Chapter 6: Using Entropy - University of Arizonajmcgrath/ln.ch6.notes1.pdf · Chapter 6: Using Entropy ... (Mollier) Diagram. ... Temperature–entropy diagram for water (SI units)

Chapter 6: Using Entropy

Combining the 1st and the 2nd Laws of Thermodynamics

Photo courtesy of U.S. Military Academy photo archives.

ENGINEERING CONTEXT

Up to this point, our study of the second law has been concerned primarily with what it says about systems undergoing thermodynamic cycles. In this chapter means are introduced for analyzing systems from the second law perspective as they undergo processes that are not necessarily cycles. The property entropyplays a prominent part in these considerations.

The objective of the present chapter is to introduce entropy and show its use for thermodynamic analysis.

The word energy is so much a part of the language that you were undoubtedly familiar with the term before encountering it in early science courses. This familiarity probably facilitated the study of energy in these courses and in the current course in engineering thermodynamics. In the present chapter you will see that the analysis of systems from a second law perspective is conveniently accomplished in terms of the property entropy. Energy and entropy are both abstract concepts. However, unlike energy, the word entropy is seldom heard in everyday conversation, and you may never have dealt with it quantitatively before. Energy and entropy play important roles in the remaining chapters of this book.

Introducing Entropy

Corrollary of Second Law is introduced:

Clasius Inequality

Expands last chapter treatment of Two heat reservoirsto arbitrary number of heat reservoirs from which systemreceives energy by heat transfer or rejects energy by heat transfer

Provides basis for two concepts of second law for analyzingclosed and open systems:

1) Entropy as a Property2) Entropy Balance

Represents heat transfer at local system boundary location, b, at temperature T. Heat transfer may be positive (IN) or negative (OUT)

Clasius Inequality

Qδ

0b

QTδ ≤ ∫For any thermodynamic cycle

For any number of reservoirs:

T must be absolute temperature (Kelvin or Rankine)T always positive. Can NOT be negative (Celsius or Farenheit)

∫ Indicates integration over all processes and all parts of boundary

Clasius Inequality

0b

QTδ ≤ ∫

Inequality has same meaning as with Kelvin-Planck:

Equality applies when no INTERNAL IRREVERSIBILITES are present

Inequality applies when INTERNAL IRREVERSIBILITES are present

Example of Multiple Heat Transfers & Ts

1Qδ

2Qδ3Qδ

jQδ

T1Tj

T3T2

2 0Qδ <

Note:

2

2

0QTδ

<

Because heat transfer is OUT

We will see later that this is and entropy transfer OUT

Reversible Heat Transfers

1Qδ

2Qδ3Qδ

jQδ

T1Tj

T3T2

Note: Ti for reservoir AND at boundary; ∆T must approach zero

Developing Clasius Inequality

“Proof” provided in text

cycleb

QTδ σ = − ∫

0b

QTδ ≤ ∫

0cycleb

QTδ σ + = ∫

Alternate Statements of Clasius Inequality

0generatedb

QTδ σ + = ∫

cycleb

QTδ σ = − ∫

0 no irreversibilities present within the system0 irreversibilities present within the system0 impossible

cycle

cycle

cycle

σ

σ

σ

=

>

<

Alternate Statements of Clasius Inequality

0generatedb

QTδ σ + = ∫

Unlike mass and energy, which are conserved in every process, entropy in the presence of

irreversibilities, is always produced.

Defining Entropy Change

Quantity is a Property if and only if change in valuebetween two states is independent of process (path)

Will now show that the quantity is a PropertyReversible

QTδ

∫

Meaning: For a REVERSIBLE process, the integral of the heat transfer divided by the local absolute temperature doesNOT depend on the process

This integral is therefore a PROPERTY

We will give this property the name “ENTROPY”


1

2C

B A

2 1

1 2cycle

A C

Q QT Tδ δ σ

+ = −

∫ ∫

2 1

1 2cycle

B C

Q QT Tδ δ σ

+ = −

∫ ∫

Cycle A-C:

Cycle B-C:

0

0


Processes A, B and C are INTERNALLY REVERSIBLE:

0cycle generatedσ σ= =


2 2

1 1A B

Q QT Tδ δ

= ∫ ∫

2 1

1 2cycle

A C

Q QT Tδ δ σ

+ = −

∫ ∫

2 1

1 2cycle

B C

Q QT Tδ δ σ

+ = −

∫ ∫

Subtract:

From:

0

0


Yielding:


1

2C

B A

Since A and B are arbitrary, it follows that the integral is same for ANYInternally Reversible process.

