Hydrostatics: Pressure & Manometry

Hydrostatics: Pressure & Manometry

$Learning Objectives$

Define pressure in a fluid at rest and distinguish pressure isotropy from Pascal's transmission principle.
Derive and apply the hydrostatic pressure-elevation relation for constant-density liquids.
Distinguish absolute, gage, atmospheric, and vacuum-pressure references.
Convert pressure to an equivalent head of a specified fluid.
Explain barometer and piezometer operation and their limitations.
Analyze single-fluid and multi-fluid manometers using a systematic pressure walk with explicit elevation changes.

$p$

Pressure is the compressive normal force intensity acting at a point in a fluid. A fluid at rest cannot sustain a shear stress, so the local stress state is purely normal.

Pressure as Normal Force Intensity

Defines average pressure over a small surface area and its limiting point value.

p=\lim_{\Delta A\to0}\frac{\Delta F_n}{\Delta A}

Variables

Symbol	Description	Unit
$p$	Pressure	Pa
$\Delta F_n$	Normal compressive force over the area element	N
$\Delta A$	Area element	$m^2$

$Pressure Isotropy$

At a point in a fluid at rest, pressure has the same magnitude in every direction. This follows from force equilibrium of an infinitesimal fluid element and the absence of static shear stress.

$Pascal's Transmission Principle$

A pressure change applied to a confined incompressible fluid is transmitted throughout the fluid and to the containing boundaries, apart from predictable hydrostatic differences caused by elevation.

$Pressure Isotropy versus Pascal Transmission$

These statements are related but not identical. Pressure isotropy describes the directional stress state at one point in a static fluid. Pascal's transmission principle describes how an imposed pressure increment is communicated through a confined fluid. In a hydraulic press, the same transmitted pressure increment acts on pistons of different areas and therefore produces different forces.

Ideal Hydraulic Press

Relates piston forces when elevation differences and losses are negligible.

\frac{F_1}{A_1}=\frac{F_2}{A_2}

Variables

Symbol	Description	Unit
$F_1$	Input-piston force	N
$A_1$	Input-piston area	$m^2$
$F_2$	Output-piston force	N
$A_2$	Output-piston area	$m^2$

Pascal transmission in a hydraulic press

Connected pistons show pressure transmission and area-based force scaling.

$Force Multiplication Does Not Create Energy$

An ideal hydraulic press trades displacement for force. Conservation of displaced volume gives $A_1s_1=A_2s_2$ , so a larger output force is accompanied by a smaller output displacement when losses are neglected.

$Hydrostatic Pressure Gradient$

The hydrostatic pressure gradient describes how pressure changes with elevation in a fluid at rest under gravity.

Hydrostatic Differential Equation

Relates vertical pressure gradient to fluid density under uniform gravity.

\frac{dp}{dz}=-\rho g

Variables

Symbol	Description	Unit
$p$	Pressure	Pa
$z$	Elevation measured positive upward	m
$\rho$	Fluid density	$kg/m^3$
$g$	Gravitational acceleration magnitude	$m/s^2$

Pressure Difference in a Constant-Density Liquid

Integrates the hydrostatic equation between two points in the same static liquid.

p_2-p_1=\rho g(z_1-z_2)=\gamma(z_1-z_2)

Variables

Symbol	Description	Unit
$p_1$	Pressure at point 1	Pa
$p_2$	Pressure at point 2	Pa
$z_1$	Elevation of point 1	m
$z_2$	Elevation of point 2	m
$\gamma$	Fluid specific weight	$N/m^3$

Pressure at Depth below a Free Surface

Computes pressure at vertical depth h below a boundary of known pressure.

p=p_0+\gamma h

Variables

Symbol	Description	Unit
$p$	Pressure at depth h	Pa
$p_0$	Pressure acting on the reference free surface	Pa
$\gamma$	Liquid specific weight	$N/m^3$
$h$	Vertical depth below the reference surface	m

Hydrostatic pressure with depth

Static-liquid pressure increases with vertical depth.

$Use the Correct Surface Pressure$

For an open tank, $p_0=p_{\text{atm}}$ on the absolute-pressure scale and $p_0=0$ on the gage-pressure scale. For a sealed tank, the gas pressure above the liquid must be included rather than automatically substituting atmospheric pressure.

