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Modern Engineering Thermodynamics

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19.4 Thermoelectric Coupling 773<br />

The Peltier and Kelvin coefficients are not independent from the Seebeck coefficient, so they need not be<br />

introduced into the phenomenological coefficients. For example, the Peltier heat _Q P is the heat transfer rate that<br />

occurs when there is no temperature difference, dT/dx = 0 = X Q . Therefore, from Eq. (19.12),<br />

or<br />

Also, from Eq. (19.13),<br />

J Q<br />

<br />

XQ<br />

=0 = _ Q P<br />

A = _q P = L QEX E<br />

<br />

_Q P = AðαT 2 k e Þ − 1 T<br />

dϕ<br />

dx<br />

<br />

<br />

dϕ<br />

= −αATk e<br />

dx<br />

or<br />

J E<br />

<br />

XQ<br />

=0 = I/A = L EEX E<br />

<br />

I = ATk e − 1 T<br />

dϕ<br />

dx<br />

Then, the Peltier coefficient π given in Eq. (19.16) becomes<br />

π =<br />

_ Q P<br />

I<br />

<br />

<br />

dϕ<br />

= −Ak e<br />

dx<br />

= αT (19.33)<br />

Thus, the Seebeck and Peltier coefficients for a pure conductor are directly related by Eq. (19.33).<br />

EXAMPLE 19.1<br />

A 0.0100 m diameter copper bus bar is maintained at a constant temperature of 20.0°C and is subjected to a voltage gradient<br />

of 1.00 V/m. Determine the Peltier heat flow, _Q P . The Seebeck coefficient for the copper is α cu = 3.50 × 10 −6 V/K, and its<br />

resistivity is ρ e = 5.00 × 10 −9 ohm meters.<br />

Solution<br />

The Peltier heat flow is given by Eq. (19.33) as<br />

_Q P = πI = ðαTÞI<br />

where<br />

I = AJ E = ρ A <br />

− dϕ <br />

e dx<br />

Here, −dϕ/dx = voltage gradient = 1.00 V/m, so<br />

Then<br />

I = ðπ/4Þð0:0100 mÞ2 ð1:00 V/mÞ<br />

5:00 × 10 −9 Ω gm<br />

= 1:57 × 10 4 V/Ω = 1:57 × 10 4 A<br />

_Q P = ð3:50 × 10 −6 V/KÞð20, 0 + 273:16 KÞð1:57 × 10 4 AÞ = 16:2V.A ¼ 16:2W<br />

EXAMPLE 19.2<br />

The open circuit voltage of an iron-copper thermocouple is approximately given by<br />

ϕ fe-cu = ð−13:4 T + 0:014 T 2 + 0:00013 T 3 Þ × 10 −6 V<br />

where T is in °C, not K. At 100.°C, determine<br />

a. The relative Seebeck coefficient α fe-cu .<br />

b. The relative Peltier coefficient π fe-cu .<br />

(Continued )

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