Thus, the Integral is a Property

Property is ENTROPY, S

Entropy Change between states 1 & 2 can be calculated for ANY Internally Reversible process as:

2

2 11 ReInt v

QS STδ

− = ∫

Entropy Change

ReInt v

QdSTδ =

1

2C

B A

Differential basis defining entropy change:

Entropy, S, is an Extensive property

Intensive form is: Ssm

=

Units for Entropy:

Extensive (SI): kJ/K Intensive (SI): kJ/kg KExtensive (English): BTU/oR Intensive (English): BTU/lb oR


1

2C

B A

Important Note:

Entropy change between states 1 & 2 is SAME whether process between 1 & 2 is REVERSIBLE or IRREVERSIBLE.Just can’t calculate change for IRREVERSIBLE processes.

Entropy change between states 1 & 2 can be calculatedfor ANY Internally Reversible process as:

2

2 11 ReInt v

QS STδ

− = ∫

Once calculated, Entropy difference is known

Recognize that Entropy is still an abstract concept for you

Like enthalpy we defined earlier, to gain appreciation for Entropy, need to understand:

HOW to use it

and

WHAT it is used for

Retrieving Entropy Data

Chapter 3: Means for retrieving property data fromTables, graphs, equations (and software)

Emphasis on properties: P, v, T, h

Which are required for:

1st Law (energy conservation) and mass conservation

For application of 2nd Law, entropy values usually needed

Thus, we need to understand how to retrieve entropy data

Entropy Reference Value?Similarity to Energy with regard to absolute values of Entropy

In most cases, absolute values not important- Only difference in Entropy between states is important

1

2C

B A

S1 = ?

S2 = ?

Sref,1

Sref,2

Re

y

y xx Int v

QS STδ

= + ∫

Sx is reference value

Finding Entropy DataFor Water and Refrigerants:

Using Tables A-2 to A-18

and Tables A-2E to A-18E

Tabulations of Water Properties

T

v

Liquid

Liquid-Vapor Mixture

Superheated Vapor

Saturation Line

Table A5, A5E

Table A2, A3 (A2E, A3E)

Table A4, A4E

f g

Sat Liquid

Sat vapor

s

Two Saturated Tables for Each

One for Saturated Temperature

One for Saturated Pressure

Sometimes we know T, sometimes P

Saturated Water

Table A-2, A3, A2E, A3E

T

s

f g

Computing Entropy Value Under Dome

1

(1 )

liquid vapor

liq vap

liq f vap g

vap

liq

f g

S S S

S SSsm m mm s m s

sm mm

xm

mx

ms x s xs

= +

= = +

= +

=

= −

∴ = − +

It is not enough to know T, P in order to establish state under dome

Need T or P, and one other property

(x= Quality)

T

s

f g

( )f g f f fgs s x s s s x s= + − = + i

Superheated Vapor

Table A-4, A-4E

T

s

f g293 F

500 F

Compressed or Sub-cooled

LiquidTable A-5, A-5E

Note:sub-cooled tables are sparse because it is accurate to use incompressible liquid model

T

v

f g

Text Examples

Problem solving using property diagrams is important

Two commonly used property diagrams are:

Temperature- Entropy (T-s diagram)

Enthalpy- Entropy (Mollier diagram)

Graphical Entropy Data

Temperature-Entropy Diagram Enthalpy-Entropy (Mollier) Diagram

Note general features

See Figs A-7 & A-7E for water

Temperature-Entropy Diagram

Figure A-7Temperature–entropy diagram for water (SI units)

Figure A-7ETemperature–entropy diagram for water (English units).

