Open access peer-reviewed chapter

Performance Analysis of Composite Thermoelectric Generators

Written By

Alexander Vargas Almeida, Miguel Angel Olivares‐Robles and Henni Ouerdane

Submitted: April 6th, 2016 Reviewed: October 4th, 2016 Published: December 21st, 2016

DOI: 10.5772/66143

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Abstract

Composite thermoelectric generators (CTEGs) are thermoelectric systems composed of different modules arranged under various thermal and electrical configurations (series and/or parallel). The interest for CTEGs stems from the possibility to improve device performance by optimization of configuration and working conditions. Actual modeling of CTEGs rests on a detailed understanding of the nonequilibrium thermodynamic processes at the heart of coupled transport and thermoelectric conversion. In this chapter, we provide an overview of the linear out-of-equilibrium thermodynamics of the electron gas, which serves as the working fluid in CTEGs. The force‐flux formalism yields phenomenological linear, coupled equations at the macroscopic level, which describe the behavior of CTEGs under different configurations. The relevant equivalent quantities—figure of merit, efficiency, and output power—are formulated and calculated for two different configurations. Our results show, that system performance in each of these configurations is influenced by combination of different materials and their ordering, that is, position in the arrangement structure. The primary objective of our study is to contribute new design guidelines for development of composite thermoelectric devices that combine different materials, taking advantage of the performance of each in proper temperature range and type of configuration.

Keywords

  • thermoelectric energy conversion
  • thermoelectric devices
  • thermodynamic constraints on energy production
  • thermoelectric figure of merit
  • thermoelectric optimization
  • efficiency

1. Introduction

Thermoelectric devices are heat engines, which may operate as generators under thermal bias or as heat pumps. For waste energy harvesting and conversion, thermoelectricity offers quite appropriate solutions, when temperature difference between heat source and heat sink is not too large. The physics underlying this type of energy conversion is based on the fundamental coupling between electric charge and energy that each mobile electron carries. The coupling strength is given by the so-called Seebeck coefficient or thermoelectric power [1]. The performance of thermoelectric system is usually assessed against the so-called figure of merit [2]: a dimensionless quantity denoted ZT, which combines the system's thermal and electrical transport properties, as well as their coupling at temperature T.

To qualify as a good thermoelectric, a material (semiconductor or strongly correlated) must boast the following characteristics: small thermal conductivity and large electrical conductivity on the one hand, so that, it behaves as a phonon glass—electron crystal system [2], and large thermoelectric power on the other hand. All these properties, which can be optimized, are temperature‐dependent, so they may take interesting values only in a particular temperature range. Improvement of thermoelectric devices in terms of performance and range of applications is highly desired, as their conversion efficiency is not size‐dependent, and the typical device does not contain moving parts. Much progress in the field of thermoelectricity has been achieved since the early days, which saw the pioneering works of Seebeck [3] and Peltier [4], but decisive improvement of the energy conversion efficiency, typically 10% of the efficiency of ideal Carnot thermodynamic cycle, is still in order.

In a general manner, transport phenomena are irreversible processes: the generation of fluxes within the system, upon which external constraints are applied, are accompanied by energy dissipation and entropy production [5]. Therefore, thermoelectric effects may be viewed as the result of the mutual interaction of two irreversible processes, electrical transport, and heat transport, as they take place [6]. Not too far from equilibrium, these transport phenomena obey linear phenomenological laws; so, general macroscopic description of thermoelectric systems is, in essence, phenomenological. Linear nonequilibrium thermodynamics provides the most convenient framework to characterize the device properties and the working conditions to achieve various operation modes.

A thermoelectric generator (TEG) is under the influence of two potentials: electrochemical (μe) and thermal (T); for each of which there is a flux and a force (as shown in examples of Table 1). If force is capable of getting the system to state close to equilibrium after perturbation, then the linear regime may characterize the situation, and approximation in this case is the linear response theory (LRT). In this chapter, we will review and discuss these issues considering thermoelectric system composed of different modules: we are particularly interested in the performance analysis of composite thermoelectric generator (CTEG). For this purpose, we will use a framework based on LRT, which allows to derive a set of linear coupled equations, which contain the system's thermoelectric properties: Seebeck coefficient (α), thermal conductivity (κ), and electrical resistivity (ρ), which are combined to form the effective transport parameters of CTEG in different thermal and electrical arrangements.

VariablesTransport coefficientExpression and name
Particle flux and densityDiffusion coefficientJN=Dn Fick's law
Energy flux and temperatureThermal conductivityJE=κT Fourier's law
Electrical current density and electric fieldElectrical conductivityJ=σEσϕ Ohm's law

Table 1.

Linear thermodynamic phenomenological laws—illustrative examples of forces and fluxes.

The present chapter is organized as follows: as thermoelectric conversion results primarily from nonequilibrium thermodynamic processes, a brief overview of some of the basic concepts and tools developed by Onsager [7, 8] and Callen [6] is very instructive, and we will see, that the force-flux formalism is perfectly suited for a description of thermoelectric processes [9]. Then, we will turn our attention to the physical model of composite thermoelectric generators, deriving and analyzing the figure of merit, the conversion efficiency and maximum output power. The chapter ends with a discussion and concluding remarks.

