Definition
Pressure is the amount of force applied perpendicular to the surface of an object per unit area. The symbol for it is
p or
P.
[1] The
IUPAC recommendation for pressure is a lower-case
p.
[2] However, upper-case
P is widely used. The usage of
P vs
p depends upon the field in which one is working, on the nearby presence of other symbols for quantities such as
power and
momentum, and on writing style.
Formula
Mathematically:
where:
is the pressure,
is the normal force,
is the area of the surface on contact.
Pressure is a
scalar
quantity. It relates the vector surface element (a vector normal to the
surface) with the normal force acting on it. The pressure is the scalar
proportionality constant that relates the two normal vectors:
The minus sign comes from the fact that the force is considered
towards the surface element, while the normal vector points outward. The
equation has meaning in that, for any surface
S in contact with the fluid, the total force exerted by the fluid on that surface is the
surface integral over
S of the right-hand side of the above equation.
It is incorrect (although rather usual) to say "the pressure is
directed in such or such direction". The pressure, as a scalar, has no
direction. The force given by the previous relationship to the quantity
has a direction, but the pressure does not. If we change the orientation
of the surface element, the direction of the normal force changes
accordingly, but the pressure remains the same.
Pressure is distributed to solid boundaries or across arbitrary sections of fluid
normal to these boundaries or sections at every point. It is a fundamental parameter in
thermodynamics, and it is
conjugate to
volume.
Units
The
SI unit for pressure is the
pascal (Pa), equal to one
newton per
square metre (N/m
2, or kg·m
−1·s
−2). This name for the unit was added in 1971;
[3] before that, pressure in SI was expressed simply in newtons per square metre.
Other units of pressure, such as
pounds per square inch and
bar, are also in common use. The
CGS unit of pressure is the
barye (Ba), equal to 1 dyn·cm
−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm
2 or kg/cm
2)
and the like without properly identifying the force units. But using
the names kilogram, gram, kilogram-force, or gram-force (or their
symbols) as units of force is expressly forbidden in SI. The
technical atmosphere (symbol: at) is 1 kgf/cm
2 (98.0665 kPa, or 14.223 psi).
Since a system under pressure has the potential to perform work on
its surroundings, pressure is a measure of potential energy stored per
unit volume. It is therefore related to energy density and may be
expressed in units such as
joules per cubic metre (J/m
3, which is equal to Pa). Mathematically:
Some
meteorologists prefer the hectopascal (hPa) for atmospheric air pressure, which is equivalent to the older unit
millibar
(mbar). Similar pressures are given in kilopascals (kPa) in most other
fields, where the hecto- prefix is rarely used. The inch of mercury is
still used in the United States. Oceanographers usually measure
underwater pressure in
decibars (dbar) because pressure in the ocean increases by approximately one decibar per metre depth.
The
standard atmosphere (atm) is an established constant. It is approximately equal to typical air pressure at Earth
mean sea level and is defined as
101325 Pa.
Because pressure is commonly measured by its ability to displace a column of liquid in a
manometer, pressures are often expressed as a depth of a particular fluid (e.g.,
centimetres of water,
millimetres of mercury or
inches of mercury). The most common choices are
mercury
(Hg) and water; water is nontoxic and readily available, while
mercury's high density allows a shorter column (and so a smaller
manometer) to be used to measure a given pressure. The pressure exerted
by a column of liquid of height
h and density
ρ is given by the hydrostatic pressure equation
p = ρgh, where
g is the
gravitational acceleration.
Fluid density and local gravity can vary from one reading to another
depending on local factors, so the height of a fluid column does not
define pressure precisely. When millimetres of mercury or inches of
mercury are quoted today, these units are not based on a physical column
of mercury; rather, they have been given precise definitions that can
be expressed in terms of SI units.
[citation needed] One millimetre of mercury is approximately equal to one
torr. The water-based units still depend on the density of water, a measured, rather than defined, quantity. These
manometric units are still encountered in many fields.
Blood pressure is measured in millimetres of mercury in most of the world, and lung pressures in centimetres of water are still common.
