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02 August 2018

The Nested Black Body Shells Model and Extreme Greenhouse Warming

And Lessons from this Model that Show Us How Limited the Greenhouse Effect Actually Is


In a previous post, Critique of The Steel Greenhouse by Willis Eschenbach, I wrote about the Willis Eschenbach thought experiment model of a perfectly conducting sphere closely surrounded by a concentric perfectly conducting shell in which these bodies behave like black body radiators and the only energy loss mechanism is thermal radiation.  That black body thought experiment was not presented as a good model of the greenhouse gas effect for the Earth.  It was presented to illustrate that there is a warming effect due to thermal radiation absorption in the atmosphere, which many call the greenhouse gas effect.  I pointed out that Eschenbach is right that the surrounding shell causes the inner sphere to have a higher temperature than it would if it were only surrounded by vacuum at T = 0 K because it loses radiant energy more slowly.

But Eschenbach makes a very serious error in common with almost all scientists because he believes the surrounding shell radiates the surface of the inner sphere with the same radiation that the outer shell radiates from its outer surface toward T = 0 K surroundings.  I have explained why this is an error in my post Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science is Wrong.  By making this error, he multiplies the photon energy density by a factor of 3 in the space between the sphere and the shell, making this one of many ways that the catastrophic man-made global warming advocates greatly strengthen the greenhouse gas effect.  I pointed out that the current NASA Earth Energy Budget amplifies the photon energy density in the atmosphere even more by a factor of 12.8!  This is critical because the warming effect due to greenhouse gases is proportional to the photon energy density at a wavelength times the absorption cross section at that wavelength.  Consequently, increasing the photon energy density by a factor of 12.8 increases the warming energy by a factor of 12.8 at all wavelengths at which absorption occurs for the gas molecule.

In this post, I will provide the radiative equilibrium solution for a black body sphere with 2 surrounding black body shells.  This might seem important because the mean free path length for absorption of longwave infrared photons that can be absorbed by water vapor and carbon dioxide is very short at the primary absorption wavelengths, though it can be much longer in the wings of those primary absorption wavelengths.  The solution will then be generalized to N surrounding black body shells since the absorption of photons of different wavelengths by a greenhouse gas in the atmosphere can take different numbers of spheres in a model.  In fact, for a given infrared-active molecule, the number of shells is a function of the absorption cross-section as a function of the wavelength because the mean free path length varies over orders of magnitude from the main absorption wavelengths out into the tails or wings of those principal absorption peaks.

Having developed this nested shell model for black body absorbers/radiators, I will then generalize this model to account for a sphere surface which has an emissivity different from the emissivity of the shell surfaces.  Because the absorption of greenhouse gases is over a substantially smaller range of frequencies than is that of the black body material, one might expect a major change in the result for the inner sphere equilibrium temperature based on the effective emissivity/absorptivity of the inner sphere surface and of the nested shell surfaces that might be an analog to greenhouse gases.  This result proves to be very interesting.

The results will inform us of interesting properties of an energy loss problem dominated by radiation loss and absorption.  However, this model is not a good model of the Earth’s greenhouse gas effect.  Energy loss and transport from the Earth’s surface is not dominated by radiant energy loss and absorption.  It is dominated instead by the effects of the Earth’s gravitational field moderated by convection and the evaporation-condensation cycle of water.  The temperature of the Earth’s surface and of each successively higher layer of air in which an infrared-active molecule will absorb the longwave radiation from the Earth’s surface or a lower layer of air is not determined primarily by radiation transport of energy, but by gravity, convection, and the water cycle.

One has to remember that there are many competing effects in determining the Earth’s climate, including many cooling effects by both water vapor and carbon dioxide that are too often underestimated.  Radiative cooling of the surface is much less than NASA and the UN IPCC claim it is.  The alarmist rendition of greenhouse gas warming does everything it can to amplify that effect, to ignore the many cooling effects, and to over-emphasize the role to carbon dioxide relative to that of water vapor.  Finally, I will adapt a portion of the shell model to make an estimate of the total greenhouse gas effect in the real world of the Earth’s climate.  I will then proceed to make an estimate for the size of the greenhouse effect for the first 400 ppm of carbon dioxide in the atmosphere and discuss briefly what one can expect as one adds higher concentrations of carbon dioxide to the atmosphere.  These estimates are not precise, but they are of the proper scale and will inform us that the warmer Earth surface compared to its overall radiative temperature as seen from space is in very little part due to the absorption of longwave radiation by carbon dioxide.  It also makes it clear that the infrared-absorbing effect due to water vapor is also a small fraction of the 33K effect normally attributed to greenhouse gases, though that effect is many times the effect due to carbon dioxide.



