# What exactly is the polytropic index and what definition do I use to describe gas flow when it's choked?

I'm trying to model the depressurising of a pressure vessel, and there is such thing as a choked flow equation. The equation is as follows:

$$\dot{m}_{max}= \gamma^{\frac{1}{2}} \left ( \frac{2}{\gamma+1}\right )^{(1/2)(\gamma+1)/(\gamma-1)} A^{\ast}\rho_0(RT_0)^\frac{1}{2}$$

Equation 9.46(a) White. Fluid Mechanics (SI Units), McGraw-Hill UK Higher Ed, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/nottingham/detail.action?docID=6422450. Created on 2023-09-25 13:53:39.

You will notice that the above choked mass flow rate is dependant on the polytropic index $$\gamma$$ (gamma) and that is said to be 1.4 for most diatomic molecules at low temperature, but gamma is also defined as $$\frac{C_p}{C_v}$$ specific heat capacity at constant pressure over specific heat capacity at constant volume, which comes to around the same number for diatomic molecules

However, the problem arises when the gas I'm modelling for is Ammonia NH3. I cannot use 1.4 as its not a diatomic molecule. And looking at the NIST webbook for thermophysical properties (link), the $$C_p$$ and $$C_v$$ are changing values for the pressures I'm interested in (the choked flow from 100 bar to 10 bar) and they seem to change pretty substantially. Which would then give rise to a non-constant polytropic index $$\gamma$$.

What is the Polytropic index exactly, and is it actually defined as $$\frac{C_p}{C_v}$$ or is it a constant for a fluid irrespective of pressure and temperature? I know that it is related to adiabatic processes, and is a sign of no increase in entropy.

However, how do I use it in this situation? Any advice would be helpful. The goal is to model the depressurising of a pressure vessel(blowdown if you will) as accurately as possible.

• Commented Sep 26, 2023 at 9:35
• There is nothing that demands that the polytropic index is independent of other parameters or a constant. You have cited a link that defines what is meant by the polytropic index - for an adiabatic process, the polytropic index is the adiabatic index = the ratio of specific heats. Commented Sep 26, 2023 at 14:11

In the formula, the first $$k$$ should be $$\gamma$$ and the exponent needs a slash between $$\gamma +1$$ and $$\gamma -1$$. With $$\gamma = 1.4$$ the exponent is 3. $$\gamma$$ is the ratio of specific heats; polytropic exponent is a different thing for non isentropic flows. You can bet that equations that have $$\gamma$$ are valid only if you expect $$\gamma$$ to be constant and ideal gas $$p=\rho RT$$ holds. Otherwise, only approximate.
• ... with a changing adiabatic index? I guess the question now becomes how accurate is the $PV=mRT$ equation to use highly accurate $C_p$ and $C_v$ values, which would give me a highly realistic $\gamma$ value. What would be the alternative $PV=mRT$ for real gases? Commented Sep 27, 2023 at 9:40