## 3 Results and Discussion
### Vapour and Liquid Water Heat Capacities
The absolute value of \(b\), 2398.9 J/kg/K, is within a maximum relative error of 12% from the literature values and isobaric reference values. However, \(b\) is greater than reference values in all the temperature range (Figure 6a), which show a maximum of 2335.5 J/kg/K at 0 \({}^{\circ}\)C. Something similar occurs with the absolute value of \(b^{\prime}\), 2841.5 J/kg/K, which is greater than the maximum isochoric reference value, 2799.0 J/kg/K at 0 \({}^{\circ}\)C (Figure 6b). This result shows that using Equations (18)-(21) with a constant pair of specific heat capacities for liquid water and vapour equal to reference values at the same temperature (Figure 5), for any temperature between 0 and 150 \({}^{\circ}\)C, leads to underestimating the slope of the linear laws of Equations (22) and (24) plotted in Figure 4. Thus, it implies underestimating the decrease in \(\Delta h_{\text{Ly}}\) or \(\Delta u_{\text{Ly}}\) with temperature in the modelling. Assuming that the enthalpy of vaporisation, or the internal energy difference between vapour and liquid water, for the  
Figure 6: Difference in specific heat capacity between liquid water and vapour as a function of temperature. Solid lines: reference values [13] for (**a**) isobaric and (**b**) isochoric conditions; dotted and dashed lines: identified \(b\) and \(b^{\prime}\) coefficients and the literature values in Table 2 [7; 12; 14; 15; 16; 17]. reference temperature, \(\Delta h_{\text{LV,T0}}\) or \(\Delta u_{\text{LV,T0}}\), is accurately set in the model, \(\Delta h_{\text{LV}}\) or \(\Delta u_{\text{LV}}\) will be accurately computed at \(T_{0}\) and lose accuracy as temperature deviates from \(T_{0}\). To inspect this loss of accuracy, the evolution of \(\Delta h_{\text{LV}}\) and \(\Delta u_{\text{LV}}\) computed with Equations (22) and (24), respectively, in a case that would maximise this loss, has been studied and plotted in Figure 7. \(T_{0}\) is set at one extreme of the temperature range considered, 0 \({}^{\circ}\)C, and \(\Delta h_{\text{LV,T0}}\) and \(\Delta u_{\text{LV,T0}}\) have been accurately taken as the reference values for 0 \({}^{\circ}\)C. Then, the maximum error will be made at the other extreme of the range, 150 \({}^{\circ}\)C, as a consequence of using a too small slope. To study the magnitude of the error at 150 \({}^{\circ}\)C, two values of the slopes (\(c^{\text{V}}_{\text{p}}-c^{\text{L}}_{\text{p}}\)) and (\(c^{\text{V}}_{\text{v}}-c^{\text{L}}_{\text{v}}\)) have been used. First, the values in the inspected literature furthest from the identified values for \(b\) and \(b^{\prime}\), from references [12; 14] for \(\Delta h_{\text{LV}}\) (Figure 6a) and [17] for \(\Delta u_{\text{LV}}\) (Figure 6b), have been used (solid grey lines in Figure 7). In this case, the relative error at 150 \({}^{\circ}\)C is 0.7% for \(\Delta h_{\text{LV}}\) and 1.9% for \(\Delta u_{\text{LV}}\). Second, the reference specific heat capacity pairs in Figure 6 furthest from \(b\) and \(b^{\prime}\) have been used: 2138 J/kg/K at 99 \({}^{\circ}\)C for isobaric values and 2027.3 J/kg/K at 150 \({}^{\circ}\)C for isochoric values (dashed orange lines in Figure 7). The relative error at 150 \({}^{\circ}\)C then increases to 1.7% for \(\Delta h_{\text{LV}}\) and 6.1% for \(\Delta u_{\text{LV}}\). Using the same approach, the relative error at 150 \({}^{\circ}\)C can, in addition, be computed for specific heat capacity pairs from the reference values [13] corresponding to all the temperature range between 0 and 150 \({}^{\circ}\)C (Figure 8). All errors are below the maximum errors mentioned in the previous paragraph (1.7% for \(\Delta h_{\text{LV}}\) and 6.1% for \(\Delta u_{\text{LV}}\)). If the relative error is to be further limited, the reference pairs used should be limited to lower temperatures.