Multiple RO's in series 2

For choked flow of an ideal gas with constant specific heats, the mass flux W/A at the orifice throat is directlyproportional to the upstream stagnation pressure and inversely proportional to the square root of the stagnation temperature.

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.

there are special restrictors for that service, check with your compressor vendor

CatBoy,

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.

For any flow rate, you can calculate the pressure upstream from the third orifice, and then the pressure upstream from the second, and finally the pressure upstream from the first. (This is possible because for each orifice there is a relation between flow rate, downstream pressure, and upstream pressure.)

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.

the greater issue is not so much how to size a compound restriction, as the suitability of such a design for that service, but that is just an opinion...

Ditto to what ETG01 suggested. It's going to be trial and error.

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.

I am still waiting for comments on how choked flow will occur on the 3 same size orfices (assuming Cd is approx the same for each).

You will not have choked flow in all three orifices. If any is choked, it will only be the last one (as EGT01 guessed would be the case).

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.

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.

It's true that I simplified a little. The truth is that if the gas has a high velocity in the pipe (I'm not talking about in the orifices), its temperature will be lowered a bit, and so the temperature at the end (where the velocity is highest) will be a bit lower than at the beginning.

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.

P.S. The fact that the temperature goes down somewhat as the velocity increases after each orifice does not change the validity of the argument. The speed of the gas if it gets to Mach 1 in an orifice is determined by the stagnation temperature, not by the actual temperature. And the stagnation temperature, for an ideal gas going through an constant-enthalpy pressure drop, doesn't change.

that makes better sense...you are right the density issue was an oversight

EricKvaalen (Chemical)states: And the stagnation temperature, for an ideal gas going through an constant-enthalpy pressure drop, doesn't change.

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.

I agree that the temperature will go down a bit (perhaps even a lot in this case--I haven't done the calculation). In fact that's what I said in my post of March 27th.

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.

well said

CatBoy, you haven't told us how you made out with your analysis of this problem. Just wondering.

For information with regard to the choked flow of a perfect gas, constant specific heats.

For an isothermal process, the choking occurs not at M=1, but at M= 1/sqrt(gamma) where gamma= Cp/Cv

I think you're talking about a situation in which the heat transfer between the gas and the environment is very high so that the temperature stays the same even as the gas accelerates to a high speed. This would require some combination of very narrow tubing and a slow decrease in diameter to give time for the gas to pick up heat from the environment. A rather theoretical situation!

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.

For isothermal choking to occur, the pipeline need not be long or narrow.
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)