Enthalpy-Entropy (Mollier) DiagramNote general features

See Figs A-8 & A-8E for water

Figure A-8Enthalpy–entropy diagram for water (SI units)

Figure A-8EEnthalpy–entropy diagram for water (English units)

Examples as per text here

Using T dS EquationsAlthough entropy changes can be determined from:

2

2 11 ReInt v

QS STδ

− = ∫

This requires knowledge of how heat transfer and Q vary during a process.

In practice, entropy changes calculated from changes in other properties.

Let’s see how….

Using T dS Equations

Note that T dS equations are also important with respect to:

1) Deriving other important properties

2) Constructing property tables (Chapter 11)

Simple Compressible Substance

( ) ( )Re ReInt v Int vQ dU Wδ δ= +

( ) ReInt vW PdVδ =

TdS dU PdV= +

( ) ReInt vQ TdSδ =

Energy balance in differential form: IN = STORED + OUT

Neglecting KE and PE

First TdS Equation:

Now consider second TdS equation

TdS dU PdV= +

H U PV= +

( )dH dU d PV dU PdV VdP= + = + +

Second TdS Equation:

Recall that, by definition:

Form differential:

dU PdV dH VdP+ = −Rearrange:

Substitute:

TdS dH VdP= −

TdS dH VdP= −Per Unit Mass: Tds dh vdP= −

Per Mole: Tds dh vdP= −

TdS dU PdV= +Tds du Pdv= +Per Unit Mass:

Per Mole: Tds du Pdv= +

Although internally reversible processes used to derive these equations, they apply generally- do not require reversible processes.

Since all terms are properties, applies to irreversibleand reversible processes (path-independent)

Change in Entropy is independent of details of process

Text Example

Entropy Determination For Incompressible Substances

P vc c c= =

Incompressible substance model is generally used with liquids and solids only, and assumes constant specific volume.

( ) ( ) ( )2

1

2 2 1 1

T

T

dTs T s T c TT

− = ∫

( )c T dTdu P duds dvT T T T

= + = =

Specific heat depends on Temperature only. No difference between cP and cv

Specific heat may be constant: ( ) ( ) 22 2 1 1

1

ln Ts T s T cT

− =

Entropy Change of an Ideal Gas:TdS equations used to compute entropy change between 2 states of Ideal Gas

For an Ideal Gas:

Equation of State

Tds dh vdP= −

Tds du Pdv= +

dh vds dPT T

= −

du Pds dvT T

= +

( ) ( )p vc T c T R= +( )( )

v

p

du c T dTdh c T dT

Pv RT

==

=

Substitute Ideal Gas Relations into TdS equations:

( )( )

v

p

du c T dTdh c T dT

Pv RT

==

=dh vds dPT T

= −

du Pds dvT T

= +

( )pdT dPds c T RT P

= −

( )vdT dvds c T RT v

= + ( ),s s T v=

( ),s s T P=

Can integrate these equations:

( )vdT dvds c T RT v

= +

( )pdT dPds c T RT P

= −

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

vT

dT vs T v s T v c T RT v

− = +

∫

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

PT

dT Ps T P s T P c T RT P

− = −

∫

Using Ideal Gas Tables:

Thus, in the equation:Where is an arbitrary reference temperatureT ′

( ) ( )Tpo

T

c Ts T dT

T′

= ∫Where

Let’s define a new variable: ( )os T

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

PT


− = −

∫

( ) ( ) ( ) ( ) ( )2 2 1

1

2 1

T T To o

P P PT T T

dT dT dTc T c T c T s T s TT T T′ ′

= − = −∫ ∫ ∫

This integral can be tabulated (Ideal Gas Tables)Tables A-22, A22E:

( ) ( ) ( )2

1

0 02 1

Tp

T

c Ts T s T dT

T− = ∫ (kJ/kg K or BTU/lb oR)

