Guest post by Robert G. Brown
Duke University Physics Department
The Problem
In 2003 a paper was published in Energy & Environment by Hans Jelbring that asserted that a gravitationally bound, adiabatically isolated shell of ideal gas would exhibit a thermodynamically stable adiabatic lapse rate. No plausible explanation was offered for this state being thermodynamically stable – indeed, the explanation involved a moving air parcel:
An adiabatically moving air parcel has no energy loss or gain to the surroundings. For example, when an air parcel ascends the temperature has to decrease because of internal energy exchange due to the work against the gravity field.
This argument was not unique to Jelbring (in spite of his assertion otherwise):
The theoretically deducible influence of gravity on GE has rarely been acknowledged by climate change scientists for unknown reasons.
The adiabatic lapse rate was and is a standard feature in nearly every textbook on physical climatology. It is equally well known there that it is a dynamical consequence of the atmosphere being an open system. Those same textbooks carefully demonstrate that there is no lapse rate in an ideal gas in a gravitational field in thermal equilibrium because, as is well known, thermal equilibrium is an isothermal state; nothing as simple as gravity can function like a “Maxwell’s Demon” to cause the spontaneous stable equilibrium separation of gas molecules into hotter and colder reservoirs.
Spontaneous separation of a reservoir of gas into stable sub-reservoirs at different temperatures violates the second law of thermodynamics. It is a direct, literal violation of the refrigerator statement of the second law of thermodynamics as it causes and maintains such a separation without the input of external work. As is usually the case, violation of the refrigeration statement allows heat engines to be constructed that do nothing but convert heat into work – violating the “no perfectly efficient heat engine” statement as well.
The proposed adiabatic thermal lapse rate in EEJ is:
where g is the gravitational acceleration (presumed approximately constant throughout the spherical shell) and cp is the heat capacity per kilogram of the particular “ideal” gas at constant pressure. The details of the arguments for an adiabatic lapse rate in open systems is unimportant, nor does it matter what cp is as long as it is not zero or infinity.
What matters is that EEJ asserts that 
in stable thermodynamic equilibrium.
The purpose of this short paper is to demonstrate that such a system is not, in fact, in thermal equilibrium and that the correct static equilibrium distribution of gas in the system is the usual isothermal distribution.
The Failure of Equilibrium
In figure 1 above, an adiabatically isolated column of an ideal gas is illustrated. According to EEJ, this gas spontaneously equilibrates into a state where the temperature at the bottom of the column Tb is strictly greater than the temperature Tt at the top of the column. The magnitude of the difference, and the mechanism proposed for this separation are irrelevant, save to note that the internal conductivity of the ideal gas is completely neglected. It is assumed that the only mechanism for achieving equilibrium is physical (adiabatic) mixing of the air, mixing that in some fundamental sense does not allow for the fact that even an ideal gas conducts heat.
Note well the implication of stability. If additional heat is added to or removed from this container, it will always distribute itself in such a way as to maintain the lapse rate, which is a constant independent of absolute temperature. If the distribution of energy in the container is changed, then gravity will cause a flow of heat that will return the distribution of energy to one with Tb > Tt . For an ideal gas in an adiabatic container in a gravitational field, one will always observe the gas in this state once equilibrium is established, and while the time required to achieve equilibrium is not given in EEJ, it is presumably commensurate with convective mixing times of ordinary gases within the container and hence not terribly long.
Now imagine that the bottom of the container and top of the container are connected with a solid conductive material, e.g. a silver wire (adiabatically insulated except where it is in good thermal contact with the gas at the top and bottom of the container) of length L . Such a wire admits the thermally driven conduction of heat according to Fourier’s Law:
where λ is the thermal conductivity of silver, A is the cross-sectional area of the wire, and ΔT=Tb–Tt . This is an empirical law, and in no way depends on whether or not the wire is oriented horizontally or vertically (although there is a small correction for the bends in the wire above if one actually solves the heat equation for the particular geometry – this correction is completely irrelevant to the argument, however).
As one can see in figure 2, there can be no question that heat will flow in this silver wire. Its two ends are maintained at different temperatures. It will therefore systematically transfer heat energy from the bottom of the air column to the top via thermal conduction through the silver as long as the temperature difference is maintained.
