Cole equation of state: Difference between revisions

Latest revision as of 13:16, 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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is large, in the following sense. The fluctuations of the density are related to the speed of sound as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta \rho}{\rho} = \frac{v^2}{c^2} ,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is the largest velocity, and $c$ is the speed of sound (the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v/c} is Mach's number). The speed of sound can be seen to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2 = \frac{\gamma B}{\rho_0}. }

Therefore, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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[edit]

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dW= -p dV = dE .}

Taking differences on the EOS,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dE = \frac{1}{\gamma-1} [(p+p^*) dV + V dp ] , }

so that the first law can be simplified to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - (\gamma p + p^*) dV = V dp.}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( p + p^* / \gamma) V^\gamma = ( p_0 + p^* / \gamma) V_0^\gamma , }

which can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_0 + ( p_0 + p^* / \gamma) ( (V_0/V)^\gamma - 1 ) . }

Going back to densities, instead of volumes,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_0 + ( p_0 + p^* / \gamma) ( (\rho/\rho_0)^\gamma - 1) . }

Comparing with the Cole EOS, we can readily identify

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = p^* / \gamma }

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

p=A\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-B,

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = p^* / \gamma + p_0 . }

This difference is negligible for liquids but for an ideal gas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^*=0} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2=\frac{dp}{d\rho} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2= ( p_0 + p^* / \gamma) \gamma /\rho_0 . }

From this expression a value of $p^{*}$ can be deduced. For water, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^*\approx 23000} bar, from which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\approx 3000} bar. If the speed of sound is used in the EOS one obtains the rather elegant expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_0 + ( \rho_0 c^2 / \gamma) ( (\rho/\rho_0)^\gamma - 1) . }

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)

[1] Robert H Cole "Underwater explosions", Princeton University Press, Princeton (1948)

[2] G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN 0521663962

[3] 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)

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-The '''Cole equation of state''' <ref>R. H. Cole "Underwater Explosions", Princeton University Press (1948) ISBN 9780691069227</ref><ref>
+The '''Cole equation of state'''
-G. K. Batchelor "An introduction to ﬂuid mechanics", Cambridge University Press (1974) ISBN  0521663962</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>
-has the form
+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>
@@ Line 30: / Line 31: @@
   \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]]

Cole equation of state: Difference between revisions

Latest revision as of 13:16, 5 December 2015

Derivation[edit]

References[edit]

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