Simulation-Based Analysis of a Novel CO₂ Ground Source Heat Pipe (GSHP) to Reduce Temperature Fluctuations in Pavements in Different Climatic Conditions

Abstract

A fully wicked ground source heat pipe (GSHP) is numerically employed and simulated to transfer thermal energy from the subsurface to ground surface pavements to reduce the environmental temperature fluctuations in the pavement, thereby reducing thermal stresses and increasing pavement life. The GSHP can also reduce or eliminate snow and ice buildup on pavement surfaces. Each single GSHP was modeled in a two-dimensional axisymmetric cross-section using COMSOL software, which employs a finite element method. The modeled GSHP consisted of two parts: a disk shape buried below the pavement surface, connected to a cylindrical part embedded in a vertical underground borehole. The GSHP finite element model was validated against published experimental heat pipe data. The simulation results demonstrated that the thermal behavior of the heat pipe system during the cold season could reduce the temperature fluctuations on the pavement surface in six various climate zones. The addition of insulation along the vertical length of the heat pipe was found to significantly reduce heat loss between the heated and unheated pavement surfaces. The low thermal conductivity of the pavement material decreases the performance of the GSHP system. Finally, the maximum-minimum normalization method was applied to the parametric analysis to normalize and compare results for future use.

1. Introduction

Temperature fluctuations on pavement surfaces vary over tens of degrees Celsius daily and seasonally. In contrast, subsurface temperature fluctuations decrease significantly with depth, equilibrating to the annual average air temperature at approximately 5–10 m below the ground surface [1]. Thus, ground source heat pump technologies have been employed in horizontal configurations [2] and vertical configurations [3] and high thermal conductivity pile configurations [4] to extract heat from the underground and release it to pavements. Still, their drawback is the requirement of heat pumps, active circulation pumps, and electrical load to operate the heat pumps. On the contrary, a ground-source heat pipe (GSHP) utilizes a passive heat transfer technology with high thermal conductivity. Previous experiments and implementations of GSHPs used a thermosyphon heat pipe configuration, which only operates if the evaporator section (the heat source) is below the condenser section (the heat sink), usually only occurring in the heating season. In the cooling season, the thermosyphon heat pipe will not operate because the liquid remains at the bottom of the pipe due to gravitational force.

A thermosyphon heat pipe is an evacuated pipe filled partially with a working fluid and consists of three sections: evaporator, condenser, and adiabatic section. In the evaporator of a buried heat pipe, the liquid absorbs heat from the surrounding soil through the wall of the heat pipe. The vapor generated by the evaporation process flows through the vapor channel up to the condenser, where the pressure drop condenses the vapor to liquid at the top part of the heat pipe. The condensed liquid then returns to the evaporator at the bottom of the heat pipe due to the gravitational force. Advantages of utilizing a thermosyphon heat pipe instead of the closed-loop ground heat exchanger in the ground source heat pump during the heating season were observed by [5]. The advantages of using the heat pipe rather than the ground loop heat exchanger are that heat pipes are passive, and higher energy transfers are inherent with phase change. The advantage of a closed-loop heat exchanger is that it is appropriate for heating and cooling seasons due to an increased number of operating hours.

Many potential applications use GSHPs to extract thermal energy stored in the ground and release it below the surface, such as snow melting, soil/turf warming, space heating by integrating the heat pipe with a heat pump, and temperature control on pavement applications. The performance of a CO₂ heat pipe for deicing and snow melting systems was investigated by [6]. They installed twenty heat pipes embedded into five boreholes. They added small cylindrical fins along each condenser pipe to improve the heat distribution below the surface to cover a 165 m² surface area. Whereas [7] designed a ground source heat pipe for soil warming below turf grass. Each heat pipe was built with an evaporator, had a cylindrical shape embedded in the underground, and was constructed for turf warming.

Using the GSHP in cooling applications may require technology to actively transport the liquid from the bottom to the top of the heat pipe if not by capillary pumping. Pavement cooling with the proposed GSHP offers a promising means to mitigate urban heat island effects, which are predicted to increase due to rapid climate change. For example, [8,9] describe extreme heat effects and future impacts on populations in Africa and Asia. The proposed GSHP technology could also offer a passive cooling technology for ground surface cooling.

