## 5 Modelling  
Step 1: Power generation calculation of the total-flow (TF) expansion stage.  
As to the combined total flow and single flash (TF-SF) system, modelling of the total-flow part system is straightforward since an average turbine efficiency is pre-set as constant. The power output (\(P_{TF}\), kW) in TF part from point 1 to point 2 comes from the multiplication of the enthalpy difference (\(h\), kJ/kg) and mass flowrate (\(W\)HM, kg/s),  
\[P_{TF}=WHM\times(h_{1}-h_{2}) \tag{1}\]  
The turbine efficiency is used to get the vapor quality of point 2 and the enthalpy of this point is thereafter gotten from REFPROP.  
Step 2: Wet efficiency calculation of steam turbine in the single flash (SF) stage.  
As to the single flash part modelling, the key process is the expansion process taking place in the steam turbine, i.e., from point 4 to 5 as shown in Figure 2. According to the Baumann rule [4], a 1% average moisture causes roughly a 1% drop in turbine efficiency. Although the inlet flow is saturated vapor, the steam turbine operates mostly in the wet region, degradation in performance caused by moisture should be taken into account. Adopting the Baumann rule, the turbine efficiency \(\eta_{t}\) is given by  
\[\eta_{t}=\eta_{td}\times(x_{4}+x_{5})/2, \tag{2}\]  
where \(\eta_{td}\) is the dry turbine efficiency which is conservatively assumed to be constant at 85%; \(x_{4}\) (=1) and \(x_{5}\) denote the vapor qualities of inlet and outlet flows of the steam turbine.  
To get the thermodynamic state of point 5, which is in return determined by turbine efficiency as shown in Figure 2, fluid properties at state 5 s, i.e., the ideal turbine outlet state would be used,  
\[\eta_{t}=(h_{4}-h_{5})/(h_{4}-h_{5s}), \tag{3}\]  
where \(h_{4s}\) is enthalpy of state 4, which is easily calculated from the known pressure and entropy values (\(s_{5s}=s_{4}\)).  
\[h_{5s}=h_{6}+(h_{7}-h_{6})\times[\frac{s_{5s}-s_{6}}{s_{7}-s_{6}}] \tag{4}\]  
Adopting the Baumann rule, enthalpy of the turbine outlet state 4 can be obtained as  
\[h_{5}=\frac{h_{4}-\left(\frac{h_{4s}}{2}\right)\times(h_{4}-h_{5s})\times \left(1-\frac{h_{6}}{h_{7}-h_{6}}\right)}{1+\left(\frac{h_{4s}}{2}\right) \times(h_{7}-h_{6})\times(h_{4}-h_{5s})} \tag{5}\]  
Then the vapor quality \(x_{5}\) is obtained from the condensing pressure and entropy value \(h_{5}\).  
Step 3: Optimal separating temperature of the two stage TF-SF system. For a traditional single flash system designed for a saturated liquid heat source, there's a 'rule of thumb' about the optimal temperature of state 2 which determines the separation temperature as well as the inlet flow state of turbine [4]. However, calculations show that this rule is no longer applicable with two-phase heat source as studied in this paper [24]. In order to reasonably determine the exit temperature at which the total-flow expansion ends and the steam turbine starts to work, a segmenting modellingmethod is applied by decreasing the separating temperature of state 2 from wellhead temperature to condensing temperature by small temperature steps, and the optimal separation temperature and system performance is therefore accurately obtained.  
Step 4: System performance. The power output (\(P_{SF}\), kW) in single flash part from point 4 to point 5 also comes from the multiplication of the enthalpy difference and mass flowrate,  
\[P_{SF}=WHM\times x_{2}\times(h_{4}-h_{5}) \tag{6}\]  
Therefore, total power generation \(P\) is the sum of power of both the TF and SF parts. In addition, since the freshwater comes entirely from the final condensation process, i.e., the total flow introduced to the steam turbine, the generation of freshwater (\(m_{FW}\), kg/s) is gotten by  
\[m_{FW}=WHM\times x_{2} \tag{7}\]