Modeling thermal conductivity of a vacuum space

[blueandwhiteg3]

I'm trying to work out the thermal conductivity of a vacuum. It is proving surprisingly problematic.

The theory is this: pressure has very little impact on the thermal conductivity of air. Thus, the intensive measurement of thermal conductivity does not vary with pressure. However, in terms of extensive properties, thermal conductivity in a large space of vacuum/near vacuum begins to drop with pressure due to the kinetic nature of gases. I believe this is around 10-15 pascals where this becomes notable for air.

However, I'm stuck at solving the math. I am looking to construct essentially a panel, one side will be hot, the other will be cold. I wish to use a vacuum to enhance the insulating properties of this space.

I want to model my panel's vacuum space thermal conductivity at varying thicknesses and varying pressures. From here, I can determine what is most appropriate for my application.

There will be no special gases or chemicals involved. Temperature range is 250-300 K typically. It's not rocket science, I just don't have the right math

Can anybody help?

 

[IRstuff]

Schoolwork is not allowed.

TTFN

FAQ731-376

 

[blueandwhiteg3]

IRstuff, this is not school work. I am not in school. This is simply something that I am working on which is outside the usual realm of work.

If anybody wishes to share some constructive information, it would be appreciated.

I am working out all the variables, but it seems quite convoluted, due to the blend of conductance and convection heat transmission that happens. And all the math needs to be re-done for each pressure and thickness I'm evaluating.

It would be easier if I could just moved to a vacuum level below which convection is an issue, but I can't. Preliminary math is showing that I really need to find the lowest acceptable vacuum, because the reality is that creating extreme vacuums can be challenging (or expensive!) in real-world situations.

I've not yet quite put all the math together, and I'm sure there has to be something a bit more... put together. Perhaps a unified formula? Or perhaps a calculator or spreadsheet somewhere?

 

[IRstuff]

Parts of this question have been answered before. Do a search. At low vacuum, the thermal conductivity drops, as expected. A big missing portion of your assumptions is radiation. You should consult a decent heat transfer text. A downloadable one is located here:

http://web.mit.edu/lienhard/

http://www/ahtt.html

TTFN

FAQ731-376

 

[MintJulep]

thread391-212672

thread391-210603

 

[willard3]

A vacuum contains nothing, no gas, no solid and etc. by definition.

If there is no matter, there is no conduction.

 

[blueandwhiteg3]

I scanned the heat transfer text. It does not address these vacuum scenarios that I'm laying out. Also,I read the other threads suggested, even prior to posting. None of them really address the issue I'm trying to solve.

I know what kind of a vacuum is in a thermos. But I can't get that low in my conditions, but if I could, I know the conductivity would be so near zero, it doesn't matter. Obviously, a true vacuum has nothing and thus does not conduct any heat. However, this IS NOT a true vacuum, just exceedingly low pressure.

Yes, I am aware of radiative heat. That is a separate issue I am dealing with in another fashion that does not affect the vacuum. My concern in this particular situation is quite squarely conductive heat loss.

To be even more concrete, I want to model the thermal conductivity of a near-vacuum space at pressures between 12 and 100 microns, with a thickness of 1 to 50 cm. The temperature range will be 250 to 300 K.

In this situation, I'm right at the threshold where convection becomes irrelevant. I can see this from working out mean free path distance at those pressures. But since I'm right on the line, it has to be modeled. And for practical reasons, I don't want to be fighting for the last few microns of vacuum if I don't need it. That's why modeling it is so important.

I don't know how to do this, and unless I'm being exceptionally thick today, none of the links or resources suggested as of yet provide a means through which to do this.

Insight would be greatly appreciated!

 

[hacksaw]

try

http://books.google.com/books?id=uj...+conductivity+of+air+low+pressure&output=html

and subsequent pages

easier to search than enquire, but it is all fun

"Proceedings of the Twenty-first International Thermal Conductivity Conference, held October 15-18, 1989, in Lexington, Kentucky"--T.p. verso.

Thermal Conductivity 21
By Clifford J. Cremers, H. Alan Fine
Contributor Clifford J. Cremers, H. Alan Fine
Published by Springer, 1989
ISBN 0306436728, 9780306436727
728 pages

 

[Twoballcane]

Your best options is to do this by ratio. Percent down on pressure may equal percent down on your K. You may want to calcualte Watt's per area.

