Advanced Energy Modeling and Prediction of Integrated Micro-Generator System for Useful Heat Harvesting

Abstract

Theoretical modeling and numerical simulation of an integrated micro-thermoelectric generator system for thermal power generation are carried out. The system measures 4.2 × 4.2 × 5 mm and consists of a micro-thermoelectric module (bismuth telluride) and two finned heat sinks (aluminum). The system can be used to convert thermal energy to electricity in Seebeck effect-based micro-applications. This work aims to improve an advanced model to effectively predict the thermal performance of the system and to develop thermal and flow simulations to accurately evaluate real micro-thermoelectric generator systems. The advanced model solves the thermoelectric module’s energy equations, incorporating heat balance in the heat transfer calculations. The thermal and flow simulations take into account the dynamic calculations under the thermal loads occurring in the system. This innovative aspect can considered separately for the different materials (ceramics, semiconductors and copper strips) of the micro-thermoelectric module for heat transfer enhancement. The results predicted that when the temperature difference of the thermoelectric module was increased from 18 K to 58 K, the power output and the conversion efficiency of the system increased by about 0.5 W and 50%, respectively. Also, the transfer of useful heat to electrical power was achieved at 83%, with 11% saved heat and thermal losses of 6% W at maximum temperature difference of the module. In terms of overall energy consumption, the integrated micro-thermoelectric generator system has a little environmental impacts. Validation of the model with particular experimental works was accomplished for dependability. Comparisons with different modeling strategies demonstrate that the accuracy and performance of the advanced model can be used to reliably study the thermal performance of real micro-thermoelectric generator systems.

1. Introduction

Thermoelectric modules can precisely transform thermal energy into electrical power and vice versa [1,2,3]. They have no working fluids and moving components and are generally reliable, noiseless, practically small, easily regulated, and environmentally friendly [4,5]. However, awareness of thermoelectricity has recently been encouraged because of the concerns about energy conservation [6]. Generally, there are two main physical phenomena related to thermoelectricity. The first phenomenon is the Seebeck effect (sized from regular generators to micro-generators [7]). The second one is the Peltier effect (sized from regular coolers to micro coolers [8]). The dimensions of industrial thermoelectric modules (Bi₂ Te₃) differ from about 50 × 50 × 5 mm to a very smaller size of about 4 × 4 × 3 mm, which is stability operated in a wide range of thermal engineering applications [9]. Thermoelectric modules can be used in microelectronics to stabilize the temperature of solid-state lasers and charge-coupled devices; control infrared detectors; increase operating speed; and reduce dropout noise in integrated circuits [10]. There are gaps in knowledge of the modeling and simulation of a micro-thermoelectric generator regarding accurate computation of the heat transfer process between the components based on useful and lost heat, as well as sustainable assessment of the environmental impacts to produce a micro-generator. This work focuses on designing, modeling, and simulating a micro-thermoelectric generator system for thermal energy harvesting by using an advanced model to solve the thermoelectric module’s energy equations and incorporating the heat transfer processes in the micro-generator.

In general, researchers concentrate on remodeling simulation approaches and tools ranging from the earliest principles simulations of a small-scale thermoelectric generator system [11] to a large-scale system by simplified models [12]. The modeling strategies of different thermoelectric generator systems are comparatively well considered. Incomplete simulations could be performed using computing power and computational procedures, among other techniques.

