Week 13. Isentropic Efficiency Entropy Balance

Week 13. Isentropic Efficiency

Entropy Balance

GENESYS Laboratory

Objectives

1. Derive the reversible steady-flow work relations

2. Develop the isentropic efficiencies for various steady-flow devices

GENESYS Laboratory

Isentropic Efficiencies of Steady-Flow Devices

The isentropic process

• involves no irreversibilities and serves as the ideal process for adiabatic

devices

• The ideal process that can serve as a suitable model for adiabatic steady-flow

devices (e.g. turbine, compressors, nozzle)

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Isentropic Efficiencies of Turbines

The ratio of the actual work output of the turbine to the work output that would be

achieved if the process between the inlet state and the exit pressure were isentropic

1 2

1 2

Actual turbine work

Isentropic turbine work

( & )

aT

s

a

s

w

w

h h

h h

ke h pe h

η = =

−≅

−

∆ << ∆ ∆ << ∆∵

GENESYS Laboratory

EX 1) Isentropic Efficiency of a Steam Turbine

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Isentropic Efficiencies of Compressors and Pumps

The ratio of the work input required to raise the pressure of a gas to a specified value in

an isentropic manner to the actual work input

( )

2 1

2 1

2 1

2 1

Isentropic compressor work

Actual compressor work

( & )

sC

a

s

a

sP

a a

w

w

h h

h h

ke h pe h

v P Pw

w h h

η

η

= =

−≅

−

∆ << ∆ ∆ << ∆

−= ≅

−

∵

A realistic model process for compressors that

are intentionally cooled during the

compression process is the reversible

isothermal process, defined as isothermal

efficiency

a

tC

w

w=η

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EX 2) Effect of Efficiency on Compressor Power Input

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Isentropic Efficiencies of Nozzles

The ratio of the actual kinetic energy of the fluid at the nozzle exit to the kinetic energy

value at the exit of an isentropic nozzle of the same inlet state and exit pressure

s

a

aa

s

aN

hh

hh

Vhh V V

V

V

21

21

2

22121

2

2

2

2

2 if

exit nozzleat KE Isentropic

exit nozzleat KE Actual

−

−≅

+=⇒<

==η

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EX 3) Effect of Efficiency on Nozzle Exit Velocity

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Entropy Balance

Total Total Total Change in the

entropy entropy entropy total entropy

entering leaving generated of the system

in out gen system

S S S S

− + =

− + = ∆

= Entropy Balance

Increase of entropy principle for any system

the entropy change of a system during a process is equal

to the net entropy transfer through the system boundary

and the entropy generated within the system

final initial 2 1

V

When the properties of the system are not uniform

V

where V is the volume of the system and is density.

systemS S S S S

S s m s dδ ρ

ρ

∆ = − = −

= =∫ ∫

Entropy Change of a System

CVgen

ki i e e

k

Q dSm s m s S

T dt+ − + =∑ ∑ ∑

ɺɺɺ ɺ

GENESYS Laboratory

Mechanisms of Entropy Transfer, Sin and Sout

By Heat Transfer

By Mass Flow

0

constant)(T

constant)(T

work

2

1heat

heat

=

≠≅=

==

∑∫

S

T

Q

T

QS

T

QS

k

kδ

mass

mass mass mass and

where is the cross-sectional area of the flow,

and is the local velocity normal to

cn c

A t

c

n c

S ms

S s V dA S s m S dt

A

V dA

ρ δ∆

=

= = =∫ ∫ ∫ɺ ɺ

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Summary

zzz

zzz

1)

2)

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Summary

zzz

3)

4)

Whew

Q

Q

Q

Q

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Summary

zzz

No. Don’t go.

Please!!!

I’ll be back!!1)

2)

3)

4)

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Summary

zzz

1)

2)

GENESYS Laboratory

Entropy Generation, Sgen

�

�(kW/K)

form rate in the or,

(kJ/K)

entropyin change of Rate

system

generationentropy of Rate

gen

mass andheat by transfer entropy net of Rate

outin

entropyin Change

system

generationEntropy

gen

mass andheat by ansferentropy trNet

outin

�����ɺ

�����ɺɺ

��������

dtSSSS

SSSS

∆=+−

∆=+−

smST

QS ɺɺ

ɺɺ == massheat ,

Entropy balance for any system undergoing any process

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Closed Systems

(kJ/K) 12systemgen SSSST

Q

k

k −=∆=+∑The entropy change of a closed system during a process is equal to the sum of the net

entropy transferred through the system boundary by heat transfer and the entropy

generated within the system boundaries

adiabatic process : k

k

Q

T∑ gen adiabatic systemS S+ = ∆

( )gen system surroundings

system 2 1

surroundings

system surroundings :

where,

surr

surr

S S S S

S m s s

QS

T

+

= ∆ =∆ + ∆

∆ = −

∆ =

∑

Since no mass flow across its boundaries

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Control Volumes

( )gen 2 1 CV

CVgen

(kJ/K)

the rate form

(kW/K)

ki i e e

k

k

i i e e

k

Qm s m s S S S

T

Q dSm s m s S

T dt

+ − + = −

+ − + =

∑ ∑ ∑

∑ ∑ ∑ɺ

ɺɺ ɺ

The rate of entropy change within the control volume

during a process is equal to the sum of the rate of

entropy transfer through the control volume boundary

by heat transfer, the net rate of entropy transfer into the

control volume by mass flow, and the rate of entropy

generation within the boundaries of the control volume

as a result of irreversibilities

The general entropy balance relations

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Control Volumes (Continue)

Steady-flow process

CVgen

steady-flow

k

i i e e

k

Q dSm s m s S

T dt+ − + =∑ ∑ ∑

ɺɺɺ ɺ

( )

( )

gen

gen

gen

0

steady-flow,single stream

steady-flow,single stream,adiabatic

ke e i i

k

ke i

k

e i

QS m s m s

T

QS m s s

T

S m s s

=

= − −

= − −

= −

∑ ∑ ∑

∑

ɺɺ ɺ ɺ

ɺɺ ɺ

ɺ ɺ

If the flow through the device is reversible and adiabatic, then the entropy

remains constant, regardless of the changes in other properties

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EX 4) Entropy Generation in a Wall

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EX 5) Entropy Generation during a Throttling Process

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EX 6) Entropy Generated when a Hot Block Is Dropped in a Lake

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EX 7) Entropy Generation in a Mixing Chamber

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EX 8) Entropy Generation Associated with Heat Transfer

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