heat transferred due to the Peltier effect; QR – Joule heat emitted due to the electric current in a closed circuit of the generator.
Figure. 2.2 Heat flow diagram for generating mode.
If to neglect temperature dependences of physical parameters, then heat balance equations in the generator mode can be written as the following:
where Th, Tc – temperatures of the hot and cold sides of generator module, respectively; N – number of pairs of thermoelements (pellets) in the module; α – average value of thermal motive force (thermoEMF), other words – Seebeck coefficient of pair of pellets n- and p- type; k -average heat conductivity of the pellet pair; ΔT – temperature difference (ΔT=Th-Tc); I – electric current through the module.
Here and below average values α, k, R are used – average values of paired thermoelements (pellets) of n-and p-type. For example, α=(αn+αp)/2. And these parameters refer to average operating temperature of pellets, in situations where the temperature dependency properties can generally be neglected.
Main parameters
Thermoelectromotive force (ThermoEMF) E of a thermoelectric generator depends on the temperature difference ΔT, number of thermoelements N and Seebeck coefficient α as the following
We introduce the notation for internal resistance of thermoelectric generator ACR=2NR, and for external load resistance – Rload.
Electric current through the generator is determined by thermoEMF E and total resistance of closed circuit (Fig. 2.2):
Here we introduce important parameter m – the ratio of the external load Rload to internal resistance of generator module ACR.
From the balanced equation (2.5) and (2.6) one can find that:
First member EI of the difference (2.10) is the total heat that is converted into electric current by the thermoelectric generator. The second member I2ACR is what part which comes back into heat due to the Joule effect.
– When the difference (2.10) is equally to zero (short circuit mode, external load resistance equal to zero) then all the converted energy comes back into heat.
– When the difference is positive (there is non-zero external load), i.e. the generator converts heat into electricity.
With given ratios (2.7)-(2.9) the balance equations (2.3) and (2.4) can be rewritten as the follows:
where Z – Figure-of-Merit of thermoelectric module; Qλ – heat flow due to thermal conductivity through the module (Qλ=2NkΔT).
Here the heat transferred through the Peltier effect QP
Joule heat Qλ
Voltage U in external circuit, taking into account (2.8), is the following.
Power P in the external circuit (converted power).
Here an important dimensionless factor F is appeared.
Formula (2.16) has a maximum defined by the dimensionless factor F, its dependence on the ratio m (2.9) of load resistance Rload and internal resistance ACR of generator module (Fig. 2.3).
Figure 2.3 Dependence of dimensionless factor F from ratio m=Rload/ACR.
Maximum F and, respectively (2.16), maximum power P are achieved when m=1, i.e.
Important note – maximum power Pmax that can be obtained by a thermoelectric generator module is only one quarter of the power converted by the module in short circuit mode.
Electric current through the module in maximum power mode Imax at m=1 is the following
The thermoelectric generator efficiency η
Taking into account formulas (2.7), (2.11) and (2.16) one can find that
Investigating the formula for maximum at fixed temperature Th and ΔT (Th, ΔT=const), we obtain formula for maximum efficiency η (optimal mode for efficiency):
Where the optimum ratio of load and internal resistances mopt (mopt=Rload/ACR) can be expressed as the following.
Note, please, the formula (2.24). If maximum power Pmax converted by generator can be achieved when m=1, then maximum efficiency η – at other value of this ratio – mopt (2.24). In the thermoelectric generator, as in any heat engine, maximum power mode operation differs from mode of maximum efficiency.
Effective thermal conductivity and thermal resistance
Heat Q passed through a media, which is the generator, one can write in general using the effective thermal conductivity K’ of this media and temperature difference ΔT as the following.
In working generator the heat is Qc (2.12), which differs from the heat transported due to “simple” thermal conductivity Qλ:
Effective thermal conductivity K’ differs from conventional thermal conductivity K of agenerator due to the additional Peltier and Joule heat flows, that appear in the working generator, and which are superimposed on the conventional thermal conductivity (Fig. 2.2).
Thermal resistance of the working generator Ȓ’TEG is the following
To a first approximation, at small temperature