$Equal-Elevation Rule$

Two points at the same elevation in the same continuous static fluid have the same pressure. This rule does not permit a horizontal jump across a solid wall or between disconnected fluids; continuity of the static fluid path matters.

$p_{\text{abs}}$

Absolute pressure is measured relative to a perfect vacuum.

$p_g$

Gage pressure is measured relative to the local atmospheric pressure. It is positive above atmosphere and negative below atmosphere.

$p_{\text{vac}}$

Vacuum pressure is the positive magnitude by which a system absolute pressure lies below the local atmospheric pressure.

Pressure Reference Relations

Converts between absolute, gage, atmospheric, and vacuum pressure.

p_{\text{abs}}=p_{\text{atm}}+p_g

p_{\text{vac}}=p_{\text{atm}}-p_{\text{abs}}=-p_g \qquad \text{when } p_g<0

Variables

Symbol	Description	Unit
$p_{\text{abs}}$	Absolute pressure	Pa
$p_g$	Gage pressure	Pa
$p_{\text{atm}}$	Local atmospheric pressure	Pa
$p_{\text{vac}}$	Vacuum-pressure magnitude	Pa

Absolute, gage, and vacuum references

Pressure levels referenced to absolute vacuum and atmosphere.

$Negative Gage Pressure Is Not Negative Absolute Pressure$

A negative gage pressure only means the system pressure is below the local atmosphere. Absolute pressure remains referenced to a vacuum and is the pressure scale required for thermodynamic relations and vapor-pressure comparisons.

$Pressure Head$

Pressure head is the height of a specified fluid column that represents a pressure or pressure difference.

Pressure Head

Converts pressure to an equivalent column height of a specified fluid.

h_p=\frac{p}{\gamma}

Variables

Symbol	Description	Unit
$h_p$	Pressure head in the selected fluid	m
$p$	Pressure on the chosen reference scale	Pa
$\gamma$	Specific weight of the head fluid	$N/m^3$

$Head Must Name Its Reference Fluid$

The same pressure corresponds to different column heights for fluids with different specific weights. State whether a head is, for example, metres of water or millimetres of mercury, and keep the pressure reference—gage or absolute—consistent.

$Mercury Barometer$

A mercury barometer measures local atmospheric pressure by balancing it against the hydrostatic pressure of a mercury column whose upper end is closed.

Mercury Barometer Relation

Relates atmospheric pressure to the supported mercury column and the pressure above it.

p_{\text{atm}}=p_v+\gamma_{Hg}h

Variables

Symbol	Description	Unit
$p_{\text{atm}}$	Local atmospheric pressure	Pa
$p_v$	Mercury vapor pressure above the column	Pa
$\gamma_{Hg}$	Mercury specific weight	$N/m^3$
$h$	Vertical mercury-column height	m

$Why Mercury Vapor Pressure Is Often Neglected$

At ordinary room temperatures, mercury vapor pressure is very small compared with atmospheric pressure, so elementary calculations commonly use $p_{\text{atm}}\approx\gamma_{Hg}h$ . The exact relation still includes the pressure above the column.

$Piezometer$

A piezometer is an open vertical tube connected to a liquid system so that the liquid-column elevation above the connection indicates positive gage pressure head.

$Piezometer Limitations$

A simple open piezometer is suited to liquids at positive gage pressure. It is impractical for very high pressures because the required column can be long, and it cannot directly provide a stable open-column reading for a gas or for a liquid pressure below atmosphere.

$Manometer$

A manometer determines pressure or pressure difference by balancing static columns of one or more fluids with known specific weights.

$Interactive Manometer Exploration$

Use the simulation to vary fluid density and elevation difference. Interpret every displayed pressure from the same hydrostatic sign rule used in the analytical pressure walk.

U-Tube Manometer Simulator

Learning objective: Track pressure changes through connected fluid columns and see how density and elevation differences control measured pressure difference.

