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]

${\displaystyle {\frac {V_{0}-V}{PV_{0}}}={\frac {A}{\Pi +P}}}$

where ${\displaystyle P}$ is the hydrostatic pressure in addition to the atmospheric one, ${\displaystyle V_{0}}$ is the volume at atmospheric pressure, ${\displaystyle V}$ is the volume under additional pressure ${\displaystyle P}$, and ${\displaystyle A,\Pi }$ are experimentally determined parameters. A very detailed historical study on the Tait equation with the physical interpretation of the two parameters ${\displaystyle A}$ and ${\displaystyle \Pi }$ is given in reference.^[2]

Tait-Tammann equation of state

In 1895,^[3][4] the original isothermal Tait equation was replaced by Tammann with an equation of the form

${\displaystyle {K}=-{V_{0}}\left({\frac {\partial P}{\partial V}}\right)_{T}={V_{0}}{\frac {(B+P)}{C}}}$

where ${\displaystyle K}$ is the isothermal mixed bulk modulus. This above equation is popularly known as the Tait equation. The integrated form is commonly written

${\displaystyle V=V_{0}-C\log \left({\frac {B+P}{B+P_{0}}}\right)}$

where

• ${\displaystyle V\ }$ is the specific volume of the substance (in units of ml/g or m³/kg)
• ${\displaystyle V_{0}}$ is the specific volume at ${\displaystyle P=P_{0}}$
• ${\displaystyle B\ }$ (same units as ${\displaystyle P_{0}}$) and ${\displaystyle C\ }$ (same units as ${\displaystyle V_{0}}$) are functions of temperature

Pressure formula

The expression for the pressure in terms of the specific volume is

${\displaystyle P=(B+P_{0})\exp \left(-{\frac {V-V_{0}}{C}}\right)-B\,.}$

A highly detailed study on the Tait-Tammann equation of state with the physical interpretation of the two empirical parameters ${\displaystyle C}$ and ${\displaystyle B}$ is given in chapter 3 of reference.^[2] Expressions as a function of temperature for the two empirical parameters ${\displaystyle C}$ and ${\displaystyle B}$ are given for water, seawater, helium-4, and helium-3 in the entire liquid phase up to the critical temperature ${\displaystyle T_{c}}$. 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]

Tait-Murnaghan equation of state

Specific volume as a function of pressure predicted by the Tait-Murnaghan equation of state.

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

${\displaystyle {\frac {V}{V_{0}}}=\left[1+{\frac {n}{K_{0}}}\,(P-P_{0})\right]^{-1/n}}$

where ${\displaystyle V}$ is the specific volume at pressure ${\displaystyle P}$, ${\displaystyle V_{0}}$ is the specific volume at pressure ${\displaystyle P_{0}}$, ${\displaystyle K_{0}}$ is the bulk modulus at ${\displaystyle P_{0}}$, and ${\displaystyle n}$ is a material parameter.

Pressure formula

This equation, in pressure form, can be written as

${\displaystyle P={\frac {K_{0}}{n}}\left[\left({\frac {V_{0}}{V}}\right)^{n}-1\right]+P_{0}={\frac {K_{0}}{n}}\left[\left({\frac {\rho }{\rho _{0}}}\right)^{n}-1\right]+P_{0}.}$

where ${\displaystyle \rho ,\rho _{0}}$ are mass densities at ${\displaystyle P,P_{0}}$, respectively. For pure water, typical parameters are ${\displaystyle P_{0}}$ = 101,325 Pa, ${\displaystyle \rho _{0}}$ = 1000 kg/cu.m, ${\displaystyle K_{0}}$ = 2.15 GPa, and ${\displaystyle n}$ = 7.15.^{[citation needed]}

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

${\displaystyle K=K_{0}\left({\frac {V_{0}}{V}}\right)^{n}\,.}$

Tumlirz–Tammann–Tait equation of state

Tumlirz-Tammann-Tait equation of state based on fits to experimental data on pure water.

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).^[4][10] This relation has the form

${\displaystyle V(P,S,T)=V_{\infty }-K_{1}S+{\frac {\lambda }{P_{0}+K_{2}S+P}}}$

where ${\displaystyle V(P,S,T)}$ is the specific volume, ${\displaystyle P}$ is the pressure, ${\displaystyle S}$ is the salinity, ${\displaystyle T}$ is the temperature, and ${\displaystyle V_{\infty }}$ is the specific volume when ${\displaystyle P=\infty }$, and ${\displaystyle K_{1},K_{2},P_{0}}$ are parameters that can be fit to experimental data.

