# Difference between revisions of "Cole equation of state"

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The '''Cole equation of state''' | 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 " | + | <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]]. | + | is the adiabatic version of the [[stiffened equation of state]] for liquids. (See ''Derivation'', below.) |

It has the form | It has the form | ||

Line 34: | Line 34: | ||

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). | 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== | ||

<references/> | <references/> | ||

[[category: equations of state]] | [[category: equations of state]] |

## Latest revision as of 14:16, 5 December 2015

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

In it, is a reference density around which the density varies, is the adiabatic index, and 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 is large, in the following sense. The fluctuations of the density are related to the speed of sound as

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

Therefore, if , 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:

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

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

Taking differences on the EOS,

so that the first law can be simplified to

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

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 .

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

which can be written as

Going back to densities, instead of volumes,

Comparing with the Cole EOS, we can readily identify

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

with

This difference is negligible for liquids but for an ideal gas 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

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

From this expression a value of can be deduced. For water, bar, from which bar. If the speed of sound is used in the EOS one obtains the rather elegant expression

## References[edit]

- ↑ Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)
- ↑ G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962
- ↑ 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)