The inner core sphere section of unit area has a power input of Q, the temperature of the sphere is TS and the power per unit area of surface it emits is PS.  The first closely surrounding concentric shell has no power input except that radiated by the sphere.  Its own temperature is TO1, which when the power to the sphere is turned on is 0 K and the shell is only then warmed by radiation from the sphere which travels at the speed of light to it.  It radiates no photons toward the sphere, but does radiate photons as its temperature rises toward the second shell with power PO1.  The second concentric shell has the same parameters but with a 2 in the subscript, rather than a 1.  It also starts from T=0K.  Only vacuum exists between the sphere and the planes so that there are no heat losses except by means of thermal radiation.

When the power Q to the sphere is first turned on, the sphere has an initial temperature of TSI, given by the Stefan-Boltzmann Law, since the sphere is at that instant surrounded by T = 0K.

Q = PSI = σTSI4

Let the thermal equilibrium values of each parameter be denoted with the addition of an E in the subscript, then at thermal equilibrium:

Q = PSE = PO1E = PO2E

Q = σ TSE4σ TO1E4 = σ TO1E4σ TO2E4 = σ TO2E4

We can see that

TSI = TO2E

From the right side of the equilibrium equation we see that

TO1E4 = 2 TO2E4

Then plugging this value in the part of the equilibrium equation involving TSE, we have

TSE4 – TO1E4 = TO2E4

TSE4 – ( 2 TO2E4 ) = TO2E4

TSE4 = 3 TO2E4 or TSE = 30.25 TO2E = 30.25 TSI  = 1.3161 TSI, since TO2E = TSI

With one surrounding shell, the equilibrium temperature of the enclosed sphere was given by

TSE = 20.25 TSI = 1.1892 TSI

Thus the second shell causes a sufficient reduction in the cooling rate that the equilibrium temperature of the sphere is 1.067 times higher than it would be with one surrounding shell.  The rise in the sphere temperature is less with each added shell.

The Nth surrounding shell results in a radiative equilibrium sphere temperature of

TSE = (N+1)0.25 TSI

Thus,

For 10 shells: TSE = 1.8212 TSI

For 100 shells: TSE = 3.1702 TSI

For 1000 shells: TSE = 5.6282 TSI

Imagine modeling the absorption of surface radiation by water vapor using such a model.  Of course, water vapor is not a black body absorber or radiator of longwave infrared radiation.  For most of the radiation that it absorbs, it has a mean free path for absorption which is short, though in the wings of an absorption maximum, the absorption cross section can be much lower and the corresponding mean free path is much longer.  But for a crude model, one might say that the absorption mean free path is something like 10 meters.  One might then say that to account for water vapor absorption out to 8000 meters altitude, one needs 800 shells.  Of course, one would think each shell would absorb only a fraction of the power that a black body would and it would emit only a fraction of that energy also.  Carbon dioxide has a longer mean free path and it absorbs a smaller fraction of the power that a black body would compared to water vapor, but it also does not have a relatively sharp cut-off in its density at higher altitudes in the atmosphere.  One might model it with a much smaller fraction of absorption compared to water with a shell every 40 meters but with 300 shells to get to an altitude of 12000 meters.

Let us see what happens in this simple model if we assign an emissivity to the sphere surface, ɛS, and an emissivity to shells representing absorptions by a greenhouse gas, ɛG.  The equations for a two-shell model then become:

Q = ɛS σ TSI4

At equilibrium,

Q = ɛS σ TSE4 - ɛG σ TO1E4 = ɛG σ TO1E4 - ɛG σ TO2E4 = ɛG σ TO2E4

ɛS σ TSI4 =  ɛG σ TO2E4, so TO2E = (ɛS/ɛG)0.25 TSI

TO1E4 = 2 TO2E4

ɛS TSE4 = 3 (ɛG TO2E4) = 3 (ɛS TSI4)

TSE = 30.25 TSI, the same solution for the equilibrium sphere temperature we had for the black body emitters and absorbers.

