Week 12. Entropy Change

Week 12. Entropy Change

Tds Equations

Objectives

1. Calculate the entropy changes that take place during processes for pure substances, incompressible substances, and ideal gases

2. Examine a special class of idealized processes, called isentropic

processes, and develop the property relations for these processes

The Tds Relations

dU W

Q

− δ

= δ

The differential form of the conservation of energy equation for a closed stationary system (a fixed mass) containing a simple compressible substance

PdV W

TdS Q

=

=

out rev, int

rev int

δ δ

(kJ)

(kJ/kg) Td

TdS dU PdV s = du + Pdv

= +

The 1^stT ds, or Gibbs, equation

The 2^nd T ds equation

vdP dh

T Pdv T

ds du

−

=

+

=

h = + u Pv

Entropy Change of Liquids and Solids

Liquids and solids can be approximated as incompressible substances

( dv c c c )

T cdT T

Pdv T

ds = du + = ∵ ≅ 0 ,

=

=

K) (kJ/kg

ln

1 2 avg

2 1 1

2

− = ∫ ≅ ⋅

T c T

T s cdT

s

Isentropic case:

0

ln

1 2 avg

1

2

T T

T c T

s

s − = = ⇒ =

EX 1) Effect of Density of a Liquid on Entropy

EX 2) Economics of Replacing a Valve by a Turbine

1

VS 2

Laws of Thermodynamics

•

Direction of a process

•

Quality point of view - In terms of “Entropy”

•

Entropy generation always increases

- If not, it violates 2

law of thermodynamics

•

A process increase Entropy high

Irreversibility high

•

A process increase Entropy low

Irreversibility low

•

Energy conservation

•

Quantity point of view - In terms of “Energy”

•

Energy cannot be created or destroyed, but it always

conserves

- If not, it violates 1

law of thermodynamics

•

Energy input – Energy output = Energy stored

•

Hot
Cold (O); Cold
Hot (X)

•

100% output w/o any loss

The first Laws The second Laws

Entropy Change of Ideal Gases

T vdP T

dh

T Pdv T

ds du

−

=

+

=

dT c

dh

dT c

du

RT Pv

p v

=

=

=

v

p

dT dv

ds c R

T v

dT dP

c R

T P

= +

= −

1 2 2

1

1 2 2

1 1 2

ln )

(

ln )

(

P R P

T T dT c

v R v

T T dT c

s s

p v

−

=

+

=

−

∫

∫

The differential entropy change of an ideal gas

The entropy change for a process obtained by integrating

Constant Specific Heats (Approximate Analysis)

2 2

2 1 avg

1 1

2 2

,avg

1 1

, ln ln

ln ln (kJ/kg K)

v

p

T v

s s c R

T v

T P

c R

T P

− = +

= − ⋅

2 2

2 1 avg

1 1

2 2

,avg

1 1

, ln ln

ln ln (kJ/kmol K)

v u

p u

T v

s s c R

T v

T P

c R

T P

− = +

= − ⋅

Entropy changes can also be expressed on a unit mole basis

The entropy change relations for ideal gases under the constant specific heat assumption

Variable Specific Heats (Exact Analysis)

K) (kJ/kg

ln

1 2 1

2 1

2

− = − − ⋅

P R P

s s

s

s

It is expressed on a unit-mole basis

The entropy change relations for ideal gases under the variable specific heat assumption

∫

=

T T dT c

s

( )





1 2

2

1

( ) s s

T T dT

c

= −

∫

K) (kJ/kmol

ln

1 2 1

2 1

2

− = − − ⋅

P R P

s s

s

s

EX 3) Entropy Change of an Ideal Gas

Isentropic Processes of Ideal Gases (Approximate Analysis)

cv

R

v

v

v T

T v

v c

R T

T 





 

= 

− ⇒

=

2 1 1

2 1

2 1

2

ln ln ln

ln

1

, = ⇒ = −

−

= k

Rc c

k c c c R

v v

p v

p

1

2 1

1 const. 2

(ideal gas)

k

s

T v

T v

−

=

   

  = 

   

1^st isentropic relation

2 2

2 1 avg

1 1

2 2

,avg

1 1

, ln ln

ln ln

v

p

T v

s s c R

T v

T P

c R

T P

− = +

= −

( 1)

2 2

1 const. 1

(ideal gas)

k k

s

T P

T P

−

=

   

  = 

   

2^nd isentropic relation

2 1

1 const. 2

(ideal gas)

k

s

P v

P v

=

   

  =  

   

3^rd isentropic relation

1 1

constant

constant (ideal gas) constant

k

( -k) k k

Tv TP

Pv

−

=

=

=

Compact forms

Isentropic Processes of Ideal Gases (Exact Analysis I)

0 ln

1 2 1

2 1

2

− = − − =

P R P

s s s

s

) exp(s R P_r = ^

1 2 1

2

ln

P R P

s s

=

+

 

 





 

 





− =

=

R s

R s R

s s P

P









1 2 1

2 1

2

exp exp exp

Relative pressure

1 2 const.

