# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 6 Similarity Analysis of the Manufacturer's Experimental Data for Air  
The first application of the similarity model uses the manufacturer's experimental data for air [17] (Table 2).  
The measurements of \(P_{muff}\) are in Table 2 for \(p_{in}=7...16\) bar(a) and \(p_{out}=1\) bar(a). The assumed inlet temperature is \(T_{in}=25\)\({}^{\circ}\)C, which is not given by the manufacturer. The applied rotational speed is \(n=750\), \(1000...3000\) rpm because \(1000\) rpm is the recommended minimum rotational speed.  
The manufacturer's power \(P_{muff}\) is approximated by the similarity power \(P_{sim}\) for the corresponding inlet/outlet pressure difference \(\Delta p_{in,out}\) and the rotational speed \(n\).  
The similarity power \(P_{sim}\) is calculated from (21):  
\[P_{sim}\ =\ n\cdot\Delta V_{geom}\cdot\left(\Delta p_{in,out}-\Delta p_{loss, sim}\right) \tag{36}\]  
where the similarity pressure loss \(\Delta p_{loss,sim}\) is calculated by numerically solving Equation (35) with the inlet viscosity \(\mu_{in}\) and density \(\rho_{in}\) for the inlet pressure \(p_{in}\) and temperature \(T_{in}\):  
\[\frac{\Delta p_{loss,sim}}{\mu_{in}^{\gamma}\cdot\rho_{in}^{1-\gamma}\cdot \frac{\left(1-\Delta p_{loss,sim}/p_{in}\right)^{2-\gamma}}{1-\Delta p_{loss, sim}/\left(2\cdot p_{in}\right)}\cdot n^{2-\gamma}}\ =\ B \tag{37}\]  
The parameters \(\gamma\) and \(B\) are approximated from the manufacturer's experimental power \(P_{muff}\) via minimization of the function \(\phi(\gamma,B)\) (38).  
This function is the maximum absolute value of the relative difference between the similarity \(P_{sim}\) and the manufacturer's power \(P_{muff}\). It is determined using the all-inlet pressures \(p_{in}\) and rotational speeds \(n\):  
\[\phi(\gamma,B)\ =\ Max\Big{[}\Big{|}\left(P_{sim}-P_{muff}\right)/P_{muff} \Big{|}:p_{in}\ =\ 7...16 bar(a),n\ =\ 750...3000\ rpm\Big{]} \tag{38}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(n\) & \multicolumn{5}{c}{\(P_{muff}\)**kW**} \\ \cline{2-6}
**rpm** & **7 Bar(a)** & **9 Bar(a)** & **11 Bar(a)** & **13 Bar(a)** & **16 Bar(a)** \\ \hline
0 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ \hline
250 & 1.830 & 2.440 & 3.08  
The minimization of \(\phi(\gamma,B)\) is the approximation in the maximum norm [22] of \(P_{mmf}\) by \(P_{sim}\). It is different from the least squares minimization [22] and it minimizes the maximum relative difference between \(P_{sim}\) and \(P_{mmf}\).  
The Nelder-Mead method of minimization of \(\phi(\gamma,B)\) gives \(\gamma\) = 0.10886 (\(-\)) and B = 119.8676, which is not dimensionless. The units of \(B\) are complicated; as in (31), they depend on the value of \(\gamma\).  
The similarity power \(P_{sim}\) and the ratio \(P_{sim}/P_{mmf}\) are presented in Table 3 and Figure 6. The ratio \(P_{sim}/P_{mmf}\) is calculated to compare \(P_{sim}\) and \(P_{mmf}\) and to present the relative errors of the approximation of \(P_{mmf}\) by \(P_{sim}\). The maximum relative error of \(P_{sim}\) vs. \(P_{mmf}\) in the analyzed ranges of the pressure and the rotational speed is about \(\pm 5\%\) (Table 3 and Figure 6b). The obtained similarity power \(P_{sim}\) is in rather good agreement with the manufacturer's experimental power \(P_{mmf}\).