Evaluation of Thermoelectric Generators under Mismatching Conditions

Abstract

Due to the wide usability of thermoelectric generators (TEG) in the industry and research fields, it is plausible that mismatching conditions are present on the thermal surfaces of a TEG device, which induces negative-performance effects due to uneven surface temperature distributions. For this reason, the objective of this study is to characterize numerically the open-circuit electric output voltage of a TEG device when a mismatching condition is applied to both the cold and hot sides of the selected N and P-type semiconductor material ${\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3$. A validated numerical simulation paired with a parametric study is conducted using the Thermal-Electric module of ANSYS 2020 R1, for which different thermal boundary and mismatching conditions are applied while considering the temperature-dependent thermoelectrical properties of the N and P-type material. The results show an inverse relationship between the open-circuit voltage and the mismatching temperature difference. When a mismatching condition is applied on the hot side of the TEG device, the temperature-dependent electrical resistance has lower values, deriving in higher voltage results (linear tendency) compared to a mismatching condition applied to the cold side (non-linear tendency).

1. Introduction

A single thermoelectric generator (TEG) is a device composed of semiconductor materials (N and P-type) connected electrically in series and thermally in parallel (More than two modules can be electrically connected either in series or parallel) that produces electric power from a temperature difference (Seebeck effect), or cooling from an electric potential source (Peltier effect) [1]. These devices are used in a wide range of applications i.e., bio-integrated wearable devices [2], pipe heat energy waste [3], automobile exhaust heat [4,5], heat exchangers [6], combustion engines [7,8], photovoltaic systems [9,10,11], and space exploration [12,13,14].

Thermoelectric devices are subject to non-uniform temperature distributions, i.e., mismatching conditions due to environmental-operating conditions. It has been reported that the performance of working thermoelectric arrangements is lower than expected under mismatched temperature conditions [15]. Tang et al. [16] found a power loss of 11% in the performance of a TEG module under mismatched temperature conditions for automobile exhaust heat recovery, while asserting that proper isolation could reduce the losses down to 2.3%. Hakim & Lim [17] compared the performance of two interconnected thermoelectric modules, one of them under temperature mismatching conditions and the other one without them. They found an electric power difference of 45.73% at a temperature difference of $\mathsf{\Delta }T=340.15 \mathrm{K}$, with the non-mismatched module producing the higher power.

Experimental studies on coupled TEG-Photovoltaic (TEG-PV) systems reported negative effects of mismatching temperature conditions, hindering efficient heat transfer, and thus lowering the expected design performance [18,19].

Material-wise, Bismuth Telluride $\left({\mathrm{Bi}}_2{\mathrm{Te}}_3\right)$ is used widely for industrial applications due to cost-benefit reasons. However, the scarcity of these materials in the earth’s crust [20] compelled research laboratories to find other types of usable materials for thermoelectric applications [21]. Therefore, new materials for this application e.g., half-Heusler, skutterudites, Calcium/Manganese oxides, Magnesium silicide, and tetrahedrites are currently being used for temperature difference ranges of $\left(300\le \mathsf{\Delta }T\le 750\right) \mathrm{K}$ and withstanding maximum temperatures of $\left(573\le T_{max}\le 1073\right) \mathrm{K}$ [22,23].

Furthermore, numerical studies on TEG devices have been performed underlining simulation methodologies [24,25], while others allow visualization that the methodology to mathematically model the mismatching temperature conditions for both photovoltaic cells and thermoelectric devices are similar [26,27].

Wang et al. [28] proposed a mathematic model which takes into account the temperature-dependent thermoelectrical properties and effects of convection but did not consider the temperature mismatching. Montecucco et al. [15] performed an experimental study of interconnected thermoelectric devices in series and parallel considering the temperature mismatching effect over the circuital model, not directly on the thermoelectric couples but on the whole arrangement. Wee [29] developed a theoretical analysis to solve the differential equations governing the thermoelectric devices based on the assumption of the linear variation of the temperature over P-N couples, but he neglected real physical phenomena such as temperature mismatching conditions. Ju et al. [30] performed a similar investigation considering only a linear variation of the Seebeck coefficient and electrical resistivity; no mismatching conditions were considered in the analysis. Therefore, current models neglect the combined effect of physical phenomena like thermal dependence of properties and the mismatching boundary conditions.

In literature, there are no reported models to evaluate the thermal mismatching on individual thermoelectrical couples considering the thermal variation of physical properties like Seebeck coefficient, thermal conductivity, electrical resistivity, and the dimensionless figure of merit. Therefore, this paper addresses such a problem by introducing the following contributions: first, a mathematical model to evaluate the mismatching conditions even on the cold and hot side; second, a thermoelectric simulation including the effect of temperature gradient on the thermoelectrical properties; and third, the evaluation of the efficiency of thermocouples under mismatching conditions.

The objective of this study is to characterize by means of numerical simulation the open-circuit electric output available power of a TEG device when a mismatching condition is applied to both the cold and hot sides of the TEG device considering temperature-dependent properties of the N and P-type semiconductor material ${\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3$.

