Pressure is defined as the physical force exerted on an object per unit area. In fluid mechanics, it refers to the force exerted by a fluid (liquid or gas) against the walls of its container or an object submerged within it.
Mathematically, pressure (\(P\)) is expressed as: \[P = \frac{F}{A}\]
Where: - \(P\): Pressure (Pascal, \(N/m^2\)) - \(F\): Normal force (\(N\)) - \(A\): Area (\(m^2\))
It is important to distinguish between the different reference points used in measurement.
\[P_{abs} = P_{atm} + P_g\] (Note: Use a negative sign for \(P_g\) if it is a vacuum pressure).
Common units used in industry and science:
| Unit | Symbol | Equivalent in Pascals (Pa) |
|---|---|---|
| Pascal | Pa | \(1 \text{ N/m}^2\) |
| Bar | bar | \(10^5 \text{ Pa}\) |
| Atmosphere | atm | \(101,325 \text{ Pa}\) |
| Millimeters of Mercury | mmHg (Torr) | \(\approx 133.32 \text{ Pa}\) |
| Pounds per Square Inch | psi | \(\approx 6,894.76 \text{ Pa}\) |
In a static fluid, pressure increases with depth due to the weight of the fluid column above.
\[P = \rho \cdot g \cdot h\]
Where: - \(\rho\) (rho): Density of the fluid (\(kg/m^3\)) - \(g\): Acceleration due to gravity (\(\approx 9.81 \text{ m/s}^2\)) - \(h\): Depth or height of the fluid column (\(m\))
A manometer uses a column of liquid to measure pressure differences. - U-Tube Manometer: The simplest form, where the height difference (\(h\)) between two arms represents the pressure difference. - Formula: \(\Delta P = (\rho_m - \rho_f)gh\) (where \(\rho_m\) is manometric fluid density and \(\rho_f\) is process fluid density).
Figure Placeholder: [Diagram of a U-Tube Manometer showing height displacement h]
Used specifically to measure atmospheric pressure. - Mercury Barometer: Invented by Torricelli; the height of mercury in a glass tube is supported by air pressure. - Aneroid Barometer: Uses a flexible metal box (capsule) that expands or contracts with air pressure changes.
A mechanical device consisting of a coiled tube. As pressure enters the tube, it tends to uncoil, moving a needle on a calibrated dial. This is the most common gauge in industrial settings.
Let’s calculate the pressure at the bottom of a swimming pool that is 3 meters deep.
# Variables
density_water <- 1000 # kg/m^3
gravity <- 9.81 # m/s^2
depth <- 3 # meters
p_atm <- 101325 # standard atmospheric pressure in Pa
# Hydrostatic Gauge Pressure (Pa)
p_gauge <- density_water * gravity * depth
# Absolute Pressure (Pa)
p_absolute <- p_atm + p_gauge
# Convert to kPa for readability
p_gauge_kpa <- p_gauge / 1000
p_absolute_kpa <- p_absolute / 1000
cat("The Gauge Pressure is:", round(p_gauge_kpa, 2), "kPa\n")## The Gauge Pressure is: 29.43 kPacat("The Total (Absolute) Pressure is:", round(p_absolute_kpa, 2), "kPa\n")## The Total (Absolute) Pressure is: 130.76 kPaSummary Checklist - [ ] Understand the difference between \(P_{abs}\) and \(P_g\). - [ ] Memorize the hydrostatic formula \(P = \rho gh\). - [ ] Be able to convert between units like \(psi\), \(bar\), and \(Pa\). ```