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Lecture Notes on Thermodynamics 2008 - PowerPoint PPT Presentation

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.

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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

• 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.

• 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

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

• Isentropic Process

or

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

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

• see Example 7–6

Figure 7–23

Figure 7–22

Figure 7–21

Figure 7–20

Figure 7–27

Figure 7–24

Figure 7–25

Figure 7–26

• 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

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.

• 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

• 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

Examples (1/3)

• Turbine : Example 7–14

Examples (2/3)

• Compressor : Example 7–14

Examples (3/3)

• Nozzle : Example 7–16

Homework #7

Solve the Examples 7–1 ~ 7–23