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 Pressure Walk

Relates boundary pressures by summing hydrostatic changes along a connected path.

p_{\text{start}} +\sum\gamma_i\Delta h_{i,\text{down}} -\sum\gamma_j\Delta h_{j,\text{up}} =p_{\text{end}}

Variables

Symbol	Description	Unit
$p_{\text{start}}$	Pressure at the starting boundary	Pa
$p_{\text{end}}$	Pressure at the ending boundary	Pa
$\gamma_i$	Specific weight of a traversed fluid segment	$N/m^3$
$\Delta h$	Vertical elevation change within that segment	m

$Manometer Analysis Procedure$

STEP-BY-STEP

Choose a starting boundary with known or unknown pressure and a path through the connected static fluids to the other boundary.
Add $\gamma h$ each time the path moves downward within a fluid and subtract $\gamma h$ each time it moves upward.
At an interface, pressure is continuous across the interface when surface-tension effects are neglected.
A horizontal move at one elevation produces no hydrostatic pressure change within the same continuous static fluid.
Set the completed pressure walk equal to the pressure at the ending boundary and solve for the unknown.
Check that the final sign and magnitude are physically consistent with the heavier-fluid column displacement.

Manometer pressure walk

Signs for moving through connected columns of different fluids.

$Do Not Apply a Memorized U-Tube Formula without Its Geometry$

Expressions such as $\Delta p=(\gamma_m-\gamma_f)h$ are valid only for specific arrangements, such as appropriate equal-elevation taps and a clearly defined level difference. When elevations, connected fluids, or interfaces differ, use the full pressure walk instead of forcing a shortcut.

$Hydrostatic Paradox$

For the same static liquid, free-surface pressure, and vertical depth, pressure at the bottom is independent of container shape. This does not mean bottom force must equal the total liquid weight: bottom force also depends on bottom area, and vertical components of pressure on sloping walls complete the force balance.

Key Takeaways

Pressure in a static fluid is isotropic at a point, while Pascal's principle concerns transmission of an applied pressure increment through a confined fluid.
In a constant-density liquid under uniform gravity, pressure increases linearly downward according to $dp/dz=-\rho g$ .
The relation $p=p_0+\gamma h$ requires the actual pressure $p_0$ at the reference surface; an open atmosphere is only one common boundary condition.
Absolute pressure is referenced to vacuum, whereas gage pressure is referenced to local atmosphere.
Pressure head must identify the fluid whose specific weight is used.
Barometers measure atmospheric pressure; piezometers indicate positive liquid gage head; manometers compare pressures through static-column balances.
A reliable manometer solution follows the connected fluid path: add pressure moving down, subtract pressure moving up, and use only vertical elevation differences.
Container shape does not alter hydrostatic pressure at a specified depth, but it can alter total force because area and wall-force components differ.

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Learning Objectives

Static Fluid Pressure, ppp

Pressure as Normal Force Intensity

Pressure Isotropy

Pascal's Transmission Principle

Pressure Isotropy versus Pascal Transmission

Ideal Hydraulic Press

Force Multiplication Does Not Create Energy

Hydrostatic Pressure Gradient

Hydrostatic Differential Equation

Pressure Difference in a Constant-Density Liquid

Pressure at Depth below a Free Surface

Use the Correct Surface Pressure

Equal-Elevation Rule

Absolute Pressure, pabsp_{\text{abs}}pabs​

Gage Pressure, pgp_gpg​

Vacuum Pressure, pvacp_{\text{vac}}pvac​

Pressure Reference Relations

Negative Gage Pressure Is Not Negative Absolute Pressure

Pressure Head

Pressure Head

Head Must Name Its Reference Fluid

Mercury Barometer

Mercury Barometer Relation

Why Mercury Vapor Pressure Is Often Neglected

Piezometer

Piezometer Limitations

Manometer

Interactive Manometer Exploration

U-Tube Manometer Simulator

General Manometer Pressure Walk

Manometer Analysis Procedure

Do Not Apply a Memorized U-Tube Formula without Its Geometry

Hydrostatic Paradox

$Learning Objectives$

$p$

$Pressure Isotropy$

$Pascal's Transmission Principle$

$Pressure Isotropy versus Pascal Transmission$

$Force Multiplication Does Not Create Energy$

$Hydrostatic Pressure Gradient$

$Use the Correct Surface Pressure$

$Equal-Elevation Rule$

$p_{\text{abs}}$

$p_g$

$p_{\text{vac}}$

$Negative Gage Pressure Is Not Negative Absolute Pressure$

$Pressure Head$

$Head Must Name Its Reference Fluid$

$Mercury Barometer$

$Why Mercury Vapor Pressure Is Often Neglected$

$Piezometer$

$Piezometer Limitations$

$Manometer$

$Interactive Manometer Exploration$

$Manometer Analysis Procedure$

$Do Not Apply a Memorized U-Tube Formula without Its Geometry$

$Hydrostatic Paradox$