# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 3 Limiting Case of No-Viscosity Fluid
The inlet pressure is \(p_{in}=7...16\) bar(a) and the atmospheric outlet pressure is \(p_{out}=1\) bar(a). The limiting power \(P_{0}\) of a no-viscosity fluid (dashed lines) is drawn over the original plot. In this case, the limiting power \(P_{0}\) is given by the following equation:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]1  
where \(\Delta h_{0}\) is the specific enthalpy change in the case of no viscosity. The combination of (4), (7), and (10) gives the formula for the specific enthalpy change \(\Delta h_{0}\) in the case of no viscosity, independent of the rotational speed \(n\):  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]2  
The value of \(\Delta h_{0}\) (11) allows calculation of the corresponding outlet temperature \(T_{out,0}\), by solving the equation for \(T_{out,0}\):  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]3  
The corresponding outlet density \(\rho_{out,0}\) is given by the following equation:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]4  
The highest available power \(P_{s}\) from the isentropic expansion between the inlet and outlet, for the same mass flow rate \(\dot{m}_{0}\) as in (10) and the same inlet and outlet pressures, is given by the following equation:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]5  
where the isentropic change in the specific enthalpy \(\Delta h_{s}\) is given by the following equation:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]6  
with the inlet-specific entropy:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]7  
A combination of (10) and (14) gives the isentropic efficiency \(\eta_{s,0}\) for the limiting case of no viscosity, independent of the rotational speed \(n\):  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]8  
In Table 1, the calculation results for the limiting case of no viscosity are presented for the expander working with compressed air and with the other fluids R245 fa, SES36, and HFE7100 in the Organic Rankine Cycle (ORC). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fluid**} & \(p_{in}\) & \(T_{in}\) & \(\rho_{in}\) & \(p_{out}\) & \(T_{out,0}\) & \(\rho_{out,0}\) & \(P_{0}\) & \(P_{s}\) & \(\eta_{s,0}\) & \(m_{0}\) \\  & **Bar(a)** & \({}^{\circ}\)**C** & **kg/m\({}^{3}\)** & **Bar(a)** & \({}^{\circ}\)**C** & **kg/m\({}^{3}\)** & **kW** & **kW** & **\%** & **kg/s** \\ \hline Air & 16.0 & 25.0 & 18.78 & 1.0 & \(-\)57.8 & 1.62 & 58.7 & 119.1 & 49.3 & 0.736 \\ \hline R245 fa & 16.0 & 115.8 & 91.09 & 1.0 & 74.1 & 4.74 & 58.7 & 187.3 & 31.4 & 3.567 \\ \hline SES36 & 16.0 & 149.8 & 125.9 & 1.0 & 94.1 & 6.22 & 58.7 & 192.8 & 30.5 & 4.931 \\ \hline HFE7100 & 16.0 & 179.6 & 175.9 & 1.0 & 153.8 & 7.21 & 58.7 & 233.5 & 25.2 & 6.889 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Limiting case of no viscosity for the expander working with compressed air and with the other fluids in the ORC. \(P_{0}\), \(P_{s}\), and \(m_{0}\) are calculated for the rotational speed \(n=3000\) rpm. The inlet pressure is \(p_{in}=16\) bar(a), the atmospheric outlet pressure is \(p_{out}=1\) bar(a) and the rotational speed is \(n=3000\) rpm. In the ORC, the inlet temperature \(T_{in}\) is 5 \({}^{\circ}\)C higher than the boiling temperature of the working fluid at the inlet pressure \(p_{in}\). Each no-viscosity power \(P_{0}\) (4) is the same as it depends only on the pressure difference \(\Delta p_{in,out}\) and the rotational speed \(n\). The isentropic power \(P_{s}\) (14) depends on the fluid, which makes the values of the isentropic efficiency \(\eta_{s,0}\) (17) different. Even in the ideal case of no viscosity, the isentropic efficiency \(\eta_{s,0}\) is far below 100% due to the expander's principle of operation with an unproductive gas expansion to the outlet, which strongly decreases the power. The rapid expansion of the gas to the outlet is a real viscous dissipation process of mechanical energy. The intention of this basic thermodynamic analysis is to show the thermodynamic differences among the working fluids without viscosity effects being taken into account.