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Thursday, November 17, 2011

Simple Thought Experiment for the Climate Change Gurus

UPDATE: I think I found not only an error in a calculation below but possibly an interesting reason for the error.

You have two flat disks filled with CO2. The containers, or shell of the disks, are transparent to all electromagnetic wavelengths and assumed to have no impact of the skin temperature of the disks. The two disks are placed in a vacuum. One at temperature T and the other at temperature 0.75T. What would be the radiant emission between the two disks?

Since no concentration is given, the only relationship available is the Stefan-Boltzmann equation, 5.67e-8(T)^4 for the disk at T and 5.67e-8(0.75T)^4. If we think in terms of the ratio of energy transfer, then R(T/0.75T)=T^4/0.75T^4 which we could reduce to 1/0.75 = 1.33 for R in one direction and 0.75 for R in the direction of the cooler to the warmer disk.

That is about as simple as math gets, a ratio.

If the T is 288, then 0.75T would be 216, the temperature impact of T on 0.75T would be 1.33 something. The temperature impact of 0.7T on T would 0.75 or less something. The net impact of the temperature of T returned to T from 0.75T would be what? Unity? I don't think so.

The temperature of the disk at T is 1.33 greater than the second disk, but the second disk can only return 50% of the energy it receives from the warmer disk. So 1.33/2 or 0.665 would be returned, but the temperature of the cooler disk limits the energy it can transfer. Also, the warmer disk is in a vacuum, it emits in both directions so only one half of the impact of T would be transmitted to the cooler disk from the energy it receives from the cooler disk.

1.33 emitted while 0.75 is received initially. This one of the conundrums of black body radiation. The flux for the two disks would be 390Wm-2 for the warmer and 123Wm-2 the cooler. At time zero, the total emission of the warmer would be 390+.5*123 = 451Wm-2 and for the cooler disk 123+.5*390=318Wm-2, so in effect, T would actually be 298K and the cooler would actually be 274K degrees.

It should be obvious, that the warmer body may be warmer due to the cooler body, but the total energy of both bodies cannot increase simultaneously. So how is conservation of energy maintained?

The net radiation from the warmer to the cooler is 390-123=267 Half of that is returned, 133 of which half is radiated to the other side. So the initial impact of placing the two bodies together in a vacuum would be that the warmer body energy would increase by 1/2 the net energy it transferred to the cooler body. The cooler body energy would increase by the net energy received by the warmer body, and the two bodies would rapidly decrease to the total energy of the two bodies initially in the same ratio. Since the mass and thermal capacity of the two bodies are unknown. then final values may not be accurately determined. A clue to is the difference in the original energy flux and the returned energy flux. The return is 133Wm-2 or 10 Wm-2 more than the initial value of the cooler body. Once the situation stabilizes, the warmer body would increase to a temperature equivalent to 400Wm-2, and the cooler body, emitting from both faces would increase by 20Wm-2 to 143Wm-2. The warmer body would increase to 289.8K and the cooler to 224K degrees, on the warm faces of the disks. Since energy must be conserved, the cooler faces of the disk would have to decrease until the original energy balance is restored.

While that may have been hard to follow, the total energy remains the same. The cooler body gains energy and the warmer body loses energy if there is no other energy source applied.

For the temperature impact, (289.8-288)/288 = 1.8K or 0.625 % increase in the warmer body on the warmer side and (224-216)/216=8K or 3.7% increase in the cooler body's warm side.

As a rough estimate, 1.8 degrees of warming at a surface of temperature T would be expected if we added a body at 0.75T of the surface temperature in a vacuum. We don't live in a vacuum.

There is one relationship that may be useful, 8:1.8 and 20:10 for 0.75T, due to 50% of the returned minus the initial net flux.

Since CO2 filled perfect disks are cheap in thought experiment world, let us insert a new disk between the two which are now somewhat stabilized but in no way completely described, i.e. no thermal mass or exact energy value. The new disk is at 0 K degrees. That would be a pretty expensive disk if it were real.

The new disk is receiving radiant energy from the warmer surface of the warm disk with an apparent temperature of 289.8K and on the other side it is receiving energy from the warm side of the cooler disk with an apparent temperature of 224K degrees.

