Sizing of orifice? 2

[ddkm]

Hello all. I'm a newbie here, searching for opinions on orifice sizing.

For a vapor through a hole (in this case, saturated steam), I was recommended the following equation (from Crowl/Louvar):


Qm (choked)
= Co A Po SQRT{(k gc M / Rg To)*(2/k+1)^[(k+1)/(k-1)]}

(Equation is provided in the old units, not SI)

Then you vary the orifice diameter - which changes the cross sectional area A - and thus, changes the maximum flow Qm (choked). Keep varying the diameter til you get the max flow that is required.

Is this correct? I'm asking because I'm getting a very low calculated result compared to the expected result. Example I'm getting only 11kg/hr instead of, say, 1100kg/hr.

 

[mbeychok]

ddkm:

Your equation has an extra g in it. This is the correct version for the indicated USA engineering units:

Qm (choked)
= Co A Po SQRT{(k gc M / R To)*(2/k+1)^[(k+1)/(k-1)]}

where:
Qm = lb/s
Co = discharge coefficient = approximately 0.72
A = ft[sup]2[/sup]
gc = gravitational conversion constant = 32.17 ft/s[sup]2[/sup]
k = c[sub]p[/sub] / c[sub]v[/sub] of the gas
M = molecular weight of the gas
R = 1545.3 (ft-lb) / (lbmol)(°R)
Po = absolute upstream gas pressure, lb/ft[sup]2[/sup]
To = upstream gas temperature, °R

(Note that the pressure is in pounds per square foot rather than pounds per square inch)
---------------------------------------------------------
And here is the same equation for the indicated SI units:

Qm (choked)
= Co A Po SQRT{(k gc M / R To)*(2/k+1)^[(k+1)/(k-1)]}

where:
Qm = kg/s
Co = discharge coefficient = approximately 0.72
A = m[sup]2[/sup]
gc = gravitational conversion constant = 1 (kg-m) / (N-s[sup]2[/sup])
k = c[sub]p[/sub] / c[sub]v[/sub] of the gas
M = molecular weight of the gas
R = 8314.5 (Pa)(m[sup]3[/sup]) / (kgmol)(°K)
Po = absolute upstream gas pressure, Pascals
To = upstream gas temperature, °K
----------------------------------------------------------
I hope this helps you,

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

 

[Latexman]

ddkm:

I'm at home and my references are at the office, but I'll take a stab at it. Is R[sub]g[/sub] a gas constant that is specific to a certain gas? In other words, it has the molecular weight built into it. If so, I don't see why you'd need M in the equation.


Good luck,
Latexman

 

[Latexman]

ddkm:

Oh yeah, give us your data to look at. That may spark an idea in someone.

Also the term you wrote as (2/k+1) is most clearly written as (2/(k+1)).


Good luck,
Latexman

 

[ddkm]

Wow, you guys are really fast and helpful. Let's keep the momentum.


Mbeychok:
The equation is virtually the same. The universal gas constant was quoted as Rg, where g is a subscript. (i don't know how to do a subscript in the text).

More importantly, your comment of: "the pressure should be in lb/ft2 instead of lb/in2" is precisely the missing link! Darn, I hate (sorry) to use the non-SI units and again, here it has caused me some calculation error. I've redone the calculation and the result looks much more reasonable! I'll show the calc in the next reply.

Even better, you have attached an SI-unit equivalent of the equation, which I was looking for. I half-suspected that it would be virtually the same, with the exception of gc and R constants. In particular, I was doubtful that we could just replace the gc as 1 for the SI units. But thanks anyway.

 

[ddkm]

Latexman:

- Thanks for the heads-up: yeah, it should be 2/(k+1)

- I think the M (molecular weight) is necessary because the PV=nRT equation applies to a molar basis (i.e. where "n" is represented and R is quoted as 8.314J/molK). As such, since we need the calculation of the max flow to be on a mass basis, therefore the M comes into play.
Alternatively, we could look at n=m/M which gives PV=mRT/M and somehow this equation must have been fitted into another equation.

