Principles of energy conversion by thermoelectric effect
For cooling or generating electricity by thermoelectric effect, a "module" is made up of electrically connected "pairs". Each of these pairs is formed by a P-type semiconductor material (S>0) and an N-type material (S<0). Both materials are joined by a conductive material whose thermoelectric power is assumed to be zero. The two branches (P and N) of the pair and all of the other pairs that make up the module are electrically connected in series, and in parallel from a thermal point of view (see the diagram on the right). This arrangement allows optimizing the thermal flow that passes through the module and its electrical resistance. For simplicity, all the development that follows will be carried out for a single pair, formed by materials of constant section.
The figure on the right presents the basic schematic of a P-N couple used for thermoelectric cooling.
The electric current is imposed in such a way that the electric charge carriers (electrons and holes) move from the cold source to the hot source (in the thermodynamic sense) in the branches of the pair. By doing so they contribute to a transfer of entropy from the cold to the hot source, and therefore to a heat flow that opposes that of thermal conduction.
If the materials used have good thermoelectric properties (we will see below which are the most important parameters), this thermal flow created by the movement of the charge carriers will be more important than that due to thermal conductivity, which will allow heat to be evacuated from the cold source to the hot one, acting as a refrigerator.
In the case of electricity generation, it is the flow of heat that implies a displacement of charge carriers and therefore, the appearance of an electric current.
Conversion performance and important parameters
The calculation of the conversion efficiency carried out by a thermoelectric system is carried out by determining the relationship between the heat flow and the electric current in the material. For this, the Seebeck, Peltier and Thomson relations are used (see above), but also the laws of heat transfer and electric current.
The following example presents the calculation of the conversion efficiency in the case of refrigeration (the case of electricity generation can be done using analogous reasoning). Return to the previous scheme. In each of the branches of the pair, the heat flow generated by the Peltier effect opposes the thermal conductivity. The total flow in branches P and N will be:.
with x being the spatial coordinate (see diagram), λ and λ being the thermal conductivities of the materials and A and A being their sections.
Heat is extracted from the cold source with a flow Q:.
At the same time, the current that runs through the two branches is initially the result of the Joule effect heat Iρ/A per unit length of the branches. Using the Domenicali equation")[6] and assuming that the Thomson coefficient is zero (this suggests that S is independent of temperature, see the Thomson relation), the conservation of energy in the system is written in the two branches:
Considering the conditions at the boundaries, T=T at x=0 and T=T at x=L or x=L with L and L the lengths of the branches P and N, T and T the temperatures are those of the sources of cold and heat, Q is written:.
with K and R the total thermal conductivity and electrical resistance of each of the branches of the pair.
The electrical power W dissipated in the torque due to the Joule effect and the Seebeck effect is:.
The performance of the thermoelectric cooling system corresponds to the ratio between the heat extracted from the cold source and the electrical power dissipated, that is:.
For a given ΔT, the performance depends on the electric current flowing. Two particular values of current make it possible to maximize either the conversion efficiency η or the heat extracted from the cold source Q_f.
By similar reasoning, the performance of a P-N pair used to generate electricity will be given by the useful electrical power consumed by a load resistor R with a thermal flow passing through the material:.
In this case there are also two particular values of I that maximize the conversion performance or the electrical power delivered by the system.
By maximizing these two conversion performances, it can be shown that they depend only on the temperatures T and T and on a dimensionless number (without units) ZT called the "factor of merit" (T is the average temperature of the system, T=(T+T)/2) whose expression is:.
It should be noted that for any thermoelectric couple, the value of Z is not an intrinsic property of the material, but depends on the relative dimensions of the module, given the relationship between the dimensions and R and K (electrical resistance and thermal conductivity). The conversion efficiency of the system (operating as an electrical generator or as a cooling device) is maximum when Z is maximum, that is, when the product RK is minimum, which happens when:.
In this case, the merit factor Z becomes an exclusive function of the intrinsic parameters of the materials:.
Thus, to achieve optimal conversion performance, it is advisable to choose the materials that form the pair in such a way that Z is maximized. As a general rule, this is not limited to simply optimizing the individual merit factors of each material that forms the pair Z=S/(ρλ). At most temperatures used in practice, and especially those used for electricity generation, the thermoelectric properties of the best P and N type materials are similar. In these cases, the merit factor of the pair is close to the average value of the individual merit factors, and it is reasonable to optimize the merit factors of each of the materials independently.
The optimization of materials for their use in the conversion of energy through the thermoelectric effect necessarily involves the optimization of their electrical and thermal conduction properties, so that the factor of merit is maximized:
Thus, a good thermoelectric material will simultaneously have a high Seebeck coefficient, good electrical conductivity, and low thermal conductivity.
The figure opposite shows the evolution of the conversion performance of a thermoelectric system under ideal conditions as a function of the factor of merit ZT.
For example, if ZT=1 and the temperature difference is 300 °C, the conversion efficiency will be 8%, which means that depending on the case considered (electricity generation or refrigeration) that 8% of the heat that passes through the material will be converted into electricity, or that the heat extracted by the cooling element will correspond to 8% of the electrical power used.
Thermoelectric modules
It has been seen that the conversion properties of the pair of thermoelectric materials that constitute a module are not exclusively intrinsic, they also depend on the geometry of the system (length and section of the module branches) which in turn influences the electrical resistance R and the thermal conductivity K of the branches. Indeed, it is necessary that K be small enough so that a thermal gradient can be maintained, but it must also be of sufficient value so that heat can travel through the module: if K is zero, no heat will travel through the module and then there is no conversion. Likewise, R must be chosen so that the best possible compromise between electrical power and electrical potential difference is achieved. Once the materials that make up the module have been chosen (thanks to the ZT merit factor), it is necessary to optimize the geometry of the system in order to achieve the conversion performance, the electrical power or the highest possible heat extraction depending on the application of the module.
In general, the materials used in the manufacture of thermoelectric conversion modules are only effective in a certain temperature range. Thus, for example, the SiGe alloy used to power the Voyager probe is only effective at temperatures above approximately 1000K. In applications in which the working temperature range is very large, it is interesting to use several thermoelectric materials in each branch, each with a temperature range in which their performance is maximized. In these cases it is said that the thermoelectric module is segmented.
The figure to the side illustrates the concept of a segmented thermoelectric module. In this case there is a very significant temperature gradient (700K difference between the hot and cold zone), and no known material is effective in this entire temperature range. Each of the two branches of the pair is then made up of several materials (in the case shown two for the N branch and three for the P branch). The length of each of these materials is chosen so that they are used in the temperature range in which they are most effective. Therefore, a module constructed in this way would achieve a higher conversion performance, electrical power or heat extraction than if each branch were composed of a single material. In this way, the best performances achieved in the laboratory with this type of modules are currently close to 15% (which means that 15% of the heat that passes through the material is converted into electrical power). However, segmented modules are much more expensive than "simple" modules, which restricts their use to applications in which cost is not a decisive factor when choosing.