User:RaoulV/Isentropic exponent

Isentropic exponent

The isentropic exponent or isentropic expansion coefficient $k$ is a dimensionless number that relates a relative change in specific volume of a fluid to a corresponding relative change in pressure, during an isentropic expansion or compression.

The isentropic exponent $k$ is defined by :

$k=-{\frac {V}{p}}{\partial p \over \partial V}|_{s=cte}={\frac {\rho }{p}}{\partial p \over \partial \rho }|_{s=cte}\,\!$ (1)

If $k$ is constant along an isentropic path, then the pressure and specific volume of the gas along this path are related by

$pV^{k}=cte\,\!$ (2)

Relation to other thermodynamic quantities

symbol	quantity
$k$	isentropic exponent
$\gamma$	heat capacity ratio
$V$	specific volume
$\rho$	density
$p$	pressure
$s$	entropy
$u_{s}$	velocity of sound

The isentropic exponent is related to the velocity of sound by :

$k={\frac {\rho }{p}}u_{s}^{2}\,\!$ (4)

Ideal gas

For an ideal gas the isentropic exponent is equal to the heat capacity ratio $\gamma$

$\gamma ={\frac {C_{p}}{C_{V}}}\,\!$ (3)

As a consequence, it is always larger than one.

Real gases, liquids and vapor-liquid mixtures

For a real gas, $k$ varies with temperature and pressure and is different from $\gamma$. The difference becomes larger as the fluid deviates more from an ideal gas (for instance at higher pressures, close to the saturation pressure, in the 2-phase region, in the liquid state).

In the two-phase region, k can be lower than 1.

Application in fluid flow

If the isentropic exponent is constant along an isentropic path from the valve inlet to the valve throat, the mass flux at the throat can be calculated from :

$q_{m}={\sqrt {p_{0}\rho _{0}}}{\sqrt {k({\frac {2}{k+1}})^{\frac {k+1}{k-1}}}}\,\!$ (5)

To apply this equation without conversion factors, use consistent units (pressure in Pa, density in kg/m3, mass flux in kg/m2s).

This equation is used in the calculation of the capacity or required size of relief valves .

Examples

Values of k can be calculated by using an equation of state.

In the table below, some values of $k$ and $\gamma$ are given for water (calculated with the IAPWS-97 equation of state). Water was selected because a very accurate equation of state is available. Similar differences between $k$ and $\gamma$ exist for other fluids, even at lower pressures (the critical pressure of water is much higher than that of most hydrocarbons).

temperature	pressure	isentropic exponent	heat capacity ratio
(°C)	(bar a)	$k$	$\gamma$
101	1	1.317	1.337
180	10	1.219	1.406
270	50	1.267	1.670
311	100	1.237	2.297

See also

• heat capacity ratio
• Speed of sound

External links

IAPWS-97

Category:Thermodynamics Category:Physical quantities Category:Ratios