Underwater divers use the
metre sea water (msw or MSW) and
foot sea water (fsw or FSW) units of pressure, and these are the standard units for pressure gauges used to measure pressure exposure in
diving chambers and
personal decompression computers. A msw is defined as 0.1 bar (= 100000 Pa = 10000 Pa), is not the same as a linear metre of depth. 33.066 fsw = 1 atm
[4]
(1 atm = 101325 Pa / 33.066 = 3064.326 Pa). Note that the pressure
conversion from msw to fsw is different from the length conversion:
10 msw = 32.6336 fsw, while 10 m = 32.8083 ft.
[5]
Gauge pressure is often given in units with "g" appended, e.g.
"kPag", "barg" or "psig", and units for measurements of absolute
pressure are sometimes given a suffix of "a", to avoid confusion, for
example "kPaa", "psia". However, the US
National Institute of Standards and Technology
recommends that, to avoid confusion, any modifiers be instead applied
to the quantity being measured rather than the unit of measure.
[6] For example,
"pg = 100 psi" rather than
"p = 100 psig".
Differential pressure is expressed in units with "d" appended; this
type of measurement is useful when considering sealing performance or
whether a valve will open or close.
Presently or formerly popular pressure units include the following:
- atmosphere (atm)
- manometric units:
- centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury,
- height of equivalent column of water, including millimetre (mm H
2O), centimetre (cm H
2O), metre, inch, and foot of water;
- imperial and customary units:
- non-SI metric units:
- bar, decibar, millibar,
- msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression,
- kilogram-force, or kilopond, per square centimetre (technical atmosphere),
- gram-force and tonne-force (metric ton-force) per square centimetre,
- barye (dyne per square centimetre),
- kilogram-force and tonne-force per square metre,
- sthene per square metre (pieze).
Pressure units
|
|
Pascal |
Bar |
Technical atmosphere |
Standard atmosphere |
Torr |
Pounds per square inch |
| (Pa) |
(bar) |
(at) |
(atm) |
(Torr) |
(psi) |
| 1 Pa |
≡ 1 N/m2 |
10−5 |
1.0197×10−5 |
9.8692×10−6 |
7.5006×10−3 |
1.450377×10−4 |
| 1 bar |
105 |
≡ 100 kPa
≡ 106 dyn/cm2
|
1.0197 |
0.98692 |
750.06 |
14.50377 |
| 1 at |
9.80665×104 |
0.980665 |
≡ 1 kp/cm2 |
0.9678411 |
735.5592 |
14.22334 |
| 1 atm |
1.01325×105 |
1.01325 |
1.0332 |
1 |
≡ 760 |
14.69595 |
| 1 Torr |
133.3224 |
1.333224×10−3 |
1.359551×10−3 |
≡ 1/760 ≈ 1.315789×10−3 |
≡ 1 Torr
≈ 1 mmHg
|
1.933678×10−2 |
| 1 psi |
6.8948×103 |
6.8948×10−2 |
7.03069×10−2 |
6.8046×10−2 |
51.71493 |
≡ 1 lbf /in2 |
Examples
The effects of an external pressure of 700 bar on an aluminum cylinder with 5 mm wall thickness
As an example of varying pressures, a finger can be pressed against a
wall without making any lasting impression; however, the same finger
pushing a
thumbtack
can easily damage the wall. Although the force applied to the surface
is the same, the thumbtack applies more pressure because the point
concentrates that force into a smaller area. Pressure is transmitted to
solid boundaries or across arbitrary sections of fluid
normal to these boundaries or sections at every point. Unlike
stress, pressure is defined as a
scalar quantity. The negative
gradient of pressure is called the
force density.
Another example is a knife. If we try to cut a fruit with the flat
side, the force is distributed over a large area, and it will not cut.
But if we use the edge, it will cut smoothly. The reason is that the
flat side has a greater surface area (less pressure), and so it does not
cut the fruit. When we take the thin side, the surface area is reduced,
and so it cuts the fruit easily and quickly. This is one example of a
practical application of pressure.
