Talk:Carnot's theorem (thermodynamics)

Modifications
This article is about to be greatly modified by me because of significant errors. --Netheril96 (talk) 05:03, 3 October 2010 (UTC)
 * 1) The Carnot theorem is a product, rather than a stepstone towards establishment, of the second law.I plan to add a section proving it with second law in a few days.
 * 2) Modern engines are not operating between two reservors whose temperatures are constant, let alone performing Carnot cycles.So the statement in section Description "Carnot's theorem sets essential limitations on the yield of a cyclic heat engine such as steam engines or internal combustion engines, which operate on the Carnot cycle. " are actually wrong.In addition, the section Example is misleading because it oversimplifies the problem.The model of inner combustion engines are mostly Diesel cycle in which the temperatures are varying.So I delete the entire Example section.Anyone is welcome to write another example,but don't refer to the REAL ENGINES because they almost never meet the condition of "two reservoirs".
 * 3) Also planning to write a section clarifying the distinction between Carnot engine and Carnot cycle.

New section: applicability to fuel cells
As the fuel cell industry continues to grow, it's important to examine the question of whether Carnot's theorem applies to fuel cells; hence the creation of this new section. If anyone can add definitive information that resolves the controversy one way or the other (does it or doesn't it?), please do. 71.221.123.158 (talk) 03:59, 21 February 2011 (UTC)
 * Kbrose, you summarily deleted the new section on the grounds of "controversial topic, no context, and no valid reliable sources."
 * Since when are controversial topics off-limits for Wikipedia? It has an article on abortion, for example, and I hardly think Carnot's theorem is as controversial as abortion.
 * The first source I cited is Google's browser-friendly transliteration of a PowerPoint file published by Case Western Reserve University itself. Why do you feel this is unreliable?  Would you prefer if I cited the PowerPoint file directly?
 * The second source was an abstract written by K. T. Jacob and Saurabh Jain, found on the web site of the Institut de l’Information Scientifique et Technique. Do you feel that the Institut did not reliably reproduce this abstract? 71.221.123.158 (talk) 05:23, 22 February 2011 (UTC)

Hello, I have come across an article that seems to prove that the Carnot efficiency DOES apply to fuel cells, after all, and thus, the maximum efficiency of a fuel cell is limited by this Carnot efficiency. It is:

"Thermodynamic comparison of fuel cells to the Carnot cycle", International Journal of Hydrogen Energy, Volume 27, Issue 10, October 2002, Pages 1103-1111, Andrew E. Lutz, Richard S. Larson, Jay O. Keller.

Would anyone like to read the relevant sections, and see if it's worth modifying this section of the article? Thanks. 217.127.0.107 (talk) 11:32, 3 June 2012 (UTC)

The statement "the second law of thermodynamics still provides restrictions on fuel cell and battery energy conversion" is cited to a source that, in my reading, says the opposite. The cited document, "Fuel Cell Efficiency Redefined: Carnot Limit Reassessed," which can be found in full at https://iopscience.iop.org/article/10.1149/200507.0629PV/pdf (as opposed to the archive.org link provided in the wiki article) says, in the last sentence of its summary, "These examples, illustrated with the aid of appropriate Ellingham diagrams, clearly establish that fuel cells are not limited by the Carnot factor." It also says, earlier in the text right before the summary, "Within the practical limits, fuel cells have the potential to generate electrical energy with higher thermodynamic efficiencies than heat engines." I could be misreading this, and the rest of the article, but this seems to be saying that fuel cells are not restricted by the second law of thermodynamics, not that they are. I think someone with more experience in this should look at the article in question and determine whether or not the wiki article needs to be changed.McNeuman (talk) 18:14, 14 November 2023 (UTC)

Removal of sections
I removed the sections entitled "'Proof' of D. ter Haar and H.N.S. Wergeland (Elements of Thermodynamics, Addison-Wesley, 1960)" and "What is wrong with the 'proof'".

The sections present an argument of ter Haar and Wergeland, and then highlight an alleged flaw in the argument. Although it can occasionally be helpful, in articles containing proofs, to present a facially appealing but incorrect "proof" of the result (see Cayley-Hamilton theorem), this is not one of those cases. The argument of ter Haar and Wergeland is much more technical and complicated than the presumably correct proof that the article already contains. Therefore, right or wrong, I do not think it has much facial appeal. It need not be included.

In addition, no reliable source is cited for the section "What is wrong with the 'proof'". Without a source, that section violates WP:OR.

