Optimization and comparison of two-stage thermoelectric generators considering the influence of temperature variation on materials for waste heat utilization

Abstract

Thermoelectricity technology, as a kind of cost-effective and pollution-free power generation solution, is often used for waste heat recovery and utilization. In this paper, the temperature distribution of a Two-stage Thermoelectric Generator (TTEG) under constant temperature conditions has been studied using a one-dimensional heat conduction model. Moreover, by combining the obtained temperature distribution with the three-dimensional size of TTEG, a calculation formula of resistance and voltage was developed based on the calculus method. When the sum of cross-sectional areas of all the PN-type thermoelectric arms respectively in high- and low-temperature layers is constant, the optimal ratio between cross-sectional areas of a single PN-thermoelectric arm respectively in high- and low-temperature layers can be calculated using the proposed formula in this study to achieve the maximum output power. Results also showed the relationship between the heights of PN-type thermoelectric arms and the temperature distributions in high- and low-temperature layers. Using PbTe as the medium temperature thermoelectric material and Bi ${}_{2}$ Te ${}_{3}$ as the low temperature thermoelectric material, a case study was conducted on the PN-type thermoelectrics with the same total height and the same total cross-sectional area. The theoretical calculation results showed that the bigger of maximum output power between the two-stage thermoelectric generator and that of the Segmented Thermoelectric Generator (STEG) is related to the hot and cold end temperature.

Keywords

Output power two-stage thermoelectric generator calculus

Nomenclature

$Q$	transferred heat
$A$	the cross-sectional area of the material
$T_{h},T_{c}$	the temperatures of the hot end and the cold end of the material, respectively
$L$	the length of the material
$Z$	the figure of merit
$A_{h},A_{c}$	the cross-sectional areas of a single PN-type thermoelectric arm in high- and low-temperature layers, respectively
$A_{Nh},A_{Ph}$	the cross-sectional areas of N-type and P-type thermoelectric arms in the high temperature layer, respectively
$A_{Nc},A_{Pc}$	the cross-sectional areas of the N-type and P-type thermoelectric arms of the low temperature layer, respectively
$A_{\textit{Nht}},A_{\textit{Pht}}$	the sums of the cross-sectional areas of all the N-type and all the P-type thermoelectric arms in the high-temperature layer, respectively
$A_{\textit{Nct}},A_{\textit{Pct}}$	the sums of the cross-sectional areas of all the N-type and all the P-type thermoelectric arms in the low-temperature layer, respectively
$A_{ht},A_{ct}$	the sums of the cross-sectional areas of all the PN-type thermoelectric arms in the high-temperature layer and in the low-temperature layer, respectively
$A_{m}$	the effective heat conduction area of the middle ceramic
$A_{NS},A_{PS}$	the cross-sectional areas of the N-type and P-type thermoelectric arms in the STEG respectively
$R$	resistances
$R_{h},R_{c}$	the resistances of a single PN-type thermoelectric arms in the high- and low-temperature layer, respectively
$R_{Nh},R_{Ph}$	the resistances of N-type and P-type thermoelectric arms of the high-temperature layer respectively
$R_{Nc},R_{Pc}$	the resistances of N-type and P-type thermoelectric arms of the low-temperature layer respectively
$R_{NS},R_{PS}$	the resistances of N-type and P-type thermoelectric arms in the STEG
$R_{t}$	the total resistance of the TTEG
$R_{tS}$	the resistance of a single PN-type thermoelectric arm in the STEG
$H_{S}$	the heights of PN-type thermoelectric arms in the STEG
$H_{m}$	the height of the intermediate ceramic
$H_{h},H_{c}$	the heights of the PN-type thermoelectric arms in the high-temperature layer and the low-temperature layer, respectively
$U_{c},U_{h}$	the voltages generated by a single PN-type thermoelectric arm in the high-temperature layer and the low-temperature layer respectively
$P$	output power
$P_{\textit{avg}}$	the output power generated by the TTEG in a unit area
$A_{mt}$	the total area of the intermediate ceramic
$q$	The rate of heat flow
$q_{Nh},q_{Ph}$	The rate of heat flow of the high-temperature layer passing through the N-type and P-type thermoelectric arms, respectively

$q_{Nc},q_{Pc}$	the rate of heat flow of the low-temperature layer passing through the N-type and P-type thermoelectric arms, respectively
$q_{m}$	the rate of heat flow through the middle ceramic
$q_{t}$	the rate of heat flow through the TTEG
$T_{b},T_{a}$	the hot end and the cold end temperature of the inter ceramic, respectively
$T_{NS},T_{PS}$	the temperature at the junction of the different N-type and P-type thermoelectric arm materials, respectively
$N_{h},N_{c}$	the numbers of PN-type thermoelectric arms in the high-temperature layer and the low-temperature layer, respectively

Greek symbols

$\alpha$	seebeck coefficient
$\kappa$	thermal conductivity
$\sigma$	electrical conductivity
$\rho$	resistivity
$\alpha_{Nh},\alpha_{Ph}$	the seebeck coefficient of N-type and P-type thermoelectric arms in the high-temperature layer, respectively
$\alpha_{Nc},\alpha_{Pc}$	the seebeck coefficient of N-type and P-type thermoelectric arms in the low-temperature layer, respectively
$\kappa_{Nh},\kappa_{Ph}$	the thermal conductivities of N-type and P-type thermoelectric arms in the high-temperature layer, respectively
$\kappa_{Nc},\kappa_{Pc}$	the thermal conductivities of N-type and P-type thermoelectric arms in the low-temperature layer, respectively
$\kappa_{\textit{NhS}},\kappa_{\textit{PhS}}$	the thermal conductivities of N-type and P-type thermoelectric materials at the high-temperature side, respectively
$\kappa_{\textit{NcS}},\kappa_{\textit{PcS}}$	the thermal conductivities of N-type and P-type thermoelectric materials at low temperature, respectively
$\kappa_{m}$	the thermal conductivity of the inter ceramic
$\sigma_{Nh},\sigma_{Ph}$	electrical conductivity of N-type and P-type thermoelectric arms in the high-temperature layer, respectively
$\sigma_{Nc},\sigma_{Pc}$	electrical conductivity of N-type and P-type thermoelectric arms in the low-temperature layer, respectively
$\sigma_{\textit{NhS}},\sigma_{\textit{PhS}}$	the electrical conductivity of N-type and P-type thermoelectric materials at the high-temperature side, respectively
$\sigma_{\textit{NcS}},\sigma_{\textit{PcS}}$	the electrical conductivity of the low-temperature layer of N-type and P-type thermoelectric materials, respectively
$\rho_{Nh},\rho_{Ph}$	the resistivities of the N-type and P-type thermoelectric arms of the high-temperature layer respectively
$\rho_{Nc},\rho_{Pc}$	the resistivities of the N-type and P-type thermoelectric arms of the low-temperature layer respectively
$\rho_{\textit{NhS}},\rho_{\textit{PhS}}$	the resistivities of N-type and P-type thermoelectric materials at the high-temperature side, respectively
$\rho_{\textit{NcS}},\rho_{\textit{PcS}}$	the resistivities of the low-temperature layer of N-type and P-type thermoelectric materials, respectively

Abbreviations

TEG	Thermoelectric Generator
TTEG	Two-stage Thermoelectric Generator
STEG	Segmented Thermoelectric Generator
Bi ${}_{2}$ Te ${}_{3}$	Bismuth telluride
PeTe	lead telluride

Subscripts

$h$	Hot side
$c$	Cold side
$P$	P-type Thermoelectric arms
$N$	N-type Thermoelectric arms
$S$	Segmented Thermoelectric arms

1. Introduction

Thermoelectric power generation technology is a kind of power generation technology based on Seebeck effect, which has the advantages of simple structure, long life and no pollution. At present, the technology is commonly used in the utilization of solar energy, waste heat of internal combustion engines and waste heat in other places. In the research of thermoelectric technology, the research of TTEG is less. The reason is that the TTEG may effectively increase the voltage value of the Thermoelectric Generator (TEG), but the resistance value increases due to the increase of the contact point between the thermoelectric material and the copper plate, which leads to the decrease of output power. The STEG can effectively avoid these, so the research of TTEG is relatively small. Compared with the traditional single-stage TEG, the TTEG can place different thermoelectric materials in a more appropriate temperature range, so it has higher performance. The manufacturing method of the TTEG is similar to that of the traditional TEG, and the production cost of the TTEG is lower than that of the STEG.

