# Carnots theorem corollary

1. Jun 13, 2009

### 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 didnt work. If you have any idea, please share with me!
Thanks!

2. Jun 13, 2009

### Andy Resnick

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

3. Jun 14, 2009

### 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=Cv(T4-T1)
Q32=Cv(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...

4. Jun 14, 2009

### 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.

5. Jun 14, 2009

### botee

Youre 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 cant find at the moment :)

6. Jun 14, 2009

### Mapes

Or the corollary is wrong.

7. Jun 14, 2009

### botee

I dont think so, I saw it in many books but without proof.

8. Jun 14, 2009

### 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.

Last edited: Jun 14, 2009
9. Jun 15, 2009

### botee

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, thats 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.

10. Jun 15, 2009

### 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?

11. Jun 15, 2009

### Mapes

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

12. Jun 15, 2009

### RonL

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

13. Jun 16, 2009

### botee

All opinions are welcome! Thank you for your interest!

Last edited: Jun 16, 2009
14. Jun 17, 2009

### RonL

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

RonL

15. Jun 17, 2009

### botee

Hey, I wanted to read that!!!

16. Jun 17, 2009

### RonL

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.