Multiple RO's in series
Multiple RO's in series
(OP)
Hi!
I've been asked to check the design of a line with three restriction orifices in series. That line is connecting the discharge of a recycled H2 compressor discharge line at 2350 psia to a 14.7 psia gas seal drain pot. The line is 0.434in ID and each RO has a bore of 0.1in.
I'm a bit lost. Sonic dP for the first RO would be 1107.4psi with a Q of 480.4SCFM. I won't be able to handle 480.4SCFM in the next RO as the dP would have to be much higher than the sonic dP. So I need to do this backwards, starting with the last RO. Should I assume sonic dP for the last RO, calculate the flow and upstream pressure and use this upstream pressure as the downstream pressure for the 2nd RO... all the way to the first one and try to reach an upstream of 2350 psia?
Is that correct?
Thanks in advance!
I've been asked to check the design of a line with three restriction orifices in series. That line is connecting the discharge of a recycled H2 compressor discharge line at 2350 psia to a 14.7 psia gas seal drain pot. The line is 0.434in ID and each RO has a bore of 0.1in.
I'm a bit lost. Sonic dP for the first RO would be 1107.4psi with a Q of 480.4SCFM. I won't be able to handle 480.4SCFM in the next RO as the dP would have to be much higher than the sonic dP. So I need to do this backwards, starting with the last RO. Should I assume sonic dP for the last RO, calculate the flow and upstream pressure and use this upstream pressure as the downstream pressure for the 2nd RO... all the way to the first one and try to reach an upstream of 2350 psia?
Is that correct?
Thanks in advance!





RE: Multiple RO's in series
W/A =const. (Po/To^.5)
Further assuming adiabatic flow
Then To remains constant.
You have stated orifice area and SCFM remain constant. Thus W/A remains constant.
The only remaining factors are effect of the Po and orifice coefficent.
--if coef of disch does not vary significantly---- then Po must remain constant.
Po upstream of each orifice cannot remain constant.
RE: Multiple RO's in series
RE: Multiple RO's in series
If your situation is as Hacksaw noted, then its probably best to check with the compressor vendor. Otherwise, if the system was designed in-house, start with trying to get the orifice spec sheets, maybe they can help in your analysis.
Off hand, I can't think of a direct way to solve for the flow capacity and would resort to an iterative approach as you indicate. I don't think it matters which end you start with, just keep adjusting the intermediate pressures (or pressure drops) until you achieve a mass balance between each orifice and can satisfy the overall pressure drop (2350 - 14.7).
I think what you will find is that you may initially have sonic flow through your first then second orifice but as flow is established completely throughout the system, most likely the last orifice will be in a sonic flow condition and the 2 upstream orifices will be sub-sonic. I would take that approach and then test for sonic flow conditions at orifices 1 & 2 during your iteration and adjust as needed.
RE: Multiple RO's in series
So then you just need to adjust the flow rate until you get the desired value for the initial pressure (2350 psia). This is easily done in Excel for example.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
RE: Multiple RO's in series
You could also calculate the equivalent K of each orifice, add them up and then do a separate check using the method in Crane.
It would be a reasonable cross check against a trial and error calc for each orifice to determine the intermediate pressures that gives the same flow rate through each orifice.
RE: Multiple RO's in series
RE: Multiple RO's in series
The temperature in the pipe will be constant (except in the vicinity of each orifice). This means that the speed of sound is the same for each section--in other words, if any orifice is choked, the velocity of gas going through it is a certain number that depends only on the one temperature which is found in all four sections of pipe. But the mass flow rate depends on both the velocity and the density (and the orifice area which doesn't vary). So to have the same mass flow rate in all three, the velocity has to be less where the density is higher. So only in the low-pressure orifice can the velocity be sonic. In the others, the density is higher so the velocity has to be sub-sonic.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
The stagnation temperature may be constant, but in general, the fluid temperature and thus the gas density at each stage is not. That is just my biased and humble opinion.
Is your argument restricted to the case for three identical restrictions? There are too many examples of multiple criticals in high pressure let-down to believe that critical flow is only possible in the last restriction.
RE: Multiple RO's in series
I didn't say that the gas density is the same at each stage. It is definitely not constant. (In fact that was part of my argument.)
My argument was based on the idea that the three orifices are identical. But it also holds if they are almost identical, or (obviously) if the upstream ones are bigger! The only way you could have multiple critical flows is if the upstream orifices are considerably smaller.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
http://Eric.Kvaalen.com
RE: Multiple RO's in series
that makes better sense...you are right the density issue was an oversight
RE: Multiple RO's in series
I don't believe one is analyzing an isothermal process.
If the analysis is for adiabatic steady flow, then the stagnation enthalpy is constant. For perfect gas, const. specific heats, it follows that the stagnation temp is constant and the static temp and sound speed will vary from source to exit.
RE: Multiple RO's in series
The speed of sound is of course a function of the temperature. But what matters for my argument is not the speed of sound in the pipe, but rather the speed of the gas at the point where it reaches its own speed of sound (in the orifice). The speed of the gas at this point (one can call this the point where the Mach number becomes 1) is less than the speed of sound in the pipe because the gas will be cooler at this point. But I maintain that the speed of the gas at this point depends only on the stagnation temperature of the gas. (I am assuming that the speed of sound depends on temperature but not on pressure.)
Think of the gas moving through a pipe that is getting slowly narrower. The speed of the gas starts very low but goes up and up until it reaches sonic velocity. The velocity at this point is a function of the temperature at the beginning, where the gas is moving slowly--in other words, the stagnation temperature. It's not important what the temperature is at some intermediate point. (I am not saying that the speed of the gas when it reaches sonic velocity is the same as the speed of sound at the stagnation temperature, but it is related to it.)