The researchers improved the performance by optimizing the mechanism of the TEG. Liu et al. [1] improved the output power and conversion efficiency of TEG by combining the optimization method with computer-aided analysis by means of multi-objective optimization. Liu et al. [2] applied the profile-based genetic algorithm to the optimal fin parameters, which can increase the average temperature and decrease the pressure drop by approximately 20%. Min et al. [3] studied the figure of merit of the TEG with a large temperature differential, and concluded that the Thomson effect has a greater impact on a STEG than on the cascaded TEG when the figure of merit is measured.

Manikandan et al. [4] studied the thermodynamic model of TTEG considering Thomson effect, Joule heat and Fourier heat conduction. The number of thermocouples is optimized according to the maximum output power and energy/exergy efficiency conditions. Sun et al. [5] studied the performance optimization of TTEG structure by using multi-objective genetic algorithm, which considered the influence of different thermal effects and analyzed the energy destruction ratio. Based on a thermodynamic model of a TTEG, Zhang et al. [6] Studied the performance of TTEG by using Latin hypercube sampling method to simulate the change of TEG parameters. It is found that the large temperature difference between the cold and hot ends of TTEG will not only improve the output power of the structure, but also reduce its stability. Liu et al. [7] proposed a novel model for TTEG from vehicle exhaust and set up an experiment to test. The experimental results show that the maximum output power is about 250 W and the system thermal efficiency can reach 5.35%. Arora et al. [8] simultaneously optimized the maximum output power, minimizing entropy generation and thermal efficiency of the proposed TTEG system based on the second version of non-dominated sorting genetic algorithm with working current, temperatures and number of thermoelectric arms in the two stages of the TTEG as design variables, which shows that the optimization result of three objectives is better than that of two objectives. By comparing the performance of two- and three-stage cascaded TEGs, Kanimba et al. [9] found that the output power of three-stage cascaded TEG is 21% higher than that of TTEG when the maximum hot side temperature and the fixed cold side temperature are 973 K and 473 K, respectively. Sun et al. [10] established a model of TTEG and parallel TEG, and concluded that under the conditions defined by the authors, the performance of a single-stage TEG is superior to that of a TTEG when the hot side temperature is below 600 K. Liang et al. [11] studied the application of TTEG in the waste heat of internal combustion engine, and optimized the structure of TTEG, and obtained the optimal ratio range of the numbers of thermocouples in high-temperature layer and low-temperature layer. Atouei et al. [12] proposed a TTEG with Phase Change Material (PCM) applied between two thermoelectric generators to improve the performance of the thermoelectric generator by using the characteristic that the PCM can absorb heat energy as latent heat. The results show that the electrical potential generated by the TTEG with PCM was 27% higher than that of the single-stage TEG. Maduabuchi et al. [13] proposed and studied a two-stage solar TEG with P-type and N-type thermoelectric arms as X-type. This new structure raised the load requirements than a traditional design.

In addition to the above structural optimization methods, researchers also combine TEG with other energy systems to form hybrid systems in order to achieve better performance. Lin et al. [14] established a new hybrid system model consisting of an Alkali Metal Thermalelectric Converter (AMTEC) and a TTEG, and derived the mathematical expressions of out power and efficiency of AMTEC, TTEG and the hybrid system. Guo et al. [15] put forward a new hybrid system composed of a High-temperature Proton Exchange Membrane Fuel Cell (HT-PMEFC), a regenerator and a TTEG, and The results show that the maximum power density and the corresponding electric efficiency of the hybrid system are 11.8% and 17.5% higher than those of the sole HT-PEMFC system, respectively. Liu et al. [16] proposed a new hybrid system model that integrates a Direct Carbon Fuel Cell (DCFC), a TTEG and a regenerator, and found that the maximum attainable power density of the new hybrid system allows about 50% larger than that of the stand-alone DCFC. Guo et al. [17] proposed a hybrid system consisting of a Molten Carbonate Fuel Cell (MCFC) and a TTEG, and concluded that the hybrid system is not only superior to the single MCFC, but also superior to the hybrid system consisting of a MCFC and a single-stage TEG. The above studies on TEG were all based on using the average values of $\alpha$ , $\kappa$ and electrical conductivity ( $\sigma$ ) as the basis of material properties for structural optimization. However, in practical application, these parameters are all changed with the temperature variation.

In addition to the above studies, many researchers established one-dimensional physical models to study the performance of TEG. Barry et al. [18] optimized the structural parameters of thermoelectric arms by using the one-dimensional thermal resistance network model, and the maximum output power and conversion efficiency of the TEG were increased by 10.4% and 3.2%, respectively. Zabrocki et al. [19] extended the linear thermodynamics of irreversible processes by assuming the material parameters of $\alpha$ , $\kappa$ and $\sigma$ , and gave the analytical solution of one-dimensional thermal energy balance by using Bessel function. Zhu et al. [20] Proposed a one-dimensional numerical model combined with genetic algorithm to optimize the performance of STEG, which was verified by the experimental results in other literatures. Cheng et al. [21] Established a one-dimensional physical model of TEG in certain conditions of heat source and heat sink temperature, and verified the model through experiments and ANSYS simulation. finite difference methods [22], electrical analogy [23] and finite element [24] were employed to the study of TEG, and these methods are characterized in that the thermoelectric arm is divided into a plurality of sections along the height direction of the TEG [25]. In the above paper, a simplified relation for the temperature distribution of the TEG is not obtained.

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 temperatures on the TEG thermoelectric arm is considered. Based on a one-dimensional heat conduction model, the heat distribution of PN-type thermoelectric arms in TTEG was analyzed. The relationship between the maximum output power, the height and the cross-sectional area of PN-type thermoelectric arms in high- and low-temperature layers has been derived. For TTEG, when the temperatures of the hot end and the cold end and the average temperature of the middle ceramic are known, the optimal ratio between the heights of the upper and lower PN type thermoelectric arms and the ratio between the cross-sectional areas of a single PN type thermoelectric arm respectively in upper and lower layers can be calculated using a mathematical expression. The mathematical expression provides a new idea for the structure optimization of TTEG. By optimizing the structure of TTEG, in addition to the advantages of lower production cost, the power generation can also be significantly improved to exceed that of STEG.

2. Principle of calculation

Heat conduction is caused by the temperature gradient inside the object, and solid temperature conduction is caused by the collision of molecules and the migration of free electrons. In the process of analysis, if the scattering of temperature in the air is not considered, it can be simplified as a one-dimensional heat conduction process. The mathematical expression for the quantity of heat transferred $Q$ can be derived using Fourier’s law of heat conduction [26], as shown in Eq. (1).

