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Lecture Notes on Thermodynamics 2008

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Lecture Notes on Thermodynamics 2008

Chapter 7 Entropy

Prof. Man Y. Kim, Autumn 2008, [email protected], Aerospace Engineering, Chonbuk National University, Korea

Entropy (1/3)

- The Inequality of Clausius

- The inequality of Clausius is a corollary or a consequence of the 2nd law of thermodynamics.
- It is valid for all possible cycles, including both reversible and irreversible ones
- The entropy is defined from this formulation, i.e.,

and

Entropy (2/3)

- Proof of the Inequality of Clausius

Consider first a reversible (Carnot) heat engine cycle :

From the definition of absolute temperature scale ( )

If , and

Finally, we conclude that for all reversible heat engines,

and

Now consider an irreversible cycle heat engine :

Consequently, for the irreversible cycle engine,

and

If we make the engine become more and more irreversible, but keep , , and fixed,

and

Finally, we conclude that for all irreversible heat engine cycles,

and

Similarly, the same procedure can be applied for both reversible and irreversible refrigeration cycles.

Entropy (3/3)

- Entropy – A Property of a System

Reversible process along path A-B

Reversible process along path C-B

Subtracting the second equation from the first, we have

is independent of the path → point function → property

This property is called entropy

and

Principle of the Increase of Entropy (1/2)

- Increase of Entropy Principle

From the Clausius Inequality

or

Here, you can find that

Entropy generation

- Entropy Generation

where, :entropy generation due to irreversibility occurring inside the system ( because of friction, unrestricted expansion, internal energy transfer over a finite temp. difference, etc.)

Reversible process : and

Irreversible process :

1st law :

Thermodynamic property relation :

Lost Work → Exergy (Chapter 8)

Thus we have an expression for the change of entropy for an irreversible process as an equality, whereas in the last slide we had an inequality.

Principle of the Increase of Entropy (2/2)

- Discussions on Entropy Generation

Discussion 1 : There are 2 ways in which the entropy of a system can be increased by

(1) transferring heat to the system

(2) having an irreversible process

Note : There is only one way in which entropy can be decreased by transferring heat from the system

Discussion 2 : For an adiabatic system, the increase of entropy is always associated with the irreversibility

Discussion 3 : The presence of irreversibility will cause the work to be smaller than the reversible work

Entropy Change of a Pure Substance

- see Examples 7–3 (p.326) and 7–4 (p.327)

- Isentropic Process

or

Isentropic Relations

Consider the case of an ideal gas undergoing an isentropic process,

However,

, where : specific heat ratio

Finally we can obtain

, and : Isentropic Relation

Note : constant is a special case of a polytropic process in which the polytropic exponent n is equal to the specific heat ratio k

T–s Diagram of the Carnot Cycle

Consider the Carnot cycle, i.e.,

① → ② : reversible isothermal heat addition process

Area 1-2-b-a-1 : heat transferred to the working fluid during the process

② → ③ : reversible adiabatic process

→ isentropic process

③ → ④ : reversible isothermal heat rejection process

Area 3-4-a-b-3 : heat transferred from the working fluid to the low-temperature reservoir.

④ →① : reversible adiabatic process

→ isentropic process

Area 1-2-3-4-1 : net work of the cycle

Efficiency

Comments on efficiency :

- see Example 7–6

What is Entropy ?

Figure 7–23

Figure 7–22

Figure 7–21

Figure 7–20

Figure 7–27

Figure 7–24

Figure 7–25

Figure 7–26

Thermodynamic Property Relations

- Gibbs Equations (T–ds Relations)

and

For the simple compressible substance with no motion or gravitational effects, the 1st law becomes

For a reversible process of a simple compressible substance,

and

Since enthalpy is defined as

For a unit mass,

and

Entropy Change during Irreversible Process

Reversible cycle : reversible process along path A-B

Irreversible cycle : irreversible path C and reversible path B

Subtracting the second equation from the first, we have

As path C was arbitrary, the general result is (both reversible and irreversible cases)

and

This is one of the most important equations of thermodynamics !

and

Therefore, we can find that the entropy change for an irreversible process is larger than the change in a reversible process for the same and T.

Entropy Change for a Solid(Liquid) and Ideal Gas

- For a Solid or Liquid

specific volume is very small, and

- For an Ideal Gas

We know that , and

and

Similarly, , and

and

If we assume that the specific heat is constant,

and

Reversible Polytropic Process for an Ideal Gas

- Polytropic Process

constant

If n is a constant,

and : Polytropic Relation

- Work done during a reversible polytropic process

and constant

constant

- Isobaric process (P=constant) : n=0
- Isothermal process (T=constant) : n=1
- Isentropic process (s=constant) : n=k
- Isochoric process (v=constant) : n=∞

for any value of n except n=1

- The reversible isothermal process

constant

constant

or

Heat Transfer and Entropy Generation

Examples (1/3)

- Turbine : Example 7–14

Examples (2/3)

- Compressor : Example 7–14

Examples (3/3)

- Nozzle : Example 7–16

Saemangum @ Jellabukdo

Homework #7

Solve the Examples 7–1 ~ 7–23