Hydrostatics: Pressure & Manometry

Hydrostatics: Pressure & Manometry

Learning Objectives

Define pressure and demonstrate Pascal's Law.
Learn how pressure varies with vertical depth in a static fluid.
Distinguish between absolute, gage, and atmospheric pressures.
Analyze pressure using equivalent pressure heads.
Understand the operation of barometers for measuring atmospheric pressure.
Calculate pressure differences using manometers and the general manometer equation.

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, which represents the compressive force exerted by a fluid per unit area.

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 the X and Y directions shows that $P_x = P_y = P_s$ .

Barometers

Barometers are instruments used for measuring atmospheric pressure by balancing atmospheric force against the weight of a fluid column.

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 Equation

Calculates the atmospheric pressure based on the height of a supported mercury column.

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

Variables

Symbol	Description	Unit
$P_{atm}$	Atmospheric pressure	$Pa \text{ or } N/m^2$
$\gamma_{Hg}$	Specific weight of mercury (approx. $133 \text{kN}/\text{m}^3$ )	$N/m^3$
$h$	Height of the mercury column	m
$P_v$	Vapor pressure of mercury (negligible at room temperature, $P_v \approx 0$ )	Pa

Atmospheric Pressure Insights

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 supports a mercury column of approximately $h \approx 760\text{ mm}$ (29.92 in).

Variation of Pressure with Depth

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

Hydrostatic Pressure Equation

The equation defining how pressure changes vertically in an incompressible fluid under gravity.

Hydrostatic Pressure Equation

Calculates the absolute or gage pressure at a given depth within a static incompressible fluid.

P = P_{atm} + \gamma h

Variables

Symbol	Description	Unit
$P$	Absolute pressure at depth $h$	$Pa \text{ or } N/m^2$
$P_{atm}$	Atmospheric pressure at the free surface	$Pa \text{ or } N/m^2$
$\gamma$	Specific weight of the fluid ( $\gamma = \rho g$ )	$N/m^3$
$h$	Vertical depth below the free surface	m

Gage Pressure in Static Fluids

The gage pressure ( $P_{\text{gage}}$ ) at any depth $h$ is given directly by:

P_{\text{gage}} = \gamma h = \rho g h

Pressure Head ( $h$ )

The equivalent height of a column of fluid that would produce a given pressure. It is a common alternative way of expressing static pressure in fluid mechanics and hydraulics.

Pressure Head Equation

Converts static pressure into an equivalent column height of a specific fluid.

h = \frac{P}{\gamma}

Variables

Symbol	Description	Unit
$h$	Pressure head (column height)	$m \text{ of fluid}$
$P$	Fluid pressure	$Pa \text{ or } N/m^2$
$\gamma$	Specific weight of the fluid	$N/m^3$

Pressure Head Units & Usage

Often expressed in "meters of water" ( $\text{m H}_2\text{O}$ ) or "millimeters of mercury" ( $\text{mm Hg}$ ).
Facilitates calculations in piping systems and pump selections.

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.

Types of Pressure

Understanding the difference between absolute, gage, and atmospheric reference frames is vital in pressure calculations.

Pressure Reference Frames

Absolute Pressure ( $P_{\text{abs}}$ ): Measured relative to a perfect vacuum (absolute zero pressure). It is always positive.
Gage Pressure ( $P_{\text{gage}}$ ): Measured relative to the local atmospheric pressure. It can be positive (above atmospheric) or negative (vacuum/below atmospheric).
Atmospheric Pressure ( $P_{\text{atm}}$ ): The pressure exerted by the ambient atmosphere. Standard atmospheric pressure at sea level is $P_{\text{atm}} = 101.325\text{ kPa}$ or $14.7\text{ psi}$ .

Absolute and Gage Pressure Relation

Relates absolute pressure to gage pressure using the local atmospheric pressure as a baseline reference.

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

Variables

Symbol	Description	Unit
$P_{\text{abs}}$	Absolute pressure	$Pa \text{ or } psi$
$P_{\text{gage}}$	Gage pressure	$Pa \text{ or } psi$
$P_{\text{atm}}$	Local atmospheric pressure	$Pa \text{ or } psi$

Manometers

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

Interactive Simulation

Experiment with fluid density and height difference to see the resulting pressure in the manometer simulation below.

U-Tube Manometer Simulator

Positive $h$ means the right column is higher, so $P_1>P_2$ .

Manometer arrangement

Manometer fluid

Signed level difference (

h

)0.20 m

Pressure difference,

P_1-P_2

26.68 kPa

\Delta p=\rho_m g h

The left side has the higher pressure.

The differential equation assumes equal-elevation taps, the same line fluid above both interfaces, negligible capillary effects, and hydrostatic equilibrium. More general arrangements require walking pressure continuously through every fluid column.

General Manometer Equation

The systematic method used to analyze manometers by starting at one pressure boundary and summing pressure changes along the fluid columns to the other boundary.

General Manometer Equation

Formulates the pressure relationship by walking through a series of continuous fluid columns.

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

Variables

Symbol	Description	Unit
$P_{\text{start}}$	Starting pressure at one end of the manometer	$Pa \text{ or } N/m^2$
$\gamma_{\text{down}}$	Specific weight of the fluid in columns where movement is downward	$N/m^3$
$h_{\text{down}}$	Vertical height of the downward column	m
$\gamma_{\text{up}}$	Specific weight of the fluid in columns where movement is upward	$N/m^3$
$h_{\text{up}}$	Vertical height of the upward column	m
$P_{\text{end}}$	Ending pressure at the other end of the manometer	$Pa \text{ or } N/m^2$

Rules for Manometer Analysis

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

Key Takeaways

Pascal's Law: Pressure acts equally in all directions inside a static fluid.
Barometers: Measure atmospheric pressure using the supported height of a fluid (typically mercury) column ( $P_{\text{atm}} \approx \gamma_{\text{Hg}} h$ ).
Hydrostatic Depth Relation: Pressure increases linearly with depth in a static incompressible fluid ( $P = P_0 + \gamma h$ ).
Manometry Method: Apply the systematic equation ( $P_{\text{start}} + \sum \gamma_{\text{down}} h_{\text{down}} - \sum \gamma_{\text{up}} h_{\text{up}} = P_{\text{end}}$ ) to solve any multi-fluid manometer configuration.
Pressure vs. Head: Pressure represents force per unit area ( $\text{Pa}$ or $\text{psi}$ ), whereas head represents the equivalent column height of fluid ( $\text{m}$ or $\text{ft}$ ).