# Long mach nozzle shape

Hi y'all, thanks in advance to anyone who knows a bit more about compressible/isentropic flow than me for the help,

I have an interesting problem which I've been working on for a bit. Basically, I would like to measure species coming out of a reactor using molecular beam mass spectrometry. The reactor should be at medium (1000-1200 K) temperatures and atmospheric or higher pressure. Because I want to measure the chemistry, I would like to transport molecules "instantly" from the sampling point to the mass spec.

However, the issue is that because of the steep pressure and temperature gradients as we leave the reactor, there is about an inch between the exit of my reactor and the stage 1 skimmer of the MBMS (see MS paint masterpiece below).

So what I have there is a converging mach nozzle, which goes into a low pressure region, so the gas expands/accelerates and forms a beam at the skimmer. The problem with this setup is there is a lot of cooling as the reactor goes through the unheated wall, so we are not getting a good picture of the species exactly as the reaction ends.

Now I am not an isentropic flow expert, and know nothing about nozzle geometries. My question is: what happens if we change the nozzle geometry to be more like this? (Brace yourself for more MS masterpiece....)

So, is the flow close to M=1 through the whole thin section of the nozzle? Or does that geometry cause shocks or something which will kill the isentropic-ness of it? Is there a geometry that help me to "stretch" the nozzle so that the time that the gas spends at low temperature is very short, but that it still expands very close to the stage 1 skimmer? The key here is that the time that the gas spends in the area with uncertain temperature must be very low, but at the same time, the nozzle exit/expanding part must be within about 5 mm of the skimmer, due to the location of the mach disk. Would something like this be much better?

Thanks for any help! Science!

Gold Member
Is there any reason this has to be sonic or supersonic? If not then you can remove the Mach disk limitation. I'll also note that without hearing your nozzle to be at least nearly adibatic, this process will not be isentropic no matter what you do here.

With such high operating temperatures, you will always lose substantial heat to the surroundings if they aren't preheated. You may be able to actually utilize supersonic expansion to minimize this since the expanded temperature will be much closer to ambient and thus incur fewer losses, but wgether that is still a useful state for you is unclear to me.

Regarding your nozzle proposal, whether the throat remains sonic through its length and whether a shock forms will depend on your pressure ratio between your reactor and ambient. With such a long, narrow throat, viscous losses may become meaningful, so you may need extra pressure to overcome that. Otherwise you will need to know your pressure ratio to determine whether the nozzle "starts" or not (i.e. does not develop a standing shock). A supersonic nozzle is designed for a single Mach number and operates ideally at a single pressure ratio. A lower reservoir pressure means some efficiency loss, and too low causes the nozzle to "unstart."

gpsimms
Is there any reason this has to be sonic or supersonic? If not then you can remove the Mach disk limitation. I'll also note that without hearing your nozzle to be at least nearly adibatic, this process will not be isentropic no matter what you do here.

With such high operating temperatures, you will always lose substantial heat to the surroundings if they aren't preheated. You may be able to actually utilize supersonic expansion to minimize this since the expanded temperature will be much closer to ambient and thus incur fewer losses, but wgether that is still a useful state for you is unclear to me.

Regarding your nozzle proposal, whether the throat remains sonic through its length and whether a shock forms will depend on your pressure ratio between your reactor and ambient. With such a long, narrow throat, viscous losses may become meaningful, so you may need extra pressure to overcome that. Otherwise you will need to know your pressure ratio to determine whether the nozzle "starts" or not (i.e. does not develop a standing shock). A supersonic nozzle is designed for a single Mach number and operates ideally at a single pressure ratio. A lower reservoir pressure means some efficiency loss, and too low causes the nozzle to "unstart."

1.) It is best for the flow to undergo supersonic expansion. The chamber between the nozzle and the skimmer will be held at approximately 0.5 torr. As such, molecular collisions can cause the beam intensity to drop. If, instead, there is supersonic expansion, then we will establish a 'zone of silence' in front of the mach disk. If we sample from this region we have directly sampled the products as they exited the reactor, and the reaction was "instantly" quenched.