The Tumlirz–Tammann version of the Tait equation for fresh water, i.e., when ${\displaystyle S=0}$, is

${\displaystyle V=V_{\infty }+{\frac {\lambda }{P_{0}+P}}\,.}$

For pure water, the temperature-dependence of ${\displaystyle V_{\infty },\lambda ,P_{0}}$ are:^[10]

${\displaystyle {\begin{aligned}\lambda &=1788.316+21.55053\,T-0.4695911\,T^{2}+3.096363\times 10^{-3}\,T^{3}-0.7341182\times 10^{-5}\,T^{4}\\P_{0}&=5918.499+58.05267\,T-1.1253317\,T^{2}+6.6123869\times 10^{-3}\,T^{3}-1.4661625\times 10^{-5}\,T^{4}\\V_{\infty }&=0.6980547-0.7435626\times 10^{-3}\,T+0.3704258\times 10^{-4}\,T^{2}-0.6315724\times 10^{-6}\,T^{3}\\&+0.9829576\times 10^{-8}\,T^{4}-0.1197269\times 10^{-9}\,T^{5}+0.1005461\times 10^{-11}\,T^{6}\\&-0.5437898\times 10^{-14}\,T^{7}+0.169946\times 10^{-16}\,T^{8}-0.2295063\times 10^{-19}\,T^{9}\end{aligned}}}$

In the above fits, the temperature ${\displaystyle T}$ is in degrees Celsius, ${\displaystyle P_{0}}$ is in bars, ${\displaystyle V_{\infty }}$ is in cc/gm, and ${\displaystyle \lambda }$ is in bars-cc/gm.

Pressure formula

The inverse Tumlirz–Tammann–Tait relation for the pressure as a function of specific volume is

${\displaystyle P={\frac {\lambda }{V-V_{\infty }}}-P_{0}\,.}$

Bulk modulus formula

The Tumlirz-Tammann-Tait formula for the instantaneous tangent bulk modulus of pure water is a quadratic function of ${\displaystyle P}$ (for an alternative see ^[4])

${\displaystyle K=-V\,{\frac {\partial P}{\partial V}}={\frac {V\,\lambda }{(V-V_{\infty })^{2}}}=(P_{0}+P)+{\frac {V_{\infty }}{\lambda }}(P_{0}+P)^{2}\,.}$

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^[11] a more appropriate form of equation of state to describe high pressures (>1 kbar) by expressing the isentropic compressibility coefficient as

${\displaystyle -{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}={\frac {1}{n(B+P)}}}$

where ${\displaystyle S}$ represents here the entropy. The two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ are now function of entropy such that

• ${\displaystyle n\ }$ is dimensionless
• ${\displaystyle B\ }$ has the same units as ${\displaystyle P}$

The integration leads to the following expression for the volume ${\displaystyle V(P,S)}$ along the isentropic ${\displaystyle S}$

${\displaystyle {\frac {V}{V_{0}}}=\left(1+{\frac {P_{0}}{B}}\right)^{1/n}\left(1+{\frac {P}{B}}\right)^{-1/n}}$

where ${\displaystyle V_{0}=V(P_{0},S)}$.

Pressure formula

The expression for the pressure ${\displaystyle P(V,S)}$ in terms of the specific volume along the isentropic ${\displaystyle S}$ is

${\displaystyle P=(B+P_{0})\,\left({\cfrac {V_{0}}{V}}\right)^{n}-B\,.}$

A highly detailed study on the Modified Tait equation of state with the physical interpretation of the two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ is given in chapter 4 of reference.^[2] Expressions as a function of entropy for the two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ are given for water, helium-3 and helium-4.

See also

• Equation of state

References

1. Tait, P. G. (1888). "Report on some of the physical properties of fresh water and of sea water". Physics and Chemistry of the Voyage of H.M.S. Challenger. Vol. II, part IV.
2. Aitken, Frederic; Foulc, Jean-Numa (2019). From Deep Sea to Laboratory 3:From Tait's Work on the Compressibility of Seawater to Equations-of-State for Liquids. London, UK: ISTE - WILEY. ISBN 9781786303769.
3. Tammann, G. (1895). "Über die Abhängigkeit der volumina von Lösungen vom druck". Zeitschrift für Physikalische Chemie. 17: 620–636.
4. 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
5. Aitken, F.; Volino, F. (November 2021). "A new single equation of state to describe the dynamic viscosity and self-diffusion coefficient for all fluid phases of water from 200 to 1800 K based on a new original microscopic model". Physics of Fluids. 33 (11): 117112. arXiv:2108.10666. Bibcode:2021PhFl...33k7112A. doi:10.1063/5.0069488. S2CID 237278734.
6. Aitken, Frédéric; Denat, André; Volino, Ferdinand (24 April 2024). "A New Non-Extensive Equation of State for the Fluid Phases of Argon, Including the Metastable States, from the Melting Line to 2300 K and 50 GPa". Fluids. 9 (5): 102. arXiv:1504.00633. doi:10.3390/fluids9050102.
7. Thompson, P. A., & Beavers, G. S. (1972). Compressible-fluid dynamics. Journal of Applied Mechanics, 39, 366.
8. Kedrinskiy, V. K. (2006). Hydrodynamics of Explosion: experiments and models. Springer Science & Business Media.
9. Macdonald, J. R. (1966). Some simple isothermal equations of state. Reviews of Modern Physics, 38(4), 669.
10. 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
11. Cole, R. H. (1965). Underwater Explosions. New York: Dover Publications.