Section 2 of the paper describes the governing equations, the thermoelectrical properties, and the methodology for the simulation. Section 3 presents the results of temperature and voltage contours and trend curves. Finally, Section 4 concludes the present work highlighting the main results.

2. Materials and Methods

2.1. Governing Equations

Figure 1 shows a thermoelectric generator (TEG) device, which is composed of thermocouples of P-type (positive) and N-type (negative) materials, and copper electrodes. A set of $n$-th materials can be connected electrically in series, and thermally in parallel to form a thermoelectric array. If a temperature difference $\mathsf{\Delta }{T}_{TEG}=T_{hot}-T_{cold}$ is applied to the TEG device, then an open-circuit voltage $V_{oc}$ is generated due to the Seebeck effect. When an electrical load $R_L$ is connected to both ends of the TEG device, an electric current $I$ flows from the N to the P material. Equation (1) defines the open-circuit voltage $V_{oc}$ produced by the TEG device. The term $\alpha ={\alpha }_P-{\alpha }_N$ is the combined Seebeck coefficient, which relates the Seebeck coefficient for both the N and P-type materials [31].

$$V_{oc}=\alpha \Delta T_{TEG}$$

(1)

Figure 1 also shows the heat energy $Q$ inputs and outputs present in a leg of the TEG device, whose energy balance equation must follow that $Q_h=Q_e+Q_{conv}+Q_s$, where $h, e,conv$, and $s$ represent the heat input, heat to electrical energy conversion, heat loss due to convection, and heat energy on the heatsink, respectively. Then, the general steady-state heat flow equation is defined by (2) [32].

$$\nabla \cdot \overrightarrow{q}=\dot{q}$$

(2)

The left side of Equation (2) represents the divergence of the heat flux vector $\overrightarrow{q}$, which is defined in terms of the Fourier’s law of thermal conduction $-k\nabla T$ involving the thermal conductivity $k$, the heat generation due to the Peltier effect $\alpha \overrightarrow{J}T$, and the electric current density vector $\overrightarrow{J}$, given by (3). According to [33], the electric current density can be computed as (4), where $\overrightarrow{E}=-\nabla \varphi$ is the electric field intensity vector, and $\rho$ is the electrical resistivity. In addition, the electric scalar potential $\varphi$ is defined by (5).

$$\overrightarrow{q}=-k\nabla T+\alpha T\overrightarrow{J}$$

(3)

$$\overrightarrow{J}=\frac{1}{\rho }\left(\overrightarrow{E}-\alpha \nabla T\right)$$

(4)

$$-\nabla \varphi =\rho \overrightarrow{J}+\alpha \nabla T$$

(5)

The right side of Equation (2) is the heat generation rate per unit volume $\dot{q}$, which is defined as (6).

$$\dot{q}=\rho {\left|\overrightarrow{J}\right|}^2+\alpha \overrightarrow{J}\cdot \nabla T-q_{conv}$$

(6)

where $\rho {\left|\overrightarrow{J}\right|}^2$ is the Joule heating, $\alpha \overrightarrow{J}\nabla T$ is the work done against the Seebeck field, and $q_{conv}=\left[hP\left(T-T_0\right)\right]/A$ is the heat loss on the side of the TEG’s legs due to convection, where the term $h \left[\mathrm{W}/{\mathrm{m}}^2\mathrm{K}\right]$ is the heat transfer coefficient, $T_0$ the ambient temperature, $P$ and $A$ are the perimeter and area of the TEG’s leg cross-section, respectively [28]. The divergence of Equation (3) using the product rule yields (7).

$$\nabla \cdot \overrightarrow{q}=-\nabla \cdot \left(k\nabla T\right)+\alpha T\left(\nabla \cdot \overrightarrow{J}\right)+\alpha \overrightarrow{J}\cdot \nabla T+T\overrightarrow{J}\cdot \nabla \alpha$$

(7)

For a steady-state analysis, the divergence of the electric current density vector is $\nabla ·\overrightarrow{J}=0$, which ensures the continuity of the current density [34]. Furthermore, if the Seebeck coefficient is a function of temperature $\alpha \left(T\right)$, then the Thomson coefficient is defined as $\beta =T\left(\partial \alpha /\partial T\right)$ [35]. Hence, substituting Equations (6) and (7) into (2), and accounting for the temperature dependency of the properties of the materials, Equation (2) can be rewritten for a three-dimensional case as a second-order partial differential equation with variable coefficients as (8).

$$k\left(T\right){\nabla }^{2}T-\beta \left(T\right)\overrightarrow{J}\cdot \nabla T+\rho \left(T\right){\left|\overrightarrow{J}\right|}^2-\frac{P}{A}h\left(T-T_o\right)=0$$

(8)

The voltage distribution throughout a control volume of a TEG device considering the electric current density vector $\nabla ·\overrightarrow{J}=0$ for a steady-state condition, can be derived using Equation (5) resulting in (9), [34].

$$\nabla \cdot \left[\sigma \left(T\right)\nabla V\right]+\nabla \cdot \left[\sigma \left(T\right)\alpha \left(T\right)\nabla T\right]=0$$

(9)

where $\sigma \left(T\right)=1/\rho$ is the electrical conductivity as a function of temperature $T$. The first and second terms of Equation (9) represent the electric conduction, and the distortion on the electric field caused by the thermoelectric effect, respectively [34]. Therefore, Equations (8) and (9) describe the existing thermoelectric phenomena in a TEG device.

