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Thursday, February 14, 2013

Back to the Carnot Efficiency with a Little Entropy

One of the bloggers mentioned Carnot Efficiency would be a good way for the climate science gang to validate their estimates.  I have mentioned to him in a post that Carnot Efficiency wouldn't likely be much good since absolute temperatures were not up to snuff.  I have a fairly decent estimate of Tc or the cold sink at 184.5 K degrees +/- a touch.  Th, or the hot source temperature is a lot harder to guestimate.  Since the TOA emissivity is close to 0.61, the efficiency pretty much has to be close to 39%, but I am not sure how much good that would do since before long we will need to add more significant digits.
Part of a real cycle though is the Clausius inequality.  There is always going to be some energy that cannot be used and part of that is source temperature changes don't have time to be fully utilized.  Once you get a system in a true steady state with stable source and since, efficiency can improve, like a gas turbine.

Since Earth has a diurnal energy cycle, some portion of that energy will never be utilized.  This brings us back to whether the RMS value of the Ein cycle should be used or the Peak value.  Since the average energy value of the oceans appears to "set" DWLR or lower atmospheric Tmin, it would be more likely the steady state "source" for the atmospheric effect.  Tmax, which is a bit elusive with the SST issues, is little help right now, but Tave, using the Aqua and BEST data should be close to 400 Wm-2 or 17.1 C degrees.  With 400 Wm-2 as Fave, or flux average and 334.5 Wm-2 as Fmin, or really Fconstant, we have about 65Wm-2 of energy flux which is transient or cyclic.  Some portion but not all of that energy would be available to do something.

Using the Carnot Efficiency, (Th-Tc)/Th  With my rough values, (303-184.5)/303=39.1% with Th-Tc= 118.5 K degrees.

I can also consider the ocean efficiency.  Since the ocean sink is much warmer, ~271.25K (-1.9C) with the same source of 303 K, (303-271.25)/303=10.5 percent with Th-Tc=31.35K degrees.  With the ocean "engine" having a lower efficiency, the ~65 Wm-2 would have less chance of providing as much useful whatever.

Making a large assumption, I am going to assume that the ocean can only use 10.5/39.1 or 26% of the transient energy.  That would be 17.4 Wm-2 of possibly usable transient leaving 47.5 Wm-2 as a rough estimate of unusable energy or entropy.  

Whether you followed all that or not, it just means that the ocean "engine" is much less efficient because of the smaller difference in source and sink temperature.  Since it has a lower efficiency and exposed to the same transient energy, it would have higher entropy at the surface.  The atmosphere may make use of more, but with a lower thermal mass, it would be unlikely to retain what it does use.

How this amount of entropy would manifest itself in a open system like Earth climate may not be that difficult to figure out.  The "atmospheric window" energy is the likeliest candidate and these rough numbers are not too far from the measured values.  There is one issue with the "window" energy and that is what surface(s) emit that energy.

It should be reasonable to assume that water, with its larger radiant spectrum releases most of the "window" energy and that since water is in the atmosphere, that both the surface and cloud bases would have free energy or entropy in the window.  If that is indeed the case, they there is an interesting situation.  As Tc increases with CO2 forcing, the efficiency of the atmosphere Carnot Engine would decrease more quickly than the oceans.  Back to the tale of two greenhouses.

Depending on what temperature CO2 impacts most, the change could be pretty significant.  At the lower sink of 184.5K, an additional 4Wm-2 would increase the sink to 187.25K which would reduce the efficiency of the internal Work done to 38% from 39.1%.

The punch line there is that more atmospheric forcing would reduce energy flow internally and externally allowing the increase in Entropy via the atmospheric window instead of closing the window if the CO2 effective radiant layer is close to the lower Tc sink.

I doubt that would lower my personal "sensitivity" estimate, but it does make me more confident in my tolerance of +/- 0.2 degrees.

This needs a lot of work and probably has a few errors, so take it with a grain of salt.  The Carnot Efficiency might be useful after all though if enough "engines" can be isolated, like NH versus SH.

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