Physics · Electricity and magnetism · Year 2

Thermoelectric phenomena

If we heat the junction of two different wires, why does a current appear?

The situation

In a teahouse yard, before baking flatbread, the tandoor baker holds his palm close to feel whether the tandoor has heated enough. An engineer uses a thermocouple for this: the ends of two wires of different metals are soldered together, the hot junction goes into the tandoor and the other into iced water — and a millivoltmeter shows a voltage of about ten millivolts. The reverse of the same phenomenon cools a car cool-box with an electric current, and in distant space probes it supplies hundreds of watts of electricity. Why does a temperature difference turn into an electric voltage, and how can a thermocouple be made into an accurate thermometer?

Topic

Thermoelectric phenomena. Measuring tandoor temperature with a thermocouple and a Peltier cooler

Key ideas

Seebeck effect
Peltier effect
Calibrating a thermocouple

Formulas

ℰ = \int_{T_{c}}^{T_{h}} S d T \approx S (T_{h} - T_{c})

Seebeck effect: the thermo-emf is proportional to the temperature difference of the junctions; for a type K thermocouple $S \approx 41$ µV/K.

Q_{P} = Π I, Π = S T

Peltier effect: at a junction, heat is carried along with the current; Kelvin's second relation.

q_{T} = - μ_{T} \vec{j} \cdot \nabla T, μ_{T} = T \frac{d S}{d T}

Thomson effect: in a uniform conductor with a temperature gradient, a current releases or absorbs heat.

J_{i} = \sum_{k} L_{i k} X_{k}, L_{i k} = L_{k i}, σ_{s} = \sum_{i} J_{i} X_{i} \geq 0

Onsager relations and entropy production: the fluxes depend linearly on the forces, and the cross coefficients are equal.

η = (1 - \frac{T_{c}}{T_{h}}) \frac{\sqrt{1 + Z T} - 1}{\sqrt{1 + Z T} + T_{c} / T_{h}}, Z T = \frac{S^{2} σ T}{κ}

Maximum efficiency of a thermoelectric generator; $Z T$ is the figure of merit of the material.

Theory

Work function and contact potential difference

To take an electron out of a metal into a vacuum, the work function $W$ (a few eV) is needed; it is the energy difference between the highest level occupied by electrons — the Fermi level — and the vacuum level. When two different metals are brought into contact, electrons pass from the metal with the smaller $W$ to the one with the larger, until the Fermi levels are equal. As a result a contact potential difference $U_{k} = (W_{2} - W_{1}) / e$ appears between the metals — from a few tenths of a volt to a few volts (diagram below). In a closed circuit at the same temperature the sum of the contact potentials is zero: there is no current, otherwise we would have a perpetual motion machine. The work function itself and thermionic emission are covered in detail in the “Electric current in vacuum and gases” experiment.

The Seebeck effect and the thermocouple

If one of the junctions is heated, the electrons at the hot end diffuse towards the cold end with greater energy, and in the two metals this process differs — a thermo-emf $ℰ \approx S Δ T$ appears in the circuit. For pairs of metals $S$ is tens of µV/K, so a 300 °C tandoor gives only 12 mV. Calibration (lab work): the cold junction is in iced water (0 °C), and the hot junction is in water being heated together with a reference thermometer; $ℰ$ is recorded at several temperatures, and the gradient of the $ℰ (t)$ graph is $S$ . Then an unknown temperature is found as $t = ℰ / S$ . The junction “catches” the temperature of the water within a few seconds — wait until the reading is steady. If the cold junction is not at 0 °C, its temperature must be added (“cold-junction compensation”). Model: NIST ITS-90 table functions for types K, J and T.

The Peltier effect and the refrigerator

When current passes through the junction of two different conductors, heat $Q_{P} = Π I$ is absorbed or released there — depending on the direction of the current. This is different from Joule heat: Joule heat $I^{2} R$ is always released and does not depend on the direction of the current. A semiconductor Peltier module consists of p-type and n-type Bi₂Te₃ pillars: charge carriers of both kinds (holes and electrons) carry heat from the cold plate to the hot plate. The cooling power is $Q_{c} = α T_{c} I - \frac{1}{2} I^{2} R - K Δ T$ : if the current is too large, Joule heating predominates, and the greatest $Q_{c}$ is at $I = α T_{c} / R$ . If a heat sink and fan do not carry heat away from the hot side, the module quickly overheats. In 1838 E. Lenz used this effect to freeze a drop of water.

The Thomson effect and Kelvin's relations

If there is a temperature gradient in a uniform conductor and current passes through it, then in addition to Joule heat, Thomson heat $q_{T} = - μ_{T} \vec{j} \cdot \nabla T$ is released or absorbed — its sign changes when the direction of the current changes. W. Thomson (Lord Kelvin) linked all three effects through thermodynamics: $Π = S T$ and $μ_{T} = T d S / d T$ . In “Onsager” mode compare the temperature distribution along the pillar at $+ I$ and $- I$ : Joule heating is symmetric, while the Thomson term shifts the curve to one side.

Semiconductors and the RTG

In semiconductors $S$ is hundreds of µV/K — ten times greater than in metals: there are few charge carriers and their number depends strongly on temperature, and in p-type and n-type material the sign of $S$ is opposite. That is why generators and coolers are made of semiconductors (Bi₂Te₃, PbTe, SiGe). The figure of merit is $Z T = S^{2} σ T / κ$ : the material must conduct electricity well and heat poorly. A radioisotope thermoelectric generator (RTG) converts the decay heat of plutonium-238 into electricity using SiGe thermocouples: among the distant planets, where sunlight is very weak, it is the only reliable source. The power falls with the half-life of 87.7 years; in practice noticeably faster — the temperature of the hot junction falls and the thermocouples age: on the Voyagers, from about 470 W at launch ≈ 220 W remains now (falling by about 4 W a year), whereas from decay alone ≈ 320 W would remain.

Onsager relations and entropy production

Near equilibrium the fluxes (electric current $J_{e}$ , heat flux $J_{q}$ ) depend linearly on the thermodynamic forces ( $X_{e} = E / T$ , $X_{q} = \nabla (1 / T)$ ): $J_{i} = \sum L_{i k} X_{k}$ . From the time-reversibility of the microscopic laws follow the Onsager reciprocal relations $L_{12} = L_{21}$ ; in thermoelectricity this is exactly $Π = S T$ . The rate of entropy production is $σ_{s} = \sum J_{i} X_{i} = κ {(\nabla T)}^{2} / T^{2} + j^{2} / (σ T) \geq 0$ . Prigogine's principle of minimum entropy production: if one force ( $Δ T$ ) is fixed and the other is free, the steady state (here an open circuit, $J_{e} = 0$ ) is the state with the smallest $σ_{s}$ — on the graph, the minimum of the parabola. Far from equilibrium the linear laws break down, and ordered dissipative structures can arise spontaneously in the system (for example Bénard cells in a heated layer of liquid) — this is the subject of non-linear non-equilibrium thermodynamics.

Did you know?

In 1821 Thomas Seebeck noticed that when one junction of a closed circuit made of two different metals was heated, a compass needle nearby was deflected. The reverse phenomenon was discovered by Jean Peltier in 1834.
Each of the Voyager probes launched in 1977 has three RTGs: at launch together they supplied about 470 W of electrical power. The half-life of plutonium-238 is 87.7 years.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Radiation from a dipole antenna — Grade 11
Oscillating circuit (LC) — Grade 11
Alternating current and RLC — Grade 11
Radio communication and radar — Grade 11
Charged-particle accelerator — Grade 11
Vector fields: div and curl — Year 3