One now has a choice:
- If EEJ is correct, the heat added to the top will redistribute itself to maintain the adiabatic lapse rate. How rapidly it does so compared to the rate of heat flow through the silver is irrelevant. The inescapable point is that in order to do so, there has to be net heat transfer from the top of the gas column to the bottom whenever the temperature of the top and bottom deviate from the adiabatic lapse rate if it is indeed a thermal equilibrium state.
- Otherwise, heat will flow from the bottom to the top until they are at the same temperature. At this point the top and the bottom are indeed in thermal equilibrium.
It is hopefully clear that the first of these statements is impossible. Heat will flow in this system forever; it will never reach thermal equilibrium. Thermal equilibrium for the silver no longer means the same thing as thermal equilibrium for the gas – heat only fails to flow in the silver when it is isothermal, but heat only fails to flow in the gas when it exhibits an adiabatic lapse in temperature that leaves it explicitly not isothermal. The combined system can literally never reach thermal equilibrium.
Of course this is nonsense. Any such system would quickly reach thermal equilibrium – one where the top and bottom of the gas are at an equal temperature. Nor does one require a silver wire to accomplish this. The gas is perfectly capable of conducting heat from the bottom of the container to the top all by itself!
One is then left with an uncomfortable picture of the gas moving constantly – heat must be adiabatically convected downward to the bottom of the container in figure 1 in ongoing opposition to the upward directed flow of heat due to the fact that Fourier’s Law applies to the ideal gas in such a way that equilibrium is never reached!
Of course, this will not happen. The gas in the container will quickly reach equilibrium. What will that equilibrium look like? The answer is contained in almost any introductory physics textbook. Take an ideal gas in thermal equilibrium:
where N is the number of molecules in the volume V, k is Boltzmann’s constant, and T is the temperature in degrees Kelvin. n is the number of moles of gas in question and R is the ideal gas constant. If we assume a constant temperature in the adiabatically isolated container, one gets the following formula for the density of an ideal gas:
where M is the molar mass, the number of kilograms of the gas per mole.
The formula for that describes the static equilibrium of a fluid is unchanged by the compressibility (or lack thereof) of the fluid – for the fluid to be in force balance the variation of the pressure must be:
(so that the pressure decreases with height, assuming a non-negative density). If we multiply both sides by dz and integrate, now we get:
Exponentiating both sides of this expression, we get the usual exponential isothermal lapse in the pressure, and by extension the density:
where P0 is the pressure at z=0 (the bottom of the container).
This describes a gas that is manifestly:
- In static force equilibrium. There is no bulk transport of the gas as buoyancy and gravity are in perfect balance throughout.
- In thermal equilibrium. There is no thermal gradient in the gas to drive the conduction of heat.
If this system is perturbed away from equilibrium, it will quickly return to this combination of static and thermal equilibrium, as both are stable. Even in the case of a gas with an adiabatic lapse rate (e.g. the atmosphere) remarkably small deviations are observed from the predicted P(z) one gets treating the atmosphere as an ideal gas. An adiabatically isolated gas initially prepared in a state with an adiabatic lapse rate will thermally equilibrate due to the internal conduction of heat within the gas by all mechanisms and relax to precisely this state.
Conclusion
As we can see, it is an introductory physics textbook exercise to demonstrate that an adiabatically isolated column of gas in a gravitational field cannot have a thermal gradient maintained by gravity. The same can readily be demonstrated by correctly using thermodynamics at a higher level or by using statistical mechanics, but it is not really necessary. The elementary argument already suffices to show violation of both the zeroth and second laws of thermodynamics by the assertion itself.
In nature, the dry adiabatic lapse rate of air in the atmosphere is maintained because the system is differentially heated from below causing parcels of air to constantly move up and down. Reverse that to a cooling, like those observed during the winter in the air above Antarctica, and the lapse rate readily inverts. Follow the air column up above the troposphere and the lapse rate fails to be observed in the stratosphere, precisely where vertical convection stops dominating heat transport. The EEJ assertion, that the dry adiabatic lapse rate alone explains the bulk of so-called “greenhouse warming” of the atmosphere as a stable feature of a bulk equilibrium gas, is incorrect.