$\displaystyle Q=\frac{\kappa A(T_{h}-T_{c})}{L}$ (1)

Equation (1) can be rewritten as follows

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

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 are uniform and constant, Eq. (3) can be obtained.

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

Where $X_{1}$ and $X_{2}$ are the positions of the two ends of the plate; $T_{1}$ and $T_{2}$ are the temperatures at $X_{1}$ and $X_{2}$ , respectively.

Equation (3) can be extended to the case of a two-layer composite plate as 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 [26].

$\displaystyle q=\frac{T_{1}-T_{2}}{(\chi_{2}-\chi_{1})/\kappa_{1}A}=\frac{T_{2% }-T_{3}}{(\chi_{3}-\chi_{2})/\kappa_{2}A}$ (4)

By extending Eq. (4) to n-layer composite plates, the following equation is obtained

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

For thermoelectric materials in TEG, the thermal conductivity varies with temperature. Taking $\Delta T$ as a standard, a single thermoelectric arm can be divided into a flat plate structure with n layers on average as shown in Fig. 1c, and Eq. (6) can be derived by using Eq. (5).

$\displaystyle q=\frac{\Delta T}{(\chi_{2}-\chi_{1})/\kappa_{1}A}=\frac{\Delta T% }{(\chi_{3}-\chi_{2})/\kappa_{2}A}=\ldots=\frac{\Delta T}{(\chi_{n+1}-\chi_{n}% )/\kappa_{n}A}$ (6) $\displaystyle\Delta T=\frac{T_{h}-T_{c}}{n}$ (7)

Equation (6) can be simplified to obtain Eq. (8).

$\displaystyle\frac{\kappa_{1}}{\chi_{2}-\chi_{1}}=\frac{\kappa_{2}}{\chi_{3}-% \chi_{2}}=\ldots=\frac{\kappa_{n}}{\chi_{n+1}-\chi_{n}}$ (8)

When the total height of the material is $H_{n}$ , according to Eq. (8), the height of the flat plate of the layer at the temperatures $T_{w}$ and $T_{w+1}$ can be expressed as follows:

$\displaystyle\chi_{w+1}-\chi_{w}=\frac{\kappa_{w}}{\sum\limits_{{i}=1}^{n}{% \kappa_{i}}}H_{n}$ (9)

Figure 2.

One-dimensional steady-state heat conduction diagram of TTEG.

For the TTEG, the thermal conductivity of the copper plate and the middle ceramic is obviously higher than that of the thermoelectric material. Therefore, in the high-temperature layer, the cold end temperature difference between N-type and P-type thermoelectric arms is not large. Therefore, in the analysis process, the hot end temperature of the middle ceramic can be assumed to be a constant value $T_{b}$ , and the cold end temperature of the middle ceramic can also be assumed to be a constant value $T_{a}$ , if we don’t consider the influence of the copper plate and the ceramic plate at the high temperature and low-temperature end of TTEG on the temperature. The heat conduction diagram is shown in Fig. 2, and the thermal conductivity of the high-temperature layer of N-type and P-type thermoelectric materials can be assumed to be $\kappa_{Nh}$ and $\kappa_{Ph}$ , and the thermal conductivity of the low-temperature layer of N-type and P-type thermoelectric materials can be assumed to be $\kappa_{Nc}$ and $\kappa_{Pc}$ . The thermal conductivity of the middle ceramic is assumed to be $\kappa_{m}$ . Equation (10) can be derived by using Eq. (6).

$\displaystyle\left\{{{\begin{array}[]{l}{q=n_{h}(q_{Nh}+q_{Ph})=q_{m}=n_{c}(q_% {Nc}+q_{Pc})}\\ \\ {q_{Nh}=\frac{\Delta T}{(\chi_{Nh(E+1)}-\chi_{\textit{NhE}})/\kappa_{\textit{% NhE}}A_{Nh}},q_{Ph}=\frac{\Delta T}{(\chi_{Ph(E+1)}-\chi_{\textit{PhE}})/% \kappa_{\textit{PhE}}A_{Ph}}}\\ \\ {q_{Nc}=\frac{\Delta T}{(\chi_{\textit{NcJ}+1}-\chi_{\textit{NcJ}})/\kappa_{% \textit{NcJ}}A_{Nc}},q_{Pc}=\frac{\Delta T}{(\chi_{\textit{PcJ}+1}-\chi_{% \textit{PcJ}})/\kappa_{\textit{PcJ}}A_{Pc}}}\\ \\ {q_{m}=\frac{\Delta T}{(\chi_{mF+1}-\chi_{mF})/\kappa_{mF}A_{m}}}\\ \\ {e=\frac{T_{h}-T_{b}}{\Delta T},f=\frac{T_{b}-T_{a}}{\Delta T},j=\frac{T_{a}-T% _{c}}{\Delta T}}\\ \end{array}}}\right.$ (10)

Where $E$ , $F$ , $J$ are integers from 1 to $e$ , $f$ , $j$ , respectively.

Because the thermal conductivity of the thermoelectric material is obviously less than that of the ceramic and the copper sheet, it can be assumed that the surface temperature of the middle ceramic is the same. Therefore, it can be assumed that the hot end temperature of the middle ceramic is $T_{b}$ and the cold end temperature is $T_{a}$ .

The P-type and N-type thermoelectric arms of the thermoelectric generator are cuboids. When the conductivity of the material is determined, the expression of its resistance can be derived as shown in Eq. (11).

$\displaystyle R=\frac{L}{\sigma A}=\frac{\rho L}{A}$ (11)

For a TTEG, the heights of the P-type and N-type thermoelectric arms in the same layer should be equal. Therefore, we can calculate the resistance values of the P-type and N-type thermoelectric arms in the high-temperature layer and low-temperature layer of the TTEG by combining Eqs (9) and (11). The expressions are as shown in Eq. (12).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{Nh}=\frac{\sum\limits_{{i}=1}^{e}{% \frac{\kappa_{\textit{Nhi}}}{\sigma_{\textit{Nhi}}}}}{\sum\limits_{i=1}^{e}{% \kappa_{\textit{Nhi}}}}\frac{H_{h}}{A_{Nh}},R_{Ph}=\frac{\sum\limits_{{i}=1}^{% e}{\frac{\kappa_{\textit{Phi}}}{\sigma_{\textit{Phi}}}}}{\sum\limits_{i=1}^{e}% {\kappa_{\textit{Phi}}}}\frac{H_{h}}{A_{Ph}}}\\ {R_{Nc}=\frac{\sum\limits_{{i}=1}^{j}{\frac{\kappa_{\textit{Nci}}}{\sigma_{% \textit{Nci}}}}}{\sum\limits_{i=1}^{j}{\kappa_{\textit{Nci}}}}\frac{H_{c}}{A_{% Nc}},R_{Pc}=\frac{\sum\limits_{{i}=1}^{j}{\frac{\kappa_{\textit{Pci}}}{\sigma_% {\textit{Pci}}}}}{\sum\limits_{i=1}^{j}{\kappa_{\textit{Pci}}}}\frac{H_{c}}{A_% {Pc}}}\\ \end{array}}}\right.$ (12)

The resistance and cross-sectional area of the PN-type thermoelectric arms are shown in Eq. (13).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{h}=R_{Nh}+R_{Ph}}\\ {R_{c}=R_{Nc}+R_{Pc}}\\ {A_{c}=A_{Pc}+A_{Nc}}\\ {A_{h}=A_{Ph}+A_{Nh}}\\ \end{array}}}\right.$ (13)

Equation (12) can also be simplified into Eq. (14) by calculus.