The TEG electrical output available power $P_e$ can be approximated using the open-circuit voltage and the temperature-dependent total internal electrical resistance $R_{total}\left(T\right)$, as Equation (10) shows, [36].

$$P_e=\frac{V_{oc}{}^2}{4R_{total}\left(T\right)}$$

(10)

The maximum efficiency of a thermoelectric material, either when the TEG device is generating electrical power or cooling, can be determined using the factor of merit $Z$, defined by (11), with ${\alpha }_P$, ${\alpha }_N$; $k_P$, $k_N$; and ${\rho }_P, {\rho }_N$ being the Seebeck coefficients, the thermal conductivities, and the electrical resistivities of the P and N materials, respectively [37]. If $Z$ is multiplied by the average temperature $T=\left(T_h-T_c\right)/2$, then the dimensionless figure of merit $ZT$ is obtained, given by (12), [38].

$$Z=\frac{{\left({\alpha }_P-{\alpha }_N\right)}^2}{{\left[{\left(k_P{\rho }_P\right)}^{1/2}+{\left(k_N{\rho }_N\right)}^{1/2}\right]}^2}$$

(11)

$$ZT=\frac{{\alpha }^{2}T}{\rho k}$$

(12)

The efficiency of the thermoelectric generator device can be determined by relating the heat transfer rate on the hot side ${\dot{Q}}_h$ and the electric output power generated $P_e$ by the TEG device (13). The efficiency expression can be written in terms of the temperature difference $\mathsf{\Delta }T=T_h-T_c$, the dimensionless figure of merit $ZT$, and the cold and hot temperatures $T_c$, $T_h$, respectively [37].

$${\eta }_{TEG}=\frac{P_e}{{\dot{Q}}_h}=\frac{\Delta T}{T_h}\frac{\sqrt{1+ZT}-1}{\sqrt{1+ZT}+\frac{T_c}{T_h}}$$

(13)

2.2. Thermoelectric Properties of Materials

For this article, a total of five materials constituted the studied thermoelectric module: copper, solder $\left({\mathrm{Sn}}_{96.5}{\mathrm{Ag}\mathrm{Cu}}_{0.5}\right)$, filler (silicone elastomer), thermal interface layer (TIL), and the N and P-type material $\left({\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3\right)$. On the one hand,

Table 1. Thermoelectric properties of the material used for the numerical simulation.

Material	$\mathit{k}$ $\left(\mathbf{W}/\mathbf{m}\mathbf{K}\right)$	$\mathit{\alpha }$ $\left(\mathsf{\mu }\mathbf{V}/\mathbf{K}\right)$	$\mathit{\rho }$ $\left(\mathbf{\Omega }\mathbf{m}\right)$
Copper	400 [34]	1.80 [39]	Variable, see Figure 2
Solder $\left({\mathrm{Sn}}_{96.5}{\mathrm{Ag}\mathrm{Cu}}_{0.5}\right)$	64 [40]	$-$	1.25 $\times$ 10−7 [41]
Filler (silicone elastomer)	0.27 [42]	$-$	$-$
Thermal interface layer (TIL)	4 [43]	$-$	$-$

$k:$ thermal conductivity; $\alpha :$ Seebeck coefficient; $\rho :$ electrical resistivity.

shows the isotropic properties of thermal conductivity $k$, the Seebeck coefficient $\alpha$, and the electrical resistivity $\rho$ of the temperature-independent materials, except for $\rho$ of the copper.

Figure 2 shows the copper electrical resistivity $\rho$ as a function of the temperature $T$, which is defined in the range of (273.15 $\le T\le$ 373.15) $\mathrm{K}$. However, for the present numerical simulation purposes, the copper $\rho$ can be extrapolated for higher values of $T$.

On the other hand, Figure 3a shows the thermoelectrical isotropic temperature-dependent properties of the N and P-type material as the thermal conductivity $k$, the Seebeck coefficient $\alpha$, and the electrical resistivity $\rho$. A range of temperature of (300 $\le T\le$ 500) $\mathrm{K}$ was selected based on the experimental study of Chen et al. [45] on the ${\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3$ material. Figure 3b shows the dimensionless figure of merit $ZT$ calculated using Equation (12), which reached a maximum value of $ZT=$ 1.52 at $T=$ 350 $\mathrm{K}$.

Table 2. Polynomial regressions of the temperature-dependent thermoelectrical properties of the N and P-type ${\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3$ material for a temperature range of $\left(300\le T\le 500\right) \mathrm{K}$.

