Modern Engineering Thermodynamics

770 CHAPTER 19: Introduction to Coupled Phenomena
This effect gave rise to Fourier’s law of heat conduction,
_Q F = −k t A dT
dx
or
_q F =
_ Q F
A
= −k t
dT
dx
(19.18)
where _Q F is the Fourier heat transfer rate, _q F is the Fourier heat flux, and k t is the Fourier coefficient (known
today as the thermal conductivity).
19.4.5 The Joule Effect 7
An electrical current passing through a homogeneous isothermal conductor produces an internal heating of the
conductor which is independent of the direction of current flow (see Figure 19.6).
This effect gave rise to the Joule heating formula
or
_Q J = k J I 2 R e (19.19)
_q J = k J I 2 R e /A
where _Q J is the Joule heating rate, _q F is the Joule heat flux, R e is the electrical resistivity of the conductor, and k J
is the Joule coefficient,
k J = 778:17 ft.lbf/Btu = 0:293 W.h/Btu = 1 WLs/joule
Today k J is recognized as being merely a units conversion factor and it is not normally written in the equation.
Therefore, from now on we write Eq. (19.19) simply as
_Q J = I 2 R e (19.20)
T
19.4.6 The Ohm Effect 8
When an electrical current is passed through a homogeneous isothermal conductor, a voltage drop occurs in the
direction of current flow (see Figure 19.7).
Current
Internal conductor
heating causes a heat
loss for an isothermal
conductor
FIGURE 19.6
A schematic of the Joule effect.
Current I
φ 1 φ 2 < φ 1
T = constant
FIGURE 19.7
A schematic of the Ohm effect.
T
This effect gave rise to Ohm’s law,
ϕ 1 − ϕ 2 = R e I (19.21)
where R e is Ohm’s coefficient, today called the electrical resistance of the
conductor.
All six of these effects occur simultaneously when thermal and electrical
energy simultaneously flow through a system. It should be apparent by now
that it is easier to comprehend thermoelectric effects through the coupling
Eqs. (19.12) and (19.13) than to master the six interrelated effects just discussed.
However, these six effects are now an integral part of our engineering
jargon and technical literature, and they cannot be dispensed with so easily.
Therefore, we develop formulae for the thermoelectric primary and secondary
coefficients in terms of the coefficients of the six empirically discovered thermoelectric
effects discussed. This provides continuity between the old and
the new interpretations of thermoelectricity.
Since the primary electrical coefficient L EE is the easiest to deal with, we begin
with it. Primary coefficients result from noncoupled phenomena, and pure
electrical effects are described by Ohm’s law.Ohm’s lawcanbecastintoa
number of forms; for example,
ϕ 1 − ϕ 2 = R e I = Iðρ e L/AÞ = ρ e ðI/AÞL = ρ e J E L
7 Discovered in 1841 by the English physicist James Prescott Joule (1818–1889).
8 Discovered in 1827 by the German physicist Georg Simon Ohm (1787–1854).