“”””” Robert Brown says:
January 27, 2012 at 2:04 pm
PS Prof Bob’s silver wire is too ductile to ring nicely, which is why I chose Aluminium.
Nice post, although by now I’m convinced that it is fruitless. Even Caballero weighing in today hasn’t cut off the nonsense. “””””
So sad Professor Brown.
I was impressed with the elegant simplicity of your figures 1 and 2, that pretty much settle the question, virtually without discussion.
And people have to believe that (thermal) conduction along a solid (even ductile) wire involves the physicsl transport of molecules from one end to another.
They probably think that the Tsunami wave that slams into Hawaii from Japan, actually was water that travelled from Japan.
But think of it this way Robert, if just one curious person got some understanding from your post, that encourages him/er to try and learn some more, then you have extended your class room in a very useful way.
I got out of academia before I got stuck in it, and just looking back on my accomplishments in industry, gives me a lot of satisfaction and no regrets.
ZP says:
January 27, 2012 at 6:19 am
Be glad to. The difference is, the vortex tube actually operates in the real world. On the other hand, the “gravito-thermal” hypotheses violate the Second Law, so they don’t operate at all.
Any other questions, I’ll be happy to help.
w.
Publius Maximus says:
January 27, 2012 at 6:48 am
That was where I went wrong too, Publius. Like you I thought the maximum entropy was when each molecule of air has the same average energy.
Instead, the maximum entropy is when each volume of air has the same average energy. That is the isothermal state. The molecules up high have more total energy because they have more potential energy. But in exactly that proportion, there are fewer molecules up high, so the total energy by volume is unchanged with altitude.
Jelbring makes the same mistake, which is why he, like you (and I when I was wrong) concludes that the equilibrium distribution is adiabatic. The true answer is isothermal, as proven by Dr. Brown above.
w.
An issue I see on this thread is falling in love with a model and therefore believing it to be true. There is no doubt in my mind that the model being described here does exactly what it says, but there is a failure to acknowledge that some of the basic assumptions might not be valid.
That’s why there are so many objections. Everybody agrees on the underlying tenets of conservation of energy, but some are questioning the physical outcome. The “defenders” of the isothermal outcome go back to their model assumptions to argue that their MODEL works (for its assumptions).
This model appears in textbooks.
I have a few shelves full of textbooks. They contain many proofs, very powerful techniques, and give valuable insight into behaviour of real world systems relevant to the field of engineering dynamics. But perhaps the dominant requirement running through them is the assumption of linearity. This allows the analysis to reach great heights, including Laplace Transforms and tests of stability. But all of this needs great care in the real world where there is no such thing as a linear system. Even a simple pendulum has a nasty non linear ordinary differential equation.
That’s the lesson many posters are bringing to the discussion. They are not challenging “are your physics wrong”. They are challenging “are your assumptions justified” and would the conclusion change if you were prepared to budge on some of them.
Why not get off the pot and challenge your assumptions. The following two should be near the front of the queue:
Is the assumption of Fourrier acting as a constraint and misdirecting the analysis? Others have pointed to the micro mechanical nature of conduction. Can this possibly operate exactly the same with or without a uniform force field?
Collisions distribute energy and the most energetic collisions are at certain elevations. What if these are not perfectly elastic. What if each collision is more variable its allocation of energy transfer between momentum and other forms of energy.
If you detach yourself from some key assumptions, you might find that reasonable points are being raised in the objections. If you can’t do that, you might be losing an opportunity to learn something worthwhile.
To simplify this question: Which formalization of the energies of the particles of the atmosphere is correct?
Tim Folkerts postulated two simple states. The first was one where the gas in a box, in Earth’s gravity, have different temperatures (average kinetic energies) at the top of the box and at the bottom of the box. The total average energies of the gases at the top of the box and the gases at the bottom of the box are the same, the higher average potential energy and lower average kinetic energy for gases at the top of the box, and the lower average potential energies and the higher average kinetic energy for gases at the bottom of the box sum to make the same total average.The second box, in Earth’s gravity, is one where the gases at the top of the box have the same average kinetic energies as the gases at the bottom of the box have. In this second box the total average energies, sum of the kinetic and potential energies are different between the gases at the top of the box and the gases at the bottom of the box.
