2.2.5. The Initial and Boundary Conditions

• The initial conditions are:

$$T\_s = T\left(\mathbf{x}, t\right), \; t = 0 \tag{9}$$

*Energies* **2020**, *13*, 6598

$$T\_{\mathcal{R}} = T\_{s\prime} \ t = 0 \tag{10}$$

• The boundary conditions of the computational domain of the horizontal GHE are:

$$\frac{\partial T\_s}{\partial \mathbf{x}} \mid\_{\mathbf{x}=0} = 0 \tag{11}$$

$$\frac{\partial T\_s}{\partial \mathbf{x}} \mid\_{\mathbf{x} = \mathbf{x}\_{\text{max}}} = \mathbf{0} \tag{12}$$

$$Q\_{ca} \; y = 0 \tag{13}$$

$$\frac{\partial T\_s}{\partial y}|\_{y=y\_{\text{max}}} = 0\tag{14}$$

• The boundary conditions of the computational domain of the vertical GHE are:

$$Q\_{cf} \, r = r\_{\text{in}} \tag{15}$$

$$\frac{\partial T\_s}{\partial r}|\_{r=r\_{\text{max}}} = 0\tag{16}$$

$$Q\_{ca\prime}z = 0\tag{17}$$

$$\frac{\partial T\_s}{\partial z}|\_{z=z\_{\text{max}}} = 0\tag{18}$$

The system of differential Equations (1), (2), and (4)–(6) were solved by using an explicit finite difference scheme, which provided the accurate solution for the temperature distribution in the soil, grout, pipe, and fluid. As an example, Figure 4 shows the computational domain of the vertical GHE. The working fluid exchanges the heat energy with the pipe's surface, which then flows through the pipe wall in a 1D way. The heat is then transferred through the grout and surrounding soil in a 2D manner. Two different boundary conditions are applied at the boundary of the soil domain. The convection heat transfer is considered at the ground surface, while the adiabatic conditions are applied at the bottom and lateral edges of soil domain boundaries.

**Figure 4.** Computational domain of the vertical GHE.

To achieve the stability of the explicit numerical calculations, the time step Δ*t* must be within the Courant–Friedrichs–Lewy stability range [26], which is given by:

$$\left|\psi\right| \le 1, \ \Delta t \le \frac{\Delta z}{v\_f} \tag{19}$$

Total of heat exchange by the GHE was calculated in accordance with the following equation:

$$Q = \dot{m}c\_p (T\_{fi} - T\_{fo})\Delta t\tag{20}$$

For the split flow mode, the final temperature of mixed fluid from both the horizontal and the vertical GHE was found as:

$$T\_{f\text{mix}} = \frac{\left(\dot{m}\_{fh}\,\text{c}\_f\,\,T\_{f\text{oh}} + \dot{m}\_{fv}\,\,\text{c}\_f\,\,T\_{f\text{av}}\right)}{\left(\dot{m}\_{fh}\,\,\text{c}\_f + \dot{m}\_{fv}\,\,\text{c}\_f\right)}\tag{21}$$

The described GHE's models were validated against the experimental data [21–23] and will be used to simulate the performance of a combined GHE.

#### **3. A Hypothetical Case Study**

In this study, each arrangement of the GHEs (horizontal or vertical) was set to have the same parameters including the pipe length, pipe diameter, and fluid flow rates. The soil domain around the GHEs was assumed to be a single soil layer, which was homogeneous and isotropic. Table 1 presents the parameters of the reference case used in the simulation to be described next.


**Table 1.** The parameters of the reference case.

The initial temperature of the ground was estimated using the analytical equation (4) suggested by Baggs [24]. The Baggs' equation was determined based on the input parameters of the climate conditions data. The amplitude data for annual air temperature and the average air temperature was obtained from the Australian Bureau of Meteorology [29]. The variable ground temperature for the local site was based on the data given by Baggs [24]. Table 2 summarises the parameters of the reference used to estimate the soil temperature conditions. Figure 5 shows the typical changes of soil temperatures in Adelaide and Brisbane at the end of winter (in August) and summer (in February). As observed from the figure that the soil temperature in Adelaide was lower than in the Brisbane. This tendency could be affected by mild, and generally a warm and temperate climate of Adelaide compared to humid subtropical climate of Brisbane as represented by the parameter of average annual air temperature (*Tm*) in Equation (11). Besides the local site variable of the ground temperature (Δ*Tm*), which varied with geographic location, it also affects the soil temperature. As an example, Figure 6 shows the ambient temperature in Adelaide and Brisbane on three consecutive summer days. These were used in the case study. In general, it can be seen from the figure that the pattern of air temperature was almost the same. The profile of air temperature fluctuated during day and night. For these three randomly selected summer days it was found that the air temperature in Adelaide was slightly higher than that in Brisbane.

**Table 2.** The parameters reference used to estimate the soil temperature.


**Figure 5.** Typical soil temperature in Adelaide and Brisbane.

**Figure 6.** An example of ambient temperature in Adelaide and Brisbane on 3 consecutive summer days.

The effect of seasonal changes on the soil temperature is modelled by incorporating the internal source term concept into the GHE model [21]. The value of the internal source term varied with the soil depths. It was higher in a shallow region and lower in a deeper zone. At a depth of 12 m below the ground surface, the value of the internal source term was assumed to be zero (this corresponded to the depth at which the effect of the ambient temperature on the soil temperature was negligible). Figure 7 shows the absolute value of the internal source term at various depths for two reference locations, namely Adelaide and Brisbane. As is shown in Figure 7, at the upper layer (0–3.5 m depth), the internal source term of the soil in Brisbane was relatively higher than in Adelaide. However, at a deeper layer, the condition was in the opposite. This phenomenon could be affected by meteorological, terrain, and subsurface conditions. The value of the internal heat source was positive when the ground temperature was warm (August to February) and conversely it was negative when the ground temperature was cool (February to August), as is shown in Figure 5.

**Figure 7.** The absolute value of the internal heat source at various depths.

#### **4. Results and Discussion**

This section shows the simulation outcomes of the GHE subjected to five operational modes, as described above. The analysis and discussion on continuous operation, intermittent operation, split flow operation, climate condition, and variations of the fluid mass flow rate will be discussed in the following section.
