# ORC Optimal Design through Clusterization for Waste Heat Recovery in Anaerobic Digestion Plants
## 3 Methodology
### Preliminary Design
#### 3.1.2 Design Optimization Problem
The modified objective function is equal to (Equation (13)):  
Figure 5:2  
The feasible region \(\Phi\) introduced in Equations (2) and (3) is defined through the optimization problem constraints. For each of the optimization variables, lower bounds (_lb_) and upper bounds (_ub_) define the search domain as Equations (14)-(19):  
Figure 5:3  
Figure 5:4  
Figure 5:5  
Figure 5:6  
Figure 5:7  
Figure 5:8  
where \(T_{crit,orc}\) is the ORC fluid critical temperature; \(T_{cool,out}\) is the cooling water temperature exiting the ORC condenser, which is set to 25 \({}^{\circ}\)C; \(T_{source,in}\) is the temperature of the flue gases at the inlet of the waste heat recovery heat exchanger; and \(\Delta T_{pp,source}\) is the minimum allowable pinch point between the flue gases and the thermal oil. The ORC operational limitations were imposed through linear and non-linear constraints. A minimum allowable pinch point was set for each heat exchanger. Both \(\Delta T_{pp,source}\) (pinch point between exhaust gases and thermal oil) and \(\Delta T_{pp,ew}\) (pinch point between thermal oil and sewage) were equal to 30 \({}^{\circ}\)C. \(\Delta T_{pp,re}\) (ORC evaporator pinch point) was equal to 15 \({}^{\circ}\)C. \(\Delta T_{pp,reg}\) (ORC regenerator pinch point) was equal to 10 \({}^{\circ}\)C. Finally, \(\Delta T_{pp,cl}\) (ORC condenser pinch point) was equal to 10 \({}^{\circ}\)C.  
Maximum ORC temperature \(T_{max,orc}\) (turbine inlet) must be at least 5 \({}^{\circ}\)C lower than that of the organic fluid degradation. The ORC condensation pressure \(p_{cd,orc}\) must be higher than 0.5 bars. A minimum superheating degree of 5 \({}^{\circ}\)C was set to prevent liquid injection into the ORC turbine. Similarly, to avoid pump cavitation, a subcooling \(\Delta T_{sc,orc}\) of 5 \({}^{\circ}\)C was set. Other considerations are related to the thermal oil loop, for which a minimum design temperature \(T_{loop,min}\) of 70 \({}^{\circ}\)C was imposed due to the high oil viscosity at low temperatures. For exhaust gases, a maximum value for the oil heat exchanger effectiveness \(\varepsilon_{source}\) of 0.8 was imposed to reflect the performance of commercially available heat exchangers. Finally, a minimum mGT exhaust gas temperature \(T_{source,out}\) of 70 \({}^{\circ}\)C was set to prevent acid condensation. All the previous considerations (and some others) can be translated into constraints, both linear and non-linear. As for the linear constraints, they can be defined as in Equation (20)-(26):  
Figure 5:9\[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]0 \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]1 (22) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]2 (23) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]3 (24) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]4 (25) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]5  
where:  
* \(\Delta T_{loop,min}\) is the minimum achievable temperature drop of thermal oil across the ORC evaporator. * \(T_{sev,out}\) is the sewage temperature at the outlet of the sewage heating heat exchanger, set at 37.2 \({}^{\circ}\)C to consider the thermal losses before entering the digester. As for the non-linear constraints, the equation \(c(x)\leq 0\) can be translated into a non-linear inequality set as defined in Equation (27)-(38):  
\[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]6 \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]7 (28) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]8 (29) \[\Delta h_{exp}=\eta_{is,exp}(h_{3,orc}-h_{4 is,orc}) \tag{4}\]9 (30) Figure 5.0 (31) Figure 5.1 (32) Figure 5.2 (33) Figure 5.3 (34) Figure 5.4 (35) Figure 5.5 (36) Figure 5.6 (37) Figure 5.7  
where:  
* \(\chi_{exp,out}\) is the vapor quality at the end of the expansion. * \(\Delta T_{reg,min}\) is the minimum temperature difference between the temperature at the end of the regeneration \(T_{4,orc}\) and ORC condensation temperature \(T_{cd,orc}\).