π Lesson 2
D2
Pressure Variation in Static Fluids & Manometry
Pressure in a still fluid increases steadily with depth because the weight of the fluid above pushes down.
π― Learning Objectives
- β Calculate pressure differences between two elevations in a static fluid using the hydrostatic equation
- β Analyze multi-fluid manometer systems to determine unknown pressures or densities
- β Design U-tube and inclined manometer configurations for field pressure measurement within Β±2% accuracy
- β Explain how fluid density and g-force variations affect pressure gradients in mining ventilation shafts or slurry transport lines
- β Apply manometric principles to interpret pressure readings from blasthole monitoring systems
π Why This Matters
In underground mines and open-pit operations, accurate pressure measurement is critical for ventilation system design, slurry transport in paste fill systems, and monitoring of explosive gas buildup in confined spaces. Misinterpreting static pressure gradients can lead to under-ventilated headings, over-pressurized pipelines, or false alarms in gas detection β all with safety and productivity consequences. Understanding how pressure changes with depth and fluid interfaces forms the foundation for reliable instrumentation in harsh mining environments.
π Core Principles
Fluid statics begins with the concept that a fluid at rest exerts pressure equally in all directions (Pascalβs principle). For an incompressible, homogeneous fluid under uniform gravity, the pressure gradient is purely vertical: dp/dz = βΟg, where z increases upward. Integrating yields the hydrostatic equation: Pβ = Pβ + Οg(zβ β zβ). When multiple immiscible fluids (e.g., water, oil, air) are layered, pressure adds across interfaces β each fluid contributes its own Οgh term. Manometers exploit this by balancing unknown pressures against known column heights of reference fluids (e.g., mercury, water, or gauge oil), converting pressure into measurable length. Absolute vs. gauge pressure distinction is essential: gauge pressure ignores atmospheric contribution, while absolute includes it β vital when calibrating sensors in high-altitude mines where P_atm drops ~1 kPa per 100 m elevation gain.
π Hydrostatic Pressure Equation
The fundamental relationship for pressure change with depth in a static fluid. Used to compute pressure at any point given a reference pressure, fluid density, and vertical distance.
Hydrostatic Pressure Difference
ΞP = Ο g hCalculates pressure difference due to vertical column of static fluid.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ΞP | Pressure difference | Pa (N/mΒ²) | Difference in pressure between two points (e.g., top and bottom of fluid column) |
| Ο | Fluid density | kg/mΒ³ | Mass per unit volume of the static fluid (e.g., water = 998 kg/mΒ³ at 20Β°C; mercury = 13,534 kg/mΒ³) |
| g | Local gravitational acceleration | m/sΒ² | Varies by latitude and elevation; typically 9.78β9.83 m/sΒ² in mining regions |
| h | Vertical height difference | m | Depth or elevation difference between two points measured perpendicular to gravity |
Typical Ranges:
Water column in ventilation duct: 0.1 β 5.0 m
Mercury column in calibration lab: 0.01 β 1.0 m
Slurry (SG 1.8) in tailings pipe: 2.0 β 20.0 m
π‘ Worked Example
Problem: A venturi-based pressure tap in a dewatering sump measures gauge pressure at 3.2 m below water surface. Water density = 998 kg/mΒ³, local g = 9.79 m/sΒ². Atmospheric pressure = 95.6 kPa (high-elevation mine site). Calculate absolute pressure at the tap.
1.
Step 1: Identify knowns β P_ref = P_atm = 95.6 kPa, Ο = 998 kg/mΒ³, g = 9.79 m/sΒ², Ξz = 3.2 m (depth, so zββzβ = +3.2 m)
2.
Step 2: Apply P_abs = P_atm + ΟgΞz = 95.6 kPa + (998)(9.79)(3.2) Pa = 95.6 kPa + 31,280 Pa = 95.6 kPa + 31.28 kPa
3.
Step 3: Sum β P_abs = 126.88 kPa. Compare to typical range for sump depths (120β135 kPa); result is physically consistent.
Answer:
The absolute pressure at the tap is 126.9 kPa, which falls within the expected range of 120β135 kPa for 3β4 m submergence at high-altitude sites.
ποΈ Real-World Application
At the Cadia East underground mine (NSW, Australia), engineers use inverted U-tube manometers filled with fluorocarbon liquid (Ο = 1,750 kg/mΒ³) to monitor differential pressure across filter bags in dust suppression systems. By measuring a 12.4 cm height difference between limbs, they calculate ΞP = Ο_fluoroΒ·gΒ·h = (1750)(9.81)(0.124) β 2.12 kPa β triggering maintenance when ΞP exceeds 2.5 kPa (indicating filter clogging). This avoids unplanned shutdowns and ensures compliance with NSW OHS Regulation 2022 Β§78 on airborne respirable dust control.