Question Bank • QB_SWCE_FM_004 NAT • 2 Marks

Section 4: Soil and Water Conservation Engineering

Fluid Mechanics

Subtopic: Bernoulli's theorem

Question Statement:

A horizontal pipeline carries water and contracts smoothly from diameter $D_1 = 0.20\text{ m}$ to $D_2 = 0.10\text{ m}$. The steady discharge is $Q = 0.031416\text{ m}^3\text{/s}$ ($= 10\pi\text{ L/s}$). If the static pressure at the larger section is $P_1 = 200\text{ kPa}$, neglecting friction head losses and taking water density $\rho = 1000\text{ kg/m}^3$, the static pressure at the throat section $P_2$ in $\text{kPa}$ is ________ (round off to 1 decimal place).

Answer Key:

192.5

Step-by-Step Solution & Explanation

1. Cross-sectional areas: $A_1 = \frac{\pi}{4} (0.20)^2 = 0.031416\text{ m}^2 \implies v_1 = \frac{Q}{A_1} = 1.0\text{ m/s}$ $A_2 = \frac{\pi}{4} (0.10)^2 = 0.007854\text{ m}^2 \implies v_2 = \frac{Q}{A_2} = 4.0\text{ m/s}$ 2. Applying Bernoulli's equation for a horizontal streamline ($z_1 = z_2$): $P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2$ $P_2 = P_1 + \frac{1}{2} \rho (v_1^2 - v_2^2) = 200000 + \frac{1}{2}(1000)(1.0^2 - 4.0^2)$ $P_2 = 200000 + 500(1 - 16) = 200000 - 7500 = 192500\text{ Pa} = 192.5\text{ kPa}$
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