1. 1
Entropy
The 2nd law states that process occur in a certain direction, not in any
direction.
It often leads to the definition of a new property called entropy,
which is a quantitative measure of disorder for a system.
Entropy can also be explained as a measure of the unavailability of
heat to perform work in a cycle.
This relates to the 2nd law since the 2nd law predicts that not all
heat provided to a cycle can be transformed into an equal amount of
work, some heat rejection must take place.
2. 2
Entropy Change
The entropy change during a reversible process is defined
as
For a reversible, adiabatic process
dS
S S
=
=
0
2 1
The reversible, adiabatic process is called an isentropic process.
3. 3
Entropy Change and Isentropic Processes
The entropy-change and isentropic relations for a process
can be summarized as follows:
i. Pure substances:
Any process: Δs = s2 – s1 (kJ/kg⋅K)
Isentropic process: s2 = s1
ii. Incompressible substances (liquids and solids):
Any process: s2 – s1 = cav T2/T1 (kJ/kg
Isentropic process: T2 = T1
4. 4
iii. Ideal gases:
a) constant specific heats (approximate treatment):
s s C
T
T
R
v
v
v av2 1
2
1
2
1
− = +, ln ln
2 2
2 1 ,
1 1
ln lnp av
T P
s s C R
T P
− = −
for isentropic process
2 1
1 2.
k
s const
P v
P v=
=
for all process
7. 7
Example
Steam at 1 MPa, 600°C,
expands in a turbine to 0.01
MPa. The isentropic work
of the turbine is 1152.2
kJ/kg. If the isentropic
efficiency of the turbine is
90 percent, calculate the
actual work. Find the
actual turbine exit
temperature or quality of
the steam.
Solution:
( )
1 2
,
1 2
,
0.9 1153
1037.7
a a
isen T
s s
a isen T s
kJ
kg
w h h
w h h
w w
η
η
−
= =
−
= ×
=
=
Theoretically:
8. 8
11
1 1 .
2
2
2 1 .
2
1
3698.61
600 8.0311
2
.
0.01
0.984
8.0311
2545.6
kJ
kg
o kJ
kg K
skJ
s kg K kJ
s kg
State
hP MPa
T C s
State s
sat mixture
P MPa
x
s s
h
==
= =
=
=
= = =
Obtain h2a from Wa
1 2
2 1
2660.9
a a
a a
kJ
kg
w h h
h h w
= −
= −
=
2
2 2
2
0.01 sup
2660.9 86.85okJ
a akg
State a
P MPa erheated
h T C
=
= =
11. where the equality holds for an internally reversible process and the inequality
for an irreversible process. We may conclude from these equations that the
entropy change of a closed system during an irreversible process is greater than
the integral of δQ/T evaluated for that process. In the limiting case of a reversible
process, these two quantities become equal. We again emphasize that T in these
relations is the thermodynamic temperature at the boundary where the differential
heat δQ is transferred between the system and the surroundings.
Note that the entropy generation Sgen is always a positive quantity or zero. Its
value depends on the process, and thus it is not a property of the system. Also, in
the absence of any entropy transfer, the entropy change of a system is equal to
the entropy generation
12. Equation 2 has far-reaching implications in thermodynamics. For an isolated
system (or simply an adiabatic closed system), the heat transfer is zero, and Eq. 2
reduces to
∆Sisolated ≥ 0
This equation can be expressed as the entropy of an isolated system during a
process always increases or, in the limiting case of a reversible process, remains
constant. In other words, it never decreases. This is known as the increase of
entropy principle. Note that in the absence of any heat transfer, entropy change is
due to irreversibility's only, and their effect is always to increase entropy.
13. Entropy is an extensive property, and thus the total entropy of a system is equal to
the sum of the entropies of the parts of the system. An isolated sys-tem may consist
of any number of subsystems . A system and its surroundings, for example,
constitute an isolated system since both can be enclosed by a sufficiently large
arbitrary boundary across which there is no heat, work, or mass transfer .
Therefore, a system and its surroundings can be viewed as the two subsystems of
an isolated system, and the entropy change of this isolated system during a process
is the sum of the entropy changes of the system and its surroundings, which is equal
to the entropy generation since an isolated system involves no entropy transfer.
That is, Sgen = ∆Stotal = ∆Ssys + ∆Ssurr ≥ 0 ------ Eq. 3
14. • where the equality holds for reversible processes and the inequality for
irreversible ones. Note that Ssurr refers to the change in the entropy of the
surroundings as a result of the occurrence of the process under consideration.
• Since no actual process is truly reversible, we can conclude that some entropy is
generated during a process, and therefore the entropy of the universe, which can
be considered to be an isolated system, is continuously increasing. The more
irreversible a process, the larger the entropy generated during that process. No
entropy is generated during reversible processes (Sgen _ 0).
15. Entropy increase of the universe is a major concern not only to engineers but
also to philosophers, theologians, economists, and environmentalists since
entropy is viewed as a measure of the disorder (or “mixed-up-ness”) in the
universe.