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2. Basic notions of linear nonequilibrium thermodynamics

2.1. Instantaneous entropy

The thermodynamic formulation presented here is that of Callen [10]. To each set of extensive variables associated to a thermodynamic system, there is a counterpart, that is, a set of intensive variables. The thermodynamic potentials are constructed from these variables. At the macroscopic scale, the equilibrium states of a system may be characterized by a number of extensive variables Xi macroscopic by nature. As one may assume that a macroscopic system is made of several subsystems, which may exchange matter and/or energy among themselves, the values taken by the variables Xi correspond to these exchanges, which occur as constraints are imposed and lifted. When constraints are lifted, relaxation processes take place until the system reaches a thermodynamic equilibrium state, for which a positive and continuous function S differentiable with respect to the variables Xi can be defined as follows:

S: XiS(Xi).E1

The function S, called entropy, is extensive; its maximum characterizes equilibrium as it coincides with the values that the variables Xi finally assume after the relaxation of constraints. Note, that extensive variables Xi differ from microscopic variables because of typical time scales, over which they evolve: the relaxation time of microscopic variables is extremely fast, while the variables Xi are slow in comparison. To put it simply, relaxation time toward local equilibrium τrelax is much smaller than the time necessary for the evolution toward the macroscopic equilibrium τeq. Hence, one may define an instantaneous entropy, S(Xi), at each step of the relaxation of the variables Xi. The differential of the function S is as follows:

dS=iSXi dXi=iFidXi,E2

where each quantity Fi is the intensive variable conjugate of the extensive variable Xi.

2.2. Thermodynamic forces and fluxes

Examples of well‐known linear phenomenological laws are given in Table 1. These laws establish a proportionality relationship between forces, which derive from potentials, and fluxes. Proportionality factors are transport coefficients, as fluxes are the manifestation of transport phenomena. Indeed, the system's response to externally applied constraints is transport, and when these are lifted, the system relaxes toward an equilibrium state.

Following the introductory discussion of this section, we now see in more detail how these forces and fluxes appear. The notions, which follow, are easily introduced considering the case of a discrete system like, for instance, two separate homogeneous systems initially prepared at two different temperatures and then put in thermal contact through a thin diathermal wall. The thermalization process triggers a flow of energy from one system to the other. So, assume now an isolated system composed of two weakly coupled subsystems, to which an extensive variable taking the values Xi and Xi, is associated. One has Xi+Xi=Xi(0)=constant and S(Xi)+S(Xi)=S(Xi(0)). Then, the equilibrium condition reads:

S(0)Xi|Xi(0)=(S+S)Xi dXi|Xi(0)=SXiSXi=FiFi=0,E3

as it maximizes the total entropy. Therefore, if the difference Fi = Fi – F′i is equal to zero, the system is in equilibrium; otherwise, irreversible process takes place and drives the system to equilibrium. The quantity Fi is the affinity or generalized force allowing the evolution of the system toward equilibrium. Further, we also introduce the variation rate of the extensive variable Xi, as it characterizes the response of the system to the applied force:

Ii=dXidt.E4

The relationship between affinities and fluxes characterizes the changes due to irreversible processes: non‐zero affinity yields non‐zero conjugated flux, and a given flux cancels, if its conjugate affinity cancels.

In local equilibrium, fluxes depend on their conjugate affinity, but also on the other affinities; so, we see, that there are direct effects and indirect effects. Therefore, the mathematical expression for the flux Ii, at a given point in space and time (r,t), shows a dependence on the force Fi, but also on the other forces Fj≠i:

Ii(r,t)Ii(F1,F2,).E5

Close to equilibrium Ii(r,t) can be written as Taylor expansion:

Ik(r,t)=jIkFj Fj+12!i,j2IkFiFj FiFj+=kLjkFk+12i,jLijkFiFj+.E6

The quantities Ljk are the first-order kinetic coefficients; they are given by the equilibrium values of intensive variables Fi. The matrix [L] of kinetic coefficients characterizes the linear response of the system. Onsager put forth the idea that there are symmetry and antisymmetry relations between kinetic coefficients [6, 7]: the so-called reciprocal relations must exist in all thermodynamic systems, for which transport and relaxation phenomena are well described by linear laws. The main results can be summarized as follows [5]: (1) Onsager's relation: Lik=Lki; (2) Onsager‐Casimir relation: Lik=εiεkLki; (3) generalized relations: Lik(H,Ω)=εiεkLki(H,Ω), where H and Ω denote, respectively, the magnetic field and angular velocity associated with Coriolis field; the parameters εi denote the parity with respect to time reversal: if the quantity studied is invariant under time reversal transformation, it has parity +1; otherwise, this quantity changes sign, and it has parity 1.