( ) ( ) ( ) ( ) 22 2 2 1 1 1 2 1

1

, , lno o Ps T P s T P s T s T RP

− = − −

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

PT


− = −

∫

Becomes:

Where:

Table A-22: Ideal Gas Properties of Air

Table A-22E: Ideal Gas Properties of Air

Using Ideal Gas Tables:

Similar approach for per mole basis

( ) ( ) 22 2 1 1 2 1

1

, , ( ) ( ) ln Ps T P s T P s T s T RP

− = − −

This integral can be tabulated (Ideal Gas Tables)Tables A-23, A23E:

What if we have T and v information, rather than T and P?

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

vT


− = +

∫

Ideal Gaseswith Constant Specific Heats

( ) ( ) 2 22 2 1 1

1 1

, , ln lnpT Ps T P s T P c RT P

− = −

( ) ( ) 2 22 2 1 1

1 1

, , ln lnvT vs T v s T v c RT v

− = +

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

vT


− = +

∫

( ) ( ) ( )2

1

22 2 2 1 1 1

1

, , lnT

PT


− = −

∫

Use Specific heat data in Tables A-20 and A-21

Table A-20: Ideal Gas Specific Heats of Some Common Gases (kJ/kg · K)

Entropy Change for Closed Systems Internally Reversible Processes

ReInt v

QdSTδ =

For Internally Reversible Processes,Entropy change can be calculated from:

This links heat transfer and entropy transfer

Note that for Closed, Internally Reversible system:

Entropy INCREASES when heat transfer INEntropy DECREASES when heat transfer OUTEntropy DOES NOT CHANGE when no heat transfer occurs

Entropy transfer ACCOMPANIES heat transfer

Entropy Change for Closed Systems Internally Reversible Processes

Q & Entropy IN Q & Entropy OUTQ=0∆S=0

Process where Q = 0 is: ADIABATIC (Insulated)

For process that is ADIABATIC AND REVERSIBLE,The ENTROPY DOES NOT change.

A constant Entropy process is called ISENTROPIC

Thus, an ADIABATIC AND REVERSIBLE processis an ISENTROPIC process

Entropy Change Internally Reversible Processes

( ) ReInt vQ TdSδ =

ReInt v

QdSTδ =

For internally reversible processes, the area under the process curve represents the process Heat Transfer.

2

Re1

Int vQ TdS= ∫

1) Heat transfer is entire area under T-S diagram

2) Temperature must be absolutescale (Kelvin or Rankine), >0

3) Not valid for IRREVERSIBLEprocesses. Then area is NOTequal to heat transfer


QdSTδ

≥

Q TdSδ ≤QdSTδ

≥


Internally reversible processes are idealizations, but are found in all Carnot cycles.

Carnot cycles on the T-S diagram. The diagram on the left represents a power cycle, and on the right, a refrigeration/heat pump cycle.

Internally reversible processes are idealizations, but are found in all Carnot cycles.

Areas (+ or -) is magnitude of work: W = QIN –QOUT

1cycle C

IN H

W TQ T

η = = −

Power Cycle Refrigeration Cycle

Note how “separation” of TH and TC related to more work

Example in text

Entropy Balance: Closed Systems

1

2Rev

Irrev, b

Cycle consists of process, Irrev, where Internal Irreversibilities are presentFollowed by process, Rev, which isInternally Reversible

2 1

1 2 Re

cycle

b Int v

Q QT Tδ δ σ

+ = −

∫ ∫

Subscript, b, designates evaluation of integral at boundary.

