Compressibility

Inthermodynamicsandfluid mechanics,thecompressibility(also known as thecoefficient of compressibilityor, if the temperature is held constant, theisothermal compressibility) is ameasureof the instantaneous relative volume change of afluidorsolidas a response to apressure(or meanstress) change. In its simple form, the compressibility$\kappa$(denotedβin some fields) may be expressed as

$\beta =-{\frac {1}{V}}{\frac {\partial V}{\partial p}}$,

whereVisvolumeandpis pressure. The choice to define compressibility as thenegativeof the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermalbulk modulus.

Definition

The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process isisentropicorisothermal.Accordingly,isothermalcompressibility is defined:

$\beta _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T},$

where the subscriptTindicates that the partial differential is to be taken at constant temperature.

Isentropiccompressibility is defined:

$\beta _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{S},$

whereSis entropy. For a solid, the distinction between the two is usually negligible.

Since thedensityρof a material is inversely proportional to its volume, it can be shown that in both cases

$\beta ={\frac {1}{\rho }}\left({\frac {\partial \rho }{\partial p}}\right).$

Relation to speed of sound

Thespeed of soundis defined inclassical mechanicsas:

$c^{2}=\left({\frac {\partial p}{\partial \rho }}\right)_{S}$

It follows, by replacingpartial derivatives,that the isentropic compressibility can be expressed as:

$\beta _{S}={\frac {1}{\rho c^{2}}}$

Relation to bulk modulus

The inverse of the compressibility is called thebulk modulus,often denotedK(sometimesBor$\beta$).). Thecompressibility equationrelates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.

Thermodynamics

Main article:Compressibility factor

Theisothermalcompressibility is generally related to theisentropic(oradiabatic) compressibility by a few relations:

${\frac {\beta _{T}}{\beta _{S}}}={\frac {c_{p}}{c_{v}}}=\gamma,$

$\beta _{S}=\beta _{T}-{\frac {\ Alpha ^{2}T}{\rho c_{p}}},$

${\frac {1}{\beta _{S}}}={\frac {1}{\beta _{T}}}+{\frac {\Lambda ^{2}T}{\rho c_{v}}},$

whereγis theheat capacity ratio,αis the volumetriccoefficient of thermal expansion,ρ=N/Vis the particle density, and$\Lambda =(\partial P/\partial T)_{V}$is thethermal pressure coefficient.

In an extensive thermodynamic system, the application ofstatistical mechanicsshows that the isothermal compressibility is also related to the relative size of fluctuations in particle density:

$\beta _{T}={\frac {(\partial \rho /\partial \mu )_{V,T}}{\rho ^{2}}}={\frac {\langle (\Delta N)^{2}\rangle /V}{k_{\rm {B}}T\rho ^{2}}},$

whereμis thechemical potential.

The term "compressibility" is also used inthermodynamicsto describe deviations of thethermodynamic propertiesof areal gasfrom those expected from anideal gas.

Thecompressibility factoris defined as

$Z={\frac {pV_{m}}{RT}}$

wherepis thepressureof the gas,Tis itstemperature,and$V_{m}$is itsmolar volume,all measured independently of one another. In the case of an ideal gas, the compressibility factorZis equal to unity, and the familiarideal gas lawis recovered:

$p={\frac {RT}{V_{m}}}$

Zcan, in general, be either greater or less than unity for a real gas.

The deviation from ideal gas behavior tends to become particularly significant (or, equivalently, the compressibility factor strays far from unity) near thecritical point,or in the case of high pressure or low temperature. In these cases, a generalizedcompressibility chartor an alternativeequation of statebetter suited to the problem must be utilized to produce accurate results.

