We don't have any electricity, so we're thinking of purchasing this Koolatron cooler.
Most of the drinks I put in it (refreshment cans, water bottles, etc) will be warm/room-temperature.
Will it make these things cold?
What else do you suggest?
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The answer is clearly no due to the Second Law of Thermodynamics. One consequence of this law is that heat always transfers spontaneously from a hot body to a cold one and never the other way round. If you want to reverse the flow, you have to apply work to the system.
Given that you have no electricity you are not applying work to the system. As that box is insulated, the drinks inside will be a bit warmer or a bit cooler that the surrounding environment for a while(time lag for heat to trasfer). As soon as you open the box inside and outside temperatures will reach equilibrium.
They cool slow. So overnight yes they will be cool. Do you have water & a down hill slope? If so a drop cooler. Run water slow let it drop pool drop pool drop. 3 foot at a time. The more drops it makes the colder it gets. At 3 drops in 10 foot water will be 50f if shaded. In the tropics. Colder water will be colder. This is a very primitive way of cooling used only in a few locations in mountains.