Problem Statement
A pipe contains an oil of sp. gr. 0.9. A differential manometer connected at two points A and B shows a difference in mercury level of 15 cm. Find the difference of pressure at the two points.
Given Data
- Specific gravity of oil, \(S_{oil} = 0.9\)
- Difference in mercury level, \(h = 15 \, \text{cm} = 0.15 \, \text{m}\)
- Specific gravity of mercury, \(S_{Hg} = 13.6\)
Solution
1. Define Densities
Density of oil (\(\rho_{oil}\)):
Density of mercury (\(\rho_{Hg}\)):
2. Calculate the Pressure Difference (\(p_A - p_B\))
The pressure difference between points A and B can be found using the differential manometer formula.
Converting to N/cm²:
The difference of pressure is \( 18,688 \, \text{N/m}^2 \) or \( 1.869 \, \text{N/cm}^2 \).
Explanation of the Differential Manometer
A differential manometer is a device used to measure the pressure difference between two points in a pipe or between two different pipes. It does not measure the absolute pressure at any point, only the relative difference.
The principle is based on balancing a column of a heavy, immiscible manometric fluid (like mercury) against the pressure difference of the flowing fluid (oil). The height difference (\(h\)) in the manometric fluid is directly proportional to the pressure difference between points A and B. The greater the pressure difference, the larger the displacement of the mercury.
Physical Meaning
The calculated pressure difference of 18,688 N/m² indicates that the pressure at point A is higher than the pressure at point B. This pressure drop between two points in a pipe is a direct result of fluid friction (viscous losses) as the oil flows from A to B.
This measurement is fundamental in fluid mechanics and engineering. It allows engineers to calculate head loss, which is a measure of the energy lost due to friction in a pipe system. Understanding and calculating pressure drops are essential for correctly sizing pumps, designing efficient piping networks, and predicting flow rates in any fluid transport system.
