The document discusses the efficiency of real engines compared to ideal Carnot engines. It proves that the efficiency of any real engine must be less than a Carnot engine operating between the same temperatures. The proof involves hypothetically coupling an engine with higher efficiency than a Carnot engine to a Carnot refrigerator, which would result in a violation of the second law of thermodynamics by extracting heat from a low temperature reservoir and discharging it to a high temperature reservoir without work. Therefore, the hypothesis must be false, and no engine can exceed the efficiency of a Carnot engine operating between the same temperatures. Real engines have lower efficiency than Carnot engines due to irreversible processes.

1. Efficiency of Real Engines
BY
UMAIR ISLAM

2. Efficiency of Real Engines
SECOND LAW OF THERMODYNAMICS ( CLAUSIUS FORM )
The efficiency of any heat engine operating between two specified
temperatures can never exceed the efficiency of a Carnot engine operating
between the same two temperatures.

3. Efficiency of Real Engines
REAL ENGINES
The engines having efficiency less than the efficiency of Carnot Engine are
called Real Engines.

4. Efficiency of Real Engines
HYPOTHESIS
Let us suppose we have an engine, which we call engine X, whose efficiency
𝜺 𝑿 exceeds the Carnot efficiency 𝜺 𝑪.

5. Efficiency of Real Engines
PROOF
Let us couple engine X to a Carnot engine operating backward as a
refrigerator.
As shown in figure

6. Efficiency of Real Engines
PROOF

7. Efficiency of Real Engines
PROOF
Engine X extracts heat 𝑸 𝑯 from the high-temperatu rereservoir
And
Discharges heat 𝑸 𝑳 to the low-temperature reservoir, doing work W in the
process.

8. Efficiency of Real Engines
PROOF
Let this work W drive the Carnot refrigerator, which
Extracts heat 𝑸 𝑳
′
from the low-temperature reservoir
And
Discharges heat 𝑸 𝑯
′
to the high-temperature reservoir.

9. Efficiency of Real Engines
PROOF
The net heat that flows from the low-temperature reservoir due to the
combination of the two devices is 𝑸 𝑳
′
− 𝑸 𝑳
And
The net heat delivered to the high-temperaturereservoir is 𝑸 𝑯
′
− 𝑸 𝑯

10. Efficiency of Real Engines
PROOF
Applying the first law to each device separately,
For Engine X
𝑾 = 𝑸 𝑯 − 𝑸 𝑳
For Carnot Refrigrator
𝑾 = 𝑸 𝑯
′
− 𝑸 𝑳
′

11. Efficiency of Real Engines
PROOF
Equating these two expressions, we find
𝑾 = 𝑸 𝑯 − 𝑸 𝑳 = 𝑸 𝑯
′
− 𝑸 𝑳
′
or, defining 𝑄 to be the difference between 𝑸 𝑯
′
and 𝑸 𝑯 ,
𝑸 = 𝑸 𝑯
′
− 𝑸 𝑯 = 𝑸 𝑳
′
− 𝑸 𝑳

12. Efficiency of Real Engines
PROOF
Our hypothesis is
𝜀 𝑋 > 𝜀 𝐶

13. Efficiency of Real Engines
PROOF
By using definition of efficiency 𝜀 =
𝑊
𝑄 𝑖𝑛
, our hypothesis is equivalent to
𝑊
𝑸 𝑯
>
𝑊
𝑸 𝑯
′

14. Efficiency of Real Engines
PROOF
𝑸 𝑯
′
> 𝑸 𝑯
This is the consequence of hypothesis

15. Efficiency of Real Engines
PROOF
𝑸 = 𝑸 𝑯
′
− 𝑸 𝑯
So
𝑸 > 𝟎

16. Efficiency of Real Engines
PROOF
• Consequence shows that we can build an engine that violates the Carnot
theorem.

17. Efficiency of Real Engines
PROOF
Thus the combination of engine X and the Carnot refrigerator is equivalent
to the perfect refrigerator
As shown in

18. Efficiency of Real Engines
PROOF

19. Efficiency of Real Engines
PROOF
Hear Q is transferred from the low-temperature reservoir at 𝑇𝐿 to the high-
temperature reservoir at 𝑇 𝐻 without external work. This clearly violates the
Clausius form of the second law, and so our original hypothesis 𝜀 𝑋 > 𝜀 𝐶
must be false. The Carnot theorem is therefore a necessary consequence of the
second law.

20. Efficiency of Real Engines
PROOF
How would this argument differ if X were a real engine?
If 𝜺 𝑿 < 𝜺 𝑪, then 𝑸 𝑯
′
> 𝑸 𝑯 would change to
𝑸 𝑯
′
< 𝑸 𝑯
and 𝑸 = 𝑸 𝑯
′
− 𝑸 𝑯 would deduce to Q < 0.

21. Efficiency of Real Engines
PROOF
In this case, we would reverse the directions of the arrows in slide 17, which
would then no longer be a refrigerator. Instead, heat Q would be flowing
from the high-temperature reservoir to the low-temperature reservoir, which
is a natural process and violates no basic law.

22. Efficiency of Real Engines
PROOF
REVERSIBLE PROCESS
If engine X operates on a cycle composed entirely of reversible processes,
then its efficiency is equal to the Carnot efficiency.
𝜺 𝒙 = 𝜺 𝒄

23. Efficiency of Real Engines
PROOF
PARTIALLY IRREVERSIBLE PROCESS
If the cycle partially irreversible, then in effect a portion of the energy
transferred in each cycle is lost, perhaps to friction, and cannot be regained
as useful work. Then
𝜺 𝒙 < 𝜺 𝒄

24. Efficiency of Real Engines
PROOF
PARTIALLY IRREVERSIBLE PROCESS
EXAPLE
It would not be true that all the work W produced by a partially irreversible
engine X would be available to run the refrigerator; some would be lost to
friction or to another cause.