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rel="nofollow" href="#ulink_5bf77ad8-8dfa-5e51-a89b-29cf63156229">Figure 1.20 Steady temperature and voltage profiles of a thermoelectric leg under (a) open‐circuit and (b) short‐circuit conditions.

It should be noted that ΔT is lowered at the short circuit condition in comparison with the open‐circuit condition because of the Peltier effect. According to the Peltier effect, the current flow cools down the hot side and heats up the cold side. This will lower the J _t.

In the normal case, there are both drift current density and thermal diffusion current density. In terms of the conservation law, the total amount of charge carriers in a material must be conserved. Thus, the variation of the charge carrier density (n) over time is related to the gradient of the current density,

(1.40) q StartFraction d n Over d t EndFraction equals minus StartFraction partial-differential upper J Over partial-differential x EndFraction period

By replacing J with the Eq. (1.38), it is given,

(1.41) q StartFraction d n Over d t EndFraction equals minus sigma StartFraction partial-differential squared upper V Over partial-differential x squared EndFraction minus sigma upper S StartFraction partial-differential squared upper T Over partial-differential x squared EndFraction period

This indicates that the variation of charge carrier density over time can be achieved by varying the voltage or the temperature gradient over position.

The energy flux in a thermoelectric material includes the heat transport and thermoelectric conversion. J _Q is used for the heat flux that is the amount of energy crossing a cross‐section area per unit time,

(1.42) upper J Subscript normal upper Q Baseline equals minus kappa StartFraction partial-differential upper T Over partial-differential x EndFraction plus upper S upper T upper J period

The first term in the right side of Eq. (1.42) is the heat conduction from the hot to cold part within a material. The second term is the heat related to the thermoelectric effects. Derivation of the Eq. (1.42) with position leads to

(1.43)

It is important to understand the terms in the right side of the Eq. (1.43). The second term where S varies with position is related to the Peltier and Thomson effects. As shown in Figure 1.21a, the system has a uniform temperature, and there is current flow through the system. This can be the case, right after applying a current to a system which was initially in equilibrium. Because the Seebeck coefficient (S _M) of the metal electrode is different from that (S) of the thermoelectric material, the transport of the current from the left metal electrode through the contact into the thermoelectric material leads to the energy absorption (cooling) by (S _M‐S)TJ due to the Peltier effect. On the other hand, the same amount of energy is released at the right contact, when the current flows from the thermoelectric material into the right metal electrode. The energy release is also owing to the Peltier effect.

Figure 1.21b shows the case when the temperature is not uniform. In addition to the Peltier effect at the two contacts, STJ is not uniform inside the thermoelectric material because S varies with temperature. The Seebeck coefficient (S _a) at the position a is different from that (S _b) at the position b. As a result, heat transfer takes place between the two positions because of the Thomson effect. The heat transfer (Q _T) due to the Thomson effect is given by

(1.44) upper Q Subscript normal upper T Baseline equals minus upper T StartFraction d upper S Over d upper T EndFraction upper J StartFraction partial-differential upper T Over partial-differential x EndFraction period

Schematic illustration of scheme of the different heat fluxes of an n-type thermoelectric material connected to metallic contacts under the flow of an electrical current density J.

Figure 1.21 Scheme of the different heat fluxes of an n‐type thermoelectric material connected to metallic contacts under the flow of an electrical current density J. (a) Only Peltier effect takes place at the junctions and there is no temperature difference. (b) Under a temperature gradient, S varies inside the thermoelectric material, which produces heat generation in the bulk (Thomson effect), the thermoelectric effects at the positions a and b are indicated. Peltier effect also takes place at the junctions.

The third term SJ ∂T/∂x in the right side of Eq. (1.43) can be converted to the potential gradient in terms of the Eq. (1.38),

(1.45) upper S upper J StartFraction partial-differential upper T Over partial-differential x EndFraction equals minus upper J StartFraction partial-differential upper V Over partial-differential x EndFraction minus StartFraction upper J squared Over sigma EndFraction equals upper J upper E minus StartFraction upper J squared Over sigma EndFraction comma

where upper E equals minus StartFraction partial-differential upper V Over partial-differential x EndFraction is the electric field. The first term in the right side of Eq. (1.45) is the electrical work, and the second term is the Joule heat generated in the material by the electrical current.

The last term upper S upper T StartFraction partial-differential upper J Over partial-differential x EndFraction at the right side of Eq. (1.43) is the energy variation due to the variation of the charge carrier concentration. At steady‐state condition the current becomes uniform, and this term is thus zero.

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,

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