Modern Engineering Thermodynamics

238 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
10 3
10 2
10 1
Silicon
10 0 0 100 200 300 400 500 600 700 800 900 1000
Resistivity (Ω.m)
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
10 −9
10 −10
Germanium
Tungsten
Temperature (K )
Copper
FIGURE 7.22
The variation of electrical resistivity with temperature. (Source: Reprinted by permission of the author from Lenert, L. H., 1968.
Semiconductor Physics, Devices, and Circuits. Charles E. Merrill Publishing, Columbus, OH.)
Ohm’s law is a simple mathematical model used for the resistive dissipation of electrical work mode energy. In
this model, the entropy production rate per unit volume due to electrical resistance is
ðσ W Þ elect = Je 2 ρ e/T
where J e = I/A is the electrical current per unit area (i.e., the electrical current flux), ρ e = R e A/L is the electrical
resistivity, R e is the total electrical resistance of the conductor, L and A are the length and cross-sectional area of
the conductor, and T is the local internal absolute temperature of the conductor. Then, Eq. (7.70) gives
Z
_S p W =
elect V
The electrical current flux J e is often expressed by the one-dimensional Ohm’s law as
dϕ
J e = −k e
dx
Je 2ρ e
dV (7.73)
T
where k e = 1/ρ e is the electrical conductivity and ϕ is the electrical potential (i.e., voltage). The relation between
ρ e and T is shown in Figure 7.22 for various materials.
For the special case of an isothermal system with uniform properties and a constant current density, Eq. (7.73) reduces
to this special case:
Entropy production rate due to electrical resistance:
_S P
W
elect
ðspecialÞ
= J2 e ρ eV
T
= ðI/AÞ2 × ðR e A/LÞ × LA
= I2 R e
T
T
(7.74)
EXAMPLE 7.12
A new high-temperature silicon computer chip 1.00 × 10 −3 m by 5.00 × 10 −3 m by 10.0 × 10 −3 m long with uniform properties
operates isothermally at 600. K and draws a constant electrical current of 0.10 A. Determine the entropy production
rate of the chip.
Solution
First, draw a sketch of the system (Figure 7.23).