Hi Philip

First of all, there should be no controversy about efficiency: it's work out (what you get) over heat in (what you pay for). Heat exchange with regenerators is free (internal exchange, you don't have to provide the heat from, say, burning a fuel), hence is not considered in the efficiency.

From all I know, the Stirling cycle, that is the ideal cycle as well as all real life approximations in Stirling engines, always include the regenerator. The ideal cycle has no irreversibilities, exchanges heat externally only with the reservoirs, at reservoir temperature, hence has Carnot efficiency.

 When you consider a cycle of isotherms and isochors without regenerator (I would not call it Stirling), then you have two options:

(a) you still exchange heat only with the two reservoirs at TH, TL, which leads to heat transfer over temperature differences. Hence, the cycle is irreversible (entropy generation in heat transfer), and, quite naturally, the efficiency is lower than that of the--fully reversible--Carnot cycle running between the same two temperatures. 

(b) You use, as you suggest, an infinite series of reservoirs at different temperatures during the isochoric heating and cooling process. Then, the process can still assumed to be reversible. But this cycle cannot be compared to the Carnot engine, because the set-up is different: Carnot exchanges heat only with two reservoirs, while this cycle uses a (infinite?) number of reservoirs at different temperatures. It should be possible, indeed, to slice this process with many reservoirs into pieces, and consider it as a combination of Carnot cycles, each between a different pair of temperatures. 

The Carnot engine is a rather limited model, since it assumes a process just between TWO reservoirs, nothing else. If there are more reservoirs to consider, or heat exchange with a gradually cooling flow, etc, the analysis of cycles based on first and second law becomes more involved, since all reservoirs have to be included (you get a taste in Chap 11 of my book, as linked before).

Carnot cycle is isothems connected with adiabatic  (which is neither isochor nor isobar)

I think Christophe has misunderstood my question. Is there any way I can make it clearer?

Ales: I'm only talking about heat engines.

My point is that according to Carnot's theorem all reversible heat engines with the same Th and Tc should have the same efficiency as the carnot process. For example an engine with two isotherms and two isobars should have the same efficiency as the Carnot (which has two isotherms and two adiabats (yes, I know that)). If I calculate the efficiency, however, I don't get that result in general.

The Carnot cycle equation gives the maximum efficiency achievable (within the constraints given). Any real existing machines may have efficiencies below that valuebut none will exceed it.

Dear U. Dreher: I would be fine with that. The problem is that the textbook on thermodynamics by Blundell gives a corollary to Carnot's theorem (with proof) stating that all revesible processes between the same Th and Tc have the same efficiency as Carnot's process. My calculations, however, say otherwise. My calculations only apply to reversible machines. I don't even consider real machines.

I know about Carnot, not about Blundell. And yes - now your question becomes clearer.

Obviously your calculations go to the extremes, thus for me an infinite pressure doesn,t come unexpected. Would your formulas give a non-infinite pressure solution for an infinite volume?

No, you have to read carefully. The outcome of my calculation is an efficiency. The infinite compression ratio is a limiting case I consider. In that case the efficiency becomes the same as for the Carnot process.

I think I already got you "dilemma": you're considering isobaric/isochoric instead of the adiabatic transitions. Which is quite 'exotic' :)

According to my math understanding, you should get If you set the "Carnot efficiency numbers"  for a non-infinity volume and infinity pressure or for non-infinity pressure and infinity volume.

I'm only a dumb E/E, so I have difficulty to imagine how to get mechanical work done by isobaric or isochoric transitions...

Your machine is not running between two temperatures. If you consider an isobaric process for instance, and you want it to be reversible (necessary for the maximum efficiency)  you need many heat baths at different temperatures. Textbooks are often inaccurate  in this respect. For example P. Atkins (Physical Chemistry, 6th edition, p 103) claims: "All reversible engines have the same efficiency" which, as you point out, is not the case. This is not required either for the second law to hold, as Atkins also claims.

@U. Dreher: you can't get mechanical work done in an isochoric process (the volume remains constant), but the work done by the system in  an isobaric process is just p∆V.

