1. Chapter: 07
ENTROPY

2. Objectives Apply the second law of thermodynamics to processes.
Define a new property called entropy to quantify the second-law effects.
Establish the increase of entropy principle.
Calculate the entropy changes that take place during processes for pure substances, incompressible substances, and ideal gases.
Examine a special class of idealized processes, called isentropic processes, and develop the property relations for these processes.
Derive the reversible steady-flow work relations.
Develop the isentropic efficiencies for various steady-flow devices.
Introduce and apply the entropy balance to various systems.

3. Clausius inequality. Inequalities in the 2nd Law
An irreversible (i.e., actual) heat engine, for example, is less efficient than a reversible one operating between the same two thermal energy reservoirs.
An irreversible refrigerator or a heat pump has a lower coefficient of performance (COP) than a reversible one operating between the same temperature limits.
## Another important inequality that has major consequences in thermodynamics is the :
Clausius inequality.

4. Clausius inequality. Inequalities in the 2nd Law
This inequality is valid for all cycles, reversible or irreversible.
The symbol (integral symbol with a circle in the middle) is used to indicate that the integration is to be performed over the entire cycle.
The cyclic integral of dQ/T can be viewed as the sum of all differential amounts of heat transfer divided by the respective temperature at the boundary.

5. The system considered in the development of the Clausius inequality.
Proof of Clausius Inequality
Clasius inequality
Formal definition of entropy
The system considered in the development of the Clausius inequality.

6. Proof of Clausius Inequality

7. Clausius inequality. Inequalities in the 2nd Law
This inequality is valid for all cycles, reversible or irreversible.
The equality in the Clausius inequality holds for totally or just internally reversible cycles
The inequality for the irreversible ones.

8. Internally and Externally Reversible Processes
Internally reversible process: If no irreversibilities occur within the boundaries of the system during the process.
Externally reversible: If no irreversibilities occur outside the system boundaries.
Totally reversible process: It involves no irreversibilities within the system or its surroundings.
A totally reversible process involves no heat transfer through a finite temperature difference, no nonquasi-equilibrium changes, and no friction or other dissipative effects.
A reversible process involves no internal and external irreversibilities.
Totally and internally reversible heat transfer processes.

9. Inequalities in the 2nd Law
Clausius inequality.

10. Entropy is an extensive property of a system.
Considering the equality part of CLAUSIUS inequality…
A quantity, whose cyclic integral is zero (i.e., a property like volume), must be a property…
Entropy is an extensive property of a system.

11. Prove that ENTROPY is a property !!!

12. TEMPERATURE - ENTROPY plot

13. TEMPERATURE - ENTROPY plot

14. TEMPERATURE - ENTROPY plot

15. ISENTROPIC PROCESSES
A process during which the entropy remains constant is called an isentropic process.
During an internally reversible, adiabatic (isentropic) process, the entropy remains constant.
The isentropic process appears as a vertical line segment on a T-s diagram.

16. TEMPERATURE - ENTROPY plot

17. TEMPERATURE - ENTROPY plot
What about ISOBARS and ISOCHORS?

18. TEMPERATURE - ENTROPY plot

19. ENTROPY CHANGE OF PURE SUBSTANCES
Entropy is a property, and thus the value of entropy of a system is fixed once the state of the system is fixed.
Schematic of the T-s diagram for water.
The entropy of a pure substance is determined from the tables (like other properties).
Entropy change

20. ENTROPY
Entropy change for Reversible process:
Entropy change for Irreversible process:
The entropy (since a property) change between two specified states is the same whether the process is reversible or irreversible.

21. ENTROPY
How to find the entropy change in case of an irreversible process?
 We simply replace the irreversible process by a reversible one between same states; and hence we proceed…
What is the relation between dS and dQ/T for an irreversible process?

22. What is the relation between dS and dQ/T for an irreversible process?
A cycle composed of a reversible and an irreversible / general process.
LHS: Entropy change for reversible / irreversible process.
The equality holds for an internally reversible process and the inequality for an irreversible process.

23. The entropy generation Sgen is always a positive quantity or zero.
For an irreversible / general process...
Equality sign holds…
Some entropy is generated or created during an irreversible process, and this generation is due entirely to the presence of irreversibilities.
The entropy generation Sgen is always a positive quantity or zero.

24. ENTROPY CHANGE for a natural process
For an irreversible / general process...
Entropy change of a system has two components:
1. Entropy transfer, and
2. Entropy generation.

25. ENTROPY of UNIVERSE Irreversible Reversible
In general for a process...
Reversible
Irreversible
These equations have far-reaching implications in thermodynamics.
For an isolated system (or simply an adiabatic closed system), the heat transfer is
zero, and the first equation reduces to…
Can the entropy of a system during a process decrease?

26. Can the entropy of a system during a process decrease?
THE INCREASE OF ENTROPY PRINCIPLE
Can the entropy of a system during a process decrease?
ΔSTER = -1 kJ/K
ΔSCYL = kJ/K
ΔSCOM = ΔSTER + ΔSCYL
= +1.5 > 0
ΔSUNIVERSE > 0

27. Can the entropy of a system during a process decrease?
THE INCREASE OF ENTROPY PRINCIPLE
Can the entropy of a system during a process decrease?
Therefore, Entropy may decrease locally during a process, but more rise somewhere else leads to net positive change of entropy during the same process.
ΔSCOM = ΔSTER + ΔSCYL
= +1.5 > 0
Sgen = +1.5 > 0
And this is because of IRREVERSIBILITY only!!!

