Irreversible Cycle

1. UNIT – II
SECOND LAW OF THERMODYNAMICS
1

2. Entropy and the Clausius Inequality
The second law of thermodynamics leads to the definition of a new property called
entropy, a quantitative measure of microscopic disorder for a system. Entropy is a
measure of energy that is no longer available to perform useful work within the
current environment. To obtain the working definition of entropy and, thus, the
second law, let's derive the Clausius inequality.
Consider a heat reservoir giving up heat to a reversible heat engine, which in turn
gives up heat to a piston-cylinder device as shown below.

3. 3
E E E
Q W W dE
in out c
R rev sys c
− =
− + =
∆
δ δ δ
( )
δ δ δ
δ δ
W W W
Q W dE
c rev sys
R c c
= +
− =
We apply the first law on an incremental basis to the combined system composed of
the heat engine and the system.
where Ec
is the energy of the combined system. Let Wc
be the work done by the
combined system. Then the first law becomes
If we assume that the engine is totally reversible, then
δ δ
δ
δ
Q
T
Q
T
Q T
Q
T
R
R
R R
=
=
The total net work done by the combined system becomes
δ
δ
W T
Q
T
dE
c R c
= −

4. 4
Now the total work done is found by taking the cyclic integral of the incremental work.
If the system, as well as the heat engine, is required to undergo a cycle, then
and the total net work becomes
If Wc is positive, we have a cyclic device exchanging energy with a single heat
reservoir and producing an equivalent amount of work; thus, the Kelvin-Planck
statement of the second law is violated. But Wc can be zero (no work done) or
negative (work is done on the combined system) and not violate the Kelvin-Planck
statement of the second law. Therefore, since TR > 0 (absolute temperature), we
conclude

5. 5
Let’s look at a simple
irreversible cycle on a p-v
diagram with two processes
P
υ
1
2
.
.
A
B
Let A be
irreversible and B
be reversible

6. 6
Irreversible cycle
0
)
T
Q
AB ≤
δ
∫
By Clausius Inequality
Evaluate cyclic integral
0
T
Q
T
Q
T
Q
2
1 B
2
1 A
cycle
≤



δ
−



δ
=



δ
∫
∫
∫
(non-rev) (rev)

7. 7
Irreversible cycle
For the reversible process, B, dS=δQ/dT,
thus:
0
dS
T
Q
T
Q
2
1
2
1 A
cycle
≤
−



δ
=



δ
∫
∫
∫
Rearranging and integrating dS:
∫ 


δ
≥
∆
2
1 A
T
Q
S

8. 8
Second Law of Thermodynamics
Entropy is a non-conserved property!
∫ 


δ
≥
−
=
∆
2
1 A
1
2
T
Q
S
S
S
This can be viewed as a mathematical
statement of the second law (for a
closed system).

9. 9
We can write entropy change as an
equality by adding a new term:
gen
2
1 A
1
2 S
T
Q
S
S +



δ
=
− ∫
entropy
change
entropy
transfer
due to
heat
transfer
entropy
production
or
generation

10. 10
Entropy generation
• Sgen > 0 is an actual irreversible process.
• Sgen = 0 is a reversible process.
• Sgen < 0 is an impossible process.

11. 11
Entropy transfer and production
• What if heat were transferred from the
system?
• The entropy can actually decrease if
gen
2
1 A
S
T
Q
>



δ
∫
and heat is being transferred away
from the system so that Q is negative.

12. 12
Entropy Production
Sgen quantifies irreversibilities. The
larger the irreversibilities, the greater
the value of the entropy production, Sgen
.
A reversible process will have no entropy
production.

13. 13
Entropy transfer and production
• S2 – S1
> 0, Q could be + or –; if –,
because Sgen is always positive.
< 0, if Q is negative and
= 0 if Q = 0 and Sgen = 0.
= 0 if Q is negative and
gen
2
1 A
S
T
Q
>



δ
∫
gen
2
1 A
S
T
Q
<



δ
∫
gen
2
1 A
S
T
Q
=



δ
∫

14. 14
Isentropic processes
• Note that a reversible (Sgen = 0),
adiabatic (Q = 0) process is always
isentropic (S1 = S2)
• But, if the process is merely isentropic
with S1 = S2, it may not be a reversible
adiabatic process.
• For example, if Q < 0 and gen
2
1 A
S
T
Q
=



δ
∫

15. 15
Entropy generation
• Consider
• What if we draw our system boundaries
so large that we encompass all heat
transfer interactions? We would
thereby isolate the system.
gen
2
1 A
1
2 S
T
Q
S
S +



δ
=
− ∫

16. 16
Entropy changes of isolated systems
• And then
gen
2
1 A
1
2 S
T
Q
S
S +



δ
=
− ∫
0
gen
1
2 S
S
S =
−
•But Sgen≥0. So, the entropy of an
isolated system always increases. (This is
the source of the statement, ‘The world is
running down.’)

