Kitabı oku: «Thermoelectric Microgenerators. Optimization for energy harvesting», sayfa 2

Chapter 5. Optimization of thermal resistance

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.

Chapter 6. Design optimization

Summary. In this Chapter dependences of parameters of thermoelectric generators on elements of their design are considered. The analysis is provided on the example of the known wide nomenclature range of microgenerator modules. On these examples it is given an idea, what parameters of generators and what role they play in their efficiency and in consumer properties.

Introduction

Useful formulas are illustrated on the example of series of standard TE microgenerators developed earlier [5] in the TEC Microsystems GmbH company in relation to tasks of “low power” – energy harvesting applications.

The nomenclature of thermoelectric modules of TEC Microsystems GmbH is developed with use of classification system of thermoelectric micromodules [6]. This classification allows systematizing thermoelectric micromodules on series in compliance with their parameters and features of a design (see also Chapter 12).

Thus, logical ranks of micromodules convenient for their choice for practical applications are created.

Number of thermoelements

The number of thermoelements 2N in the generator module at the specified temperature difference ∆T and Seebeck coefficient αdetermines the key characteristic – total thermoEMF E.

The value of thermoEMF E provided by the generator determines the output voltage U in the load circuit.

Depending on value of load resistance R_load the working voltageU range of generator could vary widely.

In maximum power mode

If to increase load resistance Rload in the limit we have

Value of thermoEMF E and corerspoindingly output voltage U of generators are variable; depend on the value of temperature drop ∆T. It is not so convenient for consumers of such non-stable power supply – to electronic devices.

Always it is necessary to use electronic DC-DC converter to transform the generator variable voltage to the standard supply voltage of electronic devices.

The DC-DC converters have restrictions on the minimum input voltage which they can transform. It needs to be taken into account. And thermoelectric generators selected for practical applications must be capable to give working voltage not below the minimum threshold of the applied DC-DC converter. In more detail about the choice of a DC-DC converters see Chapter 10.

Zones of applicability of modern DC-DC converters with the minimum input voltage are given in Fig. 6.1. It is, for instance, 20 mV (Linear Technology) [3], 80 mV and 250 mV (Texas Instruments) [4].

Figure. 6.1 ThermoEMF E depending on the number of pellet pairs N under various temperature differences ΔT.

Besides, low input voltage of the DC-DC converter requires to pay additional “cost”. It is the converter efficiency. The input voltage is lower – the efficiency of transformation of the DC-DC electronic scheme is lower.

In this regard it is more preferable to use generators giving the bigger thermoEMF, i.e. generators with a large number of thermoelements (see Fig. 6.1). Besides, practical tasks sometimes force to leave absolutely optimal solutions for generator. I.e. to use electric loads with a resistance bigger, than ACR, to increase the output voltage of the generator, though by reduction of generator’s efficiency.

It means to consider efficiencies of thermoelectric generator and DC-DC electrical circuit in a combination to obtain maximal output value (see Chapter 10).

Form-factor of thermoelements

The form-factor f of thermoelements of the generator module determines its total electric resistance.

where f – form-factor of a thermoelement.

Here we neglect the additional resistance of the generator module construction (conductors, places of soldering of thermoelements, resistance of barrier layers). In most cases it is valid, as this additional resistance is insignificant usually in comparison of the sum of resistance of thermoelements (6.5).

Then

The formula for maximum power P_max through the generator number of thermoelements N and their form-factor f has a finite shape

Thermal resistance

Thermal resistance Ȓ_TEG of thermoelectric generator determines its overall performance.

Formula (6.8) can be converted to a dependence of maximum power P_max vs thermal resistance Ȓ_TEG of generator.

Figure. 6.2 Power P_max of generators of different series (developed by TEC Microsystems) vs their thermal resistance Ȓ_TEG at different temperature differences ΔT: Points mark the boundaries of the applicability of these series (1MD02, 1MD03, 1MD04 and 1MD06).

