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Saturday, December 26, 2015

Lapse rate

From Wikipedia, the free encyclopedia

The lapse rate is defined as the rate at which atmospheric temperature decreases with an increase in altitude.[1][2] The terminology arises from the word lapse in the sense of a decrease or decline. While most often applied to Earth's troposphere, the concept can be extended to any gravitationally supported parcel of gas.


A formal definition from the Glossary of Meteorology[3] is:
The decrease of an atmospheric variable with height, the variable being temperature unless otherwise specified.
In the lower regions of the atmosphere (up to altitudes of approximately 12,000 metres (39,000 ft), temperature decreases with altitude at a fairly uniform rate. Because the atmosphere is warmed by convection from Earth's surface, this lapse or reduction in temperature is normal with increasing distance from the conductive source.

Although the actual atmospheric lapse rate varies, under normal atmospheric conditions the average atmospheric lapse rate results in a temperature decrease of 6.4 °C/km (3.5 °F or 1.95 °C/1,000 ft) of altitude above ground level.

The measurable lapse rate is affected by the moisture content of the air (humidity). A dry lapse rate of 10 °C/km (5.5 °F or 3.05 °C/1,000 ft) is often used to calculate temperature changes in air not at 100% relative humidity. A wet lapse rate of 5.5 °C/km (3 °F or 1.68 °C/1,000 ft) is used to calculate the temperature changes in air that is saturated (i.e., air at 100% relative humidity). Although actual lapse rates do not strictly follow these guidelines, they present a model sufficiently accurate to predict temperate changes associated with updrafts and downdrafts. This differential lapse rate (dependent upon both difference in conductive heating and adiabatic expansion or compression) results in the formation of warm downslope winds (e.g., Chinook winds, Santa Ana winds, etc.). The atmospheric lapse rate, combined with adiabatic cooling and heating of air related to the expansion and compression of atmospheric gases, present a unified model explaining the cooling of air as it moves aloft and the heating of air as it descends downslope.

Atmospheric stability can be measured in terms of lapse rates (i.e., the temperature differences associated with vertical movement of air). The atmosphere is considered conditionally unstable where the environmental lapse rate causes a slower decrease in temperature with altitude than the dry adiabatic lapse rate, as long as no latent heat is released (i.e. the saturated adiabatic lapse rate applies). Unconditional instability results when the dry adiabatic lapse rate causes air to cool slower than the environmental lapse rate, so air will continue to rise until it reaches the same temperature as its surroundings. Where the saturated adiabatic lapse rate is greater than the environmental lapse rate, the air cools faster than its environment and thus returns to its original position, irrespective of its moisture content.

Although the atmospheric lapse rate (also known as the environmental lapse rate) is most often used to characterize temperature changes, many properties (e.g. atmospheric pressure) can also be profiled by lapse rates...

Mathematical definition

In general, a lapse rate is the negative of the rate of temperature change with altitude change, thus:
\gamma = -\frac{dT}{dz}
where \gamma is the lapse rate given in units of temperature divided by units of altitude, T = temperature, and z = altitude.

Note: In some cases, \Gamma or \alpha can be used to represent the adiabatic lapse rate in order to avoid confusion with other terms symbolized by \gamma, such as the specific heat ratio[4] or the psychrometric constant.[5]

Types of lapse rates

There are two types of lapse rate:
  • Environmental lapse rate (ELR) – which refers to the actual change of temperature with altitude for the stationary atmosphere (i.e. the temperature gradient)
  • The adiabatic lapse rates – which refer to the change in temperature of a parcel of air as it moves upwards (or downwards) without exchanging heat with its surroundings. There are two adiabatic rates:[6]
    • Dry adiabatic lapse rate (DALR)
    • Moist (or saturated) adiabatic lapse rate (SALR)

Environmental lapse rate

The environmental lapse rate (ELR), is the rate of decrease of temperature with altitude in the stationary atmosphere at a given time and location. As an average, the International Civil Aviation Organization (ICAO) defines an international standard atmosphere (ISA) with a temperature lapse rate of 6.49 K/km[citation needed] (3.56 °F or 1.98 K/1,000 ft) from sea level to 11 km (36,090 ft or 6.8 mi). From 11 km up to 20 km (65,620 ft or 12.4 mi), the constant temperature is −56.5 °C (−69.7 °F), which is the lowest assumed temperature in the ISA. The standard atmosphere contains no moisture. Unlike the idealized ISA, the temperature of the actual atmosphere does not always fall at a uniform rate with height. For example, there can be an inversion layer in which the temperature increases with altitude.

