Origin and verification of e=(Th-Tc)/Th

[Tom Booth]

Can anyone tell me where this formula originated and how it was verified? (With citations, please).

Thanks in advance

Tom

 

[exchemist]

Can anyone tell me where this formula originated and how it was verified? (With citations, please).

Thanks in advance

Tom

Hahahaha, 12 years on and still flogging the same dead horse, eh, Tom?

I don't know historically who was the first to write down the efficiency formula in that form. It may have been Carnot, since it follows from Carnot's Theorem, or it may have been Thomson, Kelvin or Rankine, or someone else who first put it in that particular form. But you have an English translation of Carnot's "Réflexions sur la Puissance Motrice du Feu" - you posted a link to it on the thread in that other form. Have you checked whether it appears there?

As for verification, we went over this ad nauseam on the scienceforms.net site, before you were banned for posting in bad faith. You are unlikely to get a very different answer here, I suspect, though I'll be interested to see who else bites.

 

[exchemist]

Hahahaha, 12 years on and still flogging the same dead horse, eh, Tom?

I don't know historically who was the first to write down the efficiency formula in that form. It may have been Carnot, since it follows from Carnot's Theorem, or it may have been Thomson, Kelvin or Rankine, or someone else who first put it in that particular form. But you have an English translation of Carnot's "Réflexions sur la Puissance Motrice du Feu" - you posted a link to it on the thread in that other form. Have you checked whether it appears there?

As for verification, we went over this ad nauseam on the scienceforms.net site, before you were banned for posting in bad faith. You are unlikely to get a very different answer here, I suspect, though I'll be interested to see who else bites.

A bit more on this. As the concept of an absolute temperature scale was due to an 1848 paper by W Thomson (Lord Kelvin), the efficiency formula cast in terms of absolute temperature must be later than that. So not Carnot, who died in 1832. Clausius, perhaps?

 

[DaveC426913]

Gonna guess he's madly in pursuit of a Perpetulant Motion Machine.

 

[exchemist]

Gonna guess he's madly in pursuit of a Perpetulant Motion Machine.

Bingo!

But to be fair, he did come up with a good one: an ice engine! Quite fun to analyse. Turned out to be conceptually similar to Newcomen’s atmospheric steam engine, in that the power stroke was due to a phase change, and the engine takes in heat on the return stroke rather than the power stroke.

And then we had a rather instructive digression into Sadi Carnot’s work, which was based on the flawed concept of “caloric” but nevertheless gave the right thermodynamic answer. Carnot saw caloric as flowing from a higher to a lower temperature and thereby doing work, analogous to a water mill. That concept proved very insightful, although where he was wrong was in thinking the same amount of caloric came out as went in, i.e. like water.

I learnt a lot arguing with Tom. But then he got a bit stubborn and trollish and got chucked out.

 

[DaveC426913]

Bingo!

But to be fair, he did come up with a good one: an ice engine! Quite fun to analyse. Turned out to be conceptually similar to Newcomen’s atmospheric steam engine, in that the power stroke was due to a phase change, and the engine takes in heat on the return stroke rather than the power stroke.

Yeah, a fun idea to play with. I can see some upsides and some downsides.

+:
Ice is safer to store than combustibles.
-:
Its efficiency/effectiveness is limited by the fact that heat capacity / temp is asymmetrical (zero in one direction, unlimited in the other. It's a lot easier to double room temp from 300K to 600K, than it is to halve 300K to 150K.

 

[exchemist]

Yeah, a fun idea to play with. I can see some upsides and some downsides.

+:
Ice is safer to store than combustibles.
-:
Its efficiency/effectiveness is limited by the fact that heat capacity / temp is asymmetrical (zero in one direction, unlimited in the other. It's a lot easier to double room temp from 300K to 600K, than it is to halve 300K to 150K.

In his proposal, the work was done by the expansion of ice on freezing, which is ~8% volume increase. Alternate freezing and thawing caused a piston to reciprocate. So it was very much like the process that causes frost shattering at the top of a mountain, in which water in cracks in the rock progressively jacks the cracks open.

 

[DaveC426913]

In his proposal, the work was done by the expansion of ice on freezing, which is ~8% volume increase. Alternate freezing and thawing caused a piston to reciprocate. So it was very much like the process that causes frost shattering at the top of a mountain, in which water in cracks in the rock progressively jacks the cracks open.

And this happened at what rate? One cylinder cycle per season? I suppose with a high enough gear ratio he's got something there...

 

[exchemist]

And this happened at what rate? One cylinder cycle per season? I suppose with a high enough gear ratio he's got something there...

