Modern Engineering Thermodynamics

19.4 Thermoelectric Coupling 767
Equations (19.9) are called the linear phenomenological equations, because they resulted from a linearization of the
general coupling postulate relationship (Eq. (19.4)) via a Taylor’s series expansion around the equilibrium state
X 1 = X 2 = 0. These equations therefore are valid for systems that are only slightly nonequilibrium. Consequently,
the linear phenomenological equations have a limited range of usage, but they are sufficiently accurate to
explain many of the known coupled effects shown in Table 19.1.
In 1931, Lars Onsager (1903–1976) proved that the coupling coefficients, L ij , formed a symmetrical matrix and
established what is now called the reciprocity relationship for these coefficients as
L ij = L ji (19.10)
The total EPRD for a system near equilibrium is found by combining Eqs. (19.3), (19.9), and (19.10) to yield
where L ij = L ji .
σ total = ∑ m
i=1
J i X i = ∑ m
i=1
∑ m
L ij X j X i > 0 (19.11)
j=1
19.4 THERMOELECTRIC COUPLING
Consider the simultaneous near equilibrium flow of thermal and electrical energy in a system. Then, Eq. (19.3)
for the total EPRD is
σ total = J Q X Q + J E X E
where the fluxes J Q , J E and the forces X Q , X E are defined in Table 19.4. Also, Eq. (19.9) gives
and
Equation (19.11) then becomes
J Q = L QQ X Q + L QE X E (19.12)
J E = L EQ X Q + L EE X E (19.13)
σ total = L QQ X Q + L QE X Q X E + L EQ X Q X E + L EE X E
and combining this with Onsager’s reciprocity relationship, L EQ = L QE , gives
σ total = L QQ X Q + 2L QE X E X Q + L EE X E > 0 (19.14)
Equation (19.12) explains how a heat flow J Q can occur even without the presence of a temperature difference
(X Q = 0), if the coupling coefficient L QE is nonzero and a voltage source X E exists within the system. Similarly,
Eq. (19.13) illustrates how a flow of electrical energy J E can exist with no apparent voltage source (X E = 0),
when heat transport occurs J Q with a nonzero coupling coefficient L EQ . Considerable thermoelectric technology
has developed around this simple coupled effect.
Thermoelectric heaters, coolers, and temperature measurement instruments (thermocouples) are common
devices today. To fully understand and utilize thermoelectric coupling, we must evaluate the primary (L QQ , L EE )
and secondary or coupling (L EQ = L QE ) coefficients for any given system. Historically, many of the thermoelectric
coupling effects were discovered empirically by individuals long before an accurate thermodynamic understanding
of this phenomenon was known. These effects were originally thought to be independent phenomena and
had no adequate explanation. The important empirical thermoelectric discoveries occurred as follows.
19.4.1 The Seebeck Effect 2
Heating one junction of a bimetallic (or semiconductor) closed circuit while simultaneously cooling the other
junction produces a flow of current in the circuit without an apparent voltage source being present. If two such
materials are joined at a single junction to produce an open circuit, then heating or cooling that junction produces
an electrical potential (voltage) difference across the circuit that is directly proportional to the temperature
of the junction (see Figure 19.1).
2 Discovered in 1821 by the German physicist Thomas Johann Seebeck (1770–1831). It occurs because the free electron densities of
certain materials (metals and semiconductors) differ from one another at a given temperature. When two such materials are joined,
their junction appears as a voltage source due to electrons diffusing down the electron concentration gradient at the junction.