For gases, pressure is sometimes measured not as an
absolute pressure, but relative to
atmospheric pressure; such measurements are called
gauge pressure. An example of this is the air pressure in an
automobile tire, which might be said to be "220
kPa
(32 psi)", but is actually 220 kPa (32 psi) above atmospheric pressure.
Since atmospheric pressure at sea level is about 100 kPa (14.7 psi),
the absolute pressure in the tire is therefore about 320 kPa (46.7 psi).
In technical work, this is written "a gauge pressure of 220 kPa
(32 psi)". Where space is limited, such as on
pressure gauges,
name plates,
graph labels, and table headings, the use of a modifier in parentheses,
such as "kPa (gauge)" or "kPa (absolute)", is permitted. In non-
SI
technical work, a gauge pressure of 32 psi is sometimes written as
"32 psig", and an absolute pressure as "32 psia", though the other
methods explained above that avoid attaching characters to the unit of
pressure are preferred.
[7]
Gauge pressure is the relevant measure of pressure wherever one is interested in the stress on
storage vessels
and the plumbing components of fluidics systems. However, whenever
equation-of-state properties, such as densities or changes in densities,
must be calculated, pressures must be expressed in terms of their
absolute values. For instance, if the atmospheric pressure is 100 kPa, a
gas (such as helium) at 200 kPa (gauge) (300 kPa [absolute]) is 50%
denser than the same gas at 100 kPa (gauge) (200 kPa [absolute]).
Focusing on gauge values, one might erroneously conclude the first
sample had twice the density of the second one.
Scalar nature
In a static
gas, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant
random motion.
Because we are dealing with an extremely large number of molecules and
because the motion of the individual molecules is random in every
direction, we do not detect any motion. If we enclose the gas within a
container, we detect a pressure in the gas from the molecules colliding
with the walls of our container. We can put the walls of our container
anywhere inside the gas, and the force per unit area (the pressure) is
the same. We can shrink the size of our "container" down to a very small
point (becoming less true as we approach the atomic scale), and the
pressure will still have a single value at that point. Therefore,
pressure is a scalar quantity, not a vector quantity. It has magnitude
but no direction sense associated with it. Pressure force acts in all
directions at a point inside a gas. At the surface of a gas, the
pressure force acts perpendicular (at right angle) to the surface.
A closely related quantity is the
stress tensor
σ, which relates the vector force
to the
vector area 
via the linear relation
.
This
tensor may be expressed as the sum of the
viscous stress tensor
minus the hydrostatic pressure. The negative of the stress tensor is
sometimes called the pressure tensor, but in the following, the term
"pressure" will refer only to the scalar pressure.
According to the theory of
general relativity, pressure increases the strength of a gravitational field (see
stress–energy tensor) and so adds to the mass-energy cause of
gravity. This effect is unnoticeable at everyday pressures but is significant in
neutron stars, although it has not been experimentally tested.
[8]
Types
Fluid pressure
Fluid pressure is most often the compressive stress at some point within a
fluid. (The term
fluid refers to both liquids and gases – for more information specifically about liquid pressure, see
section below.)
Fluid pressure occurs in one of two situations:
- An open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere.
- A closed condition, called "closed conduit", e.g. a water line or gas line.
Pressure in open conditions usually can be approximated as the
pressure in "static" or non-moving conditions (even in the ocean where
there are waves and currents), because the motions create only
negligible changes in the pressure. Such conditions conform with
principles of
fluid statics. The pressure at any given point of a non-moving (static) fluid is called the
hydrostatic pressure.
Closed bodies of fluid are either "static", when the fluid is not
moving, or "dynamic", when the fluid can move as in either a pipe or by
compressing an air gap in a closed container. The pressure in closed
conditions conforms with the principles of
fluid dynamics.
The concepts of fluid pressure are predominantly attributed to the discoveries of
Blaise Pascal and
Daniel Bernoulli.
