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MAE 320 - Chapter 7

Entropy

The content and the pictures are from the text book: Çengel, Y. A. and Boles, M. A., “Thermodynamics: An Engineering Approach,” McGraw-Hill, New York, 6th Ed., 2008

Objectives• 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.

Definition of Entropy

300K 300k 300K

For the reversible engine:

0300300

10001000

=−

+=+KkJ

kkJ

TQ

TQ

L

L

H

H

1000kJ 1000kJ1000kJ

300kJ 550kJ 200kJ

For the irreversible engine:

For the impossible engine:

083.0

300550

10001000

<−=

−+=+

KkJ

kkJ

TQ

TQ

L

L

H

H

033.0

300200

10001000

>=

−+=+

KkJ

kkJ

TQ

TQ

L

L

H

H

<0 irreversible engine

=0 reversible engine

>0 impossible engine

∫∫ =+=+TQ

TQ

TQ

TQ

TQ

L

L

H

H

L

L

H

H δδδ )(

Clausius inequality: for any thermodynamic cycle, reversible or irreversible, the cyclic integral of δQ/T is always less than or equal to zero.

The equality in the Clausius inequality holds for totally or just internally reversible cycles and the inequality for the irreversible ones.

Definition of EntropyIf we extend this to a thermodynamic cycle, there is a new statement as follows:

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.

∫

can be viewed as the sum of all the differential amount of heat transfer divided by the temperature at the boundary.

For a device that undergoes an internally reversible cycle:

Entropy (S) is an extensive property of a system. Entropy per unit mass, designated s, is an intensive property with the unit kJ/kg.K.

The net change in volume (a property) during a cycle

is always zero.

A quantity whose cyclic integral is zero (i.e., a property like volume) typically depends on the state and not the process path. Thus such a quantity is a property. Therefore (δQ/T)in, rev must represent a property in the differential form.

Definition of Entropy

Clausius has realized this point and discovered a thermodynamic property named entropy

Note: Cv, Cp and R have the same unit kJ/kg.K

Definition of EntropyEntropy is a property, like all other properties, it has a fixed value at a fixed sate. The entropy change between two specified states is the same whether the process is reversible or irreversible.

In a special case (an internal reversibleprocess), the entropy change between two specified states:

For a closed system that undergoes an irreversible process, the entropy change between two specified states:

+ Δ

2

A Special Case: Internally Reversible Isothermal Heat Transfer Processes

This equation is particularly useful for determining the entropy changes of thermal energy reservoirs that can adsorb or supply heat indefinitely at a constant temperature.

Entropy

Recall that isothermal heat transfer processes are internally reversible:

Heat transfer to a system increases the entropy of a system, whereas heat transfer from a system decreases it. In fact, losing heat is the only way the entropy of a system can be decreased.

The Increase of Entropy Principle

A cycle composed of a reversible and an irreversible process.

(1).The equality holds for an internally reversible process. The entropy change becomes equal to , which represent entropy transfer with heat.(2). The inequality for an irreversible process.

Considering a cycle that is made of two processes: Process 1-2, which is arbitrary (reversible or irreversible), and Process 2-1, which is internally reversible.From the Clausius inequality:

The term is equal to the entropy change S1-S2:

The Increase of Entropy Principle

The inequality indicate that the entropy change of a closed system during an irreversible process is greater than entropy transfer. In other words, some entropy is generated or created during an irreversible process. The entropy generated is called entropy generation, Sgen.

The entropy generation Sgen is always a positive quantity or zero. Its value depends on the process path, and thus it is not a property of a system

Rewrite the above equation

For an isolated system (an adiabatic closed system), the heat transfer is zero, thus

This equation indicates that the entropy of an isolated system during a process always increases. It never decreases. – 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

The Increase of Entropy PrincipleSome entropy is generated or created during an irreversible process, and this generation is due entirely to the presence of irreversibilities.

The Increase of entropy principle can summarized as follows:

(1). The entropy generation, Sgen, is not a property of a system. Its value depends on the process path. It is always a positive quantity or zero.

(2). The entropy, S, is a property. It is independent on the path. The entropy change (S2-S1) can be zero, positive, ornegative

Some Remarks about Entropy1. 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.

2. 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.

3. 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.

