Thanks for the reply. Actually...you're right...the energy cost is higher than a typical compressor refrigeration circuit, however there are varying schools of thought on this as I'm sure you already know. My study showed it depended completely on the transformer you used. As long as you could convert 110 to 12v and 4+ amps effectively, you could run one of these for the same cost as a 60 watt light bulb. Once I ventured into use of a less efficient transformer, the figures begin to change, though.
As for a science project, I think this would be pure dynamite...and something a younger person could understand (the bi-metal concept) and then also explain adequately.
As for use in a computer, we've done that over here also. The problem most people had was humidity and condensation. If you notice the foam around the center of the one in the picture, that's partly to keep the chip/aluminum spacer from being exposed to humidity. In a humid environment, you have to do that or you'll pour water on your motherboard constantly. I'm interested to hear more about your thoughts on the PID controller. It makes sense, but honestly I used a common old (as you called it bang-bang) thermostat and never had any trouble. One of the things I always liked was the speed at which these can move temps. I have toyed with the idea of a drink cooler that either had a large chunk of aluminum with holes drilled in it to match the diameter of the cans or (more practically) aluminum bent to match the can's diameter. At any rate, a cold side which would contact the aluminum can as solidly as it could.
No question these are specialty devices for specialty applications. I've also thought a solar panel delivering the right voltage & amp output would be an interesting combination.
As for a multi-stage cooler, do you mean by stacking the Peltiers in a tower approach? You can easily do that with devices of different "powers" as I'm certain you know. About 20 years ago I saw a tower constructed such that water would freeze solid in one second or so when set on the top. This fellow was trying to build a thermoelectric cooler for his Volkswagen, but I never knew if that worked for him or not.