Oh it wasn't supposed to be a practical machine. His objective (in which he failed, naturally) was to show you could in principle devise a heat engine to run off ambient heat, implying, so he thought, no heat sink. (Perpetual motion of the 2nd kind is his bag, following our good friend of cranks everywhere Nikola Tesla , who thought, somewhere around the turn of the last century, this might be possible).

 

[James R]

Can anyone tell me where this formula originated and how it was verified? (With citations, please).

Surely the historical context isn't too hard to find, if you want it.

The formula follows from analysis of a Carnot engine, which can be shown (using the principles of thermodynamics) to be the most efficient possible heat engine. Thus, the Carnot engine places a firm theoretical limit on the efficiency of any kind of practical engine whose operation depends on a temperature differential between hot and cold "reservoirs".

The theoretical analysis can be found in most introductory texts about thermodynamics. The level of expertise you will need to follow the arguments is approximately second year university physics (with a supplementary level of mathematical expertise sufficient to understand the relevant maths).

I'm not going to go digging for particular citations, since there are so many texts that are readily available. No doubt the relevant "verifications" (derivations of the formula) are also readily available on the web.

 

[Tom Booth]

Surely the historical context isn't too hard to find, if you want it.
...

You would think, but actually no.

I researched the origins of this for months, until I exhausted all available resources at my disposal. Assuming I may have missed or overlooked something, I began asking on science and physics forums such as this with similar non-responsive answers.


So far it seems we can rule out Carnot himself as the originator. That does little to narrow things down.


Also, though I have been reporting off-topic responses the thread continues to grow without any assistance in the way of an actual answer or specific reference or citation.

Can such off-topic material be removed so that anyone caring to actually respond to these questions may do so without having to wade through alot of irrelevant material.

Thanks.

 

[exchemist]

You would think, but actually no.

I researched the origins of this for months, until I exhausted all available resources at my disposal. Assuming I may have missed or overlooked something, I began asking on science and physics forums such as this with similar non-responsive answers.


So far it seems we can rule out Carnot himself as the originator. That does little to narrow things down.


Also, though I have been reporting off-topic responses the thread continues to grow without any assistance in the way of an actual answer or specific reference or citation.

Can such off-topic material be removed so that anyone caring to actually respond to these questions may do so without having to wade through alot of irrelevant material.

Thanks.

It comes I think from Clausius inequality: https://en.wikipedia.org/wiki/Clausius_theorem

By definition, η = W/Qh where W is the work done by the engine and Qh is the amount of heat input to it from the hot reservoir.

But W = Qh-Qc , i.e. the difference between the amount of heat taken in from the hot reservoir and the amount rejected to the cold reservoir (i.e. applying the principle of conservation of energy)

So therefore η = (Qh - Qc)/Qh, or 1- Qc/Qh.

The tricky bit is justifying replacing Qh and Qc by Th and Tc and this is done by reference to Clausius's theorem.

Basically Clausius showed that for the Carnot cycle, Qh/Th + Qc/Tc =0 cf. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7516509/
N.B. he treats Qc as -ve, as it is heat leaving rather than entering the cycle.

So we have Qh/Th = -Qc/Tc, which can be rearranged as Qh/Qc = - Th/Tc

Using Clausius's sign convention, the efficiency , η = 1+ Qc/Qh (+ve because Qc is a -ve quantity in his convention).

So therefore η = 1- Tc/Th, or η = (Th-Tc)/Th for the Carnot cycle. This means η < (Th-Tc)/Th for all real engines.

Now what I can't track down is who first published that derivation. It may well have been Clausius, as his theorem is what is needed to derive it. Or perhaps Lord Kelvin (William Thomson) may have been the first to write it down, as he did a lot of work in this area around the same time.



P.S. I confess to having a bit of fun at your expense, it is true, but then you are a perpetual motion crank and you did get banned from the other forum, so you are not in a great position to demand a lot of respect, quite frankly. But as I say, I very much enjoyed the challenge of your ice engine and also our discussion about Carnot's work, from which I learned a lot. So kudos to you for that.

 

[Tom Booth]

It comes I think from Clausius inequality: https://en.wikipedia.org/wiki/Clausius_theorem

So therefore η = (Qh - Qc)/Qh.

The tricky bit is justifying replacing Qh and Qc by Th and Tc and this is done by reference to the Clausius inequality, as described in the Wiki entry.

Thank you very much.

I agree: "The tricky bit is justifying replacing Qh and Qc by Th and Tc"

I have difficulty seeing the justification for that since Q represents a measurable quantity of energy in joules whereas temperature is the average kinetic energy of an indeterminate quantity of a given substance. There are additional problems in simply substituting temperature for heat.

Heat is energy transfered whereas temperature is a state. There is a conceptual problem in simply substituting one for the other.

But pinning down the origin should help give this some context and maybe help me to put this into some kind of proper perspective.