Not necessary for second case because temperature isuniform throughout system

Note dotted line


2 1

1 2 Re

cycle

b Int v

Q QT Tδ δ σ σ

+ = − = −

∫ ∫

1

1 22 ReInt v

QS STδ

− = ∫

( )2

1 21 b

Q S STδ σ

+ − = −

∫

1

2Rev

Irrev

Cycle consists of process, Irrev, where Internal Irreversibilities are presentFollowed by process, Rev, which isInternally Reversible

Where:

( )2

2 11 b

QS STδ σ

− = +

∫

Entropy Change = Entropy + EntropyTransfer Production


Total entropy change associated with heat transfer (+ or -)AND entropy generated by IRREVERSIBILITIES (ALWAYS + or 0)

Can INCREASE or DECREASE entropy by heat transferIRREVERSIBILITY will ALWAYS INCREASE entropy

ZERO entropy generated for a REVERSIBLE process

( )2

2 11 b

QS STδ σ

− = +

∫


Entropy Balance: Closed Systems2

2 11 b

QS STδ σ − = + ∫

Entropy Change

Entropy Transfer

Entropy Production

Since σ measures the effect of irreversibilitiespresent within the system during a process,

its value depends on the nature of the process, and thus is NOT a property

2

2 11 b

QS STδ − ≥ ∫

Some Examples

Reversible, Adiabatic Process:

( )2

2 11 b

QS STδ σ

− = +

∫


0Adiabatic

0Reversible

A Reversible, Adiabatic Process:Is an Isentropic Process

Some ExamplesIs an Isentropic Process always a Reversible, Adiabatic Process?

( )2

2 11 b

QS STδ σ

− = +

∫


Suppose Entropy generation occurs by Irreversibility,Is there a way to decrease entropy to produce an Isentropic Process?

Some Examples

All Isentropic Processes are NOT Reversible and Adiabatic

( )2

2 11 b

QS STδ σ

− = +

∫


All Reversible and Adiabatic Processes are Isentropic

Some Examples

( )2

2 11 b

QS STδ σ

− = +

∫


What happens to Entropy for an Adiabatic, Irreversible Process?

Text Example

Other common forms of entropy balance:

2 1b

QS ST

σ− = +Uniform Boundary Temperature, Tb

2 1j

j j

QS S

Tσ− = +∑

T1

T2 Tn

Q1

QnQ2

For multiple heat transfers

(Note: Q could be + or -)

1 22 1

1 2

..... n

n

QQ QS ST T T

σ− = + − + +

Other common forms of entropy balance:

T1

T2 Tn

Q1

QnQ2

j

j j

QdSdt T

σ

••

= +∑

Rate basis, multiple heat transfers

(Note: could be + or -)

b

QdSTδ δσ = +

Q

Differential form:

Note inexact differential notation for non-properties: Q and σ

Evaluating Entropy Production and Transfer

Absolute value of entropy production not as usefulas relative values: Can use this information to focusattention on components where most irreversibiltiesare produced

Example 6.2: Irreversible process of water

Example 6.3: Eval Min theoretical compression work

Example 6.4: Pinpointing Irrevs

Increase in Entropy Principle:Closed Systems

Show that the summation of the entropy changes of surroundings AND system must always increase or remain the same

] 0isol

E∆ =

System

Surroundings

Energy Transfer is Zero

Mass Transfer is Zero

The larger system is an ISOLATED SYSTEM

] 0isol

E∆ = ] ] 0system surr

E E∆ + ∆ =

System

Surroundings



The larger system is an ISOLATED SYSTEM

Conservation of energy constrains possible processes:

Energy changes in system and surroundings must balance

However, not all such processes are possible (2nd Law)

System

Surroundings



] 0isolisolS σ∆ = >

] ] 0isolsystem surrS S σ∆ + ∆ = >

]2

1isolisol

b

QSTδ σ ∆ = + ∫

0Adiabatic

System

Surroundings



] ] 0isolsystem surrS S σ∆ + ∆ = >

Only processes that can occur are when entropy of Isolated System INCREASES.

Entropy is Extensive: Entropy need NOT increase for BOTH System AND Surroundings, but SUM must increase.

