Problem Statement
A differential manometer is connected at the two points A and B as shown in the figure. At B air pressure is 9.81 N/cm² (abs), find the absolute pressure at A.
Given Data
- Absolute air pressure at B, \(P_B = 9.81 \, \text{N/cm}^2\)
- Fluid at A: Oil (\(\rho_{oil} = 900 \, \text{kg/m}^3\))
- Fluid in right limb: Water (\(\rho_{water} = 1000 \, \text{kg/m}^3\))
- Manometer fluid: Mercury (\(\rho_{Hg} = 13600 \, \text{kg/m}^3\))
- Heights from problem text: \(h_{water} = 60 \text{ cm}\), \(h_{Hg} = 10 \text{ cm}\), \(h_{oil} = 20 \text{ cm}\)
Diagram
Solution
1. Convert Pressure at B to SI Units (N/m²)
The given pressure is in N/cm², so we convert it to N/m².
2. Set up the Manometric Equation
We establish a datum line at the lower mercury level, X-X. The total absolute pressure at this level must be the same in both the left and right limbs.
Pressure in the left limb is the sum of the absolute pressure at A (\(P_A\)), plus the pressure from the 20 cm oil column, plus the pressure from the 10 cm mercury column.
Pressure in the right limb is the sum of the absolute air pressure at B (\(P_B\)), plus the pressure from the 60 cm water column. (Note: The pressure from the air column itself is negligible).
3. Solve for the Absolute Pressure at A (\(P_A\))
Equate the expressions for the left and right limbs and substitute the known values.
Calculate the pressure values for each fluid column.
Now, isolate \(P_A\) to find the absolute pressure at A.
The absolute pressure at A is \( P_A \approx 88879 \, \text{N/m}^2 \) or \( 8.89 \, \text{N/cm}^2 \).
Explanation of Absolute Pressure in Manometry
This problem involves calculating absolute pressure. Absolute pressure is measured relative to a perfect vacuum (zero pressure), whereas gauge pressure is measured relative to the local atmospheric pressure. Since the pressure at B is given in absolute terms, our final calculated pressure at A will also be an absolute value.
The manometric balancing principle remains the same. We select a datum line and equate the total absolute pressure on both sides. The total pressure includes both the pressures applied at the ends of the manometer (A and B) and the hydrostatic pressures from the various fluid columns.
Physical Meaning
The result shows that the absolute pressure at A (\(\approx 8.89 \, \text{N/cm}^2\)) is lower than the absolute pressure at B (\(9.81 \, \text{N/cm}^2\)). This makes sense when observing the manometer configuration.
The higher pressure at B, combined with the weight of the 60 cm water column, is pushing down on the right side. This force is balanced by the pressure at A plus the significant weight of the dense mercury column (10 cm) and the oil column (20 cm) on the left side. The system arranges itself to a point of equilibrium where these competing pressures are perfectly balanced at the datum line.
