The Second Law of Thermodynamics

Note: for a reversible process, on p-v diagram

and If the volume is constant,

revWvPd =∫2

1

02

1

=∫ vPd

1

2

wpdv =∫2

1

P

V

W=0

21 vv =

2

1

P

V

• Similarly, for any reversible adiabatic process, Q=0

• For any other reversible process

0== ∫TdsQrev

∫=2

1

TdsQrev

T

S

1

2

∫=2

1

TdsQrev

1S

2S

21 SS =0=Q

T

S

• Clausius Inequality:

0≤∫ TQδ

• Clausius inequality is valid for all thermodynamic cycles, reversible or irreversible.

for internally reversible cycle

for irreversible cycle

0=⎟⎠⎞

⎜⎝⎛∫

revTQδ

0<⎟⎠⎞

⎜⎝⎛∫ T

Qδ

represents a property changerevT

Q⎟⎠⎞δ

∫ ⎟⎠⎞=−=∆

⎟⎠⎞=

2

112

rev

rev

TQSSS

TQdS

δ

δ

• Consider a cycle composed of a reversible and an irreversible process:

irreversibleprocess

internallyreversibleprocess

1

2

• The quantity represents the entropy change of the system.

• For a reversible process, entropy transfer with heat

• For an irreversible process

• Some entropy is generated during an irreversible process and is always positive quantity. Its value depends on the process, and thus it is not a property of the system.

12 sss −=∆

∫=∆2

1 TQs δ

gensys sTQs +=∆ ∫

2

1

δ

Entropy Change for Pure Substances

• The entropy of a pure substance is determined from the tables, just as for any other property.

T

s

1

1

TP

11 Tatss f≈

1 Saturated liquid-vapor mixture

2

2

xT

fg2f2 sxss +=

2

2

PT

3s =from superheatedtable

2

3

fgf xsss +=( )12 ssmS −=∆

• T-s diagram of properties:

)( 12

2

1

SSTQ

TdsQ

TdsQ

rev

rev

rev

−=

=

=

∫

δ

Constantpressure lines

=revQArea underthe process

1 2

T

s

• Entropy of a fixed mass can be changed by:1) Heat transfer2) Irreversibilities

• The entropy of a fixed mass will not change during a process that is internally reversible and adiabatic.

• For a reversible, adiabatic process (called Isentropic process)

210 sss =⇒=∆

• If a steam turbine is reversible, and the turbine is insulated (thus the process is reversible and adiabatic)

21

0then

ssor

s

=

=∆T

s

1

2

• The Tds relations: differential form of thefirst law:

second law:

dUwQ revrev =−δδ

TdsQrev =δ

duPdvTdsdUVPdTdS

+=+=

vdPdhTdsvdPPdvdudh

PvuhPdvduTds

−=++=

+=+=

First Tds relation

Second Tds relation

• Entropy changes for Liquids and solids:Tds=du+pdv

ds=

• Liquids and solids are incompressible substances

Tpdv

Tdu

+

0≅dv

1

22

112 ln

TTc

TdTcss

TcdT

Tduds

==−

==

∫

( )

kgKkJ

ccc vp ==

Isentropic process for liquids and solids:

the isentropic process of an incompressible substance is also isothermal.

121

212 ln0 TT

TTcss =⇒==−

Note: Tds equations are derived by considering an internally reversible process. An entropy change obtained by integrating these equations is the change for any process.

“Entropy is a property and the change in entropy between any two states is independent of the details of the process linking the states.”

Tds equations are used to evaluate the entropy change between two states of an ideal gas.

dvTv

Tdhds

dvTP

Tduds

−=

+=

Entropy change of an Ideal gas

For an ideal gas:

PdPR

TdTcds

vdvR

TdTcds

RccRTPv

dTcdhdTcdu

P

v

vp

p

v

−=

+=

+==

==

,

On integration these equations give, respectively

To integrate these relations, we must know the temperature dependence of the specific heats.

1

212

1

212

ln

ln

2

1

2

1

PPR

TdTcss

vvR

TdTcss

T

TP

T

Tv

−=−

+=−

∫

∫

Using Ideal gas Tables:

Define

Where is the specific entropy at a temperature T and a pressure of 1 atm.

Note: The specific entropy is set to zero at the state where the temperature is 0 K and the pressure is 1 atm.

dTTTc

TsT

po ∫=0

)()(

)(Tso

Note: Because So depends only on the temperature, it can be tabulated versus temperature, like h and u.

