1 Introduction

 

    The first of the thermoelectric effects was discovered, in 1821, by T. J. Seebeck. He showed that an electromotive force could be produced by heating the junction between two different electrical conductors. The Seebeck effect can be demonstrated by making a connection between wires of different metals (e.g., copper and iron). The other ends of the wires should be applied to the terminals of a galvanometer or sensitive voltmeter. If the junction between the wires is heated, it is found that the meter records a small voltage. The arrangement is shown in Fig. 1.1. The two wires are said to form a thermocouple. It is found that the magnitude of the thermoelectric voltage is proportional to the difference between the temperature at the thermocouple junction and that at the connections to the meter.

    Thirteen years after Seebeck made his discovery, J. Peltier, a French watchmaker, observed the second of the thermoelectric effects. He found that the passage of an electric current through a thermocouple produces a small heating or cooling effect depending on its direction. The Peltier effect is quite difficult to demonstrate using metallic thermocouples since it is always accompanied by the Joule heating effect. Sometimes, one can do no better than show that there is less heating when the current is passed in one direction rather than the other. If one uses the arrangement shown in Fig. 1, the Peltier effect can be demonstrated, in principle, by replacing the meter with a direct current source and by placing a small thermometer on the thermocouple junction.

    It seems that it was not immediately realised that the Seebeck and Peltier phenomena are dependent on one another. However, this interdependency was recognised by W. Thomson (who later became Lord Kelvin), in 1855. By applying the theory of thermodynamics to the problem, he was able to establish a relationship between the coefficients that describe the Seebeck and Peltier effects. His theory also showed that there must be a third thermoelectric effect, which exists in a homogeneous conductor. This effect, now known as the Thomson effect, consists of reversible heating or cooling when there is both a flow of electric current and a temperature gradient.

Fig. 1Experiment to demonstrate the Seebeck and Peltier effects

    The fact that the Seebeck and Peltier effects occur only at junctions between dissimilar conductors might suggest that they are interfacial phenomena but they are really dependent on the bulk properties of the materials involved. Nowadays, we understand that electric current is carried through a conductor by means of electrons that can possess different energies in different materials. When a current passes from one material to another, the energy transported by the electrons is altered, the difference appearing as heating or cooling at the junction, that is as the Peltier effect. Likewise, when the junction is heated, electrons are enabled to pass from the material in which the electrons have the lower energy into that in which their energy is higher, giving rise to an electromotive force.

Thomson’s work showed that a thermocouple is a type of heat engine and that it might, in principle, be used either as a device for generating electricity from heat or, alternatively, as a heat pump or refrigerator. However, because the reversible thermoelectric effects are always accompanied by the irreversible phenomena of Joule heating and thermal conduction, thermocouples are generally rather inefficient.

    The problem of energy conversion using thermocouples was analysed by Altenkirch , in 1911. He showed that the performance of a thermocouple could be improved by increasing the magnitude of the differential Seebeck coefficient, by increasing the electrical conductivities of the two branches and by reducing their thermal conductivities. Unfortunately, at that time, there were no thermocouples available in which the combination of properties was good enough for reasonably efficient energy conversion, although the Seebeck effect has long been used for the measurement of temperature and for the detection of thermal radiation. It was only in the 1950s that the introduction of semiconductors as thermoelectric materials allowed practical Peltier refrigerators to be made. Work on semiconductor thermocouples also led to the construction of thermoelectric generators with a high enough efficiency for special applications. Nevertheless, the performance of thermoelectric energy convertors has always remained inferior to that of the best conventional machines.

    In fact, there was little improvement in thermoelectric materials from the time of the introduction of semiconductor thermoelements until the end of the twentieth century. However, in recent years, several new ideas for the improvement of materials have been put forward and, at last, it seems that significant advances are being made, at least on a laboratory scale. It is reasonable to expect that this work will soon lead to much wider application of the thermoelectric effects.

