Stagnation Conditions of a Two-Phase-Flow 2

[KingKoch]

Hello,

I'm trying to calculate the stagnation conditions of a water / steam - two-phase-flow in a horizontal constant diameter pipe.

The object is to calculate the stagnation pressure p_0 and mass flow quality x_dot_0 with given diameter d, mass flow m_dot, inlet pressure p_in and inlet mass flow quality x_dot_in.

I know how to calculate the stagnation mass flow quality x_dot_0 once I've calculated p_0. This ist done with an isentropic flash:

x_dot_0 = (s - s'_0) / (s''_0 - s'_0)


Entropies will be calculated via REFPROP, where s = s(p_in,x_dot_in), but s'_0 and s''_0 are dependent only from stagnation pressure p_0

So far I've found no real formula to calculate stagnation pressure conditions within two-phase-flows.

Hope there's someone out here, who can be of help.


Thanks a lot in advance !

 

[TiCl4]

Are you referencing stagnation pressure developed on a flat surface that is perpendicular to flow whereby flow continues (i.e. a pitot tube configuration)? Or are you referencing shut-off pressure, where, say, a valve is shut and you want to calculate pressure of the stopped flow?

If it's the former situation, there is no good calculation. Depending on the type of two-phase flow (wavy, annular, slug, etc) and the position of the flat surface, the flat surface will experience a variety of pressures as different proportions of liquid/gas impact against the surface. Two-phase flow is rarely steady (annular is probably the steadiest regime),

If it's the latter situation, your pressure will be determined by the source of your pressure - boiler, etc. If you are looking for a transient pressure spike, there won't be much of one with saturated steam because of the compressibility of the steam.

 

[KingKoch]

Hello TiCl4,

thanks for your reply. I was indeed refering to a stagnation pressure which would be measurable via pitot configuration.
I've figured there was no such thing as a exact calculation for two-phase-flow.

Yet I'm trying to understand an example I've found in some Literature. I don't know if you - or anyone here - is familiar with the "VDI - Wärmeatlas" ? It's some german standard literature and there's an examplary calculation of a twophase flow through a tube, which is divided into some segments. And in each segment the above mentioned stagnation conditions are calculated. But the example does not explain how this is calculated, nor is it giving any further literature references.

My best guess would be to consider the two-phase flow as a homogenoues two-phase flow (pseudo one-phase flow). There'd be an extended bernoullis equation to calculate stagnation pressure. But the results aren't even close to the given values in the example.

Best results so far I've gotten out of a calculation for pure, ideal gases. But to use this calculation doesn't seem right. So I had hoped that there'd be someone here who could help me with this.

 

[Latexman]

Stagnation pressure is indeed measured with a pitot tube, but I don't think a stagnation pressure has merit when two-phase flow is involved. In compressible flow, stagnation pressure is the static pressure a gas retains when brought to rest isentropically from it's flowing velocity/Mach number (M). In reality, a two-phase flow brought to rest will separate, and will no longer be the same two-phase mixture of interest. Also, liquid/solid parts of a two-phase flow are not compressible, and the thermodynamic relationships applied to the gas/vapor parts to calculate stagnation pressure will not apply to the liquid/solid parts. Forcing this calculation on the liquid/solid parts will not be rigorous, and depending on how much of the flow is liquid/solid initially, results may not be close enough to be practical.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[TiCl4]

I agree with Latexman here - I don't think you can get anything that will be close to practical with theoretical treatment. The closest thing I can think of, off-hand, would be to take your quality of the stream and apply the stagnation force to the pitot tube area (i.e. a stream with 10% liquid by volume would impact 10% of the pitot tube area with liquid and 90% with vapor). I.E. calculate the stagnation force for the gas and liquid separately using the vapor quality to assume an impacted area, add the forces from each section of the pitot tube, and divide by the total area.

However, you will find you get very different stagnation pressures between the vapor and liquid, meaning the higher pressure phase will tend to push out the lower pressure phase - enough to the point that the calculation mentioned above would be meaningless. This brings up Latexman's point about the two-phase flow separating upon stagnation and no longer being the same flow of interest.

Lastly, a physical test with a pitot tube isn't likely to measure accurately, either. If the flow is annular and the pitot tube is centered, you will likely only measure gas phase stagnation pressure. If you have wavy, slug, pseudo-slug, or other phases that have transients, your pressure reading will jump all over the place as various amounts of liquid/vapor impact the pitot tube.