$\mathbf{Thermoelectric}\mathbf{Properties}\mathbf{of}\mathbf{B}{\mathbf{i}}_{0.4}\mathbf{S}{\mathbf{b}}_{1.6}\mathbf{T}{\mathbf{e}}_3$	Polynomial Regressions
$k\left(T\right) :$ Thermal conductivity	$1.4919+0.0022T-2.1333\times {10}^{-5}{T}^2+3.1168\times {10}^{-8}{T}^3$, $R^2=0.9952$
$\alpha \left(T\right) :$ Seebeck coefficient	$-546.7325+4.9569T-0.0102T^2+6.4221\times {10}^{-6}{T}^3$, $R^2=0.9821$
$\rho \left(T\right) :$ Electrical resistivity	$138.8930-1.0964T+0.0029T^2-2.4851\times {10}^{-6}{T}^3$, $R^2=0.9956$
$ZT\left(T\right) :$ Dimensionless figure of merit	$-22.5035+0.1753T-4.1546\times {10}^{-4}{T}^2+3.1614\times {10}^{-7}{T}^3$, $R^2=0.9996$

shows the polynomial regressions of the isotropic thermoelectric temperature-dependent properties of the N and P-type material ${\mathrm{Bi}}_{0.4}{\mathrm{Sb}}_{1.6}{\mathrm{Te}}_3$ and the performance $ZT$, where the coefficient of determination $R^2$ (COD), and the temperature range of the properties is specified.

According to Mackey et al. [46,47] the largest contributing sources of uncertainty on thermoelectric properties are electrical resistivity that includes the thermocouple tip radius, sample uniformity, and probe separation length. They estimated that typical samples measured with the ZEM-3 equipment, similar to ZEM-2 used by Chen et al. [45] to report the thermoelectrical properties presented in Figure 3, was about $\pm$7.0% across any measurement of temperature.

2.3. Validation Study of the Numerical Model

2.3.1. TEG Device Three-Dimensional Model and Mesh

Figure 4 shows an isometric view of the studied TEG device and the general dimensions. The model is composed of one pair of P-type material (red), and one pair of N-type material (dark blue), 5 copper electrodes (orange), which is composed of 3 large and 2 small-size copper electrodes. A filler (light blue), and the thermal interface layer TIL (purple). The 3D model was meshed using the module Mesh of ANSYS 2020 R1. Additionally, Figure 5a presents the mesh independence study of the model, in which 7 iterations were executed, and the open-circuit voltage was considered as the parameter of interest. A mesh with 32,832 elements with 141,375 nodes and a maximum mesh element size of 1 $\times$ 10⁻⁴ $\mathrm{m}$ was selected due to the percentage error of 0.06% compared to the next iteration. Figure 5b presents a three-dimensional view of the hexahedral selected mesh used for the numerical simulation.

Table 3. Hexahedral mesh quality parameters for the aggressive mechanical criterion of Mechanical APDL of ANSYS.

Mesh Quality Parameters	Minimum Value $[-]$	Maximum Value $[-]$	Average Value $[-]$	Standard Deviation $[-]$	Error Limit Threshold $[-]$
Element quality	0.9512	1	0.9981	3.9829$\times$10−3	$<$5$\times$10−4
Jacobian ratio (corner nodes)	0.8568	1	0.9965	8.6903$\times$10−3	$<$0.025

shows the quality parameters of the mesh that are relevant for the aggressive mechanical criterion of the APDL Mechanical solver of ANSYS such as the element quality and the Jacobian ratio (corner nodes), which limit values of 0 and 1 are bad and good quality, respectively [48,49]. The minimum and maximum values of the quality parameters are given, as well as the standard deviation and the error limit for the 3-D problem.

2.3.2. Comparison between Experimental Data and Numerical Model Results

Validation of the simulation was performed comparing the numerical open-circuit voltage $V_{OC}$ of the TEG device model used in the simulation with the experimental open-circuit voltage of the commercial thermoelectric generator module GM250-449-10-12 of the manufacturer (European Thermodynamics Ltd., Kibworth, UK) [50]. The boundary temperatures of the hot $T_h$ and cold $T_c$ side of the TEG device were applied uniformly on the surface (no mismatched temperatures). The thermoelectric properties of the N- and P-type materials of the thermoelectric generator module GM250-449-10-12 used for the numerical model are reported in Figure 6.

Montecucco et al. [15] studied the same commercial thermoelectric generator (GM250-449-10-12) and reported that the module was composed of 449 thermoelectric couples/pairs of N- and P-type materials. This allowed to scale up the voltage of the numerical simulation according to the couples contained in the commercial TEG module. The comparison of the numerical and experimental open-circuit voltage $V_{OC}$ is reported in Figure 7. A good agreement between the experimental and numerical values can be seen, confirms the validity of the numerical model used in the manuscript.

2.3.3. Verification of Numerical Thermal Efficiency

A comparison between the numerical and theoretical thermal efficiencies was carried out only for non-mismatching conditions cases, as shown in Figure 8. The theoretical values of thermal efficiency for non-mismatching conditions are calculated using Equation (13). That equation does not consider any uneven temperature distributions on both the hot and cold sides of the TEG device neither the thermoelectric variable properties.