Bernoulli's equation
can be used in almost any situation to determine the pressure at any
point in a fluid. The equation makes some assumptions about the fluid,
such as the fluid being ideal
[9] and incompressible.
[9] An ideal fluid is a fluid in which there is no friction, it is
inviscid [9] (zero
viscosity).
[9] The equation for all points of a system filled with a constant-density fluid is
[10]
where:
- p = pressure of the fluid,
- γ = ρg = density · acceleration of gravity = specific weight of the fluid,[9]
- v = velocity of the fluid,
- g = acceleration of gravity,
- z = elevation,
= pressure head,
= velocity head.
Applications
Explosion or deflagration pressures
Explosion or
deflagration pressures are the result of the ignition of explosive
gases, mists, dust/air suspensions, in unconfined and confined spaces.
Negative pressures
While
pressures are, in general, positive, there are several situations in which negative pressures may be encountered:
- When dealing in relative (gauge) pressures. For instance, an
absolute pressure of 80 kPa may be described as a gauge pressure of
−21 kPa (i.e., 21 kPa below an atmospheric pressure of 101 kPa).
- When attractive intermolecular forces (e.g., van der Waals forces or hydrogen bonds) between the particles of a fluid exceed repulsive forces due to thermal motion. These forces explain ascent of sap
in tall plants. An apparent negative pressure must act on water
molecules at the top of any tree taller than 10 m, which is the pressure head
of water that balances the atmospheric pressure. Intermolecular forces
maintain cohesion of columns of sap that run continuously in xylem from the roots to the top leaves.[11]
- The Casimir effect can create a small attractive force due to interactions with vacuum energy; this force is sometimes termed "vacuum pressure" (not to be confused with the negative gauge pressure of a vacuum).
- For non-isotropic stresses in rigid bodies, depending on how the
orientation of a surface is chosen, the same distribution of forces may
have a component of positive pressure along one surface normal, with a component of negative pressure acting along another surface normal.
- The stresses in an electromagnetic field are generally non-isotropic, with the pressure normal to one surface element (the normal stress) being negative, and positive for surface elements perpendicular to this.
- In the cosmological constant.
Stagnation pressure
Stagnation pressure
is the pressure a fluid exerts when it is forced to stop moving.
Consequently, although a fluid moving at higher speed will have a lower
static pressure, it may have a higher stagnation pressure when forced to a standstill. Static pressure and stagnation pressure are related by:
where
is the stagnation pressure
is the flow velocity- is the static pressure.
The pressure of a moving fluid can be measured using a
Pitot tube, or one of its variations such as a
Kiel probe or
Cobra probe, connected to a
manometer. Depending on where the inlet holes are located on the probe, it can measure static pressures or stagnation pressures.
Surface pressure and surface tension
There is a two-dimensional analog of pressure – the lateral force per unit length applied on a line perpendicular to the force.
Surface pressure is denoted by π:
and shares many similar properties with three-dimensional pressure.
Properties of surface chemicals can be investigated by measuring
pressure/area isotherms, as the two-dimensional analog of
Boyle's law,
πA = k, at constant temperature.
Surface tension is another example of surface pressure, but with a reversed sign, because "tension" is the opposite to "pressure".
Pressure of an ideal gas
In an
ideal gas, molecules have no volume and do not interact. According to the
ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume:
where:
- p is the absolute pressure of the gas,
- n is the amount of substance,
- T is the absolute temperature,
- V is the volume,
- R is the ideal gas constant.
Real gases exhibit a more complex dependence on the variables of state.
[12]
Vapour pressure
Vapour pressure is the pressure of a
vapour in
thermodynamic equilibrium with its condensed
phases in a closed system. All liquids and
solids have a tendency to
evaporate into a gaseous form, and all
gases have a tendency to
condense back to their liquid or solid form.
The
atmospheric pressure boiling point of a liquid (also known as the
normal boiling point)
is the temperature at which the vapor pressure equals the ambient
atmospheric pressure. With any incremental increase in that temperature,
the vapor pressure becomes sufficient to overcome atmospheric pressure
and lift the liquid to form vapour bubbles inside the bulk of the
substance.