A literature review revealed that improving the heat distribution below the surface by improving the structure of the top section of the heat pipe is desirable. In winter, as the heat pipe extracts thermal energy from the underground, the ground temperature decreases until the temperature difference between the evaporator and condenser approaches zero. In summer, the opposite would occur, cooling the ground surface while increasing the deeper underground temperature. Thus, Improving the geometry of the top part of the heat pipe can improve the heat transfer; leading to the design of a disk-shaped at top part of the GSHP to allow better heat transfer contact area below the pavement. Ref. [10] studied the performance of the heat pipe that consisted of two disks connected by a cylindrical pipe from the center. The heat pipe was employed in a solar concentrating system to improve the heat distribution inside the thermal storage tank. The effect of the spacing between vertical heat pipes was investigated by [11]. They found that the system’s performance decreased as the number of heat pipes per surface area increased.

The objectives of the present study are to model and optimize the performance of a heat pipe with a disk-shaped section at its top, connected to a vertical section buried in a vertical borehole. The heat pipe model is constructed in the COMSOL modeling environment to conduct numerical experiments to observe system performance to optimize design variables. Design variables to optimize include disk radius and length of the heat pipe, observer under various undisturbed ground temperatures, and ambient temperature conditions.

2. Materials and Methods

The heat pipe model has been constructed as a two-dimensional axisymmetric model in COMSOL software. The liquid in the wick was assumed to be saturated. Thus, the wick is designed to transport the liquid from the bottom to the top of the heat pipe for various pipe lengths. The heating load on the surface was applied by a convective heat flux relevant to ambient temperatures and wind speeds available by typical meteorological year (TMY) hourly data.

2.1. Governing Equations

The governing equations applied to the vapor region are energy, mass, and momentum conservation. The change in vapor density along the heat pipe is governed by the ideal gas law [12].

The density of vapor obtained from the Ideal gas law:

$$\rho =\frac{p}{RT}$$

(1)

Mass conservation:

$$\frac{\partial \rho }{\partial \mathrm{t}}+\nabla .\left(\rho u\right)=0$$

(2)

Momentum conservation:

$$\rho \frac{\partial u}{\partial t}+\rho \left(u.\nabla \right)u=\nabla .\left(-pI+\mu \left(\nabla u+{\left(\nabla u\right)}^T\right)-\frac{2}{3}\mu \left(\nabla .u\right)I\right)$$

(3)

Energy conservation:

$$\rho C_p\frac{\partial T}{\partial t}+\rho C_{p}u . \nabla T+\nabla .q=Q_p+Qvd$$

(4)

$$where: q=-k\nabla T$$

(5)

Viscos-dissipation:

$$Q_{vd}=\tau :\nabla u$$

(6)

Pressure work:

$$Q_p={\alpha }_{p}T \left(\frac{\partial p_A}{\partial t}+u . \nabla p_A\right)$$

(7)

$$where {\alpha }_p=-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial T}\right)}_p$$

(8)

Liquid flow in the wick was neglected because the values of liquid flow velocity through the voids of the porous layers are very small [12]. Thus, the Brinkman equation was employed for single-phase and incompressible flow to model the liquid flow in the wick for validation purposes. The Energy Equation and Brinkman equations can be expressed as:

Energy Equation:

$${\left(\rho C_p\right)}_{eff}\frac{\partial T}{\partial t}+\rho C_{p}u . \nabla T=\nabla .k_{eff}.\nabla T$$

(9)

where:

$${\left(\rho C_p\right)}_{eff}={\theta }_s{\rho }_s{C}_p{,}_s+\left(1-{\theta }_s\right)\rho C_p$$

(10)

For a screen wick: d is the filament diameter (d = 1 mm) of the screen wick, N is the mesh number (N = 360), C is a crimping factor taken to be 1.05, and k_eff is the effective thermal conductivity of the wick.

$$k_{eff}=\frac{k_f\left[k_f+k_s-\left(1-{ϵ}_p\right)\left(k_f-k_s\right)\right]}{k_f+k_s+\left(1-{ϵ}_p\right)\left(k_f-k_s\right)}$$

(11)

$$\epsilon =porosity=1-\frac{\pi CNd}{4}$$

(12)

$$\kappa =\frac{d^2{\epsilon }^3}{122{\left(1-\epsilon \right)}^2}$$

(13)

