Problem Statement
An inverted differential manometer containing an oil of sp. gr. 0.9 is connected to find the difference of pressures at two points of a pipe containing water. If the manometer reading is 40 cm, find the difference of pressures.
Given Data & Constants
- Manometer reading, \(h = 40 \, \text{cm} = 0.4 \, \text{m}\)
- Specific gravity of manometric oil, \(S_{\text{oil}} = 0.9\)
- Pipe fluid is water.
- Density of water, \(\rho_{\text{water}} = 1000 \, \text{kg/m}^3\) (standard value)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\) (standard value)
Solution
1. Calculate the Density of the Manometric Oil (\(\rho_{\text{oil}}\))
The density of the oil is its specific gravity multiplied by the density of water.
2. Calculate the Pressure Difference (\( \Delta P \))
For an inverted U-tube manometer, the pressure difference is calculated using the difference in densities between the pipe fluid and the lighter manometric fluid.
The difference of pressures is \( \Delta P = 392.4 \, \text{N/m}^2 \).
Principle of the Inverted Manometer
An inverted differential manometer operates on the same principle of hydrostatic balance as a standard U-tube manometer, but it is used for measuring small pressure differences in liquids. Its key features are:
- The U-tube is inverted (upside-down).
- The manometric fluid (oil in this case) must be lighter than the fluid in the pipe (water).
The higher pressure at one point in the pipe pushes the lighter oil downwards, causing the oil level in the other limb to rise. The pressure difference is directly proportional to the height difference (\(h\)) and the difference in densities between the two fluids.
Physical Meaning: Measuring Small Pressure Changes
The calculated pressure difference is very small (392.4 N/m², which is less than 0.4% of atmospheric pressure). This is the type of measurement where an inverted manometer excels.
Such a small pressure drop between two points in a water pipe could indicate very slow-moving (laminar) flow or flow through a very large diameter pipe where frictional losses are minimal. In industrial and laboratory settings, accurately measuring these small pressure differentials is essential for precisely controlling flow rates and analyzing fluid behavior under specific conditions.