You also need infinite heat capacity heat baths. Here is a nice exercise: take two equal blocks of material with heat capacity C. They are initially at different temperatures, but isolated from the rest of the universe. Run a reversible engine between the two blocks, until the temperature is equalized.  Calculate the final temperature, the amount of work done, and the efficiency. It does not come near the Carnot efficiency.

Thank You Gert! Now we're getting somewhere. I was thinking along those lines, but I needed confirmation (honestly:)). This of course leads to other thoughts. As you say isochoric and isobaric processes require an infinite series of heat baths to be reversible. The only processes that can be reversible without this construction are the isotherm and the adiabat. Therefore the Carnot process is really the only cyclic process between two heat baths that can be reversible. This means that this corollary in Blundell actually only says that all Carnot engines with the same temperatures have the same efficiency. Pretty lame! That's actually clear as soon as we've worked out that the efficiency only depends on the temperatures. My feeling is Blundell should remove this corollary from the text or state clearly how little it actually says. It saddens me that Atkins is no better. All comments welcome! Philip

I agree with your statements completely. I threw away most of the textbooks I used to own, but as far as I remember in many of them similar claims are made.

One of the few things Carnot did entirely correct is the identification of the reversible process, and his requirement that "The necessary condition of the maximum is, then, that in bodies employed tot realize the motive power of heat there should not occur any change of temperature which may not be due to a change of volume"  (S. Carnot, Reflections on the motive power of fire, p13, emphasis by Carnot himself), which indeed means that you have to use adiabats to change the temperature going from high to low, and vice versa; And isotherms in the conversion from heat to work (he does not consider it conversion though, his heat is conserved and it falling from high to low temperature gives the work).

Carnot makes a point in stating that there can be no other materials that are more efficient. All materials have the same maximum efficiency.  On page 12 of the same book he states: "We should then conclude that the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever". His argument is based on the observation that perpetual motion machines are not possible. He never said anything about other cycli as far as I can see. But it opens the way for him to use the (ideal) gas as the medium in  his cycle.

There are more than a few things that can sadden you in Atkins' textbooks. One of the older textbooks I have (by E.A. Guggenheim) does make the remark that only for the Carnot cycle the efficiency is 1-Tmin/Tmax and that "There is sometimes confusion between Carnot's cycle and reversible cycles. It will be observed that Carnot's cycle is a very special case of a reversible cycle" (Thermodynamics, p. 95). But, more important for you may be the fact that on p.94 he sort of proves your statement that adiabats are the only possibility. I can make a scan of that page if you want, but not today.

Yes that would be very interesting!

Thanks,

Philip

Here are the relevant pages from Guggenheim's Thermodynamics, p93-95. Sorry for the quality.

Thanks!

I have addressed this question in my papers: 1. viXra:1310.0200and 2. viXra:1310.0201

These are freely downloadable.

Did you publish these papers in a peer reviewed journal?

Take your clue from the first sentence in viXra:1310.0200 : "Ever since the efficiency of Carnot cycle is defined in such a way that it is less than one, troubles started." 

I attached a more interesting paper.

Dear Prof Phillip J B Koek,

I did not try publishing these and more of my other papers in peer reviewed journals. I found it is not worthwhile to pursue that route.

I hope you found my papers useful.

Dear Prof Gert Van der Zwen,

It is not clear to me what you wanted to convey. I am familiar with the paper you cited and worked extensively on Siedman and Michalik (SM) paper. Their calculation of efficiency suffers from certain drawbacks. I have a long story about my work on SM paper.

Hi there

The full statement on Carnot engines reads something like this: Any reversible engine that exchanges heat only with two reservoirs at TH and TL has a thermal efficiency of 1-TL/TH. This efficiency is independent from the details of the realization of the engine, and the working fluid. Any irreversible engine has lower efficiency. 

