**3. Moving infinite line source**

To consider the effect of groundwater flow, we evaluated the TRT measurement results in Bangkok and Hanoi by applying the Moving Infinite Line Source (MILS) theory. According to Diao et al., the temperature increase at a radial position, φ(rad), from a line source is expressed as [13]:

$$T(r,\eta,t) - T\_0 = \frac{q}{4\pi\lambda} \exp\left(\frac{u\_{\rm eff}r}{2D}\cos\left\{\eta\right\}\right)^{r^2/4Dt} \int\_0^{r^2/4Dt} \frac{1}{\eta} \exp\left[-\frac{1}{\eta} - \frac{u\_{\rm eff}^2r^2\eta}{16D^2}\right] \mathrm{d}\eta \tag{7}$$

*Effect of Groundwater Flow and Thermal Conductivity on the Ground Source Heat Pump… DOI: http://dx.doi.org/10.5772/intechopen.99061*

where, *η =* 4*D*(*t*–*t*')/*r* 2 , *ueff* (m<sup>2</sup> s �1 ) is the effective velocity of groundwater flow assuming local thermal equilibrium, calculated by:

$$
u\_{\rm eff} = \mu \frac{\rho\_w \sigma\_w}{\rho \sigma} \tag{8}$$

Here, *u* (ms�<sup>1</sup> ) is the seepage velocity, *ρ<sup>w</sup>* (kgm�<sup>3</sup> ) and *cw* (Jkg�<sup>1</sup> K�<sup>1</sup> ) are the volumetric mass density and the specific heat of water, respectively. The following relationship defines the volumetric mass density and specific heat of the medium:

$$
\rho \mathfrak{c} = (\mathbf{1} - \mathfrak{e}) \rho\_s \mathfrak{c}\_s + \mathfrak{e} \rho\_w \mathfrak{c}\_w \tag{9}
$$

Where *ρ<sup>s</sup>* (kgm�<sup>3</sup> ) and *cs* (Jkg�<sup>1</sup> K�<sup>1</sup> ) are the volumetric mass density and the specific heat of the soil matrix, respectively. Eq. (7) calculates the temperature of the soil medium at an arbitrary position adjacent to the line source. The following equation represents heat balance between average fluid temperature and borehole wall temperature [14]:

$$\overline{T}\_f(t) = \frac{1}{2\pi} \int\_0^{2\pi} T(r\_{bh}, \rho, t) \mathbf{d}\rho + qR\_{bh} \tag{10}$$

where *Rbh* (mKW�<sup>1</sup> ) is the borehole heat resistance.

The simulation using the MILS analytical solution is valid under the following assumptions:


### **4. Discussion**

**Figure 2** shows the TRT measurement results and the numerical simulation using Eq. (7) by setting the *ueff* = 0 and apparent thermal conductivity similar to those calculated using the ILS method. Additional parameters for the numerical simulations are listed in **Table 1**. The discrepancies between simulations and measurements at the beginning are likely the indication of heat transfer within boreholes, which have different thermal properties than the surrounding soil [14]. It is essential to ensure the apparent thermal conductivity adequately represents the value of soil mass. Typically, only the last few hours of results are considered for the linear fitting of Eq. (5).

#### **4.1 Estimation groundwater flow velocity and effective thermal conductivity**

By using the MILS method, the groundwater flow can be taken into account. However, the determination of groundwater velocity remains a problem. A

**Figure 2.** *The average fluid temperatures obtained from TRT measurements and the simulation results using Eq. (7).*


**Table 1.**

*Simulation parameters for TRT data using Eq. (7) (u*eff *= 0).*

numerical approach incorporating a parameter optimization method has been proposed [14, 15]. In this study, a similar numerical procedure was performed in Matlab employing the *fminsearch* function. The *fminsearch* is a pre-programmed function to search the minimum unconstrained multivariable function using the derivative-free method. The parameter estimation from the TRT results employs the *fminsearch* function to find the minimum value of Root Mean Square Error (RMSE) between the MILS model and the TRT result [14].

$$RMSE = \sqrt{\frac{\sum\_{i=1}^{N} \left(\overline{T\_f}(sim)(i) - \overline{T\_f}(meas)(i)\right)^2}{N}} \tag{11}$$

*Effect of Groundwater Flow and Thermal Conductivity on the Ground Source Heat Pump… DOI: http://dx.doi.org/10.5772/intechopen.99061*

The initial values of the fitting parameters are given; thermal conductivity, effective groundwater velocity, and borehole heat resistance. Three values are provided for each parameter, resulting in a total

Tfðsim<sup>Þ</sup>

of 27 combinations. Some of the results from different combinations yield similar results. To summarize the results, we select some representative values from these combinations, as shown in **Table 2**.

