# Toward More Efficient Large-Scale Green Hydrogen Systems via Waste Heat Recovery and ORC
## 2 Methods
### Maximizing ORC Efficiency  
In order to increase the efficiency of the process, the ORC module is optimized to maximize the produced electricity and enhance the cycle's efficiency. Therefore, the objective function is defined as  
\[f(x)=-\dot{\mathrm{W}}_{net}+Penalty_{quality^{\prime}} \tag{6}\]  
where the quality penalty is defined as  
\[Penalty_{quality}=\left\{\begin{aligned} & 10^{6}\times\left(0.90-X_{ quality}\right)\ ifX_{quality}<0.90\\ & 0\ otherwise\end{aligned}\right., \tag{7}\]  
And  
\[\dot{\mathrm{W}}_{net}=\dot{\mathrm{W}}_{turbine}-\dot{\mathrm{W}}_{pump}, \tag{8}\]  
\(\dot{\mathrm{W}}_{net}\) represents the cycle's work, \(\dot{\mathrm{W}}_{turbine}\) calculates the turbine's work, and \(\dot{\mathrm{W}}_{pump}\) measures the pump's work.  
\[\dot{\mathrm{W}}_{turbine}=\dot{\mathrm{m}}_{ref}\times\left(h_{turbine-in}- h_{turbine,out}\right)\times\eta_{turbine}, \tag{9}\]  
\[\dot{\mathrm{W}}_{pump}=\dot{\mathrm{m}}_{ref}\times\left(h_{pump,out}-h_{ pump,in}\right)/\eta_{pump}, \tag{10}\]  
For limiting feasible solutions and meeting actual physical properties, the constraints below have been used to avoid any system failures during the process:  
Superheat constraint:  
\[T_{superheated}\geq\ 5\ ^{\circ}\mathrm{C}, \tag{11}\]  
Subcooling constraint:  
\[T_{\mathit{Sukcooling}}\geq\ 3\ ^{\circ}\mathrm{C}, \tag{12}\]  
Pressure constraints:  
\[1\ bar\ \leq\ P_{cycle}\leq 40\ \mathrm{bar}, \tag{13}\]  
Net positive suction head (NPSH) constraint:  
\[NPSH_{available}\geq NPSH_{required}=3\mathrm{m}, \tag{14}\]  
\(NPSH\) is calculated using the equations below:  
\[NPSH_{available}=\left(h_{subcooled}-h_{sat}\right)/g, \tag{15}\]  
To make sure these calculations are correct, the overall energy balance for ORC is formulated as below:  
\[\dot{Q}_{hot\ water}=\dot{\mathrm{W}}_{net}+\dot{Q}_{cooling\ water,\ out}+ \dot{Q}_{loss}, \tag{16}\]  
\[\dot{Q}_{hot\ water}=\dot{m}_{hotwater}(h_{water,out}-h_{water,in}), \tag{17}\]  
\[\dot{Q}_{cooling\ water}=\dot{m}_{coolwater}(h_{water,out}-h_{water,in}), \tag{18}\]  
The evaporator heat transfer, condenser heat transfer, and recuperator heat transfer using the effectiveness-NTU method are calculated as follows:  
\[\dot{Q}_{evap}=\dot{\mathrm{m}}_{ref}\times\left(h_{evap,\ out}-h_{evap,in} \right)\times\eta_{HX}, \tag{19}\]  
\[\dot{Q}_{cond}=\dot{\mathrm{m}}_{ref}\times\left(h_{cond,in}-h_{cond,\ out} \right)\times\eta_{HX}, \tag{20}\]  
\[\dot{Q}_{recup}=\epsilon_{recup}\times\dot{\mathrm{m}}_{ref}\times C_{p,min,in }\times\left(T_{hot,out}-T_{cold,in}\right), \tag{21}\]The effectiveness-NTU is expressed as below:  
\[\varepsilon_{recup}=1-e^{-NTU(1-C_{ratio})}, \tag{22}\]  
\[NTU=\frac{UA}{C_{min}}, \tag{23}\]  
\[\text{UA}=\frac{\dot{Q}}{LMTD}, \tag{24}\]  
LMTD represents the lean mean temperature difference (LMTD) method for heat exchangers:  
\[LMTD=\frac{[(T_{houtin}-T_{coldout})-(T_{hotout}-T_{coldin})]}{ln((T_{hot,in}-T_{ coldout})/(T_{hot,out}-T_{cold,in}))}, \tag{25}\]  
In the end, the thermal efficiency of the cycle is calculated using Equation (26):  
\[\eta_{thermal}=\dot{W}_{net}/\dot{Q}_{evap}, \tag{26}\]