It appears that similar edits were made to the Russian version of the page by the same editor. The edits were more apposite there (though still original research) because the principal proof offered in that article is that of ter Haar and Wergeland. That proof did not originally appear in the English version of this article, however, and for good reason -- it is overly technical. It is therefore not necessary to point out a purported disproof. -- 05:22, 12 July 2013 (UTC)

Proposal of new figure
— Preceding unsigned comment added by Guy vandegrift (talk • contribs) 01:38, 12 December 2013 (UTC)

Forgot to sign--guyvan52 (talk) 02:04, 12 December 2013 (UTC)

I just jumped in and switched images. The difference between the images is subtle, but old image will be eventually deleted from Wikimeda: CarnotProofSimplified.jpg (talk) 15:54, 14 December 2013 (UTC)

By attempting to treat both theorems in one figure, we confuse the reader into caring which is which in the second theorem. The figure shown is for the second theorem. I will make another one dedicated to the first. Contrary to earlier claims (by me), there is no possibility that this is "original research" here -- I am just labeling the diagrams more clearly. And contrary to what I originally thought, we need two figures. My rewrite will follow the Ohanian reference shown below, and will be much more clear.

I will also clarify that when running a reversible cycle backwards as a refrigeration unit, low efficiency is a good thing. (That is why they use the COP instead of effeciency when evaluating air conditioners.) An air conditioner with a low COP dumps excess energy into the cold reservoir (i.e. your house in the summertime). But an air conditioner with a low Carnot efficiency dumps more heat into the hot reservoir (which makes it a much better air conditioner). That is why the second law 'bans' reversible engines if they have efficiency that is too low.

Here are the references I will use:


 * 1) The current Wikipedia reference introduces the concept as two theorems, the "first" and the "second" Carnot theorems
 * 2) Tipler's (very well known) text only proves that no engine can be more efficient than Carnots, although a problems at the end explore running the cycle in referse and could be used to introduce Carnot's second theorem.
 * 3) Another good textbook by Ohanian also proves the two versions of the theory separately.
 * 4) Course notes posted on the internet by Royal Holloway, University of London assert wthout proof that the Carnot cycle is the most efficient.
 * 5) Powerpoint presentation posted by a professor at UNL (Nebraska) that treats it as two different theorems.
 * 6) A 12 minute Kahn Academy video carefully presents one theorem and states the results at the other without proof.

Many excellent sources of physics on the internet fail to mention Carnot's theorem, which I find surprising because is one of the few calculations that apply to something other than the ideal gas. Without Carnot's theorem, we have no proof that entropy even exists as a state variable.--guyvan52 (talk) 03:21, 12 December 2013 (UTC)

Why the proof was rewritten
The problem with the old proof lies in the figure, which, while accurate, has a number of undesirable features.
 * 1) The prime superscript $$\eta'$$ forces every reader to re-examine the figure (often more than once). I have chosen the superscripts (M) and (L) for more and less efficiency.  (I would have preferred 'high' and 'low' efficiency, but that would have lead to confusion over whether 'h' meant hot or high.)
 * 2) The representations in a figure should be static.  The label 'Heat Engine' might mislead the reader into thinking that it is never a Carnot engine.  Its status as a reversible engine must be changed to essentially a Carnot engine (since it has the same efficiency.) In contrast, I call it the 'more' efficient engine throughout, although technically it is the 'not-less' efficient engine due to the possibility that $$\eta_M = \eta_L$$. To avoid confusion between 'Carnot' and 'reversible', I never call an engine 'Carnot' until after it is established that 'Carnot' and 'reversible' mean almost the same thing.  This reduces the vocabulary needed to understand the theorem by one word.
 * 3) By referring to the engines as 'more' or 'less' efficient, the one thing that remains static is its most important feature: The figure represents an 'impossible' situation throughout the discussion.   (In deference to Wikipedia's policy of neutrality, 'impossible' is qualified with a link.
 * 4) This is a minor point, but the new proof renders the so-called 'second' theorem more transparent than the 'first' part: Simply place the two reversible engines in the proper order and the violation of the 2nd law is apparent.  It is this part of Carnot's theorem that got me so interested in it.  Students learn that entropy is a state function only through the monatomic ideal gas, where entropy is shown to be a state function. This always left me wondering if there was another substance that would sustain reversible change with a different change in entropy.  The fact that all reversible engines are essentially Carnot cycles resolves this question.

