Lecture_8 - OoCities

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27 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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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 properties

Specific
property

Extensive
property

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

Entropy
gen rate

affinities

fluxes

NON
-
EQLBM
THERMODYNAMICS

Purely “resistive” systems

Flux

is dependent only on

affinity

at any
instant

at that
instant

System has no “memory”
-

NON
-
EQLBM
THERMODYNAMICS

Coupled Phenomenon

Since J
k

is 0 when affinities are zero,

NON
-
EQLBM
THERMODYNAMICS

where

kinetic Coeff

Postulate III

Relationship

between

affinity

&

flux

from

‘other’

sciences

NON
-
EQLBM
THERMODYNAMICS

Heat Flux :

Momentum :

Mass :

Electricity :

NON
-
EQLBM
THERMODYNAMICS

Postulate IV


Onsager theorem
{in the absence of
magnetic fields}




NON
-
EQLBM
THERMODYNAMICS

Entropy

production

in

systems

involving

heat

Flow

T
1

T
2

x

dx

A

NON
-
EQLBM
THERMODYNAMICS

Entropy gen. per unit volume

NON
-
EQLBM
THERMODYNAMICS

NON
-
EQLBM
THERMODYNAMICS

Entropy

generation

due

to

current

flow

:

I

dx

Heat transfer in
element length

NON
-
EQLBM
THERMODYNAMICS

Resulting

entropy

production

per

unit

volume


NON
-
EQLBM
THERMODYNAMICS

Total

entropy

prod

/

unit

vol
.

with

both

electric

&

thermal

gradients

affinity

affinity

NON
-
EQLBM
THERMODYNAMICS

Analysis of thermo
-
electric
circuits

Addl. Assumption : Thermo electric
phenomena can be taken as
LINEAR

RESISTIVE SYSTEMS

{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

Substituting

for

affinities,

the

expressions

derived

earlier,

we

get

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

End of Lecture

Lecture 8.2

Thermo
electric
phenomena

Analysis of thermo
-
electric
circuits

The basic equations can be written as

Substituting

for

affinities,

the

expressions

derived

earlier,

we

get

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

Analysis of thermo
-
electric
circuits

Consider

the

situation,

under

coupled

flow

conditions,

when

there

is

no

current

in

the

material,

i
.
e
.

J
e
=
0
.

Using

the

above

expression

for

J
e

we

get

Seebeck
effect

Analysis of thermo
-
electric
circuits

or

Seebeck coeff.

Using Onsager theorem

Analysis of thermo
-
electric
circuits

Further from the basic eqs for J
e
&
J
Q
, for J
e

= 0

we get

Analysis of thermo
-
electric
circuits

For

coupled

systems,

we

define

thermal

conductivity

as

This

gives

Analysis of thermo
-
electric
circuits

Substituting values of coeff. L
ee
, L
Qe
,
L
eQ

calculated above, we get

Analysis of thermo
-
electric
circuits

Using

these

expressions

for

various

kinetic

coeff

in

the

basic

eqs

for

fluxes

we

can

write

these

as

:

Analysis of thermo
-
electric
circuits

We

can

also

rewrite

these

with

fluxes

expressed

as

fns

of

corresponding

affinities

alone

:

Using

these

eqs
.

we

can

analyze

the

effect

of

coupling

on

the

primary

flows

PETLIER EFFECT

Under Isothermal Conditions

a

b

J
Q
,
ab

J
e

Heat flux

PETLIER EFFECT

Heat interaction with surroundings

Peltier coeff.

Kelvin Relation

PETLIER REFRIGERATOR

THOMSON EFFECT

Total energy flux thro
′ conductor is

J
Q
,
surr

J
e

J
Q

J
e

J
Q

dx

Using the basic
eq. for coupled
flows

THOMSON EFFECT

The

heat

interaction

with

the

surroundings

due

to

gradient

in

J
E

is

THOMSON EFFECT

Since

J
e

is

constant

thro


the

conductor

THOMSON EFFECT

Using

the

basic

eq
.

for

coupled

flows,

viz
.

above

eq
.

becomes



(for

homogeneous

material,


Thomson

heat

Joulean

heat

THOMSON EFFECT

reversible

heating

or

cooling

experienced

due

to

current

flowing

thro


a

temp

gradient

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
.

End of Lecture

Analysis of thermo
-
electric
circuits



Above equations can be written as

Substituting

for

affinities,

the

expressions

derived

earlier,

we

get

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