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3. Thermoelectric forces and fluxes

3.1. Coupled fluxes of heat and electrical charges

The thermoelectric effect results from the mutual interference of two irreversible processes occurring simultaneously in the system, namely heat transport and charge carriers transport. The Onsager force-flux derivation is obtained from the laws of conservation of energy and matter:

IE=IQ+μeIN,E7

where IE is energy flux, IQ is heat flux, and IN is particle flux. Each flux is the conjugate variable of its potential gradient. Considering the electron gas, correct potentials for particles and energy are μe/T and 1/T, and related forces are as follows: FE=(1/T) and FN=(μe/T), where μe is the electrochemical potential [1]. Then, the linear coupling between forces and fluxes may simply be described by a linear set of coupled equations involving the so-called kinetic coefficient matrix [L]:

(INIE)=(LNNLNELENLEE)((μe/T)(1/T)),E8

where LNE=LEN. Now, to treat properly heat flow and electrical current, it is more convenient to consider IQ instead of IE. Using IE=IQ+μeIN, we obtain:

(INIQ)=(L11L12L21L22)((μe/T)(1/T))E9

with L12=L21. Since (μe/T)=μe(1/T)1/T(μe), then heat flow and electrical current read:

(INIQ)=(LNNLNEμeLNNLNEμeLNN2LNEμe+LEE+μe2LNN)((μe/T)(1/T))E10

with the following relationship between kinetic coefficients:

L11=LNN,E11
L12=LNEμeLNN,E12
L22=LEE2μeLEN+μe2LNN.E13

Note, that since electric field derives from electrochemical potential, we also obtain:

E=1e μe.E14

3.2. Thermoelectric transport coefficients

The thermoelectric transport coefficients can be derived from the expressions of electron and heat flux densities depending on applied thermodynamic constraints: isothermal, adiabatic, electrically open or closed circuit conditions. Under isothermal conditions, electrical current may be written in the form:

IN=L11T(μe).E15

This is expression of Ohm's law, since with I=eIN we obtain the following relationship between electrical current density and electric field:

eIN=I=eL11T(μe)=σT((μe)e)=σTE,E16

which contains the definition for isothermal electrical conductivity expressed as follows:

σT=e2TL11.E17

Now, if we consider the heat flux density in the absence of any particle transport or, in other words, under zero electrical current, we get:

IN=0=L11(1T(μe))+L12(1T),E18

so that, the heat flux density under zero electrical current, IQI=0, reads:

IQI=0=1T2[L21L12L11L22L11](T).E19

This is Fourier's law, with thermal conductivity under zero electrical current given by:

κI=1T2[L11L22L21L12L11].E20

We can also define the thermal conductivity κE under zero electrochemical gradient, that is, under closed circuit conditions:

IQE=0=L22T2(T)=κE(T).E21

It follows, that thermal conductivities κE and κI are simply related through:

κE=Tα2σT+κI.E22

As thermal and electric processes are coupled, the actual strength of the coupling is given by Seebeck coefficient:

α1e(μe)(T)=1eTL12L11,E23

defined as the ratio of two forces that derive from electrochemical potential for one and from temperature for the other.

The analysis and calculations developed above allow to establish complete correspondence between kinetic coefficients and transport parameters:

L11=σTe2T,E24
L12=σTSIT2e2,E25
L22=T3e2σTSI2+T2κI,E26

so that, expressions for electronic current and heat flow may take their final forms:

IN=σTe2T((μe)T)+σTSIT2e2((1T)),E27
IQ=σTSIe2T2((μe)T)+[T3e2σTSI2+T2κI]((1T)).E28

Since I=eIN, it follows that:

I=σTEσTSIe(T),E29

from which we obtain:

E=ρTI+α(T),E30

where ρT is the isothermal conductivity. This is a general expression of Ohm's law.

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4. Formulation of physical model for thermoelectric generators

For TEG performance analysis, we have applied the model given by [11, 12], associating thermal circuit for heat transport and electrical circuit for charge carriers transport, see Figure 1.

Figure 1.

Circuit model for thermoelectric generator, red (thermal circuit), blue (electrical circuit), where ΔV, voltage; R, electrical resistance; K, thermal conductance; Tcold, temperature of the cold side; Thot, temperature of the hot side; ΔT, temperature difference; α, Seebeck coefficient; and T, average temperature.

Electrical current and heat flow, Ii and IQi, are functions of generalized forces [11], related to differences in voltage, ΔVi, and temperature, ΔTi, of thermoelectric generator:

(IiIQi)=(1/Riαi(1/Ri)αi(1/Ri)Tαi2(1/Ri)T+Ki)(ΔViΔTi),E31

where T is average temperature.

In this model, TEG is characterized by its internal electrical resistance, R, thermal conductance under open electrical circuit condition, K, and Seebeck coefficient, α. Physical conditions assumed for this model are as follows: (i) thermoelectric properties are independent on temperature, (ii) the only electrical resistance taken into account is that of the legs, (iii) there is no thermal contact resistance between the ends of the legs and heat source, and (iv) in this model, doping of the legs (p‐ or n‐type) is not taken into account, so that, TEG can be seen as only one leg.