2.) Low temperature in and of itself is fine. The key is the *time* during which the molecules are at a lower temperature. What I would love is to hold the reactor at 1000 K, have the entrance to the nozzle throat be held at that temperature, but then have the products accelerated through the nozzle so that they reach the skimmer on the 0.1 msec timescale. (For example, I might want to be measuring combustion products at 1000 K for 20 ms residence time. If the products are cooled at the exiting end of the reactor and take 5 ms to traverse the 1" plate separating the reactor from the first stage, then I have introduced very significant uncertainties into the time-history of the chemical system. This is what my hypothetical design seeks to do. Sample from the high temperature region, but then accelerate the flow to something near M=1, so that it traverses the plate in approximately 0.1 ms.

3.) I worry that viscous losses will be quite serious, as I am talking a very small nozzle. The entire reactor ID is 4 mm, so the nozzle will be on the order of 100 micron. We will be sampling from atmospheric pressure generally, and the stage one pressure will be 0.5 torr, or perhaps one order of magnitude less. So the pressure ratio is 760/0.5 = 1520, or 760/0.05 = 15,200 at the very most.

Gold Member
1.) It is best for the flow to undergo supersonic expansion. The chamber between the nozzle and the skimmer will be held at approximately 0.5 torr. As such, molecular collisions can cause the beam intensity to drop. If, instead, there is supersonic expansion, then we will establish a 'zone of silence' in front of the mach disk. If we sample from this region we have directly sampled the products as they exited the reactor, and the reaction was "instantly" quenched.

That 0.5 torr will help you immensely when it comes time to expand your gas to a supersonic velocity. If you tailor your chamber pressure (I don't know if this is possible in your case), then you can even devise a system where no shock diamonds form and you just get essentially smooth supersonic flow out of the nozzle. Otherwise, even if the nozzle starts, it will be under- or over-expanded and a shock diamond pattern will eventually form.

2.) Low temperature in and of itself is fine. The key is the *time* during which the molecules are at a lower temperature. What I would love is to hold the reactor at 1000 K, have the entrance to the nozzle throat be held at that temperature, but then have the products accelerated through the nozzle so that they reach the skimmer on the 0.1 msec timescale. (For example, I might want to be measuring combustion products at 1000 K for 20 ms residence time. If the products are cooled at the exiting end of the reactor and take 5 ms to traverse the 1" plate separating the reactor from the first stage, then I have introduced very significant uncertainties into the time-history of the chemical system. This is what my hypothetical design seeks to do. Sample from the high temperature region, but then accelerate the flow to something near M=1, so that it traverses the plate in approximately 0.1 ms.

You may be better off, then, to design a more traditional converging-diverging nozzle and use it for this purpose. You can essentially pick the conditions you'd like to reach and then design a nozzle that accelerates the flow to those conditions as quickly as possible. With a back pressure of 0.5 torr and a high total temperature of 1000 K, you should have no problem picking any suitably large flow velocity. The real trick will be designing the nozzle keeping the chemistry of the gas in mind. You will need an equation of state relating temperature, pressure, and density in order to do it, and depending on that equation, your life could be easy or hard. It's really easy to do with something that behaves reasonably like a ideal gas. I suspect that isn't the case here, though. On the other hand, maybe you can just design it assuming an ideal gas and the real gas effects will turn out to be very small. That's something you'd have to test.

At any rate, with a more typical converging-diverging nozzle, you could accelerate your flow to (nearly) any Mach number you'd like in order to move it quickly through the cool region. The limiting factors will be the available pressure in your reactor and whatever temperature is too low for you. In theory, you could easily achieve a low enough temperature to liquefy one or more of your reactants if your Mach number is too high, and I suspect this would not be what you want. This is a common issue in supersonic and hypersonic wind tunnels. I suspect you won't want a Mach number that high, though.

3.) I worry that viscous losses will be quite serious, as I am talking a very small nozzle. The entire reactor ID is 4 mm, so the nozzle will be on the order of 100 micron. We will be sampling from atmospheric pressure generally, and the stage one pressure will be 0.5 torr, or perhaps one order of magnitude less. So the pressure ratio is 760/0.5 = 1520, or 760/0.05 = 15,200 at the very most.

The good news is that 100 microns is still several orders of magnitude larger than what I would expect your mean free path to be, so you are at least still in the continuum regime. The bad news is that you are right to be concerned with viscous losses on such a small scale. This is another reason to consider a more traditional nozzle design rather than one with an extremely long throat. The shorter the length of extremely narrow throat, the less viscous losses you are going to incur. Given the pressure ratios involved, such a nozzle would almost certainly be underexpanded. You would end up with a series of shock diamonds and would just have to sample before those occur, I suppose.

