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
 is the specific volume 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

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 Tate 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 (sometimes called the Tammann equation and originally proposed by Tumlirz in 1909 and Tammann in 1911 for pure water).  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  (for an alternative see )

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 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.

See also 
 Equation of state

References 

Equations of state
Fluid mechanics