The fluid flow equation obtained from Brinkman equation:

$$\rho \frac{\partial u_2}{\partial t}=\nabla .\left[-p_{2}I+\mu \frac{1}{\epsilon \left(\nabla .u_2+{\left(\nabla u_2\right)}^T\right)}-\frac{2}{3}\mu \frac{1}{\epsilon }\left(\nabla .u_2\right)I\right]-\left(\mu {\kappa }^{-1}+{\beta }_F\left|u_2\right|+\frac{Q_m}{{\epsilon }^2}\right)u_2$$

(14)

$$Q_m=a mass source or sink \left(\frac{\mathrm{kg}}{{\mathrm{m}}^3·\mathrm{s}}\right)=\rho \nabla .\left(u_2\right)$$

(15)

where $u_2$ is the liquid velocity $\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$, $\epsilon$ is a wick porosity ($\epsilon$ = 0.7), $\kappa$ is a permeability of the porous medium in $\left({\mathrm{m}}^2\right),$ and ${\beta }_F$ is the Forchheimer coefficient $\left(\frac{\mathrm{kg}}{{\mathrm{m}}^4}\right)$. The term of a mass source or sink ($Q_m$) represents the mass flow out of the wick due to evaporation processes and the mass flow entering the wick by condensations processes.

Carbon dioxide (CO₂) was chosen as the working fluid due to its high operating temperature range and nontoxicity, and safety for humans and the environment. The heat of evaporation of the CO₂ at 283 K is approximately 200 kJ/kg [13] (

Table 1. Model material properties.

Material (Object)	$\mathit{k} (\mathbf{W}/\mathbf{m}/\mathbf{K})$	$\mathit{\rho } (\mathbf{kg}/{\mathbf{m}}^{\mathit{3}})$	${\mathit{C}}_{\mathit{p}} (\mathbf{J}/\mathbf{kg}/\mathbf{K})$	γ	$\mathit{\mu } (\mathbf{Pa} \mathbf{s})$
Stainless Steel	19	8030	502.48	---	---
CO2 (Liquid) at T = 283 K	0.098097	860.99	2998.8	3.1	8.25 × 10−5
CO2 (Vapor) at T = 283 K	0.024217	Governed by Ideal gas law	2559.6	2.692	1.61 × 10−5
Soil (ground)	1.5	1742	1175	---	---
Grout (borehole)	1.61	1613.01	1372	---	---
Concrete (pavement)	1.8	2300	880	---	---
Insulation	0.03	24	2500	---	---

). The operating temperature in a heat pipe is the temperature where evaporation or condensation may occur, which for CO₂ is in the operating temperature range from −56.56 °C to 30.85 °C; in the case of reducing the temperature fluctuations on the ground surface, the heat energy stored in underground needs to interact with the ambient conditions through the ground source heat pipe.

2.2. Boundary Conditions

Constant temperature boundary conditions were assumed at a radius of 10 m and a depth of 10 m from the surface. The evaporator section was represented by evaporator length (Le) and began at a depth of 10 m, where the temperature fluctuations due to the ambient climate are negligible. In Figure 1, at the evaporator section at ($r_1$ ₁ = $r_o$ and 0 < $z_1$ < Le), the heat transfers from the soil toward the heat pipe wall are by conduction. The boundary temperature is expressed as:

$$T{│}_{r=10 \mathrm{m} and depth \ge 10 \mathrm{m} }=T_{und}$$

(16)

The soil–wall interface boundary condition is given by:

$$k_{soil}\nabla T.n_r=k_{wall}\nabla T.n_r$$

(17)

At the wall–wick interface ($r_1$ = $r_w$ and 0 < $z_1$ < Le), the medium is a porous material. In the case of considering the liquid velocity in the wick to be zero, the thermal energy transferred through the wick can be solved as a conduction problem [12]. The thermal condition at the wall–wick interface is expressed by:

$$k_{wall}\nabla T.n_r=k_{wick}\nabla T.n_r$$

(18)

The liquid–vapor interface is the line between the vapor core and wick wall in the whole domain. As the liquid reaches the corresponding saturation pressure and temperature in the evaporator section, the liquid in the wick evaporates and transfers the vapor inside the vapor channel. In contrast, the vapor condenses at the condenser as the vapor pressure and temperature drop to the corresponding saturation condition. The Clapeyron Clausius equation governs the liquid-vapor interface to estimate the saturation pressure and temperature.