4. Entropy can transferred by two ways: (i) heat transfer and (ii) mass transfer. Net work can not transfer entropy.

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Energy balance & Entropy balance of closed system

KEPEUEWWQQ systemoutinoutin Δ+Δ+Δ=Δ=−+−

Conservation of energy is applied to a closed system that undergoes any processes regardless of reversible of irreversible, hence the energy balance equation:

There is no conservation of entropy for a closed system that undergoes any processes regardless of reversible of irreversible, hence the entropy balance equation:

sysoutin EEE Δ=−

12 SSSSTQ

TQ

sysgenout

out

in

in −=Δ=+−

In a special case, a closed system undergoes an internal reversible processes, then the entropy balance equation becomes:

12 SSSTQ

TQ

sysout

out

in

in −=Δ=−

The Increase of Entropy Principle

The Increase of Entropy PrincipleExample 7-2

The Increase of Entropy PrincipleExample 7-2

Entropy Change of Pure Substances

The entropy of a pure substance is determined from the tables (like other properties).

T-s diagram of water.

Entropy is a property, and thus the value of entropy of a system is fixed once the state of the system is fixed.

Especially, the T-s diagram of water is shown in Figure A-9 in Pg. 926

Entropy Change of Pure SubstancesThe value of entropy at a specific state is determined just like any other proper.

In the saturated region:

The entropy of a compressed liquid:

P=co

nsta

nt

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Entropy Change of Pure Substances Entropy Change of Pure SubstancesExample 7-3

Entropy Change of Pure SubstancesExample 7-3

Table A-13Superheated vapor

Table A-12

Entropy Change of Pure SubstancesExample 7-3

Isentropic Processes

During an internally reversible and adiabatic process, the entropy remains constant.

A process during which the entropy remains constant is called an isentropic process.

The isentropic process appears as a vertical line segment on a T-s diagram.

Isentropic Processes

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Isentropic ProcessesExample 7-5

Isentropic ProcessesExample 7-5

From Table A-5, the Tsat= 263.94 oC when P1= 5000 kPa.

T1=450 > Tsat= 263.94 oC, Therefore, it is superheated vapor under the State 1. So s1 can be obtained from Table A-6, s1=6.8210 KJ/kg K and h1= 3317.2 kJ/kg k(page 922)

Isentropic Processes

Method (2), Figure A-9 in Page 926 to get the h2

Example 7-5

From Table A-5, the Sg= 6.4675 KJ/kg K when P1= 1400 kPa.

s2=s1= 6.8210> sg@p=1400 kPa = 6.4675 kJ/kg K. Therefore, it is superheated vapor under the State 2. So h2 can be obtained from Table A-6, h2=2967.4 kJ/kg Kthrough interpolation (page 921)

Property Diagrams Involving Entropy

On a T-S diagram, the area under the process curve represents the heat transfer for internally reversible processes. But the area has no meaning for an irreversible process

For an internal reversible process:

For an internal reversible isothermal process:

Temperature-entropy diagram

or

Property Diagrams Involving Entropy

The T-s diagram of the Carnot CycleP-v diagram of the Carnot Cycle

Property Diagrams Involving Entropy

An isentropic process appears as a vertical line segment on a T-s diagram. This expected since an isentropic process involves no heat transfer, and therefore the area under the process path must be zero.

The T-s diagram serves as a valuable tool for visualizing the second law aspects of processes and cycles.

An isentropic process appears as a vertical line segment on a T-s diagram

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Property Diagrams Involving Entropy

The h-s diagram is useful analysis of adiabatic steady-flow devices, such as turbines, compressors and nozzles:

(i). The vertical distance ∆h (between the inlet and the exit states) on an h-s diagram is a measure of work.

(ii). The horizontal distance ∆s is a measure of irreversibilitiesassociated with the process.

Mollier diagram: The h-s diagramFig. A-10 in Appendix in Page 927

What is Entropy?

The level of molecular disorder (entropy) of a substance increases as it melts or evaporates.

A pure crystalline substance at absolute zero temperature is in perfect order, and its entropy is zero(the third law of thermodynamics).

Entropy is a measure of molecular disorder or molecular randomness

The T ds Relations

the first T ds, or Gibbs equation the second T ds equation

For a closed system that undergoing an internal reversible process:

Tds – PdV = du

Tds – PdV = du

The T ds RelationsFor a closed system that undergoing an internal reversible process:

the first T ds, or Gibbs equation

the second T ds equation

The T ds Relations

The T ds relations are valid for both reversible and irreversible processes and for both closed and open systems.

The T ds relations are derived from an internal reversible process. However, it is valid for both reversible and irreversible processes and for both closed and open systems, since entropy is a property and the change in a property between two states is independent on the type of process the system undergoes.

Entropy Change of Liquids and Solids

Since for liquids and solids

Liquids and solids can be approximated as incompressible substancessince their specific volumes remain nearly constant during a process.