Numerical thermal efficiency results are higher than those computed with Equation (13). A temperature-independent mean value of the dimensionless figure of merit was used $\left(\overline{ZT}=\left(T_h+T_c\right)/2\right)$, where $T_h$ and $T_c$ represent the temperature on the hot and cold sides of the TEG device. Lastly, we consider the relative error acceptable due to the assumption of ZT to compute the theoretical thermal efficiency.

2.4. Boundary Conditions and Parametric Studies

The thermoelectric device shown in Figure 4 was simulated with the commercial solver code of Mechanical APDL using the Thermo-Electric module of ANSYS 2020 R1. Two simulations, which considered two cases of mismatching conditions, were executed by a 12-core Intel ^® Xeon CPU E5-2667 at 2.90 GHz, and 32 GB of RAM workstation. Both simulations assumed no heat transfer by convection $Q_{conv}=0$ nor the effects of radiation $Q_{rad}=0$ due to the considered low temperature gradients [51].

2.4.1. Numerical Simulation of the Mismatching Condition on the Heat Side (Bottom Surfaces) of the TEG Device

Figure 9 shows the Dirichlet thermoelectric boundary conditions for this simulation, which are assigned as follows: firstly, a thermal boundary condition $T_{cold}$ is placed on the three upper surfaces of the model, which represent the ambient temperature or heatsink. Also, an electromagnetic boundary condition $V_{ref}$, which represents the reference voltage, is placed on the electrode of the P-type material, see Figure 9a. Secondly, the bottom surface has two thermal boundary conditions that represent the mismatching condition $\left(\Delta T_{mismatching}=T_{hot,max}-T_{hot,variable}\right)$, where one of them is the surface temperature $T_{hot max}$, which remain constant and is the highest temperature value; next to the previous-mentioned surface, lies the variable temperature $T_{hot variable}$, which generates the mismatching temperature gradient on the bottom surface of the TEG device, see Figure 9b. An example of the definition of the thermoelectric boundary conditions, e.g., the temperature on the cold side $T_{cold}$, the voltage reference $V_{ref}$, and the temperatures of the hot side $T_{hot,max}$ and $T_{hot,variable}$, is given in Figure 9 representing the first row of the parametric conditions described in

Table 4. Parametric study values of the thermoelectric boundary conditions of the mismatching condition on the heat side (bottom surfaces) of the TEG module.

${\mathit{T}}_{\mathit{h}\mathit{o}\mathit{t} \mathit{m}\mathit{a}\mathit{x}}$ $\left[\mathbf{K}\right]$	${\mathit{T}}_{\mathit{h}\mathit{o}\mathit{t} \mathit{v}\mathit{a}\mathit{r}\mathit{i}\mathit{a}\mathit{b}\mathit{l}\mathit{e}}$ $\left[\mathbf{K}\right]$	${\mathit{T}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{d}}$ $\left[\mathbf{K}\right]$	${\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}$ $\left[\mathbf{V}\right]$
500	(300 $\le T_{hot variable}\le$ 500), steps of 10 K	300	0
	(350 $\le T_{hot variable}\le$ 500), steps of 10 K	350
	(400 $\le T_{hot variable}\le$ 500), steps of 10 K	400
	(450 $\le T_{hot variable}\le$ 500), steps of 10 K	450

.

Table 4 describes the parametric study simulation of the mismatching condition on the heat side, see Figure 9. The temperature $T_{hot, max}$, and the electromagnetic boundary condition $V_{ref}$ remained constant throughout the simulation at a value of 500 $\mathrm{K}$ and 0 $\mathrm{V}$, respectively. The temperature $T_{hot, variable}$ changed within a range of 300 $\le T_{hot, variable}\le$ 500) $\mathrm{K}$ with steps of 10 $\mathrm{K}$ as the ambient temperature $T_{cold}$ increased from 300 to 450 $\mathrm{K}$.

2.4.2. Numerical Simulation of the Mismatching Condition on the Heatsink (Upper Surfaces) of the TEG Device

Figure 10 shows the Dirichlet thermoelectric boundary conditions for the mismatching condition on the heatsink surfaces, which are assigned as follows: firstly, the upper surface has two thermal boundary conditions that represent the mismatching condition $\left({\mathsf{\Delta }}_{mismatching}=T_{cold,min}-T_{cold,variable}\right)$, which are the surface temperature $T_{cold,min}$ and $T_{cold, variable}$ that generate the mismatching temperature gradient on the upper surface of the TEG device. Also, an electromagnetic boundary condition $V_{ref}$, which represents the reference voltage, is placed on the electrode of the P-type material, see Figure 10a. Secondly, a constant thermal boundary condition $T_{hot}$ is placed on the bottom surface of the model, which represent the heat side temperature, see Figure 10b. Additionally, an example of the definition of the thermoelectric boundary conditions e.g., the temperatures of the cold side $T_{cold,variable}$ and $T_{cold,min}$, the temperature on the hot side $T_{hot}$, and the voltage reference $V_{ref}$, is given in Figure 10 representing the first row of the parametric conditions described in

Table 5. Parametric study values of the thermoelectric boundary conditions of the mismatching condition on the heatsink (upper surfaces) of the TEG module.

