Consider steady flow of water in the series pipe system shown below, with specified discharge. The diameters of Pipes A and B are 2 m and 1 m, respectively. The lengths of pipes A and B are 100 m and 200 m, respectively. Assume the Darcy-Weisbach friction coefficient, \( f \), as 0.01 for both the pipes. The ratio of head loss in Pipe-B to the head loss in Pipe-A is ___________ (round off to the nearest integer).
Show Hint
The ratio of head losses in two pipes is proportional to the 5th power of the ratio of their diameters and the ratio of their lengths.
The head loss in a pipe due to friction is given by the Darcy-Weisbach equation:
\[
h_f = \frac{8 Q^2 L f}{\pi^2 g D^5}
\]
Where,
- \( Q \) is the discharge,
- \( L \) is the length of the pipe,
- \( D \) is the diameter of the pipe,
- \( f \) is the Darcy-Weisbach friction factor,
- \( g \) is the acceleration due to gravity.
The ratio of head loss in Pipe-B to Pipe-A is:
\[
\frac{h_B}{h_A} = \frac{\frac{8 Q^2 L_B f}{\pi^2 g D_B^5}}{\frac{8 Q^2 L_A f}{\pi^2 g D_A^5}} = \left( \frac{D_A}{D_B} \right)^5 \times \frac{L_B}{L_A}
\]
Substitute the given values:
\[
\frac{h_B}{h_A} = \left( \frac{2}{1} \right)^5 \times \frac{200}{100} = 32 \times 2 = 64
\]
Hence, the ratio of head loss in Pipe-B to the head loss in Pipe-A is \( \boxed{64} \).
