Page 483 - Advanced thermodynamics for engineers

20.6 THERMOELECTRICITY – THE APPLICATION OF IRREVERSIBLE 475

20.6.3 THE COUPLED EQUATIONS OF THERMOELECTRICITY
The Onsager relations as given by Eqn (20.8) may be applied.
i.e.

J 1 ¼ L 11 X 1 þ L 12 X 2
J 2 ¼ L 21 X 1 þ L 22 X 2
These become, in this case
L 11 dT dε
for heat flow J Q ¼ L 12 (20.29)
T d‘ d‘
L 21 dT dε
for electrical flow J I ¼ L 22 (20.30)
T d‘ d‘
J Q
From Eqn (20.18), the entropy flux J S is given by J S ¼ , and hence Eqn (20.29) may be written
T
L 11 dT L 12 dε
J S ¼ 2 (20.31)
T d‘ T d‘
It has been assumed that both the electrical and heat flow phenomena may be represented by
empirical laws of the form J ¼ LX.
dε
At constant temperature, Ohm’s law states I ¼ lA , giving
d‘
I dε
J I ¼ ¼ l (20.32)
A d‘
where l is the electrical conductivity of the wire at constant temperature.
If dT is set to zero in Eqn (20.30), i.e. the electrical current is flowing in the absence of a
temperature gradient, then
dε dε
l ¼ L 22 ; which gives L 22 ¼ l (20.33)
d‘ d‘
If the entropy flux in the wire is divided by the electrical current flowing at constant temperature
and this ratio is called the entropy of transport S*, then

J S L 12 dε dε L 12

S ¼ ¼ l ¼ (20.34)
J I T d‘ d‘ TL 22
T
The entropy of transport is basically the rate at which entropy is generated per unit energy flux in
an uncoupled process. It is a useful method for defining the cross-coupling terms in the coupled
equations.
Thus

L 12 ¼ TL 22 S ¼ lTS (20.35)
By Onsager’s reciprocal relation

L 21 ¼ L 12 ¼ lTS (20.36)