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Ws for IG, SISO, SS, Int Rev Isentropic Process

and:

Thus:

Or:

Ideal gas

SISO

SS

internally reversible

polytropic

processes with δ ≠ 1, ΔEpot = ΔEkin = 0

For

isentropic

processes on ideal gases with constant γ:

and:

Recall from Lesson 7E that if the

ideal gas heat capacities

are constant:

Lesson
8B Blog

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Lesson 8B: page 15 of 23

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Ch 8, Lesson B, Page 15 - Ws for IG, SISO, SS, Int Rev Isentropic Process

For an isentropic process, δ = γ. This only works when the heat capacities, and therefore γ, are constant.
When the heat capacities are NOT constant, we must go back to the SS, SISO energy balance equation..
This isn’t so bad because we find that WS,dot over mdot is equal to minus ΔHhat.
For real gases we must look up enthalpy in thermodynamic tables like the ones in the NIST WebBook.
For ideal gases, we can use polynomial heat capacity equations to determine ΔHhat.
But, when γ is constant, we can just replace δ with γ in the equations for Ws-dot over M-dot for both real fluids and ideal gases.
For ideal gases, further simplifications are possible.
Gamma is the heat capacity ratio. So for ideal gases, this is CPo over CVo. Either wiggles or hats will work. The difference is a factor of the molecular weight, but since the same factor is in both the numerator and the denominator the heat capacity ratio and the specific heat ratio have the same value.
We can also use the fact that CPo is equal to CVo plus R.
To get this in terms of the specific heat, we must divide by the molecular weight.
A little algebra produces the equation at the bottom of the page. WS,dot over mdot is equal to MINUS CPo times the change in temperature.
But wait, for ideal gases, CPo times ΔT is just ΔHhat !
Cool. WS,dot over mdot is just MINUS ΔHhat.
Hey, wait. This looks familiar !
This is what we would have obtained if we had applied the energy balance directly to this process.
It is good to know that the derivation presented on this page is consistent with the 1st Law, don’t you think ?
OK, now let’s go back and consider the case in which δ is equal to 1.

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