So I will study up on Clausius and his "inequality". Perhaps that is the missing piece of the puzzle that will make all this make sense.

Thanks!

Tom

 

[exchemist]

Thank you very much.

I agree: "The tricky bit is justifying replacing Qh and Qc by Th and Tc"

I have difficulty seeing the justification for that since Q represents a measurable quantity of energy in joules whereas temperature is the average kinetic energy of an indeterminate quantity of a given substance. There are additional problems in simply substituting temperature for heat.

Heat is energy transfered whereas temperature is a state. There is a conceptual problem in simply substituting one for the other.

But pinning down the origin should help give this some context and maybe help me to put this into some kind of proper perspective.

So I will study up on Clausius and his "inequality". Perhaps that is the missing piece of the puzzle that will make all this make sense.

Thanks!

Tom

Tom, I've just amended my post to include the derivation and one more reference that makes it clearer what Clausius's Theorem says about the Carnot cycle. Suggest you read my amended post before doing anything else. But you are right: Clausius's inequality (or theorem) is the key.

 

[Tom Booth]

Cutting to the chase;

Do you ("you"=anyone caring to offer their opinion here) agree with the statement that the Clausius inequality relies upon or is derived from the kelvin-plank statement?

See for example, timestamp around 3:00 in this lecture:

IMO (tentatively) I had suspected that to be the case while going over the math it appeared that certain values or equations (mathematical constraints) were introduced without explanation but which do not seem to follow from conservation of energy alone.

In other words, if I wish to constrain my math so as to avoid producing some unwanted result, I can draw a "line in the sand", so to speak, by introducing an inequality (greater than or equal to, or vice versa; less than or equal to).

This puts a constraint on my calculations insuring that the mathematics will not cross that boundary.

As such the "inequality" is essentially the mathematical embodiment of preceding conclusions.

Is that a fair statement?

 

[exchemist]

Cutting to the chase;

Do you ("you"=anyone caring to offer their opinion here) agree with the statement that the Clausius inequality relies upon or is derived from the kelvin-plank statement?

See for example, timestamp around 3:00 in this lecture:

IMO (tentatively) I had suspected that to be the case while going over the math it appeared that certain values or equations (mathematical constraints) were introduced without explanation but which do not seem to follow from conservation of energy alone.

In other words, if I wish to constrain my math so as to avoid producing some unwanted result, I can draw a "line in the sand", so to speak, by introducing an inequality (greater than or equal to, or vice versa; less than or equal to).

This puts a constraint on my calculations insuring that the mathematics will not cross that boundary.

As such the "inequality" is essentially the mathematical embodiment of preceding conclusions.

Is that a fair statement?

I took a second look at that link I provided in post 12 and noted the following passage:
QUOTE
Following a “suggestion” of Joule (no other justification was given), in 1851 Thomson took ∼−1. He then obtained, for a Carnot cycle with reservoir temperatures 1 and 2, the equivalent of what I will call the reversible reservoir condition

Q1/T1 + Q2/T2 = 0
(4)


Clausius, in 1854, using the temperature scale , his already-derived ∼−1, and valid but unusual physical reasoning, derived the above reversible reservoir condition.
UNQUOTE

So actually it looks as if both Thomson (Kelvin) in 1851 and Clausius in 1854, came up with the formula relating Q and T that is the "key" equation for the Carnot cycle that we were discussing. By the way, one thing one needs to understand is these guys had not yet come up with an absolute temperature scale,which must have made things very hard for them. In fact it was doing this work that led Thomson to devise one..... which we now call the Kelvin scale.

This text does not mention the "Kelvin-Planck Statement", possibly since that would seem to be a bit ahistorical. From the Wiki article on the 2nd Law:
QUOTE
It is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by ter Haarand Wergeland.[43] This version, also known as the heat engine statement, of the second law states that

It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.[2]
UNQUOTE

So this idea of a joint "Kelvin-Planck Statement" seems to have come along after they each had made their own individual statements of the 2nd Law. (All the many ways the 2nd Law can be stated are listed in the Wiki article.)

Frustratingly, no derivation of the Q1/T1 + Q2/T2 = 0 formula is given.

By the way, from reading that link I have the feeling the author is treading the path we are trying, in our more amateurish way, to tread. It does indeed seem genuinely unclear, with much of this stuff, exactly who stated what for the first time, in what form, and when.

So to return to your specific question, it does not look to me as if Clausius would have relied on what we now call the "Kelvin-Planck Statement" of the 2nd Law in so many words. However he did rely on the principle of the 2nd Law, however expressed. Famously, he used his new concept of the cyclic integral of dQ/T to define "entropy" and entropy change: ΔS ≥ ∫dQ/T , which he said is zero in a reversible process (theoretical only, e.g. Carnot cycle) or >0 in any irreversible (i.e. real) process.

