J upper E minus StartFraction upper J squared Over sigma EndFraction comma"/>
where
The last term
These analyses indicate that the energy flux variation in a material can include the variations of heat conduction, the Peltier effect, the Thomson effect, the Joule heating, the electrical work, and the distribution of charge carriers,
Heat flux can change the temperature of a material. Consider in the volume of Adx with A as the cross‐section area,
(1.47)
where d is the mass density and C p is the specific heat.
When a material is initially at thermal equilibrium with homogeneous temperature, its temperature can be varied only when an electrical work or heat transfer is applied to it. If an external electric field is applied to a system, the electrical work will affect the temperature of the system,
By replacing the first term in the right side of Eq. (1.48) with Eq. (1.46), the following equation is obtained,
Assume that the thermal conductivity is independent of position at steady state, (1.49) becomes
as
The input heat flux or the temperature difference between the hot and cold sides is usually constant when a thermoelectric system is evaluated. Consider that the material properties are independent of the temperature and they are identical for the p‐type and n‐type legs, the power delivered to the external load is given by
(1.51)
where I is the current through the external load and ΔV ex = IR ex is the voltage drop on the external load.
The values of I, ΔV ex, and P depend on the resistance of the external load. The resistance of the external load is usually varied from the open‐circuit voltage condition (R ex → ∞) to the short‐circuit voltage (R ex → 0) to find the optimal power on the external load.
As shown in Figure 1.22, for a system with a p‐type leg and an n‐type leg, the open‐circuit voltage is
(1.52)
when the p‐ and n‐type thermoelectric materials have the same absolute Seebeck coefficient value. When the system is connected to an external load and form a close circuit, the open‐circuit voltage dissipated on the external load and the internal resistance (R in) that includes the resistances of materials, contacts, and wires,
Figure 1.22 (a) A thermoelectric generator with n‐ and p‐type legs. Heat transfer (Qin) into the thermoelectric legs from the hot side and heat transfer (Qout) out from the cold side. (b) Temperature and voltage profiles at open‐circuit condition.