Consequently, for N greenhouse gas shells we have:  TSE = (N + 1)0.25 TSI just as with N black body shells.

So, the greenhouse gas hypothesis is looking as though it could indeed cause a disastrous increase in the Earth’s surface temperature even though greenhouse gases are far less efficient absorbers and emitters than are black body absorbers, right?

Wrong.  There is a fatal flaw in the model.  That fatal flaw is the assumption that the only way that heat is transported from the surface of the Earth is by means of thermal radiation.  In reality, much more heat is transported up through the atmosphere by means of the water evaporation and condensation cycle and by convection currents.  Let us look once again at the NASA Earth Energy Budget:




There is no back radiation of 100% and the surface radiation given as 117% in terms of the top of the atmosphere solar insolation is hugely exaggerated by NASA.  The real thermal radiation from the surface is the difference between these values or 17%.  This is perhaps a decent value for the loss from the surface itself and of this 12% is lost immediately to space.  This leaves only 5% of thermal radiation from the surface that is absorbed by the atmosphere along with 5% attributed to convection and 25% lost by water evaporation.  So, in this energy balance only 5% / (5% + 5% + 25%) = 0.143 of the surface energy is absorbed by the atmosphere as radiation from the surface.

Even this is a huge overstatement of the fraction of the surface energy carried off by means of transport by radiation from shell to shell.  This is the fraction that is absorbed by the first shell of the many-shell model.  Once that first absorption occurs, the absorbing greenhouse gas molecule passes off the absorbed energy to the nitrogen and oxygen molecules and the argon atoms that collide with it 6.9 billion times a second at sea level and 2.1 billion times a second at 10 Km altitude in the U.S. Standard Atmosphere of 1976.  Yes, the greenhouse gas molecule very quickly comes into temperature equilibrium with the molecules in the same layer of air with it.  These greenhouse gas molecules can radiate thermal energy to the layer of air just above which is slightly cooler, but the time between such radiation events is extremely long compared to the gas collision frequency.  Water molecules radiate only about every 0.2 seconds and carbon dioxide molecules radiate only about once a second.

Consequently, almost all of the 5% of surface energy that is radiated to the first shell and absorbed is thereafter transported by convection.  The assumption in the model above that all energy is transported through the atmosphere as radiation and in stages could not be more wrong.  In fact, while the little bit of further radiation transport from one layer of air to a cooler layer of air a short distance above it would be handled in the nested shell model as a further warming of the surface.  But, this should actually be viewed as a cooling effect, because any energy transported from air layer to air layer is transported to higher altitude faster compared to the alternative transport mechanism of convection.  These realizations constitute a good lesson in the need to check your premises!

Based on almost all of the 5% of surface radiation being absorbed by the first shell and then thereafter being transported by convection, about what surface temperature might we expect?  Recall that in the one shell case, the surface equilibrium temperature is given by

TSE = 20.25 TSI = 1.189 TSI

At a thermal radiation fraction of 0.143, the greenhouse gas effect temperature rise would be about 0.143 (0.189) = 0.027 times TSI.  If one takes TSI = 255 K, the radiative temperature of the Earth system as a whole with respect to space, then the change of temperature attributable to the greenhouse gas effect for our present atmosphere is 

ΔT = 6.9 K

Note that 6.9 K is only 21% of the 33K difference of the surface temperature with respect to the Earth’s effective radiative temperature as seen from space.  Consequently, the claim that this temperature difference is due to the absorption of longwave infrared radiation emitted from the Earth’s surface is exaggerated by nearly a factor of 5.

