Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Hydrostatic pressure and its measurement
Question Statement:
A U-tube manometer containing mercury (specific gravity $S_m = 13.6$) is connected to a pressurized water pipe. The mercury level in the open limb is $0.25\text{ m}$ higher than that in the limb connected to the pipe, and the water-mercury interface in the pipe limb is $0.15\text{ m}$ below the pipe centerline. Taking $g = 9.81\text{ m/s}^2$ and density of water $\rho_w = 1000\text{ kg/m}^3$, the gauge pressure at the pipe centerline in kilopascals ($\text{kPa}$) is ________ (round off to two decimal places).
Answer Key:
31.88
Step-by-Step Solution & Explanation
1. Gauge pressure balance equation from open limb to pipe center:
$P_{\text{pipe}} + \rho_w g h_w = \rho_m g h_m$
$P_{\text{pipe}} = \rho_w g [ S_m h_m - h_w ]$
2. Substituting given values:
- $S_m = 13.6$
- Mercury column difference $h_m = 0.25\text{ m}$
- Water column above datum $h_w = 0.15\text{ m}$
$P_{\text{pipe}} = 1000 \times 9.81 \times [ 13.6 \times 0.25 - 0.15 ]$
$13.6 \times 0.25 = 3.40$
$3.40 - 0.15 = 3.25\text{ m of water}$
$P_{\text{pipe}} = 9810 \times 3.25 = 31,882.5\text{ Pa} = 31.8825\text{ kPa} \approx 31.88\text{ kPa}$