Optimization of segmented thermoelectric power generators from waste heat while considering the influence of temperature on materials

Abstract

Thermoelectric technology is commonly used in waste heat utilization of automotive internal combustion engines and widely combined with solar energy units to form solar thermoelectric generator systems. The structure of the Thermoelectric Generator (TEG) needs to be optimized in order to obtain better performance for wider applications. In this paper, the influence of temperature on the height of PN-type thermoelectric arms was analyzed using an improved one-dimensional heat conduction model with the calculus method. At the same time, both the calculation formula of the maximum output power and the calculation formula of various size parameters of the TEG was derived when the influence of temperature on the performance of thermoelectric materials has been considered. In addition, the relationships among different size parameters were derived to obtain the maximum efficiency. The relationships include the most commonly used classical optimization relationship, that is, when the Seebeck coefficient, thermal conductivity and resistivity are averaged, the relationship is consistent with the classical optimization relationship. By considering the impact of temperature on the performance of thermoelectric materials, an improved calculation formula of the figure of merit (Z) was also given. The new optimization formula was compared with the classical optimization method by taking the maximum output power as the optimization index. In the case study, the temperatures of the cold end and the hot end were set at 330 K and 700 K, respectively. PbTe and PbSe were used as the materials with intermediate temperature, and Bi ${}_{2}$ Te ${}_{3}$ was used as the material with low temperature. Through theoretical analysis, it is found that the maximum output power of the new optimization formula can be higher than that of the classical optimization formula.

Keywords

Size optimization segmented thermoelectric generators calculus figure of merit

1. Introduction

About 40% of the energy of internal combustion engine vehicles is wasted in the form of exhaust heat [1]. How to realize the waste heat utilization has attracted more and more attention of researchers. Thermoelectric power generation technology is a solid power generation technology based on the Seebeck effect, which can convert waste heat into electricity [2, 3]. In addition to waste heat utilization, TEG can also be combined with solar energy systems to form solar thermoelectric generator systems [4, 5]. In these applications of TEG, how to improve the performance of TEG is the focus of the current research. The most commonly used method is the optimization of thermoelectric materials and thermoelectric plate structure optimization.

For thermoelectric materials, the dimensionless coefficient ZT is commonly used to express the performance, and its expression is shown in Eq. (1). In the process of thermoelectric research, researchers improved the thermoelectric performance of thermoelectric materials by reducing the lattice thermal conductivity and optimizing the carrier coefficient. In the low-temperature range, it is recognized that the best thermoelectric material is the semiconductor based on Bi ${}_{2}$ Te ${}_{3}$ . The ZT value of the most common commercial Bi ${}_{2}$ Te ${}_{3}$ material on the market is about 1. Bed Poudel et al. [6] made bulk material of Bi ${}_{2}$ Sb ${}_{3-x}$ Te ${}_{x}$ by hot-pressing the nano-particles produced by the ball milling process under inert conditions. This process effectively reduced the lattice thermal conductivity and significantly improved the figure of merit of the material. The peak value of ZT can reach 1.4. The PbTe based thermoelectric material is the best known in the intermediate temperature range. The peak ZT value of many PbTe based thermoelectric materials is above 1.5 [7]. The main thermoelectric material in high-temperature range is SiGe alloy. Bathula et al. [8] prepared nanostructured SiGe alloy by mechanical alloying, and its ZT value is as high as 1.5 at 900 ${{}^{\circ}}$ C. In addition to the common thermoelectric materials mentioned above, the research direction of thermoelectric materials includes skutterudite, oxide, organic and silicide thermoelectric material. In the study of silicide thermoelectric materials, Hinterleitner et al. [9] combined a layer of alloy thin-film material composed of iron, vanadium, tungsten and aluminum elements with a silicon substrate, effectively reducing the lattice thermal conductivity, so that it achieved ZT values as high as 5 to 6.

$\displaystyle\text{ZT}=\frac{\alpha^{2}\sigma}{\kappa}T$ (1)

Where ZT Figure of merit; $\alpha$ is the Seebeck coefficient; $\sigma$ is electrical conductivity; $\kappa$ is the thermal conductivity; $T$ is temperature.

The structure optimization of TEG is mainly the size optimization of its electrical components. By optimizing its structural size, the power generation efficiency can be improved. The structure optimization of thermoelectric materials is mainly focused on the application of thermoelectric refrigeration technology [10, 11], while the structure optimization of TEG is still mainly based on the method of classical optimization formula (Eq. (38)) [12] which used average values instead of the actual properties of thermoelectric materials. Daehyun Wee [13] obtained analytical expressions for the maximum power and efficiency of a TEG by constructing a theoretical model of the TEG. Segmented TEG (STEG) is a method to improve the thermoelectric efficiency by placing different thermoelectric materials in the appropriate temperature range, and there are many studies on the size optimization of STEG [14, 15]. Ge et al. [16] proposed a method to optimize the design of STEG based on three-dimensional numerical Simulation and multi-objective genetic algorithm. In this method, both the minimum semiconductor volume and the maximum output power are considered, and the optimization results are obtained. Ali et al. [17] performed the thermal analysis of STEG, and formulated its configurations. The effects of external load resistance, cold and hot junction temperature on the performance of STEG were studied. The effect of temperature on thermoelectric materials is seldom considered in these studies.

Barry et al. [18] optimized the geometric sizes of TEG through the one-dimensional thermal resistance network model, which can improve the maximum output power and conversion efficiency by 10.4% and 3.2%, respectively. Zabrocki et al. [19] Presented an analytical solution based on the Bessel function for one-dimensional thermal energy balance under constant gradient conditions. For constant conductivity, but linear distributions of Seebeck coefficient and thermal conductivity, the first results for the output power of TEG were given. Shen et al. [20] have proposed a comprehensive theoretical model for analyzing the performance of TEGs based on one-dimensional steady analysis. The paper particularly emphasized the impact of side surface heat transfer for two commercial thermoelectric materials; Zhu et al. [21] proposed a one-dimensional numerical model combined with a genetic algorithm to analyze the performance and optimize the design of STEG. Li et al. [22] present a prototype solar thermoelectric generator and a discrete numerical model for the evaluation of the whole system; In Refs [21, 22], the finite element was employed by dividing the thermoelectric arm into many segments along with its height of TEG. In the previous studies, the calculation expression of TEG temperature distribution is not given or the expression is complex.

Due to the low efficiency of the TEG, the internal temperature distribution is mainly determined by the temperatures of the hot and cold ends. Therefore, in the simplified theoretical model developed in this paper, only the influence of the hot and cold end temperature on the TEG thermoelectric arm is considered. Herein, the relationships among the size, position and the temperature of the thermoelectric material were determined by taking into account the combined effect of thermodynamic and thermoelectric performance through a calculus method, which can accurately calculate the voltage, resistance and out power at both the ends of the thermoelectric materials considering the influence of temperature on the material. The results can be used to optimize the size of thermoelectric elements in TEG. Meanwhile, by considering the impact of temperature on the performance of thermoelectric materials, an improved calculation formula of the figure of merit (Z) was also given. This method provides a new idea for the size determination and research of the thermoelectric material in the STEG at constant temperatures. This paper is based on theoretical derivation, belongs to the ideal state of inference, there is still a large error in the application. The purpose of this paper is to provide an idea for everyone in the future, and this paper considers the most important factor (temperature). The results should be used in the classical calculation formula.

2. Principle of calculation

Heat conduction is caused by the temperature gradient inside the object, and solid conduction is the energy transfer caused by the collision of molecules and the migration of free electrons. Without considering the scattering profiles of the air, it can be simplified as a one-dimensional heat conduction process. The mathematical expression of heat transfer $Q$ is obtained by using Fourier heat conduction law [12].

$\displaystyle Q=\frac{\kappa A(T_{h}-T_{c})}{H_{L}}$ (2)

Where $Q$ is the amount of heat transferred; $A$ is the cross-sectional area of the thermoelectric arm; $H_{L}$ is the height of the thermoelectric arm; $T_{c}$ is the cold side temperature; $T_{h}$ is the hot side temperature.