Clausius Clapeyron equation:

$$\frac{d{P}_v}{P_v}=\frac{d{T}_v}{T_v^2}{h}_{fg}/R$$

(19)

where $P_v$ and $T_v$ are saturated pressure and saturated temperature, respectively.

The heat of vaporization or condensation can be calculated by:

$$Q=\dot{m} h_{fg}$$

(20)

$$q={\rho }_v h_{fg} \left(u\ast n_r+w\ast n_z\right)$$

(21)

where ${\rho }_v$ is calculated by the ideal gas law, $n_r$ is the normal vector in the r-direction, and $n_z$ is the normal vector in the z-direction.

The heat flux was applied on the ground surface due to the ambient weather conditions (Figure 1). The hourly ambient temperature and wind speed data were provided by the typical meteorological year (TMY3) data. In order to determine the convection coefficient “${\mathrm{h}}_{\mathrm{c}}\left(\mathrm{t},\mathrm{T}\right)$” as a function of a surface temperature, ambient temperature, and wind speed based on the empirical formula described by [14].

$$q={\mathrm{h}}_{\mathrm{c}}\left(\mathrm{t},\mathrm{T}\right)\ast \left({\mathrm{T}}_{\mathrm{ambient}}\left(\mathrm{t}\right)-\mathrm{T}\right)$$

(22)

where:

$${\mathrm{h}}_{\mathrm{c}}\left(\mathrm{t},\mathrm{T}\right)=698.24 [0.00144 {\left(\frac{{\mathrm{T}}_{\mathrm{ambient}}\left(\mathrm{t}\right)+\mathrm{T}}{2}\right)}^{0.3} W_{sp}{\left(t\right)}^{0.7}+0.00097\ast {\left|\mathrm{T}-{\mathrm{T}}_{\mathrm{ambient}}\left(\mathrm{t}\right)\right|}^{0.3}]$$

(23)

where $W_{sp}\left(t\right)$ is the wind speed as a function of time, T is the surface temperature (K), and t is a time (s).

The pavement surface heated by the GSHP buried underneath is the heated surface. The GSHP system’s dimensions and surroundings are illustrated in

Table 2. Dimension of heat pipe and ground.

Parameter	Value
Heat pipe
Le (m)	From 1 to 40
L (m)	10 (m) + Le
Wc (m)	From 0.25 to 1
$r_v$ (m)	0.0312
δwall (m)	0.001
δwick (m)	0.00376
Hc (m)	0.0216
Ground
Pavement thickness (m)	0.064
Insulation length (m)	10
Insulation thickness (m)	0.060
Total soil length (m)	50.1
Total soil width (m)	10

. The surface adjacent to the heated surface without a GSHP beneath is unheated. In Figure 1, the heat transfer between the heated and unheated surfaces is blocked by adding an insulation wall to a depth of 10 m to avoid the effect of the ambient climate on the underground. If the depth is less than or equal to 10 m, the seasonal ground temperature fluctuations are high. Still, if the depth is greater than 10 m, the seasonal ground temperature fluctuations are very low. As a result, the surrounding area at a depth that is less than or equal to 10 m can be called a high-temperature fluctuation (HTF) zone, and at a depth is greater than 10 m, the surrounding area can be called a low-temperature fluctuation (LTF) zone.

$$q{│}_{r=W_c and depth \le 10 \mathrm{m}}=k_{insulation}\nabla T.n_r$$

(24)

2.3. Climate Locations

To examine the performance of the heat pipe under various thermal conditions, the present study included six different cities from different climate zones (

Table 3. Summary of weather data in different modeled locations.

City	1	2	3	4	5	6
	Albuquerque NM	Amarillo TX	Baltimore MD	Columbus OH	Lexington KY	Portland OR
${\mathrm{T}}_{\mathrm{ambient}}$, max. (°C)	16.1	21.6	20.6	17.2	13.3	14.4
${\mathrm{T}}_{\mathrm{ambient}}$, min. (°C)	−7.8	−11.3	−13.9	−17.8	−20	−2.2
${\mathrm{T}}_{\mathrm{g}}$(°C)	16	16.5	13	12.5	14	13
Snowfall rate (h/year)	44	64	56	92	50	15

). The ambient weather data were obtained from TMY3 hourly data, and the underground temperatures and the snowfall rates were obtained from [15]. The modeled period of transient analysis was from 2 December to 31 January.

