Sonic Flow - Density Determination 4

[kevllt]

The sonic flow calculation is not something I do very often so I'm certainly not an expert. I read through the section of the Crane book that discusses this issue. The texts that address this issue are:

"Further decrease in the outlet pressure will not be felt upstream because the pressure wave can only travel at sonic velocity, and the "signal" will never translate upstream. The "surplus" pressure drop obtained by lowering the outlet pressure after the maximum discharge has already been reached takes place beyond the end of the pipe."

"This [sonic] velocity will occur at the outlet end or in a constricted area, when the pressure drop is sufficiently high. The pressure, temperature, and specific volume are those occurring at the point in question."

From this I would expect that the pressure at the point of sonic flow is somewhere between the supply pressure and the MAWP of the tank. I'm not sure how to find this "sonic" pressure value to use for evaluating the density.

To address my problem specifically, I need to determine the relief rate of a conservation vent in an Atmospheric Tank in the event the internal heating (40# steam) coil rupture. The saturated temperature of the 40 psig steam will not vaporize the chemical in the tank. It is assumed that at the coil tube first breakage, steam pushes pad gas out at the same volumetric flowrate as the steam flow from the broken tube. The required relief rate is assumed to be limited by the sonic velocity flow of the 40# saturated steam through the 1.5" inlet heating coil line.

Ultimately, my question is what sonic flow conditions should we use to evaluate the density of the fluid (in my particular case, the 40 psig steam) at?

Thanks.

 

[mbeychok]

kevllt:

I am not sure that I understand your question precisely. However, I am quite sure that, if you will read faq798-1196 <== click here, you will find a very clear definition of when gas flows become choked and how to calculate the mass flow rate of a gas under choked flow conditions.

Milton Beychok
(Contact me at www.air-dispersion.com)
.

 

[kevllt]

Thanks Milton. We already have a program that calculates the sonic velocity flow rate. However, I am just having difficulty determining what density to use to calculate the relief flow through the coil tube rupture. If I use the density of the saturated steam at 40 psig pressure, 0.13 lb/ft^3, it would result in a way too conservative estimation of rupture rate, so I was just wondering at what conditions do I evaluate the density to determine a rupture rate which most likely reflects reality. My idea is to use the density of the 40# saturated steam evaluated at relief conditions, 0.036 lb/ft^3. Is this a valid assumption?

 

[mbeychok]

kevllt:

If the pressure inside the tank is at about 22 psia or less (i.e., about half the pressure of the 45.7 psia steam), the steam flow through the rupture will definitely be at choked conditions ... and the steam density you must use in determining the mass flow rate of the steam should be calculated at the pressure and temperature of the 45.7 psia steam upstream of the rupture (again, read that FAQ I referenced earlier).

You haven't described your "Atmospheric Tank" to us. You also haven't told us what pressure your conservation vent valve is set at nor have you told us what the MAWP of your atmospheric tank is. If the conservation vent valve is set at about 22 psia or more, then steam flow through the rupture will not be at choked conditions.

However, if your tank is an atmospheric, vertical cylindrical tank with a flat or conical roof, then it certainly does not have an MAWP or a conservation vent valve set point of 22 psia or more. Such tanks are only good for a few psig at most. The only way that your tank could have an MAWP of 22 psia or more is if it is a pressure vessel with hemi-spherical or semi-elliptical heads.

So what we need from you is more information about the tank and what internal pressure it can withstand.

You also said earlier that the liquid in the tank would not be vaporized by the steam. However, you should consider whether that liquid will cause the steam to cool and hence condense some of the steam ... at least initially.

If you would like to talk to me about this by telephone, you can find my phone number on my website and keep in mind that I am on Pacific Time.

Milton Beychok
(Contact me at www.air-dispersion.com)
.

 

[kevllt]

Thanks again Milton. Sorry for leaving ut the details of my Atmospheric Tank. You are right on the money with your statement that my tank is cone roofed with an MAWP < 22psia. The tank has dimensions of 12 ft Dia x 24 ft high. Well the conservation vent is set at 8 in. WC, while the MAWP of the Tank is 26 in WC. Yes, I have considered the steam to condense initially into the liquid, which would push out an equal amount of pad gas initially, which would be the controlling case since the air equivalent conversion will result in a greater relief load from relieving the initial pad gas. However, my problem right now is determining the required relief rate in the event of coil rupture. In my opinion, if we use the density of the upstream 40 psig saturated steam, 0.13 lb/ft^3, this would result in a way too conservative estimation of rupture rate, since the coil rupture would most definitely not rupture the full size of the steam coil. I suggest we use 40# saturated steam conditions evaluated at relief pressure (15.63 psia) to be used as the density requirements, which was simulated to be approx. 0.036 lb/ft^3. Honestly, I am not sure how practical this idea is, so I would appreciate your input as well.

Thanks again.

