# ORC Optimal Design through Clusterization for Waste Heat Recovery in Anaerobic Digestion Plants
## 3 Methodology
### Preliminary Design
#### 3.1.2 Design Optimization Problem
may be calculated and, by following the definition of turbine isentropic efficiency \(\eta_{is,exp}\), (equal to 0.8), the turbine specific work \(\Delta l_{exp}\) is determined (Equation (4)):  
\[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]  
For the interpretation of ORC-related subscripts, please refer to Figure 5.  
Depending on the condensation pressure and the temperature at the pump inlet (\(T_{cd,orc}-\Delta T_{sc,orc}\)), pump inlet enthalpy and entropy were obtained. By knowing the evaporation pressure, the pump outlet isentropic conditions can be estimated. Through the pump isentropic efficiency \(\eta_{is,pump}\), equal to 0.7, the pump specific work \(\Delta l_{pump}\) is calculated (Equation (5)):  
\[\Delta l_{pump}=\frac{(h_{2 is,orc}-h_{1,orc})}{\eta_{is,pump}} \tag{5}\]  
Working fluid mass flow is determined from the evaporator thermal input \(\dot{Q}_{ev}\) and evaporator net enthalpy difference \(\Delta h_{ev}\). The first one is calculated from \(T_{loop,1}\), \(T_{loop,2}\), and \(\dot{m}_{loop}\), which are provided by the solver in each iteration (Equation (6)):  
\[\dot{Q}_{ev}=\dot{m}_{loop}c_{p,loop}\left(T_{loop,1}-T_{loop,2}\right) \tag{6}\]  
\(\Delta l_{ev}\) is calculated by determining the regenerator hot side outlet conditions. These are calculated from the regenerator thermal effectiveness \(\epsilon_{reg}\) defined in Equation (7):  
\[T_{At,orc}=T_{4,orc}-\ E_{reg}(T_{4,orc}-T_{2,orc}) \tag{7}\]  
Regenerator hot outlet conditions may be used to define regenerator cold side outlet enthalpy \(h_{2,orc}\) (i.e., evaporator cold side inlet enthalpy) through regenerator energy balance, as reported in Equation (8):  
\[h_{2,orc}-h_{2,orc}=h_{4,orc}-h_{4,orc} \tag{8}\]  
From this, the evaporator net enthalpy difference can be calculated (Equation (9)):  
\[\Delta h_{ev}=h_{3,orc}-h_{2,orc} \tag{9}\]  
Finally, by combining Equation (6) and (9), \(\dot{m}_{arc}\) is calculated as in Equation (10):  
\[\dot{m}_{orc}=\frac{\dot{Q}_{ev}}{\Delta h_{ev}} \tag{10}\]  
From Equations (4), (5), and (10), ORC net power output \(W_{net,orc}\) is determined as in Equation (11):  
Figure 5:0  
where \(\eta_{el}\) is the electro-mechanical conversion efficiency, taken as equal to 0.95. Theoretically speaking, \(W_{net,orc}\) might be used as the optimization problem objective function (\(\dot{f}(\mathbf{x})\) in Equation (2)). Although this allows for the power output global maximum determination, it was noted that the same (maximum) \(W_{net,orc}\) value is yielded by several different cycle configurations. To univocally identify the best solution, the objective function was modified by including the thermal oil loop circulation pump electric consumption, defined as in Equation (12). The computation of the pump consumption in the loop allows for the optimum to be reached by a unique set of variables. Figure 5:1  
where:* \(\eta_{is,pump,loop}\) is the oil circulation pump isentropic efficiency (\(\eta_{is,pump,loop}=0.7\)). * \(\Delta p_{loss,i}\) is the estimation of the pressure losses in each \(i\)-th exchanger of the loop (10 kPa). * \(\rho_{loop}\) is the thermal oil density at the pump inlet. * The requirement of considering the pumping power is more related to the solver numerical behavior than to the system actual physics. The optimization algorithm may consider as equivalent two solutions that are only very similar. To help it to differentiate between them and since the threshold used to distinguish between the solutions is a very small number, it is sufficient to include the pumping power in the calculations to consistently achieve the same optimum every time, for the same set of boundary conditions. However, despite the significance for the optimization algorithm, the selection of the pressure drop value can be quite arbitrary due to its impact on performance being almost negligible.