When its cold enough outside and heat pumps need to turn on their resistive elements to keep your box warm enough, having something better than resistive would be nice.

better opex but may not overcome the capex cost.

What about NHL-sized ice arena using TEC? What's the theoretical limits/bottlenecks here? (asking for a friend)

It'll be ludicrously expensive to built, and then you'll also have a ridiculous amount of heat to get rid of. The main challenge will be keeping that from melting the ice again (this, in general, is the underlying issue with TECs: they don't have any means of stopping heat from flowing back through them, so it's a bit like trying to bail out a leaking boat, and it gets worse the larger the temperature difference you're trying to achieve. TECs max out their efficiency at a mediocre COP at very low temperature differences, in comparison to compressor based systems which can achieve better COP with more than 10x the temperature difference)

Heat loss due to resistive heating of the thermoelectric elements is a factor.

I do some work using TECs for cooling optical sensors. Most of the industry has settled on a very small range of materials, such as BiTe, meaning that we're probably at a plateau until a more efficient material comes along. Some of my applications are battery powered, and cooling consumes a fair portion of the total power budget, so a more efficient cooler would be welcome.

Also, TECs are mechanically delicate. There's a laundry list of needs such as being able to form the material into the desired shapes, and make a good mechanical and electrical connection to it, which is usually done with some kind of low melting solder.

> active cooling phone holders

Things I'd never think would exist much less that I would think of actual occasion for utility: my phone was in the sun (windshield), charging and performing wayfinding, resulting in "phone too hot" errors.

On the whole however, can someone explain how efficiency can be over 100%, if 100% represents conversion of inputs into all work and no entropy?

A 100w space heater will produce exactly 100w of heat. A 100w heat pump, properly installed, may turn that 100w of power into 300w of heat output on the hot side. This >100% language is marketing for people who do not understand thermodynamics or who dont want to bother with specifics.

The same is said of those scam "personal" AC devices that blow air over ice. They too are "more than 100% efficient".

> This >100% language is marketing for people who do not understand thermodynamics or who dont want to bother with specifics.

That is a little unfair: this is the language of economics rather than physics, so if you are paying for 100w and getting 300w, that is indeed over 100% efficient. SCOP is a more "correct" term but is less clear when communicating with the public and (IMO) is mostly for professionals or people with a technical understanding of heat pumps. Nobody thinks they defy the laws of physics though.

It's not exactly just marketing, it is indeed possible to "move" heat energy by using less energy than the amount of heat energy moved, which all modern high efficiency compressor based air conditioners/refrigerators/mini splits/heat pumps can do. It's still a useful metric for comparing similar devices' efficiencies. It's called Coefficient of Performance and if it's only 1, your compressor based system has a major issue.

10% efficient, without working fluids or moving parts, can be epicly more practical than 300% with those things. In very small or hard to reach places, peltiers are often only real option.

Yeah, pretty much. I haven't seen a compressor-based cooler that's a few cm across. But it's not something that will replace AC, fridges, or freezers, which is where they get hyped in popular media. It's a niche technology for niche applications.