Topic: Fluid Mechanics
A pipeline connecting two reservoirs has its diameter reduced by 20% due to deposition of chemicals. For a given head difference in the reservoirs with unaltered friction factor, this would cause a reduction in discharge of
- A. 42.8%
- B. 20%
- C. 17.8%
- D. 10.6%
Answer: A. 42.8%
Explanation: Head loss for pipe: $h=\frac{8fLQ^{2}}{\pi^{2}gD^{5}}$. For same head difference and friction factor: $\frac{Q_{1}^{2}}{D_{1}^{5}}=\frac{Q_{2}^{2}}{D_{2}^{5}} \implies \frac{Q_{2}}{Q_{1}}={(\frac{D_{2}}{D_{1}})}^{\frac{5}{2}}$. With $D_2 = 0.8 D_1$, $\frac{Q_{2}}{Q_{1}}=(0.8)^{\frac{5}{2}} \approx 0.572$. So $Q_{2}=57.2\%$ of $Q_{1}$. Reduction in discharge $= 100 - 57.2 = 42.8\%$.