${\mathit{T}}_{\mathit{h}\mathit{o}\mathit{t}}$ $\left[\mathbf{K}\right]$	${\mathit{T}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{d},\mathit{v}\mathit{a}\mathit{r}\mathit{i}\mathit{a}\mathit{b}\mathit{l}\mathit{e}}$ $\left[\mathbf{K}\right]$	${\mathit{T}}_{\mathit{c}\mathit{o}\mathit{l}\mathit{d},\mathit{m}\mathit{i}\mathit{n}}$ $\left[\mathbf{K}\right]$	${\mathit{V}}_{\mathit{r}\mathit{e}\mathit{f}}$ $\left[\mathbf{V}\right]$
500	(300 $\le T_{cold,variable}\le$ 500), steps of 10 K	300	0
	(350 $\le T_{cold,variable}\le$ 500), steps of 10 K	350
	(400 $\le T_{cold,variable}\le$ 500), steps of 10 K	400
	(450 $\le T_{cold,variable}\le$ 500), steps of 10 K	450

.

Table 5 describes the parametric study simulation of the mismatching condition on the heatsink (upper surface), see Figure 10. The temperature $T_{hot}$, and the electromagnetic boundary condition $V_{ref}$ remained constant throughout the simulation at a value of 500 $\mathrm{K}$ and 0 $\mathrm{V}$, respectively. The temperature $T_{cold,variable}$ changed within a range of (300 $\le T_{hot, variable}\le$ 500) $\mathrm{K}$ with steps of 10 $\mathrm{K}$ as the temperature $T_{cold,min}$ increased from 300 to 450 $\mathrm{K}$.

3. Results and Discussion

To calculate the total internal resistance of the TEG device, the geometric properties of the temperature-dependent materials are needed. In this manner,

Table 6. Geometric properties of the N and P-type, copper, and solder materials. $H:$ height; $W:$ width; $D:$ depth; $A:$ area.

Material	Number of Components	$\mathit{W}$ $\mathit{x}\mathbf{-}\mathbf{Coordinate}$ $\mathbf{\left[}\mathbf{m}\mathbf{\right]}$	$\mathit{H}$ $\mathit{y}\mathbf{-}\mathbf{Coordinate}$ $\mathbf{\left[}\mathbf{m}\mathbf{\right]}$	$\mathit{D}$ $\mathit{z}\mathbf{-}\mathbf{Coordinate}$ $\mathbf{\left[}\mathbf{m}\mathbf{\right]}$	$\mathit{A}$ $\mathbf{\left[}{\mathbf{m}}^{\mathbf{2}}\mathbf{\right]}$
N and P-type	4 (2 N and 2 P)	0.00150	0.00120	0.00150	2.25$\times$10−6
Copper (large)	3	0.00150	0.00030	0.00350	5.25$\times$10−6
Copper (small)	2	0.00150	0.00030	0.00175	2.63$\times$ 10−6
Solder	8	0.00150	0.0001	0.00150	2.25$\times$10−6

shows the geometric properties of the copper, which has two sizes (see Figure 4), the N and P-type materials, and the solder.

However, the total internal resistance in every point of the thermoelectric material is hard to calculate due to the three-dimensional temperature gradients induced by the mismatching conditions on the TEG device. For this reason, the total internal resistance of the copper and the N-P-type materials of the TEG module was calculated applying the mean value theorem to the polynomial regressions of the resistivities of the copper ${\rho }_{copper}\left(T\right)$ and the N and P-type materials ${\rho }_{N-P}\left(T\right)$, see Figure 2 and Table 2 for the polynomial regressions, respectively.

On the one hand, the mean value of the resistivity of the N and P-type materials ${\rho }_{N-P}\left(T\right)$ for the simulation of the mismatching condition on the heat side (see Section 2.4.1) is defined by the piecewise-defined functionin Equation (14).

$${\overline{\rho }}_{N-P,mismatching\mathrm{on}{T}_H}\left(T\right)=\{\begin{array}{l}{\rho }_{N-P}\left(T\right),\mathrm{if}{T}_{hot,variable}=T_{cold}\\ \left(\frac{1}{T_{hot,variable}-T_{cold}}{\displaystyle {\int }_{T_{cold}}^{T_{hot,variable}}{\rho }_{N-P}\left(T\right)dT}\right)+\left(\frac{1}{T_{hot,max}-T_{cold}}{\displaystyle {\int }_{T_{cold}}^{T_{hot,max}}{\rho }_{N-P}\left(T\right)}dT\right),\mathrm{if}{T}_{hot,variable}\ne {\mathrm{T}}_{cold}\end{array}$$

(14)

On the other hand, the mean value of the resistivity of the N and P-type materials ${\rho }_{N-P}\left(T\right)$ for the simulation of the mismatching condition on the cold side (see Section 2.4.2) is defined by the piecewise-defined functionin Equation (15).

