Vertical pressure variation
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.
. The equation is as follows:
(ΔP) = ρ*g*(Δh)
Where
The delta symbol is used to refer to the "change in" a given variable. Since g is negative, an increase in height will correspond to a decrease in pressure, and vice versa, 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. Even when density varies, but in an approximately discrete
rather than continuous way, such as for the pressure difference between points in different fluids 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.
Whether or not density and gravity can be reasonably approximated as constant depends on the level of precision
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, so given the same height difference, water's density can be more reasonably approximated as constant than that of air.
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, and gravity is a function of distance from Earth's core.
Density, on the other hand, varies more significantly with height. As noted in a Georgia State University webpage on atmospheric pressure:
ρ = (m*P)/(k*T)
Where
Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, this might seem like a case of circular definition
. However, one can use calculus
to work with this, as another section of the same page demonstrates. This then yields a more accurate formula, of the form:
Ph = P0*e((−m*g*h) / (k*T))
Where
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
, 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:
z = (−R*T/g)*Log(P/P0)
Where
And for the sake of comparison to the above, another formula derived in the same article shows a more complete picture for when constant temperature isn't assumed, and is also a formula for height as a function of pressure difference:
z = (T0/L)*((P/P0)-L*R/g - 1)
Where
Apart from that, the units are the same as those of the formula mentioned before it.
Basically, which formula is best to use depends on which variables are known, which are meant to be found, and which simplifying assumptions are valid to make.
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
as a function of elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....
. Depending on the fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
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
Pressure gradient force
The pressure gradient force is not actually a 'force' but the acceleration of air due to pressure difference . It is usually responsible for accelerating a parcel of air from a high atmospheric pressure region to a low pressure region, resulting in wind...
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 of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
. The equation is as follows:
(ΔP) = ρ*g*(Δh)
Where
- P is pressure,
- ρ is density,
- g is gravity, and
- h is height.
The delta symbol is used to refer to the "change in" a given variable. Since g is negative, an increase in height will correspond to a decrease in pressure, and vice versa, 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. Even when density varies, but in an approximately discrete
Discrete
Discrete in science is the opposite of continuous: something that is separate; distinct; individual.Discrete may refer to:*Discrete particle or quantum in physics, for example in quantum theory...
rather than continuous way, such as for the pressure difference between points in different fluids 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.
Whether or not density and gravity can be reasonably approximated as constant depends on the level of precision
Precision
Concepts* Accuracy and precision, measurement deviation from true value and its scatter* Precision , the number of digits from which a value is expressed* Precision , the percentage of documents returned that are relevant...
needed, but also on the length scale
Length scale
In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...
of height difference, as gravity and density also decrease with higher elevation. For density in particular, the fluid in question is also relevant; seawater
Seawater
Seawater is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% . This means that every kilogram of seawater has approximately of dissolved salts . The average density of seawater at the ocean surface is 1.025 g/ml...
, for example, is considered an incompressible fluid; its density can vary with height, but much less significantly than that of air, so given the same height difference, water's density can be more reasonably approximated as constant than that of air.
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 (troposphereTroposphere
The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....
alone being several kilometres tall; thermosphere
Thermosphere
The thermosphere is the biggest of all the layers of the Earth's atmosphere directly above the mesosphere and directly below the exosphere. Within this layer, ultraviolet radiation causes ionization. The International Space Station has a stable orbit within the middle of the thermosphere, between...
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, and gravity is a function of distance from Earth's core.
Density, on the other hand, varies more significantly with height. As noted in a Georgia State University webpage on atmospheric pressure:
ρ = (m*P)/(k*T)
Where
- m is average massMassMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
per air moleculeMoleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
, - P is pressure at a given point,
- k is the Boltzmann constant, and
- T is the temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
in KelvinKelvinThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
.
Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, this might seem like a case of circular definition
Circular definition
A circular definition is one that uses the term being defined as a part of the definition or assumes a prior understanding of the term being defined. Either the audience must already know the meaning of the key term, or the definition is deficient in including the term to be defined in the...
. However, one can use calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
to work with this, as another section of the same page demonstrates. This then yields a more accurate formula, of the form:
Ph = P0*e((−m*g*h) / (k*T))
Where
- 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 to 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
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
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
Exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:*Exponential function, also:**Matrix exponential, the matrix analogue to the above*Exponential decay, decrease at a rate proportional to value...
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
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...
) is only in the dozens of degrees, as opposed to difference between either and absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
, 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
CN Tower
The CN Tower is a communications and observation tower in Downtown Toronto, Ontario, Canada. Standing tall, it was completed in 1976, becoming the world's tallest free-standing structure and world's tallest tower at the time. It held both records for 34 years until the completion of the Burj...
) or for mountains of comparable size, the temperature variation will easily be within the single-digits. (See also lapse rate
Lapse rate
The lapse rate is defined as the rate of decrease with height for an atmospheric variable. The variable involved is temperature unless specified otherwise. The terminology arises from the word lapse in the sense of a decrease or decline; thus, the lapse rate is the rate of decrease with height and...
.)
An alternative derivation, shown by the Portland State Aerospace Society
Portland State Aerospace Society
The Portland State Aerospace Society , founded in 1997, is a student group at Portland State University building high powered rockets with the vision of putting nanosatellites into Earth orbit. Pursuing this vision has led to building advanced avionics, adaptable airframes, and high energy rocket...
, 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:
z = (−R*T/g)*Log(P/P0)
Where
- z is the elevation,
- R is the ideal gas constant,
- T is temperature in kelvin,
- g is gravity,
- P is pressure at a given point, and
- P0 is pressure at the reference point.
And for the sake of comparison to the above, another formula derived in the same article shows a more complete picture for when constant temperature isn't assumed, and is also a formula for height as a function of pressure difference:
z = (T0/L)*((P/P0)-L*R/g - 1)
Where
- L is the lapse rate, and
- T0 is the temperature at the same reference point for which P=P0
Apart from that, the units are the same as those of the formula mentioned before it.
Basically, which formula is best to use depends on which variables are known, which are meant to be found, and which simplifying assumptions are valid to make.