Cole equation of state

The Cole equation of state is the adiabatic version of the stiffened equation of state for liquids. (See Derivation, below.) It has the form

$p=B\left[\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-1\right]$

In it, $\rho _{0}$ is a reference density around which the density varies, $\gamma$ is the adiabatic index, and $B$ is a pressure parameter.

Usually, the equation is used to model a nearly incompressible system. In this case, the exponent is often set to a value of 7, and $B$ is large, in the following sense. The fluctuations of the density are related to the speed of sound as

${\frac {\delta \rho }{\rho }}={\frac {v^{2}}{c^{2}}},$

where $v$ is the largest velocity, and $c$ is the speed of sound (the ratio $v/c$ is Mach's number). The speed of sound can be seen to be

$c^{2}={\frac {\gamma B}{\rho _{0}}}.$

Therefore, if $B=100\rho _{0}v^{2}/\gamma$, the relative density fluctuations will be about 0.01.

If the fluctuations in the density are indeed small, the equation of state may be approximated by the simpler:

$p=B\gamma \left[{\frac {\rho -\rho _{0}}{\rho _{0}}}\right]$

It is quite common that the name "Tait equation of state" is improperly used for this EOS. This perhaps stems for the classic text by Cole calling this equation a "modified Tait equation" (p. 39).

Derivation

Let us write the stiffened EOS as

$p+p^{*}=(\gamma -1)e\rho =(\gamma -1)E/V,$

where E is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the first law reads

$dW=-pdV=dE.$

Taking differences on the EOS,

$dE={\frac {1}{\gamma -1}}[(p+p^{*})dV+Vdp],$

so that the first law can be simplified to

$-(\gamma p+p^{*})dV=Vdp.$

This equation can be solved in the standard way, with the result

$(p+p^{*}/\gamma )V^{\gamma }=C,$

where C is a constant of integration. This derivation closely follows the standard derivation of the adiabatic law of an ideal gas, and it reduces to it if $p^{*}=0$.

If the values of the thermodynamic variables are known at some reference state, we may write

$(p+p^{*}/\gamma )V^{\gamma }=(p_{0}+p^{*}/\gamma )V_{0}^{\gamma },$

which can be written as

$p=p_{0}+(p_{0}+p^{*}/\gamma )((V_{0}/V)^{\gamma }-1).$

Going back to densities, instead of volumes,

$p=p_{0}+(p_{0}+p^{*}/\gamma )((\rho /\rho _{0})^{\gamma }-1).$

Comparing with the Cole EOS, we can readily identify

$B=p^{*}/\gamma$

Moreover, the Cole EOS differs slightly, as it should read (as indeed does in e.g. the book by Courant)

$p=A\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-B,$

with

$A=p^{*}/\gamma +p_{0}.$

This difference is negligible for liquids but for an ideal gas $p^{*}=0$ and there is a huge difference, B being zero and A being equal to the reference pressure.

Now, the speed of sound is given by

$c^{2}={\frac {dp}{d\rho }}$

with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain

$c^{2}=(p_{0}+p^{*}/\gamma )\gamma /\rho _{0}.$

From this expression a value of $p^{*}$ can be deduced. For water, $p^{*}\approx 23000$ bar, from which $B\approx 3000$ bar. If the speed of sound is used in the EOS one obtains the rather elegant expression

$p=p_{0}+(\rho _{0}c^{2}/\gamma )((\rho /\rho _{0})^{\gamma }-1).$