# 11.7.12: More Examples of Fanno Flow

Example

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, $$\dfrac{4\,f\,L}{D}$$.
$\dfrac{4\,f\,L}{D} = {4 \times 0.05 \times 4 \over 0.02 } = 40 \tag{10}$
From Figure ?? for $$P_2 / P_1 = 0.1$$ $$M_1 \approx 0.13$$ etc. or accurately by utilizing the program as in the following table.

 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

• 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.