Compressibility
Encyclopedia
In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

, compressibility is a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

of the relative volume change of a 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....

or solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...

as a response to a pressure
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 :...

(or mean stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

) change.

where V is volume and p is pressure
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 :...

Note: most textbooks use the notation for this quantity

The above statement is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is adiabatic
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...

or isothermal. Accordingly isothermal compressibility is defined:

where the subscript T indicates that the partial differential is to be taken at constant temperature

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

The inverse of the compressibility is called the bulk modulus
Bulk modulus
The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to decrease the volume by a factor of 1/e...

, often denoted K (sometimes B). That page also contains some examples for different materials.

The compressibility equation
Compressibility equation
In statistical mechanics and thermodynamics the compressibility equation refers to an equation which relates the isothermal compressibility to the structure of the liquid...

relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.

Thermodynamics

The term "compressibility" is also used in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

to describe the deviance in the thermodynamic properties of a real gas
Real gas
Real gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behaviour of real gases, the following must be taken into account:* compressibility effects;...

from those expected from an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

. The compressibility factor is defined as

where p is the pressure
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 :...

of the gas, T is its temperature
Temperature
Temperature 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...

, and is its molar volume
Molar volume
The molar volume, symbol Vm, is the volume occupied by one mole of a substance at a given temperature and pressure. It is equal to the molar mass divided by the mass density...

. In the case of an ideal gas, the compressibility factor Z is equal to unity, and the familiar ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's 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 the critical point
Critical point (thermodynamics)
In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...

, or in the case of high pressure or low temperature. In these cases, a generalized compressibility chart or an alternative equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

better suited to the problem must be utilized to produce accurate results.

A related situation occurs in hypersonic aerodynamics, where dissociation causes an increase in the “notational” molar volume, because a mole of oxygen, as O2, becomes 2 moles of monatomic oxygen and N2 similarly dissociates to 2N. Since this occurs dynamically as air flows over the aerospace object, it is convenient to alter Z, defined for an initial 30 gram mole of air, rather than track the varying mean molecular weight, millisecond by millisecond. This pressure dependent transition occurs for atmospheric oxygen in the 2500 K to 4000 K temperature range, and in the 5000 K to 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 will greatly increase.

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 (non-thermal) energy if the surface catalyzes the slower recombination process.

The isothermal compressibility is related to the isentropic (or adiabatic) compressibility by the relation,

via Maxwell's relations. More simply stated,

where,
is the heat capacity ratio
Heat capacity ratio
The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...

. See here for a derivation.

Earth science

Vertical, drained compressibilities
Material β (m²/N or Pa-1)
Plastic clay 2 – 2.6
Stiff clay 2.6 – 1.3
Medium-hard clay 1.3 – 6.9
Loose sand 1 – 5.2
Dense sand 2 – 1.3
Dense, sandy gravel 1 – 5.2
Rock, fissured 6.9 – 3.3
Rock, sound <3.3
Water at 25 °C (undrained) 4.6

Compressibility is used in the Earth science
Earth science
Earth science is an all-embracing term for the sciences related to the planet Earth. It is arguably a special case in planetary science, the Earth being the only known life-bearing planet. There are both reductionist and holistic approaches to Earth sciences...

s to quantify the ability of a soil or rock to reduce in volume with applied pressure. This concept is important for specific storage
Specific storage
Specific storage , storativity , specific yield and specific capacity are material physical properties that characterize the capacity of an aquifer to release groundwater from storage in response to a decline in hydraulic head. For that reason they are sometimes referred to as "storage properties"...

, when estimating groundwater
Groundwater
Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...

reserves in confined aquifer
Aquifer
An aquifer is a wet underground layer of water-bearing permeable rock or unconsolidated materials from which groundwater can be usefully extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology...

s. Geologic materials are made up of two portions: solids and voids (or same as porosity
Porosity
Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...

). The void space can be full of liquid or gas. Geologic materials reduces 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 in settlement
Subsidence
Subsidence is the motion of a surface as it shifts downward relative to a datum such as sea-level. The opposite of subsidence is uplift, which results in an increase in elevation...

.

It is an important concept in geotechnical engineering
Geotechnical engineering
Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. Geotechnical engineering is important in civil engineering, but is also used by military, mining, petroleum, or any other engineering concerned with construction on or in the ground...

in the design of certain structural foundations. For example, the construction of high-rise structures over underlying layers of highly compressible bay mud
Bay mud
Bay mud consists of thick deposits of soft, unconsolidated silty clay, which is saturated with water; these soil layers are situated at the bottom of certain estuaries, which are normally in temperate regions that have experienced cyclical glacial cycles...

poses a considerable design constraint, and often leads to use of driven piles
Deep foundation
A deep foundation is a type of foundation distinguished from shallow foundations by the depth they are embedded into the ground. There are many reasons a geotechnical engineer would recommend a deep foundation over a shallow foundation, but some of the common reasons are very large design loads, a...

or other innovative techniques.

Aeronautical dynamics

Compressibility is an important factor in aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...

. At low speeds, the compressibility of air is not significant in relation to aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...

design, but as the airflow nears and exceeds the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

, 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 for World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

era aircraft to reach speeds much beyond 800 km/h (500 mph).

Some of the minor effects include changes to the airflow that lead to problems in control. For instance, the P-38 Lightning
P-38 Lightning
The Lockheed P-38 Lightning was a World War II American fighter aircraft built by Lockheed. Developed to a United States Army Air Corps requirement, the P-38 had distinctive twin booms and a single, central nacelle containing the cockpit and armament...

with its thick high-lift wing had a particular problem in high-speed dives that led to a nose-down condition. Pilots would enter dives, and then find that they could no longer control the plane, which continued to nose over until it crashed. Adding a "dive flap" beneath the wing altered the center of pressure distribution so that the wing would not lose its lift. This fixed the problem.

