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| The '''Cole equation of state''' | | The '''Cole equation of state''' <ref>R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227</ref><ref> |
| <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> | | G. K. Batchelor "An introduction to fluid mechanics", Cambridge University Press (1974) ISBN 0521663962</ref> |
| is the adiabatic version of the [[stiffened equation of state]] for liquids. (See ''Derivation'', below.) | | can be written, when atmospheric pressure is negligible, has the form |
| It has the form
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| :<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math> | | :<math>p = B \left[ \left( \frac{\rho}{\rho_0} \right)^\gamma -1 \right]</math>. |
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| In it, <math>\rho_0</math> is a reference density around which the density varies, | | 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. | | <math>\gamma</math> is an exponent and <math>B</math> is a pressure parameter. |
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| Usually, the equation is used to model a nearly incompressible system. In this case, | | Usually, the equation is used to model a nearly incompressible system. In this case, |
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| where <math>v</math> is the largest velocity, and <math>c</math> is the speed of | | 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 | | sound (the ratio <math>v/c</math> is [[Mach's number]]). The speed of sound can |
| be seen to be | | be seen to be |
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| Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations | | Therefore, if <math>B=100 \rho_0 v^2 / \gamma</math>, the relative density fluctuations |
| will be about 0.01. | | will be of about 0.01. |
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| If the fluctuations in the density are indeed small, the
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| [[Equations of state | equation of state]] may be approximated by the simpler:
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| :<math>p = B \gamma \left[
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| \frac{\rho-\rho_0}{\rho_0}
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| \right]</math>
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| 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).
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| ==Derivation==
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| Let us write the stiffened EOS as
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| :<math>p+ p^* = (\gamma -1) e \rho = (\gamma -1) E / V ,</math>
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| where ''E'' is the internal energy. In an adiabatic process, the work is the only responsible of a change in internal energy. Hence the
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| first law reads
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| :<math> dW= -p dV = dE .</math>
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| Taking differences on the EOS,
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| :<math> dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , </math>
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| so that the first law can be simplified to
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| :<math> - (\gamma p + p^*) dV = V dp.</math>
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| This equation can be solved in the standard way, with the result
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| :<math> ( p + p^* / \gamma) V^\gamma = C ,</math>
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| where ''C'' is a constant of integration. This derivation closely follows the standard derivation of the adiabatic law
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| of an ideal gas, and it reduces to it if <math> p^* =0 </math>.
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| If the values of the thermodynamic variables are known at some reference state, we may write
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| :<math> ( p + p^* / \gamma) V^\gamma = ( p_0 + p^* / \gamma) V_0^\gamma , </math>
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| which can be written as
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| :<math> p = p_0 + ( p_0 + p^* / \gamma) ( (V_0/V)^\gamma - 1 ) . </math>
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| Going back to densities, instead of volumes,
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| :<math> p = p_0 + ( p_0 + p^* / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>
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| Comparing with the Cole EOS, we can readily identify
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| :<math> B = p^* / \gamma </math>
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| Moreover, the Cole EOS differs slightly, as it should read (as indeed does in e.g. the book by Courant)
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| :<math>p = A \left( \frac{\rho}{\rho_0} \right)^\gamma - B ,</math>
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| with
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| :<math> A = p^* / \gamma + p_0 . </math>
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| This difference is negligible for liquids but for an ideal gas <math>p^*=0</math> and there is a huge
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| difference, ''B'' being zero and ''A'' being equal to the reference pressure.
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| Now, the speed of sound is given by
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| :<math> c^2=\frac{dp}{d\rho} </math>
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| with the derivative taken along an adiabatic line. This is precisely our case, and we readily obtain
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| :<math> c^2= ( p_0 + p^* / \gamma) \gamma /\rho_0 . </math>
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| From this expression a value of <math>p^*</math> can be deduced. For water, <math>p^*\approx 23000</math> bar,
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| from which <math>B\approx 3000</math> bar. If the speed of sound is used in the EOS one obtains the rather
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| elegant expression
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| :<math> p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . </math>
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| ==References== | | ==References== |
| <references/> | | <references/> |
| [[category: equations of state]] | | [[category: equations of state]] |