This formula (6.10) and the provided graph (Fig. 6.2) are important for understanding of features of practical applications of generator modules and the choice of optimal solutions.

The power provided by the generator depends on its Figure-of-Merit Z, thermal resistance Ȓ_TEG and temperature difference ∆T.

– Figure-of-Merit Z is defined by properties of the thermoelectric material used in the generator module and a design of the module. For different designs of generators average Figure-of-Merit Z – it a little changeable size is at the level of 2.8…3.0⨯10^—3 K^-1 (Table 7.1).

– Temperature difference ∆T for a specific case is the value set up by the heat source and the environment.

– The only changeable parameter is the thermal resistance Ȓ_TEG. It can vary widely for a particular design of TE generator just by changing the form-factor – by height and cross-section of thermoelements and number of the thermoelements.

Fig. 6.2 shows the broad range of applicability of modules of given nomenclature. It is due to an opportunity in these series to change form-factor (cross-section and height) of thermoelements and its number.

Coefficient of performance

In accordance with (2.21) the efficiency of the thermoelectric microgenerator in the modes of the maximum power (m=1) or maximum efficiency when m_opt~1.4 (4.5) is determined only by the performance of the thermoelectric material – Figure-of-Merit Z, temperature difference on the generator module ∆T and averaged working temperature (T_h+T_c) /2.

For practical estimates the efficiency values at averaged temperature 320K and typical values of Figure-of-Merit Z (Chapter 7, Table 7.1) of generator micromodules are given in Table 6.1.

Table 6.1 Efficiency of generator modules depending on temperature difference (at average temperature 320K).

For the small temperature differences it is possible with good precision to consider that every degree of temperature difference provides: in the mode of the maximum power the efficiency ~ 0.047%, and in the mode of the maximum efficiency ~ 0.048% of the efficiency.

It should be noted, at increasing of average temperature the efficiency falls down (see Chapter 7). For example, in Chapter 4 it has been shown that at average temperature 300K by one degree of temperature difference the efficiency is about 0.05% (Table 4.1).

From Table 6.1 it follows that at a temperature slightly above (320K) – efficiency per one degree is slightly lower (~ 0.047%). It is explained by temperature dependences of thermoelectric material of the generators (Chapter 7, Fig. 7.3).

Heat flow density

At given efficiency the total converted power will be determined by heat flow Q_c passing through the generator module. And it is set by the capacity of the heat source and the heat transport “capacity” (inverse of thermal resistance) of the generator itself. These characteristics must be coordinated.

For coordination it is useful to operate with value of heat flow density on unit of the heat giving surface and heat flow density which is passed by unit of generator surface.

Heat flow density (2.26) is given by the formula:

where x – packing density of pellets in TE generator module; k – thermal conductivity of the thermoelements (pellet) thermoelectric material.

where δ – distance between thermoelements, a – cross-section of the thermoelement.

For practical estimates the data for heat flow density for various types of generator micromodules are provided in Table 6.4.

Thus, at the given temperature difference and the given characteristic of heat conductivity of material of the generator module, heat flow density depends on density of packing of thermoelements in the module design (6.11). The more densely they are packed (less gap between thermoelements), the generator is more powerful.

Density of heat flow passing through the generator characterizes its specific power. This specific characteristic allows calculating of absolute values of a heat flow through the chosen generator module.

Comparison of heat flow density of the heat giving surface and density of a heat flow via the generator gives the answer to a question whether it is necessary in a design of the generator device the big surface of heat collecting. I.e. whether it is necessary to collect and concentrate heat on the generator for his effective work.

Table 6.2. The density of heat flow through different types of generator micromodules at various temperature differences.

From Table 6.2 it is seen that heat flow density via the generator micromodule can reach several units of W/cm². To provide such power densities may require heat flow concentrators from heat source (case of low power heat source).

This is important issue for energy harvesting tasks, where low-power heat sources usually have very small values of heat flow power density.

