F

Figure 5.1 A bar with a temperature gradient.

Figure 5.2 A bar heated by a current.

Heat conducting electrically insulating sheets

Figure 5.3 A simple thermocouple (left) and a test setup (right).

Heat conducting electrically insulating sheets

Figure 5.3 A simple thermocouple (left) and a test setup (right).

Here we used the directions of heat power flow indicated in the inset that accompanies Equations 5.3 and 5.4. However, in a more complicated structure, the heat power flow can surprisingly depart from expectation.

Consider a thermocouple consisting of two dissimilar materials (conductors or semiconductors) joined together at one end. The materials (arms A and B in Figure 5.3) may touch one another directly or may be joined by a metallic strip as indicated. As long as this metallic strip is at uniform temperature, it has no influence on the performance of the thermocouple (provided the strip has negligible electric resistance and essentially infinite heat conductivity). The free ends of arms A and B are connected to a current source.

Again, if the connecting wires are at uniform temperature, they exert no influence. Two blocks maintained at uniform temperature are thermally connected, respectively, to the junction and to the free ends. These blocks are electrically insulated from the thermocouple.

The block in contact with the junction is the heat source and is at the temperature TH. The other block is the heat sink and is at TC. The rate of heat flow, PH, from the source to the sink is measured as explained in the box at the end of this subsection.

Assume the thermocouple is carefully insulated so that it can only exchange heat with the source and with the sink. If we measure PH as a function of TH — TC with no current through the thermocouple, we find that PH is proportional to the temperature difference (as in Equation 5.1):

If we force a current, I, through a thermocouple with an internal resistance, R, we expect, as explained, PH to be given by (cf. Equation 5.3)

or, if the resistance of the thermocouple is 2.6 x 10 4 ohms,

where we used TH = 1500 K and TC = 1000 K, as an example.

The expected plot of PH versus I appears as a dotted line in Figure 5.4. It turns out that a change in PH, as I varies, is in fact observed, but it is not independent of the sign of I. The empirically determined relationship between PH and I is plotted, for a particular thermocouple, as a solid line. A seconds-order regression fits the data well:

In equation 5.9, we recognize the heat conduction term because it is independent of I. We also recognize the Joule heating term, 1.3 x 10-4 I2. In addition to these two terms, there is one linear in I. This means that if the current is in one direction, heat is transported from the source to the sink and, if inverted, so is the heat transport. Evidently, heat energy is carried by the electric current. This reversible transport is called the Peltier effect (Jean Charles Athanase Peltier, 1785-1845).

From empirical evidence, the Peltier heat transported is proportional to the current. We can therefore write

where n is the Peltier coefficient.

If we connect an infinite impedance voltmeter to the thermocouple instead of a current generator, we observe a voltage, V, that is dependent on the temperature difference, AT = TH -TC. The dependence is nonlinear as illustrated in Figure 5.5, which shows the relationship of the open-circuit

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