# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 4 Similarity Model for Viscous Working Fluids  
The presented limiting power \(P_{0}\) (4) of no viscosity is the basis for the similarity model of the actual expander's power \(P\) working with viscous fluids. The derived model concerns the similarity of the pressure loss \(\Delta p_{loss}\) due to the fluid's viscosity, which decreases the expander's power. The actual power \(P\) is less than the limiting power \(P_{0}\) due to the pressure loss \(\Delta p_{loss}\), mainly because of the fluid viscosity. The influence of internal leakage is not analyzed in this paper for the rotational speed \(n\) above the recommended minimum of 1000 rpm. The actual pressure difference \(\Delta p\) acting on the rotors is less than the inlet/outlet pressure difference \(\Delta p_{in,out}\) by the total pressure loss \(\Delta p_{loss}\):  
\[\Delta p\ =\ \Delta p_{in,out}-\Delta p_{loss} \tag{20}\]  
Both \(\Delta p\) and \(\Delta p_{loss}\) are mean values within one shaft's rotation. The actual expander's power \(P\) is a modification of the limiting power \(P_{0}\) (4), where \(\Delta p_{in,out}\) is replaced by \(\Delta p\) (20), which makes the actual power \(P<P_{0}\):  
\[P\ =\ n\cdot\Delta V_{geom}\cdot\Delta p\ =\ n\cdot\Delta V_{geom}\cdot\left( \Delta p_{in,out}-\Delta p_{loss}\right) \tag{21}\]  
In the presented similarity model, the pressure losses are not separated and are represented by the single total pressure loss \(\Delta p_{loss}\) between the inlet and the equivalent piston (Figure 5). Both rotors with a constant rotational speed are represented by the single piston with a constant velocity, jumping back to the beginning in every cycle. This way, in this model, the fluid flow and the pressure loss are constant between the inlet and the equivalent piston. Although the assumption of the single, constant pressure loss \(\Delta p_{loss}\) may seem rather coarse, the results of the derived similarity model are in good agreement with the manufacturer's experimental data and with the results of the numerical calculation. Figure 5: The model of the single inlet pressure loss \(\Delta p_{loss}\) representing the total pressure losses. In this simplified model, the pressure \(p_{rst,in}\) near the piston is given by the following equation:  
\[p_{rst,in}\ =\ p_{in}-\Delta p_{loss} \tag{22}\]  
Pressure drop \(\Delta p_{loss}\) in the incompressible viscous flow is a well-known phenomenon [20] and is represented by the general relationship between the Euler and Reynolds numbers:  
\[\text{Eu }=\ \text{f(Re) }=\ A\cdot\left(\frac{1}{\text{Re}}\right)^{\gamma} \tag{23}\]  
The coefficient \(A\) depends on the channel's geometry and the flow type and is reduced in further derivation. The exponent \(\gamma\) depends on the flow type: \(\gamma=1\) for laminar flows, \(\gamma=0.25\) for turbulent flows (the Blasius formula), and \(\gamma=0\) for local losses (valves, sudden channel contraction/expansion, etc.) or for very turbulent flows. These pure cases cover the range of the possible cases and generally \(0\leq\gamma\leq 1\). However, for the compressible gas, the pressure drop \(\Delta p_{loss}\) from the inlet to the piston makes the fluid's density change significant. In this case, the simplified concept of the mean gas pressure \(p_{m}\) and the corresponding mean gas density \(\rho_{m}\)[21] is used. The mean pressure \(p_{m}\) (Figure 5) is assumed to be the arithmetic mean of \(p_{in}\) and \(p_{rst,in}\):  
\[p_{m}\ =\ \left(p_{in}+p_{rst,in}\right)/2\ =\ p_{in}-\Delta p_{loss}/2 \tag{24}\]  
The explicit form of the Euler and Reynolds numbers in (23) gives the equation:  
\[\frac{\Delta p_{loss}}{\rho_{m}\cdot v_{m}^{2}/2}\ =\ A\cdot\left(\frac{\mu_{m}}{v _{m}\cdot d_{h}\cdot\rho_{m}}\right)^{\gamma} \tag{25}\]  
with the mean fluid density \(\rho_{m}\) and viscosity \(\mu_{m}\) at the mean pressure \(p_{m}\), and with the equivalent hydraulic diameter \(d_{h}\) reduced further.