Dry adiabatic lapse rate

Emagram diagram showing variation of dry adiabats (bold lines) and moist adiabats (dash lines) according to pressure and temperature

The dry adiabatic lapse rate (DALR) is the rate of temperature decrease with altitude for a parcel of dry or unsaturated air rising under adiabatic conditions. Unsaturated air has less than 100% relative humidity; i.e. its actual temperature is higher than its dew point. The term adiabatic means that no heat transfer occurs into or out of the parcel. Air has low thermal conductivity, and the bodies of air involved are very large, so transfer of heat by conduction is negligibly small.

Under these conditions when the air rises (for instance, by convection) it expands, because the pressure is lower at higher altitudes. As the air parcel expands, it pushes on the air around it, doing work (thermodynamics). Since the parcel does work but gains no heat, it loses internal energy so that its temperature decreases. The rate of temperature decrease is 9.8 °C/km (5.38 °F per 1,000 ft) (3.0 °C/1,000 ft). The reverse occurs for a sinking parcel of air.[7]

Since for adiabatic process:
P dV = -V dP / \gamma
the first law of thermodynamics can be written as
m c_v dT - V dp/ \gamma = 0
Also since :\alpha = V/m and :\gamma = c_p/c_v we can show that:
c_p dT - \alpha dP = 0
where c_p is the specific heat at constant pressure and \alpha is the specific volume.

Assuming an atmosphere in hydrostatic equilibrium:[8]
 dP = - \rho g dz
where g is the standard gravity and \rho is the density. Combining these two equations to eliminate the pressure, one arrives at the result for the DALR,[9]
\Gamma_d = -\frac{dT}{dz}= \frac{g}{c_p} = 9.8 \ ^{\circ}\mathrm{C}/\mathrm{km}

Saturated adiabatic lapse rate

When the air is saturated with water vapour (at its dew point), the moist adiabatic lapse rate (MALR) or saturated adiabatic lapse rate (SALR) applies. This lapse rate varies strongly with temperature. A typical value is around 5 °C/km (2.7 °F/1,000 ft) (1.5 °C/1,000 ft).[citation needed]

The reason for the difference between the dry and moist adiabatic lapse rate values is that latent heat is released when water condenses, thus decreasing the rate of temperature drop as altitude increases. This heat release process is an important source of energy in the development of thunderstorms. An unsaturated parcel of air of given temperature, altitude and moisture content below that of the corresponding dew point cools at the dry adiabatic lapse rate as altitude increases until the dew point line for the given moisture content is intersected. As the water vapour then starts condensing the air parcel subsequently cools at the slower moist adiabatic lapse rate if the altitude increases further.

The saturated adiabatic lapse rate is given approximately by this equation from the glossary of the American Meteorology Society:[10]
\Gamma_w = g\, \frac{1 + \dfrac{H_v\, r}{R_{sd}\, T}}{c_{p d} + \dfrac{H_v^2\, r}{R_{sw}\, T^2}}= g\, \frac{1 + \dfrac{H_v\, r}{R_{sd}\, T}}{c_{p d} + \dfrac{H_v^2\, r\, \epsilon}{R_{sd}\, T^2}}
\Gamma_w = Wet adiabatic lapse rate, K/m
g = Earth's gravitational acceleration = 9.8076 m/s2
H_v = Heat of vaporization of water, = 2501000 J/kg
R_{sd} = Specific gas constant of dry air = 287 J kg−1 K−1
R_{sw} = Specific gas constant of water vapour = 461.5 J kg−1 K−1
\epsilon=\frac{R_{sd}}{R_{sw}} =The dimensionless ratio of the specific gas constant of dry air to the specific gas constant for water vapour = 0.622
e = The water vapour pressure of the saturated air
p = The pressure of the saturated air
r=\epsilon e/(p-e) = The mixing ratio of the mass of water vapour to the mass of dry air[11]
T = Temperature of the saturated air, K
c_{pd} = The specific heat of dry air at constant pressure, = 1003.5 J kg−1 K−1