Almost all of this 6.9K warming effect is due to water vapor, not the 400 ppm of carbon dioxide in the atmosphere.  The absorption by carbon dioxide compared to that by water vapor is about 8 times less.  Thus the portion of the total infrared warming effect due to carbon dioxide is one ninth of 6.9 K, which is 0.8 K.  My recent post Using Heat Transport Powers of the NASA Earth Energy Budget to Prove that Carbon Dioxide has an Insignificant Effect on Surface Temperatures, estimated that the present 400 ppm of CO2 in the atmosphere causes a temperature increase of only 0.2 K, but this is the result of a different approach to the issue and it took into account the increased absorption of solar insolation caused by carbon dioxide in the atmosphere.  The fraction of the 6.9 K net greenhouse gas temperature increase that can be attributed to carbon dioxide overall is then about 3%.

Given that the 0.2 K net warming effect of carbon dioxide is already mostly saturated, additions of further carbon dioxide to the atmosphere will have very little effect on the surface temperature of the Earth.  What is more, the feedback effect of water vapor at present common levels of water vapor is more likely negative than positive and is certainly very small.

The reason that the surface of the Earth is about 33 K warmer than the effective radiative temperature of the Earth system as a whole as seen from space is because most of the energy dissipated by the Earth’s surface and through the troposphere (the lower atmosphere) is transported upward by a combination of water evaporation and convection.  It is very important that transport by thermal radiation is a much more minor actor in the transport of heat in the troposphere.  It is critically important that most of the Earth’s heat is radiated to space from the upper part of the troposphere which is at temperatures that are close to those of the Earth system as observed from space.  This actually forces the air at the altitudes where most of the radiation to space occurs to be near and slightly lower than this low effective radiative temperature of the Earth system.  Once that is a given, then the temperature gradient in our troposphere due to the Earth’s gravitational field dictates the surface temperature with adjustments due to such cooling mechanisms as convection and the effects of the water cycle and clouds.


17 comments:

Anonymous said...

Your solution for the case where the sphere and shells have emissivities not equal to one is obviously wrong. I can provide you with the full solution for a single shell if you like and we can discuss the differences. However, if you simply consider the case of one shell with a blackbody sphere and make the shell have a near zero emissivity and a transmissivity of close to 1, then the solution for the temperature of the sphere should approach the solution for no shells. I'm pretty sure that you will never be able to solve this problem correctly without using the fact that the shells emit in both directions (out and in) equally and according to their temperature and emissivity as indicated using the S-B law.

Steve Titcombe said...

Charles. Thank you for the clarity of the mathematics – I now understand how you determine that the power-generating sphere will become hotter as each successive shell is added around the sphere. Furthermore, I was particularly surprised to see that the ɛS / ɛG term dropped out – really sweet.

However, proper consideration of Radiant Energy Density (something that you made me aware of) remains a problem for me in your solution. I do hope that you don’t think me ignorant: I’m sincerely trying to get to the root of my misunderstanding of this issue. As I see it, the radiant energy density immediately above the surface of that blackbody, at temperature TSE , must be = a TSE^4 , and in this particular case that Radiant Energy Density entirely comprises photons that were forced from that surface then;
a) in a parallel plane model, that same Radiant Energy Density should be seen immediately above the inside surface of the first plate and it is this same Radiant Energy Density which will determine the inside surface temperature of the first plate. On the basis that all of the radiant energy arriving at the inside surface of the first plate is all absorbed (and thermalised into kinetic energy) and is all conducted away from the inside surface across to the outside surface of the first plate without developing a kinetic energy density gradient within the material of the plate then (i) the temperature of the outside surface of the first plate will be the same as the inside surface and (ii) the radiant energy density immediately above the outside surface of the first plate will be the same as the radiant energy density above the inside surface. Therefore, I can only imagine that; T01E = TSE
b) in a spherical model, a diminished Radiant Energy Density having been reduced by a factor of (RSPHERE / RSHELL)^2 should be seen immediately above the inside surface of the first shell and it is this diminished Radiant Energy Density which will determine the inside surface temperature of the first shell. On the basis that all of the radiant energy arriving at the inside surface of the first shell is all absorbed (and thermalised into kinetic energy) and is all conducted away from the inside surface across to the outside surface of the first shell without developing a kinetic energy density gradient within the material of the shell then (i) the temperature of the outside surface of the first shell will be the same as the inside surface and (ii) the radiant energy density immediately above the outside surface of the first shell will be the same as the radiant energy density above the inside surface. Therefore, I can only imagine that;
T01E = ((RSPHERE / RSHELL)^2 )^0.25 TSE = (RSPHERE / RSHELL)^0.5 TSE