 

[Tom Booth]

Anyway, in my mind this is not in persuit of anything other than improved engine design.

My issue with the Carnot efficiency limit is that on the off chance it happened to be in error, it could hamper engine development as a kind of self-fulfilling prophesy.

Looking at this equation, and designing my engine I would conclude for example that I need to make provision for the "fact" that 85% of the heat going into the engine will end up as "waste heat".

Consequently, I'll incorporate a highly conductive copper lined heat exchanger into the system to siphon off the excess heat as quickly as possible so as to prevent overheating. After all, the greater the temperature difference the higher the efficiency, so the more cooling the better.

Now if such intentional cooling for such a high rate of heat rejection were not actually necessary, what would be the consequence nevertheless?

My intention to improve engine performance and efficiency to the maximum extent possible, (according to the best available math and science), results in my tossing out 85% of the energy supplied to power my engine without ever making any attempt at greater utilization, or designing for a higher rate of conversion.

The general proliferation of this "scientific knowledge" would also tend to inhibit manufacture and marketing of any engine that happened, by chance to slip through the cracks through neglecting to design the engine according to standard mathematical modeling.

Inability to obtain a patent. Investors warned off from investing in an "impossible" engine designed by a "crackpot" etc. etc.

In fact, if such a person, ignorant of "established science" and the higher math were to wander onto a Science and Physics forum, such as this, and display actual video of an experimental model engine that appeared to operate without need for a cold sink to take away all the "excess heat", he would summarily be banned without recourse or trial or any objective, unbiased evaluation of his machine.

The "scientists" from whom he hoped to receive some help, guidance and support, all turning their backs in distain and implacable ridicule.

 

[exchemist]

Anyway, in my mind this is not in persuit of anything other than improved engine design.

My issue with the Carnot efficiency limit is that on the off chance it happened to be in error, it could hamper engine development as a kind of self-fulfilling prophesy.

Looking at this equation, and designing my engine I would conclude for example that I need to make provision for the "fact" that 85% of the heat going into the engine will end up as "waste heat".

Consequently, I'll incorporate a highly conductive copper lined heat exchanger into the system to siphon off the excess heat as quickly as possible so as to prevent overheating. After all, the greater the temperature difference the higher the efficiency, so the more cooling the better.

Now if such intentional cooling for such a high rate of heat rejection were not actually necessary, what would be the consequence nevertheless?

My intention to improve engine performance and efficiency to the maximum extent possible, (according to the best available math and science), results in my tossing out 85% of the energy supplied to power my engine without ever making any attempt at greater utilization, or designing for a higher rate of conversion.

The general proliferation of this "scientific knowledge" would also tend to inhibit manufacture and marketing of any engine that happened, by chance to slip through the cracks through neglecting to design the engine according to standard mathematical modeling.

Inability to obtain a patent. Investors warned off from investing in an "impossible" engine designed by a "crackpot" etc. etc.

In fact, if such a person, ignorant of "established science" and the higher math were to wander onto a Science and Physics forum, such as this, and display actual video of an experimental model engine that appeared to operate without need for a cold sink to take away all the "excess heat", he would summarily be banned without recourse or trial or any objective, unbiased evaluation of his machine.

The "scientists" from whom he hoped to receive some help, guidance and support, all turning their backs in distain and implacable ridicule.

On the contrary, guidance is always forthcoming. You just don't like the form it takes, which is: "Do not waste your time". Nobody is under any obligation to encourage you in delusional pursuits.

The point you still have not understood is that the Carnot cycle efficiency limit is a direct consequence of the Gas Laws. Nobody ever argues that those are wrong - apart from well understood deviations from ideal behaviour due to chemical effects. So why do we get people claiming the Carnot efficiency limit may be wrong? The only answer can be they have not followed the derivation from the Gas Laws, which is in every 6th form physics textbook.

 

[Tom Booth]

On the contrary, guidance is always forthcoming. You just don't like the form it takes, which is: "Do not waste your time". Nobody is under any obligation to encourage you in delusional pursuits.

....

Well, exactly! That too.

Which no doubt discourages 99.9% of any would be inventors from every trying to do anything.


Anyway, thank you for the assistance and the additional confirmation that the Carnot limit equation is based on circular reasoning. Assuming the second law so as to "derive" the second law.

 

[exchemist]

Well, exactly! That too.

Which no doubt discourages 99.9% of any would be inventors from every trying to do anything.


Anyway, thank you for the assistance and the additional confirmation that the Carnot limit equation is based on circular reasoning. Assuming the second law so as to "derive" the second law.

Er no. The second Law is used to derive the Carnot efficiency. The two things are quite different.