Module 8
Non equilibrium
Thermodynamics
Lecture 8.1
Basic Postulates
NON-EQUILIRIBIUM
THERMODYNAMICS
Steady State processes. (Stationary)
Concept of Local thermodynamic eqlbm
Extensive
property
Heat
conducting bar
z
define properties
Specific
property
Z
lim  
m 0 m
 
NON-EQLBM
THERMODYNAMICS
Postulate I
Although system as a whole is not in
eqlbm., arbitrary small elements of it
are in local thermodynamic eqlbm &
have state fns. which depend on state
parameters
through
the
same
relationships as in the case of eqlbm
states
in
classical
eqlbm
thermodynamics.
NON-EQLBM
THERMODYNAMICS
Postulate II
 
S
Entropy
gen rate
 F J
affinities
fluxes
NON-EQLBM
THERMODYNAMICS
Purely “resistive” systems
Flux is dependent only on affinity
at any
at that
instant
instant
System has no “memory”-
NON-EQLBM
THERMODYNAMICS
Coupled Phenomenon
J  J F0 , F1 , F2 ,  ; extensive prop.
Since Jk is 0 when affinities are zero,
1
J k   L jk F j  
2! i
j

j
Lijk Fi F j     
NON-EQLBM
THERMODYNAMICS
where L j
 J

 F
j


  2 J
 ; Lij  

 F F
0
 i j



0
kinetic Coeff
L jk  L jk F0 , F1 ,    
Postulate III
Relationship between
affinity & flux from
‘other’ sciences
NON-EQLBM
THERMODYNAMICS
Heat Flux
:
CT 
T
JQ  k
 
y
y
Momentum : J M
Mass
Electricity
:
:
Jm
Je
u
 u 
 
 
y
y
c
 D
y
E
 
y
NON-EQLBM
THERMODYNAMICS
Postulate IV
Onsager theorem {in the absence of
magnetic fields}
L jk  Lkj
NON-EQLBM
THERMODYNAMICS
Entropy production
involving heat Flow
in
systems
dx
T1
x
T2
A
NON-EQLBM
THERMODYNAMICS
T  Q 
J Q  k
 
x  A 
k T
Js 

T
T x
JQ
Entropy gen. per unit volume
J s xdx  J s , x
dx
NON-EQLBM
THERMODYNAMICS
 1
1
JQ 
 
Tx  dx Tx 


dx
J Q dT
d 1
 JQ      2 
dx  T 
T dx
J
Q dT

SQ   2 
T dx
NON-EQLBM
THERMODYNAMICS
Entropy generation due to current
flow :
dx
I
I
Je 
A
I
dE
J e   
A
dx
Heat transfer in
element length
  dE 
Q  I 
dx
 dx 
NON-EQLBM
THERMODYNAMICS
Resulting entropy production per
unit volume
Se 
Q
J e dE

T  A.dx 
T dx
NON-EQLBM
THERMODYNAMICS
Total entropy prod / unit vol. with
both electric & thermal gradients
J
Je dE
Q dT



S  SQ  S e   2 
 
T dx T dx
 J Q  FQ  J e .Fe
affinity
affinity
NON-EQLBM
THERMODYNAMICS
1 dT
FQ   2
T dx
1 dE
Fe  
T dx
Analysis of thermo-electric
circuits
Addl. Assumption : Thermo electric
phenomena can be taken as LINEAR
RESISTIVE SYSTEMS

J K   L jK Fj
J
{higher order
terms negligible}
Here K = 1,2 corresp to heat flux “Q”,
elec flux “e”
Analysis of thermo-electric
circuits
 Above equations can be written as
J Q  LQQ FQ  LQe Fe
J e  LeQ FQ  Lee Fe
Substituting
for
affinities,
the
expressions derived earlier, we get
LQQ dT
1 dE
JQ   2
 LQe 
T dX
T dX
LeQ dT
1 dE
Je   2 
 Lee
T dX
T dX
Analysis of thermo-electric
circuits
We need to find values of the kinetic
coeffs. from exptly obtainable data.
Defining electrical conductivity 
as the elec. flux per unit pot. gradient
under isothermal conditions we get
from above
Lee dE
dE
Je  
 
T dX
dX
 Lee  T
End of Lecture
Lecture 8.2
Thermoelectric
phenomena
Analysis of thermo-electric
circuits
The basic equations can be written as
J Q  LQQ FQ  LQe Fe
J e  LeQ FQ  Lee Fe
Substituting
for
affinities,
the
expressions derived earlier, we get
LQQ dT
1 dE
JQ   2
 LQe 
T dX
T dX
LeQ dT
1 dE
Je   2 
 Lee
T dX
T dX
Analysis of thermo-electric
circuits
We need to find values of the kinetic
coeffs. from exptly obtainable data.
Defining electrical conductivity 
as the elec. flux per unit pot. gradient
under isothermal conditions we get
from above
Lee dE
dE
Je  
 