**Figure 3** compares TRT results and the simulation results considering the optimized parameters listed in **Table 2** for both sites. The simulation results fit well with the TRT measurements. The higher RMSE error for the Bangkok site is due to the fluctuations of the measured data. As expected, the MILS models predict smaller thermal conductivities than those calculated using the ILS method for all cases. The smaller thermal conductivities are because the MILS models account for the effect of groundwater flow. It is interesting to note that the H1, H2 and H3 yield a similar value of thermal conductivity and RMSE but differs in groundwater velocities. The predicted velocities are low, and their effect on the calculated temperature response is not significant. In addition, each model shows the convergence of borehole thermal resistance values, except for the case of B1. Overall, the differences between the simulated temperatures are close to each other and visually difficult to be distinguished.

## **4.2 Effect of groundwater flow and ground thermal conductivity on the performance of ground heat exchanger**

In the previous discussion, the effect of different thermal conductivities and groundwater velocities over the short TRT measurement period is not distinct. To further examine the impact of these parameters on the fluid temperatures, we extend the simulation period to one year. The simulations assume constant heating rates similar to the field TRT measurement. The simulation results are shown in **Figures 4** and **5** for Bangkok and Hanoi models, respectively. The calculations considering the thermal conductivity values obtained using the ILS method are also presented. For the Bangkok cases, the discrepancies due to the various estimated thermal conductivities and effective groundwater velocities are clearly observable. The B3 and B4 cases, which have the lowest predicted effective groundwater velocities, show higher average fluid temperatures. Meanwhile, B1 and B2 cases yield lower average fluid temperatures.

Interestingly, the average fluid temperature converges into an asymptotic value for the highest effective groundwater velocity (B1 case) after few days. It reveals the critical role of the convective heat transfer to the ground heat exchanger performance over an extended period.


**Table 2.** *List of parameters from the MILS fitting simulation against the TRT data.*

**Figure 3.**

*The average fluid temperatures from TRT measurements and the best fits MILS simulation using parameters as listed in Table 2.*

**Figure 5** shows the differences over the extended period for the Hanoi numerical model. While H1, H2, and H4 yield almost similar values of average fluid temperatures, the highest predicted effective groundwater velocity, the H3 case, shows the lower average fluid temperature.

**Figures 6** and **7** show the contour plots representing the soil temperature increase after a year of constant heating. The ground heat exchanger is located at the center coordinate (0,0), and the grid intervals are shown in meter-unit. The groundwater flows to the positive x-direction (in **Figure 6 right** and **Figure 7 right**). The left-hand part in both **Figures 6** and **7** is the simulation result in case the apparent thermal conductivity calculated by the ILS method is used (*ueff* = 0). In comparison, the righthand part is the simulation result using the optimized value of thermal conductivity and groundwater velocity (B2 and H3 cases). The maximum temperature and the shape of isothermal lines for Bangkok show that the case with groundwater velocity *ueff* = 4.55E-07 ms<sup>1</sup> , despite its lower thermal conductivity, provides better thermal performance than the case with *ueff* = 0. On the contrary, the numerical results for Hanoi cases (**Figure 7**) show the opposite due to the low groundwater velocity.