 

[ddkm]

Let's look at the calculation I've done:


I'm trying to size an orifice for a high-pressure steam line with the purpose of restricting the flow (for safety reasons).

Data is as follows:
Steam supply
P = 75barG
T = 291ºC (saturated steam, extrapolated from tables)
k = Cp/Cv = 1.28 (I've used 1.28 based on someone else's data. It's usually around this figure or up to 1.35, I think)
M = 18 for Steam (H2O)

For the coeff of discharge, I've used 0.61 as the book recommends this number for sharp-edged orifices (although we are dealing with annular orifices here) so:
Co = 0.61

Objective is to restrict the flow to a maximum of 1700kg/hr.


Solution:
=========

Using the equation and converting to non-SI units, we get:

Co = 0.61
d = 0.01 m = 0.0328084 ft
A = = 0.000845396 ft2
Po = 75 bar(G)
= 76.00 bar(A) = 1102.2888 psia
= 158729.5872 lb/ft2
k = 1.28
gc = 32.17 ft lbm/lbf s2
M = 18.00
Rg = 1545 ft lbf/lbmole ºR
To = 291.00 ºC = 1015.8 ºR


Note: Using a spreadsheet, I've varied the diameter of the orifice to compute the Qm (choked). With a figure of 10mm or 0.01m for the , this gives a Qm (choked) of about 1.043lb/s or 1704kg/hr.

Any comments?

 

[Latexman]

ddkm:

Yes, Crane Technical Paper 410 has a section of gas properties that uses R[sub]g[/sub] as a gas constant that is specific to a certain gas. As you indicated, instead of PV=nRT they use PV=mR[sub]g[/sub]T where R[sub]g[/sub] = R/M.

To learn how to use subscripts and lots of other neat stuff, click on "Process TGML" above the "Submit Post" button.



Good luck,
Latexman

 

[ddkm]

Latexman: Thanks.

Wow, "process TGML" to do some kind of formatting. Well, as long as it gets the job done.

Latexman: Any comments on the calculation shown? I've used Co of 0.61, whereas Mbeychok had used 0.72. It may make some difference to the computed result, so just wondering what is the basis for the difference?

 

[ddkm]

Using the SI equivalent, I've redone the calculation as follows:

Co = 0.61

d = 0.01 m

A = 7.85398E-05 m2

Po = 75 bar(G)

= 76.00 bar(A) = 7601325 Pa

k = 1.28

gc = 1 kgm/Ns2

M = 18.00

Rg = 8314 J/kg-mol K

To = 291.00 ºC = 564.00 K


Solution:
========
The only difference in the basic equation between the two non-SI and SI equations are the values of gc and R, where gc = 1 and R = 8314 (adjusted for kg-mol instead of the usual g-mol units) in the "SI" units. Therefore, using virtually the same equation and orifice diameter of 10mm, I get:

Qm (choked) = 0.4735 kg/s i.e. 1704 kg/hr (same answer)


Thanks a lot, guys. This has been really helpful and given a higher level of confidence to my calc. Hope you guys were enlightened too.

 

[mbeychok]

ddkm:

Having done the same cross-checking (by using the equation in USA units and the equation in metric units) at least a hundred times in the last few years, I know that the two equations I gave you in my earlier response provide equivalent answers. I am glad to see that you found that out as well.

I have only one other comment. There are those who would say that C, the coefficient of friction, should be much higher than my 0.72 ... which I chose as being conservative. Therefore, I question your use of C = 0.61. What it really boils down to is how important accuracy is to you. If the mass flow through the orifice size you selected turns out to be higher than you calculated by using 0.61, will it be detrimental to what you are trying to do? Only you can answer that question.

With best regards,

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

 

[ddkm]

mbeychok: yeah, found that out by actually doing the calculation itself. It's just that I couldn't imagine that the g[sub]c[/sub] could so conveniently equate to 1.0 in the SI (metric) environment. When something looks too easy, well, you easily get doubts.

