Equation in the mechanics of liquids, relating density and pressure
In fluid mechanics, the Tait equation is an equation of state, used to relate liquid density to pressure. The equation was originally published by Peter Guthrie Tait in 1888 in the form[1]
![{\displaystyle {\frac {V_{0}-V}{(P-P_{0})V_{0}}}=-{\frac {1}{V_{0}}}{\frac {\Delta V}{\Delta P}}={\frac {A}{\Pi +(P-P_{0})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/848e63232b0d93c1aadb51224f77b7b391f34d13)
where
is the reference pressure (taken to be 1 atmosphere),
is the current pressure,
is the volume of fresh water at the reference pressure,
is the volume at the current pressure, and
are experimentally determined parameters.
Popular form of the Tait equation[]
Around 1895,[1] the original isothermal Tait equation was replaced by Tammann with an equation of the form
![{\displaystyle -{\frac {1}{V}}\,{\frac {dV}{dP}}={\frac {A}{V(B+P)}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64729c0ece5dcf44f3a9abb66246ea3789d67071)
The temperature-dependent version of the above equation is popularly known as the Tait equation and is commonly written as[2]
![{\displaystyle \beta =-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}={\frac {0.4343C}{V(B+P)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19ca94ce1f016faadaa0012396ba01bda651772)
or in the integrated form
![{\displaystyle V=V_{0}-C\log _{10}\left({\frac {B+P}{B+P_{0}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b659e85eb8a79eec3e998395b5005c3e4462d91)
where
is the compressibility of the substance (often, water) (in units of bar−1 or Pa)
is the specific volume of the substance (in units of ml/g or m3/kg)
is the specific volume at
= 1 bar
and
are functions of temperature that are independent of pressure[2]
Pressure formula[]
The expression for the pressure in terms of the specific volume is
![{\displaystyle P=(B+P_{0})\,10^{\left[-{\cfrac {V-V_{0}}{C}}\right]}-B\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c67c175982ce2c4dcc7b759d563e3ef6d727b8d)
Bulk modulus formula[]
The tangent bulk modulus at pressure
is given by
![{\displaystyle K={\frac {V(B+P)}{0.4343C}}={\cfrac {\left[V_{0}-C\log _{10}\left({\cfrac {B+P}{B+P_{0}}}\right)\right](B+P)}{0.4343C}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b04e0d449cde47551204b5e772a8076708da6ed)
Murnaghan-Tait 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"[3][4] is the Murnaghan model[5] which is sometimes expressed as
![{\displaystyle {\frac {V}{V_{0}}}=\left[1+{\frac {n}{K_{0}}}\,(P-P_{0})\right]^{-1/n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a792ef1b0a3bb29f077c3cdeffbcc0ce4a2043)
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
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5888d0ba09b07f8942856a4a5afeb4eb3e58e634)
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[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}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71e66b0237767b8a3cae3d837b4d2c0e39dbfe39)
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).[1][6] This relation has the form
![{\displaystyle V(P,S,T)=V_{\infty }-K_{1}S+{\frac {\lambda }{P_{0}+K_{2}S+P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec5f902f4962548b44d40c1772e3ff1ee092980)
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
![{\displaystyle V=V_{\infty }+{\frac {\lambda }{P_{0}+P}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/410bf760d95e9275fc0ecabec870a90fa65faa55)
For pure water, the temperature-dependence of
are:[6]
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd08584ced0275b9f2217c1498d45380b71694f)
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
![{\displaystyle P={\frac {\lambda }{V-V_{\infty }}}-P_{0}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d67ea4772caf597c5d82f1d02e0a0fd185480589)
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 [1])
![{\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}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/002ca09eb760e2eb6bfa406a84f0dc9f56be2841)
See also[]
References[]
- ^ a b c d 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
- ^ a b Li, Yuan-Hui (15 May 1967). "Equation of State of Water and Sea Water" (PDF). Journal of Geophysical Research. Palisades, New York. 72 (10): 2665. Bibcode:1967JGR....72.2665L. doi:10.1029/JZ072i010p02665.
- ^ Thompson, P. A., & Beavers, G. S. (1972). Compressible-fluid dynamics. Journal of Applied Mechanics, 39, 366.
- ^ Kedrinskiy, V. K. (2006). Hydrodynamics of Explosion: experiments and models. Springer Science & Business Media.
- ^ Macdonald, J. R. (1966). Some simple isothermal equations of state. Reviews of Modern Physics, 38(4), 669.
- ^ a b 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