Despite the advantage of MILS, the reverse analysis involving parameter optimization performed in this study results in several combinations of parameters. The *Effect of Groundwater Flow and Thermal Conductivity on the Ground Source Heat Pump… DOI: http://dx.doi.org/10.5772/intechopen.99061*

#### **Figure 4.**

*Effect of different parameters (thermal conductivity and effective groundwater velocity) to a year cycle of average fluid temperature under a constant heat rejection rate (*q *= 39.72 Wm<sup>1</sup> ) for Bangkok case.*

#### **Figure 5.**

*Effect of different parameters (thermal conductivity and effective groundwater velocity) to a year cycle of average fluid temperature under a constant heat rejection rate (*q *= 35.91 Wm<sup>1</sup> ) for the Hanoi case.*

improvement should be emphasized on the TRT method to provide data to narrow down the resulting combinations into the best possible solutions. One of the most feasible methods is providing different heating rates for a TRT site.

#### **4.3 Long-term GSHP performance in Bangkok and Hanoi**

One of our main interests in the GSHP potential application in Southeast Asia is to estimate the GSHPs long-term sustainability. Thus, it is essential to examine further the effect of different ground thermal properties on the long-term

#### **Figure 6.**

*Contour plot showing the soil temperature increase after a year of constant heating for the Bangkok site; (left:* Q *= 39.72 Wm<sup>1</sup> , λ = 1.82 Wm<sup>1</sup> K<sup>1</sup> ,* ueff *= 0 ms<sup>1</sup> ; right:* Q *= 39.72 Wm<sup>1</sup> , λ = 1.68 Wm<sup>1</sup> K<sup>1</sup> ,* ueff *= 4.55E-07 ms<sup>1</sup> ).*

**Figure 7.**

*Contour plot showing the soil temperature increase after a year of constant heating for the Hanoi site; (left:* Q *= 35.91 Wm<sup>1</sup> , λ = 1.42 Wm<sup>1</sup> K<sup>1</sup> ,* ueff *= 0; right:* Q *= 35.91 Wm<sup>1</sup> , λ = 1.34 Wm<sup>1</sup> K<sup>1</sup> ,* ueff *= 1.14E-07 ms<sup>1</sup> ).*

performance of GSHPs. So far, the MILS analytical model is only limited to a single ground heat exchanger. Here, we propose a simple modification to the described analytical models to simulate two ground heat exchangers.

The maximum number of ground heat exchangers in the numerical model depends on the symmetrical arrangement of boreholes and whether the model considers the groundwater flow. The simplest numerical model disregarding the effect of groundwater flow can simulate up to four ground heat exchangers. The numerical model can simulate a maximum of two ground heat exchangers if the groundwater flow is considered. In such a case, the boreholes must be arranged perpendicular to the groundwater flow direction.

The calculation of the temperature field uses the superposition method. Once the average fluid temperature for one of the boreholes is calculated, the total average fluid temperature flowing from/to the heat pump can be calculated owing to its symmetrical arrangement. The following additional equation is required to calculate the unknown inlet and outlet temperatures,*Tbh-in* and *Tbh-out*.

*Effect of Groundwater Flow and Thermal Conductivity on the Ground Source Heat Pump… DOI: http://dx.doi.org/10.5772/intechopen.99061*

$$T\_{bh-in} - T\_{bh-out} = \mathbb{Q}'\_{\rho\_f \mathfrak{G} \mathfrak{c}\_f} \tag{12}$$

Where *Q* (Watt) is the heat rejection rate from heat-pump to the ground, *ρ<sup>f</sup>* (kgm�<sup>3</sup> ), *ϑ* (m<sup>3</sup> s �1 ) is the volumetric flowrate of heat exchanger fluid, and *cf* (Jkg�<sup>1</sup> K�<sup>1</sup> ) is the specific heat capacity of heat exchanger fluid. The inlet and outlet temperatures can be calculated by combining Eqs. (4) and (12).

The long-term simulation model considers a GSHP system with a heat rejection rate of 5 kW. The heat pump connects to two vertical 50 m-long ground heat exchangers in a parallel flow configuration. A parallel flow configuration means that the heat exchanger fluid flows from the heat pump into each borehole at a proportional flow rate (see **Figure 8**). Thus, the heat exchange rate per unit length is identical for both ground heat exchangers. In the present study, the GSHP is assumed to be used only for cooling purposes (heat rejection) and operates only 8 hours a day (8 am to 4 pm) during weekdays. These assumptions are to represent the typical behavior of air conditioner use in standard office buildings. In addition, the simulation period is five years. The numerical models disregard the effect of ambient air temperature fluctuations on the cooling load.

