1. Lec 17: Carnot principles, entropy

2. For next time: Outline: Important points: Read: § 6-9 to 6-14 and 7-1
HW 9 due October 29, 2003
Outline:
Carnot’s corollaries
Kelvin temperature scale
Clausius inequality and definition of entropy
Important points:
Do not forget the first law of thermodynamics and the conservation of mass – we still need these to solve problems
Kelvin temperature scale helps us find maximum efficiencies for power cycles
Understand how entropy is defined as a system property

3. Carnot’s first corollary
The thermal efficiency of an irreversible power cycle is always less than the thermal efficiency of a reversible power cycle when each operates between the same two reservoirs.

4. Carnot’s first corollary
So, WI  WR, and
So th,I  th,R

5. Carnot’s second corollary
All reversible power cycles operating between the same two thermal reservoirs have the same thermal efficiencies.

6. Carnot’s second corollary
And so

7. Refrigerators and heat pumps
We can show in a manner parallel to that for the Carnot corollaries:
The COP of an actual, or irreversible, refrigeration cycle is always less than the COP for a reversible cycle operating between the same two reservoirs.
2. The COP’s of two reversible refrigerators or heat pumps operating between the same two reservoirs are the same.

8. Kelvin Temperature Scale
The thermal efficiency of all reversible power cycles operating between the same two thermal energy reservoirs are the same.
It does not depend on the cycle or the mechanism.
What can  depend upon?

9. Kelvin Temperature Scale
= (L, H)
where the ’s are temperatures.
It is also true that =1 -(QL/QH)rev, so it must be true that

10. Kelvin Temperature Scale
The previous equation provides the basis for a thermodynamic temperature scale--that is, one independent of the working fluid’s properties, of the cycle type, or any machine.
We are free to pick the function  any way we wish. We will go with the following simple choice:

11. Kelvin Temperature Scale
So,

12. Kelvin Temperature Scale
This only assigns the T ratio. We proceed by assigning Ttp K to the triple point of water. Then, if that is one reservoir,

13. (Reversible) Power Cycles’ Efficiency
With Carnot’s first corollary, the maximum efficiency one can expect from a power-producing cycle is that of a reversible cycle.
Also, recall that for any cycle (reversible or irreversible)

14. (Reversible) Power Cycles’ Efficiency
But So

15. TEAMPLAY
Many power cycles for electricity supply operate between a steam supply reservoir of about 1,000 °F and a heat rejection reservoir of about 70 °F.
What is the maximum thermal efficiency you can expect from such a system?

16. Carnot Efficiency
This maximum thermal efficiency for a power cycle is called the “Carnot Efficiency.”

17. Efficiencies
So, if an efficiency is obtained that is too large, it may be an impossible situation:

18. TEAMPLAY
Problem 6-90E

19. Refrigerators, air conditioners and heat pumps
Hot reservoir at TH
System
Cold reservoir at TL

20. Coefficient of Performance
Refrigerators/Air conditioners

21. For a Carnot refrigerator or air conditioner
Substituting for temperatures

22. Coefficient of Performance for Heat Pumps

23. For a Carnot heat pump
Substituting for temperatures

24. TEAMPLAY
Problem 6-104

25. This is going to seem pretty abstract..so hang on for the ride!

26. Clausius Inequality Another corollary of the 2nd Law.
Now we will deal with increments of heat and work, Q and W, rather than Q and W.
We will employ the symbol , which means to integrate over all the parts of the cycle.

27. Clausius Inequality
The cyclic integral of Q/T for a closed system is always equal to or less than zero.

28. Look at a reversible power cycle
Hot reservoir
System
Cold reservoir

29. Look at a reversible cycle:
We know:
And:

30. For the reversible cycle
Look at Q/T:
Since the heat transfer occurs at constant temperature, we can pull T out of integrals:

31. For the reversible cycle
This allows us to write:

32. What were signs of irreversibilities?
Friction
unrestrained expansion
mixing
heat transfer across a temperature difference
inelastic deformation

33. For an irreversible cycle
Hot reservoir
System
Cold reservoir

34. For an irreversible cycle
For the same heat input:
For both cycles we can write:
and
Apply inequality: or or

35. Apply cyclic integral
For the irreversible cycle:

36. Clausius Inequality And so, we have
Where the  goes with an irreversible cycle and the = goes with a reversible one.

37. Clausius Inequality
Why do we go through the “proof”?…The inequality will lead to a new property.

38. Let’s look at a simple reversible cycle on a p-v digram with two processes
Let A and B both be reversible

39. Find the cyclic integral:
We can then write:

40. What can we conclude? The integral is the same for all reversible
Paths between points (states) 1 and 2. This integral is only a function of the end states and is therefore a property of the system.
We’ll define a new property, entropy as:

41. Entropy
s2 - s1 =
Units are