Reversible Engines and Efficiency: Exploring Carnot's Theorem Corollary

[botee]

Hey there!

I have found an interesting corollary: All reversible engines have the same efficiency $$\eta$$_Carnot.
Well, I tried it for the Otto engine, but it didn`t work. If you have any idea, please share with me!
Thanks!

 

[Andy Resnick]

Can you provide us a synopsis of what you did to calculate the efficiency?

 

[botee]

Sure.
If 1-2 adiabatic, 2-3 isochore, 3-4 adiabatic, 4-1 izochore, so that V₁=V₄>V₂=V₃.
Then the efficiency is $$\eta$$=1-$$\frac{Q_{}41}{Q_{}32}$$, because there is heat exchange only on izochores.
For 1 kmole:
Q₄₁=C_v(T₄-T₁)
Q₃₂=C_v(T₃-T₂)

For the 2 adiabatic processes (use these only if you need the efficiency in terms of volumes):

T₂V₂^{($$\gamma$$-1)}=T₁V₁^{($$\gamma$$-1)}


T₃V₂^{($$\gamma$$-1)}=T₄V₁^{($$\gamma$$-1)}


It follows that:

$$\eta$$=1-(T₄-T₁)/(T₃-T₂)

Well, fine, but the highest and lowest temperatures are T₃ and T₁, so I expected 1-T₁/T₃ for the efficiency. I tried to prove that the two results are equal, but it seems that they are not. Maybe I made some mistakes or whatever...

 

[Mapes]

From the adiabatic equations we can show that

$$\frac{T_1}{T_4}=\frac{T_2}{T_3}$$

This can be used to simplify the efficiency equation to two temperatures.

 

[botee]

From the adiabatic equations we can show that

$$\frac{T_1}{T_4}=\frac{T_2}{T_3}$$

This can be used to simplify the efficiency equation to two temperatures.

Thanks for your reply.
You`re right, but then $$\frac{T_1}{T_4-T_1}=\frac{T_2}{T_3-T_2}$$ so $$\frac{T_4-T_1}{T_3-T_2}=\frac{T_1}{T_2}$$, but T_1 and T_2 are not the highest and the lowest temperatures. Maybe I made some obvious mistakes that I can`t find at the moment :)

 

[Mapes]

Or the corollary is wrong.

 

[botee]

Or the corollary is wrong.

I don`t think so, I saw it in many books but without proof.

 

[Mapes]

Which books?

EDIT: It is true that all reversible engines operating between the same two reservoirs have the same efficiency. But as far as I know, the Otto cycle requires an infinite number of reservoirs to be reversible. So I wouldn't depend on applying the two-reservoir case to the Otto cycle.

 

[botee]

Which books?

EDIT: It is true that all reversible engines operating between the same two reservoirs have the same efficiency. But as far as I know, the Otto cycle requires an infinite number of reservoirs to be reversible. So I wouldn't depend on applying the two-reservoir case to the Otto cycle.

Thanks for your replies. Ok, but if there are an infinite number of reservoirs, among them also should exist one with the highest and one with the lowest temperature.
One of the books I saw this corollary is: Stephen J. Blundell: Concepts in thermal physics. It is also on wikipedia (Ok, that`s not an argument), and on videos from Yale open courses. I insist on this problem so mutch, because it is used when proving that Carnot engine has maximum efficiency. For the proof is used that 1) Carnot engine is reversible and 2) all reversible engines have the same efficiency.

 

[botee]

Ok, finally I understand what you say Mapes. But can you give me an example of reversible engine which works with 2 reservoirs and it is not a Carnot engine?

 

[Mapes]

Honestly, I've never seen a reversible, two-reservoir heat engine called anything other than a Carnot cycle.

 

[RonL]

Honestly, I've never seen a reversible, two-reservoir heat engine called anything other than a Carnot cycle.

Would I be out of order to chip in at this point, with what I believe meets this goal ?

 

[botee]

Would I be out of order to chip in at this point, with what I believe meets this goal ?

All opinions are welcome! Thank you for your interest!

 

[RonL]

All opinions are welcome! Thank you for your interest!

Sorry, looks like I might have posted the wrong thing here, maybe someone else will step up.

RonL

 

[botee]

Sorry, looks like I might have posted the wrong thing here, maybe someone else will step up.

RonL

Hey, I wanted to read that!

 

[RonL]

Hey, I wanted to read that!

I might have been a little quick to delete my post, but having spent a couple of hours looking through the forum yesterday, I found the thread from 2005 that discussed dropping a forum titled, Therory Development, It confirmed my feelings about how my post seem to come across to most people that are old timers, and some new on PF.

I feel I have learned a lot on PF, but old thinking and habits die hard, I just can't find any information in the books or research documentation, that deal with putting the high temperature power system inside the low temperature heat sink. If this does not make sense look at my thread "scroll compressors" it is about the same as what I deleted here.

https://www.physicsforums.com/showthread.php?t=313199

If you have any questions or comments, you might prefer to send a PM.

Thanks
RonL