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5. Heat balance equation

The heat balance in TEG is governed by the following equations; basically, there are two extreme points: one in contact with the heat source (incoming point):

Qin=αThI12RinI2+K(ThTc),E32

the other point is point, where heat is rejected:

Qre=αTcI+12RinI2+K(ThTc),E33

where αTiI is Seebeck heat, 12RinI2 is Joule heat, and K(ThTc) is thermal conduction heat; in terms of these quantities, electrical power is defined as:

Pelectrical=QinQre=αI(ThTc)RI2.E34

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6. Composite thermoelectric generator (CTEG)

We consider a composite thermoelectric generator, which is composed of three thermoelectric elements (TEGs) in different configurations, each TEG is made of a different thermoelectric material, see Figure 2. The configurations considered are as follows: (A) two‐stage thermally and electrically connected in series (TES‐CTEG); (B) segmented TEG, conventional TEG, thermally and electrically connected in parallel (PSC‐CTEG). Also, we consider the effect of the arrangement of the materials on the performance of the composite system. Thus, for each of the systems (A, B), we have the following arrangements:

  1. TEG 1 = material one, TEG 2 = material two, TEG 3 = material three;

  2. TEG 1 = material three, TEG 2 = material one, TEG 3 = material two;

  3. TEG 1 = material two, TEG 2 = material three, TEG 3 = material one.

Figure 2.

Composite thermoelectric generator (CTEG) (components are three TEGs, each made of different material).

In the following sections, we analyze and show results for CTEG by applying the conditions listed above in order to contribute to development of new design guidelines for thermoelectric systems with news architectures and even to provide some clues to the search for new physical conditions in the area of science and engineering of thermoelectric materials.

6.1. Formulation of equivalent figure of merit for CTEG

To analyze CTEG performance, equivalent quantities are defined, which contain the overall contribution of individual properties of each TEG building up composite system. These quantities are as follows: equivalent Seebeck coefficient (αeq), equivalent electrical resistance (Req), and equivalent thermal conductance (Keq), in terms of which it is possible to have equivalent figure of merit (Zeq). We show the impact of the configuration of the system on Zeq for each of configuration (A, B) listed in Section  6, and we suggest the optimum configuration. In order to justify the effectiveness of the equivalent figure of merit, the corresponding efficiency has been calculated for each configuration.

6.1.1. Two‐stage thermally and electrically connected in series

Schematic view of this system is shown in Figure 3. The first stage (bottom stage) consists of two different thermoelectric modules (TEG), while the top stage consists of only one TEG. Each of components is characterized by proper thermoelectric properties (αi,Ri,Ki) [13].

Figure 3.

Schematic representation of thermoelectric system composed of two stages thermally and electrically connected in series (TES‐CTEG). (a) Equivalent circuit for TES‐CTEG, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, αi is the Seebeck coefficient, T is the average temperature, Rload is the load; (b) practical device related to TES‐CTEG, where ni is the ith n‐type material, pi is the ith p‐type material.

Using Eq. (31), the heat flux within any segment in TEGs is:

IQi=αiTIi+KiΔTi.E35

By continuity of the heat flux through the interface between stages of TES‐CTEG:

IQ1=IQ2+IQ3E77
K1(ThotTi)+α1TI=K2(TiTcold)+α2TI+K3(TiTcold)+α3TI,E36

from which we obtain the average temperature at the interface between stages [12]:

Ti=K1Thot+(K2+K3)Tcold+(α1α2α3)TIK1+K2+K3.E37

Since all components are electrically connected in series, the total voltage is given by:

ΔV=α1(ThotTi)α2(TiTcold)α3(TiTcold)+(R1+R2+R3)I,E38

substituting the value of Ti in the last equation, we have:

ΔV=[(α2+α3)K1α1K2α1K3K1+K2+K3][ThotTcold]++[(α1α2α3)2TK1+K2+K3+(R1+R2+R3)]I.E39

From Eq. (39), we identified the equivalent series Seebeck coefficient, αeqTES, and equivalent series electrical resistance, ReqTES, as follows:

αeqTES=(α2+α3)K1α1K2α1K3K1+K2+K3,E40
ReqTES=R1+R2+R3+Rrelax,E41

where

Rrelax=(α1α2α3)2TK1+K2+K3.E42

Considering open circuit condition for the system, I=0, we find, that equivalent thermal conductance for the whole system:

KeqTES=K1(K2+K3)K1+K2+K3.E43

We define the figure of merit in terms of equivalent quantities [12]:

Zeq=αeq2ReqKeq.E44

By replacing the results obtained in Eqs. (40)(43), we have:

ZeqTES=[(α2+α3)K1α1K2α1K3K1+K2+K3]2[(α1α2α3)2TK1+K2+K3+(R1+R2+R3)][K1(K2+K3)K1+K2+K3].E45

6.1.2. Segmented TEG‐conventional TEG thermally and electrically connected in parallel

In this section, we consider CTEG system, which is composed by segmented TEG and conventional TEG. These TEGs are thermally and electrically connected in parallel (PSCCTEG), as is shown in Figure 4.

Figure 4.

