UPDATE: Blogger appears to have eaten the original post, likely with some help from me, but in any case, the post and the draft of the post appear to be missing in action.

It was a simple post, existence is proof of continuity. Another way to look at it, in nature, zero does not exist. Neither does infinity. Those are constructs developed by mankind to help understand nature.

Both zero and infinity can be approached in nature, but unlike a bank account that can actually be zero, nature never can be.

Carnot Efficiency is a tool used to estimate the amount of work you can get out of a mechanical system It is a percentage based estimate so it has a range of zero to 100%, but can never be either. In a single stage system, a more realistic maximum efficiency is 50%. You can use the wasted energy from a single stage system to increase overall efficiency, but each stage will have its own limits typically less than 50 percent.

50% is a "sweet spot" for existence. Consider a system that is open to the universe, if it gains more than 50% of the energy it can release, something has to break at some point. If it loses more than 50% of the energy it gains, it ceases to exist as a system at some point in time.

One of the Climate Etc. denizens posed a simple problem on dissipation. One example or question was dissipation between two sets of plates, 300K to 150K and 150K to zero K degrees plates. It you use the Carnot Efficiency, the first two plates are (1-150/300)=50% efficient and the second set are (1-0/150)= 100% efficient. If the 150K plate can lose all its energy in an instance of time, it never existed.

If you consider the two sets of plate in series, then the efficiency of the first pair has to equal the efficiency of the second pair. That requires a non-zero temperature of the final plate. So the "system" cannot exist in isolation. There has to be a non-zero sink or an infinite number of system stages since zero and perfection in the case of Carnot efficiency, do not exist in nature.

There is more to physics that Carnot Efficiency, but it has proven itself to be useful, though imperfect like all models. So if you consider the plate example with constant efficiency there would be more stages;

300 to 150, 150 to 75, 75 to 37.5 etc. until the next is no longer significant for our problem. Since temperature is just another construct to relate to energy, also based on the range from perfection and zero, you can compare the first stage, 300K to 150K which would be 450 Wm-2 to 27 Wm-2 converting temperature to energy using the Stefan-Boltzmann law. Since the Carnot efficiency is 50% and the S-B law is related to temperature by the 4th power, the emissivity or transmittance of energy from the warmer to coller plates would be [1-(.5)]^4=93.75% which would soon cease to exist. To have a system that you could reasonably expect to exist for some time, the Carnot Efficiency would need to be (1-(1/2^.5)=29.3 percent. Nature is full of square laws. For the Emissivity to be 50%, then the energy transfer to the second plate would need to be 450/2=250 Wm-2 or 257.7K degrees which would produce a maximum Carnot efficiency of 42.8%. Ideal is actually 42.8 percent not 50% and not 100 percent.

Then with two equal efficiency stages, the energy emitted would be 250/2=125 Wm-2 and the temperature for the outer plate would be 216K degrees. Add another stage and you have 125/2=62.5 Wm-2 emitted with a plate temperature of 182.2 K degrees.

So if Earth had a surface temperature of 300K, it would have an effective radiant layer of 257.7K degrees and another of 216K degrees and another of 182.2K degrees, if the S-B law was perfect. It's not. How imperfect is it?

Earth has a near perfect radiant layer called the Turbopause. The Turbopause, where the energy is low enough that convective mixing nearly ceases, is ~185K degrees which has an S-B energy of ~67Wm-2. Then 62.5/67=93.2% is the approximate "effective" emissivity of the surface if the actual surface temperature is 300K degree. That is pretty close to the ~.924 correction factor used with the S-B law.

Note, there are rounding errors, the numbers are not exact which would require some absolutely accurate baseline which may not exist, but for a guestimate, 16% Carnot Efficiency per stage seems to be pretty close.

For the other parts of

the puzzle posed by the Climate Etc. denizen I will let you comment there.