In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, the Tait equation is an
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
, used to relate liquid
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
to hydrostatic
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
. The equation was originally published by
Peter Guthrie Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he co-wrote wi ...
in 1888 in the form
:
where
is the hydrostatic pressure in addition to the atmospheric one,
is the volume at atmospheric pressure,
is the volume under additional pressure
, and
are experimentally determined parameters.
A very detailed historical study on the Tait equation with the physical interpretation of the two parameters
and
is given in reference.
Tait-Tammann equation of state
In 1895,
[ the original isothermal Tait equation was replaced by Tammann with an equation of the form
:
where is the isothermal mixed bulk modulus.
This above equation is popularly known as the Tait equation.
The integrated form is commonly written
:
where
* is the ]specific volume
In thermodynamics, the specific volume of a substance (symbol: , nu) is an intrinsic property of the substance, defined as the ratio of the substance's volume () to its mass (). It is the reciprocal of density (rho) and it is related to the m ...
of the substance (in units of ml/ g or m3/kg)
* is the specific volume at
* (same units as ) and (same units as ) are functions of temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
Pressure formula
The expression for the pressure in terms of the specific volume is
:
A highly detailed study on the Tait-Tammann equation of state with the physical interpretation of the two empirical parameters and is given in chapter 3 of reference. Expressions as a function of temperature for the two empirical parameters and are given for water, seawater, helium-4, and helium-3 in the entire liquid phase up to the critical temperature . The special case of the supercooled phase of water is discussed in Appendix D of reference.
Tait-Murnaghan equation of state
Another popular isothermal equation of state that goes by the name "Tait equation"[Thompson, P. A., & Beavers, G. S. (1972). Compressible-fluid dynamics. Journal of Applied Mechanics, 39, 366.] is the Murnaghan model[Macdonald, J. R. (1966). Some simple isothermal equations of state. Reviews of Modern Physics, 38(4), 669.] which is sometimes expressed as
:
where is the specific volume at pressure , is the specific volume at pressure , is the bulk modulus at , and is a material parameter.
Pressure formula
This equation, in pressure form, can be written as
:
where are mass densities at , respectively.
For pure water, typical parameters are = 101,325 Pa, = 1000 kg/cu.m, = 2.15 GPa, and = 7.15.
Note that this form of the Tate equation of state is identical to that of the Murnaghan equation of state
The Murnaghan equation of state is a relationship between the volume of a body and the pressure to which it is subjected. This is one of many state equations that have been used in earth sciences and shock physics to model the behavior of matter u ...
.
Bulk modulus formula
The tangent bulk modulus predicted by the MacDonald–Tait model is
:
Tumlirz–Tammann–Tait equation of state
A related equation of state that can be used to model liquids is the Tumlirz equation (sometimes called the Tammann equation and originally proposed by Tumlirz in 1909 and Tammann in 1911 for pure water).[Fisher, F. H., and O. E. Dial Jr. Equation of state of pure water and sea water. No. MPL-U-99/67. SCRIPPS INSTITUTION OF OCEANOGRAPHY LA JOLLA CA MARINE PHYSICAL LAB, 1975. http://www.dtic.mil/dtic/tr/fulltext/u2/a017775.pdf] This relation has the form
:
where is the specific volume, is the pressure, is the salinity, is the temperature, and is the specific volume when , and are parameters that can be fit to experimental data.
The Tumlirz–Tammann version of the Tait equation for fresh water, i.e., when , is
:
For pure water, the temperature-dependence of are:[
:
In the above fits, the temperature is in degrees Celsius, is in bars, is in cc/gm, and is in bars-cc/gm.
]
Pressure formula
The inverse Tumlirz–Tammann–Tait relation for the pressure as a function of specific volume is
:
Bulk modulus formula
The Tumlirz-Tammann-Tait formula for the instantaneous tangent bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describ ...
of pure water is a quadratic function of (for an alternative see [Hayward, A. T. J. (1967). Compressibility equations for liquids: a comparative study. British Journal of Applied Physics, 18(7), 965. http://mitran-lab.amath.unc.edu:8081/subversion/Lithotripsy/MultiphysicsFocusing/biblio/TaitEquationOfState/Hayward_CompressEqnsLiquidsComparative1967.pdf])
:
Modified Tait equation of state
Following in particular the study of underwater explosions and more precisely the shock waves emitted, J.G. Kirkwood proposed in 1965 a more appropriate form of equation of state to describe high pressures (>1 kbar) by expressing the isentropic compressibility coefficient as
:
where represents here the entropy.
The two empirical paramaters and are now function of entropy such that
* is dimensionless
* has the same units as
The integration leads to the following expression for the volume along the isentropic
:
where .
Pressure formula
The expression for the pressure in terms of the specific volume along the isentropic is
:
A highly detailed study on the Modified Tait equation of state with the physical interpretation of the two empirical parameters and is given in chapter 4 of reference. Expressions as a function of entropy for the two empirical parameters and are given for water, helium-3 and helium-4.
See also
* Equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
References
{{reflist
Equations of state
Fluid mechanics