The CO₂-GSHP has been modeled as an axisymmetric two-dimensional model in COMSOL Multiphysics software. The model was discretized into a mesh with 525,838 elements based on the finite element method as shown in Figure 2. The element size was selected to be an extra-fine mesh in COMSOL. The convergence criteria value is set by default in COMSOL software with 0.01 relative tolerance for the time-dependent solver and 0.001 relative tolerance for the steady-state solver.

3. Results

3.1. Validation

3.1.1. Steady-State Conditions

The numerical solution was validated with two different experimental results. The high-temperature heat pipe (HTHP) experiment [16] shows the pressure drop curve along the vapor channel’s centerline similar to the pressure drop curve in the converging-diverging nozzle [17]. The HTHP experiment was completed with stainless steel heat pipe filled with sodium as a liquid in the wick and gas in the vapor channel. The lengths of the evaporator, adiabatic, and condenser sections are 0.1, 0.05, and 0.35 m, respectively. The vapor channel radius is 7 mm, and the wick and wall thicknesses are 0.5 and 1 mm, respectively. The inlet heat energy from the heater to the evaporator was 560 W. The condenser was cooling by the convection heat transfer with (58.5 W/m²/K) convection coefficient and (300 K) reference temperature. The curve in Figure 3 shows the comparison between the experiment result, the numerical result of [18], and the present numerical result. In addition, The HTHP model was examined many times with different element sizes in COMSOL software to investigate the impact of the mesh element size on the accuracy of the results. It showed that as the element size decreases, the accuracy of solving increases, as shown in Figure 4. The vapor temperature was reduced due to an increase in vapor velocity in the evaporator section and adiabatic section; then, the temperature increased due to releasing the latent heat energy by condensation in the condenser section. Thus, the prediction of the present solution has a similar thermal behavior compared to the experimental data and the previous numerical work. In the low-temperature heat pipe (LTHP) experiment completed by [19], a copper heat pipe was filled with water in the voids of wick and vapor in the vapor channel. The evaporator, adiabatic, and condenser lengths were 60.5, 616.5, and 300 mm. The vapor channel radius was 10.25 mm, and the thicknesses of each wick and wall were 0.75 and 1.7 mm, respectively. The power inlet to the evaporator was 97 W, and the condenser was cooling by a water-cooling jacket. The drop in pressure inside the LTHP has a very low value compared with the initial pressure; as a result, the vapor temperature at the centerline was approximately constant.

3.1.2. Transient Modeling Validation

The results of transient analysis completed with COMSOL software agree with the transient modeling (Figure 5) done by [20]. The input data of the numerical analysis solved by Cao and Faghri were stainless steel as pipe material. The evaporator, adiabatic, and condenser lengths were 0.105, 0.0525, and 0.5425 m, respectively. In addition, the vapor radius was 0.007 m, and the thicknesses of wick and wall were the same 0.001 m. The working fluid was sodium. The steady-state heat input was 623 W, and the convective heat flux at the condenser was applied with 39 W/m²/K as a convection coefficient and 300 K reference temperature. In the initial transient analysis problem at t = 0, the value of inlet power at the evaporator was assumed to increase from 623 to 770 W.

3.1.3. GSHP Transient Analysis with Liquid Flow in the Wick

The liquid flow through the wick is required to maintain the wick saturated. The liquid temperature rises as the heat transfer through the pipe wall increases. The evaporation process begins as the liquid pressure and temperature reach the saturation condition. However, the liquid flow velocity is very low compared to a vapor velocity. The low velocity of the liquid flow within the low thickness of the liquid layer leads to a negligible difference between both the liquid flow and static liquid cases (Figure 6). The simulation was applied in Portland, OR, for a GSHP with a 24 m length and 0.75 m condenser radius, whereas the other dimensions were based on the values in Table 1. The initial values of the system in the transient analysis were solved as a steady-state problem, then solved as a transient simulation in both cases.