Equation (3) can be rewritten as follow

$\displaystyle Q=\frac{(T_{h}-T_{c})}{H_{L}/\kappa A}$ (3)

Figure 1.

One-dimensional steady heat conduction: (a) a homogeneous plate; (b) two-layer homogeneous plate; (c) n-layer homogeneous plate.

As shown in Fig. 1a, when the thermal conductivity is constant and the temperatures of both surfaces of the homogeneous flat plate are uniform and constant. Equation (4) shows this relation:

$\displaystyle q=\kappa A\frac{T_{1}-T_{2}}{\chi_{2}-\chi_{1}}$ (4)

Where $q$ is the rate of heat flow; $X_{1}$ and $X_{2}$ are the positions of the two ends of the plate; $T_{1}$ is the temperature at $X_{1}$ ; $T_{2}$ Is the temperature at $X_{2}$ .

This theory can be extended to the composite plate shown in Fig. 1b.

Under steady-state conditions, the rate of incoming heat flow from the left surface is equal to the rate of outgoing heat flow from the right surface. Equation (5) shows this relation:

$\displaystyle q=\frac{T_{1}-T_{2}}{(\chi_{2}-\chi_{1})/\kappa_{1}A}=\frac{T_{2% }-T_{3}}{(\chi_{3}-\chi_{2})/\kappa_{2}A}$ (5)

It is extended to the state of the n-layer homogeneous plate, as shown in Fig. 1c. Equation (6) shows this relation:

$\displaystyle q=\frac{T_{1}-T_{2}}{(\chi_{2}-\chi_{1})/\kappa_{1}A}=\frac{T_{2% }-T_{3}}{(\chi_{3}-\chi_{2})/\kappa_{2}A}=\ldots=\frac{T_{n}-T_{n+1}}{(\chi_{n% +1}-\chi_{n})/\kappa_{n}A}$ (6)

For the thermoelectric material in the TEG, the thermal conductivity changes with the change of temperature. Taking $\Delta T$ as the standard, divide it into $n$ layers of flat plate structure, and use Eq. (6) to obtain the following relationship.

$\displaystyle q=\frac{\Delta T}{(\chi_{2}-\chi_{1})/\kappa_{1}A}=\frac{\Delta T% }{(\chi_{3}-\chi_{2})/\kappa_{2}A}=\ldots=\frac{\Delta T}{(\chi_{n+1}-\chi_{n}% )/\kappa_{n}A}$ (7) $\displaystyle\Delta T=\frac{T_{h}-T_{c}}{n}$ (8)

Simplified from Eq. (7), the following equation is obtained.

$\displaystyle\frac{\kappa_{1}}{\chi_{2}-\chi_{1}}=\frac{\kappa_{2}}{\chi_{3}-% \chi_{2}}=\ldots=\frac{\kappa_{n}}{\chi_{n+1}-\chi_{n}}$ (9)

When the height of the n-Layer is $H_{n}$ , according to Eq. (9), the length of the plate of the layer having the temperatures $T_{m}$ and $T_{m+1}$ can be expressed as follows.

$\displaystyle\chi_{m+1}-\chi_{m}=\frac{\kappa_{m}}{\kappa_{1}+\kappa_{2}+% \ldots+\kappa_{n-1}+\kappa_{n}}H_{n}$ (10)

Figure 2.

One-dimensional steady-state heat conduction of PN-type thermoelectric arms in STEG.

For the PN-type thermoelectric arms of STEG, after adding the heat source temperature ( $T_{H}$ ) and the cold source temperature ( $T_{C}$ ), the structure diagram is shown in Fig. 2. For P-type and N-type thermoelectric arms, the thermal conductivities of the thermoelectric materials near the high-temperature region are assumed to be $\kappa_{Nh}$ and $\kappa_{Ph}$ , respectively, and the thermal conductivities of the thermoelectric materials near the low-temperature region are $\kappa_{Nc}$ and $\kappa_{Pc}$ , respectively. Ignoring the influence of the thermal conductivity of the copper connecting piece and the ceramic on the temperature propagation, the cold end temperature and the hot end temperature of the P-type and N-type electric elements are respectively $T_{c}=T_{C}$ and $T_{h}=T_{H}$ , and finally, the following relation can be obtained.

$\displaystyle\left\{{{\begin{array}[]{l}{q_{N}=\frac{\Delta T}{(\chi_{\textit{% NhG}+1}-\chi_{\textit{NhG}})/\kappa_{\textit{NhG}}A_{N}}=\frac{\Delta T}{(\chi% _{\textit{NcJ}+1}-\chi_{\textit{NcJ}})/\kappa_{\textit{NcJ}}A_{N}}}\\ \\ {q_{P}=\frac{\Delta T}{(\chi_{\textit{PhE}+1}-\chi_{\textit{PhE}})/\kappa_{% \textit{PhE}}A_{P}}=\frac{\Delta T}{(\chi_{\textit{PcF}+1}-\chi_{\textit{PcF}}% )/\kappa_{\textit{PcF}}A_{P}}}\\ \\ {g=\frac{T_{h}-T_{a}}{\Delta T},j=\frac{T_{a}-T_{c}}{\Delta T},e=\frac{T_{h}-T% _{b}}{\Delta T},f=\frac{T_{b}-T_{c}}{\Delta T}}\\ \end{array}}}\right.$ (11)

Where $G$ , $J$ , $E$ , $F$ are any integers from 1 to $g$ , $j$ , $e$ and $f$ respectively; $T_{a}$ is the temperature at the junction of the different N-type thermoelectric arm materials; $T_{b}$ is the temperature at the junction of the different P-type thermoelectric arm material; $q_{N}$ is the rate of heat flow of the N-type thermoelectric arm; $q_{P}$ is the rate of heat flow of the P-type thermoelectric arm; $\kappa_{Nh}$ and $\kappa_{Nc}$ are the thermal conductivities of the thermoelectric materials near the heat source temperature and the cold source temperature in the N-type thermoelectric arm, respectively; $\kappa_{Ph}$ and $\kappa_{Pc}$ are the thermal conductivities of the thermoelectric materials near the heat source temperature and the cold source temperature in the P-type thermoelectric arm, respectively; $A_{P}$ and $A_{N}$ are the cross-sectional areas of the P-type and N-type thermoelectric arms, respectively.

Equations (10) and (11) can be combined to obtain Eq. (12).

$\displaystyle\left\{{{\begin{array}[]{l}{H_{Nh}=\frac{\sum\limits_{i=1}^{g}{% \kappa_{\textit{Nhi}}}}{\sum\limits_{i=1}^{g}{\kappa_{\textit{Nhi}}}+\sum% \limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}H}\\ {H_{Nc}=\frac{\sum\limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}{\sum\limits_{i=1}^% {g}{\kappa_{\textit{Nhi}}}+\sum\limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}H}\\ {H_{Ph}=\frac{\sum\limits_{i=1}^{e}{\kappa_{\textit{Phi}}}}{\sum\limits_{i=1}^% {e}{\kappa_{\textit{Phi}}}+\sum\limits_{i=1}^{f}{\kappa_{\textit{Pci}}}}H}\\ {H_{Pc}=\frac{\sum\limits_{i=1}^{f}{\kappa_{\textit{Pci}}}}{\sum\limits_{i=1}^% {e}{\kappa_{\textit{Phi}}}+\sum\limits_{i=1}^{f}{\kappa_{\textit{Pci}}}}H}\\ \end{array}}}\right.$ (12)

Where $H_{Nh}$ and $H_{Nc}$ are respectively the heights close to the thermoelectric materials at the high-temperature end and the low-temperature end in the N-type thermoelectric arm; $H_{Ph}$ and $H_{Pc}$ are respectively the heights close to the thermoelectric materials at the high-temperature end and the low-temperature end in the P-type thermoelectric arm; H is the height of PN-type thermoelectric arms.

Equation (13) can be obtained by Eq. (12).

