Conceptual thermodynamics question

[facenian]

According to Carnot's theorem η≥η' where η corresponds to a reversible machine and η' is not necessarily reversible, so far so good, the problem is that it is ussually assumed that when η' is irreversible the inequality must hold however a could not find a rigorous demostration of this within the context of classical phenomenological thermodynamics, can anyone tell if such a demostration exists and where I can find it?(obs: η is the machine efficiency)

 

[Studiot]

The proof you are seeking depends upon the fact that the machine works in a cyclic process.
Do you understand this as it is most important?

 

[facenian]

I think I do, and I do unerstand the demostration of the inequality derived fron the 2nd law of thermodynamics

 

[Studiot]

So we are equipped to compare the efficiencies of heat engines.

Consider two heat engines, A and B, working between reservoirs H and C.

Let A be a reversible engine so that it takes in Q_H units of heat from H and rejects Q_c units to C, performing W_A units of external work per cycle.

So the efficiency, e_A, = W_A / Q_H, by definition.

If A, being reversible, is now reversed then an input of Q_C units of from C and W_A units will supply Q_H units to H.

Now consider the proposal that B is more efficient than A and let B extract Q_H from H, performing W_B units of external work per cycle.

So the efficiency, e_B, = W_B / Q_H, by definition.

Then we have the condition that

e_B > e_A

$\left( {\frac{{{W_B}}}{{{Q_H}}}} \right) > \left( {\frac{{{W_A}}}{{{Q_H}}}} \right)$ : note we are going to prove this condition false

W_B > W_A

This implies that we can drive the reversed A engine from B and also use the additional external (W_B - W_A) to perform some additional task.

Since we are extracting (via B) and returning (via A) the same amount of heat from the hot reservoir H there will be zero net extraction of heat from H and so it is unecessary.

This is contrary to the experience that forms the Second Law and is based on the original statement of it (by Carnot) that

Two heat engines working together as a single self acting machine are unable to perform external work without a separately maintained heat reservoir.

This proves that at best the efficiency of B cannot be greater than that of A.

This argument can be developed further to show that equality holds for B also reversible, but for B irreversible e_B < e_A

Does this help?

 

[facenian]

This proves that at best the efficiency of B cannot be greater than that of A.

So you yust proved what I said I know at the beginning of my post η≥η'

This argument can be developed further to show that equality holds for B also reversible,

Yes, this leads to η=η' when the primed(') machine is also reversible

but for B irreversible
e_B < e_A

This is my problem you have not proved that the enequality must hold when the primed machine is irreversible, you proved that the inequality may hold when the primed machine is irreversible.

 

[Studiot]

So you yust proved what I said I know at the beginning of my post η≥η'

How do you know this?

Carnot did not propound a theorem.
He put into words what was technical experience at the time.

Ref: Carnot : Reflexions sur la Puissance motrice du Feu.

The whole point is that there is no more fundamental law or axiom of physics to appeal to in order to prove the laws of thermodynamics. They just make sound common sense and have never been observed to be broken.

Carnot's monograph was the first to offer a perpetual motion machine of the second kind and to disbar it.
Such a machine does not contravene the First Law.

Since you were enquiring deeply I thought, in my last line, that you might like to try to extend the line of reasoning to irreversible machines to deduce your inequality for yourself, before doing all the work for you.

Edit : Hint
What are the consequences of B being irreversible? ie what does that mean about the work that B generates?

 

[facenian]

How do you know this?
Edit : Hint
What are the consequences of B being irreversible? ie what does that mean about the work that B generates?

According to your demostration(and all the demostrations I know) you proved that $e_B > e_A$ can not hold when machine B is irreversible and from this we can only conclude $e_B\le e_A$ and that means that machine B can not generate more work than machine A under the same conditions. So my problem is this: can it be demostrated that when machine B is irreversible we must have $e_B < e_A$?
I'm sorry if it is simple but I yust don't see it

 

[Studiot]

OK the enxt step is to interchange the functions of A and B.

This then leads to the conclusion that

e_A ≤ e_B

But we have already shown that

e_B ≤ e_A

The only way they can both be true is if

e_B = e_A

Thus all reversible engines have the same efficiency.

Thus those engines which are not reversible have a lesser efficiency.

 

[facenian]

ok, thanks

 

[Studiot]

Studiot
Thus all reversible engines have the same efficiency.

I should have noted the condition
"Thus all reversible engines have the same efficiency, working between the same heat reservoirs (or temperatures)",
but I'm sure you know that.

 

[facenian]

That's right