A solution to the one-dimensional transient heat conduction equation for Joule heating of thermoelectric generator systems was presented by Montecucco A. et al. [13]. They created a computer program for the transient simulation of the system. This solution did not divide the thermoelectric material into thermocouples. Following this work, they developed a simulation model of a thermoelectric power generation system. Then, they considered the thermoelectric generator as a bulk to determine the temperature of the hot and cold sides. The developed model traded with the temperature difference and the output electrical power at the transient time. They found that the output electric power after 1800 sec at a temperature difference of 60 K was 2.6 W [14]. Naji M. et al. [15] used a fixed temperature as a boundary condition on both sides of a thermoelectric module, assumed that the temperature did not change, or assumed thermal insulation to study the module’s behavior. They found that the achieved figure of merit increased with time. Martínez A. et al. [16] proposed a computational model of a thermoelectric system. Using fluid dynamics software 2013 (based on the finite difference method), they obtained the temperature distribution of the system’s finned heat sink at different wind speeds and constant ambient temperature. Since the surfaces of the components of the thermoelectric system (expander, heat sink, and fan) did not complement each other, some heat transfer was lost.

Jang B. et al. [17] investigated the geometric effects of thermoelectric elements on micro-thermoelectric generator performance by applying a finite element method. In the numerical simulation, they found that the maximum temperature difference and the maximum electrical power of the micro-generator were small when the substrates were thick. It appeared that there was a suitable length of thermoelectric elements to reach the maximum power and efficiency. Lee H. [18] simplified a steady-state heat transfer analysis of a thermoelectric generator module with two heat sinks to obtain dimensionless parameters for the optimal design. The study focused on power output measurements in relation to the shape of the thermocouple and the heat sink’s thermal resistance. The researchers discovered that the specific fluid temperatures of the heat sink always produced an optimal design even within feasible engineering limits.

Li W. et al. [19] carried out simulations and comparisons of steady-state thermal parameters (fluid flow and heat transfer) in both tube and fin heat exchangers for thermoelectric generators. The thermoelectric generator was simplified by utilizing one-dimensional solid conduction and free convection effects on the exposed surfaces. They found that the finned heat exchangers were more compact and efficient than shell and tube heat exchangers. Sarhadi A. et al. [20] simulated thermoelectric generators and heat exchangers with computer models. Thermoelectric generators were studied as solids and flowing oil in heat exchangers was studied based on laminar flow. Heat flow was involved at the surface between the generator and the exchanger. They found that the cooling oil was evenly distributed from one ribbed surface to the other.

Högblom O. and Andersson R. [21] proposed a framework for characterizing and simulating thermoelectric generator systems that can efficiently predict electrical and thermal behavior under steady-state conditions. They found that the maximum electric power was less than 1 W at a temperature difference between 40 K and 60 K. Thus, the results displayed that the simulation framework reliably described the main thermo-physical properties of the system. Fraisse G. et al. [22] summarized several of the most common thermoelectric element models and compared them with an improved simplified model based on the finite element method. The accuracy of these models is assumed under constant thermoelectric properties. They found the model to be reasonably accurate in power generation mode. Many computational models have been refined to account for the behavior of 1D generators. These models only simulated heat source fluid states with convective or radiative effects on thermocouples and did not have a comprehensive thermoelectric generator model [23]. Wang X. et al. [24] addressed the low electrical output power and low energy conversion efficiency of truncated-leg thermoelectric generators based on a 3D multi-physics model at a fixed temperature. They found that modifying the shape of the thermoelectric legs to form truncated cones can significantly alter the distribution of temperature distribution in the legs and thus the physical properties. The numerical results provided the designed solution according to the maximum output power by 217.96% at the expense of 16.84% conversion efficiency.

Jouhara H. et al. [25] compared the temperature distribution of different models (rectangular leg, trapezoidal leg, Y-leg, X-leg and I-leg). They found that for all samples tested, the temperature dropped significantly from the hot side to the lower cold side. The rectangular legs showed a reasonably linear profile along the decay chain. In contrast, the other legs exhibited different types of curved chains due to variations in their respective cross-sectional areas. In order to examine the thermoelectric generator’s conversion process, Hu X. et al. [11] built a 3D model using finite elements. This model included Peltier, Seebeck and Thomson effects as well as Joule heating. To demonstrate these processes in action, they analyzed the energy balance between electricity and heat. When the module operated at its highest and lowest temperatures, it produced positive and negative Thomson heat, respectively. When operated in the middle range, the same modules produced similar results to Joule heat. When operated at its end temperature, they found that the Thomson heat was on the same order of magnitude as Joule heat.

