Latest revision |
Your text |
Line 1: |
Line 1: |
| 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 "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> | | <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.) | | is the adiabatic version of the [[stiffened equation of state]]. |
| 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]] |