Schematic representation of (PSC‐CTEG). (a) Thermal‐electrical circuit, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, αi is Seebeck coefficient, T is the average temperature, Rload is the load resistance, TM is the intermediate temperature; (b) structure design, where ni is the ith n‐type material, pi is the ith p‐type material.

In the composite system, there are two currents, Is for TEG 1 and TEG 2, Ic for TEG 3. If the electrical current is conserved, then [13]:

Ieq=Is+Ic.E46

The heat flux through the whole system is the sum of the heat flux flowing through segmented generator and the heat flux in conventional generator. Thus:

IQeq=IQs+IQc.E47

To obtain the equivalent electrical resistance, ReqPSC, using Eq. (45), the isothermal condition, ΔT=0, is required. Under this condition, we recover the usual expression of equivalent electrical resistance for an ohmic circuit. Thus, we get:

ReqPSC=RsRcRs+Rc,E48

where Rc is the internal electrical resistance of conventional TEG and Rs is the electrical resistance of the segmented TEG:

Rs=R1+R2+RrelaxE49

and

Rrelax=(α1α2)2TK1+K2.E50

Assuming the condition of closed circuit, ΔV=0, and applying Eq. (45), we have for equivalent Seebeck coefficient [13]:

αeqPSC=Rcαs+RsαcRs+Rc,E51

where

αs=K2α1+K1α2K1+K2.E52

To determine equivalent thermal conductance, Keq, we use the open circuit condition, Ieq=0, which is satisfied when Is=Ic=I, and, due to preservation of heat flow:

KeqPSC=Ks+Kc+(αsαc)TIΔT,E53

where

Ks=K2K1K1+K2.E54

Under open circuit condition, Ieq=0, so that, ΔV=αeqΔT. Applying this result, we have for I:

I=1Rs+Rc(αsαc)ΔT.E55

Using this last result in Eq. (53), we have:

KeqPSC=Ks+Kc+(αsαc)2T1Rs+Rc.E56

Now, we can write the figure of merit for this PSC‐CTEG system:

ZeqPSC=αeqPSC2ReqPSCKeqPSC.E57

Using the results obtained in Eqs. (48), (51), and (56), we have:

ZeqPSC=(Rcαs+RsαcRs+Rc)2[RsRcRc+Rs][Ks+Kc+(αsαc)2T1Rs+Rc].E58

6.1.3. Analysis of equivalent figure of merit for composite systems

Equivalent figure of merit (Zeq) is calculated in this section for TES and PSC systems. For performing calculations, the best known thermoelectric materials for commercial applications have been selected: BiTe, PbTe, and SiGe (experimental data taken from Refs. [1416] have been used as numerical values of thermoelectric parameters). It has also been calculated equivalent maximum efficiency (ηeqmax).

It is important to emphasize, that in this study we analyzed also the behavior of Zeq and ηeq, when ordering of materials in the composite system changes (i.e., change its position).

Table 2 shows, that performance of composite system is affected by the type of thermal and electrical connection, as well as ordering of materials. For example, PSC case reaches the highest value of Zeq and ηeq with the ordering TEG 1 = PbTe, TEG 2 = SiGe, TEG 3 = BiTe.

TEG 1TEG 2TEG 3ZeqTESZeqPSCηeqTESηeqPSC
BiTePbTeSiGe0.0004330.0004630.0799360.084392
PbTeSiGeBiTe0.0005080.0019050.0910450.224724
SiGeBiTePbTe0.0005740.0006220.1002170.106658

Table 2.

Numerical values of Zeq and ηeq in each equivalent thermoelectric system for different arrangements of the TE materials.

To analyze the performance of the composite system, with each of the different orderings, we have built plots (Figure 5a, b), that show variation of equivalent figure of merit with Seebeck coefficients ratio αj/αi.

Figure 5.

(a) ZeqTES vs. ratio α3/α2, maintaining α1 and α2 constant; (b) ZeqPSC vs. ratio, α2/α1, maintaining α1 and α3 constant.

6.2. Maximum efficiency

The figure of merit measures the performance of materials in thermoelectric device, but, if we measure the performance when the TEG is operating under a temperature difference, then the value called thermal efficiency quantifies the ability of TEG to utilize the supplied heat effectively.

From thermodynamics, Carnot cycle thermal efficiency is known as:

ηCarnot=ThotTcoldThot.E59

In terms of ηCarnot and Zeq, the maximum efficiency of thermoelectric device is defined by the next equation (with thermoelectric properties (α,R,κ) constant with respect to temperature) [2]:

ηmaxj=ΔTThot1+ZeqjT11+ZeqjT+TcoldThot,E60

where Zeqj with j=TES,PSC is given by Eqs. (45) and (58), respectively. Thus, we have for the maximum efficiency of TES‐CTEG system:

ηeqTES=ΔTThot1+ZeqTEST11+ZeqTEST+TcoldThot.E61

For the maximum efficiency of PSC‐CTEG system:

ηeqPSC=ΔTThot1+ZeqPSCT11+ZeqPSCT+TcoldThot.E62

Our results are shown in Figure 6.

Figure 6.

(a) ηmaxTES vs. ratio α3/α2. (b) ηmaxPSC vs. ratio α2/α1.