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

The constants of $C_{Nc}$ , $C_{Pc}$ , $C_{Nh}$ and $C_{Ph}$ are used to replace the known quantities in Eq. (14), and the expression is as shown in Eq. (15)

$\displaystyle\left\{{{\begin{array}[]{l}{C_{Nc}=\frac{\int_{T_{c}}^{T_{a}}{% \frac{\kappa_{Nc}({T})}{\sigma_{Nc}({T})}dT}}{\int_{T_{c}}^{T_{a}}{\kappa_{Nc}% ({T})dT}}=\frac{\int_{T_{c}}^{T_{a}}{\kappa_{Nc}({T})\rho_{Nc}(T)dT}}{\int_{T_% {c}}^{T_{a}}{\kappa_{Nc}({T})dT}}}\\ \\ {C_{Pc}=\frac{\int_{T_{c}}^{T_{a}}{\frac{\kappa_{Pc}({T})}{\sigma_{Pc}({T})}dT% }}{\int_{T_{c}}^{T_{a}}{\kappa_{Pc}({T})dT}}=\frac{\int_{T_{c}}^{T_{a}}{\kappa% _{Pc}({T})\rho_{Pc}(T)dT}}{\int_{T_{c}}^{T_{a}}{\kappa_{Pc}({T})dT}}}\\ \\ {C_{Ph}=\frac{\int_{T_{b}}^{T_{h}}{\frac{\kappa_{Ph}({T})}{\sigma_{Ph}({T})}dT% }}{\int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})dT}}=\frac{\int_{T_{b}}^{T_{h}}{\kappa% _{Ph}({T})\rho_{Ph}(T)dT}}{\int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})dT}}}\\ \\ {C_{Nh}=\frac{\int_{T_{b}}^{T_{h}}{\frac{\kappa_{Nh}({T})}{\sigma_{Nh}({T})}dT% }}{\int_{T_{b}}^{T_{h}}{\kappa_{Nh}({T})dT}}=\frac{\int_{T_{b}}^{T_{h}}{\kappa% _{Nh}({T})\rho_{Nh}(T)dT}}{\int_{T_{b}}^{T_{h}}{\kappa_{Nh}({T})dT}}}\\ \end{array}}}\right.$ (15)

Through calculation, it is known that when the values of $A_{Nc}$ , $A_{Pc}$ , $A_{Nh}$ and $A_{Ph}$ are in the following formula, the corresponding resistances $R_{c}$ and $R_{h}$ are the smallest.

$\displaystyle\left\{{{\begin{array}[]{l}{A_{Nc}=\frac{\sqrt{C_{Nc}}}{\sqrt{C_{% Pc}}}A_{Pc}}\\ \\ {A_{Nh}=\frac{\sqrt{C_{Nh}}}{\sqrt{C_{Ph}}}A_{Ph}}\\ \end{array}}}\right.$ (16)

By substituting Eq. (13) into Eq. (16), it can be calculated that the expressions of $R_{c}$ and $R_{h}$ are as shown in Eq. (17) when the resistance value is minimized.

$\displaystyle\left\{{{\begin{array}[]{l}{R_{c}=\frac{H_{c}(\sqrt{C_{Nc}}+\sqrt% {C_{Pc}})^{2}}{A_{c}}}\\ \\ {R_{h}=\frac{H_{h}(\sqrt{C_{Nh}}+\sqrt{C_{Ph}})^{2}}{A_{h}}}\\ \end{array}}}\right.$ (17)

It can be seen from Eq. (16) that the ratio of the cross-sectional areas of the PN-type thermoelectric arms in a single layer is not related to the height of the thermoelectric arms, but is only related to the selected temperature. Therefore, when the temperature is determined, the ratio of the cross-sectional areas of the P-type and N-type thermoelectric arms can be calculated by using Eq. (16).

Equation (18) can be derived by Eq. (6).

$\displaystyle\sum\limits_{i=1}^{n}(\chi_{(i+1)}-\chi_{i})=\frac{\sum\limits_{i% =1}^{n}{\kappa_{i}A\Delta T}}{q}$ (18)

Equation (10) is combined with the TTEG to derive Eq. (19).

$\displaystyle q=\frac{\Delta T}{(\chi_{Nh(E+1)}-\chi_{\textit{NhE}})/\kappa_{% \textit{NhE}}A_{Nh}n_{h}}+\frac{\Delta T}{(\chi_{Ph(E+1)}-\chi_{\textit{PhE}})% /\kappa_{\textit{PhE}}A_{Ph}n_{h}}=\frac{\Delta T}{(\chi_{Nc(J+1)}-\chi_{% \textit{NcJ}})/\kappa_{\textit{NcJ}}A_{Nc}n_{c}}+\frac{\Delta T}{(\chi_{Pc(J+1% )}-\chi_{\textit{PcJ}})/\kappa_{\textit{PcJ}}A_{Pc}n_{c}}$ (19)

Equation (18) can be used to further simplify Eq. (19) to obtain Eq. (20).

$\displaystyle\left\{{{\begin{array}[]{l}{q_{t}=\frac{\int_{T_{c}}^{T_{a}}{% \kappa_{Nc}({T})dT\cdot A_{\textit{Nct}}+\int_{T_{c}}^{T_{a}}{\kappa_{Pc}({T})% dT\cdot A_{\textit{Pct}}}}}{H_{c}}}\\ \\ {q_{t}=\frac{\int_{T_{b}}^{T_{h}}{\kappa_{Nh}({T})dT\cdot A_{\textit{Nht}}+% \int_{T_{b}}^{T_{h}}{\kappa_{Ph}({T})dT\cdot A_{\textit{Pht}}}}}{H_{h}}}\\ \\ {q_{t}=\frac{\int_{T_{a}}^{T_{b}}{\kappa_{m}({T})dT\cdot A_{mt}}}{H_{m}}}\\ \end{array}}}\right.$ (20)

By simplifying Eq. (20), the relationship between the temperature and the height of the thermoelectric arm respectively in the high-temperature layer and the low-temperature layer can be derived as shown in Eq. (21).

$\displaystyle\left\{{{\begin{array}[]{l}{\frac{H_{h}}{H_{c}}=\frac{\int_{T_{b}% }^{T_{h}}{\kappa_{Nh}({T})dT\cdot A_{\textit{Nht}}+\int_{T_{b}}^{T_{h}}{\kappa% _{Ph}({T})dT\cdot A_{\textit{Pht}}}}}{\int_{T_{c}}^{T_{a}}{\kappa_{Nc}({T})dT% \cdot A_{\textit{Nct}}+\int_{T_{c}}^{T_{a}}{\kappa_{Pc}({T})dT\cdot A_{\textit% {Pct}}}}}}\\ \\ \frac{H_{m}}{H_{h}}=\frac{\int_{T_{a}}^{T_{b}}{\kappa_{m}({T})dT\cdot A_{m}}}{% \int_{T_{b}}^{T_{h}}{\kappa_{Nh}({T})dT\cdot A_{\textit{Nht}}+\int_{T_{b}}^{T_% {h}}{\kappa_{Ph}({T})dT\cdot A_{\textit{Pht}}}}}\\ \\ \frac{H_{m}}{H_{c}}=\frac{\int_{T_{a}}^{T_{b}}{\kappa_{m}({T})dT\cdot A_{m}}}{% \int_{T_{c}}^{T_{a}}{\kappa_{Nc}({T})dT\cdot A_{\textit{Nct}}+\int_{T_{c}}^{T_% {a}}{\kappa_{Pc}({T})dT\cdot A_{\textit{Pct}}}}}\\ \end{array}}}\right.$ (21)

Seebeck effect is the main principle of thermoelectric power generation. When the cold end temperature and hot end temperature of a thermoelectric material are determined, the voltage value is shown in Eq. (22).