17. 17
Entropy
)
s
s
(
x
s
s f
g
f −
+
=
)
T
(
s
)
p
,
T
(
s f
≅
It is tabulated just like u, v, and h.
Also,
And, for compressed or subcooled liquids,

18. 18
The entropy of a pure substance is determined from the tables, just as for any
other property

19. 19
Ts Diagram for Water

20. 20
Ts diagrams
∫
= pdV
w
Work was the area under the curve.
Recall that the P-v diagram was very
important in first law analysis, and that

21. 21
For a Ts diagram
rev
int
T
Q
dS 




 δ
=
TdS
δQ rev
int =
∫
=
2
1
rev
int TdS
Q
Rearrange:
Integrate:
If the internally reversible process also is
isothermal at some temperature To:
S
T
dS
T
Q o
2
1
o
rev
int ∆
=
= ∫

22. 22
On a T-S diagram, the area under the process curve represents the heat
transfer for internally reversible processes
d

23. 23
Entropy change of a thermal
reservoir
For a thermal reservoir, heat transfer occurs
at constant temperature…the reservoir
doesn’t change temperature as heat is
removed or added:
∫
=
∆
T
Q
S
δ
Since T=constant:
T
Q
S =
∆

24. 24
Derivation of Tds equations:
dQ – dW = dU
For a simple closed
system:
dW = PdV
The work is given by:
dQ = dU + PdV
Substituting gives:

25. 25
More derivation….
For a reversible process:
TdS = dQ
Make the substitution for δQ in the energy
equation:
PdV
+
dU
=
TdS
Or on a per unit mass basis:
Pdv
+
du
=
Tds

26. 26
Entropy is a property. The Tds expression
that we just derived expresses entropy in
terms of other properties. The properties
are independent of path….We can use the
Tds equation we just derived to calculate
the entropy change between any two
states:
Tds = du +Pdv
Tds = dh - vdP
Starting with enthalpy, it is possible to
develop a second Tds equation:
Tds Equations

27. 27
Let’s look at the entropy change
for an incompressible
substance:
dT
T
)
T
(
c
ds =
We start with the first Tds equation:
Tds = cv(T)dT + Pdv
For incompressible substances, v ≅ const, so
dv = 0.
We also know that cv(T) = c(T), so we can
write:

28. 28
Entropy change of an
incompressible substance
dT
T
)
T
(
c
s
s
2
1
T
T
1
2 ∫
=
−
1
2
1
2
T
T
ln
c
s
s =
−
Integrating
If the specific heat does not vary with
temperature:

29. 29
Entropy change for an ideal gas
dT
c
dh p
= And
dp
p
RT
dT
c
Tds p −
=
Tds = dh - vdp
Start with 2nd Tds equation
Remember dh and v for an ideal gas?
v=RT/p
Substituting:

30. 30
Change in entropy for an ideal gas
p
dp
R
T
dT
c
ds p −
=
Dividing through by T,
Don’t forget, cp=cp(T)…..a function of
temperature! Integrating yields
1
2
T
T
p
1
2
p
p
ln
R
T
dT
)
T
(
c
s
s
2
1
−
=
− ∫

31. 31
Entropy change of an ideal gas
for constant specific heats:
approximation
• Now, if the temperature range is so
limited that cp ≅ constant (and cv ≅
constant),
1
2
p
p
T
T
ln
c
T
dT
c =
∫
1
2
1
2
p
1
2
p
p
ln
R
T
T
ln
c
s
s −
=
−

32. 32
Entropy change of an ideal gas
for constant specific heats:
approximation
• Similarly it can be shown from
Tds = du + pdv
that
1
2
1
2
v
1
2
v
v
ln
R
T
T
ln
c
s
s +
=
−

33. 33
Entropy change of an ideal gas
for variable specific heats: exact
analysis
1
2
T
T
p
1
2
p
p
ln
R
T
dT
)
T
(
c
s
s
2
1
−
=
− ∫
∫
2
1
T
T
p
T
dT
c
Integrating..
To evaluate entropy change, we’ll
have to evaluate the integral:

34. 34
∫
∫
∫ =
=
−
=
1
2
2
1
T
0
T
p
T
0
T
p
T
T
p
T
dT
c
T
dT
c
T
dT
c
)
T
(
s
)
T
(
s 1
o
2
o
−
=
And so
(T), the reference entropy, is
tabulated in the ideal gas tables for a
reference temperature of T = 0 and p = 1
atm.
Entropy change of an ideal gas
for variable specific heats: exact
analysis
Evaluation of the integral

35. 35
Entropy change of an ideal gas for
variable specific heats: exact
analysis
• Only is tabulated. The
is not.
• So,
dT
cp
∫ dT
cv
∫
1
2
1
o
2
o
1
2
p
p
ln
R
)
T
(
s
)
T
(
s
s
s −
−
=
−

36. 36
Entropy change of an ideal gas
• Note that the entropy change of an ideal
gas, unlike h and u, is a function of two
variables.
• Only the reference entropy, so
, is a
function of T alone.