Thermodynamic-based lapse rate

Robert Essenhigh developed a comprehensive thermodynamic model of the lapse rate based on the Schuster–Schwarzschild (S–S) integral equations of transfer that govern radiation through the atmosphere including absorption and radiation by greenhouse gases.[12] His solution "predicts, in agreement with the Standard Atmosphere experimental data, a linear decline of the fourth power of the temperature, T4, with pressure, P, and, as a first approximation, a linear decline of T with altitude, h, up to the tropopause at about 10 km (the lower atmosphere)."[12] The predicted normalized density ratio and pressure ratio differ and fit the experimental data well.[citation needed] Sreekanth Kolan extended Essenhigh's model to include the energy balance for the lower and upper atmospheres.[13][self-published source?][third-party source needed]

Significance in meteorology

The varying environmental lapse rates throughout the Earth's atmosphere are of critical importance in meteorology, particularly within the troposphere. They are used to determine if the parcel of rising air will rise high enough for its water to condense to form clouds, and, having formed clouds, whether the air will continue to rise and form bigger shower clouds, and whether these clouds will get even bigger and form cumulonimbus clouds (thunder clouds).

As unsaturated air rises, its temperature drops at the dry adiabatic rate. The dew point also drops (as a result of decreasing air pressure) but much more slowly, typically about −2 °C per 1,000 m. If unsaturated air rises far enough, eventually its temperature will reach its dew point, and condensation will begin to form. This altitude is known as the lifting condensation level (LCL) when mechanical lift is present and the convective condensation level (CCL) when mechanical lift is absent, in which case, the parcel must be heated from below to its convective temperature. The cloud base will be somewhere within the layer bounded by these parameters.

The difference between the dry adiabatic lapse rate and the rate at which the dew point drops is around 8 °C per 1,000 ft. Given a difference in temperature and dew point readings on the ground, one can easily find the LCL by multiplying the difference by 125 m/°C.

If the environmental lapse rate is less than the moist adiabatic lapse rate, the air is absolutely stable — rising air will cool faster than the surrounding air and lose buoyancy. This often happens in the early morning, when the air near the ground has cooled overnight. Cloud formation in stable air is unlikely.

If the environmental lapse rate is between the moist and dry adiabatic lapse rates, the air is conditionally unstable — an unsaturated parcel of air does not have sufficient buoyancy to rise to the LCL or CCL, and it is stable to weak vertical displacements in either direction. If the parcel is saturated it is unstable and will rise to the LCL or CCL, and either be halted due to an inversion layer of convective inhibition, or if lifting continues, deep, moist convection (DMC) may ensue, as a parcel rises to the level of free convection (LFC), after which it enters the free convective layer (FCL) and usually rises to the equilibrium level (EL).

If the environmental lapse rate is larger than the dry adiabatic lapse rate, it has a superadiabatic lapse rate, the air is absolutely unstable — a parcel of air will gain buoyancy as it rises both below and above the lifting condensation level or convective condensation level. This often happens in the afternoon mainly over land masses. In these conditions, the likelihood of cumulus clouds, showers or even thunderstorms is increased.

Meteorologists use radiosondes to measure the environmental lapse rate and compare it to the predicted adiabatic lapse rate to forecast the likelihood that air will rise. Charts of the environmental lapse rate are known as thermodynamic diagrams, examples of which include Skew-T log-P diagrams and tephigrams.

The difference in moist adiabatic lapse rate and the dry rate is the cause of foehn wind phenomenon (also known as "Chinook winds" in parts of North America).

Friday, December 25, 2015

Vertical pressure variation

From Wikipedia, the free encyclopedia

Vertical pressure variation is the variation in pressure as a function of elevation. Depending on the fluid in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of pressure gradient force and its effects. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point.

Basic formula

A relatively simple version [1] of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. The equation is as follows:
\Delta P=-\rho g\Delta h, where
P is pressure,
ρ is density,
g is acceleration of gravity, and
h is height.
The delta symbol indicates a change in a given variable. Since g is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid.