1 2

r r

s

P

P P

P  =





 



=

2 1 1 2 1

2 2

2 2 1

1 1

P P T T v

v T

v P T

v

P = → =

1 1

2 2

2 1 1

2 1

2

r r r

r

P T

P T P

P T T v

v = =

1 2 const

1 2

r r

s

v

v v

v  =





 



=

Relative specific volume v_r=T/Pr

Isentropic Processes of Ideal Gases (Exact Analysis II)

• The values of P_r and v_r listed for air in Table A-17 are used for calculating the final temperature during an isentropic process

EX 4) Isentropic Compression of Air in a Car Engine

EX 5) Isentropic Compression of an Ideal Gas

Reversible Steady-Flow Work

The energy balance for a steady-flow device undergoing an internally reversible process

dpe dke

dh w

q

− δ

= + + δ

q

Tds

Tds dh vdP

δ =

= −

dpe dke

vdP

w = + +

− δ

2

rev 1

2 rev,in 1

(kJ/kg) (kJ/kg)

w vdP ke pe

w vdP ke pe

= − + ∆ + ∆

= + ∆ + ∆

∫

∫

For incompressible fluid

(

) (kJ/kg)

rev

v P P ke pe

w = − − − −

( ) ( ) 0

2

2 1 2 2 1

2

− + − =

+

− V V g z z

P P

v

Bernoulli equation

In case of no work interactions,

EX 6) Compressing a Substance in the Liquid versus Gas Phases

Proof that Steady-Flow Devices Deliver the Most and Consume the Least Work when the Process Is Reversible

The energy balance for actual and reversible devices

act act

rev rev

Actual : ke pe

Reversible : ke pe

q w dh d d

q w dh d d

δ δ

δ δ

− = + +

− = + +

act rev

act rev

rev rev

act act

or

w w

q q

w q

w q

δ δ

δ δ

δ δ

δ δ

−

=

−

−

=

−

where, δ q

= Tds

( 0 )

s

0 s

act rev

act

act rev

act

≥

⇒ ≥

⇒ ≥

− ≥

=

−

T w

w

T d q

T w w

T d q

∵ δ

δ δ

δ

Minimizing The Compressor Work

The compressor work is given by

2 rev,in 1

w =

∫

2 1

(kJ/kg) =

∫

• Ways of minimizing the compressor work is

1. Minimizing the irreversibilities such as friction, turbulence, and nonquasi-equilibrium compression

2. Keeping the specific volume of the gas as small as possible

during the compression process, which can be achieved by maintaining the temperature of the gas as low as possible

• Effect of cooling during the compression process

( ) ( )

1

2 1 2

1 2

1 1 1 2

1 1 1 1 2 2

1 1

1 1

1

Isentropic process : constant

-1 1 1 1

k

k k k

k

k k

Pv

kR T T

P v dP k kRT P

w dP P v Pv P v

k k k P

P P

−

=

 

−    

= − = − = − = − = −  −   

∫ ∫

(

)

comp,in 1

1 1

1 1

1 1 1

k k

k k

kR T T kRT P k P

w Pv

k k P k P

− −

   

−        

= − = −        −    = −        −   

2^ndisentropic relation

Minimizing The Compressor Work (Continue)

( ) (

)

1 1 2 2

1

Polytropic process : constant

1 1 1 1

n

n n

Pv

nR T T nRT P w n Pv P v

n n n P

−

=

 

−    

= − − = − = −  −   

( )













 −









= −













 −









= −

−

= −

−

−

1 1 1 1

1

1

1 2 1

1

1 2 1

2 1 in

comp,

n n n

n

P Pv P

n n P

P n

nRT n

T T w nR

2 1 1 1

2 2 1

1 1 1 1

1 1 2

2 comp,in

1

Isothermal process : constant ,

ln ln

ln

Pv w vdP Pv Pv

P P

Pv dP Pv RT

P P P

w RT P

P

=

= − =

= − = − = −

=

∫

∫

Multistage Compression with Inter-cooling

• The gas is compressed in stages and cooled between each stage by passing it through a heat exchanger called an inter-cooler

• The effect of intercooling on compressor work is graphically illustrated on P-v and T-s diagrams

( ) ( )













 −







 + −













 −









= − +

=

−

−

1 1 1 1

1 2 1

1

1 1

in , comp in

, comp in

comp,

n n

x n

n x

P P n

nRT P

P n

w nRT w

w _{Ⅰ Ⅱ}

( )

comp,in

12 2

1 2 comp ,in comp ,in

value that minimizes the total work is or if so,

x

x x

P w

P P

P P P w w

P P

= =

=

EX 7) Work Input for Various Compression Processes