Thank you again for the insight/discussion! It's appreciated and very helpful.

That 0.5 torr will help you immensely when it comes time to expand your gas to a supersonic velocity. If you tailor your chamber pressure (I don't know if this is possible in your case), then you can even devise a system where no shock diamonds form and you just get essentially smooth supersonic flow out of the nozzle. Otherwise, even if the nozzle starts, it will be under- or over-expanded and a shock diamond pattern will eventually form.

This would be nice. I doubt that I can control pressure very precisely. The best I could do would be to attach a precision needle valve someplace to the chamber and adjust the valve to regulate pressure. This might contaminate my sample, but if I located it near the chamber exit to the pumps it might not. Just out of curiousity though, if I were able to do this where do I find the nozzle shape that goes with a particular pressure ratio/throat size to achieve this condition?

You may be better off, then, to design a more traditional converging-diverging nozzle and use it for this purpose. You can essentially pick the conditions you'd like to reach and then design a nozzle that accelerates the flow to those conditions as quickly as possible. With a back pressure of 0.5 torr and a high total temperature of 1000 K, you should have no problem picking any suitably large flow velocity. The real trick will be designing the nozzle keeping the chemistry of the gas in mind. You will need an equation of state relating temperature, pressure, and density in order to do it, and depending on that equation, your life could be easy or hard. It's really easy to do with something that behaves reasonably like a ideal gas. I suspect that isn't the case here, though. On the other hand, maybe you can just design it assuming an ideal gas and the real gas effects will turn out to be very small. That's something you'd have to test.

So actually equation of state/non-idealities might not be bad. The flow will be heavily diluted, 98% or more argon. So assuming the flow as pure argon will not be a bad assumption, even though there is chemistry going on in the other species.

At any rate, with a more typical converging-diverging nozzle, you could accelerate your flow to (nearly) any Mach number you'd like in order to move it quickly through the cool region. The limiting factors will be the available pressure in your reactor and whatever temperature is too low for you. In theory, you could easily achieve a low enough temperature to liquefy one or more of your reactants if your Mach number is too high, and I suspect this would not be what you want. This is a common issue in supersonic and hypersonic wind tunnels. I suspect you won't want a Mach number that high, though.

Correct. Honestly, anything around M=1 is plenty fast.

The good news is that 100 microns is still several orders of magnitude larger than what I would expect your mean free path to be, so you are at least still in the continuum regime. The bad news is that you are right to be concerned with viscous losses on such a small scale. This is another reason to consider a more traditional nozzle design rather than one with an extremely long throat. The shorter the length of extremely narrow throat, the less viscous losses you are going to incur. Given the pressure ratios involved, such a nozzle would almost certainly be underexpanded. You would end up with a series of shock diamonds and would just have to sample before those occur, I suppose.

Right. So that was part of the reason for the long throat, because I worried that I'd have a mach disk (is that the same as the first shock diamond?) in front of my skimmer, so to move the nozzle exit closer to the skimmer, I lengthened the nozzle.
The way I calculate mach disk distance is the formula
.

I guess you are suggesting I can lengthen the cone on the exit end of the nozzle? If the gas is underexpanded, does that mean there will be no shock structure until after the cone has ended? Is the angle of the cone important? Here's another picture:

So in that case, does the above mach disk calculation predict the distance of the disk from the nozzle throat, or from the nozzle exit? If nozzle exit, then I am golden, I just make a long cone nozzle exit which goes almost completely up to the skimmer, right?

Oops! I just saw the link that you included in your response. It looks helpful and I will do a detailed read of it this evening.

Gold Member
I am answering these out of order since some group together nicely.

So actually equation of state/non-idealities might not be bad. The flow will be heavily diluted, 98% or more argon. So assuming the flow as pure argon will not be a bad assumption, even though there is chemistry going on in the other species.

That's a nice situation to be in, and should simplify your design considerably.

Correct. Honestly, anything around M=1 is plenty fast.

It is actually probably going to be easier to maintain a supersonic flow rather than try to maintain exactly sonic flow. You can still design it for a low Mach number, say, ##M=1.5##, but I would think this would be more sustainable and predictable since any small changes centering around ##M=1## can potentially change your regime from subsonic to supersonic, which have very different properties.