$${\overline{\rho }}_{N-P,mismatching\mathrm{on}{T}_C}\left(T\right)=\{\begin{array}{l}{\rho }_{N-P}\left(T\right),\mathrm{if}{T}_{hot}=T_{cold,variable}\\ \left(\frac{1}{T_{hot}-T_{cold,min}}{\displaystyle {\int }_{T_{cold,min}}^{T_{hot}}{\rho }_{N-P}\left(T\right)dT}\right)+\left(\frac{1}{T_{\mathrm{hot}}-T_{cold,variable}}{\displaystyle {\int }_{T_{cold,variable}}^{T_{hot}}{\rho }_{N-P}\left(T\right)}dT\right),\mathrm{if}{T}_{hot}\ne T_{cold,variable}\end{array}$$

(15)

The temperature-dependent total internal resistance $R_{total}\left(T\right)$ of the TEG device is calculated using Equation (16), where $W_{N-P}$, $A_{N-P}$, and ${\rho }_{N-P}\left(T\right)$ are the width, area, and the variable resistivity of the N and P type materials, respectively. $R_{cooper,large}$, $R_{copper,small}$ and $R_{solder}$ are the constant resistivities of the temperature-independent materials.

$$R_{total}\left(T\right)=\frac{2W_{N-P}\left[{\rho }_{N-P}\left(T\right)\right]}{A_{N-P}}+R_{copper,large}+R_{copper,small}+R_{solder}$$

(16)

Figure 11 shows the temperature-dependent total internal resistance $R_{total}\left(T\right)$ of the TEG device compared to the mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}$. Figure 11a presents $R_{total}\left(T\right)$ for the mismatching condition on the heat side (bottom surfaces), in which can be observed the decreasing electrical resistance as the delta of mismatching $\mathsf{\Delta }{T}_{mismatching}$ increases. This is because as $\mathsf{\Delta }{T}_{mismatching}$ on the hot side is higher, the temperature on the bottom surfaces gets colder, thus decreasing the electrical resistance of the N and P-type materials. Similarly, Figure 11b shows $R_{total}\left(T\right)$ for the mismatching condition on the heatsink (upper surfaces) of the TEG device, where the increment of the mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}$ causes the increment of the temperature of the upper surfaces, then raising the electrical resistance of the N and P-type materials.

3.1. Numerical Results and Contours of the Mismatching Condition Simulation on the Heat Side (Bottom Surface) of the TEG Device

Figure 12 presents the numerical results of the mismatching condition simulation on the heat side (bottom surfaces) of the TEG device compared to the mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}$, which spans from 0 to 200 $\mathrm{K}$. Figure 12a presents the open-circuit voltage $V_{oc}$. Figure 12b shows the electric available power $P_e$, calculated with Equation (10). Figure 12c shows the thermal efficiency ${\eta }_{TEG}$, computed as the ratio of the electric power $P_e$ and the heat energy input $Q_h$, Equation (13). The above-mentioned numerical results are presented for different cold side temperatures ranging between (300 $\le T_{cold}\le$ 450) $K$. Furthermore, an inverse relation can be seen between the numerical results and the increment of $\mathsf{\Delta }{T}_{mismatching}$. In other words, the open-circuit voltage $V_{oc}$ generated by the TEG device decreases because the temperature gradient between the hot and cold surfaces gets smaller as when $\mathsf{\Delta }{T}_{mismatching}$ is higher. Thus, the electric power $P_e$ and the thermal efficiency ${\eta }_{TEG}$ also have the same behavior. Additionally, the mathematical tendency of $V_{oc}$ and $P_e$ is linear, whereas the tendency of ${\eta }_{TEG}$ is exponential.

Figure 13 presents the temperature and the voltage contours of the TEG device for the mismatching condition on the heat side (bottom surface) at an ambient temperature of $T_{cold}=$ 300 $\mathrm{K}$. On the one hand, Figure 13a–c show the two-dimensional $xy$-plane temperature contour distributions located at the middle of the $z$-coordinate of the TEG device $z=-D/2$ ($D$: depth) at a mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}=$ 0, 100 and 200 $\mathrm{K}$, respectively. On the other hand, Figure 13d–f present the three-dimensional open-circuit voltage contour distributions at $\mathsf{\Delta }{T}_{mismatching}=$ 0, 100 and 200 $\mathrm{K}$, respectively. According to the contours, the Figure 13a,d correspond the maximum electric voltage and output power production, and thus the maximum efficiency points due to the zero mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}=$ 0. Once a mismatching temperature difference is induced on the heat side (bottom-right surface), e.g., Figure 13b,c the open-circuit voltage $V_{oc}$, the electric output power $P_e$ and the thermal efficiency ${\eta }_{TEG}$ decrease. The above-mentioned behavior is generated by the mismatching condition on the right side of the temperature contours, which causes a reduction in the temperature gradient between the upper and bottom surfaces. The voltage not only decreases due to the reduction of the temperature gradient but also because of the Joule heating due to the flowing electric current that is generated by the left side of the TEG device, as can be seen in Figure 13e,f.