Direction of process and feasibility constrained

Spontaneous processes tend to reach equilibrium- increasing S

Increase in Entropy Principle:Closed Systems

The summation of the entropy changes of surroundings AND system must always increase (or remain the same for ideal)

] ]totalsystem surrS S σ∆ + ∆ =

0 ideal0 actual0 impossible

total

total

total

σσσ

=><

Examples in Text

Example 6.5: Quenching metal bar

Statistical Interpretation of Entropy

Statistical Interpretation of Entropy

Entropy Rate Balance:Control Volumes

jcvi i e e CV

j i ej

QdS m s m sdt T

σ

•• • •

= + − +∑ ∑ ∑Rate of entropy change

Rates of entropy transferRate of entropy

production

Inlets, iExits, e

jTjQ

cvσ

In + Gen = Stored + Out


( )cv VS t sdVρ= ∫ j

Aj bj

Q q dAT T

=

∑ ∫

When spatial variations occur, use Integral Form:

( ) ( )n n genV A A Ai ei eb

d qsdV dA s V dA s V dAdt T

ρ ρ ρ σ•

= + − +

∑ ∑∫ ∫ ∫ ∫

i i ni i A i

m s s V dAρ

=

∑ ∑ ∫


At Steady-State

0 CVj

i i e ej i ej

Qm s m s

Tσ

•• • •

= + − +∑ ∑ ∑

Mass:

Energy:

Entropy:

i ei e

m m• •

=∑ ∑

2 2

02 2i e

cv cv i i i e e ei e

V VQ W m h gz m h gz• • • •

= − + + + − + +

∑ ∑

Single Inlet, Single OutletSteady-State Control Volumes

i em m m= =

( ) ( ) ( )2 2

02

i ecv cvi e i e

V VQ W h h g z zm m

−= − + − + + −

( )0 CV

ji e

j j

Qm s s

Tσ

•• •

= + − +∑

Mass:

Energy:

Entropy:

( )10 CVji e

j j

Qs s

m T mσ

• • = + − + ∑


Entropy: ( ) 1 CVje i

j j

Qs s

m T mσ

• • − = + ∑

( ) CVe is s

mσ•

− =

Entropy passing from inlet to exit can: increase, decreaseor remain constant. Second term RHS is positive or zero.

Entropy/mass can only decrease if NET entropy flow OUT with heat transfer exceeds entropy Generation IN control volume

When NO heat transfer (adiabatic):


( ) CVe is s

mσ•

− =When NO heat transfer (Adiabatic):

Entropy increases when IRREVERSIBILTY present

Entropy constant when REVERSIBLE: ISENTROPIC

e is s=

http://www.visi.com/~darus/hilsch/

Examples

6.6 Entropy production in a steam turbine

6.7 Evaluating a performance claim

6.8 Entropy production in heat pump components

Isentropic ProcessesShowing isentropic processes is rapid/easy

on T-s or h-s diagrams. Vertical lines:

However, tabular data may still be used as well.

Isentropic Processes

State 1 is in superheated region

P1 and T1 used to get s1

State 2 is in superheated region

Where s2 = s1. Use P2, T2, other

Isentropic process: s3 = s2 = s1 Use P3 or T3 to get ssat.

IF s3 < ssat, then state 3 is in saturated region.

Get quality, x3, then can get other properties.

Saturated Water

Table A-2, A3, A2E, A3E

T

v

f g

Superheated Vapor

Table A-4, A-4E

T

v

f g293 F

500 F

Isentropic Processes: Ideal Gas

( ) ( ) ( ) ( ) 22 2 2 1 1 1 2 1

1

, , lno o Ps T P s T P s T s T RP

− = − −

2 1s s=

22 2 1 1

1

( ) ( ) ln Ps T s T RP

= +

If we have 3 of: T1, T2, P1, P2, then we can get the fourth

Example: T1 and two pressures, can get T2:

Use Tables A-22 and A-23for AIR and

A-22E and A23E (English)for other gases

( ) ( )2 12 1 exp

o os T s TP P

R −

=


( ) ( )2 12 1 exp

o os T s TP P

R −

=

( )

( )

2

2

1 1

exp

exp

o

o

s TRP

P s TR

=

( ) ( )expo

r

s TP T

R

=

Use Tables A-22 for Relative Pressure as function of Temperature. Not truly a pressure. Don’t confuse with Reduced Pressure (Compressibility Diagram)