At 0.0K, the new disk has no energy to radiant, it can only absorb. The total energy of all three disks combined cannot increase, I am too cheap to think up a power source, so the three will eventually reach equilibrium sharing the total available energy.

Now it would get to be fun. Since the first two disk are in some approximation of equilibrium, you can assume that there is a mass relationship between the two original disks. The apparent mass of the warmer would be on the order of 8:1.8 times greater and the apparent energy of the warmer would be 20:10 greater with respect to the interacting disk faces. With nearly 8 time the apparent mass, the actual energy would be greater than 2 to 1, but the apparent energy is only 2 to 1. That is because there are other sides to both disks not restricted in the same manners as the interactive faces.

Now we have a new disk that would not have that luxury. Its energy is completely dependent on two unknown thermal masses.

Eventually, the new disk will return to the warmer some value between, 133 and 123 t the warmer disk and some value between 133 and 200 to the cooler disk. This means that the new disk will eventually be cooler than the warmer and warmer than the cooler, but the apparent temperature of the warmer must decrease if energy is to be conserved. Since the new disk is absorbing energy that was warming the cooler, the apparent temperature of the cooler would also decrease.

If the mass of the new disk were large with respect to the warmer, the energy of the new system would remain the same for some moment in time, but the temperature of the system would approach 0, the temperature of the new disk.

If the mass of the new disk were negligibly small with respect to both disks, the apparent temperature of the new disk faces would approach the temperature of the adjacent original disk faces.

What if the mass of the new disk was the same as the cooler disk? The temperature of both the warmer and the cooler would decrease to warm the new disk, the temperature of the warmer would decrease at a ratio of 1:8 with respect to the original cooler disk. The temperature of the new disk would increase 20:10 or 2:1 with respect to the original cooler disk. Or the new disk would have twice the energy as the cooler disk since it would transfer only half the energy it receives from the warmer disk. Eventually, the new disk would be exactly the temperature of the original cooler disk less half of the energy transfer from the new disk to the cooler disk which would be 123/2 at a maximum. The maximum temperature of the original cooler disk would be 181K, or the temperature equivalent to a flux of 61.5Wm-2, the maximum of the new disk 224K or the equivalent flux of 142Wm-2 and the maximum of the original warmer would be 288K or the equivalent 390Wm-2.

So an object in space can cause the apparent temperature of another to be warmer, but it does not increase the energy of the warmer object.

Now with these rough ratios, 8:1.8, 2:1, what happens when you add energy to both the warmer disk and the new disk at a ratio of 2.85:1?

NOTE: I will be back to answer, find any mistakes and typos, but those interested may find something worthy of considering.

Before answering that question, consider what happened between the three disks. The warmer disk appears warmer on the face toward the cooler disk(s). Did the warmer disk gain energy? Yes, but at the expense of its losing energy, It appears warmer, but the total energy of that disk did not increase significantly. Half of the energy it gained is emitted on the other face of the disk. Once stable, the total of the energy emitted from both sides will equal the total emitted before the cooler disk(s) were introduced. But how can that be if the side facing the cooler disk is emitting more and the side away is emitting 1/4 more, (half returned from the warmer is half returned to the cooler and so on)? Because of emissivity.

There are perfect black bodies, but only in perfect space. Because the cooler disk can only return half and a perfect black body at absolute zero would not return any.

Space has a temperature of about 3K degrees, it is not a perfect sink for radiant energy so there is no observable black body. Only in a perfect vacuum can the paradox of a cooler body warming a warmer body exist. Space is as close to a perfect vacuum I have heard of so far, so we have to model perfection. If we lose sight of the fact that a model is not reality we have screwed up.

So what if we made the disk we inserted 3K instead of absolute zero? The results would be the same, we would just need more significant digits to see any difference.

If you make the warmer body infinitely large so it had infinite energy with respect to the cooler body at 3K degrees, you would find that the emissivity of the infinitely large body would decrease slightly. Its emissivity is dependent of the sink of its energy.

We can see that by adding a disk of equal temperature to the infinitely large disk, Each emit at the same temperature, but the less massive can only return half of the energy it receives from the more massive.

For convenience we will say both start at 1Wm-2 emitted on the common faces. Each emit 1 and receives 1 initially. Then the less massive continues to receive 1, but retains only 0.5 which it attempts to return to the more massive.

1+1/2+1/4+1/8 ... 1/infinity = 2 But once we add the third disk? Then a fourth?