When density and gravity are approximately constant, simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference. Where different fluids are layered on top of one another, the total pressure difference would be obtained by adding the two pressure differences; the first being from point 1 to the boundary, the second being from the boundary to point 2; which would just involve substituting the ρ and (Δh) values for each fluid and taking the sum of the results. If the density of the fluid varies with height, mathematical integration would be required.
Whether or not density and gravity can be reasonably approximated as constant depends on the level of accuracy needed, but also on the length scale of height difference, as gravity and density also decrease with higher elevation. For density in particular, the fluid in question is also relevant; seawater, for example, is considered an incompressible fluid; its density can vary with height, but much less significantly than that of air. Thus water's density can be more reasonably approximated as constant than that of air, and given the same height difference, the pressure differences in water are approximately equal at any height.

Hydrostatic paradox

The barometric formula depends only on the height of the fluid chamber, and not on its width or length. Given a large enough height, any pressure may be attained. This feature of hydrostatics has been called the hydrostatic paradox. As expressed by W. H. Besant,[2]
Any quantity of liquid, however small, may be made to support any weight, however large.
The Dutch scientist Simon Stevin was the first to explain the paradox mathematically.[3] In 1916 Richard Glazebrook mentioned the hydrostatic paradox as he described an arrangement he attributed to Pascal: a heavy weight W rests on a board with area A resting on a fluid bladder connected to a vertical tube with cross-sectional area α. Pouring water of weight w down the tube will eventually raise the heavy weight. Balance of forces leads to the equation
W={\frac {w\ A}{\alpha }}.
Glazebrook says, "By making the area of the board considerable and that of the tube small, a large weight W can be supported by a small weight w of water. This fact is sometimes described as the hydrostatic paradox."[4]

Demonstrations of the hydrostatic paradox have been used in teaching.[5]

In the context of Earth's atmosphere

If one is to analyze the vertical pressure variation of the Atmosphere of Earth, the length scale is very significant (troposphere alone being several kilometres tall; thermosphere being several hundred kilometres) and the involved fluid (air) is compressible. Gravity can still be reasonably approximated as constant, because length scales on the order of kilometres are still small in comparison to Earth's radius, which is, on average, about 6371 kilometres,[6] and gravity is a function of distance from Earth's core.[7]

Density, on the other hand, varies more significantly with height. It follows from the ideal gas law that:
\rho =(mP)/(kT)
m is average mass per air molecule,
P is pressure at a given point,
k is the Boltzmann constant, and
T is the temperature in Kelvin.
Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, it would be easy to get the impression that this was circular definition, but it is simply inter-dependency of different variables. This then yields a more accurate formula, of the form:


Ph is the pressure at point h,
P0 is the pressure at reference point 0, (typically referring to sea level)
e is Euler's number,
m is the mass per air molecule,
g is gravity,
h is height difference from reference point 0, and
k is the Boltzmann constant, and
T is the temperature in Kelvin.
And the superscript is used to indicate that e is raised to the power of the given ratio.

Therefore, instead of pressure being a linear function of height as one might expect from the more simple formula given in the "basic formula" section, it is more accurately represented as an exponential function of height.

Note that even that is a simplification, as temperature also varies with height. However, the temperature variation within the lower layers (troposphere, stratosphere) is only in the dozens of degrees, as opposed to difference between either and absolute zero, which is in the hundreds, so it is a reasonably small difference. For smaller height differences, including those from top to bottom of even the tallest of buildings, (like the CN tower) or for mountains of comparable size, the temperature variation will easily be within the single-digits. (See also lapse rate.)

An alternative derivation, shown by the Portland State Aerospace Society,[8] is used to give height as a function of pressure instead. This may seem counter-intuitive, as pressure results from height rather than vice versa, but such a formula can be useful in finding height based on pressure difference when one knows the latter and not the former. Different formulas are presented for different kinds of approximations; for comparison with the previous formula, the first referenced from the article will be the one applying the same constant-temperature approximation; in which case:


Where (with values used in the article)
z is the elevation,
R is the specific gas constant = 287.053 J/kg K
T is the absolute temperature in Kelvin = 288.15 K at sea level,
g is the acceleration due to gravity = 9.80665 m/s2,
P is the pressure at a given point at elevation z, and
P_{0} is pressure at the reference point = 101325 Pa at sea level.
A more general formula derived in the same article accounts for a linear change in temperature as a function of height (lapse rate), and reduces to above when the temperature is constant:


L is the atmospheric lapse rate (change in temperature / distance) = -6.5e-3 K/m, and
T_{0} is the temperature at the same reference point for which P=P_{0}
and the other quantities are the same as those above. This is the recommended formula to use.

