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Entropy Change

P M V Subbarao

Professor

Mechanical Engineering Department

A Sing

le Reason for Every Thing

That Happens!!!

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The Thermodynamics of Temperature Creation

The Gibbsian equation,defines the change in specific

entropy of any substance during any reversible process.vdpdhpdvduTds =+=

Consider a control mass

executing a constant

volume process:

pdvduTds +=

constant=

=

vs

uT

The relative change in internal energy of a control mass w.r.t.

change in entropy at constant volume is called as absolute

temperature.

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The Thermodynamics of Temperature Creation

vdpdhTds = Consider a control volume executing a reversible constantpressure process:

constant==

pshT

The relative change in enthalpy of a control volume w.r.t.

change in entropy at constant pressure is called as absolute

temperature.

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Entropy change of an ideal gas

rom the Gibbsian equations, the change of entropy of a

substance can be expressed as

dPT

v

T

dhdsdvT

P

T

duds =+= or

For an ideal gas, u=u(T) and h=h(T),

du=cv(T)dT and dh=cp(T)dT and Pv=RT

( ) ( )dP

T

v

T

dTTCdsdv

T

P

T

dTTcds

pv =+= or

!y "ntegration, the change in the entropy is

( )

+=

#

$

$

#

#$ lnv

vR

T

dTTcss v

( )

=

#

$

$

#

#$ lnp

pR

T

dTTcss por

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"deal Gas %ith constant specific heats

&hen specific heats are constant 'calorically perfect gas(,the integration can be simplified:

"f a process is isentropic 'that is adiabatic and reversible(, ds=0,s=s!,

=

#

$

#

$#$ lnln

p

pR

T

Tcss p

+

=

#

$

#

$#$ lnln

v

vR

T

Tcss v

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"sentropic )rocess %ith idea gas

*lnln

#

$

#

$ =

p

pR

T

Tcp*lnln

#

$

#

$ =

+

v

vR

T

Tcv

=

#

$

#

$lnln

v

vR

T

Tcv

=

#

$

#

$lnln

p

pR

T

Tcp

( )

=

#

$

#

$lnln

v

vcc

T

Tc pvv

( )

=

#

$

#

$lnln

p

pccT

Tc vpp

=

#

$

#

$ln#ln

v

v

c

c

T

T

v

p

=

#

$

#

$ln#ln

p

p

c

c

T

Tc

p

vp

( )

=

#

$

#

$ln#ln

v

v

T

T

=

#

$

#

$ln

##ln

p

p

T

T

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( )

=

#

#

$

#

$

v

v

T

T

=

#

#

$

#

$

p

p

T

T

( )

=

#

#

$

#

#

$

p

p

v

v

=

#

#

$

#

$

pp

vv

=

#

#

$

#

$p

p

v

v

=

#

$

#

$

v

v

p

p

=

$

#

#

$

v

v

p

p( ) ( )

##$$ vpvp =

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"sentropic )rocess by an idea gas %ith constant

propeties

( )

=

#

#

$

#

$

v

v

T

T

=

#

#

$

#

$

p

p

T

T

=

$

#

#

$

v

v

p

p

( )

#

##

Cv

T=

$

#

CT

p=

+Cpv =or or

Are the reversible Process practicable?

100% perfection is possible but may not ne practicable..!?!!?!

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)ractical )rocesses are influenced by "rreversibilities

luid friction

olid friction

Electrical resistance

Thermo-chemical eactions 'Combustion( /nrestrained motion

"eat Trans#er $ith a #inite temperature di##erence

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olid riction is an "rreversibility

)E 0E

1

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)E 0E1

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)E 0E

1

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11

everseT2" " 34T )4"!5E.

1?

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1

#$

6+

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"rreversible and eversible engines

5T

12E

15E

E

&net

7ssume that an irreversible

Engine is more efficient than

the reversible engine.

12E"

15E"

E"

&net"

12E

15E

E

&net"

revirr >

"ER

Rnet

"E%

%net

&'

&' ,, >

2T

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or same &net, 12E" 8 12E

"mplies that, 915E" 9 8 915E9

!ut a reversible engine can be completely reversed and it %ill

%or as a heat pump.

&net," &net,;

12E" 12E

rev

rev

# =

"PR"E%

rev

irr&&

#

5et us construct a compound machine using an irreversible engine

and reversed reversible engine 'reversible 2eat )ump(.

or same 9&net 9,

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12)

15)

12E"

E"

&net

12E"8 912) 9

15E

5T 'ource(

2T 'in(

915E"9 8 15)

15)- 915E" 9

912) 9 - 12E"

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"rreversible

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urther =iscussions

irr< rev

1/irr>1/rev but, rev= rev

1/irr> rev ' 1/rev

1/irr>rev'

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"ncrease of Entropy )rinciple

Entropy

changentropy eneration

The principle states that for an isolated 4ra closed adia*atic4r

+stem - +urroundings.

