Difference between revisions of "Cole equation of state"

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(Derivation --- work in progress)
(Derivation)
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first law reads
 
first law reads
  
:<math>  dW= -p dV  = dE</math>
+
:<math>  dW= -p dV  = dE .</math>
  
...
+
Taking differences on theEOS,
 +
 
 +
:<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^* / \gamma)  (V_0/V)^\gamma - p^* / \gamma . </math>
 +
 
 +
Going back to densities, instead of volumes,
 +
 
 +
:<math>  p      =  ( p_0 + p^* / \gamma)  (\rho/\rho_0)^\gamma - p^* / \gamma . </math>
 +
 
 +
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
  
 
==References==
 
==References==
 
<references/>
 
<references/>
 
[[category: equations of state]]
 
[[category: equations of state]]

Revision as of 23:58, 6 March 2015

The Cole equation of state [1][2][3] is the adiabatic version of the stiffened equation of state. (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= -p dV  = dE .

Taking differences on theEOS,

   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^* / \gamma)  (V_0/V)^\gamma - p^* / \gamma .

Going back to densities, instead of volumes,

   p      =   ( p_0 + p^* / \gamma)  (\rho/\rho_0)^\gamma - p^* / \gamma .

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

References