The 2:1 ratio continues indefinitely as long as the initial disk is infinite. Anything less than infinite does not return 100% of the energy flux.

So without an energy source, 2:1 is the maximum ratio and for T to .75T, 1.8:1 is the maximum ratio.

So what does this mean as far as global warming? The atmosphere is effectively a number of layers of disk without and internal source of energy. Since the average energy absorbed by the Earth's surface and lower atmosphere produces a surface temperature of 288K without the atmosphere, if the surface would be 255K, then the maximum temperature difference due to the atmosphere would be 66K degrees. With the tropopause at approximately -59 degrees and the surface at plus 15 degrees the temperature spread is 74 degrees. If we consider only the surface absorption at its daytime average, (174*2 or 348Wm-2 for an effective temperature of 280K degrees), then the tropopause difference is 66 degree or twice 33. We are near the maximum atmospheric effect without a change in solar forcing or a change in the ratio of atmospheric to surface solar absorption.

This poses an interesting possibility. Land use impact on albedo, the amount of solar reflected, potentially has more impact than the addition of CO2 to the atmosphere. Since the atmospheric effect is more based on surface temperature relative to the temperature of the tropopause, than one would expect, Urban Heat Island potentially is more of a factor than understood.

This of course just a thought experiment. I may change focus on which mitigation strategy is best, if there is any merit though. I may clean this up if I get around to it.

Continuing with the experiment. The 2:1 ratio is simple to explain. The basic model for radiant transfer is a disk or hole in in furnace. The observable face gets all the press. The opposite face gets a mention here and there.

While we don't know the quantum of energy in either disk, we do know that the disks all initial radiate the same energy flux from both faces. So twice the energy flux of the disk, both sides considered, is a measure of the total energy of the disk. I used the term Quantum, because I am going to call 2T an energy Quantum with the working term ThermZit (TZ) :)

Planck and all the old masters pondered what the quantum was, Hey, its a TZ! :) Seriously, 2T is a good working value. If you take two disks at the same T and place them together, no heat is is exchanged, but the adjacent faces would approach 2T flux, while the opposite faces approach zero flux. Once the system stabilizes, the opposite faces would approach T flux and the true flux of the adjacent faces, the net, would be T as well. The apparent flux of the adjacent faces would depend on your frame of reference. With an infinite number of disks, you would see a bar at temperature T radiating from all faces at TZ/2.

But if the original faces start at T/2, how can the new cylindrical wall of the bar also be at T/2? Elementary my dear Watson, the total energy of the bar is the number of disks times a ThermZit per disk!

If we change the initial disks to T and T/2, energy is transferred from the T disk to the T/2 disk. The total energy 1.5 TZ and will remain constant until energy is lost to the vacuum, which is still perfect, so I am not letting that happen. If no energy is lost or gained, the T/2 disk can only increase to T and will stabilize at (T-T/2)/2 plus its initial T/2 yielding 3T/4 or 0.75T, which is the reason my initial disks were T and 0.75T.

Hang on! I've got an error here I think is due to rounding, but I have to check.

As I mentioned before, 1+1/2+1/4+..1/infinity approaches 2. But since what is actually happening is 1+(1/2+1/4)+(1/4+1/8)+...the real solution would approach infinity which would violate the 2nd law of thermodynamics. So some factor is required between each disk to make the solution approach 2. That factor is 0.996528 to six significant digits. That is the effective emissivity between any two layers of radiant heat transfer. That would be approximately an ideal black body upon observation. It is the relativity thing. this is why examples of two balls in a vacuum appear to be violating the 2nd law, because if this elemental unit of emissivity is not considered they do. Small things matter!

Since space has a temperature of approximately 3K and the TZ factor is in terms of T in units K, then space would have and effective emissivity of 0.99652825^3 or 0.9896208 approximately. This is a touch different than the average black body emissivity noted in the more exact use of the Stefan-Boltzmann relationship of j<>0.924sigma T^{4}, Whether this is actually a more accurate approximation will require a little more research. As it is, it appears to be an accurate enough base line to estimate ideal emissivity between temperature layers in the atmosphere. That explanation, why I was confident that my estimates were sufficiently accurate for use in the the Modified Kimoto Equation, has been a bit of a stumbling block.

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