Out of deference to the old way of doing things, I retained the link to the original proof (at http://faculty.wwu.edu/vawter/PhysicsNet/Topics/ThermLaw2/ThermalProcesses.html)

Is this original research? The relabeling of the figure certainly is not. But I could not find another presentation of the theorem that focused on an energy flow diagram that is clearly paradoxical. Nor could I find a discussion that focused more on the equivalence of all reversible heat engines than on the proof that no engine is more efficient than the Carnot cycle. This might be an original way to present the subject, but it is not what I would call original research.guyvan52 (talk) 18:12, 13 December 2013 (UTC)

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More intuitive demonstration
That the maximum efficiency of a heat engine = 1 - Tc/Th can be demonstrated simply. By conservation of energy, the maximum amount of work/energy that can be extracted from a heat engine = difference between heat/energy taken from the hotter reservoir (Qh) and heat/energy lost to the colder reservoir (Qc) = Qh - Qc. The efficiency of an engine is defined as: η = work out/ energy in = (Qh - Qc)/Qh = 1 - Qc/Qh So what is Qc/Qh? By the second law, Entropy can never decrease so, as a minimum, the amount of entropy entering the colder reservoir (dS) must equal the entropy leaving the hotter reservoir (dS), i.e., dS = Qh/Th = Qc/Tc (where Th (resp. Tc) = temperature of the hotter (resp. colder) reservoir). If more heat than this Qc entered the colder reservoir (call it Qc+), less energy would be available for work, and dS = Qc+/Tc would be greater than in the minimum case so overall the entropy change would be positive. Therefore the maximum amount of work is extracted from the engine when the total entropy does not increase, i.e. when Qh/Th = Qc/Tc, which can be rearranged as Qc/Qh = Tc/Th. Therefore for the most efficient engine: η = 1 - Qc/Qh = 1 - Tc/Th. The maximum amount of work/energy is extracted (the engine is most efficient) when the least energy is lost to the cold reservoir, which is precisely when the entropy does not increase. No more work can be extracted for a given input of heat energy Qh between any two temperatures Th and Tc because entropy would then decrease. The Carnot cycle is simply a way of carrying out changes in a working fluid (1. absorption of heat at an original temperature, 2. expansion and decrease in temperature, 3. rejection of heat, 4. compression and increase in back to original temperature) so that work can be extracted cyclically and there is no increase in entropy and no losses due to, e.g., friction or leakage of heat. (The compression and expansion are reversible and adiabatic, hence isentropic. The absorption and rejection of heat are arranged to involve equal and opposite changes in the entropy of the system.) Because some of the energy extracted when the fluid expands is required to raise the fluid back to the original temperature, each cycle extracts less net work than is extracted in the case of a simple 'one-off' process, however both are maximally efficient. Wodorabe (talk) 13:26, 20 December 2017 (UTC)
 * Your comment is well written. But I don't have the time to critically review it for technical accuracy.  My guess is that if you were not a physics major, then your insights might be correct, but not likely to "fit in" with how this subject is typically taught.  If you were a physics major, then you should contact me on my Wikiversity page, because we need people like you.  wikiversity:User talk:Guy vandegrift 01:34, 23 December 2017 (UTC)

Irreversible heat pump
"First, we must point out an important caveat: the engine with less efficiency ( $$\eta _{L}$$) is being driven as a heat pump, and therefore must be a reversible engine. "

I understand that heat pumps (including irreversible) with efficiency less than Carnot's efficiency don't exist, because heat engines with Carnot's efficiency exists, and if heat pumps with with efficiency less than Carnot's efficiency exist then second law of thermodynamics violates.

Why irreversible heat pumps with efficiency grater then Carnot's efficiency don't exist? FeelUs (talk) 13:44, 14 February 2019 (UTC)

Is Carnot's theorem based on caloric theory?
The following is a paragraph from The Feynman Lectures on Physics, Vol. 1

'The results of thermodynamics are all contained implicitly in certain apparently simple statements called the laws of thermodynamics. At the time when Carnot lived, the first law of thermodynamics, the conservation of energy, was not known. Carnot’s arguments were so carefully drawn, however, that they are valid even though the first law was not known in his time! Some time afterwards, '''Clapeyron made a simpler derivation that could be understood more easily than Carnot’s very subtle reasoning. But it turned out that Clapeyron assumed, not the conservation of energy in general, but that heat was conserved according to the caloric theory, which was later shown to be false.''' So it has often been said that Carnot’s logic was wrong. But his logic was quite correct. Only Clapeyron’s simplified version, that everybody read, was incorrect.'

So, it seems Carnot's theorem is not based on Caloric theory if Feynman is correct. — Preceding unsigned comment added by Bojjasaikiran (talk • contribs) 14:11, 25 April 2020 (UTC)