Exit Conditions of a Vapor Stream though a Nozzle 1

[radaes]

Hi,

I'm trying to calculate the exit conditions of a high-pressure LPG nozzle. I have the dimensions (interior @ state 1 and exit) of the nozzle and the initial conditions of the vapor (P1, T1), as well as the mass flow rate (mdot).
P1 & T1 will give me density (rho)
since mdot=rho*A*V, i can solve for velocity, thereby defining state 1.
i need to find pressure & velocity at the exit of the nozzle.
i'm treating it as an adiabatic process and ignoring frictional effects (for now).
I know that Pexit is (supposed to be) below ambient atmospheric. I also know I can't just use Q1=Q2, since this is a compressible process and rho is a function of the pressure, which is a function of velocity, which is a function of rho, which is a function of pressure....
I'm stuck in a loop. any help, suggestions, or ideas?

cheers,

rad

 

[Latexman]

"High-pressure" is a relative term. Tell us all that you know, like the value and units of the knowns (T1, P1, and mdot) and the calculated value (rho) and the dimensions of the nozzle and piping/tubing. That way we can narrow down our responses.

Good luck,
Latexman

 

[radaes]

You're right, that may have been (aka "was) a stupid statement... i was thinking "high velocity" and somehow that morphed into "high pressure" when i wrote the post... :~/
anyway: propane vapor is my fluid
p1=240 kPa
t1=330 K
my handy property calculator spits out rho=3.9667.. call it 4.0 kg/m^3
mdot=0.0408 kg/s
area1=1.267 cm^2 (upstream of converging section)
areaThroat=0.3739 cm^2
areaExit= 0.45 cm^2

that gets me a velocity @ point1 of 80.5 m/s (V=mdot/rho*A)
taking rho as a constant, i can get velocities at my other points.... except rho is not a constant.
I tried Bernoulli for compressible gas flow, but that requires either knowing the velocities or the pressures at both points... and just guesstimating the numbers kinda blew up in my face... so now i'm stuck and outta ideas.

anyone want to take a stab at explaining it to me?

thanks and cheers,

rad

 

[Latexman]

Are the distances from "state 1" to the throat and from the throat to the exit short? Say, 1-2 cm each?

Good luck,
Latexman

 

[radaes]

point 1 to throat ~2.4cm, throat to exit ~.5cm

are you going to tell me that the distances are too short for fully developed flow?

cheers,
rad

 

[btrueblood]

You need to calculate c* (the sonic velocity of gas at the throat). Assume choked flow (Mach no. =1) at the throat.

 

[sailoday28]

It is stated that Pexit is supposed to be below ambient atmospheric. Does this mean the exit pressure is below the back pressure?
A good example of a converging-diverging nozzle with varying back pressure conditions is illustrated on pg 140, Vol 1 -of Shapiro--The Dynamics and Thermodynamics of Compressible Fluid Flow.

For the conditions you stated, supersonic flow will exist from the nozzle throat to exit---NO shocks.

 

[radaes]

well, c=sqrt(k*R*T), for propane k~1.1, R~189, T~330 -->
c~262 m/s, which gives me a M=1.04... taking into account that this entire model is "more or less", most of my numbers are "close enough", which pretty much means chocked flow @ the throat already

The vapor is supposed to exit the nozzle below atmospheric pressure, which sets up a vacuum that entrains air @ ambient into the vapor stream... since i know that the mixing takes place, i know that the exit pressure has to be < 1atm. i just want to know what the pressure/velocity of the vapor stream is at exit. the numbers i'm coming up with are too low (velocity) or too high (pressure).

any other ideas?

cheers,
rad

 

[Latexman]

My Crane TP410 said k (Cp/Cv) = 1.15 for propane. The critical pressure ratio = 0.5744. p* = 138 kPa. T* = 307 K. rho* = 2.45 kg/m^3. With the area ratio (Aexit/Athroat), I get Msubsonic = 0.5983 and Msupersonic = 1.4880.

If subsonic, I get Treceiver = 321 K and Preceiver = 196 kPa.

If supersonic, I get Treceiver = 283 K and Preceiver = 74 kPa.

You said the receiver pressure is below atmospheric (101 kPa), so whether it'll be sub or supersonic depends on what that really is and if it's controlling or not.

Btw, is your mdot is too high? I got 0.022 kg/sec in the throat.

Good luck,
Latexman

 

[Latexman]

I posted previously without reading your last post. Sorry. Since it exhausts into air at ambient T and P (101 kPa), the flow in the diverging section of the nozzle will be supersonic part of the way to the exit, then there will be a normal shock wave where the flow changes irreversibly from supersonic to subsonic. To keep it simple, I'd use the exit conditions in my previous post for subsonic flow. Actual conditions will vary due to the entropy change associated with the shock wave.