3.2. Parametric Analysis at Steady-State Conditions

In the heating season, heat transfers from the subsurface through the outer wall of the heat pipe and is then absorbed by CO₂ liquid restrained in the wick along the heat pipe. While the temperature of the CO₂ liquid reaches saturation temperature, the CO₂ evaporates and transfers the vapor into the vapor channel. As a result, the pressure in the vapor channel increases, and the vapor flows to the top of the heat pipe due to the pressure difference. The heat is absorbed in the top part during the cooling season and released at the bottom. The daily temperature fluctuations allow the recovery of heat absorbed by GSHP from the underground to transfer the heat to the cold surface. Thus, the GSHP operating hours increase beyond a thermosyphon case, improving performance.

The variation in the length of the evaporator (Le) and radius of the condenser (Wc) significantly affects the GSHP performance. The contact area between GSHP and underground soil can be increased as the evaporator’s length increases, increasing the heat released below the surface. However, the operating limits of the heat pipe and the gravitational pressure drop need to be considered. In order to transport the CO₂ liquid to the top of the heat pipe through the wick, the pore radius of the wick needs to be very small (i.e., on the order of nanometers). To maximize the heat flux without increasing the length of the evaporator, we can reduce the radius of the condenser so that the required heat flux reduces, as shown in Figure 7 and Figure 8.

3.3. Transient Analysis of GSHP in Various Climate Zones

The GSHP performance was modeled in varying climate zones to examine the effects of ambient and subsurface temperatures. During this work, it became evident that a significant heat loss occurs through the unheated zone (Figure 9a) unless the vertical length of the heat pipe is insulated (Figure 9b). As shown in Figure 9a, the simulation of a GSHP system has a 24 m pipe length and 0.75 m radius of condenser applied under Baltimore, MD, city climate conditions on the 46th day. The black arrows represent the conductive heat flux with the size proportional to the heat flux value at the tail of each arrow. In the case of the system without insulation, the value of the conductive heat flux increases in the direction of the unheated surface, and soil temperature decreases in the same direction. Whereas in the case of adding insulation, the thermal contact between the unheated surface and the edge of the top part blocked by insulation has a 0.03 W/m/K thermal conductivity and 60 mm thickness. As a result, the soil temperature remained at a reasonable value. The ambient temperature at this location on the 46th day was −10.6 °C, and the wind speed was 7.6 m/s.

The performance of the GSHP system was then investigated with different configurations to reduce the temperature fluctuations and avoid freezing conditions on the pavement, as shown in Figure 10 and Figure 11. The domain surrounding the heat pipe can be divided into three zones: (1) the soil surrounding the heat pipe at the evaporator section (0 ≤ $z_1$ ≤ Le), which we call a low-temperature fluctuation (LTF) zone at depth equal to or greater than (10 m); (2) the soil above the condenser section which represents soil and pavement zone; and (3) the soil surrounding the heat pipe from an evaporator section up to the lower wall of a disk part at (Le < $z_1$ ≤ L), which is called a high-temperature fluctuation (HTF) zone. The initial condition in transient modeling was based on a result of steady-state analysis with an ambient temperature value that was equivalent to the average ambient temperature of the past month, which was November, based on the period of the present work where the period goes from 2 December to 31 January and is equivalent to 60 simulation days. The first heat pipe configuration was (L = 25 m) total length of a heat pipe with (Le = 15 m) evaporating length and radius of the condenser (Wc = 1 m). The insulation added to prevent heat transfer between the heated and unheated surface extended 10 m depth to investigate the thermal behavior between the heat pipe and surrounding soil at the HTF zone. Moreover, the period represented the heating season with the design goal of keeping the pavement temperature above freezing.