 

2 Thermoelectric Effects

 

2.1 Seebeck Effect

    The thermometric effects which underlie thermoelectric energy conversion can be conveniently discussed with reference to the schematic of a thermocouple shown in Figure 2. It can be considered as a circuit formed from two dissimilar conductors, a and b (referred to in thermoelectrics as thermocouple legs, arms, thermoelements, or simply elements and sometimes as pellets by device manufacturers) which are connected electrically in series but thermally in parallel. If the junctions at A and B are maintained at different temperatures T₁ and T₂ and T₁ > T₂ an open circuit electromotive force (emf), V is developed between C and D and given by V =α(T₁-T₂) or α =  V/∆T, which defines the differential Seebeck coefficient α_ab between the elements a and b. For small temperature differences the relationship is linear. Although by convention α is the symbol for the Seebeck coefficient, S is also sometimes used and the Seebeck coefficient referred to as the thermal emf or thermopower. The sign of α is positive if the emf causes a current to flow in a clockwise direction around the circuit and is measured in V/K or more often in μV/K.

Fig. 2Schematic basic thermocouple

2.2 Peltier Effect

    If in Figure 2 the reverse situation is considered with an external emf source applied across C and D and a current I flows in a clockwise sense around the circuit then a rate of heating q occurs at one junction between a and b and a rate of cooling -q occurs at the other. The ratio of I to q defines the Peltier coefficient given by π = I/q, is positive if A is heated and B is cooled, and is measured in watts per ampere or in volts.

2.3 Thomson Effect

    The last of the thermoelectric effects, the Thomson effect relates to the rate of generation of reversible heat q which results from the passage of a current along a portion of a single conductor along which there is a temperature difference ∆T. Providing the temperature difference is small, q = βI∆T where β is the Thomson coefficient. The units of β are the same as those of the Seebeck coefficient V/K. Although the Thomson effect is not of primary importance in thermoelectric devices it should not be neglected in detailed calculations.

 

3 Thermoelectric Generation and the Figure-of-Merit

 

    A thermoelectric converter is a heat engine and like all heat engines it obeys the laws of thermodynamics. If we first consider the converter operating as an ideal generator in which there are no heat losses, the efficiency is defined as the ratio of the electrical power delivered to the load to the heat absorbed at the hot junction. Expressions for the important parameters in thermoelectric generation can readily be derived by considering the simplest generator consisting of a single thermocouple with thermoelements fabricated from n- and p-type semiconductors as shown in Figure 3(upper).

Fig. 3 Thermoelectric generator (upper); thermoelectric refrigerator (lower)

    The efficiency of the generator is given by

                                                   

    If it is assumed that the electrical conductivities, thermal conductivities, and Seebeck coefficients of a and b are constant within an arm, and that the contact resistances at the hot and cold junctions are negligible compared with the sum of the arm resistance, then the efficiency can be expressed as

                                                    

    Where λ` is the thermal conductance of a and b in parallel and R is the series resistance of a and b. In thermoelectric materials σ, λ`, and, α change with temperature, and in both, generation and refrigeration should be taken into account. However, the simple expression obtained for the efficiency can still be employed with an acceptable degree of accuracy if approximate averages of values are adopted for these parameters over the temperature range of interest. Appropriate allowances can also be made for contact resistance.

    Efficiency is clearly a function of the ratio of the load resistance to the sum of the generator arm resistances, and at maximum power output it can be shown that

                                                  

    while the maximum efficiency

                                                  

    where

                                            

                                            

    and

                                            

    and

                                                    

    The maximum efficiency is thus the product of the Carnot efficiency, which is clearly less than unity, and γ, which embodies the parameters of the materials.

If the geometries of a and b are matched to minimize heat absorption, then

                                                     

    In practice, the two arms of the junction have similar material constants, in which case the concept of a figure-of-merit for a material is employed and given by

                                                      

     where α²σ is referred to as the electrical power factor.

    The above relationships have been derived assuming that the thermoelectric parameters which occur in the   figure-of-merit are independent to temperature. Although generally this is not the case, assuming an average value provides results which are within 10% of the true value.

    The conversion efficiency as a function of operating temperature difference and for a range of values of the material’s figure-of-merit is displayed in Figure 4. Evidently an increase in temperature difference provides a corresponding increase in available heat for conversion as dictated by the Carnot efficiency, so large temperature differences are desirable. As a ballpark figure a thermocouple fabricated from thermoelement materials with an average figure-of-merit of 3 × 10^-3/K would have an efficiency of 20% when operated over a temperature difference of 500 K.

