CoP isn't really about engineering refinement to eek out the best efficiency. In fact, we have known how to get really close to the Carnot limit for nearly 100 years. The Carnot limit for a fridge in room temperature is a CoP of ~20.

However, getting close to that limit requires huge evaporators and condensers, big pipes, large amounts of working fluid, and multi stage pumps. All of that costs money, and historically people didn't care much about efficiency if it means spending more money.

The Carnot limit for 26C -> 5C rerigeration is around: 1 / (T_high / T_low - 1) = 1/(300/278-1) = 13.

Industrial refrigerators can get to CoP of around 4 for this temperature delta. The overall system performance will typically be much lower, due to the need to run fans and other systems.

A couple of interesting technologies to watch in this area:

1. Acoustic refrigerators that use acoustic waves in helium. They theoretically can approach the Carnot limit.

2. Supercritical CO2 cycles. They can be used for larger deltas without losing efficiency.

It's worth noting that the Carnot limit really only applies to systems that return to thermal equilibrium much faster than the characteristic interaction and/or rely on thermal interactions to transmit energy.

Electrochemical, magnetohydrodynamic and various other systems are not properly characterized by the Carnot limit and can approach much higher efficiency limits.

I think the systems being discussed are appropriately characterized by the Carnot limit, but it is frequently mischaracterized as an absolute limit on systems that it is not relevant to and for example here it should probably not be considered the absolute limit of refrigeration efficiency.

> Electrochemical, magnetohydrodynamic and various other systems are not properly characterized by the Carnot limit and can approach much higher efficiency limits.

That usually is a distinction without difference. Other effects limit the efficiency at far lower levels than pure Carnot.

This can be intuitively understood like this: Carnot limits apply to gases, that are the simplest interacting systems. You can realistically model them as just a collection of individual independent particles interacting only via simple collisions.

Anything more complicated like electrons in semiconductors, and you have way more interactions and way more possibilities for your system to have inefficiencies.

Take, for example, solar panels. Sunlight has the optical temperature of around 6000K, so a Carnot engine that uses 6000K hot part and 300K cold part will theoretically have a 95% efficiency. And an infinite stack of solar panels with each layer tuned for a specific wavelength has the theoretically maximum 87% efficiency.

Meanwhile an infinite diameter conductor has a maximum efficiency of 100% with every part at whatever temperature you want.

Carnot characterizes classical heat engines and things that work a lot like them, it's not a universal limit of process efficiency and it's not even a good first approximation for many systems.

Your example of a meaningless distinction is a factor of over two in waste energy, in the favor of Carnot in this case, but distinct none-the-less.