# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 8 Similarity Analysis of the Numerical Calculations for Air and the Other Fluids  
The second application of the similarity model uses the numerically calculated power for the various fluids: Air, R245 fa, SES36, and HFE7100.  
The numerical power of the reference fluid (Air), \(P_{ref,Air}\) (=\(P_{num,Air}\)) is used to calculate the corresponding similarity power \(P_{sim,R}\) for the other fluid \"\(R\)\".  
In this calculation, the same inlet and outlet pressures \(p_{in}\), \(p_{out}\), and the rotational speed \(n\) are used for Air and for the other fluid \"\(R\)\".  
Calculation of the similarity power \(P_{sim,R}\) from the \(P_{ref,Air}\) consists of four steps:  
1. Calculation of the pressure loss \(\Delta p_{loss,Air}\) (21) for the reference fluid Air for the known reference power \(P_{ref,Air}\): \[\Delta p_{loss,Air}\ =\ \Delta p_{in,out}-\frac{P_{ref,Air}}{n\cdot\Delta V _{geom}}\] (39)
2. Calculation of the similarity constant \(B\) (35) for the particular expander using the pressure loss \(\Delta p_{loss,Air}\) (39) for the inlet air viscosity \(\mu_{in,Air}\) and density \(\rho_{in,Air}\) at \(p_{in}\), \(T_{in}\): \[B\ =\ \frac{\Delta p_{loss,Air}}{\mu_{in,Air}^{\gamma}\cdot\rho_{in,Air}^{1- \gamma}\cdot\frac{\left(1-\Delta p_{loss,Air}/p_{in}\right)^{2-\gamma}}{1- \Delta p_{loss,Air}/(2\cdot p_{in})}\cdot n^{2-\gamma}}\] (40)
3. Calculation of the similarity pressure loss \(\Delta p_{loss,R}\) for the other fluid \"\(R\)\" by solving (numerically) the similarity equation using the constant \(B\) (40), and for the other fluid's \"\(R\)\" viscosity \(\mu_{in,R}\) and density \(\rho_{in,R}\) at the inlet \(p_{in}\), \(T_{in}\): \[\frac{\Delta p_{loss,R}}{\mu_{in,R}^{\gamma}\cdot\rho_{in,R}^{1-\gamma}\cdot \frac{\left(1-\Delta p_{loss,R}/p_{in}\right)^{2-\gamma}}{1-\Delta p_{loss,R}/ (2\cdot p_{in})}\cdot n^{2-\gamma}}\ =\ B\] (41)
4. Calculation of the similarity power \(P_{sim,R}\) (21) for the other fluid \"\(R\)\" using the pressure loss \(\Delta p_{loss,R}\) (41): \[P_{sim,R}\ =\ n\cdot\Delta V_{geom}\cdot\left(\Delta p_{in,out}-\Delta p_{loss,R}\right)\] (42)  
The obtained similarity power \(P_{sim,R}\) is further compared with the corresponding numerically calculated \(P_{num,R}\).  
The exponent parameter \(\gamma\) is determined for the particular pairs of Air and \"\(R\)\" by minimization of the sum of squares \(\phi\) for the rotational speed \(n\) = 1500 and 3000 rpm:  
\[\varphi(\gamma)\ =\ \sum_{n\ =\ 1500,3000}\left[\left(P_{sim,R}-P_{num,R} \right)/P_{num,R}\right]^{2} \tag{43}\]  
The Golden Section method of minimization of \(\phi\) gives the individual values of \(\gamma\) for the pairs of Air and R245 fa, SES36, and HFE100 (Table 8).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\gamma\), - & **Air, HFE7100** & **Air, SES36** & **Air, R245 fa** & \(\gamma_{common}\) \\ \cline{2-5}  & 0.5861 & 0.6578 & 0.5733 & 0.6029 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Exponent parameter \(\gamma\) for individual pairs of fluids and the common value.  
The minimization of \(\phi\) for all the analyzed fluids gives the common value \(\gamma_{common}\):  
\[\varphi(\gamma_{common})\ =\ \sum_{R\ =\ R245 fa,\ SES36,\ HFE7100\ n=1500,3000}[(P_{sim,R}-P_{ num,R})/P_{num,R}]^{2} \tag{44}\]  
The common value \(\gamma_{common}\) (Table 8) is more correct in the sense of similarity,  
The ratio \(P_{sim,R}/P_{num,R}\) is calculated to compare \(P_{sim,R}\) and \(P_{num,R}\) and to present the relative errors of the approximation of \(P_{num,R}\) by \(P_{sim,R}\) in the analyzed ranges of pressure and rotational speed. The maximum relative error of \(P_{sim}\) vs. \(P_{mmf}\) is about \(\pm 5\%\) (Table 9 and Figure 11), which is a rather good agreement. The same \(\pm 5\%\) as in Section 6 is probably coincidental.