I agree that, in the parallel plane model, if the surface temperatures of the power-generating body and the plate were identical then there would be no heat transferred from the power-generating body to the plate i.e. Q=0. However, I still imagine that radiant exitance would still be flowing from power-generating body to the plate (to replenish the radiant energy emitted from the outside surface of the plate). So, as I naively see it, radiant exitance will always be emitted from the power-generating body (at a rate corresponding to the power it’s generating). At the receiving surface of the receiving body, this will all be thermalised into kinetic energy but only some of it be considered as heat (if the receiving body becomes warmer). However, if the other emitting surface(s) of the receiving body are already at the same temperature as the receiving surface of the receiving body then none of this incoming power will be considered as ‘heat’ but will instead all be used to maintain the same temperature by replacing the energy being dissipated from the other emitting surfaces of that body.

I don’t know if my earlier attempts to contact you have been successful – or if you do not want to respond. If this is the case, please say so just once, and I’ll stop bothering you with my thoughts.

Best Regards,
Steve Titcombe

Alan Tomalty said...

I have looked for any references on the internet for your theory of an electromagnetic gradient preventing cooler body IR radiation to a hotter body and cannot find any. Is this your original theory or did you get it from somewhere else? The physicists on Physics forum all say that you are wrong. Since this is a key point for the possibility of back radiation, you need proof of your theory.

Alan Tomalty said...

You said "My recent post Using Heat Transport Powers of the NASA Earth Energy Budget to Prove that Carbon Dioxide has an Insignificant Effect on Surface Temperatures, estimated that the present 400 ppm of CO2 in the atmosphere causes a temperature increase of only 0.2 K, but this is the result of a different approach to the issue and it took into account the increased absorption of solar insolation caused by carbon dioxide in the atmosphere. The fraction of the 6.9 K net greenhouse gas temperature increase that can be attributed to carbon dioxide overall is then about 3%."

Whether the CO2 catches the solar energy before it hits the surface or after should not affect the numbers if there is back radiation. Therefore your 2 methods of calculating the amount of K attributed to CO2 dont agree.


Also in one of your 1st blogs you said that you disagreed with the 33 K warming provided by grenhouse gases and you put the figure at 9K. Now you are back to 33K What gives?

Ultra said...

In your June article you said "and the loss of surface energy due to the evaporation of water are not very well-established numbers." I did the numbers working backward from the hydrological cycle of total evapotranspiration and NASA is definitely in the ballpark with that figure. The only other one I trust is the incoming solar insolation 340 before reflection. . Are there other ones that have been actually measured? Is the 240 outgoing LWIR accurate?
I believe that has been measured.

Anonymous said...

I see that you still have not corrected your post.

The solution for a core with emissivity ec and reflectivity (1-ec) surrounded by a shell with emissivity es and transmissivity (1-es) is:

Ts^4 = (Q/sigma)/(2-es)
Tc^4 = es Ts^4 + Q/sigma/ec = (Q/sigma)(2-es+es*ec)/ec/(2-es)

Would you like more details on how this solution is derived?

TL Winslow said...

Linked to this article in my new climate historyscope, best on the Net.
http://www.historyscoper.com/climatescope.html

Charles R. Anderson, Ph.D. said...

Anonymous: The emissivity for a surface is really just an adjustment for the energy density of the electromagnetic field immediately outside the surface. A gray body will have a surface energy density which is less than that of a black body. The generated photon flux between two surfaces is then driven by the difference in their surface energy densities.

Charles R. Anderson, Ph.D. said...

Steve Titcombe: When the power generating body with an input power of Q emits radiation toward another body starting at T=0K, it will initially emit power at a rate given by

Q = PS = σ TSI^4

Where TSI is its initial temperature, given by the simple application of the Stefan-Boltzmann Law of thermal radiation.