Bubble
formation deeper in the liquid requires a higher pressure, and
therefore higher temperature, because the fluid pressure increases above
the atmospheric pressure as the depth increases.
The vapor pressure that a single component in a mixture contributes to the total pressure in the system is called
partial vapor pressure.
Liquid pressure
When a person swims under the water, water pressure is felt acting on
the person's eardrums. The deeper that person swims, the greater the
pressure. The pressure felt is due to the weight of the water above the
person. As someone swims deeper, there is more water above the person
and therefore greater pressure. The pressure a liquid exerts depends on
its depth.
Liquid pressure also depends on the density of the liquid. If someone
was submerged in a liquid more dense than water, the pressure would be
correspondingly greater. The pressure due to a liquid in liquid columns
of constant density or at a depth within a substance is represented by
the following formula:
where:
- p is liquid pressure,
- g is gravity at the surface of overlaying material,
- ρ is density of liquid,
- h is height of liquid column or depth within a substance.
Another way of saying the same formula is the following:
| [show]Derivation of this equation |
The pressure a liquid exerts against the sides and bottom of a
container depends on the density and the depth of the liquid. If
atmospheric pressure is neglected, liquid pressure against the bottom is
twice as great at twice the depth; at three times the depth, the liquid
pressure is threefold; etc. Or, if the liquid is two or three times as
dense, the liquid pressure is correspondingly two or three times as
great for any given depth. Liquids are practically incompressible – that
is, their volume can hardly be changed by pressure (water volume
decreases by only 50 millionths of its original volume for each
atmospheric increase in pressure). Thus, except for small changes
produced by temperature, the density of a particular liquid is
practically the same at all depths.
Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the
total pressure acting on a liquid. The total pressure of a liquid, then, is
ρgh plus the pressure of the atmosphere. When this distinction is important, the term
total pressure
is used. Otherwise, discussions of liquid pressure refer to pressure
without regard to the normally ever-present atmospheric pressure.
It is important to recognize that the pressure does not depend on the
amount
of liquid present. Volume is not the important factor – depth is. The
average water pressure acting against a dam depends on the average depth
of the water and not on the volume of water held back. For example, a
wide but shallow lake with a depth of 3 m (10 ft) exerts only half the
average pressure that a small 6 m (20 ft) deep pond does (note that the
total force
applied to the longer dam will be greater, due to the greater total
surface area for the pressure to act upon, but for a given 5-foot
section of each dam, the 10 ft deep water will apply half the force of
20 ft deep water). A person will feel the same pressure whether his/her
head is dunked a metre beneath the surface of the water in a small pool
or to the same depth in the middle of a large lake. If four vases
contain different amounts of water but are all filled to equal depths,
then a fish with its head dunked a few centimetres under the surface
will be acted on by water pressure that is the same in any of the vases.
If the fish swims a few centimetres deeper, the pressure on the fish
will increase with depth and be the same no matter which vase the fish
is in. If the fish swims to the bottom, the pressure will be greater,
but it makes no difference what vase it is in. All vases are filled to
equal depths, so the water pressure is the same at the bottom of each
vase, regardless of its shape or volume. If water pressure at the bottom
of a vase were greater than water pressure at the bottom of a
neighboring vase, the greater pressure would force water sideways and
then up the narrower vase to a higher level until the pressures at the
bottom were equalized. Pressure is depth dependent, not volume
dependent, so there is a reason that water seeks its own level.
Restating this as energy equation, the energy per unit volume in an
ideal, incompressible liquid is constant throughout its vessel. At the
surface, gravitational potential energy is large but liquid pressure
energy is low. At the bottom of the vessel, all the gravitational
potential energy is converted to pressure energy. The sum of pressure
energy and gravitational potential energy per unit volume is constant
throughout the volume of the fluid and the two energy components change
linearly with the depth.
[13] Mathematically, it is described by
Bernoulli's equation, where velocity head is zero and comparisons per unit volume in the vessel are
Terms have the same meaning as in
section Fluid pressure.
