Tikalon Blog by Dev Gualtieri

Thermoelectrics

October 26, 2020

As the First law of thermodynamics informs us, a heat engine will only provide work through a difference in temperature between two thermal reservoirs. A heat engine is not the only way to extract work from a temperature differential. Work can be extracted from a temperature differential in the form of electrical power by using the thermoelectric effect; specifically, the Seebeck effect.

The simplest thermoelectric device based on the Seebeck effect is the thermocouple, which is just a junction between two dissimilar metals. Some metal pairs offer a greater voltage change per degree than others. A popular thermocouple is the Type K, which is a junction between chromel, an alloy that's 90% nickel and 10% chromium and alumel, an alloy of 95% nickel, 2% aluminum, 2% manganese, and 1% silicon. This thermocouple has a thermopower (sensitivity) of about 41 μV/°C.

A thermopower of 41 μV/°C, combined with electronic amplification, makes thermocouples useful for temperature measurement. Additionally, electrical power can be extracted from them with a little effort. This would be done by stringing many of these in series, so their voltages add. There's one caveat, however. Each heated thermocouple junction must be matched by a cold junction, as required by the first law of thermodynamics (see figure).

A series connection of many thermocouple junctions intended as a voltage source. Useful voltages will be generated only with a large temperature difference between the hot and cold junctions and many thermocouples are used. Since the power output of such a device depends on its internal resistance as well as the voltage, there are practical problems, as explained in the text. (Created using Inkscape.)

One of my sixth grade classmates made an electrical generator this way by putting the cold junctions in an ice bath, and heating the hot junctions with a propane torch. The intended use of such a device was as a solar energy system in which the hot junctions would be at the focus of a parabolic mirror. A hundred junctions with a several hundred degree temperature difference will generate enough voltage to barely light an incandescent flashlight bulb.

The problem with this thermoelectric approach is that metals that are good electrical conductors are also good thermal conductors, as explained by the Wiedemann–Franz law. That's a consequence of electrons' being responsible for both conductivies. Fortunately, specially crafted semiconductors can be used to make somewhat useful thermoelectric generators. Junctions of n- and p-doped bismuth telluride (Bi₂Te₃) are typically used in thermoelectric cells, as shown in the figure.

A thermoelectric cell of n- and p-doped semiconductor.

A series connection of many such cells offer a means to extract useful electrical power from sources of waste heat, such as automobile exhaust gas.

(Click for larger image.)

Bismuth telluride thermoelectric generators are relatively inefficient, so there are efforts to find better materials. Obviously, good thermoelectric performance, as identified by a high Seebeck coefficient (S, volts/kelvin), and low thermal conductivity (κ,W/(m-K)) are both beneficial. These are combined into a thermoelectric figure of merit, commonly denoted as zT, as follows:

In this equation, σ is the electrical conductivity. Somewhat confusingly, zT, is used as the figure of merit, with temperature multiplied to get a dimensionless number. That's why we don't just cancel the temperature term from both sides of the equation. The figure of merit changes slightly with temperature, and the highest figures of merit have been just slightly above 1.0 with no great advances over the years. However, a figure of merit of 1.0 is far greater than my calculated value for a type-K thermocouple, 0.05, obtained by using averaged values of the properties of chromel and alumel.

An advance in thermoelectric figure of merit has been achieved by scientists and engineers from the Huazhong University of Science and Technology (Wuhan, China), Wuhan University (Wuhan, China), and the Chinese Academy of Sciences (Beijing, China).[1-2] Their device, a liquid-state thermocell, is a variation on an effect most recently studied by Japanese scientists.[3] The Japanese researchers describe what they call a tertiary battery, a new battery type additional to primary (single use) batteries and secondary (rechargeable) batteries. These tertiary batteries exhibit a temperature related voltage hysteresis that can be used for thermal energy harvesting (see graph).

Temperature related voltage hysteresis in a "tertiary battery."

This effect can be used for thermal energy harvesting.

Temperature cycling must be between a lower critical temperature (T_c^l) and an upper critical temperature (T_c^u).

(fig. 4(d) of ref. 3, licensed under a Creative Commons Attribution 4.0 International License.[3])

The Japanese tertiary battery is enabled by a phase transition at one of the electrodes.[3] The transition occurs in the cobalt Prussian blue analogue, Na_xCo[Fe(CN)₆]_y, the composition of which is tuned so that the transition happens around room temperature.[3] The entire cell is thermally cycled in that device, while the liquid-state thermocell operates as a usual thermoelectric cell in contact with cold and hot thermal reservoirs.

Unlike solid state thermoelectric cells, like those made from bismuth telluride, the liquid-state thermocell transports electrical charge by means of ions instead of electrons, and such ions are less effective at charge transport than electrons. Ions likewise still conduct heat, so the figure of merit and energy conversion efficiency are low.[2] The liquid-state thermocell overcomes these problems by using temperature sensitive crystallization and dissolution of a material.[1-2] In this fashion, a Seebeck coefficient of 3.73 mV/K was achieved with low thermal conductivity.[1]

The cell liquid is a ferricyanide (Fe(CN)₆) solution that transfers electrons to the hot electrode, and receives electrons at the cold electrode by the following reactions:

Hot: Fe(CN)₆^-4 -> Fe(CN)₆^-3 + e^-

Cold: Fe(CN)₆^-3 + e^- -> Fe(CN)₆^-4

The electric current is conducted through an external circuit to provide power (see figure).[2]

A liquid-state thermocell based on ferricyanide ions.

(Created using Inkscape. Click for larger image.)

The thermal conductivity of the liquid is reduced by adding the positively charged organic compound, guanidinium.[2] This causes the ferricyanide to crystallize at the cold electrode, thereby thermally insulating it.[2] The crystallites are precipitated by gravity down to the hot electrode, which is hot enough to liquify them.[2]

Experiments using a 50°C temperature differential gave a Carnot efficiency of about 11%, which is more than double the efficiency needed for a viable commercial device.[1-2] Stringing twenty of these cells in series provided sufficient voltage and power to charge a mobile phone.[1-2]

Thermoelectrics

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