So which box more correctly describes the steady state condition that the gases in the boxes evolve to over time?
Two formalizations have been argued here to show which box is the correct description of the evolved steady state condition of the gases in a box in Earths gravity.
One formalization invokes an imaginary quantity called ‘heat’, and an imaginary condition called ‘equilibrium’, and an imaginary mechanism called a ‘heat engine’, to show that the condition of the gases in the second box is the only and the inevitable evolution of the state of the gases that is possible. This formalization is exactly correct and internally consistent. This formalization does not include quantification of the Earth’s gravity and gravitational potential. This formalization shows that the state of the gases in the second box is the steady state condition that a box of gases in the Earth’s gravity will evolve to.
The second formalization is dependent on one major assumption; that is the assumption that the idea of equal partition of energy to the free moving pieces of the system, the gases in the box in Earth’s gravity, is a correct condition. The other quantities in this formalization is real; gravity, mass and distance and time. By extension, the other realities in this formalization are potential energy, kinetic energy, velocity and momentum. This formalization shows that the state of the gases in the first box is the the condition that a box of gases in the Earth’s gravity will evolve to.
So which of the formalization more exactly describes the state of the gases in a box in Earth’s gravity??
The problem of physics, sometimes more than one ‘correct’ answer to a question… 😉
Willis, you wrote;
“Instead, the maximum entropy is when each volume of air has the same average energy. That is the isothermal state. The molecules up high have more total energy because they have more potential energy. But in exactly that proportion, there are fewer molecules up high, so the total energy by volume is unchanged with altitude.”
Please help me here, where is volume parametized in the equipartition principle?
“Willis Eschenbach says:
January 27, 2012 at 11:13 pm
Ian W says:
January 27, 2012 at 5:34 am
It is apparent that the writer is a physicist and not an engineer.
Take the thought experiment = a very long cylinder with a gas and a silver wire (assumed to be a perfect conductor) running from low in the cylinder to the top of the cylinder and claimed to cause a problem with the lapse rate explanations.
The assumption that is false in this ‘experiment’ is that the silver to gas heat conductivity at the top of the cylinder is the same as that as at the base of the cylinder. This is an error and the assumption is false.
Here’s the key, Ian. Any conduction is sufficient to disprove the theory. Doesn’t matter if the conduction is minimal, or is limited by density. The question is not how much heat will flow. It is if heat will flow at all.”
It seems obvious that a silver wire would conduct heat. Just obvious that one can truck snow from the mountains.
But air does not conduct the heat. You can’t compare air to silver or stone. It’s not just that air is poor conductor. Air molecule is excellent conductor to itself- the molecule velocities are “transferred” at 100% effectively in less than nanosecnd, but the only way energy through air is “conducted” is via air packets- or air doesn’t conduct to itself, it transfers heat from one location to another via movement of air molecules- convection. Other than redistribution or averaging velocity of molecules, air only tranfers energy thru air via convection. So if you have a condition of not having some buoyancy difference of air packets, is no conduction of heat via air.
willis’
If maximum entropy is when each unit volume of atmosphere has the same average energy, where do you stop the atmosphere? How high do you go before you say ” Now we quit, this is where the atmosphere stops”? Does each uppermost unit of volume have one very, very fast molecule in it? What about the next higher unit volumes that don’t have any molecules in them?
I hope this isn’t what Dr Brown is saying 😉
“kuhnkat says:
January 27, 2012 at 11:42 pm
Willis Eschenbach,
You always have at least 1 G affecting the gas in the centrifuge no matter what orientation it is in if it is on the earh. The idea the there would be a lack of pressure at one end is not possible unless the centrifuge is out of the earth’snear surface gravitational field. The fact that you have a huge range of pressure does not change the 1g at the low end at right angles to the centrifuge’s action .
There is no zero pressure zone due to the earth’s gravitational field. You do accept just plain old basic gravity don’t you????”
The vomit comet can give something resembling zero gravity- the preferred term is micro-gravity, as in very little gravity.
But also centrifuge does not act as gravity. Earth gravity is causes an atmospheric weight 10 tons over your head. With centrifuge you making less than inch of air weight more. If had centrifuge which compressed say meter length of air, you might have something vaguely close. take meter length of tube with bottom capped, and rotated that, can one easily get say 10 gee. And this would better than a normal centrifuge which can give hundreds of thousand of gees.
