choked flow pv=zrt Shouldn't gamma be corrected? 1

[sailoday28]

Based on Wikipedia, Choked flow
Using pv=zRT and constant specific heats, choked flow is proportional to the term sqrt(gamma/(ZRT) where Z is stated as constant. That Z corrected for the term rho*P


The z apparantely corrects for compressibility of the gas.
However for pv=zRT where Z is constant

on a per mol basis Cp -Cv = ZR (R is univ const)
so that Cp/Cv- 1 =ZR/Cv or gamma= 1 +ZR/Cv

For low pressure conditions with Z=1 gamma = 1+R/Cv
If Cv remains constant at the new high pressure condition
shouldn't gamma be adjusted

{gamma(new)-1}/[gamma (z=1)-1] = [1+ZR/Cv]/[1 + R/Cv]

Regards

 

[Latexman]

I've seen thermodynamic functions corrected by z before*, so it makes sense that what you suggest is right.

* Properties of Liquids and Gases by Sherwood, Reid, and Prausnitz.


Good luck,
Latexman

 

[davefitz]

I think it may need to be corrected for those cases where the change in fluid properties is relatively gradual , as with frictional choking along a long pipe, or along a long nozzle.

In the case of a sudden expansion across a small orifice, I don't think such a correction is correct. In the final analysis, the choking represents the gas molecules at the end plane realizing soundspeed,( or some fraction of soundspeed if oblique shock waves form), and so the only important gamma is that ocurring one molecule upstream of the endplane.

 

[sailoday28]

davefitz (Mechanical) In going thru an orifice, whether up stream or at the vena contract, the fluid follows and equation of state and state path. And gamma will be the same for PV=ZRT, constant Z whether up or downstream.

Regards

 

[davefitz]

Once you get to a discontinuity such as a shock, you may have to go back to the orignal mechanistic model of the gases . The equation of state is a convenience that works mathematically for processes, but it may be the wrong tool for the job of understanding the process at the shock boundary.

It is my understanding that the fluid entering a shock is traveling at an average molecular , single direction, velocity equal to the current ( marginally upstream ) soundspeed. This implies that it is the properties immediately upstream that determines the soundpseed, and not the downstream properties. If no downstream signals propagate into the upstream fluid across a shock, then how can one propose to use downstream variation of gamma to adjust the process across the shock?

 

[sailoday28]

davefitz (Mechanical)What if I just have a converging nozzle that chokes. I don't see what a shock has to do with what is going on from the source to the exit.
The thrust of my original question was proper use of gamma based on an equation of state, z=constant in a choked flow formulation.
For example, If I used a real equation of state,and a specific heat, one could formulate or calculate choked flow for isentropic conditions.

Regards

 

[Latexman]

I believe you are both right.

Usually the upstream P and T are fixed. Therefore the upstream z is fixed.

Then, what I do, is assume the pressure at the end plane of the nozzle is 1 psi less than P[sub]upstream[/sub]. Then 2 psi less than P[sub]upstream[/sub] and I check for choking, i.e. has mass velocity reached a maximum? If not, I keep reducing the end plane pressure 1 psi at a time until there is choking observed or P[sub]end plane[/sub] = P[sub]surrondings[/sub].

At each end plane pressure increment, the most rigorous thermodynamic model in the world for your fluid can be used to estimate T[sub]end plane[/sub], [ρ][sub]end plane[/sub], z[sub]end plane[/sub], etc. to check for choking.

It's only the end plane pressure that determines the flow rate.

See my faq1203-1293

Good luck,
Latexman

 

[davefitz]

For a converging nozzle, or for a long length of pipe, where the pressure is continuously decreasing , then correcting for a varying gamma is the correct procedure.

In the case where there is a discontinuity ,as across a shock, the downstream gamma seems to be irrelevant, in my opinion. But I haven't studied this field for over 20 yrs, so it may just be cobwebs in the brain.

 

[torpen]

Hello,

The relations you have written hold for a mixture of perfect gases.
Z indicates the total number of particules after and before dissociation.
For example, Z = 1.2 means that 20 % of the molecules have dissociated.

For molecules the internal energy per unit mass is equal to
sensible energy = translational energy + rotational energy
+ vibrational energy + electronical energy

Internal energy is equal to n*(0.5*R*T) where n is the number of dofs of the molecule.
For temperatures below 2000 K, there is 3 dofs in translation and 2 in rotation
and hence 5 dofs.
Then the internal energy is for this type of gas:
e = 3*(0.5*R*T) + 2*(0.5*R*T) = 5/2*R*T
For the enthalpy h = e + R*T = 7/2*R*T = Cp*T
And for the specific heat and the ratio gamma :
Cp/R = 7/2 and gamma = 1/(1-R/Cp) = 1.4 = constant

For temperatures between 2000 K and 4000 K, the vibration of the molecules adds 2 dofs
and then the specific heat increases and the ratio of specific heats decreases.
e = 3*(0.5*R*T) + 2*(0.5*R*T) + 2*(0.5*R*T) = 7/2*R*T
and h = 9/2*R*T. Then Cp/R = 9/2 and gama = 1.29.

