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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In a recent work, we have reported a study on the figure of merit of a thermoelectric system composed by thermoelectric generators connected electrically and thermally in different configurations. In this work, we are interested in analyzing the output power delivered by a thermoelectric system for different arrays of thermoelectric materials in each configuration. Our study shows the impact of the array of thermoelectric materials in the output power of the composite system. We evaluate numerically the corresponding maximum output power for each configuration and determine the optimum array and configuration for maximum power. We compare our results with other recently reported studies.

Efficient use of energy is one of the challenges that occupies both the field of sustainable energy research and industry. The energy sector is currently undergoing a process of transition that demands high energy efficiency, renewable energy sources and energy harvesting. An increase has been projected in energy consumption of approximately 40 percent for the year 2035 [

Recently, there has been a surge of research in the area of power harvesting. Energy harvesters use piezoelectric, smart and semiconductors materials. In fact, there exist both vibratory-based energy harvesters (moving parts) and thermoelectric energy harvesters (no moving parts) [

Thermoelectric generators are solid-state devices with no moving parts. They are silent, reliable and scalable, making them ideal for small, distributed power generation and energy harvesting [

Thermoelectric generator (TEG), based on the Seebeck effect, is a solid-state device that converts heat into electricity and is composed of two dissimilar semiconductors, and it operates between two heat reservoirs. This type of system has the advantage of generating electrical power, without using moving parts [

TEGs have been used for electricity generation, and it is only in recent years that interest has increased regarding new applications of energy generation through thermoelectric harvesting. Thermoelectric energy harvesting has found some applications in developing thermoelectric energy harvesters to recover waste heat, for example, in vehicles [

A thermoelectric generator utilizes heat flow across a temperature gradient to power an electric load through the external circuit. The temperature difference provides the voltage (

Recent progress in the technological development of TEGs has relied on advances in material sciences: new materials and new techniques to produce specific structures have permitted the improvement of device performance through the characterization and optimization of the electrical and thermal transport properties (see the review of Di Salvio [

Thermoelectric processes that occur in thermoelectric generators are subjected to the laws of thermodynamics. Treatment of these devices is accomplished within the framework of the thermodynamics of linear irreversible processes, because the thermoelectric effects (like the Seebeck effect) can be seen as the mutual interference of two irreversible processes occurring simultaneously in the TEG [

It is observed that the heat flux, _{Q}

Using _{out}_{load}

The optimization of thermoelectric devices, in the thermodynamic framework, considering both its efficiency under different working conditions and irreversibilities inside inhomogeneous thermoelectric systems, is widely recognized by abundant studies on the subject [

It has been pointed out that the optimization of thermoelectric systems for energy conversion not only involves the improvement of the materials’ properties to enhance the so-called figure of merit, but also a strategic reflexion in device design [

This paper is organized as follows: in Section 2, we show the basic theory for the power of a thermoelectric generator. In Section 3, we calculate the output power and maximum power of coupled thermoelectric generators. In Section 4, we show and discuss our results, and finally, in Section 5, we give our conclusions.

We follow the treatment proposed by Apertet _{H}_{C}_{Q}_{0}, under the open circuit condition and a Seebeck coefficient,

For the case of a thermoelectric generator connected to a load resistor, _{load}_{load}

where we now define the load ratio _{load}/R

The load resistance, _{load}_{load}_{load}

Based on the model presented in Section 2, the electrical power and the maximum power delivered by systems composed of thermoelectric modules (conventional and segmented) is calculated in terms of equivalent amounts, i.e., the equivalent Seebeck coefficient (_{eq}_{eq}

Thus, we use in our analysis _{eq}_{eq}

The first configuration considered is a two-stage thermoelectric system (TES) composed of three TEMthermally and electrically connected in series (SC-TES) [_{i}_{i}_{i}_{hot}_{cold}

The corresponding equivalent quantities for these configurations are (1) The Seebeck coefficient series equivalent,

and (

where,

Thus, substituting _{load}

and the maximum power is given by,

In this section, we consider a TES system, which is composed of a segmented TEM and a conventional TEM. These TEMs are thermally and electrically connected in parallel (PSC-TES), as is shown in

The corresponding equivalent Seebeck coefficient is given by,

where:

and the electrical resistance equivalent,

where _{c}_{s}

and

Substituting

and using

In this case, we have considered a thermoelectric system (SSC-TES) composed of a segmented TEM and a conventional TEM, but they are thermally connected in parallel and electrically connected in series [

The corresponding equivalent Seebeck coefficient is given by,

and the electrical resistance equivalent,

where,

Thus, we have for the output power of the SSC-TES system,

and for the maximum power for the SSC-TES system,

In the following section, we show the power curves for each of the configurations of this system.

