Compressibility
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In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change.
where V is volume and p is pressure. The above statement is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal. Accordingly we define the isothermal compressibility as:
where the subscript T indicates that the partial differential is to be taken at constant temperature. The adiabatic compressibility as:
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, often denoted K (sometimes B). That page also contains some examples for different materials.
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[edit] Fluid Dynamics
Compressibility is an important notion in aerodynamics. At low speeds, the compressibility of air is not important for aircraft design, but as the airflow nears and exceeds the speed 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 for World War II era aircraft to reach speeds much beyond 800 km/h (500mph).
Some of the minor effects include changes to the airflow that lead to problems in control. For instance, the P-38 Lightning had a particular problem in high speed dives that led to a nose-heavy 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 to upset the airflow (and so increasing drag and restraining the terminal velocity) fixed the problem.
A similar problem affected some models of the Supermarine Spitfire. At high speeds the ailerons 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 what the pilot expected, and led to a number of accidents. This wasn't noticed until later model Spitfires like the Mk.IX started to appear, because earlier models weren't fast enough. This was mitigated by adding considerable tortional rigidity to the wings, and was wholly cured when the Mk.XIV was introduced.
The Messerschmitt Bf 109 and Mitsubishi Zero had the exact opposite problem, the controls were too weak. 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 manoeuvre, and at high enough speeds even less manoeuvrable aircraft could out-turn them.
Finally, another common problem that fits into this category is flutter. 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 of the control's movement, the resonance could break the control off completely. This was a serious problem on the Zero. When they first encountered problems with the poor control at high speed they addressed it with a new style of control surface with more power. However this introduced a new resonant mode, and a number of planes disappeared before this was discovered.
All of the items above are often talked about when the term "compressibility" is used, but in a manner of speaking, they are all incorrectly used. From a strictly aerodynamic point of view, the term should refer only to those effects arising as a side effect of the changes in airflow from an incompressible fluid (similar in effect to water) to compressible fluid (acting as a gas) as you approach the speed of sound. There are two effects in particular, wave drag 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. Air which was previously following the streamline around the aircraft now hits it directly. The amount of power needed to overcome this effect is considerable.
At the speed of sound the way that lift is generated changes dramatically, from being dominated by Bernoulli's principle to forces generated by shock waves. 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.
All of these effects have adverse effects on the control or performance of the plane. For this reason it's common to see references to aircraft that suffer from compressibility. The P-38 and Zero are particularly common examples, although in fact they are both bad ones.
[edit] Thermodynamics
The term "compressibility" is also used in thermodynamics to describe the deviance in the thermodynamic properties of a real gas from those expected from an ideal gas. The compressibility factor is defined as
where p is the pressure of the gas, T is its temperature, and 
is its molar volume. In the case of an ideal gas, the compressibility factor Z is equal to unity, and the familiar ideal 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 the critical point, or in the case of high pressure or low temperature. In these cases, a generalized Compressibility chart or an alternative equation of state better suited to the problem must be utilized to produce accurate results.
[edit] Earth sciences
| Material | β (m²/N) |
|---|---|
| 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 |
| Rock, fissured | 6.9×10–10 – 3.3×10–10 |
| Rock, sound | <3.3×10–10 |
| Water at 25°C (undrained)[2] | 4.8×10–10 |
Compressibility is used in the Earth sciences to quantify the ability of a soil or rock to reduce in volume with applied pressure. This concept is important for specific storage, when estimating groundwater reserves in confined aquifers. Geologic materials are made up of two portions: solids and voids (or same as porosity). 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.
It is an important concept in geotechnical engineering in the design of certain structural foundations. For example, the construction of high-rise structures over underlying layers of highly compressible bay mud poses a considerable design constraint, and often leads to use of driven piles or other innovative techniques.
[edit] References
- ^ Domenico, P.A. and Mifflin, M.D. (1965). "Water from low permeability sediments and land subsidence". Water Resources Research 1 (4): 563–576. OSTI:5917760.
- ^ ?
[edit] See also
- Avogadros number
- Mach number
- Prandtl-Glauert singularity, associated with supersonic flight.
- Shear strength