Right. So that was part of the reason for the long throat, because I worried that I'd have a mach disk (is that the same as the first shock diamond?) in front of my skimmer, so to move the nozzle exit closer to the skimmer, I lengthened the nozzle.

With the long throat, I would expect either you are going to end up just moving your shock down to the end of the tube, removing the shock entirely due to viscosity slowing it down enough to never go supersonic, or end up with a Mach disk anyway since the flow will expand and go supersonic after leaving the throat in at least a small region anyway. You are better off using a small throat to minimize losses and then designing a contoured diverging section to try to control that expansion as best you can.

I guess you are suggesting I can lengthen the cone on the exit end of the nozzle? If the gas is underexpanded, does that mean there will be no shock structure until after the cone has ended? Is the angle of the cone important? Here's another picture:

View attachment 101523
So in that case, does the above mach disk calculation predict the distance of the disk from the nozzle throat, or from the nozzle exit? If nozzle exit, then I am golden, I just make a long cone nozzle exit which goes almost completely up to the skimmer, right?

Just out of curiousity though, if I were able to do this where do I find the nozzle shape that goes with a particular pressure ratio/throat size to achieve this condition?

This is precisely what I was suggesting. Nozzle contours are generally calculated using CFD of varying degrees of complexity. The simplest method is to assume the entire thing is inviscid and use the method of characteristics to calculate the wave structure and tailor the contour to avoid oblique shocks, giving you a nice smooth expansion to your design point. This is surprisingly easy to do for a 2D nozzle (rectangular cross section). For an axisymmetric nozzle, it is tougher. Once the contour is developed this way, generally corrections are applied iteratively to account for viscosity (calculate the boundary layer, adjust the contour accordingly, recalculate, and so on). You can get away with not using the corrections if you don't mind a lower Mach number than your original design (the boundary layers decrease the effective expansion ratio of the nozzle).

There's a pretty good description of the method in https://www.amazon.com/dp/0486419630/?tag=pfamazon01-20. It is limited to the 2D case, however, and it looks like you will need something axisymmetric. I believe the axisymmetric case is discussed in https://www.amazon.com/dp/0471066915/?tag=pfamazon01-20. I don't recall which volume it is in, and that book is hard to find, so if you are at a university, I'd suggest trying to check it out from their library. https://www.amazon.com/dp/0072424435/?tag=pfamazon01-20 also has discussion on the topic in two dimensions, though it is a bit lower-level, in my opinion.

The end result of such a system is that the flow will be expanded to a given Mach number and its associated pressure (based on the pressure in your reactor). If that exit pressure is lower than ambient, the flow is over-expanded, and will be squished by the ambient pressure as it leaves until the pressure equalize, which involves a shock diamond pattern (the same thing as Mach disks). If it is too over-expanded, then shocks form in the nozzle to bring that pressure back up to ambient.

The alternative is if the exit pressure is higher than ambient, in which case the flow is called under-expanded. I suspect this would be the case you would experience. In this case, after leaving the nozzle, the flow continues expanding (much like if you didn't have a diverging section there in the first place) and you would again develop the shock diamond pattern, though it would first involve some expansions and would likely occur farther downstream than in the over-expanded case. I would bet that your empirical formula is more likely to apply to this case than for an over-expanded nozzle, but since I don't know where your formula comes from or what assumptions were made, I don't want to say for certain either way.

If you match the ambient pressure to the exit pressure, no shocks diamonds form and you get a supersonic jet. In fact, you may even want to experiment with this configuration for your experiment (if possible) since your jet would essentially push all of the ambient gas out of the way anyway. So after a brief transient period (where a shock will pass through the nozzle as it starts), you should end up with a stream of essentially undiluted test gas, and I would highly doubt that it would become diluted or collide with many ambient gas particles once the flow is established as long as your test apparatus is close enough that no turbulence can develop and the flow reaches the opening faster than diffusion can occur.

You could use just a conical diverging section, but you may get some irregular patterns of waves and/or shocks that form, and your exit flow wouldn't come out in an stream moving all the same direction. It would be slightly diverging. Maybe that is fine for your purposes. You'd have to test it to determine that.

If you don't mind me asking, is this for a school project or eventually publishable research or what?

Also, I applaud your MS Paint skills. They are far superior to mine.

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