3.2. Numerical Results and Contours of the Mismatching Condition Simulation on the Heatsink (Upper Surface) of the TEG Device

Figure 14 presents the numerical results of the mismatching condition simulation on the cold side (upper surfaces) of the TEG device compared to the mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}$, which spans from 0 to 200 $\mathrm{K}$. Figure 14a presents the open-circuit voltage $V_{oc}$. Figure 14b shows the electric output available power $P_e$, calculated with Equation (10). Figure 14c shows the thermal efficiency ${\eta }_{TEG}$, computed as the ratio of the electric power $P_e$ and the heat energy input $Q_h$, Equation (13). These numerical results are presented for different cold-side mismatching temperatures ranging between (300 $\le T_{cold,min}\le$ 450) $\mathrm{K}$. The curve tendencies of the numerical results are inversely proportional to the increment of the mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}$ due to the shortening of the temperature gradient between the upper and bottom surfaces, as the temperature boundary conditions $T_{cold,min}$ and $T_{cold,variable}$ increase. Unlike the linear tendency of $V_{oc}$, the electric output power $P_e$ and the thermal efficiency ${\eta }_{TEG}$ present a non-linear tendency.

Figure 15 presents the temperature and the voltage contours of the TEG device for the mismatching condition on the heat side (bottom surface) at a mismatching temperature $T_{cold,min}=$ 300 $\mathrm{K}$. Figure 15a–c show the two-dimensional $xy$-plane temperature contour distributions located at the middle of the $z$-coordinate of the TEG device $z=-D/2$ ($D$: depth) at a mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}=$ 0, 100 and 200 $K$, respectively. On the other hand, Figure 15d–f present the three-dimensional open-circuit voltage contour distributions at $\mathsf{\Delta }{T}_{mismatching}=$ 0, 100 and 200 $K$, respectively. The Figure 15a,d correspond the maximum electric voltage and output power production, having the maximum efficiency point due to the zero mismatching temperature difference $\mathsf{\Delta }{T}_{mismatching}=$ 0. If a mismatching temperature difference is induced on the cold side (upper-right surface), e.g., Figure 15b,c, the open-circuit voltage $V_{oc}$, the electric output power $P_e$ and the thermal efficiency ${\eta }_{TEG}$ decrease. The previous-mentioned behavior is generated by the mismatching condition on the upper-right side of the temperature contours, which causes a reduction in the temperature gradient between the upper and bottom surfaces of the TEG device. Also, the voltage not only decreases due to the reduction of the temperature gradient but also because of the Joule heating when the flowing electric current that is generated by the left side of the TEG device is dissipated as heat by the right side of the TEG device, as can be seen in Figure 15e,f.

3.3. Comparison of the Numerical and Contour Results between the Mismatching Conditions on the Heat and Cold Sides of the TEG Device Simulations

Figure 16 presents a comparison of the available electric power outputs $P_e$ for the mismatching condition on the heat side (black-color curves), and cold side (red-color curves). Figure 16a presents the variation of $P_e$ for a range of the mismatching temperature difference of (0 $\le \mathsf{\Delta }{T}_{mismatching}\le$ 200) $\mathrm{K}$. Figure 16b for a range of (0 $\le \mathsf{\Delta }{T}_{mismatching}\le$ 150). Figure 16c for a range of (0 $\le \mathsf{\Delta }{T}_{mismatching}\le$ 100). Figure 16d for a range of (0 $\le \mathsf{\Delta }{T}_{mismatching}\le$ 50). From the figures it can be seen that $P_e$ decreases linearly for the mismatching condition on the hot side $T_{hot}$, while $P_e$ decreases in a non-linear way for the mismatching condition on the cold side $T_{cold}$. Thus, the tendency of the presented curves suggests that the mismatching condition on the cold side $T_{cold}$ (curves in red color) of a TEG device causes the electric output power to decrease more rapidly than a mismatching condition applied on the heat side $T_{hot}$.

4. Conclusions

The generation of electric power decreases when the mismatching temperature difference increases because of the reduction of the temperature gradient between the hot and cold sides of the thermoelectric generator device (TEG). Although the values of the open-circuit voltage are similar for the mismatching conditions applied to both the cold and hot sides of the TEG, the generation of electric power is higher for a mismatching condition on the hot side than on the cold side due to the lower values of the temperature-dependent electrical resistance of the N and P-type materials when a mismatching condition is applied to the hot side. Additionally, the electric output power generated by the TEG device decreases when the mismatching temperature difference increases. Then, if a mismatching condition is applied to the hot side, the electric power decreases linearly and thus slower than that of a mismatching condition applied to the cold side, which decreases faster in a non-linear way.

In summary, there are three main contributions of this paper. The first one is the validated proposed model to evaluate thermoelectric couples under mismatching conditions. The second contribution is the inclusion of the behavior of thermoelectric properties as temperature functions. The last contribution is the evaluation of the thermal efficiency over TEG couples considering the mismatching conditions and thermal variable properties.