A review of the simulation results shown in Figure 10 revealed that the GSHP system with configuration 1 reduced the temperature fluctuations on the pavement for all locations and maintained the minimum surface temperature at Albuquerque and Portland above the freezing point, but not in Amarillo, Baltimore, Columbus, and Lexington. Therefore, in those four locations, the heat pipe configuration needs improvement by either increasing the heat pipe length, reducing the radius of the condenser, or both. The minimum ambient temperatures for the six locations were in Columbus and Lexington, which were -17.8 °C and −20 °C, respectively, and the ground temperatures (T_g) at Columbus and Lexington were 12.5 °C and 14 °C, respectively. The corresponding minimum surface temperatures of the heated surface during the study period were −5.3 °C and −7 °C in Columbus and Lexington cities, respectively. The temperature on the unheated surface approached the ambient temperature values with a noticeable lag in a case of a sudden change in ambient temperature. Thus, the GSHP system raised the minimum temperature on the heated surface in Columbus city by 6.3 °C from −10.9 °C on the unheated surface to −4.6 °C on the heated surface. In Lexington, the GSHP raised the minimum temperature on the heated surface by 6.66 °C from −12.9 °C on the unheated surface to −6.24 °C on the heated surface. The maximum heat pipe length in the present work is 50 m, and the minimum radius condenser was 0.25 m.

The GSHP with configuration 2 was modeled as a 50 m length, and 0.25 m radius of the condenser (

Table 4. Heat pipe configurations.

	Configuration 1	Configuration 2
Total length, m	25	50
Evaporator length (Le), m	15	40
The radius of the condenser (Wc), m	1	0.25

) applied at the four locations, with a minimum surface temperature below the freezing point with configuration 1. In Figure 11, the GSHP system maintained the minimum surface temperature above the freezing point in Amarillo. Still, in Baltimore, Columbus, and Lexington, the average minimum surface temperature was −0.82 °C, −0.8 °C, and −1.73 °C, respectively. Thus, increasing in GSHP length from 25 to 50 m and decreasing the radius of the condenser from 1 to 0.25 m enhanced the GSHP performance by raising the minimum temperature for each of the four cities in Figure 11a–d from −2.65, −3.7, −5.3, and −7 °C with configuration 1 to 2.2, −0.82, −0.8, and −1.73 °C, respectively. The maximum temperature difference between heated and unheated surfaces occurred in Columbus, OH, at 16.8 °C on the 56th day. The change in average ground temperature was observed at the point at z1 = Le out of the heat pipe wall. The low ground temperature indicated a high heat absorption rate by the GSHP system relative to a low conductive heat flux through the soil for heat recovery in the heating season. Thus, the GSHP system may need to be enhanced again. In Columbus, the average ground temperature was 7.3 °C with configuration 1; then, the ground temperature was raised to 11.6 °C with configuration 2. The ground temperature with a second configuration showed that the rate of heat absorption was low at this location because the increase in heat pipe length led to an increase in the contact area between the heat pipe and surroundings.

3.4. Parametric Analysis

A parametric analysis of key design variables was conducted to determine the effects of key design variables on GSHP performance. Parameters varied, including the heat pipe length and condenser radius for the six various climate locations mentioned above. The min-max normalization method was applied to scale the values of ambient temperature, ground temperature, the radius of the condenser, and the length of the heat pipe. Thus, the value of all variables was scaled to a value between 0% and 100%. The minimum percentage describes the minimum value for each variable, except the radius of the condenser, where the minimum percentage shows the maximum radius because as the radius increases, the convective heat flux to the ambient increases.

Figure 12 shows simulation results compiled from many transient simulations similar to those described above. As shown in Figure 12, the lowest case with a minimum surface temperature equivalent of −2 °C, which also had the lowest ambient temperature (scaled to 0%), required a higher heat pipe length to maximize the heat of absorption and lower radius of the condenser to minimize the convective heat flux to the ambient. Regarding ground temperature, the low ground temperature required a higher length and lower radius of condenser, such as the case with a minimum surface temperature equivalent to −0.8 °C. In contrast, the case with a minimum surface temperature equal to 0.6 °C, where the system was just capable of achieving a surface temperature above freezing, required a lower pipe length (scaled to 18%) and lowest radius of condenser because of a high ground temperature and high ambient temperature compared to other locations.