T dX
dX
 Lee  T
Analysis of thermo-electric
circuits
Consider the situation, under
coupled flow conditions, when
there is no current in the material,
i.e. Je=0.
Using the above
expression for Je we get
LeQ dT Lee dE
0 2

T dX T dX
LeQ
 dE dX 

 

T Lee
 dT dX  J e 0
Seebeck
effect
Analysis of thermo-electric
circuits
or
LeQ
 dE 



T Lee
 dT  J e 0
 dE 
Seebeck coeff.     
 dT  J e 0
 LeQ   T Lee    T
Using Onsager theorem
LQe  LeQ    T 2
2
Analysis of thermo-electric
circuits
Further from the basic eqs for Je &
JQ, for Je = 0
we get
LQQ dT LQe  LeQ dT 
 

JQ   2


T dX T  T Lee dX 
 Lee LQQ  LeQ LQe dT


2
Lee T
dX
Analysis of thermo-electric
circuits
For coupled systems, we define
thermal conductivity as
k 
 JQ 


 dT dX Je 0
This gives
 Lee LQQ  LQe LeQ
k 
2
Lee T
Analysis of thermo-electric
circuits
Substituting values of coeff. Lee, LQe,
LeQ calculated above, we get

LQQ  kT  T   T
2

 T k   T
2
2

2

Analysis of thermo-electric
circuits
Using these expressions for various
kinetic coeff in the basic eqs for
fluxes we can write these as :


dT
 dE 
JQ   k    T
  T 

dX
 dX 
2
dT
dE
J e   

dX
dX
Analysis of thermo-electric
circuits
We can also rewrite these with fluxes
expressed as fns of corresponding
affinities alone :
T
J Q  k
 T Je
dX
k
dE

Je  


JQ
2
2
k    T dX k    T
Using these eqs. we can analyze the
effect of coupling on the primary
flows
PETLIER EFFECT
Under Isothermal Conditions
dE
J e  
dX
b
a
Je
JQ, ab
Heat flux
J Qa   a T J e ;
J Qb   b T J e
PETLIER EFFECT
Heat interaction with surroundings
J Q a b  J Q a  J Q b   a   b T J e
Peltier eff .   ab J e
Peltier coeff.
 ab  T  a   b 
Kelvin Relation
PETLIER REFRIGERATOR
a : Cu
b : Fe
C   F  13.7 V 0 K
u
JQ ab  ?
 13.7
V
K
e
20 Amp. T ~ 270 K
 270 K 20 Amp  .074W
Semi conductors : Bi 2 Te3 N  P
 a   b  423 v K
THOMSON EFFECT
Total energy flux thro′ conductor is
JQ, surr
J  J  J E
E
Q
e
Using the basic
eq. for coupled
flows
T
J E  k
 T Je  Je E
x
T
 k
  T  E  J e
x
Je
JQ
dx
Je
JQ
THOMSON EFFECT
The heat interaction with the
surroundings due to gradient in JE
is
d J Q , surr  J E xdx  J E x
d JE

 dx
dx
d  T

  k
 T  E J e   dx
dx 
x

THOMSON EFFECT
Since Je is constant
conductor
d J Q , surr
dx
thro′ the
 T dk dT
 k 2  
x
dx dx
dT dE 
 d
 J e T



dx dx 
 dx
2
THOMSON EFFECT
Using the basic eq. for coupled flows,
viz.
dT
dE
J e   

dx
dx
above eq. becomes (for homogeneous
dT
material, k const.; const .
dx
dJ Q , surr
dx
d J e
 T Je

dx

Thomson heat
2
Joulean heat
THOMSON EFFECT
reversible heating or
cooling experienced
due to current flowing
thro′ a temp gradient
d
T Je
dx
J Q ,T
dT
  Je 
dx
Thomson coeff
Comparing we get
d
 T
dT
THOMSON EFFECT
We can also get a relationship
between Peltier, Seebeck & Thomson
coeff. by differentiating the exp. for
ab derived earlier, viz.
 ab   a   b T
d ab
 d a d b 

  a   b   T 


dT
dT 
 dT
  a   b    a   b 
End of Lecture
Analysis of thermo-electric
circuits
 Above equations can be written as
J Q  LQQ FQ  LQe Fe
J e  LeQ FQ  Lee Fe
Substituting
for
affinities,
the
expressions derived earlier, we get
LQQ dT
1 dE
JQ   2
 LQe 
T dX
T dX
LeQ dT
1 dE
Je   2 
 Lee
T dX
T dX
Analysis of thermo-electric
circuits
We need to find values of the kinetic
coeffs. from exptly obtainable data.
Defining electrical conductivity 
as the elec. flux per unit pot. gradient
under isothermal conditions we get
from above
Lee dE
dE
Je  
 
T dX
dX
 Lee  T