Earth science

Vertical, drained compressibilities

Material	$\beta _{T}$(m2/N or Pa−1)
Plastic clay	2×10−6–2.6×10−7
Stiff clay	2.6×10−7–1.3×10−7
Medium-hard clay	1.3×10−7–6.9×10−8
Loose sand	1×10−7–5.2×10−8
Dense sand	2×10−8–1.3×10−8
Dense, sandy gravel	1×10−8–5.2×10−9
Ethyl alcohol	1.1×10−9
Carbon disulfide	9.3×10−10
Rock, fissured	6.9×10−10–3.3×10−10
Water at 25 °C (undrained)	4.6×10–10
Rock, sound	<3.3×10−10
Glycerine	2.1×10−10
Mercury	3.7×10−11

TheEarth sciencesusecompressibilityto quantify the ability of a soil or rock to reduce in volume under applied pressure. This concept is important forspecific storage,when estimatinggroundwaterreserves in confinedaquifers.Geologic materials are made up of two portions: solids and voids (or same asporosity). The void space can be full of liquid or gas. Geologic materials reduce in volume only when the void spaces are reduced, which expel the liquid or gas from the voids. This can happen over a period of time, resulting insettlement.

It is an important concept ingeotechnical engineeringin the design of certain structural foundations. For example, the construction ofhigh-risestructures over underlying layers of highly compressiblebay mudposes a considerable design constraint, and often leads to use of drivenpilesor other innovative techniques.

Fluid dynamics

Main article:Navier–Stokes equations § Compressible flow

The degree of compressibility of a fluid has strong implications for its dynamics. Most notably, the propagation of sound is dependent on the compressibility of the medium.

Aerodynamics

Main article:Aerodynamics § History

Compressibility is an important factor inaerodynamics.At low speeds, the compressibility of air is not significant in relation toaircraftdesign, but as the airflow nears and exceeds thespeed of sound,a host of new aerodynamic effects become important in the design of aircraft. These effects, often several of them at a time, made it very difficult forWorld War IIera aircraft to reach speeds much beyond 800 km/h (500 mph).

Many effects are often mentioned in conjunction with the term "compressibility", but regularly have little to do with the compressible nature of air. From a strictly aerodynamic point of view, the term should refer only to those side-effects arising as a result of the changes in airflow from an incompressible fluid (similar in effect to water) to a compressible fluid (acting as a gas) as the speed of sound is approached. There are two effects in particular,wave dragandcritical mach.

One complication occurs in hypersonic aerodynamics, where dissociation causes an increase in the "notional" molar volume because a mole of oxygen, as O₂,becomes 2 moles of monatomic oxygen and N₂similarly dissociates to 2 N. Since this occurs dynamically as air flows over the aerospace object, it is convenient to alter the compressibility factorZ,defined for an initial 30 gram moles of air, rather than track the varying mean molecular weight, millisecond by millisecond. This pressure dependent transition occurs for atmospheric oxygen in the 2,500–4,000 K temperature range, and in the 5,000–10,000 K range for nitrogen.

In transition regions, where this pressure dependent dissociation is incomplete, both beta (the volume/pressure differential ratio) and the differential, constant pressure heat capacity greatly increases. For moderate pressures, above 10,000 K the gas further dissociates into free electrons and ions.Zfor the resulting plasma can similarly be computed for a mole of initial air, producing values between 2 and 4 for partially or singly ionized gas. Each dissociation absorbs a great deal of energy in a reversible process and this greatly reduces the thermodynamic temperature of hypersonic gas decelerated near the aerospace object. Ions or free radicals transported to the object surface by diffusion may release this extra (nonthermal) energy if the surface catalyzes the slower recombination process.

Negative compressibility

For ordinary materials, the bulk compressibility (sum of the linear compressibilities on the three axes) is positive, that is, an increase in pressure squeezes the material to a smaller volume. This condition is required for mechanical stability.However, under very specific conditions, materials can exhibit a compressibility that can be negative.

See also

• Mach number
• Mach tuck
• Poisson ratio
• Prandtl–Glauert singularity,associated with supersonic flight
• Shear strength