Fundamental Aspects and Advances in Thermoelectric Materials for Power Generation: A Numerical Simulation Case Study

Abstract

Power generation using thermoelectric generator technology is becoming increasingly attractive solution due to the ongoing substantial improvements in material engineering, system optimization, and novel manufacturing technologies with recent advances in nanotechnology. The design and fabrication of novel thermoelectric materials is challenging because they require co-optimization of complex properties to efficiently convert thermal energy to electricity in what is known as the Seebeck effect. Computational chemistry and machine learning offer a solution toward finding optimal thermoelectric semiconductor alloys with higher figure of merit values. In this chapter, fundamental aspects and advances in thermoelectric materials for power generation are presented and discussed. A thorough modeling and numerical simulation for a case study of a TEG device application are also presented and discussed in this chapter.

1. Introduction

Due to the increasing demands and attractions toward the use of alternative and green power generation, thermoelectric (TE) technology is considered to be promising one for its inherent merits and advantages over other alternative technologies. Thermoelectrics is the direct conversion of thermal energy (heat) due to temperature difference into electrical energy using solid-state Seebeck effect in semiconductors that would contribute to mitigating the worldwide energy crisis, and reduce air pollution and GHG emissions. The key distinct merits of using thermoelectric power generator (TEGs): they are compact and safe devices; they are flexible power sources and convenient for remote applications; they are eco-friendly; they are capable of operating at high temperatures; they are very reliable; they have no mechanical moving parts and therefore noise-free in operation with significantly less maintenance requirements [1]. The major shortcoming of the thermoelectric power generator (TEG) is its relatively low conversion and thermal efficiency. This has been a major cause in restricting their use in electrical power generation to certain specialized fields. However, the ongoing substantial improvements in TE material engineering, system optimization, and novel manufacturing technologies with recent advances in nanotechnology and machine learning/artificial intelligence (AI) bring TEG to a different level of renewed significance. Over the past couple of decades, TEG applications included industrial instruments, medical and aerospace, military, and applications for mobile and remote power supply [2, 3, 4]. More recently, industrial applications involving TEG technology are becoming more attractive, especially if the waste heat associated with these applications is dissipated to the environment and available at no cost, which could be used as heat source for operating TEG and producing power at larger amounts possible [2]. In these applications, large quantities of waste heat energy are discharged into the earth’s environment much of it at temperatures which are typically too low to recover using conventional electrical power generators. In general, the cost of a TEG mainly consists of the device cost and operating cost. Ismail and Ahmed [1] and Ismail [5] presented various interesting waste-heat industrial related applications where TEGs were successfully used. Luo et al. [6] presented recent advances in modeling and simulation of thermoelectric power generation. They performed a comprehensive review of theoretical models with a specific focus on the different modeling approaches and different application scenarios. In particular, the basic principles of theoretical models of the TEG were presented in their paper, including the thermal resistance model, thermal-electric numerical model, and analogy model. They also reviewed in detail the theoretical models of the TEG system, including the thermal resistance-based analytical model, computational fluid dynamics models, and fluid-thermal-electric multiphysics field coupled numerical model. In their work, the methods to improve the accuracy of theoretical models were also discussed.

In this chapter, some fundamental and material related aspects of TEG are introduced and discussed. A thorough modeling and numerical simulation for a case study of a TEG device application are also presented and discussed in this chapter.

