Читать книгу Thermoelectric Microgenerators. Optimization for energy harvesting - Gennady Gromov - Страница 6

Introduction. In any heat engine the mode of maximum power differs from the mode of maximum efficiency. In this Chapter the operating mode of thermoelectric generator – maximum efficiency is considered in details.

General formula

The value of efficiency η changes with variations of the load resistance R_load similarly to dependence of P vs R_load (e.g., Fig. 3.1). Namely dependence of efficiency η from the ratio m also has a maximum. But this is not the same point as the maximum P_max. The maximum of η takes place at other value of power P_opt somewhat different from the P_max.

General formula for the efficiency η is the following

Maximum efficiency and maximum power modes

Omitting the detailed math (see Chapter 2), it can be shown that in a simplified form the maximal efficiency η_opt occurs when

where T_h and T_c – correspondingly, temperatures of the hot and cold sides of generator; Z – thermoelectric Figure-of-Merit of the generator.

In practice, in applications with small temperature differences and typical Figure-of-Merit of generators, the value m_opt given by (4.2) is approximately equal to

With regard to the P_max and P_opt – they are pretty close each other.

It should be noted that the efficiency η at maximum power mode and at maximum efficiency mode are also close to each other. It can be shown if the corresponding values m_max and m_opt to apply in formula (4.1), respectively, then

Choosing of practically optimal load resistance between maximum power and maximum efficiency modes can be in the range (Fig. 4.1).

Figure. 4.1 Dependence of the efficiency η and power P related to P_scvs ratio m of resistances for two main generator operation modes – maximuum power (P_max, m_max, η_max) and maximum efficiency (P_opt, m_opt, η_opt).

In practice, use the electric load from this range (4.8) turns out to be more comfortable than to select optimal electric load with exactly the specified value.

From the formula (4.1) with use of (4.3) it can be build a useful table for estimates of the absolute value of the maximum efficiency η_opt of thermoelectric generator depending on temperature difference ΔT (Table 4.1).

Table 4.1 Dependence* of maximum efficiency η_opt vs temperature difference ΔT on a generator.

The common conclusions from formula (4.1) and Table 4.1 are the following:

– in practice, the efficiency η_opt is almost a linear function of the temperature difference ΔT.

– one degree of the temperature difference ΔT gives about 0.05% of maximum efficiency η_opt.

Efficiency and carnot cycle

Useful information on the efficiency of thermoelectric generator should be of the following formulas for two marginal efficiency modes: the maximum efficiency mode η_opt and the maximum power mode η_max.

For these modes the efficiency η can be written as the following:

– for maximum efficiency mode η_opt

– for maximum power mode η_max

In both formulas (4.9) and (4.10) the first fractional multiplier is, generally speaking, the ideal Carnot cycle efficiency (∆T/T_h). The second multiplier – thermoelectric factor reduces ideal efficiency of the Carnot cycle.

So, near room temperature (T_c≃ 300K) and typical Z≃0.003K^-1 we have an numerical expression for the maximal efficiency η_opt

As for the mode of maximum power efficiency η_max, correspondingly:

In other words, state-of-art thermoelectric microgenerators provide efficiency only 15.5—16% of the ideal Carnot cycle efficiency.

Here you can make an important note about the maximum possible efficiency for any heat engines used in waste heat recycling applications (energy harvesting).

Namely, the ideal heat engine working by Carnot cycle near room temperature provides efficiency only about 0.33% per degree of temperature difference (4.11).

Thus, this is the absolute maximum.

According to Carnot’s theorem, such wording [22]:

“Maximum efficiency of any heat engine may not exceed the efficiency of Carnot heat engine, running at the same temperatures of the heater and cooler.

This is an important point to general understand. To avoid posing unrealistic tasks to retrieve large efficiency with thermoelectric generators – larger than limited by the ideal Carnot cycle.

This issue will be discussed further in Chapter 7.