7 process can onltae place such that"gen 0 %here"gen# 0 for a

reversible process only and"gencan never be less than >ero.

Entropy

Transfer

(due to heat

transfer)

Increase of Entropy

Principle

=efine entropy generation +genas,

or a general )rocess

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"mplications of "ncrease of Entropy )rinciple

Entrop, unlie energy, is non/conservativesince it is al%ays

increasing.

The entropy of the universe is continuously increasing, in

other %ords, it is becoming disorgani>ed and is approaching

chaotic.

The entropy generation is due to the presence of

irreversibilities.

Therefore, the higher irreversibilities lead to the higher the

entropy generation and the lo%er the efficiency of a device.

The above is Engineering statement of the second la%

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Second Law & Entropy Balance

"ncrease of Entropy )rinciple is another %ay of stating the

+econd a$ o# Thermodnamics:

econd 5a% : Entropy can be created but 34T destroyed

"n contrast, the first la% states: Energy is al$asconserved.

3ote that this does not mean that the entropy of a system

cannot be reduced, it can.

2o%ever, totalentropy of a system ? surroundings cannotbe reduced.

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Entropy of /niverse

7 quantity of heat

& is spontaneousl trans#erred from the

surroundings at temperature T0to the control mass at temperature T1

et the %or done during this process be '1For this process * control mass and %rite

or the surroundings at T0, & is negative,and $e assume a reversi*le heat e2traction

so

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The total net change of entropy is therefore

ince T0 3 T, the 4uantit 5(T) / (T0)6 is positive, and$e conclude that

3et Change in Entropy of /niverse

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"f T 3 T0, the heat trans#er is #rom the control mass to the

surroundings

"t should be noted that the right-hand side of above equationrepresents an external entropy generation due to heat transfer

through a finite temperature difference.

+

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The Third 5a% of Thermodynamics

%&e entrop' c&an(e o) a s'ste* +"r,n( areers,ble ,sot&er*al process ten+s to$ar+s

ero $&en t&e t&er*o+'na*,c te*perat"reo) t&e s'ste* ten+s to$ar+s ero.

n t&e ne,(&bo"r&oo+ o) absol"te ero allreact,ons ,n a l,",+ or sol,+ ,n ,nternal

e",l,br,"* ta#e place $,t& no c&an(e ,nentrop'.

Nernst pr,nc,ple.

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)lanc@s statement of the +rd la%

"n #A##, )lanc one step further and made the hypothesis

that not only does the entropy di##erence vanish as T 7 0,

*ut that8

)lanc@s statement of the Third 5a%: The entrop o# ever

solid or li4uid su*stance in internal e4uili*rium at

a*solute 9ero is itsel# 9ero1

)lanc is Bust saying:

*lim*

=

+T

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Engineering elations from econd 5a%

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Entropy as 7 ate Equation

The second la% of thermodynamics %as used to %rite the

balance of entropy for a infinitesimal variation for a finite

change.

2ere the equation is needed in a rate form so that a given

process can be traced in time.

Tae the incremental change and divide * t1

&e get

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or a given control mass %e may have more than one source of

heat transfer, each at a certain surface temperature 'semi-

distributed situation(.

The rate o# entrop change is due to the #lu2 o# entrop into

the control mass #rom heat trans#er and an increase due to

irreversi*le processes inside the control mass1

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The econd 5a% 4f Thermodynamics

or 7 Control olume

%&e rate o) c&an(e o) propert' o) t&e s'ste*.

( ) ( ) inoutC:CM smsm

dt

d+

dt

d+ +=

5et;D Entropy of the system, +D ms.

inoutC:CM ;;

dt

d;

dt

d; +=

genCM +

T

&

dt

d+

+=

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Entropy ate Equation for C

Rate o# change in entrop o# a C: = Entrop in #lo$ rate

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The teady tate teady lo% )rocess

or the steady-state process, %hich has been defined before,%e conclude that there is no change %ith time of the property

'entropy( per unit mass at any point %ithin the control volume.

That is,

so that, for the steady-state process,

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"f in a steady-state process there is only one area over%hich mass enters the control volume at a uniform rate and

only one area over %hich mass leaves the control volumeat a uniform rate,

%e can %rite

and dividing the mass flo% rate out gives

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incesgenis al%ays greater than or equal to >ero, for an adiabatic

process it follo%s that

%here the equality holds for a reversible adiabatic process.

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Geometry of Turbine !lades for 2igh Efficiency

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Transient )rocess

or the transient process, the second la% for a control

volume, it can be %ritten in the follo%ing form:

"f this is integrated over the time interval t, $e have

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Therefore, for this period of time t, %e can %rite the second la%

for the transient process as

ince in this process the temperature is uniform throughout

the control volume at any instant of time, the integral on the

right reduces to

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and therefore the second la% for the transient process can be

%ritten

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