 

[sailoday28]

kevllt (Mechanical)If your source is at stagnation conditions then critial steady state flow will occur when the downstream pressure is about .58(sat stm press) or
31.9 psia or about 17.2psig. At higher back pressue, the flow will not be choked.
The above is based upon isentropic steady flow. I do not know the configuration of your piping. If it is a relatively long pipe rupturing at one end and fed by the saturated steam source at the other end, the other things must be considered.
For steady state flow, friction must be accounted for.
If transient conditions and a long pipe are to be considered, you must supply additional info. For example with a sudden rupture at one end, choked flow M=1 (at the rupture)will require much lower back pressures.

The 0.58 multiplier is that recommended by Spirax-Sarco to determine critical flow press ratio for a sat steam source.

 

[mbeychok]

kevllt:

There simply is no way to justify using the density of steam at the tank's relief valve set point pressure. It would be more reasonable to justify assuming that the coil does not completely rupture ... instead, it has a tear with an area equivalent to a round hole of ??? inches diameter. Just choose whatever size hole you deem to be realistic, calculate the choked mass flow rate of steam through that hole (using the density of steam at the upstream conditions), and use that as the amount that the conservation vent must relieve.

Other than that, I have no other suggestions.



Milton Beychok
(Contact me at www.air-dispersion.com)
.

 

[TD2K]

Kevllt, if you take a look at examples 4.20 and 4.21 in Crane, I think you will find they explain the 'sonic' or chocked pressure quite well.

Basically, as a gas flows through a pipe, its pressure decreases. As its pressure decreases, it's density also decreases and therefore its velocity has to increase to keep the same mass flow. The increase in velocity further increases the rate of pressure change in the gas. At some point down the pipe, the gas velocity reaches sonic and at that point, further reductions in the outlet pressure does not get you any more flow. For the cases where you have a pipe flowing to atmosphere/lower pressure system you will be choked at its outlet. The pressure at the outlet where you have sonic flow in the pipe is a function of the inlet pressure but it's always less of course than the inlet pressure. Crane goes through how to estimate this if you want to but it's not necessary using their formulas, those use the inlet pressure/density.

In your case, it sounds like you have a steam header system in the tank farm supplying this particula tank via a lateral and you have a coil in the tank.

Assuming you have a coil break, that you don't get backflow from the condensate system (or minimal backflow) and that the coil breaks right at the inside wall, the flow of steam into the tank will depend on the piping configuration to the tank. Depending on the sizes involved, you may find that the coil acts as a significant restriction to the potential flow from the steam system (say a 4" lateral with a 1" coil) or that the lengths or piping involved in your tank farm themselves restricts the flow to a reasonable value. Unfortunately, on the other hand, you might also find that the calculated maximum steam flow is very, very large.

If you find out that it's not practical to protect against this due to the flows, you might want to look at a buckling pin valve. I know they offer a shut-off valve exactly for this service: to prevent a vessel from being overpressurized due to the failure of a high pressure coil.

 

[ddkm]

The 0.58 multiplier is that recommended by Spirax-Sarco to determine critical flow press ratio for a sat steam source.

This multiplier is actually derived from the equation for critical flow behaviour. Many literature quotes this - as an example, I take from API520:

Sizing for Gas or Vapor Relief - Critical Flow Behavior

The absolute pressure ratio of the pressure in the throat at sonic velocity (Pcf) to the inlet pressure (P1) is called the Critical Pressure Ratio.

Under critical flow conditions, the actual pressure in the throat cannot fall below the critical flow pressure even if a much lower pressure exists downstream. At critical flow, the expansion from throat pressure to downstream pressure takes place irreversibly with the energy dissipated in turbulence into the surrounding fluid.

Equation for the ratio:

Pcf/P1 = [2/(k+1)]^[k/(k-1)]

where:

Pcf = critical flow throat pressure (absolute)
P1 = upstream relieving pressure (absolute)
k = ratio of specific heats for any ideal gas

Using the above equation, and a value of k=1.3 for saturated steam, we get:

Pcf/P1 = 0.546

Unfortunately, I'm unable to find a reliable source of data for the various values to be used for k, but I would imagine that it would not be too far off.

---engineering your life---

 

[mbeychok]

ddkm and all others:

Please click here ==> faq798-1196 and read the FAQ. You will find that it includes all of the equations relevant to the mass flow rate of an ideal gas through an orifice or leak at choked conditions. It also has the equation that defines the ratio of upstream pressure to downstream pressure required to achieve choked flow (i.e., the critical pressure ratio). The above referenced FAQ is in this Chemical Process Engineering Forum.

Milton Beychok
(Contact me at www.air-dispersion.com)
.

 

[katmar]

kevllt,

What you have to decide is whether you want to cover the most likely case, or the worst case. In order to focus your mind, imagine your tank has blown up killing three people and you are now standing in the dock explaining to the judge why you designed for a small tear in the coil half way along its length instead of assuming that the coil broke off completely right at the entrance.