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

Where $U_{e}$ is the voltage between the two ends of the thermoelectric material.

When it is assumed that there are $n_{h}$ pairs of PN-type electric elements in the high-temperature layer and $n_{c}$ pairs of PN-type thermoelectric arms in the low-temperature layer in the TTEG, the voltage value of TTEG is as shown in Eq. (23).

$\displaystyle\left\{{{\begin{array}[]{l}{U_{t}=n_{c}\cdot U_{c}+n_{h}\cdot U_{% h}}\\ {U_{h}=\int_{T_{b}}^{T_{h}}{(\left|{\alpha_{Ph}}\right|+\left|{\alpha_{Nh}}% \right|)dT}}\\ \\ {U_{c}=\int_{T_{c}}^{T_{a}}{(\left|{\alpha_{Pc}}\right|+\left|{\alpha_{Nc}}% \right|)dT}}\\ \end{array}}}\right.$ (23) $\displaystyle\left\{{{\begin{array}[]{l}{n_{h}\cdot A_{Ph}+n_{h}\cdot A_{Nh}=A% _{\textit{Pht}}+A_{\textit{Nht}}=A_{ht}}\\ {n_{c}\cdot A_{Pc}+n_{c}\cdot A_{Nc}=A_{\textit{Pct}}+A_{\textit{Nct}}=A_{ct}}% \\ \end{array}}}\right.$ (24)

The total resistance of the TTEG is shown in Eq. (25).

$\displaystyle R_{t}=n_{c}R_{c}+n_{h}R_{h}$ (25)

The minimum resistance and voltage of the TTEG can be calculated by Eqs (17) and (23). When the voltage and resistance are determined, the maximum power can be calculated by Eq. (26).

$\displaystyle P=\frac{U^{2}}{{4}R}$ (26)

The P-type and N-type thermoelectric arms in the TTEG are connected by a copper sheet, and the electrical conductivity of the copper is far greater than that of the P-type and the N-type thermoelectrical material. Therefore, in the calculation process, the resistance of the connecting pieces is not considered, and the resistance caused by the welding of the current guide plate and the thermoelectric material is also not considered. That is, only the resistance of the P-type and N-type thermoelectric arms is calculated in the calculation process.

When the external resistor forms a closed circuit, the temperature of the cold end and the hot end is still the main factor affecting the temperature distribution in the PN-type thermoelectric arms due to the low efficiency of the TTEG, so the resistance of the TTEG itself can be assumed to be unchanged after the external resistor is connected.

Equations (23) and (25) are substituted into Eq. (26), and the out power of TTEG in the unit area can be calculated by considering that the total cross-sectional area of the TTEG piece is $A_{mt}$ .

$\displaystyle P_{\textit{avg}}=\frac{P_{NP}}{A_{mt}}=\frac{U_{t}^{2}}{4R_{t}A_% {mt}}=\frac{(n_{c}U_{c}+n_{h}U_{h})^{2}}{4A_{mt}\cdot(n_{c}R_{c}+n_{h}R_{h})}$ (27)

Substituting Eq. (24) into Eq. (27) yields Eq. (28).

$\displaystyle P_{\textit{avg}}=\frac{\left(\frac{A_{ct}}{A_{c}}U_{c}+\frac{A_{% ht}}{A_{h}}U_{h}\right)^{2}}{4A_{mt}\cdot\left(\frac{(\sqrt{C_{Nc}}+\sqrt{C_{% Pc}})^{2}\cdot H_{c}A_{ct}}{A_{c}^{2}}+\frac{(\sqrt{C_{Nh}}+\sqrt{C_{Ph}})^{2}% \cdot H_{h}A_{ht}}{A_{h}^{2}}\right)}$ (28)

Assuming $k=A_{h}/A_{c}$ , substituting it into Eq. (28) can be reduced to Eq. (29).

$\displaystyle P_{\textit{avg}}=\frac{1}{4A_{mt}}\cdot\frac{(kA_{ct}U_{c}+A_{ht% }U_{h})^{2}}{(k^{2}(\sqrt{C_{Nc}}+\sqrt{C_{Pc}})^{2}\cdot H_{c}\cdot A_{ct}+(% \sqrt{C_{Nh}}+\sqrt{C_{Ph}})^{2}\cdot H_{h}\cdot A_{ht})}$ (29)

At the beginning of the design, in order to ensure a large out power of the TTEG, the ratio of the total cross-section area of the PN-type thermoelectric arms to the area of TTEG should be as large as possible. However, if the distance between the two thermoelectric arms is too small, thermal interference will occur, thus affecting its out power. Therefore, it can be assumed that the ratio of the total cross-sectional area of the PN-type thermoelectric arms to the area of the TEG should have a maximum value. During the design process, if $A_{m}$ is known, $A_{ht}$ and $A_{ct}$ are also known.

According to Eq. (29), the out power of the TTEG in unit area is related to three values of $H_{c}$ , $H_{h}$ and $k$ , wherein $H_{c}$ and $H_{h}$ can be calculated by using Eq. (21), and the out power of the TTEG is the maximum when the value of k is taken as shown in Eq. (30).

$\displaystyle k=\frac{U_{c}H_{h}\cdot(\sqrt{C_{Nh}}+\sqrt{C_{Ph}})^{2}}{U_{h}H% _{c}\cdot(\sqrt{C_{Nc}}+\sqrt{C_{Pc}})^{2}}$ (30)

Substituting Eq. (30) into Eq. (29) gives the maximum value of $P_{\textit{avg}}$ as shown in Eq. (31).

$\displaystyle P_{\textit{avg}}=\frac{1}{4A_{mt}}\cdot\left(\frac{A_{ct}U_{c}^{% 2}}{(\sqrt{C_{Nc}}+\sqrt{C_{Pc}})^{2}H_{c}}+\frac{A_{ht}U_{h}^{2}}{(\sqrt{C_{% Nh}}+\sqrt{C_{Ph}})^{2}H_{h}}\right)$ (31)

3. Experimental simulation

In the selection of thermoelectric materials, the dimensionless figure of merit ZT is often used to express the performance of thermoelectric materials, and its expression is as shown in Eq. (32).

$\displaystyle ZT=\frac{\alpha^{2}\sigma T}{\kappa}$ (32)

Bi ${}_{2}$ Te ${}_{3}$ -based thermoelectric materials commonly available in the market were selected as reference materials for low-temperature thermoelectric materials. PbTe in reference [27] is selected as the thermoelectric material of the high-temperature layer. The properties of four thermoelectric materials can be obtained as shown in Fig. 3.

As shown in Fig. 3d, the temperature of the ZT value crossing point of the P-type thermoelectric material is at a position of about 474 K, and the temperature of the ZT value crossing point of the N-type thermoelectric material is at a position of about 539 K.

Figure 3.