2 2

1 1

r

r

P PP P

=

Define Relative Pressure, Pr

(for s1 = s2, AIR only)

Isentropic Processes: Ideal GasRTvP

=PG equation of state:

2 2 1

1 2 1

v RT Pv P RT

= ( )

( )1 12 2

1 2 2 1

r

r

P Tv RTv P T RT

=

( ) ( ) ( )rr

RTv T f TP T

= =

Define Relative Volume, vr

Use Tables A-22 for Relative Volume as function of Temperature. Not truly a volume. Don’t confuse with Pseudo-reduced specific volume (Compressibility Diagram)

2 2

1 1

r

r

v vv v= (s1=s2, AIR only)


( )1 1

2 2 1

1 1 2

k kkT P v

T P v

− −

= =

Assume constant specific heatsand define ratio of specific heats, k:

( ) ( ) 2 22 2 1 1

1 1

, , ln lnpT Ps T P s T P C RT P

− = −

( ) ( ) 2 22 2 1 1

1 1

, , ln lnvT vs T v s T v C RT v

− = +

2 2

1 1

0 ln lnvT vC RT v

= +

2 2

1 1

0 ln lnpT PC RT P

= −

1P

v

ckc

= ≥

1vRck

=−1p

kRck

=−

2 2

1 1

kP vP v

=

Examples:

6.9 Isentropic process in Air

6.10 Air leaking from tank

Isentropic EfficienciesComparing actual (adiabatic) and isentropic devices with

- same inlet states- same exit pressure

Turbines, compressors, nozzles and pumps

Isentropic turbine efficiency:

This helicopter gas turbine engine photo is courtesy of the U.S.Military Academy.

WTurbine

2

Turbine

1

WWTurbine

2

Turbine

1

Actual versusIsentropic

Isentropic Efficiencies(Adiabatic, Reversible)

Turbines

This helicopter gas turbine engine photo is courtesy of the U.S. Military Academy.

1st Law:

2nd Law:

1 2cvW h hm

•

• = −

WTurbine

2

Turbine

1

WWTurbine

2

Turbine

1

2 1 0cv s sm

σ•

• = − ≥

0.7 0.9Tη≤ ≤Typical:

Isentropic Turbine Efficiency

( )( )

1 2

1 2

1cv

Tcv s

s

Wm h h

h hWm

η

− = = ≤

−

p2

s2

s2s

(h1 – h2)(h1 – h2s)

Accessible states: Q = 0; ∆s > 0

Isentropic expansion

Note:

Mollier chart (T-s)

Actual expansion

( )( )

1 2

1 2

1cv

Tcv s

s

Wm h h

h hWm

η

− = = ≤

−

p2

s2

s2s

(h1 – h2)(h1 – h2s)


Actual expansion

Isentropic expansion

Note:

Isentropic Turbine Efficiency (T-s)

(Mollier chart better)

Compressors and Pumps

Rotating compressors

Reciprocating compressor

Common Form of 1st Law:

( ) ( )2 2

1 21 2 1 22

W V Vh h g z zm

•

•

−= − + + −

[ W < 0 ]

s2

( )( )

2 1

2 1

1

cv

ssc

cv

Wm h h

h hWm

η

− − = = ≤

− −

p1s2s

(h2s – h1)

(h2 – h1)


Actual compression

Isentropic compression

Note:

p2

T2

T1s1

Isentropic Compressor Efficiency (Gases)

h

Liquid

Isentropic Efficiencies: Compressors (Gases) and Pumps (Liquids)

Typical Isentropic Efficiencies of Compressors and Pumps:

Pumps, assuming incompressible model:(will consider more detail later)

( )2 1

2 1

sP

v P Ph h

η−

=−

0.75 0.85cη≤ ≤

Nozzles and Diffusers

( )2 2

1 21 20

2V Vh h −

= − +

Common Form of 1st Law:

Isentropic Nozzle Efficiency

V

1

Nozzle

V

2Nozzle

V

2

222

2

/ 2/ 2nozzle

s

VV

η =

0.95nozzleη ≥Common for:

Examples:

6.11 Eval turbine work using Isentropic efficiency

6.12 Eval Isentropic turbine efficiency

6.13 Eval Isentropic nozzle efficiency

6.14 Eval Isentropic compressor efficiency

Special Cases: Evaluate Q & W for NO Internal Irreversibility

Internally Reversible, Steady-State FlowOne-inlet, One-Outlet:

( )2 1cvQ T s sm

•

• = −

Heat Transfer:

( )1 20 CVcvQ m s sT

σ•

• •

= + − +

0

2

1

Re

cv

Int v

Q Tdsm

•

•

=

∫

T=T(s)

S.S

Special Cases (Internally Reversible): Heat Transfer and Work

( ) ( ) ( )2 22

1 21 2 1 2

1Re 2cv

Int v

V VW Tds h h g z zm

− = + − + + −

∫

Internally Reversible, Steady-State FlowOne-inlet, One-Outlet:

Work Transfer for Reversible or Irreversible:

( ) ( ) ( )2 2

1 21 2 1 22

cv cvV VW Q h h g z z

m m

−= + − + + −

Special Cases: Heat Transfer and Work

Tds dh vdP= −

( )2

2

2 1 11

Tds h h vdP= − −∫ ∫( ) ( )

2 221 2

1 21Re 2

cv

Int v

V VW vdP g z zm

− = − + + −

∫

Thus,

2

1nt

cv

I rev

W vdPm

= −

∫

KE and PE changes often negligible

2

1nt

cv

I rev

W vdPm

= −

∫

Note: magnitude of work per mass for gas or liquid isdirectly related to specific volume of fluid, v

Thus, for same pressure rise, magnitude of work per mass for liquid in pump (low v) is much smaller than for gas(larger v) in compressor

Derived for Internally Reversible case, but qualitatively truefor real, irreversible processes

Special Cases: Heat Transfer and Work

Internally Reversible, Steady-State Flow:

2

1intrev

cvQ Tdsm

•

•

=

∫2

1intrev

cvW vdPm

•

•

= −

∫ (In many cases∆KE = ∆PE = 0)

Internally Reversible, Steady-State, Incompressible fluid:

( )2 1

intrev

cvW v P Pm

•

•

= − −

Steady State, Reversible,no significant CV work terms(eg nozzles & diffusers):

( )2 2 2

2 12 1

1

02

V VvdP g z z −+ + − =

∫

This equation, used commonly in fluid mechanics, is known as the Bernoulli Equation

Work in Polytropic ProcessesInternally Reversible

( )2 2 1 1

intrev

(n 1)1

cvW n P v Pvnm

•

•

= − − ≠

−

constantnPV =

( ) 21 1

1intrev

ln (n=1)cvW PPvPm

•

•

= −

( )2 2

11

1 1intrev

tan ncvn

W dPvdP cons tPm

•

•

= − = −

∫ ∫General:

Work in Polytropic Processes: Ideal Gas

( )2 2 1 1

intrev

(n 1)1

cvW n P v Pvnm

•

•

= − − ≠

−

1

2 2

1 1

kkT P

T P

−

=

constantnPV =

For Ideal Gases:

( )2 1

intrev

(ideal gas, 1)1

cvW nR T T nnm

•

•

= − − ≠

−

( )1

1 2

1intrev

1 (ideal gas, n 1)1

n ncvW nRT P

n Pm

• −

•

= − − ≠ −

Equivalent to:

2

1intrev

ln (ideal gas, n=1)cvW PRTPm

•

•

= −

Where:

Examples:

6.15 Polytropic compression of air

END

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Chapter 6: Using Entropy - University of Arizonajmcgrath/ln.ch6.notes1.pdf · Chapter 6: Using Entropy ... (Mollier) Diagram. ... Temperature–entropy diagram for water (SI units)