We show the behavior of the electrical output power delivered by each configuration of the TES. For this purpose, we have selected the following materials, BiTe, PbTe and SiGe. For showing the impact of the array of the thermoelectric materials in the output power of the TES, we consider different arrays for each configuration, in which we change cyclically the positions the three materials (one material for each module) throughout the whole system. Thus, we consider the following arrays, (BiTe, PbTe, SiGe), (PbTe, SiGe, BiTe) and (SiGe, BiTe, PbTe). Our results are given by

Notice that each figure provides the highest output power value for one of the three arrays of the TES. These results show that the output power of the TES depends both on the configuration and the arrays of thermoelectric materials. Furthermore, our results show that, in the case of the PSC system with array (PbTe, SiGe, BiTe), maximum power is generated. This result is consistent with the results obtained by Vargas-Almeida

The behavior of the output power for each array of equivalent TES is consistent with the results obtained by Apertet

Finally, The maximum power as a function of the equivalent amounts were numerically calculated for each of the three arrays of the system discussed above; see

Another comparison is made with the works of Nemir ^{−3} for electrical resistances;

In this work has been evaluated the electrical power delivered by a TES composed of thermoelectric modules. The particular aspects of this TES are combined conventional and segmented thermoelectric modules, in different configurations, namely, series (SC), parallel (PSC) and mixed (SSC) configurations. We have shown the impact of the arrays of the modules with different thermoelectric materials. Our results show the impact in the output power of the TES of both the configuration and the arrays of thermoelectric materials. We find that the PSC system with the array (PbTe, SiGe, BiTe) delivers the highest power of all possible combinations; this is due to the type of thermal and electrical connection and also to the array of materials used for each module. We suggest that this work could lead to a new scheme for the design of composite thermoelectric systems.

This work was supported in part by Instituto Politecnico Nacional (IPN) grant PIFI-20140647-IPN Mexico (Miguel Angel Olivares-Robles). Alexander Vargas-Almeida and Pablo Camacho-Medina acknowledge Consejo Nacional de Ciencia y Tecnologia-Mexico for financial support.

Miguel Angel Olivares-Robles supervised, processed and analyzed the results of this paper and wrote the paper. Pablo Camacho-Medina and Alexander Vargas-Almeida contributed to the calculations of this paper. Both authors contributed to the initial motivation of the problem, to research and to the writing. Both authors read and approved the final manuscript. Francisco Solorio-Ordaz helped in the analysis with constructive discussions, read and commented on the manuscript.

The authors declare no conflict of interest.

A thermoelectric generator and its representation as a thermal-electrical circuit.

Circuit schematic of three thermoelectric generators thermally and electrically connected in series. (

Thermally and electrically parallel circuit for the connection of a conventional module and a segmented module. (

Schematic circuit electrically in series and thermally in parallel for the connection of a conventional module and a segmented module. (

Power _{Out–eq–SC}_{load}/R

Power _{Out–eq–PSC}_{load}/R

Power _{Out–eq–SSC}_{load}/R

Power _{Out–eq–PSC}_{load}/R

Power _{Out–eq–PSC}_{load}/R

Power _{Out–eq–PSC}_{load}/R

Numerical values of the maximum power, in terms of the equivalent amounts of each of the compounds’ thermoelectric systems, evaluated for each order of the TE material.

TEM 1 | TEM 2 | TEM 3 | _{max–eq–SC} |
_{max–eq–PSC} |
_{max–eq–SSC} |
---|---|---|---|---|---|

BiTe | PbTe | SiGe | 1.27618 | 4.34854 | 4.42523 |

PbTe | SiGe | BiTe | 1.65563 | 12.2877 | 3.62461 |

SiGe | BiTe | PbTe | 2.22968 | 4.28067 | 3.81606 |