2. Evolution of thermoelectric materials: from nanostructures to wearable applications

Over the past couple of decades, there has been extensive research carried out related to advances in thermoelectric materials for various applications. In general, effective TE materials should have a low thermal conductivity but a high electrical conductivity. Most widely used TE semiconductor materials are based on Bismuth Telluride (Bi₂Te₃), Lead Telluride (PbTe), and Si-Ge alloys. The large amount of research in thermoelectric materials has focused on optimizing the nanostructure of the thermoelectric materials to specifically improve the thermoelectric properties (e.g., increasing the Seebeck coefficient and reducing the thermal conductivity) of the TEG device. For example, Weiling and Shantung [7] reported that because the electrical conductivity and thermal conductivity correlate with the charge carriers, new methods of material synthesis must be applied to conciliate the contradiction between high electrical conductivity and low thermal conductivity. Research has been focused mainly on improving the material’s figure-of-merit $Z\overline{T}$, and thus the TE conversion and thermal efficiencies, by reducing the lattice thermal conductivity [8]. Riffat and Ma [2] and Weiling and Shantung [7] reported that thermoelectric materials, such as Lead Telluride PbTe and Bismuth Telluride Bi₂Te₃, have a $Z\overline{T}$ value of approximately 1.0 (at 500–700 K for PbTe and at room temperature for Bi₂Te₃). However, at a higher $Z\overline{T}$ values (somewhere between 2.0 and 3.0), TEG would become competitive with other power generation systems (e.g., solar power, nuclear power, wind power, fuel cell power generation systems). Very recently, Cui et al. [9] reported that AgSbSe₂ (Antimony-Silver-Selenium-based alloy) has a structure similar to PbTe (Lead-Tellurium-based alloy) but does not contain toxic element Pb or expensive element Te. In addition, this new alternative TE alloy has both high Seebeck coefficient and inherently low thermal conductivity, which makes it a promising candidate for medium-temperature TE power generation. Most recently, Zulkepli et al. [10] provided an overview of the key challenges in optimizing $Z\overline{T}$ values according to their TE physical properties including the state-of-the-art of the advanced approaches in $Z\overline{T}$ optimization more particularly for TE materials at low-temperature operating applications.

Moreover, it is desirable to fabricate TE modules which can conform easily to a heat source surface which would improve the thermal contact to heat sources of arbitrary geometry. Therefore, recent research has also been focused on developing novel flexible- and cylindrical-based shapes of TEG devices. For example, Yadav et al. [11] proposed and demonstrated the use of flexible and cost-effective TEG based on thin film thermoelectric on flexible fiber substrates. Min and Rowe [12] have also recently developed a novel tube-shape thermoelectric module for power generation. Most recently, Soleimani et al. [13] reported that wearable TEG devices are becoming attractive power supply for relatively low-power electronic devices. They indicated from their research that to maximize the higher power output from these devices, the focus should be on improving TE material, such as using promising electrically conductive and flexible hybrid organic and inorganic TE material, and the configuration and arrangement of thermoelements in the TEG device. In these applications, the thermal energy sources of the human body are used to power these wearable devices. Soleimani et al. [13] provided extensive research of recent studies on wearable TEG devices. Most recently, Zhu et al. [14] indicated the challenges and outlooks toward future development of wearable TEG devices and their potential applications. Lemine et al. [15] reported a comprehensive review research highlighting the promising and future-generation TEG devices based on thin film technologies with highly flexible, transparent, non-toxic, plentiful, and light-density p-type Copper Iodide (Cul) thin films. Sanin-Villa [16] reported that research involving carbon nanotubes/polyaniline composite films has found promising results due to its low thermal conductivity.