Direction of liquid pressure
An experimentally determined fact about liquid pressure is that it is exerted equally in all directions.
[14]
If someone is submerged in water, no matter which way that person tilts
his/her head, the person will feel the same amount of water pressure on
his/her ears. Because a liquid can flow, this pressure isn't only
downward. Pressure is seen acting sideways when water spurts sideways
from a leak in the side of an upright can. Pressure also acts upward, as
demonstrated when someone tries to push a beach ball beneath the
surface of the water. The bottom of a boat is pushed upward by water
pressure (
buoyancy).
When a liquid presses against a surface, there is a net force that is
perpendicular to the surface. Although pressure doesn't have a specific
direction, force does. A submerged triangular block has water forced
against each point from many directions, but components of the force
that are not perpendicular to the surface cancel each other out, leaving
only a net perpendicular point.
[14]
This is why water spurting from a hole in a bucket initially exits the
bucket in a direction at right angles to the surface of the bucket in
which the hole is located. Then it curves downward due to gravity. If
there are three holes in a bucket (top, bottom, and middle), then the
force vectors perpendicular to the inner container surface will increase
with increasing depth – that is, a greater pressure at the bottom makes
it so that the bottom hole will shoot water out the farthest. The force
exerted by a fluid on a smooth surface is always at right angles to the
surface. The speed of liquid out of the hole is
, where
h is the depth below the free surface.
[14] Interestingly, this is the same speed the water (or anything else) would have if freely falling the same vertical distance
h.
Kinematic pressure
is the kinematic pressure, where
is the pressure and
constant mass density. The SI unit of
P is m
2/s
2. Kinematic pressure is used in the same manner as
kinematic viscosity 
in order to compute
Navier–Stokes equation without explicitly showing the density
.
- Navier–Stokes equation with kinematic quantities
See also
Notes
- The preferred spelling varies by country and even by industry. Further, both spellings are often used within a particular industry or country. Industries in British English-speaking countries typically use the "gauge" spelling.
References
Giancoli, Douglas G. (2004). Physics: principles with applications. Upper Saddle River, N.J.: Pearson Education. ISBN 0-13-060620-0.
External links
| [show]
Diving medicine, physiology, physics and environment
|
McNaught, A. D.; Wilkinson, A.; Nic, M.; Jirat, J.; Kosata, B.; Jenkins, A. (2014). IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). 2.3.3. Oxford: Blackwell Scientific Publications. ISBN 0-9678550-9-8. doi:10.1351/goldbook.P04819.
"14th Conference of the International Bureau of Weights and Measures". Bipm.fr. Retrieved 2012-03-27.
US Navy (2006). US Navy Diving Manual, 6th revision. United States: US Naval Sea Systems Command. pp. 2–32. Retrieved 2008-06-15.
"U.S. Navy Diving Manual (Chapter 2:Underwater Physics)" (PDF). p. 2.32.
"Rules and Style Conventions for Expressing Values of Quantities". NIST. Retrieved 2009-07-07.
NIST, Rules and Style Conventions for Expressing Values of Quantities, Sect. 7.4.
"Einstein's gravity under pressure". Astrophysics and Space Science. 321: 151–156. Bibcode:2009Ap&SS.321..151V. arXiv:0705.0825 
. doi:10.1007/s10509-009-0016-8. Retrieved 2012-03-27.
Finnemore, John, E. and Joseph B. Franzini (2002). Fluid Mechanics: With Engineering Applications. New York: McGraw Hill, Inc. pp. 14–29. ISBN 978-0-07-243202-2.
NCEES (2011). Fundamentals of Engineering: Supplied Reference Handbook. Clemson, South Carolina: NCEES. p. 64. ISBN 978-1-932613-59-9.
Karen Wright (March 2003). "The Physics of Negative Pressure". Discover. Retrieved 31 January 2015.
P. Atkins, J. de Paula Elements of Physical Chemistry, 4th Ed, W. H. Freeman, 2006. ISBN 0-7167-7329-5.
Streeter, V. L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.