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

The fitting equations of the thermal conductivities of the four thermoelectric materials can be obtained by using a fitting function mode, and the relation between the thermal conductivity and the temperature can be obtained by fitting the thermal conductivity as a quartic function of the temperature, as shown in Eq. (33).

$\displaystyle\kappa=f_{\kappa}(T)=b_{4}T^{4}+b_{3}T^{3}+b_{2}T^{2}+b_{1}T+b_{0}$ (33)

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

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:PbTe	N:Bi ${}_{2}$ Te ${}_{3}$	P:Bi ${}_{2}$ Te ${}_{3}$
$b_{0}$	1.0167109557E+01	7.5760606061E+00	2.9922894752E+00	7.7343621085E+00
$b_{1}$	$-$ 3.0382556300E-02	$-$ 1.1210994600E-02	$-$ 1.3736915000E-02	$-$ 5.6630615500E-02
$b_{2}$	3.2012820513E-05	$-$ 2.5369463869E-05	7.1386672151E-05	2.0554134101E-04
$b_{3}$	$-$ 7.6301476301E-09	6.6682206682E-08	$-$ 1.3289455642E-07	$-$ 3.3225284519E-07
$b_{4}$	$-$ 3.2634032634E-12	$-$ 3.7762237762E-11	8.8852324146E-11	2.1390374332E-10

The relationship between the Seebeck coefficient and the temperature of the four thermoelectric materials is obtained by the same method, and the relationship is shown in Eq. (34). The value of $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ and $a_{4}$ are shown in Table 2.

$\displaystyle\left|\alpha\right|=f_{\alpha}(T)=a_{4}T^{4}+a_{3}T^{3}+a_{2}T^{2% }+a_{1}T+a_{0}$ (34)

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

Table 2

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

Materials	N:PbTe	P:PbTe	N:Bi ${}_{2}$ Te ${}_{3}$	P:Bi ${}_{2}$ Te ${}_{3}$
$a_{0}$	$-$ 1.9196445221E+02	9.6501722611E+02	1.9739312746E+02	$-$ 3.5285881765E+02
$a_{1}$	1.7220108780E+00	$-$ 8.1350299145E+00	$-$ 5.6029977870E-01	3.4596219075E+00
$a_{2}$	$-$ 4.3442075000E-03	2.5101587400E-02	2.1536021000E-03	$-$ 7.2940518000E-03
$a_{3}$	5.7225485625E-06	$-$ 3.0538710179E-05	$-$ 1.9815165227E-06	6.0577540107E-06
$a_{4}$	$-$ 2.7347319347E-09	1.3075058275E-08	$-$ 3.9928698752E-10	$-$ 1.9481694776E-09

From the results of fitting the data, a plot of the product of resistivity $\rho$ and thermal conductivity $\kappa$ versus temperature is obtained. In the process of polynomial fitting, the thermal conductivity and resistivity are fitted by the function with the highest term of the fourth power. If the product of resistivity and thermal conductivity is accurately expressed, the function with the highest term of the eighth power should be fitted. 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. In order to reduce the amount of calculation, the fifth power fitting is selected. Thus, the relationship between the product of the thermal conductivity and the specific resistance and the temperature can be obtained as shown in Eq. (35).

$\displaystyle\kappa\cdot\rho=f_{\kappa\cdot\rho}(T)=d_{5}T^{5}+d_{4}T^{4}+d_{3% }T^{3}+d_{2}T^{2}+d_{1}T+d_{0}$ (35)

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

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 3.

Table 3

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:PbTe	N:Bi ${}_{2}$ Te ${}_{3}$	P:Bi ${}_{2}$ Te ${}_{3}$
$d_{0}$	$-$ 1.1796520740E-01	5.0805492210E-01	1.2120505450E-01	2.6703009900E-01
$d_{1}$	1.2428526000E-03	$-$ 4.6821030000E-03	$-$ 9.3609440216E-04	$-$ 3.1135527000E-03
$d_{2}$	$-$ 4.2619343676E-06	1.7217810269E-05	1.8970291250E-06	1.5026241145E-05
$d_{3}$	6.9182640285E-09	$-$ 3.0814190277E-08	2.4082954333E-09	$-$ 3.6780258647E-08
$d_{4}$	$-$ 5.4719271369E-12	2.7084470240E-11	$-$ 1.0867734363E-11	4.6271292748E-11
$d_{5}$	1.7656375701E-15	$-$ 9.3788011852E-15	8.7230856482E-15	$-$ 2.3339691708E-14

Figure 4.

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

For the TTEG, based on existing products in the market, it can be assumed that the total height of the high-temperature layer and the low-temperature layer PN-type thermoelectric arms is 2 mm. The total area of the PN-type thermoelectric arms of the high-temperature layer and the low-temperature layer is equal, and the ratio of this size to the total area of the intermediate ceramic is 3.5:1. It is assumed that the sum of the cross-sectional areas of the individual PN-type thermoelectric arms in the upper and lower layers of the TTEG is 2 mm ${}^{2}$ , and the width of the thermoelectric arms is constant at 1 mm. Copper material is used as the inducer of the PN-type thermoelectric arms, and the dimensions of length, width and height of the flow guide are 2.5 mm, 1.0 mm and 0.2 mm, respectively. The area of the intermediate ceramic was 7 mm ${}^{2}$ , and its specific dimensions of length, width and height were 2 mm, 3.5 mm and 0.2 mm, respectively. This dimension can be used to verify the rationality of the theoretical heat distribution presented above, when the temperature of the hot end is assumed to be 800 K and the temperature of the cold end is assumed to be 300 K. According to Eqs (16) and (20), when the intermediate temperature of the intermediate ceramic is 506 K, the corresponding $T_{a}$ and $T_{b}$ are 505.3 K and 506.7 K, respectively, and the corresponding dimensions of different thermoelectric materials at this condition are shown in Table 4.

Table 4

When $T_{h}=$ 800 k, $T_{c}=$ 300 K, the average temperature of the middle ceramic is 506 K, the relevant dimensions of the TTEG

Materials	Length (mm)	Width (mm)	Height (mm)
N:PbTe	0.922	1	0.847
P:PbTe	1.078	1	0.847
N:Bi ${}_{2}$ Te ${}_{3}$	0.905	1	1.153
P:Bi ${}_{2}$ Te ${}_{3}$	1.095	1	1.153

According to the dimensions in Table 4, the corresponding three-dimensional model drawings are constructed for simulation analysis. The contact resistance and the effect of the outermost ceramic layer on the thermal conductivity are not considered in the software simulation. In order to ensure the accuracy of the calculation, the resistivity of the ceramic is adjusted to 100 $\Omega\cdot$ m to simulate the insulation state. The material property parameters used in the analysis are shown in Table 5.

Table 5

Various performance parameters of different thermoelectric material

Materials	Seebeck coefficient $\alpha$ ( $\mu$ V/K)	Thermal conductivity $\kappa$ (W/mK)	Resistivity $\rho$ ( $\Omega\cdot$ m)
N:PbTe	As shown in Figs 5–8
P:PbSe
P:Bi ${}_{2}$ Te ${}_{3}$
N:Bi ${}_{2}$ Te ${}_{3}$
Cu	0	390	2.93E-9
Ceramics	0	30	100

In the calculation process, the hot end temperature is set to 800 K and the cold end temperature is set to 300 K. The cold end of the N-type thermoelectric arm is arranged in the PN-type thermoelectric arms of the low-temperature layer is 0 mV. The hot end of the P-type thermoelectric arm arranged in the PN-type thermoelectric arms of the high-temperature layer is 0 mV. In this way, the maximum value of the simulation result is the voltage generated by the PN-type thermoelectric arms in the low temperature layer, and the absolute value of the minimum value of the simulation result is the voltage generated by the PN-type thermoelectric arms in the high temperature layer.

