Linear thermodynamic phenomenological laws—illustrative examples of forces and fluxes.
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.
- thermoelectric energy conversion
- thermoelectric devices
- thermodynamic constraints on energy production
- thermoelectric figure of merit
- thermoelectric optimization
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 . The performance of thermoelectric system is usually assessed against the so-called figure of merit : a dimensionless quantity denoted , which combines the system's thermal and electrical transport properties, as well as their coupling at temperature .
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 , 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  and Peltier , 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 . 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 . 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 () and thermal (); 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 (), thermal conductivity (), and electrical resistivity (), which are combined to form the effective transport parameters of CTEG in different thermal and electrical arrangements.
|Variables||Transport coefficient||Expression and name|
|Particle flux and density||Diffusion coefficient||Fick's law|
|Energy flux and temperature||Thermal conductivity||Fourier's law|
|Electrical current density and electric field||Electrical conductivity||Ohm's law|
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  is very instructive, and we will see, that the force-flux formalism is perfectly suited for a description of thermoelectric processes . 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.
2. Basic notions of linear nonequilibrium thermodynamics
2.1. Instantaneous entropy
The thermodynamic formulation presented here is that of Callen . 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 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 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 differentiable with respect to the variables can be defined as follows:
The function , called entropy, is extensive; its maximum characterizes equilibrium as it coincides with the values that the variables finally assume after the relaxation of constraints. Note, that extensive variables 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 are slow in comparison. To put it simply, relaxation time toward local equilibrium is much smaller than the time necessary for the evolution toward the macroscopic equilibrium . Hence, one may define an instantaneous entropy, , at each step of the relaxation of the variables . The differential of the function is as follows:
where each quantity is the intensive variable conjugate of the extensive variable .
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 and , is associated. One has and . Then, the equilibrium condition reads:
as it maximizes the total entropy. Therefore, if the difference = 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 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 , as it characterizes the response of the system to the applied force:
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 , at a given point in space and time , shows a dependence on the force , but also on the other forces :
Close to equilibrium can be written as Taylor expansion:
The quantities are the first-order kinetic coefficients; they are given by the equilibrium values of intensive variables . The matrix 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 : (1) Onsager's relation: ; (2) Onsager‐Casimir relation: ; (3) generalized relations: , where and denote, respectively, the magnetic field and angular velocity associated with Coriolis field; the parameters denote the parity with respect to time reversal: if the quantity studied is invariant under time reversal transformation, it has parity ; otherwise, this quantity changes sign, and it has parity .
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:
where is energy flux, is heat flux, and is particle flux. Each flux is the conjugate variable of its potential gradient. Considering the electron gas, correct potentials for particles and energy are and , and related forces are as follows: and , where is the electrochemical potential . 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 :
where . Now, to treat properly heat flow and electrical current, it is more convenient to consider instead of . Using , we obtain:
with . Since , then heat flow and electrical current read:
with the following relationship between kinetic coefficients:
Note, that since electric field derives from electrochemical potential, we also obtain:
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:
This is expression of Ohm's law, since with we obtain the following relationship between electrical current density and electric field:
which contains the definition for isothermal electrical conductivity expressed as follows:
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:
so that, the heat flux density under zero electrical current, , reads:
This is Fourier's law, with thermal conductivity under zero electrical current given by:
We can also define the thermal conductivity under zero electrochemical gradient, that is, under closed circuit conditions:
It follows, that thermal conductivities and are simply related through:
As thermal and electric processes are coupled, the actual strength of the coupling is given by Seebeck coefficient:
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:
so that, expressions for electronic current and heat flow may take their final forms:
Since , it follows that:
from which we obtain:
where is the isothermal conductivity. This is a general expression of Ohm's law.
4. Formulation of physical model for thermoelectric generators
Electrical current and heat flow, and , are functions of generalized forces , related to differences in voltage, , and temperature, , of thermoelectric generator:
where is average temperature.
In this model, TEG is characterized by its internal electrical resistance, , thermal conductance under open electrical circuit condition, , and Seebeck coefficient, . 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.
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):
the other point is point, where heat is rejected:
where is Seebeck heat, is Joule heat, and is thermal conduction heat; in terms of these quantities, electrical power is defined as:
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:
TEG 1 = material one, TEG 2 = material two, TEG 3 = material three;
TEG 1 = material three, TEG 2 = material one, TEG 3 = material two;
TEG 1 = material two, TEG 2 = material three, TEG 3 = material one.
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 (), equivalent electrical resistance (), and equivalent thermal conductance (), in terms of which it is possible to have equivalent figure of merit (). We show the impact of the configuration of the system on 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 .
