isothermal choked flow 1

[sailoday28]

What is mass flux?
I'm looking for the derivation or basis for choked ISOTHERMAL FLOW from a large reservoir thru a nozzle. Friction should be neglected.

 

[sailoday28]

Latexman (Chemical)
My Shapiro is in storage and I don't have access to the description of use of eq. 4.11. If my memory serves me correctly, is it possible that the equation is applicable to only adiabatic flow?
Please note that for isothermal flow in ducts and I believe isothermal flow in a nozzle, choking occurs NOT at M=1,but M=1/sqrt (k).

Regards

 

[Latexman]

I keep my Shapiro at home, so I'll doublecheck tonight. There's two ways to achieve isothermal conditions. One is to have heat transfer and the other is to have a gas with k ~ 1. The equation I created is for the latter, if you are after the first, it may be different.

Good luck,
Latexman

 

[Latexman]

Also, it is my experience that isothermal conditions are modelled with k = 1 since isothermal flow with a gas with k > 1 (say k = 1.4 for air) is not likely to occur in reality due to the enormous heat transfer requirements near sonic velocity. I'm curious, can you share your application and how the heat transfer will be addressed?

Good luck,
Latexman

 

[sailoday28]

Latexman (Chemical)
Again, my Crane is also in storage, but there are curves with fl/d for isothermal flow (and pv=RT) in the handbook.
My application is to apply a similar approach with real gas.
In reality heat flow will be difficult to maintain. However, with theoretical adiabatic flow, can we believe that at M=1, there is not heat transfer?
Regards

 

[Latexman]

However, with theoretical adiabatic flow, can we believe that at M=1, there is not heat transfer?

Agreed, neither model is perfect.

Good luck,
Latexman

 

[sailoday28]

Consider horizontal, uniform steady flow of perfect gas, constant specific heats.
dU^2/2)+dH=dQ energy
TdS+ VdP=dH combined 1st and 2d law
adding and considering reversible process
dU^2/2+VdP=0

Neglecting upstream velocity to make my calc easier
U^2/2 + int VdP=0 G, mass flux U=GV
G^2V^2/2 +int VdP=0
isentropic process
substitute V from PV^gamma=PoVo^gamma
subscript o for upstream
standard isentropic critical pressure and choked flow will be obtained by taking derivative of G wrt P

For isothermal
dU^2/2+VdP=0 do same as isentropic but substitute V=RT/P with T=constant R=unive R/mol
Isothermal choked flow results will be obtained.
Regards

 

[sailoday28]

Latexman (Chemical)
MY ERROR, ANOTHER CORRECTION-NOTE THE 2UNDER U1^2
USA UNITS

Pc= e^[.5 - (u1^2/2gRT)] and G = P/sqrt(2RT/g)


With USA units, say for air, I would use R=53.3 ft/deg
g=32.2 ft/s/s P #/ft^2 J=778

For heat transfer BTU per pound flowing is
(Ut^2/2 -U1^2/2)/gJ or
[Runiversal/mol*T/2 -U1^2/2]/gJ

 

[Latexman]

The equation in Shapiro is for adiabatic flow. I looked at Crane. They do not say whether isothermal or adiabatic. You may have to track it down through the references they took the graphs from. I suspect adiabatic though.

Good luck,
Latexman