According to the described studies, the current focus of the investigated topic is the losses of thermal energy between the components of the micro-thermoelectric generators because of the components do not complement each other. Also, there is no comprehensive micro-thermoelectric generator model to deal with all types of heat transfer collectively. In addition, there is no sustainable assessment of the environmental impacts to produce a micro-generator. This work aims to propose an advanced energy model that can accept the heat flow in and out of an integrated micro-thermoelectric generator to predict it in an efficient manner. The computational model uniquely deals with the different materials of the micro-thermoelectric module (ceramic, semiconductor and copper strip) separately. This would provide an advanced thermal and flow simulation that allows micro-systems to be accounted for in a physically correct way. The model analyzes the thermal performance of all components of the micro-thermoelectric generator system under steady-state and transient conditions. Similarly, the model uses a sustainability assessment method to assess the environmental impacts to produce a micro-generator. The remainder of this study includes the following sections: the methodology of the study, which starts from the integrated system description with theoretical modeling to the thermal conditions; the important findings of the thermal performance of the system, with the validation of the model with experimental works and a comparison between different modeling strategies; and the conclusions and future works.

2. Methodology

2.1. Integrated System Description

The components of the micro-thermoelectric generator system (4.2 × 4.2 × 5 mm) include one micro-thermoelectric generator module in a closed-loop system which is sandwiched between two aluminum finned heat sinks. There is an equivalent between the surface areas of the thermoelectric module at the hot and cold sides with the finned heat sinks. In this work, the module consists of 16 rectangular legs of n-type and p-type Bi₂Te₃ semiconductors coupled electrically in series by 17 copper strips and inserted between 2 electrically insulating but thermally conducting ceramic plates. Figure 1 shows the components and dimensions of the micro-thermoelectric generator system. Basically, when a temperature difference is created between the semiconductors legs in the module, a Seebeck voltage is generated at the copper terminals, and it is directly proportional to the temperature differences. The system is operated without any working fluids because the conduction electrons are used as a working fluid. On the other hand, there is a fluid (for example, air or any gas or nano fluids, depending on the application) flow around the system components and between the thermoelectric legs. It is necessary to discover the influence of the fluid-exposed components on the convection heat transfer and thermal losses.

2.2. Energy Analysis—Theoretical Modeling

An effective model can be used to predict the thermal behavior of the system. It accounts for natural convection and Joule heating when creating theoretical models for micro-thermoelectric generators. Additionally, thermal radiation across the surface of a component can be accounted for when modeling these systems.

2.2.1. Assumptions

In the theoretical modeling, the following supporting assumptions are required: (1) The physical properties of the solid components are constant because the attractive forces between the particular molecules are larger than the energy causing them to move apart. (2) The physical properties of air across the components are constant because the pressure and temperature of the location do not change. (3) The materials of the components are considered isotropic in order to make the mechanical properties of the components the same in all directions. (4) The Thomson effect is negligible in comparison with the Joule effect because of the positive Thomson coefficient indicates a slight enhancement in the thermal performance of the thermoelectric generator, while the negative Thomson coefficient indicates a slight declination in the performance. The comparison between the solutions of the governing heat equations of the cases and the coefficient of thermal expansion over a wide range of air temperature differences proved close agreement. (5) The semiconductor properties of both n-type and p-type Bi₂Te₃ are homogeneous at a medium temperature change of 150–300 K [26,27,28].