When the temperature reaches 600 K or higher, vibrational energy is no longer negligible.
And Cv = f(T) and gamma is no longer constant.
Above 2000 K chemical reactions begin to occur (at 1 atm, O2 begins to dissociate)
and Cv experiences large variations. Cp and Cv are functions of temperature and pressure
and hence gamma = f(T,P).

Regards,

Torpen.

 

[hacksaw]

the wikipedia comment is for a perfect gas, only with constant specific heats

for real gases a thermodynamic calculation must be used using the equation of state and correlations for the heat capacities.

attempts to "modifiy" the perfect gas result to account for real gas behavior is not very reliable and indeed misleading

 

[sailoday28]

hacksaw (Mechanical)and torpen (Structural)
For a perfect gas, either Cp or Cv is determined from the thermodynamic relation of Cp-Cv=R, Knowing for example, Cp, then one determines, Cv and k=Cp/Cv. Also for a perfect gas, it can be shown that for example, Cv is a function only of temperature. If Cp shows small varitation with tempearature, then k and Cv can be approximated as constant.

If the equation of state is PV=ZRT AND Z = CONSTANT
then Cp -Cv = ZR. Similarly for this case it is easily proven that Cv is a function of only temperature.

When one uses Z, then a real gas is being reasonably approximated. When one assumes Z= constant, they are putting further restrictions on the behavior of the gas.

My original question relates to how valid is a "choking" formula based on an ideal gas and constant k for other gases with an equation of state PV=ZRT, Z= constant.

Regards

 

[Latexman]

sailoday28,

I would expect results closer to reality, than assuming a perfect gas, when using z to correct for just density effects of a real gas.

I would expect results even closer to reality when using an EOS to correct for density effects and thermodynamic departures from ideal.

Good luck,
Latexman

 

[sailoday28]

Latexman (Chemical)I am in perfect agreement, except when the gas expands into the 2 phase region. In the two phase region, flow may not be homogeneous or in thermal equilibrium.



My original question relates to making sure we as engineers do not just "plug" into equations to get an answer.
Regards

 

[hacksaw]

my experience is that it is a poor approximation, but then how often have i used a wild guess in the absence of better information.

if you are wanting a self-consistent set of fluid property data that is developed on a sound basis then you have to model the real gas properties or use correlation data.

in the plant, i've only needed it where we needed consistency

 

[torpen]

Hello,

We consider a fluid as a mixture of perfect gases.
The inputs are Z, Cp, Cv and gamma.
These parameters are defined as follows:
Cp = Cv+Z*R
gamma = Cp/Cv
Cp/R = gamma*Z/(gamma-1)
gamma = 1/(1-Z*R/Cp)
They vary at each point of the flow.

The equations of state are written as:
p/rho = Z*R*T
h = e + p/rho
h = Cp*T = gamma/(gamma-1)*Z*R*T
e = cv*T = Z*R*T/(gamma-1)
The speed of sound a is equal to (gamma*p/rho)^1/2 = (gamma*Z*R*T)^1/2

You can use the shock equations to obtain donwstream conditions (rho2, p2, h2,v2 and teta2)
in function of upstream conditions, the shock angle teta1 and the density ratio rho2/rho1.

This ratio is unknown and defined as:
rho2/rho1 = gamma1/(gamma1-1)*gamma2/(gamma2-1)*h2/h1*p1/p2

The Mach number and the speed of sound are calculated with the following equations:
(a2/a1)^2 = gamma2/gamma1*p2/p1*rho1/rho2
(M2/M1)^2 = v2/v1*a1/a2

(Upstream and downstream conditions are denoted by suffixes 1 and 2 respectively)

You just have to check if the level of temperature T2 is valid
according the assumption of a mixture of perfect gases.

Regards,

Torpen.

 

[sailoday28]

torpen (Structural) Your derivations are based on constant Z
And clearly Cp/Cv will not be the same as that of the perfect gas with constant specific heats. In effect the subject referenced equation
Note for Z=constant, if one specific heat shows little variation with temperature, then both specific heats can be approximated as constant.
based on Wikipedia, Choked flow should be corrected.

Please also note, I am not advocating use of a constant Z, since in reality Z is a function of two variables such aa P and T.

Regards

 

[maxh]

Dear sirs,

I can't enter this interesting debate, but I have a question in this vein.

Can anyone say if you get a different sound speed in a pipe than you would in free field conditions for a gas at the same conditions.

Would the pipe dimensions with respect to frequency be important ?