Plots in Figure 6 show typical dependences of CTEGs efficiency on the properties of component materials. The presented results of maximum efficiency reached by the thermoelectric device approach the limit established by Bergman's theorem for composite materials [17]: the efficiency of composite thermoelectric system cannot be greater than the module's component with highest efficiency.

The maximum efficiencies achieved by studied CTEGs, see plots in Figure 6, are of similar order of magnitude as CTEG systems investigated in some works, e.g. [18], where reported efficiencies from 17 to 20%.

6.3. CTEG: maximum output power

We analyze also the maximum output power of the studied CTEG system, again, assuming configurations and physical conditions shown in Section  6. The obtained results have been compared with some analytical work and numerical simulations.

For the case of thermoelectric generator connected to load resistor Rload (Figure 7), the power delivered to Rload is given by the following equation [19]:

Poutm=[α(THTC)]2m(m+1)2R,E63

Figure 7.

Thermal‐electrical circuit for TEG delivering power to the load, where ΔV is the voltage, Ri is the electrical resistance, Ki is the thermal conductance, Tcold is the temperature of the cold side, Thot is the temperature of the hot side, ΔT is the temperature difference, αi is Seebeck coefficient, Rload is the load resistance.

The strategy consists of defining the optimal ratio m=Rload/R and then by applying the method of maximizing variable to obtain the value of the load resistance, which maximizes power. It yields Rload=R, and in this case, the maximum output power is:

Pmax=α2(THTC)24R.E64

6.3.1. Formulation of output power for CTEG

Here, in similar way as in previous sections, formulating of output power will be considered using thermoelectric equivalent quantities, see Sections 6.1.1, 6.1.2 [20]. Thus, using Eqs. (62, 63) in terms of αeq and Req, we can write:

Pouteqm=[αeq(THTC)]2Reqm(m+1)2,E65
Peqmax=αeq2(THTC)24Req.E66

Application of the formalism described above Eqs. (64, 65) give the output power for each configuration as follows.

Two‐stage thermoelectric system connected in series:

Pouteq(TESCTEG)m=([(α2+α3)K1α1K2α1K3K1+K2+K3](THTC))2[R1+R2+R3+(α1α2α3)2T¯K1+K2+K3]m(m+1)2E67

and the maximum power is given by:

Peq(TESCTEG)max=([(α2+α3)K1α1K2α1K3K1+K2+K3](THTC))24[R1+R2+R3+(α1α2α3)2T¯K1+K2+K3].E68

Segmented‐conventional thermoelectric system in parallel (PSC‐CTEG):

Pouteq(PSC)m=(Rc[K2α1+K1α2K1+K2]+[R1+R2+[(α1α2)2T¯K1+K2]]αc)2(THTC)2[[R1+R2+(α1α2)2T¯K1+K2]Rc(Rs+Rc)]m(m+1)2,E69

and using Eqs. (51, 48) and Eq. (66), the maximum power of this system obtained is:

Peq(PSC)max=14(Rc[K2α1+K1α2K1+K2]+[R1+R2+[(α1α2)2T¯K1+K2]]αc)2(THTC)2[[R1+R2+(α1α2)2T¯K1+K2]Rc(Rs+Rc)].E70

6.3.2. Analysis of output power

We show the behavior of the electrical output power delivered in each CTEG configuration using the data of Section 6.1.3. Figure 8, panels (a) and (b), shows the output power as a function of the ratio between the electrical resistance of the load and the electrical resistance of the thermoelectric system m=RloadR.

Figure 8.

(a) Plot for output power delivered by TES‐CTEG system as function of ratio Rload/R; combination, producing the highest output power, is (TEM 1 = SiGe, TEM 2 = BiTe, TEM 3 = PbTe); (b) plot for output power delivered by the PSC‐CTEG system as function of ratio Rload/R; combination, producing the highest output power, is (TEM 1 = PbTe, TEM 2 = SiGe, TEM 3 = BiTe).

Plots in Figure 8 show, that similarly to the equivalent figure of merit and equivalent efficiency (Sections 6.1.3 and 6.2), the output power of a composite system is also influenced by the type of thermal‐electrical connection and ordering of materials, and again, PSC‐CTEG case shows the highest performance quantified by generated output power. This result is consistent with the results obtained by Vargas‐Almeida et al. [20], and the behavior of the output power for each array of equivalent TES‐CTEG is consistent with the results obtained by Apertet et al. [11]. Table 3 shows the comparison of maximum output power values for different types of connections and possible arrangements.

TEG 1TEG 2TEG 3PmaxeqTESPmaxeqPSC
BiTePbTeSiGe1.276184.34854
PbTeSiGeBiTe1.6556312.2877
SiGeBiTePbTe2.229684.28067

Table 3.

Numerical values of maximum output power, in terms of equivalent amounts of each compound of CTEG, evaluated for each order of building TEGs.

To confirm the validity of our results, we have built plots for CTEG output power using ΔT values of some work: [21] (experimental) and [22, 23] (analytical). Plots in Figure 9 were produced using the temperature difference of Ref. [21].

