## 2 Mathematical Model
### Mathematical Model  
#### 2.5.1 Heat Exchanger  
Aspen plus-EDR was used to generate and draw a simple structural diagram of the heat exchanger, as shown in Figure 3.  
The evaporator heat load is  
\[Q_{r}=C\times q_{r}\times(h_{5}-h_{2}) \tag{1}\]  
Figure 3: A simple structural diagram of the heat exchanger. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Single board channel volume & 0.86 & L \\ Maximum water flow rate & 203 & m\({}^{3}\)/h \\ Plate thickness & 0.0004 & m \\ Board spacing & 0.0023 & m \\ Effective heat exchange area of a single plate & 0.45 & m\({}^{2}\) \\ Single channel cross-sectional area & 0.0014 & m\({}^{2}\) \\ Process length & 0.63 & m \\ Plate width & 0.42 & m \\ Number of installed boards in the evaporator & 41 & – \\ Number of installed boards in the condenser & 80 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The structural parameters of the selected heat exchanger. (1) Water volume flow \[V_{hwv}=m_{hwv,in}\times 3600/\rho_{hw}\] (2) Number of flow channels and heat exchange area \[n_{e}=V_{hw}/(s_{e}\times u_{hw}\times 3600)\] (3) \[A_{total}=(2\times n_{e}-1)\times f_{e}\] (4) Working fluid pressure drop and phase area ratio (1) Two-phase area Gas- and liquid-phase working fluid flow rates: \[u_{rl,rg}=m_{r}\times(1-x_{m})/(n_{e}\times\rho_{rl,rg}\times s_{e})\] (5) Reynolds number of the gas- and liquid-phase working fluid: \[\text{Re}_{rl,rg}=u_{rl,rg}\times d_{e}/v_{rl,rg}\] (6) Friction coefficients along the liquid phase and gas phase: \[\left\{ \begin{array}{ll}f_{rl}=38/\text{Re}_{rl},\ f_{rg}=38/\text{Re}_{ rg}&(\text{Re}_{rl}\leq 1000,\ \text{Re}_{rg}\leq 1000,\ C_{2}=5)\\ f_{rl}=38/\text{Re}_{rl},\ f_{rg}=1.22/\text{Re}_{rg}&(\text{Re}_{rl}\leq 1000, \ \text{Re}_{rg}\text{>}1000,\ C_{2}=12)\\ f_{rl}=38/\text{Re}_{rl},\ f_{rg}=1.22/\text{Re}_{rg}&(\text{Re}_{rl}\text{>} 1000,\ \text{Re}_{rg}\leq 1000,\ C_{2}=10)\\ f_{rl}=1.22/\text{Re}_{rl},\ f_{rg}=1.22/\text{Re}_{rg}&(\text{Re}_{rl}\leq 1000, \ \text{Re}_{rg}\leq 1000,\ C_{2}=10)\end{array} \right.\] (7) The respective proportions of the three-phase areas: \[l_{etp,esc,esh}=\Delta h_{etp,esc,esh}/\Delta T_{etp,esc,esh}\] (8) \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\] (9) Length of each phase area: \[L_{esc,esh,etp}=c_{esc,esh,etp}\times L_{e}\] (10) Liquid- and gas-phase pressure drops: \[\Delta P_{rl,rg}=\left(4\times f_{rl,rg}\times L_{etp}\times u_{rl,rg}^{2} \right)\times 10^{-3}/(2\times d_{e})\] (11) Martinelli parameters: \[X_{e}=\sqrt{\frac{\Delta P_{rl}}{\Delta P_{rg}}}\] (12) The square of the liquid-phase coefficient of friction: (\(C_{2}=12\)) \[\varphi_{rl}^{2}=1+C_{2}/X_{e}+1/X_{e}^{2}\] (13) Pressure drop in the two-phase area: \[\Delta P_{etp}=1.25\times\Delta P_{rl}\times\varphi_{rl}\] (14) (2) Supercooled zone and superheated zone  
Water side pressure drop: \[\Delta P_{hw}=1843\times\text{Re}_{hw}^{-0.25}\times\rho_{hw}\times u_{hw}^{2} \times 10^{-3}\] (29) (6) Boiling heat transfer coefficient on the working fluid side \[X_{pb}=\left(\frac{G_{l}}{G_{g}}\right)^{0.9}\times\left(\frac{\rho_{g}}{\rho_{ l}}\right)^{0.5}\times\left(\frac{\mu_{l}}{\mu_{g}}\right)^{0.1}\] (30) Correction coefficient: (\(C_{2}=21\)) \[F=\varphi_{pb}^{0.89}\] (31) Pool boiling heat transfer coefficient: \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\]0 (32) (7) Total working fluid side heat transfer coefficient \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\]1 (33) (8) Total heat transfer coefficient of the evaporator \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\]2 (34) (9) Logarithmic mean temperature difference \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\]3 (35) (10) Actual heat exchange area \[c_{etp,esh,esc}=l_{etp,esh,esc}/(l_{etp}+l_{esc}+l_{esh})\]4 (36)  
The principles, methods, and calculating model and evaporator model are essentially the same and will not be described in detail here.