In my opinion, you must assume the worst - the coil has broken off completely right at the tank wall, and you have the full steam pressure and density.

It may mean that you have to install a larger vent on your vessel, but as the design engineer you are responsible for making it safe.

In similar situations, I have seen a simple water seal used to cover this eventuality. Use your (small) conservation vent to cope with the normal breating required as the tank is filled and emptied, and then rely on the water seal to blow if there is a coil rupture. Of course, if water is not compatible with your tank contents you could use an oil of some sort.

regards
Harvey

 

[mbeychok]

Harvey (aka Katmar):

You deserve the star I just gave you for summarizing kevllt's options so succinctly and clearly.

Milt Beychok

P.S.: Just one small point ... unless kevllt has a background in British law, he may not understand that "standing in the dock" means standing in a court of law. :-D

 

[sailoday28]

mbeychok (Chemical)
The 0.58 approximation is based on a gamma of 1.135 as recommended by Spirax-Sarco.

With reference to the text"Thermodynamics, Principals and Applications to Engineering", By Dr. Ernst Schmidt-translated by J. Kestin Dover Publications

Pg 208 "......The isentropic expansion of wet steam above x=0.75 can be represented by .....PV^n = ..const.. in which the exponent depends on the dryness fraction x at the beginning of expansion, according to the equation
n=1.035+0.1x......"

The paragraph goes on ending with
"In practice it is more convenient and more accurate to use steam tables or charts"
I don't have the most recent edition of the ASME steam tables, however, in the past, a simple graphical chart provided choked mass flux, (lbs/time/unit flow area ) for choked flow based on upstream conditions.

It seems to me that you and other engineers are resistant to using the steam tables which would provide information to solve choked steady state (isentropic) problems for steam flows. The only stipulation is that the mass flux is for homogeneous flow.

With regard to my previous discussion on the use of perfect gas, const gamma (note: Cp and Cv being undefined in the wet region)--
While one has an approximation of Pv^gamma, its use in determing flow also depends upon the variation of enthalpy during the isentropic expansion.
Stag enthalpy = static enthalpy + KE for the adiabatic process.
Ho =H +V^2/2 or Ho= H +G^2(spc vol)2/2
Where G is the mass flux.
If a simple relation of wet steam H vs press combined with the spc vol as a function of press (pv^n=const), then with substitution in the above formula the G may be calculated for no-choked and choked flow. Use of ASME steam tables for choked flow is easier.

If with use of the gammas that you recommend, the mass flux is in reasonable agreement with ASME or other documented flows or measured flows, then I would accept your recommended formulations.
As I have also stated in the past, the steady state homogeneous (wet) flows can also be higher than calculated flows. I leave that topic for another discussion.

 

[mbeychok]

Sailoday:

Thanks for the elucidation in your last response. I have re-read all of my contributions to this thread and I did not recommend any gamma values. All of my contributions to this thread dealt with telling kevllt to use the upstream pressure to determine the steam density required in the choked flow equation for gases.

The FAQ798-1196 that I asked kevllt to read says that, for "many gases", [(k+1)/2]^k/(k-1) ranges from 1.7 to 1.9 ... which translates to a gamma value range of 0.59 to 0.53 which certainly includes your value of 0.58 with which I have no quarrel.


Milt Beychok
(Visit me at www.air-dispersion.com)

 

[mbeychok]

Sailoday:

In my last response to you, I said:

which translates to a gamma value range of 0.59 to 0.53 which certainly includes your value of 0.58 with which I have no quarrel.

That sentence should have read:

"which translates to a pressure ratio range of 0.59 to 0.53 which certainly includes your value of 0.58 with which I have no quarrel."

Please excuse my typo.

Milt Beychok

 

[denniskb]

Please note that the method proposed by FAQ 798-1196 is not accurate for thin sharp edge orifice plates. Experiments done by RG Cunningham in 1951 clearly demonstrated that these devices to not choke down to P2 = 0.1 x P1. I am currently preparing a paper and software which proposes an accurate way to determine this flow.

The method proposed however works for flow from a pipe end or through a thick orifice (t > 6 x d) and so is suitable for the case being discussed here.

As suggested by several contributors if you do have a critical flow case then the density should be determined immediately upstream of the sonic flow point and this will be the source pressure less friction. The Crane method assumes low friction (<10% of P1) so uses the upstream density. An iterative solution is required if dP is high as the density changes and so do does the velocity of sound. As suggested by Katmar assume the worst case full pipe rupture and location of break in the coil where it enters the tank. Let someone else stand in the dock!

Dennis Kirk Engineering

http://www.ozemail.com.au/~denniskb

 

[sailoday28]

denniskb (Mechanical)Kudos to you for bringing up Cunningham. He has been referenced and commented upon in Crane in the past. However, in the recent editions, only the reference appears.
Is it possible to get a copy of the original paper from you?
Regards