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isothermal choked flow

isothermal choked flow

isothermal choked flow

(OP)
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.

RE: isothermal choked flow

Mass flux = mass flow rate per unit cross sectional area.  Units kg/(m^2-s)
 

RE: isothermal choked flow

Sorry.  After a moment's thought, you might need a more formal definition, depending on what you're analyzing and how.  A somewhat more techie version of mass flux would be:

M = (rho)V*n

where M is mass flux, rho is mass density, V is the velocity vector, n is the surface normal vector (dimensionless unit vector) for the surface of interest, and * denotes a vector dot-product.  This is a point value equation.  

An "average mass flux over the exit area" calculation would have you calculate the surface integral the above term over the area of intestest, and divide the result by the area.

RE: isothermal choked flow

(OP)
btrueblood (Mechanical)
Sorry, my question should have been, "What is choked mass flux for isothermal frictionless flow thru a nozzle"?

 

RE: isothermal choked flow

Ah.  I will have to go grab my old gasdynamics book from home (if I still have it).  Liepmann & Roshko, from roughly the '60s.

Or, surf to here:

http://www.potto.org/gasDynamics/node128.php

and click "next" to follow the derivation through to where he provides a table and some following examples.  Essentially, the choked mass flux depends on the upstream conditions and length of pipe vs. heat flux parameter (the 4fL/D parameter).

RE: isothermal choked flow

sailoday28,

This is not exactly on topic and you may have looked at it already, but FAQ1203-1293: A Simple Numerical Method for Gas/Vapor Flow in a Safety Valve answers the same question, except for isentropic, not isothermal, flow.  You may be able to use that as a guide and make isothermal assumptions and get to the answer you seek.

Good luck,
Latexman

RE: isothermal choked flow

(OP)
Latexman (Chemical)
I am looking at exact solutions (or near to that as possible. For example, isothermal flow (even with friction)for a perfect gas has an exact solution.

Regards

RE: isothermal choked flow

Can you take the isentropic equations and set k = 1.  That is what is needed to be isothermal.

Good luck,
Latexman

RE: isothermal choked flow

For example an isentropic case is:

Po =    100    psia
To =    25    C
MW =    29    lb/lb.mole
k =    1.4    
dnozzle =    1    inch
        
Pn =    52.0    psia
Tn =    -25.8    oC
rn =    0.316    lbm/ft3
Sum(DP/rave) =    -118.439    lbf.ft3/(in2.lbm)
Gn =    330.699    lbm/(ft2.sec)
w =    6493    lbm/hr

and for isothermal it's:

Po =    100    psia
To =    25    C
MW =    29    lb/lb.mole
k =    1    
dnozzle =    1    inch
        
Pn =    60.0    psia
Tn =    25.0    oC
rn =    0.302    lbm/ft3
Sum(DP/rave) =    -101.432    lbf.ft3/(in2.lbm)
Gn =    292.940    lbm/(ft2.sec)
w =    5752    lbm/hr
 

Good luck,
Latexman

RE: isothermal choked flow

(OP)
I think I have my answer for pv=RT  Crititcal press ratio, Pc is

Pc=   e^[.5  - (u1^2RT)]  and   G = P/sqrt(2RT)
                                   P is pressure at throat.
                                 G is mass flux

RE: isothermal choked flow

I'm not so sure.  I couldn't get the units for G to work out to mass/(area*time).

Good luck,
Latexman

RE: isothermal choked flow

(OP)
Latexman (Chemical) With USA units, g should be included.
Also note my typos where I should have a division sign.
Crititcal press ratio, Pc is

Pc=   e^[.5  - (u1^2/RT)]  and   G = P/sqrt(2RT)
                                   P is pressure at throat.
                                 G is mass flux  

 USA UNITS

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

Regards
 

RE: isothermal choked flow

I think you need the MW for the units to work out, like:

G = P/sqrt(2RT/MWg)
 

Good luck,
Latexman

RE: isothermal choked flow

(OP)
Latexman (Chemical)

 USA UNITS

Pc=   e^[.5  - (u1^2/gRT)]  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



 

RE: isothermal choked flow

OK, you are using an individual gas contant = 1545/MW.  I tend to use the universal gas constant = 1545.  Looks good now!

Good luck,
Latexman

RE: isothermal choked flow

Checking my isothermal example above:

G = 60*144/sqrt(2*53.3*537/32.2) = 205 lbm/ft2.sec

I got 293 lbm/ft2.sec

Something is not right.  ???

Good luck,
Latexman

RE: isothermal choked flow

(OP)
G = 60*144/sqrt(2*53.3*537/32.2) = 205 lbm/ft2.sec

I got 293 lbm/ft2.sec

Something is not right.  ???

Why is something wrong?????

RE: isothermal choked flow

Two methods giving different answers.  One or both are wrong.  I'll look at it more closely this weekend.

Good luck,
Latexman

RE: isothermal choked flow

205/293 = 0.7

Is it a coincident the answers differ by about 1/sqrt(2)?

Good luck,
Latexman

RE: isothermal choked flow

Equation 4.11 in Shapiro is valid for both isentropic and nonisentropic flows through a passage of varying cross section:

(w/A)*sqrt(T0)/P*1/sqrt(MW) = sqrt{kg/R')*M*sqrt(1+(k-1)/2*M2)

For isothermal (T = T0), choked flow set k = 1 and M = 1 to get (w/A)max:

(w/A)max*sqrt(T)/P*1/sqrt(MW) = sqrt{g/R')

Rearrange:

(w/A)max = P*sqrt{gMW/R'T)

R = R'/MW:

(w/A)max = P*sqrt{g/RT)

Notice there's no 2 in the equation.
 

Good luck,
Latexman

RE: isothermal choked flow

(OP)
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

RE: isothermal choked flow

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

RE: isothermal choked flow

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

RE: isothermal choked flow

(OP)
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

RE: isothermal choked flow

Quote (sailoday28):

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

RE: isothermal choked flow

(OP)
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






 

RE: isothermal choked flow

(OP)
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
 

RE: isothermal choked flow

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

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