Kmole.K

KJ ln)(

Kg.KKJ ln)(

;

,

1

21212

1

21212

1200

122

1

PPRssssor

PPRssss

then

ssTdTc

TdTc

TdTc

Also

uoo

oo

ooT

p

T

p

T

Tp

−−=−

−−=−

−=−= ∫∫∫

Entropy change of an Incompressible substance:for incompressible substance, specific heat depends solely on temperature, and

When the specific heat is constant:

Cconstant ible,incompress ln1

212 ←=−

TTCss

ibleincompress )(

)()(

)(

2

1

12 ←=−

=+=

+=

=

∫T

T TdTTCSS

TdTTCdv

TP

TdTTCds

dvTP

Tduds

dTTCdu

Isentropic Processes of an Ideal Gas

For internally reversible process:

For Isentropic process(Reversible, adiabatic process):1

22

112

1

22

112

ln)(

ln)(

VVR

TdTTcss

PPR

TdTTcss

v

p

−=−

−=−

∫

∫

012 =−ss

1

22

1

1

22

1

ln)(0

ln)(0

VVR

TdTTc

PPR

TdTTc

v

p

−=

−=

∫

∫

kk

PP

TT

similarly1

1

2

constants1

2

,−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛1

2

1

constants1

2

1

2

1

1

2

2

1

1

2

1

2

1

2

1

2

1

2

1

2

lnln

lnlnln

lnln0

lnln0

−

=

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=−=⇒

+=

−=

k

k

cR

v

v

p

VV

TT

VV

TT

VV

VV

cR

TT

VVR

TTc

PPR

TTc

v

Integrate:

heats specificconstant th wi gas, ideal process Isentropic Constant

Constant )ln(Constant lnConstant lnln

Constant lnln

lnln Constant

1

1

←=⇒

=×

=−=+

=+⇒=

+=

−

−

k

k

kv

v

PvvPv

RvPv

vcR

RPvRTVPuse

vRTc

TdvP

Tduds

PdvduTds

+=

+=

vdvR

TdTcv +=0

Ideal gas, Isentropic Process(PV=RT, ds=0)

Rss

PP

Rss

PP

PPRss

PPRssss

PPR

TdTcss

oo

oo

oo

oo

T

Tp

12

1

2

12

1

2

1

212

12

1

21212

1

212

exp

ln

ln

0ss : process Isentropic

ln

ln2

1

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

=−

=−

−−=−

−=− ∫

Relative pressure and relative volume:

The quantity is a function of temperature only and is given the symbol Pr, define as the relative pressure, and is tabulated for air and other ideal gases

)/exp()/exp(

1

2

1

2

RsRs

PP

o

o

=

)/exp( Rso

Note: Pr is not truly pressure, and also Pr should not be confused with the reduced pressure of compressibility chart.

)T(PP and )T(PP where

2rr1rr

1

2

21

1

2

==

=r

r

PP

PP Isentropic Process

(s1=s2) Ideal gas variable specific heats

Similarly a relation between specific volumes and temperatures and for two states having the same entropy can also be developed

The ratio is the relative volume

or

1

1

2

2

1

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

RTPP

RT

vv

rr

vPRT

=

1

2

1

2 ←=r

r

vv

vv Isentropic process,

Ideal gas variable specific heats

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

2

1

2

1

2

RTPPRT

vv

r

r

In general Isentropic process for ideal gas: (s2=s1)

re temperatuoffunction a isheat specific

heat specificconstant

1

2

1

2

1

2

1

2

←

⎪⎪⎭

⎪⎪⎬

⎫

=

=

←=

vv

vv

PP

PP

CPv

r

r

r

r

k

General, reversible (Polytropic) process for an Ideal gas:Pvn=c, where n is constant

∫=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

2

1

2

1

1

2

2211

PdvW

vv

PP

vPvPn

nn

12211

2

1 −−

== ∫ nvPvPdv

vc

n

1

2Slope=-n

P

Log VP

V

cPvn =1

2

T

S

n=1

n=k constant)(v =±∞=nn=0 (P=constant)

n=1

1<n<k

±∞=n

Constant pressure

n=1 (T=constant)Isothermal process

n=k (S=constant)Isentropic process

V=constant

n=0

P

V

Polytropic process on P-v and T-s diagrams

Isentropic Efficiencies of Steady-Flow Devices

• The Isentropic process involves no irreversibilitiesand serves as the ideal process for adiabatic devices.

• The actual process is irreversible and the actual device performance is less than the ideal case.

• The more closely the actual process approximates the idealized isentropic process, the better the device will perform.

• We define the efficiency (isentropic efficiency) of these devices as a measure of deviation of actual processes from the idealized one.

1) Isentropic efficiency of Turbines

00

21

21t

t

100η

workIsentropic work turbineActualη

<−−

==

=

s

a

s

a

hhhh

ww

T

s

1

1P

2P

s2a2

21 hhw −=&

2) Isentropic efficiency of Compressors and Pumps

12

12pc

pc

ηor η

ηor η

hhhh

WW

a

s

a

s

−−

=

=

T

s

T

s

a2s2

1

s2a2

2P

1P

1

3) Isentropic efficiency of Nozzles

s

a

aa

s

a

hhhh

Vhh

VV

21

21N

22

21

22

22

N

N

η

2 :Note

η

exit Nozzleat K.E. Isentropicexit Nozzleat E. K. Actualη

−−

≅

+=

=

=

T

s

1

1P

2P

s2a2