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Can somebody explain where does the heat go in such process?

Also please note that while the difference between two thermometer readings is 10C, drop they claim is actually 5C.

I suppose if the air is compressed by the bottles, the inlet is on the outer side of the cardboard membrane and discharge is on the inner side.

Excessive heat is then released into the bottle and stored in the plastic until it radiates away - mosly by the large, voluminous bottle part.

According to the DIY build instructions, the bottles are cut in half, so they provide horn-like flares, open on both ends.

(Not that this adds any plausibility.)

The technical claim is summarized in this image:

http://i.dailymail.co.uk/i/pix/2016/06/10/18/3521DBE60000057...

Compressing, then expanding air, so this is the same principle as a fridge uses with freon (ok, not freon any more), on a much gentler scale, with much-lesser-but-still-useful results.
Sorry about my earlier comment; re-reading yours now, it's clear you're writing about the same thing.
Euch. Daily mail - No thanks.
Don't shoot the messenger.
In this case the messenger is a loud-mouthed racist bigot, and I wouldn't trust him to tell me the time.
And it removes the heat through... magic? All this does is redirect the air into your house, and compress it a little (causing it to heat up very slightly). Sure, it will feel cooler because you have air circulation, but opening a window has the same effect.
I can see a theoretical way it could work - the air enters the wide end, compresses as the bottle narrows, heats up above ambient temperature due to the compression, dumps its excess heat to the plastic of the bottle, and then expands back to ambient pressure as it exits the nozzle thus cooling below ambient temperature - the bottle then radiates away the heat, either mostly outside through virtue of the shape of bottle, or just equally outwards and inwards which still results in some cooling effect.

I find it very hard to believe that you'd have any appreciable compression without gale force winds though. And you don't get to "accumulate" minute amounts of cooling either - the temperature of the "chilled" air coming in is the coldest it will get.

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I think you don't need to invoke the heat exchange with the bottle to be relevant. I also believe that you get to accumulate it but there is a negative feedback due to the temperature gradient which prevent you to reach a high temperature difference.

For what it's worth (I'm not a physicist), here is my mental model of how I think it works. We are working at equilibrium we don't care about the macroscopic flow of air as it balances out. When the wind blows in the bottle it creates a more ordered arrangement of the air molecules (imagine laminar or vortex) at the neck (zone of low entropy). The internal energy will diffuse toward the region of more order (Second principle). So internal energy of the inside air flows towards the interface where it then diffuse into the nearby flow and get carried away by the wind.

As soon as there is a temperature difference between the inside air and outside air, traditional heat exchanges also occurs at the interface which has an opposite effect.

plastic bottles are free? that seem like good country to set up recycling plant
This probably work by admitting air (as a window would) and denying solar radiation (as a window would not).

Other factors to consider include the optics associated with light passing through the bottle and how focused air might affect people's perception of temperature (wind chill).

The bottles might reflect enough light away from the structure to have a discernible effect.