# ORC Optimal Design through Clusterization for Waste Heat Recovery in Anaerobic Digestion Plants
## 3 Methodology
### Off-Design
#### 3.2.1 Turbine Off-Design Analysis  
[PERSON]'s ellipse law provides a simplified approach for predicting off-design performance in the idealized case of an infinite stage number turbines, where acoustic choking could never happen [44]. In the case described in this paper, due to the low evaporating temperature obtained, a single-stage turbine was enough to avoid flow choking. [PERSON]'s approach may be modified to account for a finite number of stages, up to single-stage turbines, as demonstrated in [44]. [PERSON]'s approach is based on the idea of assimilating an entire group of stages to a single nozzle, for which the following proportionality relation holds (Equation (46)). \[\psi\propto\sqrt{1-\left(\frac{B}{p_{0}}\right)^{2}} \tag{46}\]  
where \(B\) is the turbine outlet static pressure; \(p_{0}\) is the total inlet pressure; and \(\psi\) is the flow coefficient, defined as in Equation (47):  
\[\psi=\frac{\dot{m}}{\sqrt{\frac{p_{0}}{v}}} \tag{47}\]  
where \(\dot{m}\) is the mass flow rate through the turbine and \(v\) the specific volume at the turbine section at the pressure \(p_{0}\). Equation (46) is called [PERSON]'s ellipse as it may be represented as an ellipse in the plane. In [44], it was demonstrated how the ratio \(\psi^{*}\) (\(0\leq\psi^{*}\leq 1\)) between \(\psi\) in any condition and \(\psi\) at acoustic chocking condition may be written according to Equation (48) for a finite number of stages:  
\[\psi^{*}=\sqrt{1-\left(\frac{B-p_{0}\cdot a}{p_{0}-p_{0}\cdot a}\right)^{2}} \tag{48}\]  
where \(a\) is the acoustic chocking expansion ratio. To maintain physical significance, \(\psi^{*}\) follows Equation (48) as long as \(B>p_{0}\cdot a\). For \(B=p_{0}\cdot a\), \(\psi^{*}\) reaches its maximum value (i.e., \(\psi^{*}=1\)) and for lower B values (i.e., beyond acoustic choking), \(\psi^{*}=1\) is maintained as the flow coefficient does not vary. By following Equation (48), the proportionality relation in Equation (46) may be restated as in Equation (49) [44]:  
\[\frac{\psi^{*}}{\psi^{*}_{d}}=\frac{\psi}{\psi_{d}}=\frac{\sqrt{1-\left(\frac {B-p_{0}\cdot a}{p_{0}-p_{0}\cdot a}\right)^{2}}}{\sqrt{1-\left(\frac{B_{d}-p _{0}\cdot a}{p_{0}\cdot a}\cdot a\right)^{2}}} \tag{49}\]  
and:  
\[p_{0}=p_{0,d}\left(\frac{\dot{m}^{2}\cdot v}{\dot{m}_{d}^{2}\cdot v_{d}}\right) \tag{51}\]  
Equation (29), which is valid for chocked operations, stems from the fact that \(\psi^{*}\) and \(\psi\) are constant for \(B\leq p_{0}\cdot a\). Both Equation (50) and (51) link off-design pressures to design specifications and allows for the determination of off-design operating conditions. In the case of real gases, like ORC working fluids, \(a\) must be calculated numerically, as the ideal gas approach does not provide a good estimation of this parameter. Concerning the ORC design provided in the previous section, the expansion ratio is well below the acoustic chocking one, even considering strong off-design conditions. Therefore, in the following analysis, Equation (51) will be used, instead of Equation (50), which is used for non-chocked conditions. In addition to Equation (51), some additional assumptions must be made to define off-design operation completely. In this case, turbine outlet pressure and inlet temperature are assumed to be constant and equal to their design values. Of course, different assumptions could be made for different control strategies. By assuming some inlet and outlet conditions and by exploiting Equation (51), off-design isentropic enthalpy difference across the turbine (isentropic specific work) may be calculated. However, some further corrections must be considered. As suggested in [45], turbine isentropic efficiency should be correct in off-design conditions by considering two parameters, \(CF_{1}\) and \(CF_{2}\), to modify \(\eta_{is,d}\).