A similar problem affected some models of the Supermarine Spitfire
Supermarine Spitfire
The Supermarine Spitfire is a British single-seat fighter aircraft that was used by the Royal Air Force and many other Allied countries throughout the Second World War. The Spitfire continued to be used as a front line fighter and in secondary roles into the 1950s...

. At high speeds the aileron
Aileron
Ailerons are hinged flight control surfaces attached to the trailing edge of the wing of a fixed-wing aircraft. The ailerons are used to control the aircraft in roll, which results in a change in heading due to the tilting of the lift vector...

s could apply more torque than the Spitfire's thin wings could handle, and the entire wing would twist in the opposite direction. This meant that the plane would roll in the direction opposite to that which the pilot intended, and led to a number of accidents. Earlier models weren't fast enough for this to be a problem, and so it wasn't noticed until later model Spitfires like the Mk.IX started to appear. This was mitigated by adding considerable torsional rigidity to the wings, and was wholly cured when the Mk.XIV was introduced.

The Messerschmitt Bf 109
Messerschmitt Bf 109
The Messerschmitt Bf 109, often called Me 109, was a German World War II fighter aircraft designed by Willy Messerschmitt and Robert Lusser during the early to mid 1930s...

and Mitsubishi Zero
A6M Zero
The Mitsubishi A6M Zero was a long-range fighter aircraft operated by the Imperial Japanese Navy Air Service from 1940 to 1945. The A6M was designated as the , and also designated as the Mitsubishi A6M Rei-sen and Mitsubishi Navy 12-shi Carrier Fighter. The A6M was usually referred to by the...

had the exact opposite problem in which the controls became ineffective. At higher speeds the pilot simply couldn't move the controls because there was too much airflow over the control surfaces. The planes would become difficult to maneuver, and at high enough speeds aircraft without this problem could out-turn them.

These problems were eventually solved as jet aircraft reached transonic and supersonic
Supersonic
Supersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...

speeds. German scientists in WWII experimented with swept wing
Swept wing
A swept wing is a wing planform favored for high subsonic jet speeds first investigated by Germany during the Second World War. Since the introduction of the MiG-15 and North American F-86 which demonstrated a decisive superiority over the slower first generation of straight-wing jet fighters...

s. Their research was applied on the MiG-15 and F-86 Sabre
F-86 Sabre
The North American F-86 Sabre was a transonic jet fighter aircraft. Produced by North American Aviation, the Sabre is best known as America's first swept wing fighter which could counter the similarly-winged Soviet MiG-15 in high speed dogfights over the skies of the Korean War...

and bombers such as the B-47 Stratojet
B-47 Stratojet
The Boeing Model 450 B-47 Stratojet was a long-range, six-engined, jet-powered medium bomber built to fly at high subsonic speeds and at high altitudes. It was primarily designed to drop nuclear bombs on the Soviet Union...

used swept wings which delay the onset of shock waves and reduce drag. The all-flying tailplane which are common on supersonic planes also help maintain control near the speed of sound.

Finally, another common problem that fits into this category is flutter
Aeroelasticity
Aeroelasticity is the science which studies the interactions among inertial, elastic, and aerodynamic forces. It was defined by Arthur Collar in 1947 as "the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural...

. At some speeds the airflow over the control surfaces will become turbulent, and the controls will start to flutter. If the speed of the fluttering is close to a harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

of the control's movement, the resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

could break the control off completely. This was a serious problem on the Zero. When problems with poor control at high speed were first encountered, they were addressed by designing a new style of control surface with more power. However this introduced a new resonant mode, and a number of planes were lost before this was discovered.

All of these effects are often mentioned in conjunction with the term "compressibility", but in a manner of speaking, they are incorrectly used. 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
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...

and critical mach.
Wave drag is a sudden rise in drag on the aircraft, caused by air building up in front of it. At lower speeds this air has time to "get out of the way", guided by the air in front of it that is in contact with the aircraft. But at the speed of sound this can no longer happen, and the air which was previously following the streamline
Streamlines, streaklines and pathlines
Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, streaklines and pathlines are field lines resulting from this vector field description of the flow...

around the aircraft now hits it directly. The amount of power needed to overcome this effect is considerable. The critical mach is the speed at which some of the air passing over the aircraft's wing becomes supersonic.

At the speed of sound the way that lift is generated changes dramatically, from being dominated by Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

to forces generated by shock wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

s. Since the air on the top of the wing is traveling faster than on the bottom, due to Bernoulli effect, at speeds close to the speed of sound the air on the top of the wing will be accelerated to supersonic. When this happens the distribution of lift changes dramatically, typically causing a powerful nose-down trim. Since the aircraft normally approached these speeds only in a dive, pilots would report the aircraft attempting to nose over into the ground.

Dissociation absorbs a great deal of energy in a reversible process. This greatly reduces the thermodynamic temperature of hypersonic gas decelerated near an aerospace vehicle. In transition regions, where this pressure dependent dissociation is incomplete, both the differential, constant pressure heat capacity and beta (the volume/pressure differential ratio) will greatly increase. The latter has a pronounced effect on vehicle aerodynamics including stability.

• Poisson ratio
• Mach number
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...

• Prandtl-Glauert singularity
Prandtl-Glauert singularity
The Prandtl–Glauert singularity is the prediction by the Prandtl–Glauert transformation that infinite pressure conditions would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not...

, associated with supersonic flight.
• Shear strength
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is...