$\displaystyle\left\{{{\begin{array}[]{l}{\frac{H_{Nc}}{H_{Nh}}=\frac{\sum% \limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}{\sum\limits_{i=1}^{g}{\kappa_{% \textit{Nhi}}}}}\\ {\frac{H_{Pc}}{H_{Ph}}=\frac{\sum\limits_{i=1}^{f}{\kappa_{\textit{Pci}}}}{% \sum\limits_{i=1}^{e}{\kappa_{\textit{Phi}}}}}\\ \end{array}}}\right.$ (13)

When $\Delta T$ approaches infinity to zero, $g$ , $j$ , $e$ and $f$ will approach infinity. When the numerator and denominator of the two formulas in Eq. (13) are multiplied by $\Delta T$ , Eq. (14) is obtained according to the knowledge of calculus.

$\displaystyle\left\{{{\begin{array}[]{l}{\frac{H_{Nc}}{H_{Nh}}=\frac{\int_{T_{% c}}^{T_{a}}{\kappa_{Nc}({T})dT}}{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})dT}}}\\ \\ {\frac{H_{Pc}}{H_{Ph}}=\frac{\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})dT}}{\int_{T% _{b}}^{T_{h}}{\kappa_{Ph}({T})dT}}}\\ \end{array}}}\right.$ (14)

Expressions of $H_{Nc}$ , $H_{Nh}$ , $H_{Pc}$ , and $H_{Ph}$ can be obtained by combining Eqs (12) and (14) as follow.

$\displaystyle\left\{{{\begin{array}[]{l}{H_{Nc}=\frac{\int_{T_{c}}^{T_{a}}{% \kappa_{Nc}({T})dT}}{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})dT+\int_{T_{c}}^{T_{% a}}{\kappa_{Nc}({T})dT}}}H}\\ \\ H_{Pc}=\frac{\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})dT}}{\int_{T_{b}}^{T_{h}}{% \kappa_{Ph}({T})dT+\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})dT}}}H\\ \\ H_{Nh}=H-H_{Nc},H_{Ph}=H-H_{Pc}\\ \end{array}}}\right.$ (15)

According to Eq. (15), the ratio of the heights of different thermoelectric materials in the PN-type thermoelectric arms can be obtained, so that the actual heights of different thermoelectric materials can be calculated. At the same time, the formula can also be extended to the PN-type thermoelectric arm composed of three, four, or even multi-layer thermoelectric materials. P-type and N-type thermoelectric arms mainly have three parameters of length, width and height, so the values of the other two parameters also need to be optimized. The height of the thermoelectric material is optimized for a large voltage value, while the length and width are optimized for smaller resistance values.

In the actual design and production, to ensure the rationality of thermoelectric space, the sum of the cross-sectional area of P-type and N-type thermoelectric arm should be fixed values. When the electrical conductivity of the material is known, the expression for the resistance is as follows.

$\displaystyle R=\frac{L}{\sigma A}=\frac{\rho L}{A}$ (16)

Where $R$ is the resistance of the material; $\rho$ is the resistivity of the material; $A$ is the cross-sectional area of the material.

The resistance of the thermoelectric material at the high-temperature end portion and the low-temperature end portion of each of the N-type and P-type thermoelectric arms in the thermoelectric sheet can be calculated as shown in the following equations in combination with Eqs (10), (15) and (16).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{Nh}=\frac{\sum\limits_{i=1}^{g}{% \frac{\kappa_{\textit{Nhi}}}{\sigma_{\textit{Nhi}}}}}{\sum\limits_{i=1}^{g}{% \kappa_{\textit{Nhi}}}+\sum\limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}\frac{H}{A% _{N}}}\\ {R_{Nc}=\frac{\sum\limits_{i=1}^{j}{\frac{\kappa_{\textit{Nci}}}{\sigma_{% \textit{Nci}}}}}{\sum\limits_{i=1}^{g}{\kappa_{\textit{Nhi}}}+\sum\limits_{i=1% }^{j}{\kappa_{\textit{Nci}}}}\frac{H}{A_{N}}}\\ {R_{Ph}=\frac{\sum\limits_{i=1}^{e}{\frac{\kappa_{\textit{Phi}}}{\sigma_{% \textit{Phi}}}}}{\sum\limits_{i=1}^{e}{\kappa_{\textit{Phi}}}+\sum\limits_{i=1% }^{f}{\kappa_{\textit{Pci}}}}\frac{H}{A_{P}}}\\ {R_{Pc}=\frac{\sum\limits_{i=1}^{f}{\frac{\kappa_{\textit{Pci}}}{\sigma_{% \textit{Pci}}}}}{\sum\limits_{i=1}^{e}{\kappa_{\textit{Phi}}}+\sum\limits_{i=1% }^{f}{\kappa_{\textit{Pci}}}}\frac{H}{A_{P}}}\\ \end{array}}}\right.$ (17)

Where $R_{Nh}$ and $R_{Nc}$ are respectively the resistance of the thermoelectric materials close to the high-temperature end and the low-temperature end in the N-type thermoelectric arm; $R_{Ph}$ and $R_{Pc}$ are respectively the resistance of the thermoelectric materials close to the high-temperature end and the low-temperature end in the P-type thermoelectric arm; $\sigma_{Nh}$ and $\sigma_{Nc}$ are respectively the electrical conductivity of the thermoelectric materials close to the high-temperature end and the low-temperature end in the N-type thermoelectric arm; $\sigma_{Ph}$ and $\sigma_{Pc}$ are respectively the electrical conductivity of the thermoelectric materials close to the high-temperature end and the low-temperature end in the N-type thermoelectric arm.

Multiply the numerator and denominator in Eq. (17) by $\Delta T$ , and the resistance of P-type and N-type thermoelectric arms can be obtained. The resistance values are given as shown in Eq. (18).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{N}=\frac{\int_{T_{a}}^{T_{h}}{% \frac{\kappa_{Nh}({T})}{\sigma_{Nh}({T})}dT}+\int_{T_{c}}^{T_{a}}{\frac{\kappa% _{Nc}({T})}{\sigma_{Nc}({T})}dT}}{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}(T)dT+\int_{% T_{c}}^{T_{a}}{\kappa_{Nc}({T})dT}}}\frac{{H}}{A_{N}}}\\ \\ R_{P}=\frac{\int_{T_{b}}^{T_{h}}{\frac{\kappa_{Ph}({T})}{\sigma_{Ph}({T})}dT}+% \int_{T_{c}}^{T_{b}}{\frac{\kappa_{Pc}({T})}{\sigma_{Pc}({T})}dT}}{\int_{T_{b}% }^{T_{h}}{\kappa_{Ph}({T})T+\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})T}}}\frac{H}{% A_{P}}\\ \\ R_{t}=R_{N}+R_{P}\\ \end{array}}}\right.$ (18)

Where $R_{t}$ is the resistance of PN-type thermoelectric arms; $R_{N}$ , $R_{P}$ are the resistances of the N-type and P-type electrical elements, respectively.

$R_{t}$ is the sum of the resistances of the N-type and P-type thermoelectric arm in the thermoelectric material, and the other parts in the formula are fixed values except $A_{N}$ , $A_{P}$ and $H$ , and $C_{N}$ and $C_{P}$ can be used to replace the parts except $A_{N}$ , $A_{P}$ and $H$ in the formula.