But as the energy density builds up between the power generating body and the first shell plane, the emission from the power generating body decreases continuously until an equilibrium condition is achieved. When an equilibrium condition is achieved, the temperature of the first shell is higher than it was initially (I assumed it was at T=0K to start), so

Q = PSE = σ TSE^4 – σ TO1^4

Now TO1 is greater than zero, so the only way this power of emission can still be equal to Q is if TSE is greater than TSI.

With multiple shell plates, the power absorbed by each is the same and the power absorbed by each is also equal to the power that the last shell outer surface emits to the T=0K environment outside it.

Note that because the temperature of the power generating surface is greater when equilibrium is achieved, the emission of photons from that surface is enough to maintain the energy density in the space between it and the first shell plane without any emission of photons from the surface of the shell plane facing the generating surface. The first shell plane only emits photons toward the second shell if there is one or to T=0K if there is no second shell.

Charles R. Anderson, Ph.D. said...

Alan Tomalty First Comment: I am not aware of anyone offering a similar proof that the prolific and wasteful generation of photons in the model that every surface emits thermal radiation as though it were surrounded only by a T=0K environment. Of course, some scientists have claimed that this is not the case based on the 2nd Law of Thermodynamics, though that really requires the addition of a minimum energy solution. Such minimum energy solutions are very often correct for physics problems, so it should not surprise us that it turns out to be correct here also. However, my approach is from the direct consideration of the energy density of electromagnetic fields and attends to the principal characteristic of a black body cavity, namely that it has a constant energy density given by Stefan's Law.

I provided proof of my theory in the prior post entitled Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science Is Wrong, which is referred to in the article above. If your friends at the Physics Forum say I am wrong, then let them address my proof with an argument rather than just an opinion or some claim of consensus. I clearly showed that their consensus misuse of the Stefan-Boltzmann equation causes a doubling of the energy density of a black body cavity in violation of Stefan's Law. One can also show many instances in which their misuse of the Stefan-Boltzmann Law leads to violations of The Conservation of Energy.

Charles R. Anderson, Ph.D. said...

Alan Tomalty (Second Comment): First of all, there is no back radiation except in the occasional cases when the air at a higher altitude is warmer than that at a lower altitude. In the normal equilibrium case, this is not so. Second, if there were back radiation, the absorption of solar radiation would still matter, since the absorbing molecules would then supposedly radiate the absorbed energy in all directions equally, including toward space and much more importantly, those molecules would transfer almost all of that energy to infrared-inactive molecules which would then primarily transport that energy upward by convection. The interception of solar energy in the atmosphere keeps it from the surface, from which the important cooling mechanisms are all working to transport the surface energy upward through the entire atmosphere, rather than just a portion of the atmosphere.

There is a 33K difference between the Earth system radiative temperature as seen from space and the Earth's average surface temperature, at least approximately. Some people call that entire difference the greenhouse effect. Others call the greenhouse effect just that warming that occurs directly due to infrared active molecules absorbing long wave infrared radiation emitted from the Earth and its atmosphere. There are actually many warming and cooling mechanisms and one can parse them in many ways to include or exclude them from being part of the greenhouse effect. So, consider the way I have done that in each instance in which I have dealt with this issue. One of the key issues is whether one counts the way the Earth's gravitational field creates a temperature gradient in the atmosphere as separate though dependent upon the role of infrared-active gases or one only talks about energy transported through the atmosphere by thermal radiation between layers of air as the greenhouse effect.

Charles R. Anderson, Ph.D. said...

Reply to Ultra: I expect that the top of the atmosphere incoming and outgoing radiation numbers should be about the best numbers we have in terms of direct measurements. I am curious about how you did your calculations on the surface water latent energy of evaporation and what you mean by NASA being in the ballpark.

Anonymous said...

Charles,

Your response is yet again incorrect. I provided you with the solution for the case where the surface is reflective and the shell is transmissive. If the shell is perfectly transmissive then the solution for the core temperature must be the same as if the shell is not there. Hence, your solution is obviously wrong. Why don't you post my solution so that your readers can see the difference?

This is a very simple problem where you are claiming that the correct temperature for the core and shell are DIFFERENT from those determined from radiative transfer theory. Furthermore, when something is not a blackbody it does matter if it is transmissive, reflective, or some combination of the two. If there were a single shell with emissivity of zero and REFLECTIVITY of one, then the temperature of the core would go to infinity.

