Cole equation of state

2015-12-05T12:16:25Z

103.15.217.222: space inserted between 'the' and 'EOS' for friendly reading

The '''Cole equation of state'''
<ref>[http://www.archive.org/details/underwaterexplos00cole Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)]</ref><ref>G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref><ref>[http://www.archive.org/details/supersonicflowsh00cour Richard Courant "Supersonic flow and shock waves a manual on the mathematical theory of non-linear wave motion", Courant Institute of Mathematical Sciences, New York University, New York (1944)]</ref>
is the adiabatic version of the [[stiffened equation of state]] for liquids. (See ''Derivation'', below.)
It has the form

:<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>

In it, <math>\rho_0</math> is a reference density around which the density varies,
<math>\gamma</math> is the [[Heat capacity#Adiabatic index | adiabatic index]], and <math>B</math> 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 <math>B</math> is large, in the following
sense. The fluctuations of the density are related to the speed of sound as

:<math>\frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,</math>

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

:<math>c^2 = \frac{\gamma B}{\rho_0}. </math>

Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations
will be about 0.01.

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

:<math>p = B \gamma \left[
\frac{\rho-\rho_0}{\rho_0}
\right]</math>

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

:<math>p+ p^* = (\gamma -1) e \rho = (\gamma -1) E / V ,</math>

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

:<math> dW= -p dV = dE .</math>

Taking differences on the EOS,

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

so that the first law can be simplified to

:<math> - (\gamma p + p^*) dV = V dp.</math>

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

:<math> ( p + p^* / \gamma) V^\gamma = C ,</math>

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 <math> p^* =0 </math>.

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

:<math> ( p + p^* / \gamma) V^\gamma = ( p_0 + p^* / \gamma) V_0^\gamma , </math>

which can be written as

:<math> p = p_0 + ( p_0 + p^* / \gamma) ( (V_0/V)^\gamma - 1 ) . </math>

Going back to densities, instead of volumes,

:<math> p = p_0 + ( p_0 + p^* / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>

Comparing with the Cole EOS, we can readily identify

:<math> B = p^* / \gamma </math>

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

:<math>p = A \left( \frac{\rho}{\rho_0} \right)^\gamma - B ,</math>

with

:<math> A = p^* / \gamma + p_0 . </math>

This difference is negligible for liquids but for an ideal gas <math>p^*=0</math> 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

:<math> c^2=\frac{dp}{d\rho} </math>

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

:<math> c^2= ( p_0 + p^* / \gamma) \gamma /\rho_0 . </math>

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

:<math> p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>

==References==
<references/>
[[category: equations of state]]

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Cole equation of state