$\displaystyle\left\{{{\begin{array}[]{l}C_{N}=\frac{\int_{T_{a}}^{T_{h}}{\frac% {\kappa_{Nh}({T})}{\sigma_{Nh}({T})}dT}+\int_{T_{c}}^{T_{a}}{\frac{\kappa_{Nc}% ({T})}{\sigma_{Nc}({T})}dT}}{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})dT+\int_{T_{% c}}^{T_{a}}{\kappa_{Nc}({T})dT}}}\\ \\ \quad=\frac{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})\rho{}_{{Nh}}(T)dT}+\int_{T_{% c}}^{T_{a}}{\kappa_{Nc}({T})\rho_{{Nh}}(T)dT}}{\int_{T_{a}}^{T_{h}}{\kappa_{Nh% }({T})dT+\int_{T_{c}}^{T_{a}}{\kappa_{Nc}({T})dT}}}\\ \\ C_{P}=\frac{\int_{T_{b}}^{T_{h}}{\frac{\kappa_{Ph}({T})}{\sigma_{Ph}({T})}dT}+% \int_{T_{c}}^{T_{b}}{\frac{\kappa_{Pc}({T})}{\sigma_{Pc}({T})}dT}}{\int_{T_{b}% }^{T_{h}}{\kappa_{Ph}({T})dT+\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})dT}}}\\ \\ \quad=\frac{\int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})\rho_{Ph}(T)dT}+\int_{T_{c}}^% {T_{b}}{\kappa_{Pc}({T})\rho_{Pc}(T)dT}}{\int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})% dT+\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})dT}}}\\ \\ R_{t}=R_{N}+R_{P}=\frac{C_{N}H}{A_{N}}+\frac{C_{P}H}{A_{t}-A_{N}}\\ \end{array}}}\right.$ (19)

Where $\rho_{Nh}$ and $\rho_{Nc}$ are respectively resistivities of the thermoelectric materials close to the high-temperature end and the low-temperature end in the N-type thermoelectric arm; $\rho_{Ph}$ and $\rho_{Pc}$ are respectively resistivities of the thermoelectric materials close to the high-temperature end and the low-temperature end in the P-type thermoelectric arm; $A_{t}$ are the cross-sectional areas of the PN-type thermoelectric arms.

As can be seen from Eq. (19), when the values of $A_{N}$ and $A_{P}$ are represented by Eq. (20), the obtained resistance $R_{t}$ is minimum.

$\displaystyle\left\{{{\begin{array}[]{l}{A_{N}=\frac{\sqrt{C_{N}}}{\sqrt{C_{N}% }+\sqrt{C_{P}}}A_{t}}\\ \\ {A_{P}=\frac{\sqrt{C_{P}}}{\sqrt{C_{N}}+\sqrt{C_{P}}}A_{t}}\\ \end{array}}}\right.$ (20)

The formula for calculating the areas can be obtained. The area calculation formula can also be extended to the PN-type thermoelectric arms composed of multilayer thermoelectric materials. By substituting Eq. (20) into Eq. (16), the minimum total resistance of the PN-type thermoelectric arms can be calculated as shown in Eq. (21).

$\displaystyle R_{t}=\frac{H\cdot(\sqrt{C_{N}}+\sqrt{C_{P}})^{2}}{A_{t}}$ (21)

Seebeck effect is the main principle of thermoelectric power generation. The voltage generated by thermoelectric materials is related to the Seebeck coefficient and the temperature at both ends, and the relationship is shown in Eq. (22).

$\displaystyle{U}_{e}=\int_{T_{C}}^{T_{H}}{\alpha dT}$ (22)

Where $U_{e}$ is the voltage across the thermoelectric material; $T_{C}$ is the cold side temperature; $T_{H}$ is the hot side temperature.

Equation (23) can be derived from Eq. (22).

$\displaystyle{U}_{e}=\int_{T_{C}}^{T_{H}}{\alpha dT}{=}\int_{T_{C}}^{T_{O}}{% \alpha_{C}dT}+\int_{T_{O}}^{T_{H}}{\alpha_{H}dT}$ (23)

Where $T_{0}$ is the temperature at the material interface of the thermoelectric materials; $\alpha_{C}$ and $\alpha_{H}$ are the Seebeck coefficients of the thermoelectric material near the cold side and the heat side, respectively.

So that, when the temperature value of the cold side and the hot side of different thermoelectric materials of the functionally graded thermoelectric material are determined, the voltage value of the two ends of the thermoelectric material can be obtained by using Eq. (23). According to Eq. (23), when the temperature of the hot end and the cold end is constant, the voltage at the two ends of the PN-type thermoelectric arms has no relation with the cross area of the PN-type thermoelectric arms, namely, when the temperature at the two ends of the PN-type thermoelectric arms is constant as well as the height dimension of the PN-type thermoelectric arms, the voltage generated by the PN-type thermopile arms is constant.

When an external resistor is connected, a closed-loop circuit is formed. Because the efficiency of TEG is very low, only the influence of the temperature of the cold and hot ends on the temperature distribution of the TEG thermoelectric arm is considered. That is, the resistance of the TEG in the closed-loop circuit is also shown in Eq. (21). The maximum out power is related to the voltage and electrical resistance, as shown in Eq. (24).

$\displaystyle{P}=\frac{{U}^{2}}{{4R}}$ (24)

Where $P$ is the power of PN-type thermoelectric arms; $U$ is the voltage across the PN-type thermoelectric arms; $R$ is the resistance of the PN-type thermoelectric arms.

The area of one PN-type thermoelectric arm on the thermoelectric generating piece is the sum of the cross-section areas of the PN-type thermoelectric arms and the average pore areas between all the thermoelectric arms. Therefore, by using Eqs (21) and (24), it can be obtained that the power generated by a PN-type thermoelectric arm in a unit area is shown as the following Eq. (25).

$\displaystyle P_{\textit{avg}}=\frac{P_{NP}}{A_{\textit{total}}}=\frac{U_{NP}^% {2}\cdot A_{t}}{4(\sqrt{C_{N}}+\sqrt{C_{P}})^{2}\cdot H_{L}\cdot A_{\textit{% total}}}$ (25)

Where $P_{\textit{avg}}$ is the power generated per unit area by the PN-type thermoelectric arms; $A_{\textit{total}}$ is the area of PN-type thermoelectric arms in the thermoelectric plate; $P_{NP}$ is the power generated by a PN-type thermoelectric arm; $U_{NP}$ is the voltage generated by a PN-type thermoelectric arm.

According to Eq. (25), the maximum output power of a single PN-type thermoelectric arm is inversely proportional to H and directly proportional to $A_{t}/A_{\textit{total}}$ based on considering only the influence of the temperature of the cold and hot ends on the temperature distribution of the thermoelectric arm of TEG. When the external resistance is connected, the thermal effect of the current and the Peltier effect of thermoelectric material itself will have an impact on the temperature. Because the impact is more complex, this part is not involved.

3. Experimental simulation

Bi ${}_{2}$ Te ${}_{3}$ is generally recognized as the best low-temperature thermoelectric material, and for low-temperature materials, the data of thermoelectric materials on the market can be directly used as a data source. For the medium temperature thermoelectric material, the material properties given in Ref [23] are used as a reference to construct the data of the medium temperature thermoelectric material in the simulation process.

According to the references and relevant data, the relevant properties of the following four thermoelectric materials can be obtained, in which the thermal conductivity is shown in Fig. 3a, the Seebeck coefficient is shown in Fig. 3b, the resistivity is shown as Fig. 3c, and the ZT value is shown as Fig. 3d.

Figure 3.

The material properties of different thermoelectric materials: (a) thermal conductivity; (b) Seebeck coefficient; (c) resistivity; (d) the value of ZT.

As shown in Fig. 3d, the temperature of the ZT value crossing point of the N-type thermoelectric material is about 499 K, and the temperature of the ZT value crossing point of the P-type thermoelectric material is about 576 K. Therefore, when $T_{a}$ is 499 K and $T_{b}$ is 576 K, all thermoelectric materials can be guaranteed to be in the position with high ZT value.

The thermal conductivity fitting equations of the four thermoelectric materials can be obtained by using the fitting function. The relationship between thermal conductivity and temperature can be obtained by fitting the quartic function of thermal conductivity and temperature, as shown in Eq. (26).

$\displaystyle\kappa=f_{\kappa}(T)=b_{4}T^{4}+b_{3}T^{3}+b_{2}T^{2}+b_{1}T+b_{0}$ (26)

Where $b_{0}$ , $b_{1}$ , $b_{2}$ , $b_{3}$ and $b_{4}$ are all constant.

Through fitting, the corresponding $b_{0}$ , $b_{1}$ , $b_{2}$ , $b_{3}$ and $b_{4}$ in the four thermoelectric materials can be obtained as shown in Table 1.

