Post on 27-Mar-2015
Module 8
Non equilibrium Thermodynamics
Lecture 8.1
Basic Postulates
NON-EQUILIRIBIUM THERMODYNAMICS
Steady State processes. (Stationary)
Concept of Local thermodynamic eqlbm
Heat conducting bar
define propertiesSpecific property
Extensive property
z
m
Zm 0lim
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
JFS Entropy gen rate
affinities fluxes
NON-EQLBM THERMODYNAMICS
Purely “resistive” systems
Flux is dependent only on affinityat any instant
at that instant
System has no “memory”-
NON-EQLBM THERMODYNAMICS
Coupled Phenomenon
.;,,, 210 propextensiveFFFJJ
Since Jk is 0 when affinities are zero,
jiijkji
jj
jkk FFLFLJ!2
1
NON-EQLBM THERMODYNAMICS
where0
2
0
;
ji
ijj
j FF
JL
F
JL
kinetic Coeff ,, 10 FFLL jkjk
Postulate IIIRelationship between affinity & flux from ‘other’ sciences
NON-EQLBM THERMODYNAMICS
Heat Flux :
Momentum :
Mass :
Electricity :
y
TC
y
TkJQ
y
u
y
uJM
y
cDJm
y
EJ e
NON-EQLBM THERMODYNAMICS
Postulate IV
Onsager theorem {in the absence of magnetic fields}
kjjk LL
NON-EQLBM THERMODYNAMICS
Entropy production in systems involving heat Flow
T1 T2x
dx
A
NON-EQLBM THERMODYNAMICS
x
T
T
k
T
JJ Qs
A
Q
x
TkJQ
Entropy gen. per unit volume
dx
JJ xss dxx ,
NON-EQLBM THERMODYNAMICS
dx
TTJ
xdxxQ
11
dx
dT
T
J
Tdx
dJ QQ
2
1
dx
dT
T
JS QQ
2
NON-EQLBM THERMODYNAMICS
Entropy generation due to current flow :
Idx
A
IJ e
dx
dE
A
IJ e
Heat transfer in element length
dxdx
dEIQ
NON-EQLBM THERMODYNAMICS
Resulting entropy production per unit volume
dx
dE
T
J
dxAT
QS ee
.
NON-EQLBM THERMODYNAMICS
Total entropy prod / unit vol. with both electric & thermal gradients
dx
dE
T
Je
dx
dT
T
JSSS QeQ
2
eeQQ FJFJ .
affinity affinity
NON-EQLBM THERMODYNAMICS
dx
dT
TFQ 2
1
dx
dE
TFe
1
Analysis of thermo-electric circuits
Addl. Assumption : Thermo electric phenomena can be taken as LINEAR RESISTIVE SYSTEMS
J
jjKK FLJ {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
eQeQQQQ FLFLJ
eeeQeQe FLFLJ
Substituting for affinities, the expressions derived earlier, we get
dX
dE
TL
dX
dT
T
LJ Qe
QQQ
12
dX
dE
TL
dX
dT
T
LJ ee
eQe
12
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
dX
dE
dX
dE
T
LJ eee
TLee
End of Lecture
Lecture 8.2
Thermoelectric phenomena
Analysis of thermo-electric circuits
The basic equations can be written as
eQeQQQQ FLFLJ
eeeQeQe FLFLJ
Substituting for affinities, the expressions derived earlier, we get
dX
dE
TL
dX
dT
T
LJ Qe
QQQ
12
dX
dE
TL
dX
dT
T
LJ ee
eQe
12
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
dX
dE
dX
dE
T
LJ eee
TLee
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
dX
dE
T
L
dX
dT
T
LeeeQ
20
ee
eQ
JLT
L
dXdT
dXdE
e
0
Seebeck effect
Analysis of thermo-electric circuits
or
ee
eQ
J LT
L
dT
dE
e
0
0eJdT
dESeebeck coeff.
2TLTL eeeQ
Using Onsager theorem2TLL eQQe
Analysis of thermo-electric circuits
Further from the basic eqs for Je & JQ, for Je = 0
we get
dX
dT
LT
L
T
L
dX
dT
T
LJ
ee
eQQeQQQ 2
dX
dT
TL
LLLL
ee
QeeQQQee
2
Analysis of thermo-electric circuits
For coupled systems, we define thermal conductivity as
0
eJ
Q
dXdT
Jk
This gives
2TL
LLLLk
ee
eQQeQQee
Analysis of thermo-electric circuits
Substituting values of coeff. Lee, LQe, LeQ
calculated above, we get
22 TTkTLQQ
TkT 22
Analysis of thermo-electric circuits
Using these expressions for various kinetic coeff in the basic eqs for fluxes we can write these as :
dX
dET
dX
dTTkJQ 2
dX
dE
dX
dTJe
Analysis of thermo-electric circuits
We can also rewrite these with fluxes expressed as fns of corresponding affinities alone :
eQ JTdX
TkJ
Qe JTkdX
dE
Tk
kJ
22
Using these eqs. we can analyze the effect of coupling on the primary flows
PETLIER EFFECT
Under Isothermal Conditions
a b
JQ, ab
JedX
dEJ e
Heat flux
ebQbeaQa JTJJTJ ;
PETLIER EFFECT
Heat interaction with surroundings
ebabQaQbaQ JTJJJ
eab JeffPeltier .
baab T
Peltier coeff.
Kelvin Relation
PETLIER REFRIGERATORFebCua ::
KVeu FC
07.13
KTAmpJ baQ 270~.20?
WAmpKK
V074.202707.13
Kv
PNTeBi
ba 423
:32
conductorsSemi
THOMSON EFFECT
Total energy flux thro′ conductor is
EJJJ eQE JQ, surr
Je
JQ
Je
JQdx
Using the basic eq. for coupled flows
EJJTx
TkJ eeE
eJETx
Tk
THOMSON EFFECT
The heat interaction with the surroundings due to gradient in JE
is
dxdx
JdJJJd EEEsurrQ xdxx
,
dxJETx
Tk
dx
de
THOMSON EFFECT
Since Je is constant thro′ the conductor
dx
dT
dx
dk
x
Tk
dx
Jd surrQ
2
2,
dx
dE
dx
dT
dx
dTJe
THOMSON EFFECT
Using the basic eq. for coupled flows, viz.
dx
dE
dx
dTJ e
above eq. becomes (for homogeneous material, ..; const
dx
dTconstk
2
, ee
surrQ J
dx
dJT
dx
dJ
Thomson heat Joulean heat
THOMSON EFFECT
dx
dJT e
reversible heating or cooling experienced due to current flowing thro′ a temp gradient
dx
dTJJ eTQ ,
dT
dT
Thomson coeff
Comparing we get
THOMSON EFFECT
We can also get a relationship between Peltier, Seebeck & Thomson coeff. by differentiating the exp. for ab derived earlier, viz.
Tbaab
dT
d
dT
dT
dT
d baba
ab
baba
End of Lecture
Analysis of thermo-electric circuits
Above equations can be written as
eQeQQQQ FLFLJ
eeeQeQe FLFLJ
Substituting for affinities, the expressions derived earlier, we get
dX
dE
TL
dX
dT
T
LJ Qe
QQQ
12
dX
dE
TL
dX
dT
T
LJ ee
eQe
12
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
dX
dE
dX
dE
T
LJ eee
TLee