Page 422 - Mechanical Engineers' Handbook (Volume 4)

3 Thermal Control Techniques 411

2
1
q ⁄2IR (111b)
j
and minus the heat regained at the cold junction (known as the Fourier heat or Fourier effect)
due to the temperature difference
T T T c
h
q K
T K(T T ) (111c)
F
h
c
Thus, the net heat absorbed at the cold junction is
2
1
q IT ⁄2IR K
T (112)
c
where the total resistance of the couple is the series resistance of material A and material B
having areas A and A , respectively (both have length L),
B
A
R A

B
A A L (113)
A B
and where the overall conductance K is the parallel conductance of the elements A and B:
1
K (kA kA ) (114)
A
A
B
B
L
To power the device, a voltage equal to the sum of the Seebeck voltages at the hot and
cold junctions plus the voltage necessary to overcome the resistance drop must be provided:
V T T RI
T RI
c
h
and the power is
P VI (
T RI)I (115)
The coefficient of performance (COP) is the ratio of the net cooling effect to the power
provided:
q TI ⁄2IR K
T
2
1
COP c (116)
2
P
TI IR
Optimizations
The maximum possible temperature differential
T T T will occur when there is no
c
h
net heat absorbed at the cold junction:
1

T ⁄2zT 2 c (117)
m
where z is the figure of merit of the material
2
z (118)
KR
The current that yields the maximum amount of heat absorbed at the cold junction can
be shown to be 42
T
I I c (119)
m
R
and the coefficient of performance in this case will be