Tobalcane
"If you avoid failure, you also avoid success."

 

[hacksaw]

Look at the google books search, nice graph, nice correlations all the way down. It is not a linear dependence on pressure.

The thermal conductive changes predictably and is even used for vacuum gages down to a few microns Hg

good luck

 

[blueandwhiteg3]

Thanks for the comments. This is starting to head in the right direction.

What makes this messy is that there are two factors at play I need to model, which are not fully being touched upon by any posts to date.

Air is really actually a pretty good insulator, with a thermal conductivity of around 0.024 W/m-K.

Problem is that air suffers from a ton of convection. So when I have a space of air between two panels, the heat differential causes the air to circulate within the space. The net effect is a dramatic increase in effective insulative properties - like on an order of magnitude.

When planning insulating between two panels at sea level, there's a sweet spot around 1.6 cm thickness that provides maximum insulation without letting convection get out of control. Wider spaces tend to be actually less insulating, at least until they are vastly, vastly wider.

There is also the effect wherein dropping pressure reduces the apparent thermal conductivity of the gas. At some point, the molecules become so sparse that they can't move heat nearly so efficiently, and the thermal conductivity drops.

As we reduce pressure, the mean free distance for air molecules increases, and causes convection to be less and less of an issue. For example, at 0.002 kPa, 273 K, and assuming a molecular diameter of 0.3 angstroms, we get a mean free path of about 47 cm. This means that if I had a vacuum in a cube form of 47 cm, half the time the air would hit other air, half the time it would hit a wall.

When we have very low pressure and/or low spacing, the mean free distance is so great there's almost no air-to-air thermal transmission and, quite obviously, almost no convection effect. This is where I can use the readily available data (linked above) on how dropping pressure lowers apparent thermal conductivity.

The problem is that given my spacing and pressure range, the convection model is still potentially relevant. And convection is strongly affected by the thickness of the vacuum space. There is where I run into my problem; I need to model how significant a factor convection is going to be at a given pressure and thickness.

I may be over-thinking this. I'm going to do some more reading, but I don't think anything cited to date really addresses how to model the convection of air at varying pressures and spaces.

I'd certainly appreciate more comments / ideas / direction here.

 

[blueandwhiteg3]

FYI: I am aware that many pressure gauges use thermal conductivity to measure pressure. This is why I am even more convinced that there's some kind of very reliable, workable formula somewhere. I just haven't found it!

 

[blueandwhiteg3]

Another URL I found that I really like in terms of charting air's thermal conductivity versus pressure is here:

http://www.alma.nrao.edu/memos/html-memos/alma554/memo554.pdf

Scroll to the second page. It's on a log-10 graph. Also lets you zoom in a lot more than with the prior link.

From what I can tell, I can't easily attain equipment for a vacuum below 0.012 mbar. At 0.01 mbar, we're looking at a thermal conductivity of around 0.015 W/m-K. This isn't really appreciably less than at room temperature.

At the higher end of the pressure spectrum, 0.13 mbar, we're looking at thermal conductivity that's maybe 0.018-0.020 W/m-K - really very close to atmospheric pressure.

Clearly, the question is convection. I just noticed that wikipedia was updated with a good new section on calculating this:

http://en.wikipedia.org/wiki/Insulated_glazing#Calculation

The problem is that the examples presented are for atmospheric pressure and it's not entirely clear what variables would be altered (aside from thermal conductivity) at lower pressure .

 

[IRstuff]

If it's that tight a problem, then perhaps the approach just isn't suitable. Consider a VIP:

http://www.glacierbay.com/products/ultra_R_specs.html

TTFN

FAQ731-376

 

[blueandwhiteg3]

Space isn't the problem. I could use as many cm of thickness as I want.

I don't think the convection issue is going to be a show-stopper, but at the same time, I do need to figure it out, and determine workable pressure and thickness ranges.

I have looked at vacuum insulated panels, they're certainly interesting, but not quite what I need here.