An isentropic process of an incompressible substance is also an isothermal process:

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The Entropy Change of Ideal GasesFrom the first T ds relation From the second T ds relation

for ideal gas

Constant Specific Heats (Approximate Analysis)

Under the constant-specific-heat assumption, the specific heat is assumed to be constant at some average value.

Assuming constant specific heats for idea gases is a common approximation:

Variable Specific Heats (Exact Analysis)We choose absolute zero as the reference temperature and define a function s° as

The entropy of an ideal gas depends on both T and P. The function s represents only the temperature-dependent part of entropy.

On a unit–mole basis

On a unit–mass basis

Variable Specific Heats

Variable Specific HeatsExample 7-9

Variable Specific HeatsExample 7-9

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Isentropic Processes of Ideal GasesConstant 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.

Isentropic Processes of Ideal GasesVariable Specific Heats (Exact Analysis)

Relative Pressure and Relative Specific Volume

T/Pr is the relative specific volume vr

exp(s°/R) is the relative pressure Pr

The use of Pr data for calculating the final temperature during an isentropic process.

The use of vr data for calculating the final temperature during an isentropic process

S2- S1 =

Isentropic Processes of Ideal GasesVariable Specific Heats (Exact Analysis)

The use of Pr data for calculating the final temperature during an isentropic process.

The use of vr data for calculating the final temperature during an isentropic process

Table A-17 Table A-17

Isentropic Processes of Ideal Gases

Isentropic Processes of Ideal GasesExample 7-10

Isentropic Processes of Ideal Gases

If we use k=1.400 at the initial temperature 295 K

T2 = 667.7 K

Thus we can estimate the average temperature is 481 K

At 481k, K=1.389 based on Table A-2b

T2= 662.4 K

T2=(295K) (8)1.4-1

Example 7-10

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Isentropic Efficiencies of Steady-flow Devices

The isentropic process involves no irreversibilities and serves as the ideal process for adiabatic devices.

Isentropic process: internal reversible, adiabatic

Isentropic Efficiency of Turbines:

Isentropic efficiency is also called the second law efficiency, which is a measure of the deviation of actual processes from the corresponding idealized ones.

Thermal efficiency is called the first law efficiency.

Typically a turbine under the steady operation, the inlet state of the working fluid and the exhaust pressure is fixed

Isentropic Efficiencies of Steady-flow Devices

The h-s diagram for the actual and isentropic processes of an adiabatic turbine.

Isentropic Efficiency of Turbines:

Isentropic Efficiencies of Steady-flow Devices Isentropic Efficiencies of Steady-flow Devices

the inlet state of the working fluid and the exhaust pressure is fixed.3. For a turbine under the steady operation,

Isentropic Efficiencies of Steady-flow DevicesExample 7-14

, State 2s is in the liquid-vapor region

Isentropic Efficiencies of Steady-flow DevicesExample 7-14

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Isentropic Efficiencies of Compressors

The h-s diagram of the actual and isentropic processes of an

adiabatic compressor

When kinetic and potential energies are negligible

Isentropic Efficiencies of CompressorsIsentropic Efficiencies of Compressors

Compressors are sometimes intentionally cooled to minimize the work input.

Can you use isentropic efficiency for a non-adiabatic compressor? Can you use isothermal efficiency for an adiabatic compressor?

Isentropic Efficiencies of Compressors

For a reversible isothermal process, then we can define an isothermal efficiency:

Wt and Wa are the required work inputs to the compressor for the reversible isothermal and actual processes.

Isentropic Efficiencies of PumpsIsentropic Efficiencies of Pumps:

When kinetic and potential energies of a liquid are negligible

Isentropic Efficiencies of Pumps

Isentropic Efficiencies of PumpsExample 7-15

Isentropic Efficiencies of PumpsExample 7-15State 1:

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Isentropic Efficiencies of PumpsExample 7-15

Isentropic Efficiency of Nozzles

The h-s diagram of the actual and isentropic processes of an adiabatic nozzle.

If the inlet velocity of the fluid is small relative to the exit velocity, the energy balance is

Isentropic Efficiency of Nozzles is defined as:

outin EE••

=

)2

()2

(2

22

21

1a

aVhmVhm +=+

••

0,0,0 ≅Δ==••

peWQSince

Then

Isentropic process of Nozzles Isentropic process of Nozzles

Isentropic process of Nozzles

h1 = 988 kJ/kg = 988,000 J/kg at 950K

h2s = 766 kJ/kg = 766,000 J/kg at 748K

V1 = 0 Attention: unit conversion

(Table A-17)