The Carnot cycle--two isothermal and two adiabatic processes--is one realization of a Carnot engine, but there are others. Here is one: The ideal Stirling cycle with ideal gas as working fluid, that is a cycle of two isothermal and two isochoric processes. The heat exchange during the isochoric processes cannot be with the reservoirs, since the heat exchange would be irreversible (heat transfer over finite temperature difference between reservoirs and working fluid creates entroy).

The trick with Stirling cycle is to use a regenerator, that stores heat in one of the isochoric processes, and releases it in the other. Then, the heat exchange for the isochoric processes is inside the engine, and heat exchange with the reservoirs is only in the isothermal processes.

Note that for an ideal gas the internal energy depends only on temperature, but not on volume. The isochoric processes run between the same temperatures (TL->TH, or TH->TL), but at different volumes. Since energy depends only on T, the heat exchanged with the regenerator is equal for the two isochoric processes...and in the end, you find Carnot efficiency.

The Ericsson cycle uses a regenerator in a process of isotherms and isobars, and gives Carnot effiicency as well--also for ideal gas as working fluid.

You should find discussions of both cycles in most thermo textbooks. The trick is regeneration, and use of the right working fluid.

Hope this helps,

Henning

Dear Henning, Gert and Radhakrishnamurty.

This regenerator business certainly adds a twist to things! I haven't seen it in textbooks so far, but I see it's discussed in the articles mentioned in this thread. I'll have to have a closer look at this. Thank you very much for your help!

Regenerators are one example of internal heat exchange, which is done widely in power plants, to increase thermal efficiency. You find discussion of all of this in textbooks for Mechanical Engineering Thermodynamics. If your institution subscribes to Springer Link, you might be able to download one from the link below (see Chap 5 for Carnot engines, Chap 8 for Carnot cycle, and Chap 13for Stirling and Ericsson cycles).

http://link.springer.com/book/10.1007%2F978-3-662-43715-5

Dear Radhakrishnamurty.

I've had a look at your publications now.

I think the reason why you get the Carnot-efficiency for all cycles you consider is simply that they are all linked Carnot cycles. Your TS-diagrams only contain isotherms and adiabats. You don't consider any processes that absorb heat other than the isotherms so you cannot get anything but the Carnot-efficiency. Obviously you could approximate an isochor (for example) by a sequence of adiabats and isotherms, but that doesn't change the fact that you are only using adiabats to move from one temperature to the next. You sort of lose the essence of the isochor, which is change of temperature due to heat transfer. I'm not completely clear on this, however. In any case the Salter paper makes a good case that your conclusions are wrong.

I will come to Salter paper that you refer, later. In fact, Seidman and Michalik paper that inspired Salter would be more informative, if you care. But before that, I may bring to your attention (to look into) the method of proof of Classius inequality.

Take an arbitrary reversible cycle and prove delta S cycle is zero for the system.

That perhaps would enable you to see the strength of methods employed in my papers.

Coming to Salter paper:

If you read Salter paper more carefully, you would find in Figure 3 in the appendix, the two cycles consist of only reversible isotherms and adiabats (linked Carnot cycles) and yet the efficiency, Salter finds, to be lowered not because of the presence of reversible isochores or the like, but due to the participation of an intermediate temperature reservoir!

I have more to share on your statement of linking Carnot cycles. Kelvin faulted on linking Carnot cycles!

By the way, what is your stand on efficiency of Stirling heat engine (SHE) ? Is it less than or equal to Carnot heat engine (CHE), when highest and lowest T's of SHE are the same as the T's of CHE? 

Hi.

When I asked my original question I never imagined that there would be any controversy on how to calculate efficiencies of non-Carnot engines. I also posted the same question on Physics Forums with the same result. Scary but interesting!

I'm far from an expert, but I would calculate the efficiency of the Stirling cycle as work per cycle divided by total heat in per cycle. Total heat in is from the higher isotherm and from the isochoric pressure increase. I assume that both these heat contributions come from a series of external thermal reservoirs, so no regenerators in the engine. In that case I get an efficiency that is lower than Carnot between the same limiting temperatures for any finite compression ratio. Only for infinite compression ratio I get the Carnot efficiency.

