Problem Statement
An inverted differential manometer is connected to two pipes A and B which convey water. The fluid in the manometer is oil of sp. gr. 0.8. For the manometer readings shown in the figure, find the pressure difference between A and B.
Given Data
- Fluid in pipes A and B: Water (\(\rho_w = 1000 \, \text{kg/m}^3\))
- Manometer fluid: Oil (\(S_{oil} = 0.8 \implies \rho_{oil} = 800 \, \text{kg/m}^3\))
- Height of water in left limb above datum: \(h_A = 30 \, \text{cm} = 0.3 \, \text{m}\)
- Height of water in right limb above datum: \(h_B = 30 \, \text{cm} = 0.3 \, \text{m}\)
- Height of oil column (difference): \(h_{oil} = 20 \, \text{cm} = 0.2 \, \text{m}\)
Diagram
Solution
1. Set up the Manometric Equation
For an inverted manometer, it's easiest to start at one pipe (A), move through the manometer to the other pipe (B), and sum the pressure changes. Pressure decreases as we move up and increases as we move down.
Let's re-examine this by balancing pressures at the datum line X-X, which is the higher water-oil interface.
Pressure in the left limb at X-X is the pressure at A minus the pressure from the 30 cm water column above it.
Pressure in the right limb at X-X is the pressure at B minus the pressure from the 30 cm water column and the 20 cm oil column above it.
2. Solve for the Pressure Difference (\(P_A - P_B\))
Now, we set the left and right pressure equations equal.
The term for the 30 cm water column (\(\rho_w g (0.3)\)) appears on both sides and can be cancelled out.
Rearrange the equation to find the pressure difference, \(P_A - P_B\).
The negative sign indicates that the pressure at A is lower than the pressure at B. The magnitude of the pressure difference is \(1569.6 \, \text{N/m}^2\).
The pressure difference between A and B is \( P_A - P_B = -1569.6 \, \text{N/m}^2 \).
This means the pressure at B is greater than the pressure at A by \(1569.6 \, \text{N/m}^2\).
Explanation of Inverted Manometry
An inverted U-tube manometer is used for measuring small pressure differences between two points. It is typically used for liquids and employs a manometer fluid (in this case, oil) that is lighter than the fluid in the pipes (water). The U-tube is inverted, with the lighter fluid trapped in the top portion.
The principle is the same as a standard manometer: we balance the pressures at a common datum line. For an inverted manometer, it is convenient to choose the datum at one of the fluid interfaces. The pressure difference between the pipes causes the lighter fluid to be displaced, and the magnitude of this displacement is used to calculate the pressure difference.
Physical Meaning
The result shows that the pressure at point B is higher than at point A. This is evident from the manometer reading itself: the higher pressure at B has pushed the lighter oil column upwards and towards the left limb, which is connected to the lower-pressure point A.
The pressure difference between the two pipes is exactly balanced by the hydrostatic pressure of the 20 cm column of the lighter oil. This is why inverted manometers are particularly useful for measuring small pressure differences—even a slight difference in pipe pressure can cause a significant, easily measurable displacement of the light manometer fluid.
