Problem Statement
Water is flowing through two different pipes to which an inverted differential manometer having an oil of sp. gr. 0.8 is connected. The pressure head in pipe A is 2 m of water. Find the pressure in pipe B for the manometer readings as shown in the figure.
Given Data
- Pressure head at A, \(h_A = 2 \, \text{m of water}\)
- Specific gravity of oil, \(S_{oil} = 0.8\)
- Height of water in left limb, \(h_1 = 30 \, \text{cm} = 0.3 \, \text{m}\)
- Height of oil difference, \(h_{oil} = 12 \, \text{cm} = 0.12 \, \text{m}\)
- Height of water in right limb, \(h_2 = 10 \, \text{cm} = 0.1 \, \text{m}\)
Solution
1. Define Pressures and Densities
Pressure at A (\(p_A\)):
Density of oil (\(\rho_{oil}\)):
2. Set up the Manometer Equation
We start at point A and move through the manometer to point B. We will set the datum line X-X at the level of A.
Pressure in the left limb at the oil-water interface:
Pressure in the right limb at the same level:
Equating the pressures at the same level:
Rearranging to solve for \(p_B\):
3. Calculate the Pressure at B (\(p_B\))
Now we substitute the known values into the equation.
Converting to N/cm²:
The pressure in pipe B is approximately \( 18,600 \, \text{N/m}^2 \) or \( 1.86 \, \text{N/cm}^2 \).
Explanation of the Inverted Manometer
An inverted U-tube differential manometer is used to measure small pressure differences between two points in a liquid system. It is "inverted" because it is placed above the pipes, and the manometric fluid (in this case, oil) is lighter than the fluid in the pipes (water).
The principle involves balancing the hydrostatic pressures. We start at a point of known pressure (A) and move through the manometer, subtracting pressure for upward movements and adding for downward movements, until we reach the point of unknown pressure (B). Since the pressure at any two points at the same horizontal level in a continuous fluid is the same, we can equate the pressure expressions for the left and right limbs at a common interface.
Physical Meaning
The initial pressure head at A is 2 m of water, which corresponds to 19,620 N/m². The calculated pressure at B is approximately 18,600 N/m². This shows that the pressure at B is slightly lower than the pressure at A.
This pressure difference (\(p_A - p_B \approx 1020\) N/m²) is what causes the 12 cm difference in the oil levels in the manometer. The higher pressure at A pushes the lighter oil up into the right limb of the manometer. This type of instrument is very sensitive and is ideal for accurately measuring small pressure drops in pipelines, which is essential for monitoring flow and calculating energy losses due to friction.
