Modern Engineering Thermodynamics

772 CHAPTER 19: Introduction to Coupled Phenomena
where _Q i is the isothermal heat transport rate “induced” by the presence of the electrical current I. From
Eqs. (19.24) and (19.26), we see that
α =
Q _ i
IT = S _ i
I
(19.27)
where _S i = _Q i /T is the isothermal entropy transport rate “induced” by the presence of the electrical current I.
This is the entropy transport rate due to the flow of electrons in the conductor. 10
Fourier’s law represents pure thermal effects, but it is somewhat more difficult to interpret since it corresponds
to a condition of zero electron flow (J E = 0) but not necessarily a zero voltage difference. From Eq. (19.13) with
J E = 0, we find that
then, Eq. (19.12) gives
X E = − L EQ
L EE
X Q
J JE Q =0 = L QQX Q + L QE − L
EQ
X Q = L 2
QQL EE − L QE
X Q
L EE
L EE
where we use Onsager’s reciprocity relationship, L EQ = L QE . Now, by definition,
and Table 19.4 gives
so
consequently,
J Q
JE
=0 =
X Q = 1 T 2
_ Q F
A
= _q F
dT
dx
J JE Q =0 = _q F = −k dT
t = − L
2
QQL EE − L
QE dT
dx
L EE T 2 dx
k t = L QQL EE − L 2 QE
L EE T 2 (19.28)
Therefore, the thermal conductivity of a substance is not a simple quantity. It is composed of all three phenomenological
coefficients combined with the inverse of the absolute temperature squared. Substituting the formulae
for L EE and L QE from Eqs. (19.23) and (19.25) into Eq. (19.28) and solving for the remaining coefficient,
L QQ , gives
L QQ = T 2 ðk t + α 2 Tk e Þ (19.29)
Substituting Eqs. (19.23), (19.25), and (19.29) along with the formulae for X Q and X E from Table 19.4 into
Eqs. (19.12) and (19.13) produces the linear phenomenological equations (in terms of the pure conductor
Seebeck coefficient) as
J Q = −ðk t + α 2 Tk e Þ dT
dϕ
− αTk e
(19.30)
dx dx
and
dϕ
dT
J E = −k e − αk e
dx dx
Also, substitution into the thermoelectric total EPRD formula of Eq. (19.14) gives
σ thermoelectric = 1
T ðk 2 t + α 2 Tk e Þ dT dϕ
+ αTk dT
e + 1
dx dx dx T k dϕ
dT
e + αk e
dx dx
dϕ
dx
(19.31)
> 0 (19.32)
10 Curiously, though work mode transport of energy cannot produce a direct entropy transport, it can induce a secondary entropy
transport.