Wednesday, December 23, 2015


Common symbols
p, P
SI unit Pascal (Pa)
In SI base units N/m2 or 1 kg/(m·s2)
Derivations from
other quantities
p = F / A
A figure showing pressure exerted by particle collisions inside a closed container. The collisions that exert the pressure are highlighted in red.
Pressure as exerted by particle collisions inside a closed container.

Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled gage pressure)[a] is the pressure relative to the local atmospheric or ambient pressure.

Various units are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the SI unit of pressure, the pascal (Pa), for example, is one newton per square metre; similarly, the pound-force per square inch (psi) is the traditional unit of pressure in the imperial and US customary systems. Pressure may also be expressed in terms of standard atmospheric pressure; the atmosphere (atm) is equal to this pressure and the torr is defined as 1760 of this. Manometric units such as the centimetre of water, millimetre of mercury and inch of mercury are used to express pressures in terms of the height of column of a particular fluid in a manometer.


Pressure is the amount of force acting per unit area. The symbol for pressure 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 on 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.


Pressure force area.svg
p = \frac{F}{A}
p is the pressure,
F is the normal force,
A 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:
d\mathbf{F}_n=-p\,d\mathbf{A} = -p\,\mathbf{n}\,dA
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 transmitted 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.


Mercury column

The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (N/m2 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/cm2 or kg/cm2) 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/cm2 (98.0665 kPa or 14.223 psi).

Since a system under pressure has 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 measured in units such as joules per cubic metre.

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, and is not the same as a linear metre of depth, and 33.066 fsw = 1 atm.[4] 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
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[5] 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, and millimetre of mercury (torr)
    • Height of equivalent column of water, including millimetre (mm H
      ), centimetre (cm H
      ), 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 0.980665×105 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 1.315789×10−3 1/760 atm
≈ 1 mmHg
1 psi 6.8948×103 6.8948×10−2 7.03069×10−2 6.8046×10−2 51.71493 ≡ 1 lbF /in2


The effects of an external pressure of 700bar on an aluminum cylinder with 5mm 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 of a common knife. If we try to cut a fruit with the flat side it obviously will not cut. But if we take the thin side, 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.[6]

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 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 \vec{F} to the vector area \vec{A} via
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.[7]


Fluid pressure

Fluid pressure is the pressure at some point within a fluid, such as water or air (for more information specifically about liquid pressure, see section below).

Fluid pressure occurs in one of two situations:
  1. an open condition, called "open channel flow", e.g. the ocean, a swimming pool, or the atmosphere.
  2. 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[8] and incompressible.[8] An ideal fluid is a fluid in which there is no friction, it is inviscid,[8] zero viscosity.[8] The equation for all points of a system filled with a constant-density fluid is
p = pressure of the fluid
γ = ρg = density·acceleration of gravity = specific weight of the fluid.[8]
v = velocity of the fluid
g = acceleration of gravity
z = elevation
\frac{p}{\gamma} = pressure head
\frac{v^2}{2g} = velocity head


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

low pressure chamber in Bundesleistungszentrum Kienbaum, Germany

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.[10]
  • 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 the 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:
p_{0} = \frac{1}{2}\rho v^2 + p
p_0 is the stagnation pressure
v is the flow velocity
p 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.
\pi = \frac{F}{l}
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.
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.[11]

Vapor pressure

Vapor pressure is the pressure of a vapor 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 vapor 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:
p=\rho gh
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 this same formula is the following:
p = \text{weight density} \times \!\, \text{depth}
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 10ft deep water will apply half the force of 20ft 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.[12] 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.[13] 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.[13] 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 \scriptstyle \sqrt{2gh}, where h is the depth below the free surface.[13] 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 p is the pressure and \rho_0 constant mass density. The SI unit of P is m2/s2. Kinematic pressure is used in the same manner as kinematic viscosity \nu in order to compute Navier–Stokes equation without explicitly showing the density \rho_0.
Navier–Stokes equation with kinematic quantities
 \frac{\partial u}{\partial t} + (u \nabla) u = - \nabla P + \nu \nabla^2 u