Good luck,
Latexman

 

[sailoday28]

Latexman (Chemical)If a shock occurs, then the exit pressure will match the back pressure. I believe the exit pressure will correspond to that of the exit Mach no which will be supersonic.

 

[Latexman]

Sailoday,

I'm glad you rejoined.

I agree. The exit pressure = the back pressure, which is ambient pressure of about 101 kPa.

I disagree the exit flow will be supersonic.

On page 140 and using Fig. 5.21 (a), the exit pressure is between the subsonic design curve (condition 2) and the supersonic design curve (condition 4). It is as condition 3. Therefore the following applies, "As regime II is entered, a normal shock wave appears downstream of the throat and the process aft of the shock comprises subsonic deceleration."

So, the flow is supersonic from the throat to the shock wave, but after the shock wave the flow is subsonic and remains subsonic to the exit and outside to the air.

That's how I see it.

I wimped out and did not calculate the changes in properties across the shock wave. I hope using the subsonic solutions' properties are close enough. Can you comment on that? Thanks!

Good luck,
Latexman

 

[sailoday28]

Latexman
As I see it, refering again to Shapiro, with path 4, exit and back pressure are equal only if the shock occurs at the exit plane.
The original statement of the problem is that exit pressure is less than the back pressure. Doesn't this indicate that the flow conditions are either in region III or IV just below path (4)? and that oblique shock waves occur downstream of the exit??

Sorry for those without the bible of Shapiro?

Regards

 

[Latexman]

Sailoday,

Since radAES said "I know that Pexit is (supposed to be) below ambient atmospheric." I assumed by the "(supposed to be)" he was not so sure, so I did not put much faith in that and used my own judgement. If it is true, you are right. If it is not true, then I am right.

radAES, can you clarify?

Good luck,
Latexman

 

[radaes]

I know for a fact that the mixing takes place, and I know that the explanation for the mixing is that Pexit is lower than Pair, which sets up a vacuum that "pulls" the air into the vapor stream; therefore, I am very comfortable with saying that Pexit is less than 1atm.

On a different note: I'm very sure of my mass flowrate - that's just based on the rating of the nozzle (in BTU/hr) and the energy content of LP (BTU/kg) - yes, I know I'm mixing unit systems, but I despise standard - so sue me



cheers,
rad
"Remember, if you leave it to the last minute, it'll only take a minute"

 

[Latexman]

radAES,

When the propane leaves the nozzle, it is a "free jet" if it enters an area whose cross-sectional area is > 5 times the area of the nozzle exit. I bet this is your case. When a "free jet" leaves its outlet and goes into the surrounding fluid, the jet will entrain the surrounding fluid, expand, and decelerate. The momentum of the jet is transferred to the surrounding fluid being entrained. While there may be some fluid movement caused by static pressure differences, the resulting mixing due to just this component will be minimal. In your case, the vast majority of the “jet mixing” will be due to momentum transfer.

The 2X difference in flow could be due to many, many things.


Good luck,
Latexman

 

[Latexman]

Sailoday,

Do you have access to Perry’s Chemical Engineers’ Handbook (7th Edition)? It speaks to our little debate on this case very well. On page 6-25 under Convergent/Divergent Nozzles (De Laval Nozzles) it discusses the A/A* equation, “With A set equal to the nozzle exit area, the exit Mach number, pressure and temperature can be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure; shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result.”

I thought this was well said.

Good luck,
Latexman

 

[sailoday28]

Latexman
I don't have Perry's Handbook.
"Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle."
Refering again to Shapiro pg 140 Vol 1. If path is on (4),(5),or (6) of Filg 5.21a----Then I believe the exit plane Mach nos for each path will be the same.

"If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result.”--I have to digest this last statement.
Regards

 

[aviat]

In supersonic flow the relationship between pressure and velocity is opposite to subsonic flow. If the exit pressure is less than ambient there, there should be an oblique wave beyond the nozzle exit, and nozzle is overexpanded. This would occur if the the diverging section is supersonic. M1 occurs at the throat of a converging-diverging nozzle. See site below.

http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html

As far as determining pressure and velocity, it involves expansion waves and obilique shock waves, which are two and three dimensional in nature.

 

[sailoday28]

aviat (Aerospace) states"As far as determining pressure and velocity, it involves expansion waves and obilique shock waves, which are two and three dimensional in nature"

I don't disagree.--However, as Aviat further states this is downstream of the exit plane----
This will not affect mass flow rate in the nozzle.