# Similarity Model of a Rotary Lobe Expander Working with Various Fluids
## 7 Numerical Calculation of the Power for Air and the Other Fluids
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Fluid**} & \multirow{2}{*}{\(\mathbf{p}_{in}\)} & \multirow{2}{*}{\(\mathbf{T}_{in}\)} & \multirow{2}{*}{\(\mathbf{\rho}_{in}\)} & \multirow{2}{*}{\(\mathbf{\mu}_{in}\)**\(\mathbf{10^{5}}\)**} & \multirow{2}{*}{\(\mathbf{p}_{out}\)} & \multirow{2}{*}{\(\mathbf{p}_{cr}\)} & \multirow{2}{*}{\(\mathbf{T}_{cr}\)} & \multirow{2}{*}{\(\mathbf{\rho}_{cr}\)} \\  & & \({}^{\circ}\)**C** & & & & & & \\ \hline Air & 16.0 & 25.0 & 18.70 & 1.831 & 1.0 & 37.86 & \(-\)140.6 & 342.9 \\ \hline R245 fa & 16.0 & 115.8 & 91.09 & 1.440 & 1.0 & 36.51 & 154.0 & 516.1 \\ \hline SES36 & 16.0 & 149.8 & 125.9 & 1.841 & 1.0 & 28.49 & 177.6 & 517.6 \\ \hline HFE7100 & 16.0 & 179.6 & 175.9 & 1.553 & 1.0 & 22.29 & 195.3 & 555.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values used in the numerical calculation and the critical parameters of the working fluids. Figure 8: Instantaneous output power as a function of the rotational angle of the upper rotor (the expander’s shaft) for the various computational scenarios of the rotational angle and the numerical grid density, given the inlet pressure \(p_{in}=16\) bar(a) of air, the outlet pressure \(p_{out}=1\) bar(a), and the rotational speed \(n=3000\) rpm. The same expander type, but with the lower rotor on the expander's shaft, was numerically analyzed for air in [11]. They used the same Immersed Solid method in ANSYS/CFX; however, for the rotational speed of the upper rotor, they used \(n=2500\) and 3750 rpm. The open question is the difference between the manufacturer's power \(P_{mmf,Air}=38.940\) kW and the numerically calculated \(P_{mm,Air}=44.550\) kW for the same \(n=3000\) rpm, \(p_{in}=16\) bar(a), and \(p_{out}=1\) bar(a). It may be partly caused by the assumed lack of clearance among the rotors and the housing in the calculation and by the possibly different inlet temperatures \(T_{in}\) for \(P_{mm}\) and \(P_{mmf}\), which are not presented in the manufacturer's data. The main reason seems to be the assumed (forced) outlet pressure \(p_{out}=1\) bar(a) at the end of the fluid domain, i.e., at the end of the outlet chamber. In reality, the fluid (Air) cannot go out fast enough at the high rotational speed of 3000 rpm to achieve \(p_{out}=1\) bar(a) in this place. The actual outlet pressure is higher and the actual power is lower. The same issue is analyzed in [11] for the rotational speed of 3750 rpm for the upper rotor, where the pressure 1 bar(a) at the outlet makes the calculated numerical power too high in comparison with the manufacturer's data. After increasing this pressure to 1.4 bar(a), the authors of [11] obtained good agreement between the calculated power and the manufacturer's power. For \(n=1500\) rpm, the values \(P_{mmf,Air}=24.490\) kW and \(P_{mm,Air}=24.000\) kW are in good agreement. Because of this discrepancy, the manufacturer's \(P_{mmf}\) and the numerical \(P_{mm}\) are used separately in the similarity analysis and are not combined. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(n\) & & & \(P_{num}\), kW & & & \(\eta_{num}=P_{num}/P_{0}\), \(-\) & & \\ \cline{2-9} rpm & **HFE7100** & **SES36** & **R245 fa** & **Air** & \(P_{0}\) & **HFE7100** & **SES36** & **R245 fa** & **Air** \\ \hline
0 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\ \hline
1500 & 20.255 & 21.142 & 21.905 & 24.000 & 29.373 & 0.6896 & 0.7198 & 0.7458 & 0.8171 \\ \hline
3000 & 32.379 & 34.946 & 36.891 & 44.550 & 58.745 & 0.5512 & 0.5949 & 0.6280 & 0.7584 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical power \(P_{num}\) and efficiency \(\eta_{num}=P_{num}/P_{0}\) for \(p_{in}=16\) bar(a) and \(p_{out}=1\) bar(a). Figure 9: (**a**) Numerical power \(P_{num}\) vs. the rotational speed \(n\) for various working fluids and (**b**) efficiency \(\eta_{num}=P_{num}/P_{0}\). The dashed line shows the limiting power \(P_{0}\) of the no-viscosity fluid.