## 3 Validation of Line-Source Theory and Uncertainty of System  
### Line-Source Theory  
The line-source theory for calculating the thermal performance of a hermetic heat exchanger is well established, and it was applied to thermal performance testing for decades [57,58]. The theory was extended to comparisons and studies of energy losses in different heat exchanger arrangements [59,60]. In this study, the line-source theory is applied to the TRT of a closed source heat exchanger, and the performance of an open underground heat exchanger is extended to the measurement.  
The equation for the temperature domain as a function of time (\(t\)) and radius (\(r\)) around the line-source model with a constant heat injection rate (\(Q\)) can be used as the approach value of heat injection from closed and open geothermal heat exchangers as follows:  
\[T(r,\ t)-T_{o}=\frac{Q}{4\pi kH}\int_{x}^{\infty}\frac{e^{-u}du}{u}. \tag{1}\]  
The exponential integral of Equation (1) can be approximated for a large value of a parameter (\(\alpha t/r^{2}\)) with the following simple relationship:  
\[E_{1}\left(\frac{r^{2}}{4\alpha t}\right)=\ln\left(\frac{4\alpha t}{r^{2}} \right)-\gamma. \tag{2}\]  
If the infinite series is used for Equation (1), the following summary is obtained:  
\[T(r,t)-T_{o}=\frac{Q}{4\pi kH}\left\{\ln\frac{4\alpha t}{r^{2}}-\gamma+\frac{ r^{2}}{4\alpha t}\left(1-\frac{r^{2}}{16\alpha t}\right)\right\}, \tag{3}\]  
where \(\gamma\) is a Euler constant of 0.5772. The solution of the heat source to the bore hole can be applied only to \(r\geq r_{b}\) and the heat resistance \(R_{b}\) can be used in the bore hole.  
\[T_{f}=\frac{Q}{4\pi kH}\ln(t)+\frac{Q}{4\pi kH}\left(\ln\left(\frac{4\alpha}{r _{b}^{2}}\right)-\gamma\right)+\frac{Q}{L}R_{b}+T_{o}. \tag{4}\]  
Thus, a time-dependent fluid temperature equation can be described as shown in Equation (4).  
Figure 5: Schematic diagrams of a geothermal heat exchanger system: (**a**) closed loop; (**b**) SCW with return pipe location; (**c**) single-well SCW with bleeding rate; (**d**) two-well SCW system.  
The only variable in Equation (4) is \(\ln t\). It can be expressed by transforming constant \(\ln t\) in a general linear equation as follows:  
\[y=kx+c, \tag{5}\]  
where \(k=\frac{Q}{4\pi\lambda H}\), \(x=\ln(t)\), \(c=k(\ln\frac{4\alpha}{r_{o}^{2}}-0.5772)+\frac{Q}{H}R_{b}+T_{o}\).  
Therefore, a linear equation can be obtained by plotting the borehole fluid temperature over time obtained through the heat exchanger system as a measured value, and the fluid temperature and \(\ln\) t relationship as coordinates. The thermal conductivity (\(\lambda\)) can be determined by calculating using the slope (a) of Equation (5). The effective thermal conductivity and thermal resistance can be obtained from TRT data as given in Equations (6) and (7), respectively [45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55].  
\[k=\frac{Q}{4\pi\lambda H}. \tag{6}\]  
Thermal conductivity (\(\lambda\)) can be expressed using the slope (\(k\)) calculated in Equation (5), the length of the borehole (\(H\)), and heat energy (\(Q\)). In Equation (6), \(k\) is proportional to the effective thermal conductivity, and  
\[R_{b}=\frac{H}{Q}\Big{(}T_{f}-T_{\text{sur}}\Big{)}-\frac{1}{4\pi\lambda} \left(\ln(t)+\ln\left(\frac{4\alpha}{r_{o}^{2}}\right)-\gamma\right). \tag{7}\]