Heat Transfer Between Heated Upper Plate and Cooled Lower Plate

[skygerator]

The question relates specifically to US Patent 4624113, Passive-Solar Directional-Radiative Cooling System, University of Chicago representing Argonne National Laboratory on behalf of assignee, USA, DOE, 1986, in a 3D compound parabolic reflector assembly with a 3D compound parabolic housing, a circular plate for the radiator, an HDPE cover and a vacuum in the cavity between the radiator and cover.

The patent claims that the invention can achieve 40-80°C below ambient at the radiator without forming condensation on the cover. It does not quantify the vacuum, but let's assume that it can't be high vacuum, due to practical construction materials, as described by the patent. The patent also does not describe the model for determining this performance potential. The patent claims that dry air will not achieve the goal of preventing condensation.

The upper HDPE plate is heated by the ambient heat of the environment. This seems to be addressed in Eq. 47 at Convection from a Rectangular Plate. Eq. 47 deals with a heated circular downward facing plate. Given that the cover is well below the lip of the compound parabolic aperture, for the purpose of this question let's assume that the radiator and cover are the same size.

The lower plate is cooled by net radiative transfer with the sky through the compound parabolic aperture. That calculus is easy with GB Smith-2009, Berdahl-Martin, etc.

For estimating non-radiative losses, Smith is using air or dry air in the cavity and also looking at insulative losses of the compound parabolic housing but makes no mention of using a vacuum instead of air. Of course, Smith was looking at solving a different problem than US4624113.

I've looked at the answers to similar questions here, but the model eludes me.

The question: How does one model the heat transfer between the radiator and the cover with a reduced pressure in the cavity, to determine the necessary vacuum to achieve the aforementioned goal of preventing the cover from reaching the external dew point for a set of ambient conditions?

 

[skygerator]

btrueblood,

I've come to the conclusion that the group is hazing me.

 

[btrueblood]

sky,

I'd hold off on making that judgment. You could've posted the link that I gave, or at least some sketches so people could see what you are talking about. It's not a conventional design, and uses some principles that a lot of people never get exposed to, except those of us that got to play a bit as spacecraft/rockit engineers. It's also a concept that, as you are finding, kinda works on the dusty corners of what is physically possible...

Back to your earlier posts,

I'd think a PE cover would be fairly simple to keep clean, so I'm guessing the level of insulation required to the flask, or the vacuum sealing ability of the cover, became the limiting constraints. I spent some time years back trying to keep the working elements of a thermoelectric refrigerator frost/moisture free, and it is a b%tch. Typical polymers have ridiculously high permeabilities to water vapor, so I'd bet that any transmissive (i.e. thin) window you put on the thing would eventually pass enough moisture vapor through to fog the mirrors. Realize, that the cold plate makes a very effective moisture trap - i.e. the vapor pressure of water on the inside of the cavity (note that the differential vapor pressure is what drives the diffusion of water inwards) will always be to the low side, since moisture is continually being sucked out of the space (condensed and/or frozen) by the cold plate. The setup might work if the lens was glass? Also, any gasket you use should be made of butyl rubber, as it has the best vapor/moisture sealing of any squishable material. Finally, making the cover removable (to wipe the mirrors clean) might be a good idea, though it may mean you have to put a vacuum pump on the thing to regenerate the right conditions for it to operate.

"The question is about modeling the heat transfer in an enclosed volume at a reduced pressure between a cooled lower horizontal circular plate (radiator) and a heated upper horizontal circular plate, to determine the pressure reduction that would be required to prevent condensation on the cover, external to the enclosed volume."

Right. Lower the pressure enough, and you enter the regime of free molecular flow, where convection virtually disappears. And things like the "thermal accomodation coefficient" come into play. This is how vacuum flasks (aka Thermos bottles) work.

 

[IRstuff]

I'm not sure why you think you are being hazed, but whatever.

TTFN
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Of course I can. I can do anything. I can do absolutely anything. I'm an expert!

 

[skygerator]

btrueblood,

Oh, great. Now, I have a semipermeable membrane. That's funny, because I almost joked about moisture diffusion.

Glass would be great, if only it were more transparent to infrared in the atmospheric window? Does Depot carry tellurium?

I'll do the math and worry about materials later.

 

[skygerator]

| It's also a concept that, as you are finding, kinda works on the dusty corners of what is physically possible...

Didn't they use the same principle for the power pods on Tatooine? Although, they must have had something more efficient than a Stirling engine for the generator.

Melbourne found good use for radiative cooling through an aperture for reducing the carbon footprint at the airport.

The idea isn't that far out on the fringe. It'll just require some engineering ingenuity.

 

[btrueblood]

There is discussion of permeabilities of polymers over in the polymer engineering section from time to time. Using multiple layers of different polymers is one approach used for things like blood transfusion packaging, IIRC. Also, aluminum vapor deposit layering can help cut down the diffusion also, and may help reject solar wavelengths, if that is ever an issue.

The good news for your radiator is that you don't have Bismuth Telluride chips with electric current flowing through them at the core of the cold-making bits. So, make the cover out of whatever plastic you want, but make it easily removed for periodic cleaning/wiping.

The dusty corners and fuzzy edges of your design is more a comment about modeling low-density heat flux, especially if the mean free path is only on the order of, or some single digit fraction of, the nominal cavity dimension. In that regime, there is some level of both natural convection and pure conduction occurring, and the accommodation coefficient comes into play...basically the math gets hairy and you really need to sharpen your modeling and testing skills.