So that's why I say that in any orifice along a pipe where the velocity becomes sonic, that velocity is the same value. And therefore, the mass flow is proportional to the area of the orifice and the density. And therefore, if all the orifices have the same size, only the one where the density is lowest can be sonic.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
well said
RE: Multiple RO's in series
http://Eric.Kvaalen.com
RE: Multiple RO's in series
For an isothermal process, the choking occurs not at M=1, but at M= 1/sqrt(gamma) where gamma= Cp/Cv
RE: Multiple RO's in series
But I hope you understand that I wasn't referring to any such scenario when I spoke about the temperature being (more or less) constant except in the vicinity of the orifices. In the orifices the temperature definitely goes down as the gas reaches high velocity, but then the temperature pops back up as the high-speed gas slams into the slowly moving gas on the downstream side of the orifice. Its kinetic energy is converted back into internal heat.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
Similarly, For adiabatic flow with choking the pipeline also need not be long and narrow. Place an orifice and have the needed pressure drop and choking will occur.
For isothermal flow of perfect gas, const specif heats, the heat input (+ or -)needed \
is
Q/a^2= 0.5(1/gamma- Mi^2) where a= sonic speed
gamma =ratio of specific heats
Mi= upstream Mach number.
Place an orifice, add or deduct the necessary heat input and choking will occur at M=1/sqrt(gamma)
RE: Multiple RO's in series
But achieving the isothermal case would be tricky. If you simply add the required heat before the orifice, then the temperature will increase in the part where you add the heat, but then as the gas goes through the orifice, it will nevertheless be almost adiabatic, its temperature will go down, and it will achieve sonic velocity at the narrowest point. Then, just after the orifice, its temperature will go back up as its kinetic energy is converted into enthalpy.
To achieve an isothermal critical flow situation, you'd have to add the heat "au fur et à mesure" (French for "as you go along"). In other words, you have to add the heat during the acceleration. And then take heat away during the deceleration, which would be impossible with an orifice.
http://Eric.Kvaalen.com
RE: Multiple RO's in series
In previous responses I noted statements in relation to adiabatic flow of a perfect gas that indicated
isothermal flow
constant stagnation temp (without considering specific heats)
const enthalpy.
Can any of the above be true for any adiabatic process? Other readers might use those type statements in other problems and get erroneous results.
For isothermal flow, I believe the heat added or taken away would be upstream between the source and the orifice. How close to reality this is will have to be determined by knowing heat transfer coef's on the inside/outside and other thermal resistance for the piping.
For adiabatic flow, the flow distribution does change just upsteam of the orifice even though one generally assumes a one dimensional analysis.
RE: Multiple RO's in series
I have been following this thread out of curiosity because it is a boundary value problem with possible discontinuities, and therefore a challenge to solve. An excel based itterative solution seems a reasonable approach, but I don't feel that the information provided in this thread has actually educated me enough to solve such a problem. I would love to see a case actually worked out with some real numbers and orifice formulas. Thanks, sshep
RE: Multiple RO's in series
RE: Multiple RO's in series
I suggest we responders look at the problem with 2 identical orifices, each with a Cd=1. (with the last orifice have a short distance of frictionless piping downstream of it)
I would start the problem with the second orifice choking using the formula of my original post
W/A =const. (Po/To^.5)= mass flux from second orifice.
Po the stag pressure downstream of first orifice and To for adiabatic conditions remaining constant.
Po upstream of first orifice has to be greater than Po just upstream of second orifice.
I hope there will be some responce.
RE: Multiple RO's in series
Now some comments on the calculation of a series of orifices. When an orifice is in critical flow (or "choked"), then the flow is related simply to the upstream conditions, as sailoday has stated:
W = A const. (Pb/Tb^.5)
(I use Tb and Pb for the stagnation temperature and pressure before the orifice.)
This can be solved for any unknown if the others are known.
But when the flow is not critical, the situation is more complicated. The flow is
W = A rho v = A v M Pa / (R To) = A v M Pa / (R Tb) (Pb/Pa)^(R/Cp)
(M is molecular weight, Pa is pressure after the orifice (and in the orifice), To is temperature in orifice.)
The velocity is related to the pressures by
M v^2 / 2 = Cp Tb [1 - (Pa/Pb)^(R/Cp)]
Combining the above, we have
W^2 = 2 A^2 M Pa^2 (Cp/R) [(Pb/Pa)^(R/Cp) - 1] / (R Tb)
This cannot be solved for Pa explicitly, but it can be solved for other things. For instance,
Pb = Pa [1 + W^2 R Tb (R/Cp) / (2 A^2 M Pa^2)]^(Cp/R)
In the two-orifice case, we can calculate the flow W either from P0 and P1 (that is, before and after the first orifice), or from P1 alone using the simpler equation for the second orifice (assuming it is critical). Equating these two expressions for W gives an equation for P1 in terms of P0, A, etc. This can be solved for P1, but requires an iterative method (it is not a quadratic equation).
In the case of three orifices, one can also use an iterative method, as I wrote on March 25. We know P0 and P3, but not P1 or P2. We can assume a W, find P2 (simple equation assuming third orifice is critical), then find P1 (complicated equation), then find the P0 which would give our assumed W. We compare this with the desired P0 and iterate.
If the last orifice is not critical for the assumed W, then we just have to use the complicated formula to find P2 instead of the simple formula. It doesn't change anything essential.
There are other variants, such as assuming P1, finding W from this and P0, then finding P2 from W (using choked orifice 3), then finding P1 from W and P2, and comparing with the assumed P1.
But if you do it wisely, there is only one iteration to do--no nesting or solving for more than one variable simultaneously.
http://Eric.Kvaalen.com