Dear Prof. Philip J B Koeck

Many people who claim they are experts don't really understand thermodynamics well, so don't worry if you are far from an expert.

When I asked a simple question about efficiency of SHE in comparison with CHE, you gave an elaborate statement concluding that the equality of efficiencies is conditional -

"In that case I get an efficiency that is lower than Carnot between the same limiting temperatures for any finite compression ratio. Only for infinite compression ratio I get the Carnot efficiency." 

But, the answer is unconditionally,EQUAL!

You did not comment on fig 3 of Salter's paper. Two reversible cycles involving only isothermal and adiabatic processes where in more than two heat reservoirs suffer change are shown. Efficiency of  each is lower than that of the corresponding CHE, according to Salter's calculation. You seem to have concluded that my results gave equality of efficiencies of all reversible cycles with a given pair of TH and TL because I considered cycles involving only isothermal and adiabatic processes (which is what Salter, too, did - but did not get equality of efficiencies).

Are you biased?

One final note. If you consider a reversible isochoric process then implicitly you are assuming a regenerator. In other words, it is impossible to carry out a reversible isochoric (or reversible isobaric etc) process without implying a regenerator!

There are philosophical connotations to the issue as well. In a reversible process there are no wastages. Therefore, those processes lead to maximum utilization of the input. 

Let me not prolong this unless you be interested more deeply on the issue.

Hi Philip

First of all, there should be no controversy about efficiency: it's work out (what you get) over heat in (what you pay for). Heat exchange with regenerators is free (internal exchange, you don't have to provide the heat from, say, burning a fuel), hence is not considered in the efficiency.

From all I know, the Stirling cycle, that is the ideal cycle as well as all real life approximations in Stirling engines, always include the regenerator. The ideal cycle has no irreversibilities, exchanges heat externally only with the reservoirs, at reservoir temperature, hence has Carnot efficiency.

 When you consider a cycle of isotherms and isochors without regenerator (I would not call it Stirling), then you have two options:

(a) you still exchange heat only with the two reservoirs at TH, TL, which leads to heat transfer over temperature differences. Hence, the cycle is irreversible (entropy generation in heat transfer), and, quite naturally, the efficiency is lower than that of the--fully reversible--Carnot cycle running between the same two temperatures. 

(b) You use, as you suggest, an infinite series of reservoirs at different temperatures during the isochoric heating and cooling process. Then, the process can still assumed to be reversible. But this cycle cannot be compared to the Carnot engine, because the set-up is different: Carnot exchanges heat only with two reservoirs, while this cycle uses a (infinite?) number of reservoirs at different temperatures. It should be possible, indeed, to slice this process with many reservoirs into pieces, and consider it as a combination of Carnot cycles, each between a different pair of temperatures. 

The Carnot engine is a rather limited model, since it assumes a process just between TWO reservoirs, nothing else. If there are more reservoirs to consider, or heat exchange with a gradually cooling flow, etc, the analysis of cycles based on first and second law becomes more involved, since all reservoirs have to be included (you get a taste in Chap 11 of my book, as linked before).

Dear Radhakrishnamurty.

Now I think I've sorted things out for myself. In the paragraphs following your figures 2 to 5 of the paper called "part A" you actually show that if all input heat comes from an external source the efficiency gets lower the more you slide the lower rectangle to the right. This agrees completely with what Salter writes. Then you say that this is not true because the change to the environment is the same for all 4 cases. This last statement I can accept if the heat exchange at the intermediate temperature is with a regenerator. In that case (if everything is reversible) you actually do get the Carnot efficieny. Here you also agree with Salter. As I see it a reversible Stirling cycle without a regenerator has a lower effeciency than Carnot, but with a reversible regenerator it has the same.

My previous comment on your paper was nonsense, I think.

Thank you prof. Philip J B Koek,

Your post (I'm sure without your knowledge) suffers from internal inconsistency, with regard to reversible Stirling engine without a regenerator.

Let me reiterate what I wrote in my previous post, if it had missed your attention, -  "In other words, it is impossible to carry out a reversible isochoric (or reversible isobaric etc) process without implying a regenerator!"

