Читать книгу Thermoelectric Microgenerators. Optimization for energy harvesting - Gennady Gromov - Страница 4

Preface. Key aspects of applications of thermoelectric microgenerators derived from fundamental physical principles underlying their work. These are effects of Seebeck, Peltier, Thompson, and Joule. Accordingly, we start with the physical basics of thermoelectric modules. This Chapter introduces key concepts and parameters of thermoelectric generators.

Heat balance equations

Cooling mode

To date, the widest use of thermoelectric micromodules is connected with cooling of miniature objects – thermoelectric cooling applications.

For comparison, let us give the equations of thermal balance (2.1) – (2.2) of thermoelectric module in cooling mode (Fig. 2.1).

Figure. 2.1 Heat flow diagram for cooling mode.

Generating mode

Heat balance equations of thermoelectric module in generating mode (Fig. 2.2) are the following.

where Q_h – heat flow absorbed by the hot side of the module; Q_c – heat flow emitted by the cold side of the module; Q _λ – heat conducted due to thermal conductivity; Q_P – heat transferred due to the Peltier effect; Q_R – 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 T_h, T_c – 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=T_h-T_c); 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= (α_n+α_p) /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 – R_load.

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 R_load 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; Q_λ – heat flow due to thermal conductivity through the module (Q_λ=2NkΔT).

Here the heat transferred through the Peltier effect Q_P

Joule heat Q_λ

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 R_loadand internal resistance ACR of generator module (Fig. 2.3).

Figure 2.3 Dependence of dimensionless factor F from ratio m=R_load/ACR.

Maximum F and, respectively (2.16), maximum power P are achieved when m=1, i.e.

Important note – maximum power P_max 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 I_maxat 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 T_hand ΔT (T_h, ΔT=const), we obtain formula for maximum efficiency η (optimal mode for efficiency):

Where the optimum ratio of load and internal resistances m_opt (m_opt=R_load/ACR) can be expressed as the following.

Note, please, the formula (2.24). If maximum power P_max converted by generator can be achieved when m=1, then maximum efficiency η – at other value of this ratio – m_opt (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 Q_c (2.12), which differs from the heat transported due to “simple” thermal conductivity Q_λ:

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 differences ΔT the 3-rd member (Joule) in the sum in brackets (2.27) can be ignored. Indeed, it can be shown that contribution of this term at small temperature differences is small, no more than 0.5—1%.

Then

Exclusion of a member, depending on ΔT dramatically simplifies analysis of a thermoelectric generator in the tasks of complex ambient. Where the generator is placed between other media and interfaces with different thermal resistance, and it is desirable to optimize the thermal resistance Ȓ’_TEG of the working generator (see Chapter 5).

When open electrical circuit takes place in the generator, then there is no Peltier and no Joule heat flows. Only thermal conductivity heat flow takes place. In other words, then R_load=∞ and m=∞ then the formula (2.26) is simplified to:

Temperature difference ΔT at a generator module when an open circuit takes place is associated with heat flow of thermal contuctivity Q_λ as the following.