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# Isentropic Exponent not = cp/cv ? Steam

## Isentropic Exponent not = cp/cv ? Steam

(OP)
Dear colleagues,
I'm confused because I thought that kappa = cp /cv generally.
Now I found in different property-tools, that for example water-steam 30barA, 245°C,cp=3.1992, cv=2.1134, cp/cv=1.513 BUT kappa=1,2804.
Why that? How can I get to this kappa-value. In my EXCEL-tools I use the water97_v13.xla which provides cp and cv only.

Thanks!
Johann

### RE: Isentropic Exponent not = cp/cv ? Steam

Sounds like one reference is providing cp and cv, and a different reference is providing k. Is that right? A deep dive into both references, their basis, and calculations is the only way to answer that.

Good Luck,
Latexman

### RE: Isentropic Exponent not = cp/cv ? Steam

The isentropic exponent k is = Cp/Cv for ideal gases and for real gas at low pressures only, where z=1.0. Due to the real gas compressiblility of the gas z <1.0, real gas k is obtained by some other means, which involves partial derivative terms.

k = - (Cp/Cv).(δp/δv)T..(v/p) for real gas, where v = specific molar volume, all terms in consistent units.

In any case, if your value for Cp is correct, then the value for Cv, based on semi ideal gas behaviour, ought to be ( from Cp-Cv=R)

Cv = ((3.2 x 18) - 8.314) / 18 = 2.74 kJ/kg/degC - higher than your value of 2.1kJ/kg/degC

and semi ideal Cp/Cv = 3.2 / 2.74 = 1.17

which is somewhat closer to the real value of 1.28 than 1.5

### RE: Isentropic Exponent not = cp/cv ? Steam

The value of k = Cp/Cv is used in the calculations related with the flow of compressible fluids through pipes, ROs, nozzles etc. and appears in the equation Pv**k = Constant.
See for example the Crane Technical Paper 410.
More realistic would be to consider the expansion of the fluid as isentropic, isenthalpic or intermediate between both. In a nozzle, for example, that usually has a negligable friction pressure drop, the expansion approachs to the isentropic and in a pipe is more close to intermediate.
If you change k by a, being a the expansion coefficient in the same equation as before, Pv**a = Constant, the values of a of the reheated steam are 1.3 for the isentropic expansion and a = 1 for the isenthalpic expansion. In the case of the hydrogen, the values of a, are 1.4 and 1
In any case the use of k = Cp/Cv is correct to solve the fluid calculations

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