For the question on value of C[sub]o[/sub], obviously it could be critical since the calculation is done on purposes of safety. I guess I'll look around for more interpretation of this coefficient.

By the way, how is it that you call it coeff of friction? In the literature, they simply call it "discharge coeff"

 

[Latexman]

ddkm:

I got about the same answer with your numbers.

Perry's Handbook speaks to flow coefficients of annular orifices and has references.


Good luck,
Latexman

 

[mbeychok]

ddkm:

My copy of Perry's is the 6th edition and the pages I will cite in this message are from that edition.

(1) Page 5-3 defines the gravitational conversion constant g[sub]c[/sub] (which Perry's calls the dimensional constant) to be 1 (kg-m)/(N-s[sup]2[/sup]).

(2) Page 5-3 also correctly makes a distinction between the gravitational conversion constant g[sub]c[/sub] and the local gravitational acceleration g. That is needed because some equations (other than the choked flow equation we are discussing) involve both g[sub]c[/sub] and g.

In the SI metric units, g[sub]c[/sub] = 1 (kg-m)/(N-s[sup]2[/sup]) and g = 9.807 m/s[sup]2[/sup]. In the customary USA units, both g[sub]c[/sub] and g have the same numeric value of 32.17 ft/s[sup]2[/sup].

(3) Page 5-16 discusses the case of critical flow (i.e., choked flow) through a square-edged or sharp-edged orifice. It states that if the pressure ratio r, which Perry's defines as P[sub]2[/sub]/P[sub]1[/sub], is at the critical pressure pressure ratio of r[sub]c[/sub] (i.e., at choked flow), then the discharge coefficient is "about 0.75". Perry's also states that if the downstream pressure P[sub]2[/sub] drops below that corresponding to the critical pressure ratio, the coefficient of discharge will increase ... and as r[sub]c[/sub] approaches zero, the coefficient of discharge increases to "about 0.84".

(4) I would like to encourage you to use R (without any subscript) and to always make sure to define it as the universal gas constant. That is because many authors in the technical literature often use a gas constant that is only applicable to a specific gas molecular weight. I call that specific gas constant R[sub]s[/sub] where the subscript s denotes "specific". As pointed out by Latexman, R[sub]s[/sub] = R/M. Unfortunately, those same authors often do not explain which gas constant they are using and that leads to much confusion.

(5) You might find it useful to visit www.air-dispersion.com/msource/html and read the section on "Gas Discharge To The Atmosphere From A Pressurized Source Vessel" using SI metric. The same section is available in the customary USA units at www.air-dispersion.com/usource/html.

Please excuse me for being so long-winded but I thought you might find the above to be useful.

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

 

[sailoday28]

FYI
For those use the Qm choked equation , American units

The the product PoA in the referenced formula has to be in lbs units. Po can be psia as long as A is in square inches.

Also note, that Po has to be stagnation and not static upstream pressure and more important

The referenced formulas are for perfect gas, constant specific heats.
For choked flow of saturated steam, which will pass thru wet states, I would refer to the graphs for choked flow that are illustrated in the ASME steam tables (at least they were illustrated in the past)

 

[Latexman]

ddkm:

If there is significant pipe and fittings in the high-pressure steam line, that can restrict the flow too, especially if the velocity in the pipe is quite high during maximum flow. Some folks like to ignore the piping when sizing a restriction orifice, but I've seem a few cases when you can't.

For reason #3 in Milton Beyckok's post above and to make the flow restriction a little more difficult to change in the field, the last flow restriction I put in was a spool piece with a reducer, 1 foot length of smaller piping and an expander. This acted as a "thick plate orifice" and once the critical pressure ratio was reached, the flow did not increase if downstream pressure decreased further.