Sorry Charles, but this is a case where you can't just handwave away your error with words. Your solution gives a nonsensical result.

Charles R. Anderson, Ph.D. said...

Addressing anonymous again: Of course if you have only one shell and it does not absorb infrared radiation emitted by the power generating surface, then it is as though that shell is not there for this problem. However, this model specifically makes the shells either black bodies or gray bodies. For the model to apply, the shell does have to have a temperature which can respond to the radiation from the powered surface through some absorption of energy. Otherwise, as in the case you are claiming, the shells have no interaction at all with the power generating surface and this model in the equations does not address that case.

Charles R. Anderson, Ph.D. said...

To Anonymous who says: "I see that you still have not corrected your post.

The solution for a core with emissivity ec and reflectivity (1-ec) surrounded by a shell with emissivity es and transmissivity (1-es) is:

Ts^4 = (Q/sigma)/(2-es)
Tc^4 = es Ts^4 + Q/sigma/ec = (Q/sigma)(2-es+es*ec)/ec/(2-es)"

No photons are incident upon the core, so its reflectivity is irrelevant. You are thinking in terms of the willy nilly emission of photons model as though every surface emits photons as though it is surrounded only by T=0K environment. That does not apply in the model I am presenting. The only parameter that matters for the gray body is the energy density that it sets up at its surface when it is at temperature T, which is es a Ts^4 in where a is Stefan's Constant.

Anonymous said...

Yes Charles, I know that the shells can be gray bodies. A shell with an emissivity APPROACHING zero is gray body. In that limit the solution to the problem for the temperature of the core must approach the case where there are no shells. Your solution does not agree with this limit. Furthermore, your solution makes no distinction between a gray reflecting shell and a gray transmissive shell.

"No photons are incident upon the core, so its reflectivity is irrelevant."

Come again? Matter at finite temperature emits radiation in the form of photons in a manner that is only dependent on the temperature of the emitting matter. What are you claiming is happening to the photons that are emitted by the inner surface of the shell?

"You are thinking in terms of the willy nilly emission of photons model as though every surface emits photons as though it is surrounded only by T=0K environment."

So, does that mean that you think that the following description is incorrect?

"Although when I say two things at the same temperature, which means you have balance, it doesn't mean they have the same energy in them. It just means it's just as easy to pick energy off of one as to pick it off the other, so as you put them next to each other, nothing apparently happens. They pass energy back and forth equally. The net result is nothing."

Are you trying to claim that when two objects are at equal temperature they are NOT passing energy back and forth equally?

"That does not apply in the model I am presenting."

Right. You have made up a model that does not agree with established physics. The reason that matter emits photons is due to the "vibration" of charge at fintite temperature. How that matter vibrates and emits does not depend on the temperature of the matter that the object is emitting to.

Again, this is a diversion. The simple fact is that your formula produces an absurd result for the case where the emissivity of the shell is 10^-10.

Charles R. Anderson, Ph.D. said...

Reply to Anonymous: I was not clear in the sentence you quoted stating that no photons were incident upon the core. I was intending to say that no photons generated by the inner surface of the shell were incident upon the core surface, since the inner surface of the shell does not generate photons because the energy density at the surface of the inner shell is lower than that at the surface of the core. But even this qualification on my part is not adequate. I will further correct myself below.

As for the case in which a shell has an emissivity that approaches zero, then if the energy Q is to flow through it and beyond it, the temperature of the shell becomes very high. One can introduce a direct transmission of energy radiated from the core by making the shell perforated, but that was not built into the model I was developing.

Neither did the model I was developing assume any core absorption and re-emission of photons from the core to compensate for any absorption, but one could easily add that as well. It does not even matter whether photons are emitted from the shell actually, as long as the energy density at the core surface is es aT^4 and that at the inner and outer walls of the shell are eg aT^4. In other words, one can have a certain amount of photon generation by the shell, but any photon generation by the shell is offset by a decrease of photon generation from the core. The overall energy density between core and shell is fixed by the electromagnetic field gradient between them and we know the energy density of that field at the surfaces of both the core and the shell surfaces.