Thermodynamics IChapter 6Entropy

Mohsin Mohd SiesFakulti Kejuruteraan Mekanikal, Universiti Teknologi Malaysia

Entropy (Motivation)

The preferred direction implied by the 2nd Law can bebetter understood and quantified by the concept ofentropy.Entropy quantifies the 2nd Law and imposes thedirection of processes.Here we will study entropy and how processes willproceed only in the direction of increasing entropy.

Clausius Inequality

Erev Eirrev

QH QH

QL QL,irrev

W Wirrev

TH

TL

Consider two heat engines between the same temperatures

Consider of the whole cycle

T

Q

(a) Erev

0

0

11

=

=

−=

−=

−=

∫

∫∫

∫∫∫

rev

L

L

H

H

LL

HH

L

L

H

H

rev

T

Q

T

Q

T

Q

QT

QT

T

Q

T

Q

T

Q

(b) Eirrev

ΔQ+Q=Q

Q>Q

LirrevL,

LirrevL,

Combined to be:

= 0 if reversible< 0 if irreversible> 0 impossible

ENTROPY

1

2B

A

∫

∫∫

∫∫∫

=

−=

=

+

=

2

1

1

2

2

1

1

2

2

1

0

B

BA

BArev

T

Q

T

Q

T

Q

T

Q

T

Q

T

Q

Does not depend onpath; thermodynamicproperty!

Let's call this property entropy, S[kJ/K]

revT

QdS

≡

Entropy , S = extensive property [kJ/K]

Specific entropy, s = intensive property [kJ/kg.K]

∫∫

=−=

2

112

2

1revT

QSSdS

We have actually defined entropy change

From statistical thermodynamics, entropy ~ measure of molecular disorderWork is an ordered form of energy (High quality)

No entropy transfer along with work transferHeat is a disordered form of energy (Low quality)

Entropy is transferred along with heat transfer

“Entropy = zero for pure crystal at 0 K”Third Law of Thermodynamics

Entropy with a 0 K reference is called absoluteentropy

T-ds Relations

To find the relationship between δQ and T so that theintegration of δQ/T can be performed

- From the 1st Law (neglecting KE & PE)

pdvduTds

pdVdUTdS

dUpdVTdS

+=+==−

(a)

dU=δWδQU=WQ

revrev −−

pdV=δWTdS=δQ

rev

rev

- From the definition of enthalpy

vdppdvdhdu

vdppdvdudh

pvuh

−−=++=

+=

From (a)

vdpdhTds

pdvvdppdvdh

pdvduTds

−=+−−=

+=

(b)

T

vdp

T

dhds

T

pdv

T

duds

−=

+=&

vdpdhTds

pdvduTds

−=+=

Can be used for ∆S of any process (rev. & irrev.)Can be used for open & closed system

T-ds Relations

Entropy Change of Pure Substances

Can be obtained from Tds Relations in conjunction with otherthermodynamic relationsFor 2 phase substances like water and R-134a, use property tablesS = S2 – S1

For ideal gas, from Tds relations

1

2

1

212 lnln

v

vR

T

TCsss

v

dvR

T

dTCds

v

dvR

T

dTCds

T

pdv

T

duds

v

v

v

+=−=∆

+=

+=

+=

∫∫∫

v

R

T

pRTpv

dTCdu v

=→=

=

Assume Cv=constant& integrate

1

2

1

212 lnln

p

pR

T

TCsss

p

dpR

T

dTCds

p

dpR

T

dTCds

T

vdp

T

dhds

p

p

p

−=−=∆

−=

−=

−=

∫∫∫

Another relation,

v

R

T

pRTpv

dTCdh p

=→=

=

Assume Cp =constant& integrate

1

2

1

212

1

2

1

212

lnln

lnln

p

pR

T

TCsss

v

vR

T

TCsss

p

v

−=−=∆

+=−=∆

Both equations give the same resultChoose according to available data..

Entropy change

Isentropic Processes, S = 0

For ideal gases, with constant Cp & Cv

1

2

1

1

2

1

2

1

1

2

2

1

1

2

1

2

1

2

1

2

1

2

1

2

lnln

lnlnln

lnln

0lnln

−

−

=

=→

=−=

−=

=+=∆

k

k

CR

v

v

v

v

v

T

T

v

v

T

T

v

v

v

v

C

R

T

T

v

vR

T

TC

v

vR

T

TCs

v

1−=∴

=

=−

kC

R

C

Ck

RCC

v

v

p

vp

k

k

k

k

CR

p

p

p

p

T

T

p

p

p

p

p

p

C

R

T

T

p

pR

T

TCs

p

1

1

2

1

2

1

1

2

1

2

1

2

1

2

1

2

1

2

ln

ln

lnln

0lnln

−

−

=

=

=

=

=−=∆

Another one

k

k

C

R

kC

C

C

C

C

R

RCC

p

p

v

p

p

p

vp

1

11

−=∴

−=−=

=−

Combined to become

1

2

1

1

1

2

1

2

−−

=

=

kk

k

v

v

p

p

T

T Isentropic process forideal gas,constant Cp , Cv

( ) ( )