George E Smith wrote;
“And people have to believe that (thermal) conduction along a solid (even ductile) wire involves the physical transport of molecules from one end to another.
They probably think that the Tsunami wave that slams into Hawaii from Japan, actually was water that travelled from Japan.”
Thermal conduction in matter involves the physical transport of kinetic energy by molecules banging into one another.
The tsunami easily propagated across an equal potential surface, the ocean surface. Try propagating the tsunami 600 miles vertically against gravity 😉
Jim Z, I’m finding your comments refreshing! Somebody is actually thinking.
Have you ever approached this from the pressure-density side? I have been giving some time on that aspect. Since T = P/ρ × ~k, and when a lapse is present, this pressure and density curves diverge. Makes sense, that is just the IGL rearranged. If you notice the squiggle before the ‘k’, it is there because gravity itself decreases as the squared inverse of altitude and if there is any influence concerning the heat capacity that also changes with both temperature and possibly pressure. But here I am going to assume ‘k’ is a constant for simplicity.
In order for a tall column to be isothermal the density must identically follow pressure, there can be –no- deviation of the P/ρ ratio, and that gives me some intuitive heartburn. Have you ever thought from that direction? It seems to me if in the space between collisions if the exact tracks of a molecule is assumed to be a straight line, you would have an isothermal column. However, if tracks between collisions have an always –z curvature from gravity, it seems this would have to be some degree of a lapse rate. Can you see it from that direction?
Robert Brown says:
January 27, 2012 at 9:29 am
The assumption that is false in this ‘experiment’ is that the silver to gas heat conductivity at the top of the cylinder is the same as that as at the base of the cylinder. This is an error and the assumption is false.
No, silly beanie. The assumption is that the silver conducts heat to the gas at allfrom the bottom to the top. I don’t really give a damn what the conductivity in the silver is or how good the thermal contact between the silver or the gas is as long as it is nonzero throughout the wire. And bear in mind that I can increase the thermal contact of the silver with the gas at both ends by means of adding a simple heat sink. Good engineering, no?
I do so find this sort of mistake to be embarrassing, don’t you?
====================================================
Actually Robert you have proved my point. To transfer heat from the bottom to the top of the cylinder you need to have more surface area (heat sink) at the top of the cylinder than at the bottom. You can cut this either way – the same conductive area, in which case the transfer will be limited to the lapse rate due to the difference in the number of molecules hitting the silver at the bottom in the dense gas compared to the number of molecules hitting the silver at the top in the less-dense gas; OR, you can try to equalize the number of molecules hitting the silver at the top in the less-dense gas by increasing the surface area of the silver in a ‘heat sink’. In which case, you will find that the ratio of the surface area in contact with the gas at the bottom to the surface area in contact with the gas at the top is the same as the lapse rate.
You have no need to be embarrassed to learn something 😉
Robert Brown says at 1/27 2:04pm:
“At the moment, I think the score is this: 100% of the Ph.D. physicists on list think that equilibrium is isothermal.”
100% of the Ph.D. physicists here are proven incorrect about the column in Fig 1 in top post which is non-isothermal according to part b of the Verkley paper cited by the poster Rodrigo Caballero at 1/27 6:07am, linked by guinganbresil at 1/27 9:38am.
To be correct on the column being isothermal, Fig. 1 needs to be modified to show work is allowed to be done on the atmosphere above and below the column as proved in part a of the Verkley paper. Consistent with experimental evidence.
The count is 2 PhD physicist posters in the thread voting Fig. 1 in top post being isothermal; they are being out voted as incorrect by 2 authors of the Verkley paper, 3 authors of the Velasco paper and an on-line text which all prove Fig. 1 as shown is non-isothermal.
Robert Brown at 1/27 1:48pm:
“You did read Caballero’s actual post on the list earlier today, right? Where he directly stated that an isolated atmosphere in a gravitational field reaches isothermal equilibrium, as first demonstrated by Gibbs some 120 years ago?”