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 constraint conditions according to the above requirements, the results are shown in Fig. 5.

Figure 5.

When $A_{m}=$ 7 mm ${}^{2}$ , $T_{h}=$ 800 k, $T_{c}=$ 300 K, the voltage value of a single PN-type thermoelectric arm in TTEG.

Using Eqs (21) and (23) and the data in Table 3, it can be calculated that the voltage of the high-temperature layer of PN-type thermoelectric arms is 116.749 mV, and the voltage of the low-temperature layer of PN-type thermoelectric arms is 76.908 mV. As a result of the comparison, the voltage error of the high-temperature layer of PN-type thermoelectric arms was 0.39 mV, and the voltage error of the low-temperature layer of PN-type thermoelectric arms was 0.344 mV. The error value may be caused by the following reasons: 1. In the process of calculation, the thermal resistance of the current inducer is not considered, which leads to the difference between the temperature at both ends of the PN-type thermoelectric arms and the set temperature; 2. The heat conduction process in the middle of the ceramic piece is not an ideal one-dimensional heat conduction state, resulting in ceramic piece thermal resistance being greater than the calculated results; 3. In the process of simulation calculation, the parameters are set in the form of points, not in the form of function, so there will inevitably be errors in the calculation process.

Figure 6.

When $A_{m}=$ 2.5 mm ${}^{2}$ , $T_{h}=$ 800 k, $T_{c}=$ 300 K, the voltage value of a single PN-type thermoelectric arm in TTEG.

In order to verify the conjecture, the length, width and height of the intermediate ceramic can be changed to 2.5 mm, 1.0 mm and 0.2 mm. Because the area of the intermediate ceramic is equal to the area of the current inducer plate, the heat conduction process is closer to an ideal heat conduction state. The thermal conductivity of the copper material is changed to 3.9 $\times$ 10 ${}^{7}$ W/m $\cdot$ K, so as to ignore the influence of the current inducer on the temperature conduction, and the simulation result is shown in Fig. 6. Under this condition, $T_{a}=$ 504.84 and $T_{b}=$ 507.16 can be deduced through calculation. At this time, the voltage of the corresponding high-temperature layer of PN-type thermoelectric arms is 116.522 mV, and the voltage of the low-temperature layer of PN-type thermoelectric arms is 76.634 mV. The voltage error of the PN-type thermoelectric arms of the high-temperature layer is 0.019 mV, and the voltage error of the PN-type thermoelectric arms of the low-temperature layer is 0.024 mV. Therefore, it can be assumed that the theoretical derivation of the thermal distribution of TTEG is reasonable, and the temperature distribution results can be used to calculate the resistance of the PN-type thermoelectric arms of TTEG. In practice, the effective heat transfer area of the intermediate ceramic should be greater than the contact area with the current inducer plate and less than the actual area of the ceramic. The difference of the effective heat conduction area of the middle ceramic can influence the values of $T_{a}$ and $T_{b}$ , but has little influence on the values of $T_{a}$ and $T_{b}$ . According to Eq. (21), when the change of $T_{a}$ and $T_{b}$ is not big, the influence on the ratio of the cross-section areas of P-type and N-type thermoelectric arms is not big. Therefore, this method can still be used to calculate under ideal conditions.

4. Analysis and discussion

For the TTEG, it can be assumed that the number of PN-type thermoelectric arms in the low-temperature layer is 98, the cross-sectional area of a single PN-type thermoelectric arm is 2 mm ${}^{2}$ . the sum of the heights of the thermoelectric arms in the high-temperature layer and the low-temperature layer is constant at 2 mm, and the cross-sectional area and height of the current inducer plate are 2.5 mm ${}^{2}$ and 0.2 mm, respectively. The thickness of the middle ceramic plate is 0.2 mm, and the ratio of the cross-sectional area of the ceramic and the cross-section area of PN-type thermoelectric arms is 3.5: 1, so that the value range of $A_{m}/A_{c}$ is between 1.25 and 3.5. On this premise, when $A_{m}/A_{c}=$ 3.5 and $A_{m}/A_{c}=$ 1. 25, the power corresponding to the TTEG can be calculated respectively.

When $T_{h}=$ 800 k and $T_{c}=$ 300 K, the corresponding maximum out power value can be quickly calculated by adjusting the height PN-type thermoelectric arms of the high and low-temperature layer, respectively. Finally, under the set conditions, the relationship between the heights of thermoelectric arms in different temperature layers and the corresponding maximum output power is shown in Fig. 7.

As shown in Fig. 7, it can be calculated that the corresponding optimal size at the maximum output power is shown in Table 6, when $A_{m}/A_{c}$ is 3.5 and 1.25 respectively. Compared with the data in Table 6, it can be seen that the value of $A_{m}$ has little effect on the optimization result of size. Under ideal conditions, when $H_{h}=$ 0.494 mm, $H_{c}=$ 1.506 mm, $n_{h}=$ 109, $A_{h}=$ 1.7982 mm ${}^{2}$ , the corresponding out power is maximum.

Table 6
The relevant dimensions corresponding to the maximum output power of the TTEG

$A_{m}/A_{c}$	3.5		1.25
Materials	Cross-sectional area (mm ${}^{2}$ )	Height (mm)	Cross-sectional area (mm ${}^{2}$ )	Height (mm)
N:PbTe	0.8378	0.4943	0.8380	0.4940
P:PbTe	0.9604	0.4943	0.9602	0.4940
P:Bi ${}_{2}$ Te ${}_{3}$	0.8954	1.5057	0.8953	1.5060
N:Bi ${}_{2}$ Te ${}_{3}$	1.1046	1.5057	1.1047	1.5060

Figure 7.

Relationship between $H_{h}$ and maximum output power of TTEG.

The four thermoelectric materials can be used for manufacturing the STEG, and in order to make a comparison with the STEG, the size of the STEG can also be assumed to be as follows: the cross-sectional area of a single PN type thermoelectric arms is 2 mm ${}^{2}$ , the height of the PN-type thermoelectric arms is 2 mm, and The single STEG has 98 PN-type thermoelectric arms.

By further deriving Eq. (12), it can be calculated that the resistance of the P type and N type thermoelectric arm in the segmented thermoelectric generator is calculated as shown in Eq. (36).

$\displaystyle\left\{{{\begin{array}[]{l}{R_{NS}=\frac{\int_{T_{NS}}^{T_{h}}{% \frac{\kappa_{\textit{NhS}}({T})}{\sigma_{\textit{NhS}}({T})}dT}+\int_{T_{c}}^% {T_{NS}}{\frac{\kappa_{\textit{NcS}}({T})}{\sigma_{\textit{NcS}}({T})}dT}}{% \int_{T_{NS}}^{T_{h}}{\kappa_{\textit{NhS}}({T})dT+\int_{T_{c}}^{T_{NS}}{% \kappa_{\textit{NcS}}({T})dT}}}\frac{H_{S}}{A_{NS}}}\\ \\ R_{PS}=\frac{\int_{T_{PS}}^{T_{h}}{\frac{\kappa_{\textit{PhS}}({T})}{\sigma_{% \textit{PhS}}({T})}dT}+\int_{T_{c}}^{T_{PS}}{\frac{\kappa_{\textit{PcS}}({T})}% {\sigma_{\textit{PcS}}({T})}dT}}{\int_{T_{PS}}^{T_{h}}{\kappa_{\textit{PhS}}({% T})dT+\int_{T_{c}}^{T_{PS}}{\kappa_{\textit{PcS}}({T})dT}}}\frac{H_{S}}{A_{PS}% }\\ \\ R_{tS}=R_{NS}+R_{PS}\\ \end{array}}}\right.$ (36)

By further derivation of Eq. (36), It can be obtained that the output power is maximum when the cross-sectional area of P-type and N-type thermoelectric arms is as shown in Eq. (37).