( ) ( )

dpv=w

vdp=δw

ped+ked+dh=δwvdpdh

ped+ked+dh=δwδq

rev

rev

rev

revrev

∫−

−

−−

−

2

1

Work for Open Systems

Reversible boundary work forclosed systems

∫= pdvwB

Open system reversible work, from 1st Law (single inlet/outlet)

Reversible work foropen system

neglecting ke & pe, & rearrange;vdpdh=δq

vdpdf=Tds

Tds=δq

rev

rev

−→−

Increase in Entropy Principle

• For reversible processes, entropy of universe (system surrounding)doesn't change, it is just transferred (along with heat transfer)

• In real processes, irreversibilities cause entropy of universe to increase• Heat reservoirs don't have irreversibility• To simplify analysis, assume irreversibilities to exist only inside the

system under study.• Inside the system, entropy transfer and generation occur.

irrev

1

2

rev

Irreversible cycle since one of theprocesses is irreversible

From Clausius Inequality;

gen

irrev

irrev

S+T

δQ=SS

T

δQSS

T

δQ>SS

T

δQ<SS

∫

∫

∫

∫

−

≥−

−

−−

2

1

12

2

1

12

2

1

12

2

1

12

Rearrange

Generalize for all processes

= when rev> when irrev< impossible

Sgen=entropy generated= 0 when rev.> 0 when irrev.< 0 impossible

genb

ST

QSS +

=− ∫

2

112

Change ofsystementropy

Transfer of entropy(T where Q istransferred, usuallyat boundary

Entropygeneration

Entropy Generation for Closed System

Sgen is the measure of irreversibilities involved during process

S surrounding or system may decrease (-ve), but S total(surrounding + system) must be > 0

Entropy of universe always increases (Entropy increase principle)

No such thing as entropy conservation principle

For a process to occur, Sgen> 0

0≥= systemgsurroundinuniversetotal,gen ΔS+ΔS=ΔSS

Entropy Generation for Closed System

gen

n

j j

j

genb

genb

ST

QSS

ST

QSS

ST

QSSS

+=−

+=−

+=−=∆

∑

∫

=112

12

12

For multiple heat transfers at different temperatures

gen

n

j j

j

gen

n

j j

j

ST

QdS

ST

Q

dt

dS

+=

+=

∑

∑

=

=

1

1

In rate form

In differential form

Sgen - depends on the process (friction, ∆T etc.)- not a property

Entropy Generation for Closed System

Entropy Generation for Open Systems

Apart from heat, mass flows also carry entropy along with them

Qin

min

sin

mout

soutCV

genoutinj j

jCV

genoutinj j

jCV

SmsmsT

QS

SSST

QS

+−+=∆

+−+=∆

∑∑∑

∑∑∑

=

=

1

1

genoutinj j

jCV SsmsmT

Q

dt

dS

+−+= ∑∑∑=1

Rate ofchange ofsystementropy

Rate of entropytransfer

Rate ofentropygeneration

Qin

min

sin

CV

Entropy Generation for Open Systems

Steady Flow Process

genooiij j

j

genooiij j

jCV

CV

outinCV

SsmsmT

Q

SsmsmT

Q

dt

dS

dt

dS

mmdt

dm

+−+=

+−+=

=

=→=

∑∑∑

∑∑∑

∑∑

=

=

1

1

0

0

0

m

S

Tm

Q

ss

SssmT

Q

gen

j jio

genoij j

j

j

+=−

+−+=

∑

∑

=

=

1

1

)(0

single inlet/outlet;

Steady Flow Process

Reversible adiabatic process = isentropic process

Isentropic is not necessarily reversible adiabatic

Reversible process Sgen = 0

Produces Wmax

Consumes Wmin

30

ISENTROPIC EFFICIENCIES OF STEADY-FLOW DEVICES

The isentropic process involves noirreversibilities and serves as the ideal

process for adiabatic devices.

The h-s diagram forthe actual andisentropic processesof an adiabaticturbine.

IsentropicEfficiencyof Turbines

Also known as Adiabatic Efficiencyor 2nd Law Efficiency

31

Isentropic Efficiencies of Compressors and Pumps

The h-s diagram of the actualand isentropic processes of an

adiabatic compressor.

Compressors aresometimesintentionallycooled tominimize thework input.

When kinetic andpotential energiesare negligible

32

Isentropic Efficiency of Nozzles

The h-s diagramof the actual and

isentropicprocesses of an

adiabatic nozzle.

If the inlet velocity of thefluid is small relative tothe exit velocity, theenergy balance is

Then,

A substance leavesactual nozzles at a

higher temperature(thus a lower velocity)as a result of friction.