Example 7-16Isentropic process of Nozzles

)/(666)766000998000(2)(2 212 smhhV s =−=−=

Alternatively, use a constant Cp

76698898892.0 2

−−

= ahh2a = 783.8 kJ/kg

From table A-17: h=800.03kJ/kg at 780K, and h=778.18 kJ/Kg at 760K

18.7788008.78303.800

760780780 2

−−

=−− aT

T2a = 765KInterpolation:

Example 7-16

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Isentropic process of Nozzles

Alternatively, use a constant Cp

Example 7-16Entropy Balance

Energy and entropy balances for a system

Entropy Change of a System, ∆Ssystem

When the properties of the system are not uniform

Entropy Generation, SgenAttention: the unit for following three forms:

(1). For an irreversible process, the entropy generation (Sgen) is greater than zero(2). For a reversible process, the entropy generation (Sgen) is zero and thus the entropy change of a system is equal to the entropy transfer

Mechanisms of Entropy TransferEntropy can be transferred to or from a system by two mechanisms: heat transfer and mass flow.

Entropy transfer by work:Heat transfer is always accompanied by entropy transfer in the amount of Q/T, where T is the boundary temperature.

No entropy accompanies work as it crosses the system boundary. But entropy may be generated within the system as work is dissipated into a less useful form of energy.

Entropy transfer by heat:

Entropy transfer by mass:

Mass contains entropy as well as energy, and thus mass flow into or out of system is always accompanied by energy and entropy transfer.

When the properties of the mass change during the process

Mechanisms of Entropy Transfer Entropy Change of a System

Mechanisms of entropy transfer for a general system.

The entropy change between the final state and the initial state of system that undergoes a process:1) The entropy transfer by heat2) The entropy transfer by mass flow3) The irreversibility extent of the process

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Entropy Balance of Closed SystemsA closed system involves no mass transfer across its boundary

For an adiabatic process, no heat transfer across the boundary of a closed system, the entropy change of the system only depends on the irreversibility of the process:

Entropy Balance of Closed SystemsNoting that any closed system and its surroundings can be treated as an adiabatic system, and then:

Entropy generation outside system boundaries can be accounted for by writing an entropy balance on an extended system that includes the system and its immediate surroundings.

surr

Balance Equations of Closed SystemsA closed system involves no mass transfer across its boundary

Energy balance:

PEKEUWWQQ outinoutin Δ+Δ+Δ=−+− )()(

For a closed system without change in the KE and PE

UWQ outnetinnet Δ=− ,,

Entropy balance:

Entropy Balance of Control VolumesAs compared with a closed system, a control volume involves mass transferacross its boundary. Thus the entropy balance:

The entropy change within a control volume during a process is equal to the sum:

(1) Entropy transfer by heat(2) the net entropy transfer by mass flow(3) the entropy generation as result from

irreversibility

Entropy Balance of Control VolumesFor a general steady-flow process:

(kW/K)

For a single-stream, steady-flow device:

0=+−+•••

•

∑ ∑ ∑ geneeii

k

k SsmsmTQ

0)( =+−+••

•

∑ geneik

k SssmTQ

0)( =+−••

genei Sssm

For an adiabatic, single-stream, steady-flow device:

For an adiabatic, single-stream, steady-flow device that undergoes a reversible process:

si = se

Balance Equations of Control Volumes

Mass balance :

For a general control volume:

Energy balance :

Entropy balance :

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Balance Equations of Steady-flow Control VolumesFor a steady-flow control volume:Mass balance :

Energy balance :

Entropy balance : 0=+−+•••

•

∑ ∑ ∑ geneeii

k

k SsmsmTQ

Entropy Generation

Entropy GenerationExample 7-18

Entropy Generation

(Table A-6)

0=+−+•••

•

∑ ∑ ∑ geneeii

k

k SsmsmTQ

Example 7-18

(Table A-6)

Interpolation

Entropy Generation Entropy GenerationExample 7-19

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Entropy Generation

Cavg=

UWQ outnetinnet Δ=− ,,

For the iron block:

Entropy Generation

The heat transfer form the iron block to the lake while the lake keeps a constant temperature (285K). Therefore, we can consider this isothermal process is an internal reversible. Thus Sgen =0

Now we take the lake as the closed system.Example 7-19

Entropy GenerationExample 7-19

Entropy Generation

Entropy Generation

0=+−+•••

•

∑ ∑ ∑ geneeii

k

k SsmsmTQ

Example 7-20Entropy Generation

Energy Balance:

h1 = ?s1= ?

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Summary• Entropy

• The Increase of entropy principle

• Entropy change of pure substances

• Isentropic processes

• Property diagrams involving entropy

• The T ds relations

• Entropy change of liquids and solids

• The entropy change of ideal gases

• Isentropic efficiencies of steady-flow devices

• Entropy balance