Cole equation of state

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The Cole equation of state [1][2][3] 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[edit]

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= -p dV  = dE .

Taking differences on the EOS,

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

so that the first law can be simplified to

   - (\gamma p + p^*)  dV  = V dp.

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) .

References[edit]