Modern Engineering Thermodynamics

19.4 Thermoelectric Coupling 771
where R e = ρ e L/A is the electrical resistance, ρ e is the electrical resistivity, L is the length of the conductor, A is the
cross-sectional area of the conductor, and J E = I/A is the electron flux in the conductor. Now, let
ϕ 1 − ϕ 2
= − ϕ
2 − ϕ 1
= − dϕ
L
L dx = ρ eJ E
and let the electrical conductivity k e be defined as
k e = 1/ρ e
then we can rearrange Ohm’s law as
dϕ
J E = I/A = − k e
dx
For purely electrical effects (no thermal effects, i.e., no temperature differences), Eq. (19.13) reduces to
(19.22)
and Table 19.4 gives
Combining these equations produces
consequently, we find that
J E
T¼constant = L EE X E
J E
T¼constant = − L EE
T
X E = − 1
dϕ
T dx
dϕ
dx
dϕ
= −k e
dx
L EE = Tk e = T/ρ e (19.23)
The next easiest term to deal with is the coupling coefficient (L EQ = L QE ), because it is simply related to the
Seebeck coefficient. 9 The absolute Seebeck coefficient α is defined as
α = − dϕ
dT = − dϕ
I =0
dT
Equation (19.13) then gives, for zero current flow,
JE =0
L E = L EQ X Q + L EE X E = 0
and introducing the formula for X Q and X E from Table 19.4 provides
or
So that
L EQ − 1
dT
I + L
T 2 EE − 1
dϕ I
= 0
dx =0
T dx =0
dϕ/dx I
= dϕ
dTdx =0
dT = − L EQ
= −α (19.24)
I =0
TL EE
L EQ = L QE = αTL EE
and introducing L EE from Eq. (19.23) gives the coupling coefficient as
L EQ = L QE = αT 2 k e = αT 2 /ρ e (19.25)
Combining Eqs. (19.12) and (19.13) with X Q = dT/dx = 0 (i.e., isothermal conditions) gives
J QT
= L QE Q
= _ i
J E =constant
L EE T
(19.26)
9 The phenomenological coefficients have been developed in terms of the Seebeck coefficient, because it is the easiest thermoelectric
coefficient to measure.