 

[Latexman]

The closest thing I can think of, off-hand, would be to take your quality of the stream and apply the stagnation force to the pitot tube area (i.e. a stream with 10% liquid by volume would impact 10% of the pitot tube area with liquid and 90% with vapor). I.E. calculate the stagnation force for the gas and liquid separately using the vapor quality to assume an impacted area, add the forces from each section of the pitot tube, and divide by the total area.

Or, one could try to calculate average properties for the two-phase mixture and use those as a psuedo-component to estimate stagnation pressure. Though I'm not sure at the moment how to handle average Cp, Cv, and Cp/Cv with a liquid/solid and gas/vapor component.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[KingKoch]

Hello Latexman and TiCl4,

and thanks again for your contribution.

In compressible flow, stagnation pressure is the static pressure a gas retains when brought to rest isentropically from it's flowing velocity/Mach number (M). In reality, a two-phase flow brought to rest will separate, and will no longer be the same two-phase mixture of interest

Yes in the case of one-phase flows, either compressible or incompressible, the stagnation conditions can be achieved in an experiment (or could be, if it'd be possible to stop the flow isentropically)
Many calculations in one-phase flow depend on those stagnation conditions. So I kind of figured that there'd be an comparable (theoretical) approach to deal with this in a two-phase-flow. (And obviously the author of the beforementioned book has done so - I'm really considering contacting him about this subject)

However, you will find you get very different stagnation pressures between the vapor and liquid, meaning the higher pressure phase will tend to push out the lower pressure phase - enough to the point that the calculation mentioned above would be meaningless. This brings up Latexman's point about the two-phase flow separating upon stagnation and no longer being the same flow of interest.

Yes, I see that already when I compare my results the homogeneous two phase bernoulli stagnation pressure calculation and the one with an ideal gas. Relative increase in pressure when comparing stagnation pressure with static pressure is much higher in the case of a compressible flow.

Or, one could try to calculate average properties for the two-phase mixture and use those as a psuedo-component to estimate stagnation pressure. Though I'm not sure at the moment how to handle average Cp, Cv, and Cp/Cv with a liquid/solid and gas/vapor component.

which would be kind of my approach with a homogeneous two-phase-flow and extended bernoullis equation ? (where there would be no need of Cp and Cv either)

 

[Latexman]

Yes, though the HEM is conservative, it is not difficult to apply.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[hacksaw]

stagnation even for ideal gases is not necessarily isentropic and there is the stagnation temperature that must be addressed.

 

[KingKoch]

Hello Hacksaw,
thanks for your contribution.

stagnation even for ideal gases is not necessarily isentropic

The definition of stagnation conditions I've found is: "the conditions that occur, when an flow is brought to rest isentropically"
And the object is to perform an isentropic flash calculation. So I think thats the one thing, I can be really certain about.

and there is the stagnation temperature that must be addressed

Why does it need to be adressed ?
Of course I could use the stagnation temperature T_0 to calculate bulk properties of s' & s'' via s = s(T_0,x_dot=0 v 1), but thats just the same information as s = s(p_0,x_dot=0 v 1), isn't it ? Both p_0 and T_0 are unknown. But it's p_0 I'm looking for.

With known T_0 and x_dot_0, I'd be able to calculate my corresponding p_0 (and vice versa for that matter)
But unfortunately I don't know how to calculate T_0 either

 

[hacksaw]

Your fluid comes to rest on impacting the probe, with an "adiabatic increase in temperature" (in the absence of condustion) as the fluid stagnates and forms boundary layer flow.

This stagnation can result in a considerable temperature increase at extreme velocities. Granted it is a bit more complex when dealing with two-phase flows, given the various phase categories being considered.

 

[KingKoch]

Hello Hacksaw,

yes, that's all true. But this change in temperature should be covered in an equation of state via the joule thompson coefficient, should it not ? and thus s = s(p,x) is equal to the information content in s = s(T,x) (or any other thermodynamic state variable) ?

So what I don't understand about your post: What (new) information does the knowlegde of this stagnation temperature has in store for my problem ? Why is it neccesary to know this temperature ?

I guess it would be helpful, if you'd knew a way to calculate this temperature in order to then obtain the information of interest: the stagnation pressure p_0 ?

 

[hacksaw]

KingKoch, you can easily calculate the stagnation for steam, but with two phases, things get complicated. There are a half dozen forms of multiphase flow, so you have to identify which one is causing the greatest concern.