Table 1

When the thermal conductivity is fitted by a quartic function, the values of each coefficient corresponding to different thermoelectric materials

Materials	N:PbTe	P:PbSe	P:Bi ${}_{2}$ Te ${}_{3}$	N:Bi ${}_{2}$ Te ${}_{3}$
$b_{0}$	1.6298907343E+01	1.7013738345E+01	7.7343621080E+00	2.9922894750E+00
$b_{1}$	$-$ 8.8746941000E-02	$-$ 8.7934489000E-02	$-$ 5.6630615000E-02	$-$ 1.3736915000E-02
$b_{2}$	2.1346809440E-04	2.0434804780E-04	2.0554134100E-04	7.1386672150E-05
$b_{3}$	$-$ 2.3951048950E-07	$-$ 2.2285353540E-07	$-$ 3.3225284520E-07	$-$ 1.3289455640E-07
$b_{4}$	1.0227272730E-10	9.2365967370E-11	2.1390374330E-10	8.8852324150E-11

Using the same method, the temperature dependence of the Seebeck coefficient of these four thermoelectric materials is obtained. And the relationship is shown in Eq. (27). The corresponding $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ and $a_{4}$ are shown in Table 2.

$\displaystyle\left|\alpha\right|=f_{\alpha}(T)=a_{4}T^{4}+a_{3}T^{3}+a_{2}T^{2% }+a_{1}T+a_{0}$ (27)

Where $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ and $a_{4}$ are all constant.

Table 2

When the Seebeck coefficients are fitted by a quartic function, the values of each coefficient corresponding to different thermoelectric materials

Materials	N:PbTe	P:PbSe	P:Bi ${}_{2}$ Te ${}_{3}$	N:Bi ${}_{2}$ Te ${}_{3}$
$a_{0}$	3.826923077E+01	1.397727273E+01	$-$ 3.528588177E+02	1.973931275E+02
$a_{1}$	1.531177000E-02	1.652535000E-01	3.459621910E+00	$-$ 5.602997800E-01
$a_{2}$	2.819055940E-04	$-$ 9.960664340E-04	$-$ 7.294050000E-03	2.153600000E-03
$a_{3}$	6.585081590E-07	2.569930070E-06	6.057754010E-06	$-$ 1.981516520E-06
$a_{4}$	$-$ 8.741258740E-10	$-$ 1.573426570E-09	$-$ 1.948169480E-09	$-$ 3.992869880E-10

Using the same method, the temperature dependence of the Seebeck coefficient of these four thermoelectric materials is obtained. And the relationship is shown in Eq. (28). The corresponding $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ and $a_{4}$ are shown in Table 3.

$\displaystyle\rho=f_{\rho}(T)=c_{4}T^{4}+c_{3}T^{3}+c_{2}T^{2}+c_{1}T+c_{0}$ (28)

Where $c_{0}$ , $c_{1}$ , $c_{2}$ , $c_{3}$ and $c_{4}$ are all constant.

Table 3

When the resistivity is fitted by a quartic function, the values of each coefficient corresponding to different thermoelectric materials are given

Materials	N:PbTe	P:PbSe	P:Bi ${}_{2}$ Te ${}_{3}$	N:Bi ${}_{2}$ Te ${}_{3}$
$c_{0}$	$-$ 5.4557109560E-01	$-$ 2.0259207459E+00	$-$ 3.5890756300E-01	7.8791088323E+00
$c_{1}$	2.7838384000E-03	1.6684887300E-02	1.4151128300E-02	$-$ 7.6994564300E-02
$c_{2}$	9.3240093240E-07	$-$ 4.2454545455E-05	$-$ 8.4902646373E-05	2.8363746058E-04
$c_{3}$	$-$ 7.0551670552E-09	4.6604506605E-08	2.0656519951E-07	$-$ 4.4445907034E-07
$c_{4}$	9.3240093240E-12	$-$ 1.3986013986E-11	$-$ 1.6168929110E-10	2.5624571507E-10

A plot of the product of resistivity $\rho$ and thermal conductivity $\kappa$ versus temperature can be obtained by fitting the data. In the process of polynomial fitting, the thermal conductivity and resistivity are fitted by a quartic function, if the product of resistivity and thermal conductivity is accurately expressed, it should be fitted by a function with the highest term as the eighth power. However, in the actual fitting process, it is found that the 5th power fitting can be well in line with the actual curve results as shown in Fig. 4. Hence, the relationship between the product of the thermal conductivity and the resistivity and the temperature can be obtained as shown in Eq. (29).

$\displaystyle\kappa\cdot\rho=f_{\kappa\cdot\rho}(T)=d_{5}T^{5}+d_{4}T^{4}+d_{3% }T^{3}+d_{2}T^{2}+d_{1}T+d_{0}$ (29)

Where $d_{0}$ , $d_{1}$ , $d_{2}$ , $d_{3}$ , $d_{4}$ and $d_{5}$ are all constant.

The values of $d_{0}$ , $d_{1}$ , $d_{2}$ , $d_{3}$ , $d_{4}$ and $d_{5}$ in different thermoelectric materials can be obtained using the above method. The results are shown in Table 4.

Table 4

When the product of thermal conductivity and resistivity is fitted by a quintic function, the values of various coefficients corresponding to different thermoelectric materials

Materials	N:PbTe	P:PbSe	P:Bi ${}_{2}$ Te ${}_{3}$	N:Bi ${}_{2}$ Te ${}_{3}$
$d_{0}$	$-$ 1.2987423500E-01	$-$ 1.5350611400E-01	2.6703011400E-01	1.2120510300E-01
$d_{1}$	1.3797930000E-03	1.6372040000E-03	$-$ 3.1135530000E-03	$-$ 9.3609498220E-04
$d_{2}$	$-$ 5.5013629290E-06	$-$ 6.3118462040E-06	1.5026241980E-05	1.8970318540E-06
$d_{3}$	1.1006901090E-08	1.1866567070E-08	$-$ 3.6780260560E-08	2.4082890920E-09
$d_{4}$	$-$ 1.0886782160E-11	$-$ 1.0887174210E-11	4.6271294920E-11	$-$ 1.0867727080E-11
$d_{5}$	4.2720394200E-15	3.9487126960E-15	$-$ 2.3339692690E-14	8.7230823400E-15

For the TEG, based on existing products on the market, it can be assumed that the height of the PN-type thermoelectric arms is 2 mm, the sum of the cross-sectional areas of both is 2 mm ${}^{2}$ , and the width of its electrical element is constant at 1 mm. We choose the copper sheet as the connecting sheet between the P-type and N-type thermoelectric arm, and the length, width and height of the copper sheet are respectively replaced by 2.5 mm, 1.0 mm and 0.2 mm. When the hot end temperature is assumed to be 700 K and the cold end temperature is assumed to be 330 K, the length and height parameters of different thermoelectric materials can be optimized by the above method. The dimension of each thermoelectric material can be calculated according to Eqs (15) and (20), as shown in Table 5.

Table 5

When $T_{c}=$ 330 K and $T_{h}=$ 700 K, the corresponding size of different thermoelectric materials

Materials	Length (mm)	Width (mm)	Height (mm)
N:PbTe	0.934	1	0.771
P:PbSe	1.066	1	0.473
P:Bi ${}_{2}$ Te ${}_{3}$	1.066	1	1.527
N:Bi ${}_{2}$ Te ${}_{3}$	0.934	1	1.229

Figure 4.

Product of thermal conductivity and resistivity of different thermoelectric materials is fitted by a quintic function.

According to the dimensions in Table 5, the corresponding three-dimensional model drawings are constructed for simulation analysis. The contact resistance is not considered in the software simulation, and the influence of the outermost ceramic layer on the thermal conductivity is also ignored. The values of Seebeck coefficient, thermal conductivity and Resistivity of the copper sheet were 0 $\mu$ V/K, 390 W/m $\cdot$ K and 2.93 $\times$ 10 ${}^{9}\Omega\cdot$ m, respectively. The material property parameters used in the analysis are shown in Fig. 3.