Figure 9.

Output power POuteqPSC delivered by composed PSC system vs ratio Rload/R. At temperature difference ΔT = 20 K, curves behave similarly to the plots shown in Ref. [21]. This figure is consistent with the result obtained by Abdelkefi [21]. Our results have also been compared to other published works [22, 23].

The results for comparisons with [22, 23] are shown in [24].

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7. Opportunity analysis to improve CTEG design by varying configuration

In this section, we generalize results shown in previous sections by formulating corollary and including some results with realistic approaches, for example, consideration of contact thermal conductance. To achieve this goal, we combine physical conditions imposed in Section  6 with the next options: (1) the whole system is formed of the same thermoelectric material (α1,K1,R1=α2,K2,R2=α3,K3,R3); (2) the whole system is constituted by only two different thermoelectric materials (αi,Ki,Ri=αj,Kj,Rjαl,Kl,Rl), where i,j,l can be 1, 2 or 3, [25].

7.1. Case A: homogeneous thermoelectric properties, configuration effect

We consider configurations of CTEG with the same thermoelectric material, (α1,K1,R1)=(α2,K2,R2)=(α3,K3,R3). In this case, equivalent figure of merit Zeqh is as follows,

for homogeneous TES‐CTEG:

ZeqTESh=(4αi3)2((αi)2T3Ki+3Ri)(2Ki3),E71

for homogeneous PSC‐CTEG:

ZeqPSCh=(αi)2(2Ri3)(3Ki2),E72

where i=(BiTe,PbTe,SiGe).

Table 4 shows numerical values of equivalent figure of merit Zeqh obtained by us for CTEG with considered configurations.

MaterialZeqTEShZeqPSCh
BiTe0.002121330.00305269
PbTe0.000551090.000657238
SiGe0.0002875620.00033337

Table 4.

Numerical values of Zeqh, for each of three configurations with different materials.

It is important to note, that fulfillment condition TEG 1 = TEG 2 = TEG 3 evidences the fact, that although composite system is made of single material, the figure of merit reaches different values depending on type of connection.

7.2. Case B: two different materials in CTEG

CTEG is made of two same materials and the other one different. Thus, two TEGs include same semiconductor material and the other one different semiconductor material. In this case, equivalent figure of merit Zeqh is as follows, for heterogeneous TES‐CTEG:

ZeqTESInh=((αj+αl)Kiαi(Kj+Kl)Ki+Kj+Kl)2((αiαjαl)2TKi+Kj+Kl+Ri+Rj+Rl)(Ki(Kj+Kl)Ki+Kj+Kl),E73

for heterogeneous PSC‐CTEG:

ZeqPSCInh=(Rl(Kjαi+KiαjKi+Kj)+(Ri+Rj+(αiαj)2TKi+Kj)αlRi+Rj+Rl+(αiαj)2TKi+Kj)2(Rl(Ri+Rj+(αiαj)2TKi+Kj)Rl+Ri+Rj+(αiαj)2TKi+Kj)(KjKiKi+Kj+Kl+((Kjαi+Kiαj)Ki+Kjαl)2TRi+Rj+Rl+(αiαj)2TKi+Kj).E74

Eqs. (72) and (73) are applied with condition TEGi=TEGj, that is, two TEGs are made of the same thermoelectric material, and third TEGl is made of different thermoelectric material. Thus, we have three possibilities (TEG 1 = TEG 2 ≠ TEG3, TEG 1 = TEG 3 ≠ TEG 2, TEG 2 = TEG 3 ≠ TEG 1) for each configuration [25]. Note that, each arrangement has six different combinations, if the cyclical order of the material is taken into account.

The behavior of the equivalent figure of merit as a function of the ratio of the thermal conductivities of the two component materials is shown in Figure 10. This step is important, because it shows numerical values, that CTEG maker must meet for both component materials to reach the highest value of Zeq.

Figure 10.

(a) Equivalent figure of merit for heterogeneous TES‐CTEG, under condition TEG 2 = TEG 3 ≠ TEG 1, the highest numerical value is corresponding to TEG 2 = TEG 3 = BiTe ≠ TEG 1 = PbTe; (b) Equivalent figure of merit for heterogeneous PSC‐CTEG under condition TEG 1 = TEG 2 ≠ TEG 3, the highest numerical value is corresponding to TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe.

Table 5 shows maximum values of equivalent figure of merit of CTEG with material arrangements in every configuration, when TEGi=TEGjTEGl.

TEG 1TEG 2 = TEG 3ZeqTESmaxInh
BiTePbTe0.00168734
BiTeSiGe0.0012388
PbTeBiTe0.00273649
PbTeSiGe0.00118802
SiGeBiTe0.00150947
SiGePbTe0.000994534
TEG 3TEG 1 = TEG 2ZeqPSCmaxInh
BiTePbTe0.0055567
BiTeSiGe0.00325841
PbTeBiTe0.00445846
PbTeSiGe0.0011157
SiGeBiTe0.00392902
SiGePbTe0.00172358

Table 5.

