Cole equation of state: Difference between revisions
(Derivation --- work in progress) |
|||
| Line 43: | Line 43: | ||
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 22:58, 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
![{\displaystyle p=B\left[\left({\frac {\rho }{\rho _{0}}}\right)^{\gamma }-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e250e61d89c636d24e9bd5fb17f4140aac0989fb)
In it, 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 \rho_0} is a reference density around which the density varies, 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} 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 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 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} 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
Let us write the stiffened EOS 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^* = (\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 theEOS,
- 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 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 } .
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^* / \gamma) (V_0/V)^\gamma - p^* / \gamma . }
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^* / \gamma) (\rho/\rho_0)^\gamma - p^* / \gamma . }
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
References
- ↑ 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)