3. Accelerating the discovery of TE materials using computational chemistry and machine learning

The design of novel thermoelectric materials is challenging because they require co-optimization of complex properties to efficiently convert thermal energy to electricity in what is known as the Seebeck effect. As stated previously, a TE material should possess high electric conductivity to allow sufficient movement of electrons while simultaneously having low thermal conductivity to prevent heat transfer or loss. This fine balance between thermal and electrical conductivity is captured in the dimensionless $Z\overline{T}$, which is calculated using the Seeback coefficient, internal electric resistivity, and thermal conductivity of the TE material. Optimizing $Z\overline{T}$ typically requires tuning the electronic structure and various properties of the material making the design process complicated. Additionally, the quest to identify TE materials with desired properties is amplified by the sheer size of the chemical space. For a long time, discovery of TE materials has been led by Edisonian methods of experimental error-and-trial that have been employed to navigate the chemical space making material exploration slow and expensive. However, lately, alternative paradigms such large-scale computations have been widely adopted to accelerate the process and move beyond serendipitous discovery. Many of the research studies in this area have focused on high-throughput density functional theory (DFT) calculations to compute desired properties from a myriad selection of input atomic structures. The materials are then ranked based on their computed target property such as $Z\overline{T}$ or the power factor and the most promising candidates are sent for experimental testing. While this approach has shown significant speed ups in guiding material design, the heavy computational cost associated with DFT still limits discovery. Due to how large and diverse the search space for optimal material selection is, recently, this issue has been tackled by utilizing machine learning models. The models could learn from large DFT-curated or experimental datasets enabling access to uncharted chemical spaces while requiring only a fraction of the computational cost. Gorai et al. [17] reported that the DFT method offers an efficient and feasible route of calculating material properties replacing explicit atomic quantum calculations without significantly compromising accuracy. It works by estimating the electronic density ρ(r) of the multiple electrons in the atomic structure instead of calculating the individual wave functions of electrons. This becomes increasingly important as the number of electrons and the complexity of the electron-electron interactions increase in large systems. DFT captures the effect of electron-electron interactions on the energy of the system using exchange-correlation functionals. The energy of the system could be obtained by solving Kohn-Sham’s equation shown below:

$$E\left(\rho \left(r\right)\right)=\sum _j{E}_j+E_{\mathit{xc}}\left(\rho \left(r\right)\right)-\int \mathit{dr}{v}_{\mathit{xc}}\left(r\right)\rho \left(r\right)-\frac{1}{2}\int \mathit{drd}{r}^{\prime }\frac{\rho \left(r\right)\rho {\left(r\right)}^{\prime }}{\mid r-r^{\prime }\mid }$$E1

where E_j is the energy of Kohn-Sham orbitals, E_xc is the exchange-correlation functional, and v_xc is the exchange-correlation potential = $\frac{{\mathit{\delta E}}_{\mathit{xc}}\left(\rho \left(r\right)\right)}{\mathit{\delta \rho }\left(r\right)}$. The equation is solved by starting with an initial guess for the charge density and then iteratively calculating the energy and updating the value of the charge density through a self-consistent scheme until convergence is achieved. The component that is most crucial to accuracy in this equation is exchange-correlation functionals. These functionals factor in for the exchange and correlation energies of the electron interactions. While their energy values make up a small portion of the total energy of the system, careful choice of functionals is a key to predict the electronic structure and properties of different classes of materials with sufficient accuracy for experimental verification.

Local density approximation (LDA) is one of the simplest functionals that were implemented for DFT calculations. It neglects spatial variations of the electron density across the system and only considers local correlation effects by treating charge density as a uniform gas of electrons, which offers a reasonable accuracy for some applications [18]. To incorporate non-local correlations, generalized gradient approximation (GGA) improves on LDA by considering not only the electron density at a given point but also considers the gradient of the electron density across the space. This provides a more accurate model accounting for electron density variations that are important in many systems. There has been a growing interest in developing more accurate functionals such as hybrid functionals and machine learning (ML)-based functionals capture complex correlations beyond the scope of LDA and GGA [19]. One of the most popular functionals used for screening of thermoelectric materials using DFT is PBE-GGA [20] and van der Waals-corrected functionals DFT-D2 [21]. However, despite selecting a suitable functionals, there are still inherit challenges that stem from the nature of DFT calculations, which do not account for temperature effects. Additionally, the lack of standardization of functionals prevents the direct comparison and transfer of results in the literature. This in turn widens the gap between computational predictions and experimental validations. Choubisa et al. [22] addressed both issues listed above regarding computational cost and accuracy of predictions by utilizing an error-correction learning (ECL) based on a neural network to model the error correction function from experimental data. This approach is implemented in two basic steps: (1) The model learns from prior experimental or computational datasets and then (2) it utilizes new experiments to provide feedback and refine the model to improve its accuracy. The second step allows the model to implicitly account for disparities in synthesis methods, material morphology, and defects, which vary from one lab setting to another and are normally not captured by DFT calculations. This improved model is then used to screen the material space, particularly Materials Project [23], a large open computational dataset of materials, with the purpose of discovering new promising chemistries. The authors focused their search on low-temperature thermoelectric materials <300°C. Notably, a new chemical family based on PbSe:SnSb (lead selenide:tin antimony) of thermoelectric materials was identified using this approach. The best composition exhibited double the power factor of a standard PbSe (lead selenide). This study encourages the development of new hybrid strategies guided by computational and experimental results, as well as machine learning models to shorten the cycle of developing thermoelectric materials.