The increase of entropy principle does not imply that the entropy of a sys-tem
cannot decrease. The entropy change of a system can be negative during a
process (Fig. 3), but entropy generation cannot. The increase of entropy
principle can be summarized as follows:
Sgen > 0 Irreversible process
Sgen = 0 Reversible process
Sgen < 0 Impossible process
18. • The specific heats of ideal gases, with the exception of monatomic gases, depend
on temperature, and the integrals in Eqs. 3 and 4 cannot be performed unless the
dependence of cv and cp on temperature is known. Even when the cv(T ) and cp(T )
functions are available, performing long integrations every time entropy change
is calculated is not practical. Then two reasonable choices are left: either
perform these integrations by simply assuming constant specific heats or
evaluate those integrals once and tabulate the results. But here we are going to
present variable specific heats (Exact Analysis)
19.
20.
21.
22. Available Energy
• There are many forms in which an energy can exist. But even under ideal
conditions all these forms can not be converted completely into work. This indicates
that energy has two parts:
-Available part
-Unavailable part
• ‘Available energy’ or‘Exergy’: is the maximum portion of energy which could be
converted into useful work by ideal processes which reduce the system to a dead
state(a state in equilibrium with the earth and its atmosphere).
-There can be only one value for maximum work which the system alone could do
23. • ‘Unavailable energy’ orAnergy’: is the portion of energy which could not be
converted into useful work and is rejected to the surroundings
A system which has a pressure difference from that of surroundings, work can be
obtained from An expansion process, and if the system has a different
temperature, heat can be transferred to a cycle and work can be obtained. But
when the temperature and pressure becomes equal to that of the earth, transfer of
energy ceases, and although the system contains internal energy, this energy is
unavailable
•The theoretical maximum amount of work which can be obtained from a system
at any state p1 and T1 when operating with a reservoir at the constant pressure
and temperature p0 and T0 is called ‘availability’.
24. First Law of Thermodynamics (law of energy conservation) used for may analyses
performed Second Law of Thermodynamics simply through its derived property -
entropy (S) Other ‘Second Law’ properties my be defined to measure the maximum
amounts of work achievable from certain systems This section considers how the
maximum amount of work available from a system, when interacting with
surroundings, can be estimated All the energy in a system cannot be converted to
work: the Second Law stated that it is impossible to construct a heat engine that does
not reject energy to the surroundings
25. For stability of any system it is necessary and sufficient that, in all possible
variations of the state of the system which do not alter its energy, the variation of
entropy shall be negative
• This can be stated mathematically as ∆S < 0
It can be seen that the statements of equilibrium based on energy and entropy,
namely ∆E > 0 and ∆S < 0
26. Availability for a Closed System (non-steady)
All the displacement work done by a system is available to do useful work This
concept will now be generalized to consider all the possible work outputs from a
system that is not in thermodynamic and mechanical equilibrium with its
surroundings (i.e. not at the ambient, or dead state, conditions)
The maximum work that can be obtained from a constant volume, closed system
WS +WR = - (dU – TodS)
Hence, the maximum useful work which can be achieved from a closed
system is WS + WR = -(dU + PodV -TodS)
27. This work is given the symbol dA
Since the surroundings are at fixed pressure and temperature (i.e. po and To are
constant) dA can be integrated to give
A = U + po V - TOS
A is called the non-flow availability function
It is a combination of properties
A is not itself a property because it is defined in relation to the arbitrary datum
values of po and To It is not possible to tabulate values of A without defining both
these datum levels The datum levels are what differentiates A from Gibbs energy
28. The maximum useful work achievable from a system changing state from 1 to 2 is
given by
Wmax = ∆A= -(A2 - Al) = Al - A2
The specific availability, a , i.e. the availability per unit mass is
a = u + pov - Tos
If the value of a were based on unit amount of substance (i.e. kmol) it
would be referred to as the molar availability
The change of specific (or molar) availability is
∆a = a2 - a1 = (u2 + pov2 - Tos2)- (u1 + pov1-Tos1)
= ( h2 + v2(Po-P2) - (h1+ V1(Po – P1)) -To(S2 - S1)
29. Availability of a Steady Flow System
Consider a steady flow system and let it be assumed that the flowing fluid has the
following properties and characteristics; Internal energy u, specific volume v,
specific enthalpy h, pressure p, velocity c and location z
System delivers a work output W units
Normally, P2 &T2 ambient or state dead condition
30. Heat Q rejected by the system may be made to run a reversible heat engine, the
output from the engines equal to
=Q – To (S1 – S2)
Maximum available useful work or net work Wnet = Ws + Wengine
Irreversibility
The entropy of a system plus its surroundings (i.e. an isolated system) can never
decrease (2nd law).
• The second law states: Ssystem + Ssurr. = 0
where, = final - initial > 0 irreversible (real world)
=0 reversible (frictionless, ideal)