 

[prex]

What math in fact are you speaking about? You din't even tell us whether the enclosed space is horizontal or vertical.
Personally I would come back to the treatment of heat transmission in enclosed spaces in a heat transfer book: an example is McAdams's Heat Transmission.
That treatment is based on Grashof and other numbers that you can calculate with the properties of air at low pressures (I use for this chapter 10 of Perry's Chemical Engineers' Handbook). Not sure where this could lead, but I can't see other practical routes, besides finding applicable experimental correlations.

prex

http://www.xcalcs.com

: Online engineering calculations

http://www.megamag.it

: Magnetic brakes for fun rides

http://www.levitans.com

: Air bearing pads

 

[blueandwhiteg3]

Hmm, those are interesting possibilities in terms of reading.

However, I think I was not entirely clear. I'm not asking "let's work through the fluid dynamics of this particular design" - I'm asking "let's get an idea as to how thick a design needs to be when convection will be a significant factor in designing at a given pressure."

I would be very, very happy if I could even just get thermal conductivity into the 0.03-0.02 W/m-K range. It's just that the big question mark around convection is what kills me.

If I can write off convection at doable pressures, or define a maximum amount convection could increase the conductivity at a given pressure, I will have made substantial progress in working things out. Regardless of the design, I can determine the maximum possible thermal conductivity right off the bat. It's basic smart problem solving: use heuristics, don't solve things that aren't practically relevant.

I'm specifically looking at two panels, positioned either horizontally or vertically. Infinite plane model with one hot plane and one cold plane is plenty. Max 250 / 300 kelvin range. Pressure 0.002 kpa to 0.004 kpa. Yes, this is vague, but the point here is a direction, not an exact fluid dynamics simulation.

Best case, we're looking at a scenario where we have a mean free path of 47 cm and a thermal conductivity of around 0.15 W/m-K. So if we had 47 cm spacing between the walls, many of the collisions would occur with other air molecules, but a fair percentage would still be with the direct walls. (It would be 50/50 only if we were using a 47 cm cube model, rather than infinite planes)

Common sense leads me to ask whether that's a point where the convection currents would be significant or not. On one hand, I can see how there's enough air to air contact to lead to convection. On the other hand, I can see how the impacting the walls even 10% of the time could create a sort of "friction" and largely eliminate convection.

I also see how as the molecules per unit of volume drops, convection will become increasingly insignificant, as soon, all conduction is through molecules hitting both the hot and cold panels. It's almost like conductance and convection are one and the same at this point; all heat is transmitted through the same molecules are hitting the hot and cold sides.

So clearly, the potential for convection to increase heat transfer has to be dropping quite a bit with pressure (fewer molecules), long before convection is essentially impossible. If this wasn't the case, we'd see thermal conductivity rising as pressure drops, but it doesn't.

As you can tell from the mean free distance values, we're right in the range where convection is becoming increasingly irrelevant as described. But how irrelevant, given the temperature, pressure and vague thickness ranges I'm working with?

That's the question I want to answer!

 

[Compositepro]

You seem to be a little confused about the distinction between thermal conductivity and convection in a gas. Thermal conductivity is due to random molecular motion and it does not matter if the molecules are hitting each other or walls. Convection is heat transfer that occurs due to mass transfer (non-random)often cuased by density differences. At your low pressures convection is negligilble.

Most common insulators like fiberglass work by reducing convection so that thermal conductivity is reduced to approach the thermal conductivity of air (as you said air is a poor conductor of heat). I think you are trying to solve a problem that is not there.

 

[IRstuff]

What I meant by tight, is that when you're worrying about the transition between convection and conduction, you do not have sufficient margin in the analysis. There is no way that you're going to get a sufficiently "correct" analysis, since many of the numbers that go into the analysis are difficult to get right for normal conditions, and are probably impossible to get right for near-vacuum conditions.

If you are that worried about convection, then you should baffle the air space, essentially converting it to a closed-cell insulator layer.

TTFN

FAQ731-376

 

[blueandwhiteg3]

Baffling the air space is not an option here. Fiberglass is not an option here.

I agree that my use of the terms convection and conductance is not consistent with their general use. Speaking more properly, you are correct, convection diminishes at low pressures.

I know what you are saying about "if it's too close, don't do it" but I am trying to define an approximation of the maximum convection I could be seeing at a given pressure. By modeling a pressure range, I can get an idea whether I'm too close or not.

If I had pressures of 0.0002 kpa, I'd not be talking about this. But the problem is that we're in a transitional place (at least in my head) and I want to work it out because I can't intuitively work it out.

So, back to my question... how can I model or estimate the degree to which convection will play a role in the stated conditions?