Thermoelectric Microgenerators. Optimization for energy harvesting

© Gennady Gromov, 2018

ISBN 978-5-4493-4334-5

Created with Ridero smart publishing system

Preface

The idea for this book came from the experience. Our team is in thermoelectric business for over 20 years. Firstly, for many years it was thermoelectric cooling solutions for miniature applications. In recent years, thermoelectrics moves in microgenerators trend. Green energy, energy harvesting…

Promising directions – applications of thermoelectric microgenerators for tasks of low energy – recycling waste heat surrounding thermal sources, of human body energy harvesting and so on.

This direction only began its development. Of course, there are many ideas for development come from inventors and scientists. Often an absence of sufficient knowledge and understanding of the subject of thermoelectricity is compensated by enthusiasm, the broadest fields of promising applications, interesting tasks.

Numerous appeals of such potential consumers to assist them in thermoelectric solutions usually are accompanied by neediness of detailed technical consulting from fundamental physical base of the thermoelectricity phenomenon till fine nuances in their applications.

It has given rise to the idea of writing of the reference book FAQ (Frequently Asking Questions) on the most important questions. Gradually this idea has developed into the separate book on the volume of the reference material.

The structure of this book contains detailed explanations addressed to a wide range of readers, which for the most part are not specialists in the field of thermoelectricity, of the basic ideas, important aspects of the practical application of thermoelectric microgenerators in the tasks in energy harvesting. I will be glad, if this book will serve as a reference tool in developing appropriate solutions.

Every Chapter was considered as a separate small manual, easy for an inexperienced user to find answers to their particular questions without resorting to detailed studying of the all material of the book. Therefore, the repetition takes place in formulas and expressions from Chapter to Chapter, for example.

Professionals may also say about excessively detailed clarification of “simple” basics of thermoelectricity. Please, these moments can be simply ignored.

But exactly simple and detailed explanations of important aspects of the application of thermoelectric microgenerators and was the purpose of this book. Many formulas, but their detailed explanations, I hope, will help users understand the issues of interest to them.

Chapter 1. Introduction

Creation of alternative, first of all renewable, power sources is the perspective direction of development of power engineering in the world now. The whole direction under definition “green power” (green energy) was appead.

It is the power engineering based on renewable natural sources of heat, causing the minimum damage to the environment, safe and eco-friendly. The big section of new power engineering is the direction of utilization of waste heat of the sources surrounding the person. Including, very interestingly, – thermal emission of a human body.

It should be noted that due to the nature of these heat sources, this is small power engineering. It received a general definition- energy harvesting.

To transform the energy of such energy sources it is possible to use devices that are based on different physical principles: electrodynamic, photovoltaic (small solar panels), piezoelectric and thermoelectric (microgenerators).

Every of the above types of energy converters has its advantages and disadvantages at the same time. And mostly they still compete for the tasks of energy harvesting at the level of concepts and laboratory projects.

Thermoelectric (TE) converters, thermoelectric generators (TEG) have several advantages over other mentioned:

– “omnivore” – convert any weak heat flow, all sources, including the heat of the human body.

– all-weather is not dependent on the daylight cycle (as solar panels).

– latent installations, due to the very small size, which is important for masking.

– highest reliability and long service life (over 25 years).

– no moving parts, no noise, do not need periodic maintenance.

In accordance with recent forecasts [1], [2], the direction of thermoelectric converters of thermal energy in the coming years will be developed into a large segment of the world market with a volume of about 1 billion dollars to 2024 (see Chapter 17).

Thermoelectric generators have, as we know, very low conversion efficiency, which is no more than 15—20% of the Carnot cycle efficiency. It is only few percent of conversion of energy from a heat source.

In energy harvesting applications, where natural heat sources have a slight temperature, generator is working at very low temperature differences and therefore, a fortiori, with very low levels of efficiency.

Although it is most commonly “waste” energy and low efficiency seems to be tolerated, but efficiency remains a key challenge of thermoelectric generators

In this regard, optimization of all aspects of device design and materials of thermoelectric generator is a very pressing task of extracting maximum effectiveness in their application.

The primary task is to develop an efficient thermoelectric material. This is a key component of the device as a whole, determining its main parameters. It is important to obtain material with not only high thermoelectric efficiency, but also sufficiently high mechanical properties. It is the task of the semiconductor materials science where there are their approaches, achievements and expectations.

The second task is the optimum design parameters to ensure the effectiveness of the device in general. It includes development of methods of constructing of optimal thermoelectric microgenerators with available thermoelectric materials.

This book discusses a range of aspects of optimization of thermoelectric microgenerators, their applications in directions of energy harvesting. Interrelations of various parameters are demonstrated, various nuances of operation of thermoelectric microgenerators are discussed.

We will discuss thermoelectric generators in whole, but meaning their miniature (micro – is a slang) types suitable for energy harvesting, small-scale power engineering.

Chapter 2. Physical basics

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.

Chapter 3. Optimization of electrical circuit

Preface. Thermoelectric generator transforms thermal energy and gives it to external electric circuit. Here coordination of elements of the electric circuit with parameters of the generator is essential for extraction of maximum power. In this Chapter questions of optimization of the electric circuit are considered.

Basic formulas

Simplified electrical circuit of a generator module is shown in Fig.3.1.

Figure. 3.1 Schematics of thermoelectric generator.

Maximum electric power transformed by a generator from heat source is defined by thermoEMF E and internal resistance ACR of the generator.

The thermoEMF E is found as the following

where α – thermoelectric coefficient (Seebeck coefficient) of pair of thermoelements n- and p-types; 2N – number of elements in generator module; ΔT -working temperature difference on generator module (ΔT=T_h-T_c).

If to short the electric circuit of the generator (R_load=0), the short-circuit current I_sc is

At short-circuit the power P_sc allocated in the electric circuit is maximal.

However, all this power will be converted into Joule heat in thermoelements of the generator. In fact, heat converted into electric current returns again into the heat. There is no useful work.

A generator performs useful work when converted power is given out to the external load that has electrical resistance different from zero (R_load> 0).

Then the working current I in the electric circuit (Fig. 3.1) is

Voltage U in the electric circuit, correspondingly

Formula of the net power P has the following form:

Maximum power

From equation (3.6) it follows that the generated power Pnonlinearly dependends on the load resistance R_load.

Figure. 3.2 Sample dependence of generator power (four variants by ACR) from the load resistance. Temperature difference 5°C.

This dependence has maximum power P_max when external load resistance R_load is equal to internal resistance ACR of the generator (R_load=ACR):

– At given internal resistance ACR of generator, there is an optimal load resistance R_load in terms of maximum power conversion.

– On the other side also follows from the equation (3.6) that at the given loading resistance R_load, the less internal resistance ACRof generator, then more power P (an example, Fig. 3.2 – an arrow up).

– At the same time, for such generator with smaller resistance ACR, there is even more optimal loading (Fig. 3.2 – arrow sideways) with smaller resistance R_load (Fig. 3.2 – red arrow down) which provides even more power.

Maximum power P_max conversion occurs when:

And corresponding maximum electric current I_max

Comparing (3.2) and (3.3) and (3.7) (3.8) respectively, we get:

– Maximum power P_max that can be obtained by thermoelectric generator is only one quarter of the maximum available transformed power by the generator (short circuit capacity).

– Electric current I_max through the module in maximum power output is half of short-circuit electric current I_sc (3.2).

Chapter 4. Optimization of efficiency

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.