During the analysis, $T_{h}=$ 700 K, $T_{c}=$ 330 K, and the potential at the low-temperature end of the N-type thermoelectric arm were set to 0 mV. In the practical application process, each thermoelectric generating piece comprises a plurality of PN-type thermoelectric arms, so that a thermoelectric generating piece can be assumed to comprise n PN-type thermoelectric arms; and according to the series-parallel connection law, obviously, the voltage generated by the thermoelectric piece is n times of that of the PN-type thermoelectric arms, and the resistance is also n times of that of the PN-type thermopile arms. According to Eq. (24), the generated power is also n times the power generated by PN type thermoelectric arms. Therefore, in the analysis process, only one of the PN-type thermoelectric arms needs to be simulated, and the result multiplied by the corresponding multiple is the overall result of the thermoelectric sheet.

In this paper, the thermoelectric analysis module of the finite element simulation software COMSOL Multiphysics was used for the finite element analysis. After adding the boundary conditions according to the above requirements, the results are shown in Fig. 5.

Figure 5.

Potential of PN-type thermoelectric arms at $T_{c}=$ 330 K and $T_{h}=$ 700 K.

At this temperature, the thermal conductivity of copper is much higher than that of thermoelectric materials, so the effect of copper on temperature is ignored. Using Eq. (23), Eq. (27) and the data in Table 2, the module produces a voltage value of 127.954 mV at a hot end temperature of 700 K and a cold end temperature of 330 K. The final result is 0.119 mV more than the COMSOL calculation. There may be two main reasons. One is that the impact of copper on temperature is not omitted in the calculation process of COMSOL software. Secondly, in the calculation process of COMSOL software, the material properties are input in the form of multiple points, rather than in accordance with the formula about temperature, and the curve in COMSOL is not exactly the same as the fitted curve.

Assuming that the cold and hot end temperatures of the thermoelectric material are two groups of 300 K/800 K and 400 K/700 K respectively, the dimensions of each thermoelectric material can be calculated as shown in Table 6 according to Eqs (15) and (20).

Table 6

The corresponding sizes of different thermoelectric materials were measured when the hot and cold temperatures were 300 K, 800 k and 400 K, 700 K, respectively

$T_{h}$ / $T_{c}$ (K)	800/300			700/400
Materials	Length (mm)	Width (mm)	Height (mm)	Length (mm)	Width (mm)	Height (mm)
N:PbTe	0.953	1	0.861	0.944	1	1.020
P:PbSe	1.047	1	0.635	1.056	1	0.580
P:Bi ${}_{2}$ Te ${}_{3}$	1.047	1	1.365	1.056	1	1.420
N:Bi ${}_{2}$ Te ${}_{3}$	0.953	1	1.139	0.944	1	0.980

The simulation is carried out by COMSOL. According to Eqs (23) and (27, when $T_{c}=$ 300 K and $T_{h}=$ 800 K, the voltage between two ends of PN-type thermoelectric arms is 175.720 mV, where the voltage between two ends of an N-type thermoelectric arm is 91.867 mV, and the voltage across two ends of a P-type thermoelectrical arm is 83.852 mV. The results of the COMSOL calculation are shown in Fig. 6a. By comparing the data of the two, it is found that the voltage difference between the two ends of the P-type thermoelectric arm is 0.004 mV, while the voltage difference between the two ends of the N-type thermoelectric arm is significantly higher than that of the P-type thermoelectric arm, about 0.008 mV. This may be because the height of the N-type PbTe thermoelectric material is 0.635 mm, while under the assumed conditions, its ideal height corresponds to 0.6354 mm. The temperature of $T_{a}$ corresponding to 0.635 mm should be 576.1 K, and the voltage across the corresponding N-type thermoelectric arm should be 83.860 mV, and compared with the simulation result of COMSOL, the voltage difference is less than 0.001 mV. The voltage difference of the PN-type thermoelectric arms connected by the copper sheet is obviously larger than that of a single thermoelectric arm, and the reason may be that the thermal resistance generated by the added copper sheet leads to the increase of the difference between the two calculation results. Even so, the difference is only 0.154 mV.

Figure 6.

(a) Corresponding potential when $T_{c}=$ 300 K and $T_{h}=$ 800 K; (b) Corresponding potential when $T_{c}=$ 400 K and $T_{h}=$ 700 K (the left figure is the potential of individual P-type and N-type thermoelectric arms, and the right figure is the potential of PN-type thermoelectric arms connected by copper sheets).

According to Eqs (23) and (27), when $T_{c}=$ 400 K and $T_{h}=$ 700 K, the voltage between the two ends of a PN-type type thermoelectric arms is 101.812 mV, wherein the voltage between the two ends of an N-type type thermoelectric arm is 53.614 mV, and that of a P-type thermoelectric arm is 48.199 mV. The results obtained in COMSOL are shown in Fig. 6b. By comparison, it is found that under this condition, the error of the voltage value of the single thermoelectric arm is less than 0.002 mV, and the difference of the voltage value of the PN-type thermoelectric arms connected by copper sheets is only about 0.1 mV. Therefore, it can be concluded that the theory proposed in this paper is reasonable for the analysis of temperature distribution for I $=$ 0 case. The voltage value obtained by the method is effective. Therefore, it can also be considered that the inference of the resistance value obtained based on the analysis result of the temperature distribution is also reasonable.

4. Analysis and discussion

If we do not consider the heat energy produced by current and the influence of Peltier effect on temperature distribution of TEG, the resistance value of PN-type thermoelectric arms can be calculated according to Eq. (18). The resistivity of the copper sheet is much smaller than that of the thermoelectric material, and the thermal conductivity of the copper sheet is much larger than that of the thermoelectric materials. Therefore, the resistivity and thermal resistance of the copper sheet is not considered in the analysis process.

The low-temperature thermoelectric material is a Bi ${}_{2}$ Te ${}_{3}$ -based semiconductor, and the melting point of the material is only about 700 K, so in the application process, to ensure the service life and performance of the material, the temperature change range is controlled below 600 K. Base on that premise, the resistance value and the voltage value of the PN-type thermoelectric arms can be calculated through Eqs (21) and (23, and then the power of the PN-type thermoelectric arms can be calculated through Eq. (24).

When $T_{h}=$ 700 K and $T_{c}=$ 330 K, the relationship between the maximum power of PN-type thermoelectric arms and $T_{a}$ , $T_{b}$ can be calculated as shown in Fig. 7a.

It is found that the resistance is 37.163 m $\Omega$ and the maximum out power is 0.1101 W when the temperature of $T_{a}$ is 499 K and the temperature of $T_{b}$ is 576 K. Obviously, the size of the PN-type thermoelectric arms corresponding to this position is not the optimal size. As can be seen from Fig. 7a, when the temperature of $T_{b}$ is 600 K, the maximum power position is not reached yet, and the temperature of $T_{b}$ is selected to be 600 K in consideration of the performance of the material. The relationship between $T_{a}$ and its minimum resistance and maximum power can be obtained as shown in Fig. 7b.

According to Fig. 7a and in combination with Fig. 7b, when the temperature of $T_{b}$ is 600 K and the temperature of $T_{a}$ is 573 K, the minimum resistance corresponding to the PN-type thermoelectric arm is 37.433 m $\Omega$ , the voltage is 130.820 mV, and the maximum power is 0.1143 W. At this time, the corresponding power is the optimal solution within the constraints. The size of the PN-type thermoelectric arms corresponding to this condition is shown in Table 7.

Table 7
Dimensions of different thermoelectric materials at maximum power

Materials	Length (mm)	width (mm)	height (mm)
N:PbTe	0.924	1	0.391
P:PbSe	1.076	1	0.352
P:Bi ${}_{2}$ Te ${}_{3}$	1.076	1	1.648
N:Bi ${}_{2}$ Te ${}_{3}$	0.924	1	1.608

Figure 7.

(a) Relationship between $T_{a}$ , $T_{b}$ and a maximum power of PN-type thermoelectric arms; (b) Relation between $T_{a}$ and the maximum power and the minimum resistance of PN-type thermoelectric arms when $T_{b}=$ 600 K.

