11.7.12: More Examples of Fanno Flow
- Page ID
- 830
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To demonstrate the utility in Figure ?? consider the following example. Find the mass flow rate for \(f=0.05\), \(L= 4[m]\), \(D=0.02[m]\) and pressure ratio \(P_2 / P_1 = 0.1, 0.3, 0.5, 0.8\). The stagnation conditions at the entrance are \(300 K\) and \(3[bar]\) air.
Solution
First calculate the dimensionless resistance, \(\dfrac{4\,f\,L}{D}\).
\[
\dfrac{4\,f\,L}{D} = {4 \times 0.05 \times 4 \over 0.02 } = 40
\]
Fanno Flow | Input: \(\dfrac{P_2}{P_1}\) and \(\dfrac{4\,f\,L}{D}\) | k = 1.4 | |||
\(M_1\) | \(M_2\) | \(\dfrac{4\,f\,L}{D}\) | \(\left.\dfrac{4\,f\,L}{D}\right|_{1}\) | \(\left.\dfrac{4\,f\,L}{D}\right|_{2}\) | \(\dfrac{P_2}{P_1}\) |
0.12728 | 0.99677 | 0.99195 | 4.5910 | 0.98874 | 4.5393 |
0.12420 | 0.99692 | 0.99233 | 4.7027 | 0.98928 | 4.6523 |
0.11392 | 0.99741 | 0.99354 | 5.1196 | 0.99097 | 5.0733 |
0.07975 | 0.99873 | 0.99683 | 7.2842 | 0.99556 | 7.2519 |
Therefore, \(T\approx T_0\) and is the same for the pressure. Hence, the mass rate is a function of the Mach number. The Mach number is indeed a function of the pressure ratio but mass flow rate is a function of pressure ratio only through Mach number. The mass flow rate is
\begin{align*}
\dot{m} = P\, A\, M\, \sqrt{\dfrac{k }{ R\, T}} =
300000\, \times \dfrac{\pi \times 0.02^2 }{ 4 } \times 0.127 \times
\sqrt{\dfrac{ 1.4 }{ 287\, 300}} \approx 0.48
\left(\dfrac{ kg }{ sec} \right)
\end{align*}
and for the rest
\begin{align*}
\dot{m} \left( \dfrac{\mathbf{P_2 }{ P_1}} = 0.3 \right)
\sim 0.48 \times \dfrac{0.1242 }{ 0.1273}=0.468 \left(\dfrac{kg }{ sec}\right) \\
\dot{m}\, \left( \dfrac{\mathbf{P_2 }{ P_1}} = 0.5 \right)
\sim 0.48 \times \dfrac{0.1139 }{ 0.1273}=0.43 \left(\dfrac{kg }{ sec}\right) \\
\dot{m} \, \left( \dfrac{ \mathbf{P_2 }{ P_1}} = 0.8 \right)
\sim 0.48 \times \dfrac{0.07975 }{ 0.1273}=0.30 \left(\dfrac{kg }{ sec}\right)
\end{align*}
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.