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
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

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. The case of liquid argon between the triple point temperature and 148 K is dealt with in detail in section 6 of the reference.

Tait–Murnaghan equation of state

Another popular isothermal equation of state that goes by the name "Tait equation" is the Murnaghan model 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 Tait equation of state is identical to that of the Murnaghan equation of state.

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. 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 of pure water is a quadratic function of

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 by expressing the isentropic compressibility coefficient as
where represents here the entropy.
The two empirical parameters 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.