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Inthermodynamics andfluid mechanics, thecompressibility (also known as thecoefficient of compressibility[1] or, if the temperature is held constant, theisothermal compressibility[2]) is ameasure of the instantaneous relative volume change of afluid orsolid as a response to apressure (or meanstress) change. In its simple form, the compressibility (denotedβ in some fields) may be expressed as
whereV isvolume andp is pressure. The choice to define compressibility as thenegative of 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.
The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process isisentropic orisothermal. Accordingly,isothermal compressibility is defined:
where the subscriptT indicates that the partial differential is to be taken at constant temperature.
Isentropic compressibility is defined:
whereS is entropy. For a solid, the distinction between the two is usually negligible.
Since thedensityρ of a material is inversely proportional to its volume, it can be shown that in both cases
For instance, for anideal gas,
Consequently, the isothermalcompressibility of an ideal gas is
The ideal gas (where the particles do not interact with each other) is an abstraction. The particles in real materials interact with each other. Then, the relation between the pressure, density and temperature is known as theequation of state denoted by some function. TheVan der Waals equation is an example of an equation of state for a realistic gas.
Knowing the equation of state, the compressibility can be determined for any substance.
Thespeed of sound is defined inclassical mechanics as:
It follows, by replacingpartial derivatives, that the isentropic compressibility can be expressed as:
The inverse of the compressibility is called thebulk modulus, often denotedK (sometimesB or).).Thecompressibility equation relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.
Theisothermal compressibility is generally related to theisentropic (oradiabatic) compressibility by a few relations:[3]
whereγ is theheat capacity ratio,α is the volumetriccoefficient of thermal expansion,ρ =N/V is the particle density, and is thethermal pressure coefficient.
In an extensive thermodynamic system, the application ofstatistical mechanics shows that the isothermal compressibility is also related to the relative size of fluctuations in particle density:[3]
whereμ is thechemical potential.
The term "compressibility" is also used inthermodynamics to describe deviations of thethermodynamic properties of areal gas from those expected from anideal gas.
Thecompressibility factor is defined as
wherep is thepressure of the gas,T is itstemperature, and is itsmolar volume, all measured independently of one another. In the case of an ideal gas, the compressibility factorZ is equal to unity, and the familiarideal gas law is recovered:
Z can, 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 chart or an alternativeequation of state better suited to the problem must be utilized to produce accurate results.
| Material | (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[5] | 1.1×10−9 |
| Carbon disulfide[5] | 9.3×10−10 |
| Rock, fissured | 6.9×10−10 –3.3×10−10 |
| Water at 25 °C (undrained)[5][6] | 4.6×10–10 |
| Rock, sound | <3.3×10−10 |
| Glycerine[5] | 2.1×10−10 |
| Mercury[5] | 3.7×10−11 |
TheEarth sciences usecompressibility to quantify the ability of a soil or rock to reduce in volume under applied pressure. This concept is important forspecific storage, when estimatinggroundwater reserves 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 engineering in the design of certain structural foundations. For example, the construction ofhigh-rise structures over underlying layers of highly compressiblebay mud poses a considerable design constraint, and often leads to use of drivenpiles or other innovative techniques.
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.
Compressibility is an important factor inaerodynamics. At low speeds, the compressibility of air is not significant in relation toaircraft design, 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 II era 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 drag andcritical 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 N2 similarly 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.[7]
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.Z for 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.
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.[8] However, under very specific conditions, materials can exhibit a compressibility that can be negative.[9][10][11][12]
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