Fig. 4 Generating efficiency as a function of temperature and thermocouple material figure-of-merit.

 

4 Thermoelectric Refrigeration and the Coefficient of Performance

 

    The performance of any refrigerator is in general expressed by its coefficient of performance. This is given by the cooling power produced divided by the rate at which electrical energy is supplied. In Figure 3 (lower) is illustrated an ideal thermoelectric circuit upon which the operation of a thermoelectric cooling device is based as is the case of a generation. The circuit is identical to that of a generator discussed above, but in this case a direct current is passed through the thermocouple circuit and heat is pumped from T_C to T_H. If T_H > T_C the device operates as a refrigerator.

As a result of the Peltier effect the rate of heat pumping at the junction T_C is given by π_abI.

    Using the Kelvin relationship

                                                          α_ab= π_ab/T_c

     whereα_ab is the Seebeck coefficient of the junction we can write

                                                          π_ab I =α_ab (T_M -∆T=2)I

    T_M is the mean absolute temperature and DT the temperature difference T_H – T_C.

    The cooling effect at the source junction is opposed by Joule heating in the thermoelements and by heat conducted from the hot junctions. Half the overall Joule heating travels to each of the junctions.

Thus neglecting the Thomson effect, the rate of absorption of heat from the source is given by

                                                           Q_ab =α_ab T_C I-0.5I²R-K(T_H - T_C)

      An input power by

                                                            P =α_ab ∆T I + I²R

    where the potential difference applied to the thermocouple is used in part to overcome the electrical resistance of the thermoelements and to balance the Seebeck voltage which results from the temperature difference between the junctions.

    The energy efficiency of a refrigerator is measured by its coefficient of performance, COP, defined as

                                               

  K is the thermal conductance of the thermoelements in parallel, and R is the electrical resistance of the thermoelements in series. Evidently the coefficient of performance for a given temperature difference is dependent on the current I.

    The current I` for maximum cooling power is given by

                                                  

    with the corresponding COP

                                                  

    The maximum temperature difference is given by

                                                  

    and the current I for maximum coefficient of performance

                                                 

    with the maximum coefficient of performance

                                                

    Evidently the figure-of-merit Z determines both the maximum temperature difference that can be

achieved and also the maximum coefficient of performance. The dependence of fmax with Z value

for different temperature differences is shown in Figure 5 and the dependence of the maximum

temperature difference with Z in Figure 6.

Fig. 5Theoretical coefficient of performance of a thermoelectric module plotted against temperature difference for different Z values, mean temperature

Fig. 6 Theoretical maximum temperature difference of a thermoelectric module plotted against Z at a hot-side temperature of 298 K (25ºC).

 

5. Applications for Thermoelectric Coolers

 

5.1 Applications for thermoelectric modules cover a wide spectrum of product areas.

    These include equipment used by military, medical, industrial, consumer, scientific/laboratory, and elecommunications organizations. Uses range from simple food and beverage coolers for an afternoon picnic to extremely sophisticated temperature control systems in missiles and space vehicles.

    Unlike a simple heat sink, a thermoelectric cooler permits lowering the temperature of an object below ambient as well as stabilizing the temperature of objects which are subject to widely varying ambient conditions. A thermoelectric cooler is an active cooling module whereas a heat sink provides only passive cooling.

    Thermoelectric coolers generally may be considered for applications that require heat removal ranging from milliwatts up to several thousand watts. Most single-stage TE coolers, including both high and low current modules, are capable of pumping a maximum of 3 to 6 watts per square centimeter (20 to 40 watts per square inch) of module surface area. Multiple modules mounted thermally in parallel may be used to increase total heat pump performance. Large thermoelectric systems in the kilowatt range have been built in the past for specialized applications such as cooling within submarines and railroad cars. Systems of this magnitude are now proving quite valuable in applications such as semiconductor manufacturing lines.

 

5.2 Typical applications for thermoelectric modules include: