Choked flow: A short note about how and why?

What is choked flow?

Choked flow is a compressible flow effect.

Choked flow is a fluid dynamic condition associated with the venturi effect. When a flowing fluid at a given pressure and temperature passes through a constriction (such as the throat of a convergent-divergent nozzle or a valve in a pipe) into a lower pressure environment the fluid velocity increases.

At initially subsonic upstream conditions, the conservation of mass principle requires the fluid velocity to increase as it flows through the smaller cross-sectional area of the constriction. At the same time, the venturi effect causes the static pressure, and therefore the density, to decrease at the constriction.?Choked flow is a limiting condition where the mass flow will not increase with a further decrease in the downstream pressure environment for a fixed upstream pressure and temperature.

Thermodynamics

Role of Cp/Cv in choked flow

When a gas expands adiabatically it uses its internal energy to expand. The ability of a gas to expand comes from how much internal energy it has to supply for expansion. The more gas has internal energy the more it is compressible and therefore, the more it is prone to choked flow.?

Detail

When you push a high-pressure gas through a constriction it expands adiabatically following the polytropic equation PV ^n = C.?n = Cp/Cv = y = 1.4 for a diatomic gas.?Cp/Cv depends on the atomicity of a gas.

?Y [Gamma] = Cp/Cv = 1 + 2/[DOF], DOF is degrees of freedom, the way a molecule can store energy

Cp/Cv ratio for monoatomic, diatomic, triatomic is 1.67,1.4,1.33 respectively.

dH / dT =Cp, H is enthalpy

dU / dT = Cv, U is internal energy

Cp/Cv = dH / dU

Cp/Cv is simply the amount of adiabatic work you can extract from a molecule. The more internal energy the less Cp/Cv. The more adiabatic work you get.

Example [ y is gamma Cp/Cv]

Look at the attached table. Take the case of CO2, y = 1.3, Choked flow pressure ratio = 1.83 and helium, y = 1.66 and choked flow pressure ratio = 2.05.

?y [ gamma] for CO2 is the lowest in the list because it is a polyatomic polar molecule with a larger DOF [ degrees of freedom to store energy].?Its two oxygen atoms with two lone pairs of electrons each make CO2 more polar than any other molecules in the list of gas. It has the highest internal energy stored in the molecule for the above reasons for adiabatic work so it is the most compressible gas in this list of gases and has the maximum ability for choked flow.

He [ helium] on the other hand has the highest Cp/Cv in the list, it has the lowest internal energy to do adiabatic work because it is an inert small molecule. It has no polarity. It has very few van der Waal forces. So it is the least compressible almost like an ideal gas compared to CO2 Z < 1with the minimum choked flow potential.

But nothing is absolute. Everything is on a comparative scale.

All gases are compressible. But within the gases, some real gases are more compressible than others. Like CO2 is more compressible than He [ comparatively]. The reason is CO2 has z < 1. In CO2 because it is a big polar molecule the attractive forces dominate. On the other side helium, is a small nonpolar molecule. It has dominating repulsion forces acting between the molecules therefore it has z > 1 [ z is the compressibility factor]. This makes on a relative scale CO2 more compressible than He

Finally, it is the atomicity or degrees of freedom (DOF) which relates to y ( gamma) Cp / Cv = 1 + 2/ DOF that takes the final call. The more the degrees of freedom the less is CP/ Cv and this is a fixed number. For all diatomic molecules, it is 1.4. So all diatomic molecules have identical choked flow potential. For triatomic molecules like CO2 or H2O, because there are more degrees of freedom, y = 1.33.

Therefore, a triatomic molecule has more chances to choke than a diatomic molecule

The bottom line is if you reach a mass flow rate that generates sonic velocity at the point of constriction, MAC = 1 regardless of what the gas is there will be choked flow.

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