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target="_blank" rel="nofollow" href="#ulink_ce9bf853-63cf-5d4e-aa70-42c752e7a288">(1.46)

StartFraction partial-differential upper J Subscript upper Q Baseline Over partial-differential x EndFraction equals minus StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis kappa StartFraction partial-differential upper T Over partial-differential x EndFraction right-parenthesis plus upper T StartFraction d upper S Over d upper T EndFraction upper J StartFraction partial-differential upper S Over partial-differential x EndFraction minus StartFraction upper J squared Over sigma EndFraction plus upper J upper E plus upper S upper T StartFraction partial-differential upper J Over partial-differential x EndFraction period

Heat flux can change the temperature of a material. Consider in the volume of Adx with A as the cross‐section area,

(1.47) d upper C Subscript normal p Baseline StartFraction d upper T Over d t EndFraction equals minus StartFraction partial-differential upper J Subscript upper Q Baseline Over partial-differential x EndFraction comma

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,

(1.48) d upper C Subscript normal p Baseline StartFraction d upper T Over d t EndFraction equals minus StartFraction partial-differential upper J Subscript upper Q Baseline Over partial-differential x EndFraction plus upper J upper E period

By replacing the first term in the right side of Eq. (1.48) with Eq. (1.46), the following equation is obtained,

(1.49)

Assume that the thermal conductivity is independent of position at steady state, (1.49) becomes

(1.50) kappa StartFraction partial-differential squared upper T Over partial-differential x squared EndFraction minus upper T StartFraction d upper S Over d upper T EndFraction upper J StartFraction partial-differential upper S Over partial-differential x EndFraction plus StartFraction upper J squared Over sigma EndFraction equals 0 comma

as StartFraction d upper T Over d t EndFraction equals 0 and StartFraction partial-differential upper J Over partial-differential x EndFraction equals 0 at steady state. The Eq. (1.50) is used for the evaluation of thermoelectric efficiency.

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) upper P equals upper I upper Delta upper V Subscript e x Baseline comma

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) upper V Subscript upper O upper C Baseline equals 2 left-parenthesis upper S right-parenthesis upper Delta upper T comma

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,

Schematic illustration of (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.

Figure 1.22 (a) A thermoelectric generator with n‐ and p‐type legs. Heat transfer (Q_in) into the thermoelectric legs from the hot side and heat transfer (Q_out) out from the cold side. (b) Temperature and voltage profiles at open‐circuit condition.

(1.53) 2 StartAbsoluteValue upper S EndAbsoluteValue upper Delta upper T equals upper I upper R Subscript e x Baseline plus upper I upper R Subscript i n

The current through the circuit can be obtained as

(1.54) upper I equals StartFraction 2 StartAbsoluteValue upper S EndAbsoluteValue upper Delta upper T Over upper R Subscript i n Baseline plus upper R Subscript e x Baseline EndFraction period

Thus, the equation for the power is given by

(1.55) upper P equals upper I upper Delta upper V Subscript e x Baseline equals StartFraction left-parenthesis 2 StartAbsoluteValue upper S EndAbsoluteValue upper Delta upper T right-parenthesis squared Over left-parenthesis upper R Subscript i n Baseline plus upper R Subscript e x Baseline right-parenthesis squared EndFraction upper R Subscript e x Baseline period

By introducing a parameter Скачать книгу