Yes. Poster Rodrigo Caballero is writing about the isothermal case when work is allowed across the top and bottom control volume which is not the case for Fig. 1 in the top post. Poster Rodrigo cites the Verkley paper which proves in part b that the column in Fig. 1 in the top post is non-isothermal b/c it doesn’t allow any work across the control volume. In fig. 1 constant entropy is allowed by 2nd law & is the max. entropy for Fig 1. top post.
But think of it this way Robert, if just one curious person got some understanding from your post, that encourages him/er to try and learn some more, then you have extended your class room in a very useful way.
You’re very kind, but there is an old saying in academia — you can lead a horse to water but you can’t make it think….
I am happy to have discussions and even debates with students, provided that there at least some element of respect. I don’t mean the sort of respect that is old fashioned “teaching by authority”, where you accept and memorize everything I say because I’m the teacher and I said so; I actually tell my students not to believe what I way just because I’m the teacher, to challenge it, to test it, and then when it makes sense or corresponds to their experiences in the labs (which at least spot check the concepts, although such checks cannot be exhaustive) accept it and build a strong conceptual understanding. However, the students enter the class knowing that no matter how smart they are, no matter how clever, no matter how iconoclastic, they are children compared to the collective combined work of not one or two but hundreds, thousands, tens of thousands of the brightest minds our species has yet produced, compared to the work of millions of people that have built and continue to build our base of consistent knowledge. I am teaching not as just myself but as a representative of all the great physicists and other scientists who have built a consistent view of physics that works.
That’s the truly appalling thing about the continuing aspects of this debate. The argument I present is, as you note, simple and elegant. It is literally irrefutable. No reasonable person could read it and not agree given even a crude knowledge of things like thermal conductivity or thermodynamics. A supposedly stable thermal lapse in any isolated system, gravitationally driven or not, that can be exploited to do work violates the second law.
Yet a large number of unreasonable people take it upon themselves to completely reinvent all sorts of physics just to suit the conclusion that they wish to draw, that such a lapse is possible because then they don’t have to give credence to the GHE. They have lost any pretence of objectivity, and there is an appalling lack of respect. Physics, as it now stands, is largely consistent. A large number of very smart people have worked for centuries to make it so. It is not perfect, it is not beyond doubt, but it is beyond reasonable doubt and the onus of proof is very much on anyone who wishes to reject things like “thermodynamics” to do an ENORMOUS amount of work to show that their new “theory” is both consistent and confirmed by experiment.
None of this has been done. People assert absurdities, one after another, without even thinking about the consequences, and without ever doing any actual algebra to support it.
You can’t make them think.
rgb
Robert Brown says
” A supposedly stable thermal lapse in any isolated system, gravitationally driven or not, that can be exploited to do work violates the second law. Yet a large number of unreasonable people take it upon themselves to completely reinvent all sorts of physics just to suit the conclusion that they wish to draw, that such a lapse is possible because then they don’t have to give credence to the GHE.”
The dispute about the isothermal/adiabatic distribution is much older
that any concern over the GHE .
It has never been resolved by experiment.
It should be possible surely to discuss it without splitting into tribal groups .
I have identified people who agree with IPCC science but not necessarily accept the isothermal distribution.
I don’t think either way it has anything to do with the GHE
Nobody who proposes an adiabatic distribution believes it can produce work.
Yet others propose lossless silver cables and magic thermocouples to falsify the adiabatic distribution.
How convincing is that?
Its true that in the rare event of any physics text book mentioning this rather abstruse problem as a working assumption they will choose the isothermal distribution.
thepompousgit says:
January 27, 2012 at 9:08 am
Paul Birch said “What is this latex thingy? Is it a specific markup language?”
TeX is a markup language, LaTeX is a set of extensions to make using it easier; intro here: http://www.latex-project.org/intro.html
DeWitt Payne says:
January 27, 2012 at 1:01 pm
“LaTex … ”
Thanks to both of you. I’ll look into it.
The tsunami easily propagated across an equal potential surface, the ocean surface. Try propagating the tsunami 600 miles vertically against gravity 😉
Obviously you do not understand either waves or the conditions of adiabatic isolation. But let’s simplify it still further.
Try propagating blackbody radiation 600 miles vertically against gravity.