A standard parameter to evaluate the thermal performance of GSHP is the Coefficient of Performance (COP). The COP is a ratio between the total rate of cooling or heating to the required electrical input. While the COP is affected by various factors, a simple approximation is possible via a correlation with the heat pump's fluid temperature (*Tbh-out*) [16]. Such correlation can be obtained from the performance tables provided by the manufacturer. **Figure 9** shows the correlation between COP and *Tbh-out* for a heat pump with a rated capacity of 5.27 kW. This correlation is specific for a 15.8 L min�<sup>1</sup> fluid flowrate and 27°C dry-bulb and 19°C room air temperature at the 13.45 m<sup>3</sup> min�<sup>1</sup> flow rate (air flowrate in the fan-coil unit).

**Figure 8.** *Schematic figure of the numerical model for long-term performance evaluation.*

#### **Figure 9.**

*A relationship between* Tbh-out *and COP of a heat pump, obtained from the performance table supplied by the manufacturer.*

**Figures 10** and **11** present the fluid temperature leaving the ground heat exchanger (*Tbh-out*) and the calculated COP, respectively. The initial ground temperature and ground thermal properties are similar to those applied in the previous simulations (see **Tables 1** and **2**). The simulation results considering the thermal properties calculated using the ILS method show higher fluid temperatures than the simulations with the groundwater flow (case B4 for Bangkok and H3 for Hanoi). At the end of the five-year operation, the final fluid temperature for Bangkok are 41.77°C and 41.1°C for ILS and B4 cases, respectively. While, the final fluid temperatures for Hanoi are 41.89°C and 40.86°C for ILS and H3 case, respectively. Note that the thermal conductivities for B4 and H3 cases are lower than those calculated using the ILS. **Figure 12** compares the iso-temperature plot of B4 and H3 cases after five years. The B4 case with higher thermal conductivity and groundwater velocity provides a better heat transfer rate.

The results also suggest the significant role of groundwater convection in lowering the fluid temperature. For the application of GSHP in tropical countries with high initial ground temperature, the role of groundwater flow is ultimately essential.

*Effect of Groundwater Flow and Thermal Conductivity on the Ground Source Heat Pump… DOI: http://dx.doi.org/10.5772/intechopen.99061*

**Figure 11.** *COP of the GSHP system for a five-year GSHP operational period.*

#### **Figure 12.**

*Contour plot showing the soil temperature increase after a five-year GSHP operation; the thermal parameters for B4 and H3 cases are used for Bangkok (left) and Hanoi (right), respectively.*

## **5. Conclusion**

The application of GSHP in tropical countries, such as Thailand and Vietnam, encounters several problems. One of the main problems is the insufficient data on how GSHP could perform under such a climate. We carried out the numerical analysis of the TRT measurement results conducted in Bangkok, Thailand and Hanoi, Vietnam.

The MILS analytical method provides a better numerical analysis to evaluate the TRT result than the commonly used ILS method. The inverse analysis of TRT result using the MILS method with parameter optimization resulted in multiple solutions of unknown parameters: groundwater flow velocity, thermal conductivity, and borehole thermal resistance. The groundwater velocity and thermal conductivity are parameters with significant variations, while the borehole thermal resistance indicates a stable convergence into a single value. Simulations considering parameters obtained using the MILS and ILS method do not show a clear difference over a short-term TRT period. In the extended period of simulations, more than a year time scale, the differences are evident. The results also suggest the importance of groundwater flow in the long-term performance of GSHP, especially in tropical regions with high soil background temperatures. Over-reliance on the ILS method and the use of apparent thermal conductivity, especially for the site with high groundwater flow, can lead to a severe error.

Further, we extended the MILS simulations by incorporating two borehole heat exchangers and a simple approximation of COP to evaluate the GSHP performance over five years under different ground thermal parameters. The COP decreases over a more extended period because of the thermal imbalance resulting from the absence of heat extraction. The simulation results also suggest that the groundwater flow can effectively reduce the decreasing rate of COP.
