# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 3 Limiting Case of No-Viscosity Fluid  
The limiting case of no-viscosity fluid of the expander's power \(P_{0}\) is the basis for the further-derived similarity model of the actual power \(P\) for the real viscous fluids. The inlet/outlet pressure difference \(\Delta p_{in,out}\) is defined as follows:  
\[\Delta p_{in,out}\ =\ p_{in}-p_{out} \tag{1}\]  
In the limiting case of no viscosity, the pressure difference \(\Delta p_{,0}\) acting on the rotors is the highest and is equal to \(\Delta p_{in,out}\) because there is no pressure loss:  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]  
The limiting and maximum work \(W_{0}\) produced by the rotors in one shaft's revolution is given as follows:  
\[W_{0}\ =\ \Delta V_{geom}\cdot\Delta p_{0}\ =\ \Delta V_{geom}\cdot\Delta p_{in,out} \tag{3}\]  
where \(\Delta V_{geom}\) is the volume transferred by the rotors in one shaft's revolution. It is a constant value for the geometry of the particular rotary lobe expander. It gives the limiting power \(P_{0}\), applied in the next section of the similarity model:  
\[P_{0}\ =\ n\cdot W_{0}\ =\ n\cdot\Delta V_{geom}\cdot\Delta p_{in,out} \tag{4}\]  
The limiting power \(P_{0}\) is independent of the fluid because of no viscosity. The ratio of the upper rotor's power \(P_{0,u}\) to the total \(P_{0}\) is given by the following equation:  
\[\frac{P_{0,u}}{P_{0}}\ =\ \frac{n\cdot\Delta V_{geom,u}\cdot\Delta p_{in,out}}{n \cdot\Delta V_{geom}\cdot\Delta p_{in,out}}\ =\ \frac{\Delta V_{geom,u}}{\Delta V_{geom}} \tag{5}\]  
Substitution of the numerical values (Appendix A) gives \(P_{0,u}/P_{0}=0.6858\). The small upper rotor produces 69% of the expander's power \(P_{0}\) in the limiting case of no viscosity. Figure 2: Screenshots from the geometry simulation program. The expander’s cross-section (**a**) at an angle of 0, (**b**) just before the opening of the upper high-pressure gas pocket, and (**c**) just after its opening, with no work from this gas expansion (white arrows). The manufacturer's experimental characteristics of the expander's power \(P_{mmf}\) for compressed air [17] are compared with the power \(P_{0}\) of no-viscosity fluid. Figure 3 shows the manufacturer's plot of the power \(P_{mmf}\) (solid lines) vs. the rotational speed \(n\), for the inlet air pressure \(p_{in}=7...16\) bar(a) and the atmospheric outlet one \(p_{out}=1\) bar(a). The limiting power \(P_{0}\) (4) of no viscosity (dashed lines) is drawn over the original manufacturer's plot. At the low rotational speed, \(n\), the influence of viscosity is almost negligible, and the actual experimental power \(P_{mmf}\) is close to the limiting power \(P_{0}\). The expander's power \(P\) depends on the mass flow rate and the specific enthalpy change:  
\[P\ =\ \dot{m}\cdot\Delta h\ =\ \dot{m}\cdot\left[h(p_{in},T_{in})-h(p_{out},T_{ out})\right] \tag{6}\]  
where the mass flow rate is given by the inlet volumetric flow rate \(\dot{V}_{in}\):  
\[\dot{m}\ =\ \rho_{in}\cdot\dot{V}_{in} \tag{7}\]  
In the case of no viscosity, the inlet volumetric flow rate \(\dot{V}_{in,0}\) is equal to the volumetric rate \(\dot{V}_{geom}\) taken and transferred by the rotors, \(\dot{V}_{in,0}\ =\ \dot{V}_{geom}\) because there is no pressure loss or density change from the inlet to the rotors. For the real viscous fluids, the inlet pressure \(p_{in}\) and the density \(\rho_{in}\) are higher than those near the rotors, which means the volumetric flow rates \(\dot{V}_{in}<\dot{V}_{geom}\). The geometric flow rate transferred by the rotors is given by the following equation:  
\[\dot{V}_{geom}\ =\ n\cdot\Delta V_{geom} \tag{8}\]  
The mass flow rate taken by the rotors in the case of no viscosity is given by the following equation:  
\[\dot{m}_{0}\ =\ \rho_{in}\cdot\dot{V}_{in,0}\ =\ \rho_{in}\cdot\dot{V}_{geom} \tag{9}\]  
\[\Delta p_{0}\ =\ \Delta p_{in,out} \tag{2}\]0 Manufacturer’s experimental plot of the expander’s power \(P_{mmf}\) vs. the rotational speed \(n\), for compressed air [17] (solid lines).