Question 8.11: The average friction factor for a 50 mm diameter pipe is 0.0......

Given: M_1=0.25 ; D=0.05  m ; f=0.004 ; M_2=0.49

Refer Figure 8.22. The flow is from an initial Mach number of 0.25 to a final Mach number of 0.49. The length of the pipe can be obtained from the flow resistance parameter (4fL/D) required to accelerate the flow from M = 0.25 to M = 0.49. The parameter can be calculated from the maximum resistance parameter (tabulated in Fanno tables) at the two sections.

From Fanno tables (γ = 1.4) corresponding to M_1 = 0.25

\left(\frac{4 f L_{\max }}{D}\right)_{M=0.25}=8.537

The length of the pipe required to achieve sonic condition at exit is:

L_{\max 1}=\frac{8.537 D}{4 f}=\frac{8.537 \times 0.05}{4 \times 0.004}=26.678   m

Also, corresponding to M_2 = 0.49, from Fanno tables,

\left(4 f \frac{L_{\max }}{D}\right)_{M-0.49}=1.157

The length required to achieve sonic condition is:

L_{\max 2}=\frac{1.157 D}{4 f}=\frac{1.157 \times 0.05}{4 \times 0.004}=3.616   m

Length of the pipe between the two sections:

\begin{aligned}L &=L_{\max 1}-L_{\max 2}\\&=26.678-3.616 \\&=23.062  m\end{aligned}