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.

 

[btrueblood]

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

 

[btrueblood]

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.

 

[sailoday28]

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

 

[btrueblood]

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).

 

[Latexman]

sailoday28,

This is not exactly on topic and you may have looked at it already, but faq1203-1293 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

 

[sailoday28]

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

 

[Latexman]

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

Good luck,
Latexman

 

[Latexman]

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

 

[sailoday28]

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

 

[Latexman]

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

Good luck,
Latexman

 

[sailoday28]

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

 

[Latexman]

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

G = P/sqrt(2RT/MWg)


Good luck,
Latexman

 

[sailoday28]

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

 

[Latexman]

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

 

[Latexman]

Checking my isothermal example above:

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

I got 293 lb_m/ft².sec

Something is not right. ???

Good luck,
Latexman

 

[sailoday28]

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?????

 

[Latexman]

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

Good luck,
Latexman

 

[Latexman]

205/293 = 0.7

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

Good luck,
Latexman

 

[Latexman]

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