not considered; the heat transfer study is performed considering rectangular coordinates; and 2D and steady state and different types of material for the GAHE piping are not analyzed.

The main lines for future work are related to implementing turbulence models and techniques to the developed computational software. It is also intended to collect more experimental data and/or measure the physical and chemical properties of different soils in Mexico, to model the boundary conditions at the soil surface, as well as the initial temperature distribution. Additionally, in the short term, it is necessary to build a modular experimental bench of a GAHE, since it would be possible to control some variables such as the inlet air flow.