A reversible Stirling engine without a regenerator is a myth - it is a concept with inbuilt inconsistencies.  

We may talk about an irreversible Stirling engine without a regenerator but not a reversible Stirling engine without a regenerator. 

I may add the following:

Stirling cycle (b) shown in of Fig 1. of Salter's paper corresponds to a reversible cycle. Yet, it has a lower efficiency compared to the Carnot cycle (a) of Fig 1. So, lower efficiency of Stirling cycle according to Salter is not due to irreversibility (or absence of regenerator).

The argument in my paper was: Stirling cycle shown in Salter paper, (b) of Fig 1. and the Carnot cycle shown in (a) of Fig 1 have equal efficiencies.

Take the hybrid cycle (c) of Fig 1. of Salter's paper. That too is a reversible cycle. Its efficiency according to Salter is lower than Carnot and higher than Stirling. This is not true.

All 3 cycles in Fig 1. of Salter's paper are equal.

It would be interesting if you have a look into Seidman and Michalic paper referred in Saltor's paper. I wish Saltor, too, was there in this discussion.

When in the Carnot cycle adiabates are replaced by isochors, the efficiency of the cycle will not change. When calculating the efficiency of a heat engine only the heat exchanged with external heat sources should be considered. The heat of isochors is exchanged reversibly within the engine (heat regeneration). The increase of entropy principle applid for the Carnot gives

0 - Qin/Thigh + Qout/Tlow = 0

hence

Qout/Qin = Tlow/Thigh

I was thinking of engines that only exchange heat with the surroundings when I posted my original question. I didn't even know about regenarators. I agree completely with your analysis of a Sterling engine that uses a regenerator for the isochors. If you look at Salter's paper you'll see that he confirms your result. He also points out that a reversible Stirling engine with regenerators can be regarded as a Carnot engine if one includes the regenerator in the system (the isochors become adiabats). In that sense it's not surprising that the efficiency is the same.

Dear Dr. Koeck,

Stirling cycle efficiency was not much of a problem. It has been well recognized that it has same efficiency as Carnot engine.

The real issue is with respect to Fig. 3 of Salter's paper. It is a cycle with reversible isotherms and reversible adiabats only. The cycles are three temperature cycles. According to Salter those cycles have lower efficiency compared to Carnot cycle. What I showed in my papers was that those cycles have Carnot efficiency! You felt (May 16) that I got it because I combined Carnot cycles. Salter Fig 3 also has such such combination only.

You would find it interesting to find why was the difference arising, if you read Salter paper and my paper. It is a problem worth spending time upon, since you seem to be concerned with this interesting question.

This article may be helpful for you.

Thanks, I'll read it.

Carnot made the most efficient heat engine. Net work taken from the Carnot cycle is the largest possible for a given amount of heat supplied.

• Carnot’s theorem: No real (irreversible) engine operating between 2 heat reservoirs can be more efficient than a Carnot (reversible) engine operating between the same 2 reservoirs.

• Carnot used an ideal gas contained in a cylinder with a movable piston at one end. The cylinder walls and the piston are thermally non–conducting.

• The Carnot cycle consists of 2 adiabatic and 2 isothermal processes, all reversible:

A ---> B

Isothermal expansion of a gas placed in thermal contact with a heat reservoir at temp. Th.

During the process, the gas absorbs heat Qh from the base of the cylinder and does a work WAB in raising the piston.

B ---> C

Adiabatic expansion by replacing the base of the cylinder by a thermally non–conducting wall.

During the process, T falls from Th to Tc and the gas does work WBC in raising the piston.

C ---> D

Isothermal compression by placing the gas in thermal contact with a heat reservoir at temp. Tc.

During the process, the gas expels heat Qc to the reservoir and the work WCD is done on the gas by external agent.

D ---> A

Adiabatic compression by replacing the base of the cylinder by a non–conducting wall.

During the process, T increases from Tc to Th and the work WDA is done on the gas by external agent.