Gauge and Absolute Pressure in a Liquid
What are the gauge pressure and absolute pressure at a point 4 m below the free surface of a liquid of specific gravity 1.53, if atmospheric pressure is equivalent to 750 mm of mercury?
Given Data
- Depth in liquid (h) = 4 m
- Specific Gravity of liquid (S.G.liquid) = 1.53
- Atmospheric pressure = 750 mm of Mercury (Hg)
- Specific Gravity of Mercury (S.G.Hg) = 13.6 (standard value)
- Density of water (ρwater) = 1000 kg/m³ (standard value)
Formulas Used
1. Density from Specific Gravity: ρ = S.G. × ρwater
2. Gauge Pressure: Pgauge = ρgh
3. Absolute Pressure: Pabsolute = Pgauge + Patmospheric
Step-by-Step Solution
Step 1: Calculate Density of the Liquid
First, we find the density of the given liquid using its specific gravity.
ρliquid = S.G.liquid × ρwater
ρliquid = 1.53 × 1000
ρliquid = 1530 kg/m³
Step 2: Calculate Gauge Pressure (Pgauge)
Gauge pressure is the pressure due to the liquid column only.
Pgauge = ρliquid × g × h
Pgauge = 1530 kg/m³ × 9.81 m/s² × 4 m
Pgauge = 60049.2 N/m² or 60.05 kPa
Step 3: Calculate Atmospheric Pressure (Patm) in N/m²
We need to convert the atmospheric pressure from 750 mm of mercury into N/m² (Pascals).
First, find the density of mercury (ρHg).
ρHg = S.G.Hg × ρwater = 13.6 × 1000 = 13600 kg/m³
Next, convert the height of the mercury column to meters.
hHg = 750 mm = 0.75 m
Now, calculate the pressure.
Patm = ρHg × g × hHg
Patm = 13600 kg/m³ × 9.81 m/s² × 0.75 m
Patm = 100062 N/m² or 100.06 kPa
Step 4: Calculate Absolute Pressure (Pabsolute)
Absolute pressure is the sum of gauge pressure and atmospheric pressure.
Pabsolute = Pgauge + Patm
Pabsolute = 60049.2 N/m² + 100062 N/m²
Pabsolute = 160111.2 N/m² or 160.11 kPa
Final Answer
The Gauge Pressure is 60,049.2 N/m² (60.05 kPa).
The Absolute Pressure is 160,111.2 N/m² (160.11 kPa).
Explanation & Key Concepts
Gauge vs. Absolute Pressure
Understanding the difference between gauge and absolute pressure is crucial in engineering and physics.
- Gauge Pressure: This is the pressure measured relative to the local atmospheric pressure. It is the pressure exerted by the fluid column itself. A standard tire pressure gauge, for example, reads gauge pressure; it reads zero when the tire is flat because the pressure inside is equal to the atmospheric pressure outside.
- Atmospheric Pressure: This is the pressure exerted by the weight of the air in the atmosphere. It varies with altitude and weather conditions. In this problem, it was given in “mm of mercury,” a common manometric unit that we converted to Pascals (N/m²).
- Absolute Pressure: This is the pressure measured relative to a perfect vacuum (zero pressure). It is the true total pressure at a point. It is calculated by adding the gauge pressure to the local atmospheric pressure.
Applications
These calculations are fundamental to many practical applications:
- Diving: A diver must know the absolute pressure to calculate decompression times and understand the forces on their body. The pressure they feel is the gauge pressure of the water plus the atmospheric pressure at the surface.
- Hydraulic Systems: Engineers design hydraulic lifts and brakes based on gauge pressure, as it’s the pressure above atmospheric that does the work.
- Weather Forecasting: Barometers measure atmospheric pressure. Changes in this pressure help predict weather patterns.
- Process Engineering: In chemical plants and refineries, many vessels are kept at pressures above or below atmospheric. Both gauge and absolute pressure measurements are critical for safety and process control.