In the classical formula [24] for size optimization of TEG, the resistance and voltage are calculated as shown in Eq. (30).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{t}=\frac{\overline{\rho_{N}}\cdot H% _{N}}{A_{C}}+\frac{\overline{\rho_{P}}\cdot H_{P}}{A_{P}}}\\ {{U}_{{PN}}=\int_{T_{c}}^{T_{h}}{(\left|{\alpha_{P}}\right|+\left|{\alpha_{N}}% \right|)dT}}\\ \end{array}}}\right.$ (30)

Where $\rho_{P}$ , $\rho_{N}$ represent the resistivity of the P-type and N-type thermoelectric arms, respectively; $\overline{\rho_{P}}$ , $\overline{\rho_{N}}$ are the average values of the resistivities of the P-type and N-type thermoelectric arm in the temperature range between $T_{h}$ and $T_{c}$ , respectively; $\alpha_{P}$ , $\alpha_{N}$ represent the Seebeck coefficient of the P-type and N-type thermoelectric arms, respectively; $U_{PN}$ is the voltage across the PN-type thermoelectric arms.

For a function containing $T$ , the average value is calculated as Eq. (31)

$\displaystyle\overline{f(T)}=\frac{\int_{T_{c}}^{T_{h}}{f(T)dT}}{T_{h}-T_{c}}$ (31)

By substituting Eq. (31) into Eq. (5), it is found that the relationship between the height of different thermoelectric materials and the temperature of the STEG in the classical optimization method is consistent with Eq. (14). Meanwhile, by analyzing Eq. (30), it can be known that when the relationship between $A_{N}$ and $A_{P}$ is as shown in Eq. (32), the corresponding output power is maximum.

$\displaystyle\frac{A_{P}}{A_{N}}=\frac{\overline{\rho_{P}}\cdot H_{P}}{% \overline{\rho_{N}}\cdot H_{N}}$ (32)

The relationship between $T_{a}$ , $T_{b}$ and the corresponding maximum output power of the TEG calculated in the classical optimization mode can be obtained by using Eq. (30), as shown in Fig. 8.

From Fig. 8, it can be seen that when $T_{a}=$ 590 K and $T_{b}=$ 600 K, the calculated output power is the largest in the classic optimization mode. The corresponding dimensions of the thermoelectric material are shown in Table 8.

Table 8

The size parameters of different thermoelectric materials at maximum output power are obtained by the classical method

Materials	Length (mm)	width (mm)	height (mm)
N:PbTe	0.924	1	0.326
P:PbSe	1.076	1	0.352
P:Bi ${}_{2}$ Te ${}_{3}$	1.076	1	1.648
N:Bi ${}_{2}$ Te ${}_{3}$	0.924	1	1.674

Figure 8.

Relationship between $T_{a}$ , $T_{b}$ and the maximum output power of PN-type thermoelectric arms obtained by the classical method.

The parameters in Table 8 are selected as the design parameters. Under the theoretical calculation condition, when the relationship between each parameter of the thermoelectric material and the temperature is considered, the output power of a single PN-type thermoelectric arms under this condition can be calculated as 0.1140 W according to Eq. (25). When $T_{h}=$ 700 K, $T_{c}=$ 330 K, the results of the proposed optimization method are 0.22% higher than those of the classical optimization method. It can be seen that the maximum output power calculated by the classical optimization method also has a certain error because the relationship between the performance of the material and the temperature is not considered.

When considering the Peltier effect and Joule heating effect to calculate the maximum efficiency of TEG, the expression can be known from Ref 12 as shown in Eq. (33).

$\displaystyle\eta=\frac{T_{h}-T_{c}}{T_{h}}\cdot\frac{\frac{m}{m+1}}{1+\frac{% KR_{t}}{\alpha_{PN}}\cdot\frac{m+1}{T_{h}}-\frac{1}{2}\cdot\frac{T_{h}-T_{c}}{% T_{h}}\cdot\frac{1}{m+1}}$ (33)

Where $m$ is the ratio of external resistance to internal resistance; $\alpha_{PN}$ is the average value of the Seebeck coefficient of the PN-type thermoelectric arms in the temperature range between $T_{h}$ and $T_{c}$ ; $K$ is the thermal conductance; $\eta$ is the power generation efficiency.

As can be seen from Eq. (33), only $Z$ (as shown in Eq. (34)) is determined depending on the property and size parameters of the thermoelectric material. At the same time, the efficiency increases as $Z$ increases.

$\displaystyle Z=\frac{\alpha_{PN}^{2}}{KR_{t}}$ (34)

The calculation expression of $K$ is as shown in Eq. (35)

$\displaystyle q_{t}=K(T_{h}-T_{c})$ (35)

Where $q_{t}$ is the rate of heat flow of the PN-type thermoelectric arms.

Equation (36) is derived from Eqs (11) and (35).

$\displaystyle K=K_{P}+K_{N}=\frac{\int_{T_{c}}^{T_{h}}{\kappa_{P}dT}\cdot A_{P% }}{H_{P}(T_{h}-T_{c})}+\frac{\int_{T_{c}}^{T_{h}}{\kappa_{N}}dT\cdot A_{N}}{H_% {N}(T_{h}-T_{c})}$ (36)

Where $\kappa_{P}$ , $\kappa_{N}$ represent the thermal conductivity of the P-type and N-type thermoelectric arms, respectively.

Substituting Eqs (18) and (36) into Eq. (34, it can be obtained that the corresponding efficiency is maximum when the dimension parameter value is as shown in Eq. (37).

$\displaystyle\frac{H_{N}A_{P}}{A_{N}H_{P}}=\frac{\int_{T_{c}}^{T_{h}}{\kappa_{% N}dT\cdot\sqrt{\int_{T_{c}}^{T_{h}}{\kappa_{P}\cdot\rho_{P}dT}}}}{\int_{T_{c}}% ^{T_{h}}{\kappa_{P}dT\cdot\sqrt{\int_{T_{c}}^{T_{h}}{\kappa_{N}\cdot\rho_{N}dT% }}}}$ (37) $\displaystyle\quad=\frac{(\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})dT+\int_{T_{c}}% ^{T_{a}}{\kappa_{Nc}({T})dT}})\cdot\sqrt{\int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})% \rho_{Ph}(T)dT}+\int_{T_{c}}^{T_{b}}{\kappa_{Pc}({T})\rho_{Pc}(T)dT}}}{(\int_{% T_{a}}^{T_{h}}{\kappa_{Ph}({T})dT+\int_{T_{c}}^{T_{a}}{\kappa_{Pc}({T})dT}})% \cdot\sqrt{\int_{T_{a}}^{T_{h}}{\kappa_{Nh}({T})\rho_{{Nh}}(T)dT}+\int_{T_{c}}% ^{T_{a}}{\kappa_{Nc}({T})\rho_{{Nh}}(T)dT}}}$

At the same time, it can be seen from Eq. (37) that if $\rho$ and $\kappa$ are calculated by taking the average value, the obtained result is as shown in Eq. (38)

$\displaystyle\frac{H_{N}A_{P}}{A_{N}H_{P}}=\frac{\sqrt{\overline{\kappa_{N}}% \cdot\overline{\rho_{P}}}}{\sqrt{\overline{\kappa_{P}}\cdot\overline{\rho_{N}}}}$ (38)

Where $\overline{\kappa_{P}}$ , $\overline{\kappa_{N}}$ are the average values of the resistivities of the P-type and N-type thermoelectric arms in the temperature range between $T_{h}$ and $T_{c}$ , respectively.

Equation (38) is the most commonly used classical formula in the optimization process of TEG, which does not take into account the effect of temperature on the performance of thermoelectric material. When the influence of temperature on thermoelectric materials is considered, Eq. (37) can be obtained. Obviously, the classical formula (Eq. (38)) is only a special solution of Eq. (37) when the influence of temperature on thermoelectric materials is not considered.

By substituting the result of Eq. (37) into Eq. (34), a method for calculating the $Z$ value in consideration of the influence of temperature on the performance of the thermoelectric material can be obtained, and the result is shown in Eq. (39).