Using Eq. (31), the heat flux within any segment in TEGs is:
By continuity of the heat flux through the interface between stages of TES‐CTEG:
from which we obtain the average temperature at the interface between stages :
Since all components are electrically connected in series, the total voltage is given by:
substituting the value of in the last equation, we have:
From Eq. (39), we identified the equivalent series Seebeck coefficient, , and equivalent series electrical resistance, , as follows:
Considering open circuit condition for the system, , we find, that equivalent thermal conductance for the whole system:
We define the figure of merit in terms of equivalent quantities :
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 (PSC
In the composite system, there are two currents, for
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:
To obtain the equivalent electrical resistance, , using Eq. (45), the isothermal condition, , is required. Under this condition, we recover the usual expression of equivalent electrical resistance for an ohmic circuit. Thus, we get:
where is the internal electrical resistance of conventional TEG and is the electrical resistance of the segmented TEG:
To determine equivalent thermal conductance, , we use the open circuit condition, , which is satisfied when , and, due to preservation of heat flow:
Under open circuit condition, , so that, . Applying this result, we have for :
Using this last result in Eq. (53), we have:
Now, we can write the figure of merit for this PSC‐CTEG system:
6.1.3. Analysis of equivalent figure of merit for composite systems
Equivalent figure of merit () is calculated in this section for
It is important to emphasize, that in this study we analyzed also the behavior of and , 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 and with the ordering
|TEG 1||TEG 2||TEG 3|
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 .
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:
In terms of and , the maximum efficiency of thermoelectric device is defined by the next equation (with thermoelectric properties constant with respect to temperature) :
For the maximum efficiency of
Our results are shown in Figure 6.
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 : the efficiency of composite thermoelectric system cannot be greater than the module's component with highest efficiency.
The maximum efficiencies achieved by studied CTEGs
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.
The strategy consists of defining the optimal ratio and then by applying the method of maximizing variable to obtain the value of the load resistance, which maximizes power. It yields , and in this case, the maximum output power is:
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 . Thus, using Eqs. (62, 63) in terms of and , we can write:
Two‐stage thermoelectric system connected in series:
and the maximum power is given by:
Segmented‐conventional thermoelectric system in parallel (PSC‐CTEG):
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 .
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. , and the behavior of the output power for each array of equivalent TES‐CTEG is consistent with the results obtained by Apertet et al. . Table 3 shows the comparison of maximum output power values for different types of connections and possible arrangements.
|TEG 1||TEG 2||TEG 3|
To confirm the validity of our results, we have built plots for CTEG output power using values of some work:  (experimental) and [22, 23] (analytical). Plots in Figure 9 were produced using the temperature difference of Ref. .
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 (); (2) the whole system is constituted by only two different thermoelectric materials (), where can be 1, 2 or 3, .
7.1. Case A: homogeneous thermoelectric properties, configuration effect
We consider configurations of CTEG with the same thermoelectric material, . In this case, equivalent figure of merit is as follows,
for homogeneous TES‐CTEG:
for homogeneous PSC‐CTEG:
Table 4 shows numerical values of equivalent figure of merit obtained by us for CTEG with considered configurations.
It is important to note, that fulfillment condition
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 is as follows, for heterogeneous TES‐CTEG:
for heterogeneous PSC‐CTEG:
Eqs. (72) and (73) are applied with condition , that is, two TEGs are made of the same thermoelectric material, and third is made of different thermoelectric material. Thus, we have three possibilities (
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 .
Table 5 shows maximum values of equivalent figure of merit of CTEG with material arrangements in every configuration, when .
|TEG 1||TEG 2 = TEG 3|
|TEG 3||TEG 1 = TEG 2|
Table 6 shows each configuration with the most efficient material arrangements for every
|TES||TEG 2 = TEG 3 = BiTe ≠ TEG 1 = PbTe|
|PSC||TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe|
Results show again, that the most efficient system of three configurations is PSC with corresponding material arrangement, namely
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 .
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 () 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 () :
Figure 12a shows maximum output power values for PSC system as function of ratio , that is, in terms of internal thermal conductance and contact thermal conductance , under condition
To calculate the efficiency of PSC systems with TEG 1 = TEG 2 = PbTe ≠ TEG 3 = BiTe arrangement, we applied the equation:
Finally, for an ideal TEG, that is, without taking into account heat exchangers, we can analyze TEG efficiency considering intrinsic thermal conductances ratio () and electrical resistances ratio ().
Figure 12b shows contour plot for different values of as function of ratios, and . We can see, that the range of optimal values for the best efficiency of
7.3.3. Corollary: maximum efficiency 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 of CTEG:
If the material is the same in all components, CTEG reaches the maximum value of with a specific type of thermal—electrical connection.
When components of TEGs composing CTEG are made of different materials, where can be 1, 2, or 3; then, for a given thermal‐electrical connection, there exists an optimal arrangement of thermoelectric materials for which is maximum.
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
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