After all, the silver wire can be replaced by a vacuum. Heat a blackbody at the bottom. Channel/reflect the blackbody radiation at the bottom into the vertical reflecting tube. Absorb it at the top (which will clearly happen, see radiative transfer equations, this is hotter to colder). Transmit to fluid.
Now please, explain just how gravity will completely prevent the transfer of blackbody radiation up a distance of 600 miles, or 6000 miles, or 600,000 miles.
I look forward to your explanation of how Maxwell’s Equations are modified in vertical gravitational fields, so that we cannot actually see things like the sun because all those photons must experience the exact same gravitational thermal lapse that a column of adiabatic gas 150 million kilometers high would.
Now do you think you can stop being ridiculous?
Damn, I let myself get sucked in again. And for what? I’m sure that you’ll invent some other bizarre explanation that is contradicted every time one looks up during the day.
rgb
Dr Brown,
I do thank you for your explanation and effort.
Willis Eschenbach says:
January 28, 2012 at 12:34 am
“Instead, the maximum entropy is when each volume of air has the same average energy. That is the isothermal state. The molecules up high have more total energy because they have more potential energy. But in exactly that proportion, there are fewer molecules up high, so the total energy by volume is unchanged with altitude.”
Willis, please don’t replace one fallacy with another. The total energy by volume is not unchanged with altitude. The potential energy (and total energy) per molecule goes up linearly with altitude. But the number density of molecules drops off exponentially with altitude. So the energy per unit volume goes like (z+z0).exp(-z/z0), which is anything but constant. There is a lot more energy low down than high up.
Be glad to. The difference is, the vortex tube actually operates in the real world. On the other hand, the “gravito-thermal” hypotheses violate the Second Law, so they don’t operate at all.
Any other questions, I’ll be happy to help.
This answer is non-responsive. I asked for specific mathematical equations governing the equilibrium position for the gas within the tube at any specific instance in time. The key point about how physicochemical processes occur is that they are constantly striving to reach a state of equilibrium. If there is a lower energy state available, the system will shift towards that position. Inefficiencies, tubulance, etc. confound the problem, but does not change the key point.
Earlier, reference was made to a proof by Gibbs that the column in question would be isothermal if in thermodynamic equilibrium. The section where Gibbs discusses this problem is titled, “The Conditions of Equilibrium for Heterogeneous Masses under the Influence of Gravity,” with the most relevant discussion given on pages 144 and 145.
Gibbs derives the general condition of equilibrium, an equation which is of the same form as I showed earlier. He then asserts that the general condition of equilibrium can be partitioned into two parts: a condition of thermal equilibrium (i.e. dT = 0) and a condition of mechanical equilibrium (i.e. VmdP + Mgdz = 0). In summary, Gibbs invokes the temperature formulation of the zeroth law to simplify the total differential instead of rigorously proving (from a purely mathematical point of view) that the most stable state is indeed isothermal.
Robert, I just re-read your Figure 1 and this didn’t register earlier:
You said: “… internal conductivity of the ideal gas is completely neglected …”
Completely neglected? You mean completely absent.
There is your problem. There is no real conduction, just equipartition. Conduction is a transfer of energy to equalize, to a closer degree, an energy difference. There is no difference (do not forget g). The gravity levels z and z+dz are identical to the particles. The slightly slower upper ones will speed if they move to z to ‘conduct’ but then the mean velocities do match… no transfer. And, those at level z if moved to z+dz they will slow exactly the amount to match the mean velocities there. Properly, actual conduction is not being performed and there cannot be any ‘conduction’ in the field in this case, they are already in thermal equilibrium with the m.g.dh term in place in the velocity components of the z axis.
I’m surprised you never saw that.
Dr Brown says: “Damn, I let myself get sucked in again. ”
Yep — it can get addicting. Many of the same sorts of arguments have come up a year ago, and they will come up again in another year, I am sure. We could always just require some sort of math before responding. Like Jerry Maguire, we could say “Show me the derivation.”
So when someone like Ian W says “you will find that the ratio of the surface area in contact with the gas at the bottom to the surface area in contact with the gas at the top is the same as the lapse rate. ” we could simply reply “If you are sure that is the right answer, then show the mathematical derivation that supports this conclusion.” without even bothering to point out that the lapse rate is assumed to be some universal feature, but I could make the surface areas anything I want.
