Problem Statement
What are the gauge pressure and absolute pressure at a point 3 m below the free surface of a liquid having a density of 1.53 x 10³ kg/m³ if the atmospheric pressure is equivalent to 750 mm of mercury? The specific gravity of mercury is 13.6 and the density of water is 1000 kg/m³.
Given Data
- Depth in liquid, \(h = 3 \, \text{m}\)
- Density of liquid, \(\rho_{liq} = 1.53 \times 10^3 \, \text{kg/m}^3 = 1530 \, \text{kg/m}^3\)
- Atmospheric pressure head, \(h_{atm} = 750 \, \text{mm of Hg} = 0.75 \, \text{m of Hg}\)
- Specific Gravity of Mercury, \(S.G._{Hg} = 13.6\)
- Density of Water, \(\rho_w = 1000 \, \text{kg/m}^3\)
Solution
1. Calculate Atmospheric Pressure (\(p_{atm}\))
First, find the density of mercury.
Now, calculate the atmospheric pressure from the mercury column height.
2. Calculate Gauge Pressure (\(p_{gauge}\))
Gauge pressure is the pressure at the point relative to the local atmospheric pressure. It is calculated from the height of the liquid column above the point.
3. Calculate Absolute Pressure (\(p_{abs}\))
Absolute pressure is the sum of the gauge pressure and the atmospheric pressure.
Gauge Pressure: \( p_{gauge} \approx 45,028 \, \text{N/m}^2 \) or \( 4.503 \, \text{N/cm}^2 \)
Absolute Pressure: \( p_{abs} \approx 145,090 \, \text{N/m}^2 \) or \( 14.509 \, \text{N/cm}^2 \)
Explanation of Pressure Types
Atmospheric Pressure (\(p_{atm}\)): The pressure exerted by the weight of the atmosphere. It varies with altitude and weather conditions. It's often measured by how high it can support a column of mercury in a barometer.
Gauge Pressure (\(p_{gauge}\)): This is the pressure measured relative to the surrounding atmospheric pressure. It is the pressure that most gauges read, hence the name. It is zero at the free surface of the liquid and increases with depth.
Absolute Pressure (\(p_{abs}\)): This is the total pressure at a point, measured relative to a perfect vacuum (zero pressure). It is the sum of the gauge pressure and the local atmospheric pressure. Absolute pressure can never be negative.
$$ p_{abs} = p_{gauge} + p_{atm} $$Physical Meaning
The results show two different ways of expressing the pressure at the same point.
The gauge pressure (45,028 N/m²) tells us how much the pressure at that point exceeds the pressure of the surrounding atmosphere. This is the pressure a standard pressure gauge would read if placed at that depth and is what would cause stress on the walls of a submerged container relative to the outside.
The absolute pressure (145,090 N/m²) represents the total, true pressure from all sources (both the liquid and the atmosphere above it). This value is critical for scientific and engineering calculations where the full thermodynamic state of the fluid is important, such as determining boiling points or gas solubility.
