Hydrostatics: Pressure & Manometry - Theory & Concepts

Hydrostatics: Pressure & Manometry

Fluid pressure concepts, Pascal's Law, pressure variation with depth, and manometers.

Concept Overview

Hydrostatics deals with fluids at rest. The primary variable of interest is pressure.

Pascal's Law

The pressure at a point in a fluid at rest is the same in all directions.

Proof (Concept): Consider a small triangular wedge of fluid. Since the fluid is at rest, there are no shear stresses. Summing forces in X and Y directions shows that

P_x = P_y = P_s

Barometers

Instruments used for measuring atmospheric pressure.

Mercury Barometer

A simple device consisting of a glass tube closed at one end and open at the other, filled with mercury and inverted into a pool of mercury. Atmospheric pressure acting on the pool supports the mercury column in the tube.

Mercury Barometer

Instruments used for measuring atmospheric pressure.

P_{atm} = \gamma_{Hg} h + P_v

Since the vapor pressure of mercury at room temperature is extremely low (P_v \approx 0), we approximate atmospheric pressure as P_{atm} \approx \gamma_{Hg} h.
Standard atmospheric pressure at sea level is approximately h \approx 760 \text{ mm}.

Variation of Pressure with Depth

Concept Overview

In a static fluid, pressure increases linearly with depth due to the weight of the fluid above.

Hydrostatic Pressure Equation

For an incompressible fluid:

Hydrostatic Pressure Equation

P = P_{atm} + \gamma h

Variables

Symbol	Description	Unit
$P$	Absolute Pressure at depth $h$	-
$P_{gage} = \gamma h = \rho g h$	Gage Pressure	-
$\gamma$	Specific Weight of fluid	-
$h$	Depth below free surface	-

Note

Pressure Head ( $h$ ): The height of a column of fluid that would produce the given pressure.

h = \frac{P}{\gamma}

Often expressed in meters of water or mm of mercury.

The Hydrostatic Paradox

Hydrostatic Paradox

The pressure exerted by a fluid on the bottom of a container depends only on the depth of the fluid and its density, not on the shape of the container or the total volume (weight) of the fluid it holds.

For instance, if three differently shaped containers (e.g., a wide cylinder, a narrow cone, and an inverted cone) are filled with water to the exact same depth

h

, the pressure

P = \gamma h

at the bottom of all three containers is identical.

Pressure Measurement

Types of Pressure

Absolute Pressure ( $P_{abs}$ ): Measured relative to a perfect vacuum. Always positive.
Gage Pressure ( $P_{gage}$ ): Measured relative to local atmospheric pressure. Can be positive or negative (vacuum).
Atmospheric Pressure ( $P_{atm}$ ): Pressure exerted by the weight of the atmosphere. Standard $P_{atm} = 101.325 \text{ kPa}$ or $14.7 \text{ psi}$ .

Types of Pressure

P_{abs} = P_{gage} + P_{atm}

Manometers

Manometers use columns of fluids to measure pressure differences. The fundamental principle is that pressure changes with elevation in a continuous fluid.

Manometer Simulation: Experiment with fluid density and height difference to see the resulting pressure.

U-Tube Manometer Simulator

Manometer Fluid

Height Difference ($h$): 0.20 m

Differential Pressure

0.00 kPa

$P = \gamma h = \rho g h$

Adjust the height difference to see the corresponding pressure. In a real scenario, the pressure difference causes the height change.

General Manometer Equation

Start from a point of known pressure and move through the fluid to the other end.

General Manometer Equation

P_{start} + \sum \gamma_{down} h_{down} - \sum \gamma_{up} h_{up} = P_{end}

Moving Down: Add pressure ( $+\gamma h$ ).
Moving Up: Subtract pressure ( $-\gamma h$ ).
Crossing a fluid: Jump horizontally across the same continuous fluid without changing pressure.

Key Takeaways

Pascal's Law: Pressure acts equally in all directions.
Barometers: Measure atmospheric pressure using the height of a mercury column ( $P_{atm} \approx \gamma_{Hg} h$ ).
Depth Rule: Pressure increases linearly with depth ( $P = \gamma h$ ).
Manometry: Use the "start + down - up = end" method to solve any manometer configuration.
Units: Pressure is force/area ( $Pa, psi$ ). Head is length ( $m, ft$ ).

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