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Introduction. In this Chapter optimization of use of thermoelectric generator by coordination of thermal resistance of elements of design of the generator device are considered. As it appears, coordination on thermal resistance is in many respects similar to coordination of electric load resistance. Namely, there is an optimal solution with maximum efficiency at a certain ratio of thermal resistance of the generator module and other elements of a design.

Thermal resistance

Working parameters of a thermoelectric generator is determined by temperature difference ∆T that is created when heat is passing through the generator.

In basic formulas for thermoEMF E, efficiency η and net power Pthe working temperature difference ∆T is mentioned that is created directly on the sides (hot and cold) of the generator module.

In practice, however, this working ∆T is less than total temperature difference ∆T_s that is created at generator device by heat source relating to the envirnment, where heat is dissipated (Figure 5.1).

Figure. 5.1 Simplified schema of generator device in working arrangement with interfaces and heat sink.

Total temperature difference:

where T_h – temperature of heat source; T_a – ambient temperature.

Working (net) temperature difference ∆T on generator module is always less than total value ∆T_s:

This is due to the fact that in the design with thermoelectric generator inevitable parasitic thermal contact resistance at the crossings of the design. Particularly thermal resistance of heat sink is most important, the heat which dissipates into the surrounding ambient (Fig. 5.1).

In general

where Ȓ_s – total thermal resistance of generator device; Ȓ’_TEG – thermal resistance of working thermoelectric generator module; Ȓ_c -thermal resistances other items of the generator device.

Presence of parasitic thermal resistances Ȓ_c besides total thermal resistance of generator device Ȓ_s reduces working temperature difference ∆T on TE generator module in relation to the total difference ∆T_s and, consequently, reduces its effectiveness.

Taking into account formulas (2.24) and (2.25)

where T’_C – temperature on cold side of thermoelectric generator; Ȓ_TEG – thermal resistance of thermal conductivity of the thermoelectric generator.

In practical tasks, you must always strive to reduce parasitic heat resistance of construction, because it means a loss of working temperature difference and correspondingly – efficiency of generator.

However, as there is always non-zero values of such losses (Ȓ_c> 0) it is necessary an approach of optimization – the search for an optimal balance of these values Ȓ’_TEG and Ȓ_c

Net power

Consider net power P converted by thermoelectric generator.

where ACR – internal resistance of thermoelectric generator; m – ratio of resistances: external electrical load to internal resistance of the generator.

Here

where f – thermoelement form-factor (ratio of cross-section to height); p – electrical resistivity of thermoelement material; a -thermoelectric coefficient (Seebeck coefficient) of thermoelectric material of thermoelement; N – number of pairs of thermoelements.

Then

With

where k -thermal conductivity of thermoelement; Z – Figure of Merit.

Then the desired dependency of net power conversion P from the thermal resistance is:

Where given (5.6) and (5.7) the total dependence of net power Pon heat resistance for full temperature difference ∆T_s the system is as follows.

Maximum power

Maximum power P_max is a particular case of the above general formula (5.16), namely, at m=1. The expression for maximum power P_max has the following form

The formula (5.17) and graphical view (Fig. 5.2) for maximum output power P_max from the thermal resistance Ȓ'_TEG of the working generator is similar to the dependence of the power from the electrical resistance (Fig. 3.1). Namely, in both cases there are local maximums of power.

Figure. 5.2 Dependence of power P from thermal resistance of generator module Ȓ_TEG at different thermal resistances of the rest of the system Ȓ_c (paracitic thermal resistance). Here full temperature difference is ∆T_s=10°C. Dotted line – power at zero parazitic thermal resistance – when the working temperature difference is a half (∆T=5°C).

In the case of thermal resistances optimization – maximum power is achieved with equal thermal resistance of working generator Ȓ'_TEGand other (sum of parasitic) thermal resistances Ȓ_c of the generator system.

Physical sense

Optimizing of thermal resistance has a simple physical meaning.

Generator works optimally if its heat transport capacity (thermal conductivity is inverse value to thermal resistance) and heat throughput of all other structural elements (primarily is the element responsible for heat -dissipation of past heat) agreed upon, namely, equal.

For simplicity we will consider only the element of heat dissipation – heat sink. If its heat transport capacity is less than similar parameter of generator module, less heat passes through the system as a whole the generator converts less efficient.

On the other hand, the smaller thermal resistance of heat sink is better. And generator with the specified thermal resistance works better (example, Figure 5.1 – up arrow). But then already the generator will “brake” heat flow, as its thermal resistance becomes higher than of the heat sink.

For smaller thermal resistance of heat transport, there is a different, more optimal generator that will produce even more power (Figure 5.1 – red arrow sideways). But such more optimal thermal resistance of the generator should be smaller. So according to (5.17) and Fig.5.2 it should correspond to thermal resistance of the heat exchange element (Fig. 5.1 – red down arrow).

Particular solutions

Note that maximum power is achieved when there is no loss in generator on parasitic thermal resistances. This is a particular (ideal) case when the full temperature difference drops on the generator module:

In real system with finite parasitic thermal resistance (then ΔT <ΔT_s), the maximal available output power is 4 times (!) less than the maximum power output under ideal conditions.

In a case of non-zero parasitic thermal resistance the optimal case is the equality Ȓ’_TEG=Ȓ_c. This is equivalent to the ideal case of half temperature difference on the generator module (ΔT=½ΔT_s). The dependence of the power from the temperature difference is quadratic. This explains the ¼ of the optimum output power (5.18—5.19).

The construction of thermoelectric generators allows quite easily manage their thermal resistance. Namely, to obtain the optimal solution for the thermal resistance (thermal resistance change) there is a direct way to change the height and/or cross-sections of thermoelements of generator module.

Heat runs directly in thermoelectric generator through the thermoelements. Their thermal resistance is fundamental in the total thermal resistance of TE generator. Therefore, variation in height and cross section of thermoelements allows optimizing thermal resistance of the TE generator for maximum efficiency.

Thermal resistance of working generator

It must be noted that in the working generator module the effective thermal resistance, namely, the ratio of temperature difference to transported heat power differs from the thermal resistance related to the heat conductivity (Chapter 2). And approach of optimization on thermal resistance needs to be applied to the effective thermal resistance, i.e. taking into account ratios (2.24) – (2.26).

In contrast to thermal resistance of thermal conductivity Ȓ_TEG the effective thermal resistance Ȓ’_TEG of working thermoelectric generator depends on the operating mode (2.28).

– When an open electrical circuit the m=∞ and Ȓ’_TEG=Ȓ_TEG.

– At maximum efficiency mode the value m_opt≈1.4 is given by the expression (4.2). If ZT≈1, the effective thermal resistance Ȓ’_TEG turns out to be approximately 29% less then thermal resistance of thermal conductivity Ȓ_TEG.

– In maximum power mode m=1 and the effective thermal resistance of approximately a third less than Ȓ_TEG.

Since the heat resistances at maximum power (5.20) and maximum efficiency (5.21) modes differ slightly, it is often convenient to perform all calculations and modeling for maximum power mode with m=1.