4. The principle of a thermoelectric generator (TEG)

Figure 1 shows how a thermoelectric generator (TEG) works. A temperature difference is established between two junctions, namely, the hot junction and the cold junction, of two dissimilar materials made of metals or semiconductors. Due to this temperature difference, a voltage is generated using the Seebeck effect. This effect is fundamentally used in applications of thermocouples for temperature measurements. TEG devices can operate as electrical power generators using this effect. In a basic TEG thermocouple, heat transfers at a rate $\dot{Q_H}$from the heat source maintained at a high temperature $T_H$ to the hot junction. It then transfers through the thermoelectric materials A and B and reaches the cold junction maintained at a low temperature $T_H$ where heat is rejected at the heat sink at a rate $\dot{Q_L}$. The heat transferring at the hot junction causes an electric current to flow in the circuit based on Seebeck effect and an electrical power is produced when an electric load is connected to this circuit. This constitutes a thermodynamic heat engine with a power cycle established. The electrons (charge carriers) serve as the working fluid. The power output $\dot{W_e}$ is the difference between $\dot{Q_H}$ and $\dot{Q_L}$based on the first law of thermodynamics.

Figure 1.

A schematic diagram showing the principle of a TEG.

Figure 2 shows a schematic diagram of a simple TEG device with its arrangement of components [1]. The TEG device is composed of two ceramic plates (substrates) that serve as a foundation, providing mechanical integrity, and electrical insulation for n-type (heavily doped to create excess electrons) and p-type (heavily doped to create excess holes) semiconductor thermoelements. In TE materials, electrons and holes operate as both charge carriers and energy carriers. There are very few modules designed and fabricated without ceramic plates. Removing the ceramic plates from the modules would eliminate the thermal resistance associated with them but might lead to mechanical fragility of the module. The ceramic plates are commonly made from alumina, but when large lateral heat transfer is required, materials with higher thermal conductivity (e.g., beryllia and aluminum nitride) are desired. The semiconductor thermoelements (e.g., silicon-germanium, lead-telluride-based alloys) that are sandwiched between the ceramic plates are connected thermally in parallel and electrically in series to form a thermoelectric device (module). More than one pair of semiconductors are normally assembled together to form a thermoelectric module and within the module a pair of thermoelements is called a thermocouple.

Figure 2.

A schematic diagram showing arrangement of a basic TEG device.

The junctions connecting the thermoelements between the hot and cold plates are interconnected using highly conducting metal (e.g., copper) strips as shown in Figure 2. The power output for most of the commercially available TEGs ranges from microwatts to multi-kilowatts [2] (Rowe, 1999).

5. Modelling and numerical simulation: case study of the performance of a TEG device

In Figure 1, the heat transfer at the hot junction, ${\dot{Q}}_H$ includes three terms, given by [24]:

$${\dot{Q}}_H={\dot{Q}}_{\mathit{SE}}+{\dot{Q}}_{\mathit{JH}}+{\dot{Q}}_{\mathit{CON}}$$E2

where ${\dot{Q}}_{\mathit{SE}}$ is heat flow due to Seebeck effect, given by

$${\dot{Q}}_{\mathit{SE}}=\alpha T_{H}I$$E3

${\dot{Q}}_{\mathit{JH}}$ is the irreversible heat flow due to Joule heating effect that is generated as the electric current flows in the wire. There are two elements (legs) of the TEG module, so this heat on each leg accounts for one-half of Jule heating, given by [24]:

$${\dot{Q}}_{\mathit{JH}}=-\frac{1}{2}{I}^{2}R$$E4

${\dot{Q}}_{\mathit{CON}}$ is heat flow associated with conduction heat transfer, given by

$${\dot{Q}}_{\mathit{CON}}=K\left(T_H-T_L\right)$$E5

Substituting Eqs. (3)–(5) in Eq. (2), yields

$${\dot{Q}}_H=\alpha T_{H}I-\frac{1}{2}{I}^{2}R+K\left(T_H-T_L\right)$$E6

In Eq. (6), α is the Seebeck coefficient, R is the internal electrical resistance, and K is the total thermal conductance of the thermoelements, given by

$$R=\frac{\rho L}{A}$$E7

$$R=R_p+R_n$$E8

$$K=\frac{kA}{L}$$E9

$$K=K_p+K_n$$E10

where A, L,$k\text{and}\rho$ are the cross-sectional area, length, thermal conductivity, and electrical resistivity’ of the n-type and p-type thermoelements, respectively. The Seebeck coefficient α for the TE device can be rewritten in terms of the average values of Seebeck coefficients for the dissimilar n-type and p-type thermoelements as [24]

$$\alpha ={\alpha }_p-{\alpha }_n$$E11

The performance of thermoelectric materials can be expressed by figure of merit, Z, given by

$$Z={\alpha }^2/\mathit{KR}$$E12

The relationship of K and R with properties of the thermoelements is given by

$$KR={\left[{\left(k_p{\rho }_p\right)}^{1/2}+{\left(k_n{\rho }_n\right)}^{1/2}\right]}^2$$E13

Maximizing Z means that $KR$ should be minimized. The minimum $KR$ necessitates that

$$\frac{L_n}{A_n}=\frac{L_p}{A_p}\sqrt{\frac{k_n{\rho }_p}{k_p{\rho }_n}}$$E14

The power generated at the electrical load $\dot{W_e}$ can be determined using,

$$\dot{W_e}=I^2{R}_L$$E15

where $R_L$ is the load electrical resistance. The thermal efficiency of a TEG device (treated as a thermodynamic heat engine), ${\eta }_{\mathit{th}}$, is defined as

$${\eta }_{\mathit{th}}=\frac{\dot{W_e}}{{\dot{Q}}_H}$$E16

Eq. (16) can be rewritten using Eqs. (6) and (15) as

$${\eta }_{\mathit{th}}=\frac{I^2{R}_L}{\alpha T_{H}I+\frac{1}{2}{I}^{2}R+K\left(T_H-T_L\right)}$$E17

Using the first law of thermodynamics of the closed system, the heat transfer at the low-temperature side (cold junction), ${\dot{Q}}_L$ can be written as

$${\dot{Q}}_L={\dot{Q}}_H-\dot{W_e}$$E18

In Figure 1, the potential difference (or voltage V) is proportional to the temperature difference $∆T$, given by

$$V=\alpha ∆T$$E19

where,

$$∆T=T_H-T_L$$E20

Eq. (19) can also be written in terms of the open circuit voltage, $V_{\mathit{oc}}$,

$$V_{\mathit{oc}}=\alpha ∆T$$E21

Also, $V_{\mathit{oc}}$ is given by

$$V_{\mathit{oc}}=I\left(R+R_L\right)$$E22

Equating Eqs. (21) and (22) and solving for the electric current, yield

$$I=\frac{\alpha ∆T}{(R+R_{L)}}$$E23

For a thermodynamic heat engine, the maximum thermal efficiency is limited by the second law of thermodynamics given by Carnot efficiency:

$${\eta }_{\mathit{CAR}}=1-\frac{T_L}{T_H}$$E24

Introducing the dimensionless parameters, θ, ξ, and β given by

$$\mathrm{\theta }=R_L/R$$E25

$$\mathrm{\xi }=Z\overline{T}$$E26

$$\mathrm{\beta }=T_L/T_H$$E27

$${\eta }_{\mathit{CAR}}=1-\mathrm{\beta }$$E28

where the average temperature, $\overline{T}$, of the TE device is given by

$$\overline{T}=\frac{T_H+T_L}{2}$$E29

Substituting Eq. (23) into Eq. (17) with Eqs. (12), (24)-(26), yields

$${\eta }_{\mathit{th}}={\eta }_{\mathit{CAR}}\left[\frac{\mathrm{\theta }}{\frac{{\left(1+\mathrm{\theta }\right)}^2}{Z{T}_H}+\left(1+\mathrm{\theta }\right)-\left(\frac{{\eta }_{\mathit{CAR}}}{2}\right)}\right]$$E30

The product $Z{T}_H$ in Eq. (30) can be written in terms of the dimensionless parameters ξ, and β, as

$$Z{T}_H=\frac{2\mathrm{\xi }}{\left(1+\mathrm{\beta }\right)}$$E31

Substituting Eqs. (27) and (31) into Eq. (30), gives the thermal efficiency of the TE device in terms of dimensionless parameters, θ, ξ, and β, as

$${\eta }_{\mathit{th}}=\frac{\mathrm{\theta }{\eta }_{\mathit{CAR}}}{\left(1+\mathrm{\theta }\right)-\left(\frac{{\eta }_{\mathit{CAR}}}{2}\right)+\frac{1}{2\mathrm{\xi }}{\left(1+\mathrm{\theta }\right)}^2\left(1+\mathrm{\beta }\right)}$$E32

Normally, in designing and operating TEGs for various power generation applications, it would be very useful to maximize their performance. To determine the maximum power generation and conversion efficiency of the TEG module, the following set of equations is developed.

The maximum efficiency of the TE device can be determined using

$$\frac{d{\eta }_{\mathit{th}}}{d\mathrm{\theta }}=0$$E33

Which leads to

$${\mathrm{\theta }}_{\mathit{max}}=\sqrt{\left(1+\mathrm{\xi }\right)}$$E34

Using Eq. (34) in Eq. (32), yields

$${\eta }_{\mathit{th},\mathit{max}}={\eta }_{\mathit{CAR}}\left[\frac{{\mathrm{\theta }}_{\mathit{max}}-1}{{\mathrm{\theta }}_{\mathit{max}}+\mathrm{\beta }}\right]$$E35

Substituting Eq. (23) in Eq. (15) and using Eq. (26), gives

$${\dot{W}}_e=\mathrm{\theta }{\left(\frac{\alpha ∆T}{\mathrm{\theta }+1}\right)}^2$$E36

The maximum power output of the TE device can be determined using

$$\frac{d{\dot{W}}_e}{d\mathrm{\theta }}=0$$E37

This leads to the condition of maximum power of the TEG device,

$$\mathrm{\theta }=1$$E38

That is, $R_L=R$.

In this case, we have

$${\dot{W}}_{e,\mathit{max}}=I_{\mathit{mp}}^{2}R$$E39

$$I_{\mathit{mp}}=\frac{\alpha ∆T}{2\mathrm{R}}$$E40

Eq. (39) can then be rewritten as

$${\dot{W}}_{e,\mathit{max}}=\frac{1}{R}{\left(\frac{\alpha ∆T}{2}\right)}^2$$E41

Also, the maximum power output can be expressed in terms of

$${\dot{W}}_{e,\mathit{max}}=I_{\mathit{mp}}{V}_{\mathit{mp}}$$E42

where,

$$V_{\mathit{mp}}=\frac{\alpha ∆T}{2}$$E43

The maximum current can be determined by setting $R_L=0$ in Eq. (23), as

$$I_{\mathit{max}}=\frac{\alpha ∆T}{\mathrm{R}}$$E44

The maximum voltage is the open circuit voltage, using Eq (21) (or Eq. (22))

$$V_{\mathit{max}}=\alpha ∆T$$E45

The maximum power efficiency is given by

$${\eta }_{\mathit{mp}}={\eta }_{\mathit{CAR}}\left[\frac{1}{\frac{4}{Z{T}_H}-\frac{{\eta }_{\mathit{CAR}}}{2}+2}\right]$$E46