28. REVERSIBILITY causes no Entropy generation.
THE INCREASE OF ENTROPY PRINCIPLE
Consider a reversible process:.
ΔSCOM = ΔSTER + ΔSCYL
= 0
Sgen = 0
REVERSIBILITY causes no Entropy generation.

29. A system and its surroundings form an isolated system.
THE INCREASE OF ENTROPY PRINCIPLE
The entropy change of an isolated system is the sum of the entropy changes of its components, and is never less than zero.
A system and its surroundings form an isolated system.
The increase of entropy principle

30. Some Remarks about Entropy
Processes can occur in a certain direction only, not in any direction. A process must proceed in the direction that complies with the increase of entropy principle, that is, Sgen ≥ 0. A process that violates this principle is impossible.
Entropy is a nonconserved property, and there is no such thing as the conservation of entropy principle. Entropy is conserved during the idealized reversible processes only and increases during all actual processes.
The performance of engineering systems is degraded by the presence of irreversibilities, and entropy generation is a measure of the magnitudes of the irreversibilities during that process. It is also used to establish criteria for the performance of engineering devices.
The entropy change of a system can be negative, but the entropy generation
cannot.

31. WHAT IS ENTROPY? Boltzmann relation
A pure crystalline substance at absolute zero temperature is in perfect order, and its entropy is zero (the third law of thermodynamics).
The level of molecular disorder (entropy) of a substance increases as it melts or evaporates.
Disorganized energy does not create much useful effect, no matter how large it is.

32. The paddle-wheel work done on a gas increases the level of disorder (entropy) of the gas, and thus energy is degraded during this process.
In the absence of friction, raising a
weight by a rotating shaft does not
create any disorder (entropy), and thus
energy is not degraded during this
process.
During a heat transfer process, the net entropy increases. (The increase in the entropy of the cold body more than offsets the decrease in the entropy of
the hot body.)

34. THE T ds RELATIONS the first T ds, or Gibbs equation
The T ds relations are valid for both reversible and irreversible processes and for both closed and open systems.
the second T ds equation
Differential changes in entropy in terms of other properties

35. ENTROPY CHANGE OF LIQUIDS AND SOLIDS
Liquids and solids can be approximated as incompressible substances since their specific volumes remain nearly constant during a process.
Since for liquids and solids
For an isentropic process of an incompressible substance

36. THE ENTROPY CHANGE OF IDEAL GASES
From the first T ds relation
From the second T ds relation

37. Constant Specific Heats (Approximate Analysis)
Entropy change of an ideal gas on a unit–mole basis
Under the constant-specific-heat assumption, the specific heat is assumed to be constant at some average value.

38. Isentropic Processes of Ideal Gases
Constant Specific Heats (Approximate Analysis)
Setting this eq. equal to zero, we get
The isentropic relations of ideal gases are valid for the isentropic processes of ideal gases only.

39. ENTROPY
PROBLEM NO.: 01
The latent heat of fusion of water at 0°C is 335 kJ/kg. How much does the entropy of 1 kg of ice change as it melts into water in each of the following ways: (a) Heat is supplied reversibly to a mixture of ice and water at 0°C. (b) A mixture of ice and water at 0°C is stirred by a paddle wheel.
(Ans kJ/K)

40. ENTROPY
PROBLEM NO.: 02
Two kg of water at 80°C are mixed adiabatically with 3 kg of water at 30°C in a constant pressure process of 1 atmosphere. Find the increase in the entropy of the total mass of water due to the mixing process (cp of water = kJ/kg K).
(Ans kJ/K)

41. ENTROPY
PROBLEM NO.: 03
A heat engine receives reversibly 420 kJ/cycle of heat from a source at 327°C, and rejects heat reversibly to a sink at 27°C. There are no other heat transfers. For each of the three hypothetical amounts of heat rejected, in (a), (b), and (c) below, compute the cyclic integral of dQ /T. from these results show which case is irreversible, which reversible, and which impossible: (a) 210 kJ/cycle rejected (b) 105 kJ/cycle rejected (c) 315 kJ/cycle rejected.
[Ans. (a) Reversible, (b) Impossible, (c) Irreversible]

42. ENTROPY
PROBLEM NO.: 05
Calculate the entropy change of the universe as a result of the following processes: (a) A copper block of 600 g mass and with Cp of 150 J/K at 100°C is placed in a lake at 8°C. (b) The same block, at 8°C, is dropped from a height of 100 m into the lake. (c) Two such blocks, at 100 and 0°C, are joined together.
(Ans. (a) 6.69 J/K, (b) J/K, (c) 3.64 J/K)

43. Summary Entropy The increase of entropy principle
Entropy change of pure substances
Isentropic processes
Property diagrams involving entropy
What is entropy?
The T ds relations
Entropy change of liquids and solids
The entropy change of ideal gases
Reversible steady-flow work
Minimizing the compressor work
Isentropic efficiencies of steady-flow devices
Entropy balance