Thermodynamics |
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In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility [1] or, if the temperature is held constant, the isothermal compressibility [2] ) is a measure of the instantaneous relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility (denoted β in some fields) may be expressed as
where V is volume and p is pressure. The choice to define compressibility as the negative 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 isothermal bulk modulus.
The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is isentropic or isothermal. Accordingly, isothermal compressibility is defined:
where the subscript T indicates that the partial differential is to be taken at constant temperature.
Isentropic compressibility is defined:
where S is entropy. For a solid, the distinction between the two is usually negligible.
Since the density ρ of a material is inversely proportional to its volume, it can be shown that in both cases
The speed of sound is defined in classical mechanics as:
It follows, by replacing partial derivatives, that the isentropic compressibility can be expressed as:
The inverse of the compressibility is called the bulk modulus, often denoted K (sometimes B or ).). The compressibility equation relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.
The isothermal compressibility is generally related to the isentropic (or adiabatic) compressibility by a few relations: [3]
where γ is the heat capacity ratio, α is the volumetric coefficient of thermal expansion, ρ = N/V is the particle density, and is the thermal pressure coefficient.
In an extensive thermodynamic system, the application of statistical mechanics shows that the isothermal compressibility is also related to the relative size of fluctuations in particle density: [3]
where μ is the chemical potential.
The term "compressibility" is also used in thermodynamics to describe deviations of 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, all measured independently of one another. 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.
Material | (m2/N or Pa−1) |
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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 |
The Earth sciences use compressibility to quantify the ability of a soil or rock to reduce in volume under 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 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 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.
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 in aerodynamics. At low speeds, the compressibility of air is not significant in relation to 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 (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 and critical 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 factor Z, 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]
The Mach number, often only Mach, is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Czech physicist and philosopher Ernst Mach.
In thermodynamics, the specific heat capacity of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat capacity or as the specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg−1⋅K−1. For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.
In fluid mechanics, the Grashof number is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.
In fluid mechanics, the Rayleigh number (Ra, after Lord Rayleigh) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 108.
Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or the fluid's potential energy. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. 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. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At 20 °C (68 °F), the speed of sound in air is about 343 m/s, or one km in 2.91 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating. At 0 °C (32 °F), the speed of sound in air is about 331 m/s. More simply, the speed of sound is how fast vibrations travel.
The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change, liquid vapor. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One way to write this equation is:
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.
In fluid dynamics, the Euler equations are a set of
In fluid mechanics, or more generally continuum mechanics, incompressible flow refers to a flow in which the material density of each fluid parcel — an infinitesimal volume that moves with the flow velocity — is time-invariant. An equivalent statement that implies incompressible flow is that the divergence of the flow velocity is zero.
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.
The bulk modulus of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, 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 γ (gamma) for an ideal gas or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers.
In fluid dynamics, stagnation pressure is the static pressure at a stagnation point in a fluid flow. At a stagnation point the fluid velocity is zero. In an incompressible flow, stagnation pressure is equal to the sum of the free-stream static pressure and the free-stream dynamic pressure.
Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics and elasticity. These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.
In thermodynamics, the heat capacity at constant volume, , and the heat capacity at constant pressure, , are extensive properties that have the magnitude of energy divided by temperature.
In fluid mechanics, the Tait equation is an equation of state, used to relate liquid density to hydrostatic pressure. The equation was originally published by Peter Guthrie Tait in 1888 in the form
Isentropic nozzle flow describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in entropy.