Tait equation
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[1] where
is the hydrostatic pressure in addition to the atmospheric one,
is the volume at atmospheric pressure,
is the volume under additional pressure
are experimentally determined parameters.
A very detailed historical study on the Tait equation with the physical interpretation of the two parameters
[2] In 1895,[3][4] the original isothermal Tait equation was replaced by Tammann with an equation of the form where
is the isothermal mixed bulk modulus.
The integrated form is commonly written where 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
[2] Expressions as a function of temperature for the two empirical parameters
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.
[5] The case of liquid argon between the triple point temperature and 148 K is dealt with in detail in section 6 of the reference.
[6] Another popular isothermal equation of state that goes by the name "Tait equation"[7][8] is the Murnaghan model[9] which is sometimes expressed as where
is the specific volume at pressure
This equation, in pressure form, can be written as where
For pure water, typical parameters are
[citation needed] Note that this form of the Tate equation of state is identical to that of the Murnaghan equation of state.
The tangent bulk modulus predicted by the MacDonald–Tait model is 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).
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
The inverse Tumlirz–Tammann–Tait relation for the pressure as a function of specific volume is The Tumlirz-Tammann-Tait formula for the instantaneous tangent bulk modulus of pure water is a quadratic function of
(for an alternative see [4]) Following in particular the study of underwater explosions and more precisely the shock waves emitted, J.G.
Kirkwood proposed in 1965[11] a more appropriate form of equation of state to describe high pressures (>1 kbar) by expressing the isentropic compressibility coefficient as where
are now function of entropy such that The integration leads to the following expression for the volume
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
[2] Expressions as a function of entropy for the two empirical parameters
