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 thePressure 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 modelMacdonald, 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.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 tangentModified 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 stateReferences
{{reflist Equations of state Fluid mechanics