$\displaystyle Z=\frac{\alpha_{PN}^{2}}{\left(\sqrt{\int_{T_{c}}^{T_{h}}{\kappa% _{N}\cdot\rho_{N}dT}}+\sqrt{\int_{T_{c}}^{T_{h}}{\kappa_{P}\cdot\rho_{P}dT}}% \right)^{2}}(T_{h}-T_{c})$ (39)

The value of $Z$ bears no relationship to the size of the thermoelectric material, and is only related to the performance of the material itself. The value of $Z$ of the thermoelectric material in a certain temperature range can be calculated by using Eq. (39), so that further theoretical calculation can be carried out according to the value of $Z$ .

5. Conclusions

In this paper, a new calculation method is proposed taking into account the combined effect of thermodynamic and thermoelectric performance, which can calculate the appropriate size of thermoelectric materials with different properties in the application of Segmented Thermoelectric Generator (STEG). By adjusting the size of thermoelectric materials with different properties, each of them can be placed in an appropriate temperature range. Using this method, the temperature variation of each kind of thermoelectric material can be constrained in a certain range, thus enabling the thermoelectric material to be used with a higher ZT value.

It is generally believed that the thermoelectric material has better performance when the corresponding ZT value is higher. However, when the temperature at the junction of the different N-type thermoelectric arm materials ( $T_{a}$ ) and the temperature at the junction of the different P-type thermoelectric arm materials ( $T_{b}$ ) are the temperatures at the ZT intersections of P-type and N-type thermoelectric arm materials, the corresponding maximum output power is not the optimized value. Therefore, it is not feasible to simply use the magnitude of the ZT value as a reference for $T_{a}$ and $T_{b}$ . When the maximum output power is used as the optimization target, the calculation on the $T_{a}$ and $T_{b}$ could be a hotspot of future research.

At present, the ZT value is still an important basis for judging the performance of thermoelectric materials. This paper gives the calculation method of ZT value in a certain temperature range of STEG. It provides a reference for the future performance comparison of STEG. At the same time, the method also gives the corresponding size parameters of STEG when the Z value is maximum. The results of this research are based on considering only the influence of the temperature of the cold and hot ends on the temperature distribution of the TEG thermoelectric arm. Future works can be conducted on more factors to increase the accuracy of the results.

Footnotes

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Funding

This work was financially sponsored by the National Natural Science Foundation of China (51879117).

This work was also supported by the Guiding (Key) Project of Science & Technology Department of Fujian Province (2016H0024) and Natural Science Foundation of Fujian Province in China (2023J01146).

References

Liu

Deng

Wang

Liu

Wang

. Multi-objective optimization of heat exchanger in an automotive exhaust thermoelectric generator. Applied Thermal Engineering. 2016; 108: 916-926.

Wen

Wang

Zhou

. Experimental investigation of header configuration improvement in plate-fin heat exchanger. Applied Thermal Engineering. 2007; 27(11-12): 1761-1770.

Niu

Diao

Kui

Qing

Gequn

. Investigation and design optimization of exhaust-based thermoelectric generator system for internal combustion engine. Energy Conversion and Management. 2014; 85: 85-101.

Kraemer

Jie

McEnaney

Cao

Liu

Weinstein

Loomis

Ren

Chen

. Concentrating solar thermoelectric generators with a peak efficiency of 74%. Nature Energy. 2016; 1(11): 1-8.

Liu

Qin

Shen

Gong

. High-performance terrestrial solar thermoelectric generators without optical concentration for residential and commercial rooftops. Energy Conversion and Management. 2019; 196: 69-76.

Poudel

Hao

Lan

Minnich

Yan

Wang

Muto

Vashaee

Chen

liu

Dresselhaus

Chen

Ren

. High-thermoelec-tric performance of nanostructured bismuth antimony telluride bulk alloys. Science. 2008; 320(5876): 634-638.

Pei

Wang

Snyder

. Band engineering of thermoelectric materials. Advanced Materials. 2012; 24(46): 6125-6135.

Bathula

Jayasimhadri

Singh

Srivastava

Pulikkotil

Dhar

Budhani

. Enhanced thermoelectric figure-of-merit in spark plasma sintered nanostructured n-type SiGe alloys. Applied Physics Letters. 2012; 101(21): 213902.

Hinterleitner

Knapp

Poneder

Shi

Müller

Eguchi

Eisenmenger-Sittner

Stöger-Pollach

Kakefuda

Kawamoto

Guo

Baba

Mori

Ullah

Chen

Bauer

. Thermoelectric performance of a metastable thin-film Heusler alloy. Nature. 2019; 576(7785): 85-90.

10.

Wang

Leng

Wang

. Parameter analysis and optimal design for two-stage thermoelectric cooler. Applied Energy. 2015; 154: 1-12.

11.

Gong

Gao

. Numerical simulation on a compact thermoelectric cooler for the optimized design. Applied Thermal Engineering. 2019; 146: 815-825.

12.

Zhang

. Thermoelectric Technology, Tianjin China: Tianjin Science and Technology Press. 2013: 14-30.

13.

Wee

. Analysis of thermoelectric energy conversion efficiency with linear and nonlinear temperature dependence in material properties. Energy Conversion and Management. 2011; 52(12): 3383-3390.

14.

Zhang

Liao

Tang

Ming

Qiu

Bai

Shi

Uher

Chen

. Realizing a thermoelectric conversion efficiency of 12% in bismuth telluride/skutterudite segmented modules through full-parameter optimization and energy-loss minimized integration. Energy & Environmental Science. 2017; 10(4): 956-963.

15.

Ngan

Christensen

Snyder

Hung

Linderoth

Nong

Pryds

. Towards high efficiency segmented thermoelectric unicouples. Physica Status Solidi (a). 2014; 211(1): 9-17.

16.

Liu

Sun

Liu

. Optimal design of a segmented thermoelectric generator based on three-dimensional numerical simulation and multi-objective genetic algorithm. Energy. 2018; 147: 1060-1069.

17.

Ali

Yilbas

. Configuration of segmented leg for the enhanced performance of segmented thermoelectric generator. International Journal of Energy Research. 2017; 41(2): 274-288.

18.

Barry

Agbim

Rao

Clifford

Reddy

BVK

Chyu

. Geometric optimization of thermoelectric elements for maximum efficiency and power output. Energy. 2016; 112: 388-407.

19.

Zabrocki

Müller

Seifert

. One-dimensional modeling of thermogenerator elements with linear material profiles. Journal of Electronic Materials. 2010; 39(9): 1724-1729.

20.

Shen

Xiao

Yin

. Theoretical modeling of thermoelectric generator with particular emphasis on the effect of side surface heat transfer. Energy. 2016; 95: 367-379.

21.

Zhu

Chen

Tian

Kang

Jiang

Qiu

. Optimization analysis of a segmented thermoelectric generator based on genetic algorithm. Renewable Energy. 2020; 156: 710-718.

22.

Cai

Zhai

. Design of a concentration solar thermoelectric generator. Journal of Electronic Materials. 2010; 39(9): 1522-1530.

23.

Chen

Liu

Shi

. Thermoelectric materials and devices. Elsevier. 2020: 89-94.

24.

Hadjistassou

Kyriakides

Georgiou

. Designing high efficiency segmented thermoelectric generators. Energy Conversion and Management. 2013; 66: 165-172.

25.

Ponnusamy

de Boor

Müller

. Using the constant properties model for accurate performance estimation of thermoelectric generator elements. Applied Energy. 2020; 262: 114587.

26.

Ponnusamy

de Boor

Müller

. Discrepancy between constant properties model and temperature-dependent material properties for performance estimation of thermoelectric generators. Entropy. 2020; 22: 1128.

27.

Ryu

Chung

Choi

E-A

Ziolkowski

Müller

Park

. Counterintuitive example on relation between ZT and thermoelectric efficiency. Appl. Phys. Lett. 2020; 116: 193903.

Optimization of segmented thermoelectric power generators from waste heat while considering the influence of temperature on materials

Abstract

Keywords

1. Introduction

Table 7 Dimensions of different thermoelectric materials at maximum power

Footnotes

Conflict of interest

Funding

References

Table 7
Dimensions of different thermoelectric materials at maximum power