Maximum values of equivalent figure of merit of CTEG with material arrangements in every configuration, when TEMi=TEMjTEMl.

Table 6 shows each configuration with the most efficient material arrangements for every TEG.

SystemArrangement
TESTEG 2 = TEG 3 = BiTe ≠ TEG 1 = PbTe
PSCTEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe

Table 6.

Most efficient material arrangements TEGi=TEGjTEGl for TES and PSC‐CTEG systems.

Results show again, that the most efficient system of three configurations is PSC with corresponding material arrangement, namely TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe; see Figure 11.

Figure 11.

Optimal configuration corresponds to PSC‐CTEG with arrangement TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe.

Again, it is important to note, that this result proves, that although the performance of composite systems is affected by combination of different materials, it is affected by the position of such materials in the system structure as well.

7.3. Performance analysis with realistic approximations

The results of the previous sections have argued, that application of output power and efficiency as quantities to measure performance of the system is reasonable; however, in this new section, we extend the analysis of these quantities using realistic considerations. Numerical treatment is performed with ZeqPSCInh.

7.3.1. Maximum output power

In the following analysis, we consider thermoelectric modules as isolated units only. Although this is usually considered as an ideal situation, such an approach is useful to study the performance of materials in the composite system. However, for real applications, modules must be coupled to heat exchangers, which produces thermal conductance of contact (Kc) at the coupling points. This affects system performance and reflects in the output power. Here, the maximum output power is calculated using the maximum value of the equivalent figure of merit (ZeqPSCInh) [23]:

PmaxPSC=(KcΔT)24(KI=0+Kc)T¯ZeqPSCInhT¯1+ZeqPSCInhT¯+Kc/KI=0.E75

Figure 12a shows maximum output power values for PSC system as function of ratio KI=0/Kc, that is, in terms of internal thermal conductance KI=0 and contact thermal conductance Kc, under condition TEG 1 = TEG 2 ≠ TEG 3.

Figure 12.

(a) Maximum power of PSC system under condition TEG 1 = TEG 2 ≠ TEG 3, the highest numerical value corresponding to arrangement TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe. (b) Contour plot: efficiency of PSC system under condition TEG 1 = TEG 2 ≠ TEG 3, assuming the maximum value of efficiency ZeqPSCInh for arrangement TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe.

7.3.2. Efficiency

To calculate the efficiency of PSC systems with TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe arrangement, we applied the equation:

ηeqPSCInh=ΔTTH1+ZeqPSCInhT¯11+ZeqPSCInhT¯+TCTH.E76

Finally, for an ideal TEG, that is, without taking into account heat exchangers, we can analyze TEG efficiency considering intrinsic thermal conductances ratio (K3/K1,2) and electrical resistances ratio (R3/R1,2).

Figure 12b shows contour plot for different values of ηeqPSCInh as function of ratios, K3/K1,2 and R3/R1,2. We can see, that the range of optimal values for the best efficiency of PSC—CTEG lies in intervals 0.1–1.0 and 0.1–0.5 for K3/K1,2 and R3/R1,2, respectively. It is remarkable, that thermal conductances ratio shows a wider range of good values in comparison with electrical resistances ratio, which shows narrower range.

7.3.3. Corollary: maximum efficiency Zeq for composite thermoelectric generator

Based on the progress presented in this paper, we have been formulated the following corollary: two features of design must be met to ensure the maximum value of Zeq of CTEG:

  • If the material is the same in all components, CTEG reaches the maximum value of Zeq with a specific type of thermal—electrical connection.

  • When components of TEGs composing CTEG are made of different materials, TEGiTEGjTEGl where i;j;l can be 1, 2, or 3; then, for a given thermal‐electrical connection, there exists an optimal arrangement of thermoelectric materials for which Zeq is maximum.

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8. Conclusions

The main objective of this chapter was to present new ideas for designing more complex thermoelectric systems taking into account the effects of electrical and thermal connection, combination of different materials and ordering of materials in CTEG. For this purpose, we considered the framework of linear response theory for nonequilibrium thermodynamic processes, and we used the constant parameter model. Through the definition of equivalent parameters αeq, Req, and Keq, we have shown the significant impact of these parameters on the system's properties, which characterize the performance of CTEG, namely Zeq, ηeq, and Peq. The numerical results show, that the optimal configuration for CTEG considered here is the thermal and electrical connection in parallel with arrangement (PbTe, SiGe and BiTe). For completeness, we have shown the effect of contact thermal conductance on the parameter ZeqPSCInh for the most efficient case—PSC‐CTEG system, in terms of both ratio K3/K1,2 (intrinsic thermal conductances) and R3/R1,2 (intrinsic electrical resistance). Although in this study, the composite system is restricted to only three components, the results can be generalized to systems consisting of N modules, either analytically by extension of the mathematical model or through numerical simulations; guidelines for this purpose are provided by the corollary 7.3.3.

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Written By

Alexander Vargas Almeida, Miguel Angel Olivares‐Robles and Henni Ouerdane

Submitted: April 6th, 2016 Reviewed: October 4th, 2016 Published: December 21st, 2016