For a TE device with $\mathrm{\lambda }$ multicouple n-type and p-type thermoelements in a TEG device, we have

$${\left({\dot{Q}}_H\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }{\dot{Q}}_H$$E47

$${\left({\dot{W}}_e\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }{\dot{W}}_e$$E48

$${\left(V\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }V$$E49

$${\left(K\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }K$$E50

$${\left(R\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }R$$E51

$${\left(R_L\right)}_{\mathrm{\lambda }}=\mathrm{\lambda }{R}_L$$E52

$${\left(I\right)}_{\mathrm{\lambda }}=I$$E53

$${\left({\eta }_{\mathit{th}}\right)}_{\mathrm{\lambda }}={\eta }_{\mathit{th}}$$E54

Numerical simulations: case study

A TEG module is to be designed with its performance analytically and numerically evaluated to deliver a total electrical power of 1 kW using a waste heat source from hot exhaust gas at 811 K produced from an IC engine. The cold heat is rejected at a temperature of 436 K. The design specifications, material properties, and thermal conditions of the p-type and n-type of the PbTe thermoelements of the TEG device are summarized in Table 1. For this practical case study, it is required to:

1. Determine the maximum thermal efficiency, the efficiency at maximum power output, the number of thermocouples required for the TEG device, and other performance parameters.

2. Perform numerical simulations to determine the effect of lowering the cold temperature (heat sink temperature) on the key performance parameters of the TEG device.

Design specifications
Lp	8 mm
Ln	8 mm
An	0.6 cm^2
TE material properties: Lead Telluride (PbTe)
$k_n$	0.0140 W/cm K
$k_p$	0.0120 W/cm K
${\rho }_n$	0.00101 Ω cm
${\rho }_p$	0.00095 Ω cm
${\alpha }_n$	−170 × 10^−6 V/K
${\alpha }_p$	190 × 10^−6 V/K
Thermal conditions
$T_L$	436 K
$T_H$	811 K

Table 1.

Design specifications, material properties, and thermal conditions for the TEG device.

TEG device performance parameters were calculated using MS Excel program. The numerical results are shown in Table 2. For this case, 593 thermocouples are required to construct the TEG device to deliver 1 kW of electrical power. The maximum thermal efficiency and the efficiency at maximum power output of the TEG device were found to be 13.1% and 12.5%, respectively.

Parameter	Value	Equation # used
$\alpha$	360 × 10^−6 V/K	Eq. (11)
$\mathrm{\beta }$	0.538	Eq. (27)
e	0.629 cm^2	Eq. (14)
$R$	0.002555 Ω	Eqs. (7) and (8)
$KR$	5.0927 × 10^−5 ΩW/K	Eq. (13)
$Z$	0.002545 K^−1	Eq. (12)
$\overline{T}$	623 K	Eq. (29)
$\mathrm{\xi }$	1.586	Eq. (26)
${\mathrm{\theta }}_{\mathit{max}}$	1.6081	Eq. (34)
$R_L$	0.004109 Ω	Eq. (25)
$\mathrm{\theta }$	1.6082	Eq. (25)
${\eta }_{\mathit{CAR}}$	0.462	Eq. (28)
${\eta }_{\mathit{th},\mathit{max}}$	0.1309	Eq. (35)
$Z{T}_H$	2.0624	Eq. (31)
$I$	20.26 A	Eq. (23)
$\dot{W_e}$	1.6866 W	Eq. (15)
${\eta }_{\mathit{th}}$	0.1309	Eq. (30)
${\dot{Q}}_H$	12.8846 W	Eq. (16)
${\dot{Q}}_L$	11.1980 W	Eq. (18)
$\alpha ∆T$	0.135 V
$I_{\mathit{max}}$	52.84 A	Eq. (44)
$V_{\mathit{max}}$	0.135 V	Eq. (45)
$I_{\mathit{mp}}$	26.42 A	Eq. (40)
$V_{\mathit{mp}}$	0.0675 V	Eq. (43)
${\dot{W}}_{e,\mathit{max}}$	1.7833 W	Eq. (42)
${\eta }_{\mathit{mp}}$	0.1246	Eq. (46)
${\left({\dot{W}}_e\right)}_{\mathrm{\lambda }}$	1000 W	Given
$\mathrm{\lambda }$	593 thermocouples	Eq. (48)
${\left({\dot{Q}}_H\right)}_{\mathrm{\lambda }}$	7641 W	Eq. (47)
${\left({\dot{Q}}_L\right)}_{\mathrm{\lambda }}$	6641 W	Eq. (18)

