## 2 System Description and Modelling
### Mathematical Modelling
#### 2.2.2 The Heat Pump Model
The zone length fractions can range from 0 to 1, and the rate of heat transfer depends on the fluid phase of each zone. In addition, heat transfer is calculated between the fluids and the external environment. The same correlations with the condenser were also used for the Nusselt number estimation. The geometric and thermal characteristics used in the evaporator design are presented in Table 2.  
Finally, the coefficient of performance (COP) is estimated using the heating capacity of the brine at the heat pump's condenser and the power consumed by the compressor during the whole operation. \[\text{COP}=\frac{\dot{Q}_{cd}}{\dot{W}_{comp}} \tag{6}\]  
The nominal characteristics of the heat pump's components for the freshwater production of 160 L/h of water are presented in Table 3.  
The control strategy is based on the physics of the problem described in Section 2.1 and is shortly described next. The pre-selected vacuum pressure determines the saturation temperature (e.g., at 0.25 bar, the evaporation temp is 65 \({}^{\circ}\)C) of the brine, where phase change takes place. Thus, the heat pump condensation temperature is set to obtain a brine inlet temperature higher than the evaporation temperature (at the vacuum pressure), so that this temperature difference can assure that the brine is continuously evaporating. The set-point temperature is the condensation temperature of the HTHP, and it is defined by the user/operator. Therefore, the high inlet temperature of the brine leads to increased void fraction (water production), but the increase in the condensation temperature leads to the reduced COP of the HP. On the other hand, the brine temperature depends on the flash tank conditions and the recirculation ratio (the percentage of the flow that recirculates after the flash tank and the flow that comes from RO). When this temperature difference is greater than 2 degrees, the controller switches on the HTHP. When this difference is below 2, the HTHP is switched off. Since the condensation temperature is defined by the set-point and the evaporation temperature depends on the evaporation temperature at the pressure conditions of the flash tank, the load of the HTHP is not expected to vary significantly. For validation purposes, the heat pump model was compared for the steady operation at various operating points with the software provided by the compressor manufacturer (Hanbell), presenting similar results for the heating capacity and COP. Therefore, the heat pump developed model that presents acceptable accuracy to predict the whole system \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Refrigerant & R134a \\ Displacement (m\({}^{3}\)/h) & 98 \\ Design compressor speed (RPM) & 2950 \\ Compressor volume ratio & 3 \\ Compressor nominal power consumption (\(k\)W) & 35.9 \\ Nominal heating capacity (\(k\)W) & 102.6 \\ Condensation temperature (\({}^{\circ}\)C) & 80 \\ Evaporation temperature (\({}^{\circ}\)C) & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design specifications of the heat pump. operation is considered. Figure 4 presents the variation between the in-house developed model and the software developed by Hanbell, showing a deviation smaller than 2%.