—————————————
I also love the challenge that “It has never been resolved by experiment.” First of all, there is a good chance that the person making the claim simply does not know the relevant experiments.
Beyond that, often there are related experiment that do indeed show the expected results, just not in exactly the form the person want. For example, the satellite IR spectra looking down at the earth DOES show conclusively that the CO2 warms the earth, but only if you understand enough of the other science involved. It doesn’t show what ELSE might affect the surface temperature. It doesn’t show exactly how changes in CO2 would change the surface temperature. But is does clearly show a warming effect from the CO2 that is present. In this present case, there are centuries of experiments involving gases which are consistent with the laws of thermodynamics as presented in standard textbooks.
Finally, many experiments are just damn difficult to do. Here you would need a column of gas many meters tall, surrounded by walls with insulation a couple orders of magnitude better than air, left undisturbed for several days or weeks. If someone actually proposed doing such an experiment, people would complain (rightly) about wasting money on something that was either already obvious or not important.
Tim Folkerts says
“I also love the challenge that “It has never been resolved by experiment.” First of all, there is a good chance that the person making the claim simply does not know the relevant
experiments.”
Well a number of so called experts have been given every chance to produce an experiment.
Some seem initially promising but on closer reflection with real values they don’t stand up to scrutiny.
I don’t hold out much hope for Dr Browns latest suggestion of a lossless light tube.
Tim is nearer reality when he says
“Finally, many experiments are just damn difficult to do. Here you would need a column of gas many meters tall, surrounded by walls with insulation a couple orders of magnitude better than air, left undisturbed for several days or weeks. If someone actually proposed doing such an experiment, people would complain (rightly) about wasting money on something that was either already obvious or not important.”
Since when did physics investigations restrict themselves to things that are important or saving money.
Whats that story again about the discovery of the electron?
Two things are required before a topic reaches anywhere near the area of ‘settled science’.
1.The theory should fit into some accepted framework.
2. There should be experimental evidence to support the theory.
Of these the second trumps the first and unfortunately we are not there yet.
Dr Brown,
Yes, if thermal radiation is the equilibrium mechanism, I see that you are correct. I apologize for my obtustness, your derivation at the top is a mechanical one.
So radiative transport is the reason the evolved isothermal atmosphere departs from a equipartition of energy distribution? I’m sorry if i missed that earlier.
gbaikie says:
January 28, 2012 at 1:46 am
Wrong, wrong, wrong. Air conducts as well as convects. Otherwise stagnant air would be a perfect insulator. But, of course, it’s not. It’s just that convection is orders of magnitude faster than conduction because the viscosity of air is low. What you fail to consider is what happens after an air packet moves somewhere, in particular, to a surface boundary. It can only transfer energy to or from a surface by conduction. Period. Conduction works something like a Newton’s Cradle. The balls in the middle don’t move, only the balls on the end.
Jim Z says:
January 28, 2012 at 1:26 am
Equipartition only applies to a system in thermodynamic equilibrium. The (correct) argument is that the box with a lapse rate is not in thermodynamic equilibrium so equipartition does not have to apply. Equipartition is not violated in the isothermal box because it applies to the averages over the whole box.
Another point to consider for those folks who still don’t accept that the equilibrium state is isothermal and that if while a lapse rate exists heat somehow wouldn’t conduct from bottom to top is that it would apply to any real thermometer as well. What happens if you stuck a thermometer in the bottom of the column and allowed it to equilibrate and then quickly moved it to the top of the column? If no energy can be transferred to or from the thermometer to the gas at the top of the column, then, by definition, it’s at the same temperature as the bottom. You can claim all you want that the average kinetic energy is lower at the top of the column, but the only way you can know this is to measure the temperature. You don’t have a radar gun capable of measuring the velocity of the individual molecules.
Dewitt Payne,
You were ripping until:
“You can claim all you want that the average kinetic energy is lower at the top of the column, but the only way you can know this is to measure the temperature. You don’t have a radar gun capable of measuring the velocity of the individual molecules.”
Arguing that it ain’t real cause you can’t measure it is a real come down. You mean that before we had accelerators all those particles weren’t real and weren’t doing their thing?? No, we just didn’t know about it. It is called being ignorant, like me.