3.4.1. The Effect of the Climatic Conditions

A parametric analysis was conducted to examine the effects of various climatic conditions by fixing the GSHP to configuration 1 (described above) with 25 m length and 1 m radius of the condenser. The results are shown in Figure 13. The ambient temperature and relative wind speed seem to have a significant influence compared to the undisturbed ground temperature, as seen in Figure 13. For the case where the surface temperature is -1.5 °C, the undisturbed ground temperature was normalized to 100%, indicating the maximum undisturbed ground temperature, implying that the heat recovery through the soil up to a wall of heat pipe should be higher as compared to other locations. However, it has a 33% normalized temperature at the wall of GSHP because of the high amount of heat absorbed by the system. For the case of 4.44 °C surface temperature, the ambient temperature was normalized to 100%, implying that the GSHP absorbs less heat than other locations. However, the undisturbed ground temperature was normalized to 13%, and the temperature at the wall of the heat pipe was 43%, which is higher than the first case of −1.5 °C surface temperature. The thermal resistances at the pavement down to the top of a heat pipe are lower than the thermal resistances from the temperature boundary to the heat pipe wall. Thus, the ambient temperature has a considerable effect compared to the undisturbed ground temperature; the temperature at the wall of the heat pipe indicates the amount of heat absorbed by the system compared to the recovered heat either by conductive heat flux through the soil or by the higher ambient temperature during the period of study.

3.4.2. The Effect of the GSHP Size on the Performance

Figure 14 summarizes normalized heat pipe lengths, condenser radii, and heat pipe wall temperature required to meet various pavement surface temperatures at each climate location. As one might expect, both the radius of the condenser and the length of the evaporator have a significant effect on the GSHP performance to increase the pavement surface temperature through the cold season (

Table 5. Albuquerque, NM; city 2: Amarillo, TX; city 3: Baltimore, MD; city 4: Columbus, OH; city 5: Lexington, KY; city 6: Portland, OR).

City	1	2	3	4	5	6
${\mathrm{T}}_{\mathrm{ambient}}$, max. (°C)	16.1	21.6	20.6	17.2	13.3	14.4
${\mathrm{T}}_{\mathrm{ambient}}$, min. (°C)	−7.8	−11.3	−13.9	−17.8	−20	−2.2
Tg (°C)	16	16.5	13	12.5	14	13
Configuration 1
Le (m)	15	15	15	15	15	15
Wc (m)	1	1	1	1	1	1
Tsurface, max. (°C)	12.2	16.4	15.4	13.9	11.7	12
Tsurface, min. (°C)	0.42	−2.65	−3.7	−5.3	−7	4
Configuration 2
Le (m)	40	40	40	40	40	40
Wc (m)	0.25	0.25	0.25	0.25	0.25	0.25
Tsurface, max. (°C)	-	18.2	16.3	15	13.35	-
Tsurface, min. (°C)	-	2.2	−0.82	−0.8	−1.73	-
Heat pipe optimization (ΔTmin)	-	4.85	2.88	4.5	5.27	-

). In the parametric simulations, the minimum values of evaporator length and radius of condenser were 10 m and 0.25 m, respectively, and the maximum values were 50 m and 1 m, respectively. In Albuquerque, the surface temperature increased from −0.3 °C to 1.24 °C due to increased heat pipe length from 11 m to 25 m. At the same time, the temperature fluctuations reduced significantly, and the ground temperature at the wall increased from 7.7 °C to 14 °C due to the increased contact area between the outer wall and the surroundings along the heat pipe. The minimum surface temperature in Columbus and Lexington was −0.8 °C and −0.82 °C, while the ground temperature at the wall of the heat pipe was relatively high in both cities. While the pavement surface temperature drops below freezing, this was only for a small number of hours, and the temperature fluctuations in the pavement dropped significantly.

4. Conclusions

The numerical analysis of the two-dimensional axisymmetric model of a fully wicked ground source heat pipe (GSHP) system has been investigated under different climatic conditions. The vertical cylindrical is extracted the heat from the ground; then, the heat is released below the pavement through a disk part. The system efficiently reduces the temperature fluctuation on the pavement by increasing the temperature difference between the heated and unheated surface up to 16 °C. The length of the heat pipe is associated with the rate of heat absorbed from the ground, whereas the radius of the condenser is associated with the area of the target surface. Hence, increasing the length of the heat pipe and decreasing the condenser radius optimizes the GSHP system’s performance. The edge of the disk part is influenced by unnecessary heat loss from the unheated surface, which affects the performance of the GSHP system. Thus, adding insulation to the edge of the disk part leads to increased soil temperature at the edge of the disk part by 3 °C. It was also found that the pavement materials of the construction result in relatively high thermal resistance, and thus, performance can be improved by lowering pavement thermal resistance.