Good luck,
Latexman

 

[katmar]

Forgive me if this comes across as a bit of a rant, but one of my buttons has been pressed here.

ddkm has expressed surprise that the value of g[sub]c[/sub] turns out to be exactly 1.0 in SI units. We need to look at this in more detail because the units are even more surprising than the value.

mbeychok has quoted from Perry that the units of g[sub]c[/sub] in SI units are (kg.m)/(N.s[sup]2[/sup]). But N is the unit of force, and force is mass x acceleration, so we could express N as kg.m/s[sup]2[/sup]. These units all cancel out and low and behold, not only does g[sub]c[/sub] have a value of 1.0, but it is dimensionless!

What is the point of including a factor that has a value of 1.0, and has no units???

This is a perfect example of the problems involved in converting an equation from one set of units to another, which was described by zdas04 in thread378-134079. The problem here is that the SI equation referenced by mbeychok has been derived by converting the Customary Units equation, instead of from first principles.

I have found that very few engineers really understand what g[sub]c[/sub] actually is.

In the English system of units (also known as the foot-pound-second system) the unit of mass is the pound, and the unit of force is the poundal.

In the British Engineering system of units the unit of mass is the slug and the unit of force is the pound force.

Unfortunately in the US Customary system of units they have used the unit of mass (i.e. the pound) from the English set of units and the unit of force (i.e. the pound force) from the British Engineering system of units.

But
1 pound force = 32.17 poundals, and
1 slug = 32.17 pounds (mass)

So, if you mix these systems of units you have to include a conversion factor, which is the origin of the much misunderstood g[sub]c[/sub]. It is ONLY in US Customary units that there is any need for g[sub]c[/sub] and in every other consistent set of units the need for g[sub]c[/sub] simply does not exist.

 

[sailoday28]

katmar (Chemical)states".....So, if you mix these systems of units you have to include a conversion factor, which is the origin of the much misunderstood g."
Which leads misunderstanding and use of lbs in force and mass. I have called the British system, the American system. Weren't the Brits smart enough to stop using it?
I believe those of us in the US are the only ones still using a complicated system.

Go SI

 

[mbeychok]

katmar:

In principle, I agree wholeheartedly with you. If you will take a few minutes to visit www.air-dispersion.com/msource.html and read the section on Gas Discharge To The Atmosphere From a Pressurized Source, you will see that I have indeed excluded the g[sub]c[/sub] from the SI equation for choked flow.

But, like it or not, this forum and many similar ones still have participation from a great many older engineers who live in the USA and who still do not use the SI metric system. That is why the originator of this thread obtained the choked flow equation (that used USA units) from the textbook written by Crowl and Louvar who are professors in USA universities.

Until the day that the USA finally decides to go fully metric, we have to face the fact that people will have to cope with conversion of units from one system to another. I might add that we will also have to cope with the fact that, even in the metric countries, some people express pressure in Pascals, some use kg/cm[sup]2[/sup], and some use bars.

Milton Beychok

 

[katmar]

Milton,

I meant no offense to you, and I apologise unreservedly if I came across as giving that impression. I have the huge advantage of having been educated and employed in South Africa over the last forty + years, during which time we moved from English units to Metric units and finally to SI units. I believe this has given me a perspective that would be hard to obtain for an engineer operating "inside" the old system, or even for an engineer who has grown up "metric" in Europe.

I still believe that it is an abomination that a book like Perry lists g[sub]c[/sub] with the units given above. Their mistake perpetuates the lack of understanding amongst practising engineers. I appreciate the fact that people like you are able to overcome the habits of the past, in the way that you have given this formula without the g[sub]c[/sub] in the SI version. It is time mainstream engineering caught up.

In some ways the Europeans have it worse than the Americans because with their "metric" systems they were always halfway there, and it seems this gave them even less motivation to drop the old ways than the Americans. As you say, the Europeans still use a variety of units, and that is probably why over 100 people download my Uconeer units conversion program every day.

The SI system is a quantum advance over the metric system (actually there were several metric systems) and I welcome the progress, slow as it is, that is being made in accepting this excellent tool. An indication that progress is being made is that I see in my web logs that people get to the Uconeer web site by typing "How many inches in a foot" into Google. Hard to believe, but some people do not know this anymore!

regards
Harvey