Illustrative example – recycling of the human body heat. Heat flow density of such kind of heat “source” is small compared with tabular values for the microgenerators (Table 6.2) – no more than 10^—1…10^—2 W/cm². Clearly, the generator device in such tasks should include heat flow concentrators. Moreover, the area of such a hub should significantly exceed size of the generator micromodule.

Another example is soil generators which utilize heat flowing in the upper layers of soil in the daily cycle. Here the heat flows have values at the level of 100W/m² (0.01W/cm²). It also requires the concentration of heat for the supply to generator micromodule.

Another important task is to effectively dissipate heat passed through the generator. If this is not achieved, the useful heat, transported through generator system, limited to a size not more than wasted on a radiator.

Here it is also important to the specific characteristics of the density of heat flow through the radiator -" flow capacity” of the radiator. It is necessary to calculate the dimensions of the radiator (see Chapter 9). In many practical cases, dimensions of a heat sink should be also noticeably larger than the generator micromodule.

Chapter 7. Thermoelectric materials

Summary. In this Chapter properties of the thermoelectric materials applied in microgenerators for energy harvesting are considered. On the working temperature range of these tasks, such generators can be classified as “low-temperature”. Besides in this temperature range the majority of thermoelectric coolers works. Therefore also thermoelectric materials for generator applications are used same, as for thermoelectric cooling – solid-state solutions on the basis of the Bi-Sb-Te.

Typical parameters

As shown above (Chapter 6), thermoelectric material properties of the thermoelements play decisive role in the parameters of efficiency of microgenerator modules.

Therefore, when designing of microgenerator it is necessary to pay great attention to properties of thermoelectric material from which they are should be made.

Application of energy harvesting means heat recycling from heat sources of low power, small temperature difference relating to the environment. Approximately this temperature range can be defined as ~250 … 450K (-25… +180°C).

In this regard an analogy arises to other field of applications of thermoelectricity – thermoelectric cooling. Practical temperature range of thermoelectric cooling is approximately similar stated above.

It explains that for thermoelectric generation in energy harvesting tasks practically the same thermoelectric materials on the basis of solid solutions of Bi-Sb-Te, as successfully can be applied as to thermoelectric cooling. Such thermoelectric semiconductor materials have maximal efficiency in this temperature range.

Key properties of thermoelectric material are combined in expression for its Figure-of-Merit Z.

where α – thermoEMF coefficient (Seebeck coefficient) of thermoelectric material; σ – electrical conductivity; k – thermal conductivity.

Typical parameters of thermoelectric material applied for manufacturing of microgenerators and microcoolers are given in Table 7.1.

Table 7.1 Typical parameters of thermoelectric materials of p-and n-types at 300K.

Also temperature dependences of properties in the working temperature ranges are important for detailed calculations and modeling of real devices operation.

For convenience of calculations and mathematical modeling temperature dependences can be presented in the polynomial form.

Temperature dependences of properties of the applied thermoelectric materials are described well by a polynomial of the 3rd order of general view as following

where x=T/100

Detailed data for typical polynomial temperature dependences of the major properties are provided in Table 7.2. In Fig. 7.1 these typical temperature dependences are graphically presented.

Table 7.2 Polynomial coefficients of temperature dependences (7.2) of properties of thermoelectric materials.

a)

b)

c)

d)

Figure. 7.1 Temperature dependences of thermoelectric materials properties: Seebeck coefficient (a), electric conductivity (b), thermal conductivity (c) and Figure-of-Merit (d).

In calculated parameters of thermoelectric microgenerators average characteristics of pair of thermoelements of n-and p-types are applied. Therefore temperature dependences of properties of materials of both n-and p-types and these average dependences are given in Fig. 7.1. Average dependences characterize properties of p-n pair of such thermoelements. In mathematical formulas for parameters of thermoelectric generators such average properties on p-n to pair of thermoelements are just used.

On the presented temperature dependences it is necessary to make several important remarks.