## 3 Methodology
### Simulation of the Geothermal Source
#### 4.1.2 Analysis of Heat Transfer on a U-Tube Heat Exchanger  
Case 1  
The first case corresponded to the numerical heat transfer between W1 and the U-tube heat exchanger for an estimation of the maximum water heating temperature in the heat exchanger. The initial parameters used for this analysis were \(d\) = 0.0254 m [29], \(L\) = 100 m [29], and fluid velocity \(\upsilon\) =  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rock** & **Density (kg/m\({}^{3}\))** & **Heat Capacity (J/kg\({}^{\circ}\)C)** & **Thermal Conductivity (W/m\({}^{\circ}\)C)** \\ \hline \hline Pumice (0–12 m) & 2360 & 885 & 1.768 \\ Ash (12–43 m) & 2180 & 920 & 1.535 \\ Basalt (43–50 m) & 2700 & 880 & 2.200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermophysical properties of the drilled cores from W1 [29].  
Figure 6: Thermal profile of the ground temperature variation for 100 m of depth for each month of the year 2017 (from La Reforma caldera).  
1 m/s (to the inlet and outlet of the heat exchanger). The thermal properties of water corresponding to \(k\), \(\rho\), \(\mu\), and \(Cp\) were calculated using Equations (8)-(11). Under these conditions, 12 heat transfer models were simulated in the U-tube heat exchanger (the thermal ground profile obtained for each month corresponded to \(T_{W}\)), and for all models, the values of \(T_{max}\) were located between 88 and 100 m of depth with temperatures from 65.0 \({}^{\circ}\)C to 67.6 \({}^{\circ}\)C (Figure 7). If we assumed that the U-tube could be isolated from a depth of 91 m to the surface (only in the ascending section of the fluid) and assumed that the heat losses were negligible, the heat load \(Q\) (kW) from the system achieved a value of 2.029 kW (Figure 7). From these results, it was clear that it was possible to use the well W1 as a low-enthalpy geothermal energy source coupled to a half-effect absorption system for air conditioning and refrigeration applications.  
Case 2  
The second case was a thermal analysis of well W-LV5 from Las Tres Virgenes geothermal field (an inoperative well): The Comision Federal de Electricidad (CFE) uses it only as a monitoring well. W-LV5 had a depth of 1800 m, and the static formation temperature at different depths has been reported by [PERSON] et al. [30]. Figure 8 shows the temperature profile of W-LV5, which had purely conductive thermal behavior. The maximum temperature reported was 219.3 \({}^{\circ}\)C to a depth of 1800 m.  
For the numerical heat transfer between W-LV5 and the U-tube heat exchanger for the estimation of the maximum water heating temperature in the heat exchanger, we considered the diameter of the U-tube to be 0.0254 m, the total length to be 1850 m, and fluid velocity to be v = 7 m/s. The thermal properties of water (\(k\), \(\rho\), \(\mu\), and \(Cp\)) for interval temperatures between 20 \({}^{\circ}\)C and 300 \({}^{\circ}\)C were calculated using the polynomial regression in Equations (36)-(38) (these equations were obtained from thermodynamics data values of water to high temperatures). A geothermal gradient of 156.7 \({}^{\circ}\)C/km was estimated from the SFTs of W-LV5 as showed in Figure 8. The heat transfer model result in the U-tube heat exchanger is shown in Figure 9, where \(T_{max}\) at the exit was 72.8 \({}^{\circ}\)C. Theoretically, with this result, it was also possible to use well W-LV5 as a low-enthalpy source coupled to the half-effect absorption systems to produce cooling:  
\[k = 0.5561+0.0024T-1.2930\times 10^{-5}T^{2}+8.7655\times 10^{-9}T^{3}+1.007 3\times 10^{-10}T^{4}-2.5537\times 10^{-13}T^{5}, \tag{36}\] \[u = 0.0017-4.4804\times 10^{-5}T+5.5960\times 10^{-7}T^{2}-3.4932 \times 10^{-9}T^{3}+1.0418\times 10^{-11}T^{4}-1.179410^{-14}T^{5}, \tag{37}\]  
Figure 7: Temperatures simulated along the U-tube heat exchanger in well W1.  
\[\rho=1002.2885-0.1898T-0.0024T^{2}-6.8357\times 10^{-7}T^{3}, \tag{38}\] \[C_{P} = 4165.2926+7.4586T-0.2692T^{2}+0.0032T^{3}-1.4370\times 10^{-5}T^{4}+ 2.2723\times 10^{-8}T^{5}. \tag{39}\]