$\displaystyle\frac{A_{NS}}{A_{PS}}=\frac{\sqrt{\frac{\int_{T_{NS}}^{T_{h}}{% \kappa_{\textit{NhS}}({T})\rho_{\textit{NhS}}(T)dT}+\int_{T_{c}}^{T_{NS}}{% \kappa_{\textit{NcS}}({T})\rho_{\textit{NcS}}(T)dT}}{\int_{T_{NS}}^{T_{h}}{% \kappa_{\textit{NhS}}({T})dT+\int_{T_{c}}^{T_{NS}}{\kappa_{\textit{NcS}}({T})% dT}}}}}{\sqrt{\frac{\int_{T_{PS}}^{T_{h}}{\kappa_{\textit{PhS}}({T})\rho_{% \textit{PhS}}(T)dT}+\int_{T_{c}}^{T_{PS}}{\kappa_{\textit{PcS}}({T})\rho_{% \textit{PcS}}(T)dT}}{\int_{T_{PS}}^{T_{h}}{\kappa_{\textit{PhS}}({T})dT+\int_{% T_{c}}^{T_{PS}}{\kappa_{\textit{PcS}}({T})dT}}}}}$ (37)

Finally, the voltage of the single PN-type thermoelectric arms in the STEG can be obtained by using Eqs (22) and (34), and the resistance value of the single PN-type thermoelectric arms can be obtained by using Eq. (36). So that the output power of a single PN-type thermoelectric arm can be obtained, and the output power value of the STEG can be obtained by multiplying the calculation result by the number of the PN-type thermoelectric arms contained in the STEG.

Using the obtained resistance and voltage, it can be calculated that when $T_{h}=$ 800 K and $T_{c}=$ 300 K, the relationship between the corresponding maximum output power and the temperature at the junction of different thermoelectric materials in the N-type thermoelectric arm ( $T_{NS}$ ) and P-type thermoelectric arm ( $T_{PS}$ ) is shown in Fig. 8. It can be seen from the figure that when $T_{NS}=$ 599 K and $T_{PS}=$ 531 K, the output power is the maximum, and the output power is 21.488 W.

The above method can be used to calculate the maximum power corresponding to the STEG and the TTEG when $T_{h}$ and $T_{c}$ take different values, and the results are shown in Table 7.

Table 7

Maximum output power of STEG and TTEG at different temperatures

Temperature (K)		Max power (W)
$T_{h}$	$T_{c}$	STEG	TTEG ( $A_{m}/A_{c}=$ 3.5)	TTEG ( $A_{m}/A_{c}=$ 1.25)
800	300	21.488	21.250	21.125
775	325	17.322	17.184	17.079
750	350	13.560	13.518	13.432
725	375	10.242	10.273	10.206
700	400	7.395	7.467	7.416

Figure 8.

Relationship between $T_{PS}$ , $T_{NS}$ and maximum output power of STEG.

As can be seen from Table 7, under the conditions set above, when $T_{h}=$ 725 K, $T_{c}=$ 375 K, the difference between the maximum output power of the STEG and that of the TTEG is not large after optimization; When $T_{h}=$ 700 K, $T_{c}=$ 400 K, the maximum output power of the TTEG is better than that of the STEG.

5. Conclusions

Due to the low efficiency of the Two-stage Thermoelectric Generator (TTEG), the temperature of the cold end ( $T_{c}$ ) and hot end ( $T_{h}$ ) are still the main influence factors on the temperature distribution of PN-type thermoelectric arms. By considering only the influence of the temperature of hot and cold ends, a one-dimensional heat conduction model was applied to study the temperature distributions of PN thermoelectric arms for the TTEG. The relationship between the temperature distribution in the TTEG and the heights of PN-type thermoelectric arms respectively in high- and low-temperature layers ( $H_{h}$ , $H_{c}$ ) was obtained using the calculus calculation concept. When $T_{h}$ and $T_{c}$ are constant, the relationship between $H_{h}$ , $H_{c}$ and the temperature zones of high- and low-temperature layers can be calculated using the proposed mathematical formula with the verification from the simulations carried out by COMSOL software. To achieve the maximum output power, the size of the TTEG was analyzed and optimized in this paper.

When $T_{h}$ , $T_{c}$ , sums of the cross-sectional areas of all the PN-type thermoelectric arms respectively in high- and low-temperature layers ( $A_{ht}$ , $A_{ct}$ ) are constant, the optimal relationship expression between $H_{c}$ , $H_{h}$ and the cross-sectional areas of single PN-type thermoelectric arms in high- and low-temperature layers ( $A_{h}$ , $A_{c}$ ) was given.

Based on the proposed model, the maximum output power of the STEG in the same size with different materials was studied and compared by taking PbTe as the medium temperature material and Bi ${}_{3}$ Te ${}_{2}$ as the low-temperature material. It is found that under the condition of the same total cross-sectional areas of PN-type thermoelectric and $H_{h}+H_{c}=H_{S}$ (the heights of PN-type thermoelectric arms in the STEG), the maximum output power of the TTEG is higher than that of the STEG when $T_{h}=$ 700 K and $T_{c}=$ 400 K, and the maximum output power of the STEG is higher than that of the TTEG when $T_{h}=$ 750 K and $T_{c}=$ 350 K. Therefore the performance of STEG and TTEG is also related to temperature.

In view of the author’s limited ability, the calculation process of this paper also has many deficiencies. Because the efficiency of the TTEG is very low, after the external resistance is connected, the main factor affecting the internal temperature distribution of the PN thermoelectric arm in the thermoelectric generator is the temperature of the hot end and the cold end. Therefore, in the calculation process, the temperature change inside the PN thermoelectric arm after the external resistance is ignored, and only the influence of the temperature of the cold and hot end on the internal temperature distribution of the PN thermoelectric arm is considered. Although the most important factors affecting the internal temperature distribution are considered in the process of structural optimization analysis of TTEG, the influence of some secondary factors such as Peltier effect, Thomson effect, Joule heat and other heat conduction modes between thermoelectric arms on temperature is ignored. As a result, the calculation results of this paper will be different from the actual results. Therefore, a more perfect mathematical or theoretical model should be established in the future research, and more influencing factors can be taken into account in order to obtain more accurate optimization results.

Footnotes

Acknowledgments

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).

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Optimization and comparison of two-stage thermoelectric generators considering the influence of temperature variation on materials for waste heat utilization

Abstract

Keywords

Nomenclature

Greek symbols

Abbreviations

Subscripts

1. Introduction

2. Principle of calculation

Table 6 The relevant dimensions corresponding to the maximum output power of the TTEG

Footnotes

Acknowledgments

References

Table 6
The relevant dimensions corresponding to the maximum output power of the TTEG