Question

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).

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.

Answer 1

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 \( \boxed{64} \).