Table 2.

Calculations of the TEG performance parameters.

The numerical simulations were carried out using the simple computer program and the numerical results are shown in Table 3. It is interesting to see the detrimental effect of lowering the heat sink temperature of the TEG device on the various efficiency values of the device. Also, the number of multi-thermocouple required in constructing the TEG device decreased significantly with decreasing the heat sink temperature of the device. For example, by operating the device at the cold temperature of room temperature of 25°C, there are 317 thermocouples (i.e., $\mathrm{\lambda }=317$) required in constructing the device as opposed to 593 thermocouples when operating the TEG device at 163°C, as can be determined using Eq. (48). The difference of 276 reflects a significant drop in the required material for designing and building this device which would then significantly reduce the cost of the device and its maintenance cost requirements. The efficiency at the maximum power output of the TEG device increased from 12.5% to 17.5% (i.e., by 5% increase) only by means of lowering the heat sink temperature to room temperature. It should be noted that lowering the cold heat sink operating temperature T_L would increase the potential temperature difference (driving force for TEG device) $∆T$ at a fixed heat source temperature T_H, as can be seen in Eq. (20). This in turn would result in increasing the maximum power output of the TEG device (see Eq. (41)) per thermocouple. Ultimately, this would result in lowering the number of thermocouples (i.e., thermoelements $\mathrm{\lambda }$) required to form the TEG device, as can be seen in Eq. (48), for a given total power output requirement in an application where TEG device is used. One another useful operation metric is the size of the TEG device. As the number of thermocouples is decreased, the volume and weight of the device would decrease. This will make the TEG device more compact and cost-effective.

$T_L$ (°C)	163	150	125	100	75	50	25 (room temp)
${\eta }_{\mathit{CAR}}$ (%)	46.2	47.8	50.9	54.0	57.1	60.2	63.3
${\eta }_{\mathit{th},\mathit{max}}$ (%)	13.1	13.6	14.5	15.4	16.3	17.3	18.2
${\eta }_{\mathit{mp}}$ (%)	12.5	12.9	13.8	14.7	15.6	16.5	17.5
$\mathrm{\lambda }$	593	554	489	435	389	350	317

Table 3.

Numerical simulation results (case study—heat source exhaust gas temperature $T_H$= 538°C).

6. Conclusion

In this chapter, some fundamental and material related aspects of thermoelectric materials are introduced and discussed. A thorough modeling and numerical simulation for a hypothetical case of a TEG device are also presented and discussed in this chapter. The numerical simulation was carried out for a hypothetical practical case of a TEG made of PbTe semiconductors alloy. It was found that the heat sink cold temperature has a detrimental effect on the thermal and conversion efficiencies of the TEG device. More particularly, the efficiency at the maximum power output of the TEG device increased from 12.5% to 17.5% (i.e., by 5% increase) only by means of lowering the heat sink temperature to room temperature. In addition, the cost of the TEG device and its maintenance cost requirements were significantly reduced by lowering the operating cold temperature of the device for this case study.

Acronyms

DFT

density function theory

GHG

greenhouse gases

GGA

generalized gradient approximation

LDA

local density approximation

ML

machine learning

TE

thermoelectric

TEG

thermoelectric generator