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
V 0 − V P V 0 = A Π + P {\displaystyle {\frac {V_{0}-V}{PV_{0}}}={\frac {A}{\Pi +P}}}
where P {\displaystyle P} is the hydrostatic pressure in addition to the atmospheric one, V 0 {\displaystyle V_{0}} is the volume at atmospheric pressure, V {\displaystyle V} is the volume under additional pressure P {\displaystyle P} , and A , Π {\displaystyle A,\Pi } are experimentally determined parameters. A very detailed historical study on the Tait equation with the physical interpretation of the two parameters A {\displaystyle A} and Π {\displaystyle \Pi } 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
K = − V 0 ( ∂ P ∂ V ) T = V 0 ( B + P ) C {\displaystyle {K}=-{V_{0}}\left({\frac {\partial P}{\partial V}}\right)_{T}={V_{0}}{\frac {(B+P)}{C}}}
where K {\displaystyle K} is the isothermal mixed bulk modulus. This above equation is popularly known as the Tait equation. The integrated form is commonly written
V = V 0 − C log ( B + P B + P 0 ) {\displaystyle V=V_{0}-C\log \left({\frac {B+P}{B+P_{0}}}\right)}
where
V {\displaystyle V\ } is the specific volume of the substance (in units of ml/g or m3/kg)
V 0 {\displaystyle V_{0}} is the specific volume at P = P 0 {\displaystyle P=P_{0}}
B {\displaystyle B\ } (same units as P 0 {\displaystyle P_{0}} ) and C {\displaystyle C\ } (same units as V 0 {\displaystyle V_{0}} ) are functions of temperature
Pressure formula The expression for the pressure in terms of the specific volume is
P = ( B + P 0 ) exp ( − V − V 0 C ) − B . {\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 C {\displaystyle C} and B {\displaystyle B} is given in chapter 3 of reference. Expressions as a function of temperature for the two empirical parameters C {\displaystyle C} and B {\displaystyle B} are given for water, seawater, helium-4, and helium-3 in the entire liquid phase up to the critical temperature T c {\displaystyle T_{c}} . 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
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