Help with pressure gradient calculation

akus · May 6, 2016, 17:03

Hello everyone,

I am quite ignorant of fluid mechanics, so I was hoping someone on this forum would be able to at least point me in the right direction on this question. This question really relates to vacuum chambers and the gas flow between them, but the pressure regime which I'm doing this work in is in the viscous/knudsen flow range (...I think). So, I'm hoping some fluid mechanics expert will be able to help me out.

The situation I'm interested in is the following. There are two chambers held at constant pressures. For example, chamber 1 is held at 50 mTorr, while chamber 2 is held at 0.2 mTorr. The two chambers are separated by an orifice of 3mm. Now, the mean free path for air at 50 mTorr is about 1.5 mm, which puts the knudsen number, in regards to the aperature at 0.5. I read somewhere that a knudsen number < 1 is a viscous/knudsen flow, which is why I believe the physics I need is in that regime. I realize that in regards to the entire chamber it would still be molecular flow, but I'm really more interested with whats going on near the aperture.

What I'm wondering is if there is any analytical way of calculating the pressure gradient between the two chambers. There is some volumetric flow between the two chambers (at the speed of sound), so I would think that there is some P(x) that describes the pressure in each chamber. For example, my intuition would say that just past the aperture into chamber 1, the pressure is something less than what it is in the bulk of the chamber (30 mTorr). Over some distance the pressure would increase up to the bulk pressure. Is this intuition correct? Is there a way to calculate this?

If someone could at least point me in the right direction I would be very grateful.

Thanks

FMDenaro · May 6, 2016, 18:04

Unfortunately, your problem touches the limits of the continuum hypothesis and the definition of a "pressure gradient" is not well suited.
I suggest to see specific textbook for rarefied gas (e.g. see http://www.annualreviews.org/doi/abs...rnalCode=fluid)

engio · June 19, 2016, 04:46

Maybe you should try Molflow+(http://test-molflow.web.cern.ch/).
I have modeled etch chamber(process pressure around 20mTorr, chamber size is for 300mm wafer) and simulated with Molflow, compared the pressure with the measurement from the actual equipment. The simulated pressure was ~12mTorr or so, where the measurement was 11.2mTorr or so. Quite surprised to have such a close values...

May 6, 2016, 17:03	Help with pressure gradient calculation	#1
akus New Member Join Date: May 2016 Posts: 1 Rep Power: 0	Hello everyone, I am quite ignorant of fluid mechanics, so I was hoping someone on this forum would be able to at least point me in the right direction on this question. This question really relates to vacuum chambers and the gas flow between them, but the pressure regime which I'm doing this work in is in the viscous/knudsen flow range (...I think). So, I'm hoping some fluid mechanics expert will be able to help me out. The situation I'm interested in is the following. There are two chambers held at constant pressures. For example, chamber 1 is held at 50 mTorr, while chamber 2 is held at 0.2 mTorr. The two chambers are separated by an orifice of 3mm. Now, the mean free path for air at 50 mTorr is about 1.5 mm, which puts the knudsen number, in regards to the aperature at 0.5. I read somewhere that a knudsen number < 1 is a viscous/knudsen flow, which is why I believe the physics I need is in that regime. I realize that in regards to the entire chamber it would still be molecular flow, but I'm really more interested with whats going on near the aperture. What I'm wondering is if there is any analytical way of calculating the pressure gradient between the two chambers. There is some volumetric flow between the two chambers (at the speed of sound), so I would think that there is some P(x) that describes the pressure in each chamber. For example, my intuition would say that just past the aperture into chamber 1, the pressure is something less than what it is in the bulk of the chamber (30 mTorr). Over some distance the pressure would increase up to the bulk pressure. Is this intuition correct? Is there a way to calculate this? If someone could at least point me in the right direction I would be very grateful. Thanks

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May 6, 2016, 18:04		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	Unfortunately, your problem touches the limits of the continuum hypothesis and the definition of a "pressure gradient" is not well suited. I suggest to see specific textbook for rarefied gas (e.g. see http://www.annualreviews.org/doi/abs...rnalCode=fluid)

June 19, 2016, 04:46		#3
engio New Member Join Date: Jun 2016 Posts: 4 Rep Power: 9	Maybe you should try Molflow+(http://test-molflow.web.cern.ch/). I have modeled etch chamber(process pressure around 20mTorr, chamber size is for 300mm wafer) and simulated with Molflow, compared the pressure with the measurement from the actual equipment. The simulated pressure was ~12mTorr or so, where the measurement was 11.2mTorr or so. Quite surprised to have such a close values...