2.2.2. The Heat Balance

The model used the heat transfer equations to calculate the heat required for power generation and that moved by thermal processes in the micro-thermoelectric generator system, as shown in Figure 2. Firstly, energy conservation at the control volume for transient conditions is applied. Three main heat transfer modes are presented in the system; (1) the total heat transfer rate to the system ($\dot{\mathrm{Q}}$_in), (2) the saved heat transfer rate in the system ($∆\dot{\mathrm{Q}}$_s) which is associated with reversible joule heating of the thermoelectric module, and (3) the total heat transfer rate out of the system ($\dot{\mathrm{Q}}$_out). Relating the conservation of the energy, the heat balance is obtained from the following formula:

$${\dot{\mathrm{Q}}}_{\mathrm{i}\mathrm{n}}-{\dot{\mathrm{Q}}}_{\mathrm{o}\mathrm{u}\mathrm{t}}={∆\dot{\mathrm{Q}}}_{\mathrm{S}}$$

(1)

In the heat transfer analysis, the subsequent equations are established to define the saved heat transfer rate in the thermoelectric leg as a function of the average temperature of the hot and the cold sides of the thermoelectric module (${\mathrm{T}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$).

$${∆\dot{\mathrm{Q}}}_S={\dot{\mathrm{Q}}}_S{}_{\mathrm{at}}{\mathrm{T}}_{\mathrm{a}\mathrm{v}\mathrm{g}2}-{\dot{\mathrm{Q}}}_S{}_{\mathrm{at}}{\mathrm{T}}_{\mathrm{a}\mathrm{v}\mathrm{g}1}$$

(2)

$${\mathrm{T}}_{\mathrm{a}\mathrm{v}\mathrm{g}}=({\mathrm{T}}_{\mathrm{H}}+{\mathrm{T}}_{\mathrm{C}})/2$$

(3)

where T_C is the cold-side temperature and T_H is the hot-side temperature of the thermoelectric module. Two modes of heat transfer take place when a fluid (air, for example) or nano fluid flows through the hot surfaces of the heat sink and between the legs. The first mode is the natural convection rate ($\dot{\mathrm{Q}}$_conv).

$${\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}={\mathrm{h}}_{\mathrm{a}\mathrm{i}\mathrm{r}}{\mathrm{A}}_{\mathrm{h}\mathrm{s}}({\mathrm{T}}_{\mathrm{a}\mathrm{i}\mathrm{r}}-{\mathrm{T}}_{\mathrm{h}\mathrm{s}})$$

(4)

where ${\mathrm{h}}_{\mathrm{a}\mathrm{i}\mathrm{r}}$ is the heat transfer coefficient of air, ${\mathrm{A}}_{\mathrm{h}\mathrm{s}}$is the surface area of the hot surface of the heat sink, ${\mathrm{T}}_{\mathrm{a}\mathrm{i}\mathrm{r}}$ is the air temperature, and ${\mathrm{T}}_{\mathrm{h}\mathrm{s}}$ is the temperature at the hot surface of the heat sink. The second mode is the heat radiation rate at the hot surface of the heat sink ($\dot{\mathrm{Q}}$_rad).

$${\dot{\mathrm{Q}}}_{\mathrm{r}\mathrm{a}\mathrm{d}}=\mathsf{\epsilon }\mathsf{\sigma }{\mathrm{A}}_{\mathrm{h}\mathrm{s}}{T^4}_{\mathrm{h}\mathrm{s}}$$

(5)

where $\mathsf{\epsilon }$ is the emissivity at the hot surface of the heat sink; $\mathsf{\sigma }$ is the Stefan–Boltzmann constant. Both modes in Equations (4) and (5) represent the total heat transfer rate from the hot surface to the system.

$${\dot{\mathrm{Q}}}_{\mathrm{i}\mathrm{n}}={\dot{\mathrm{Q}}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}+{\dot{\mathrm{Q}}}_{\mathrm{r}\mathrm{a}\mathrm{d}}$$

(6)

Secondly, heat transfer from side to side of the thermoelectric legs is calculated by involving Fourier solid conduction, the Seebeck effect, and Joule heating for the thermoelectric generator module. Using the heat balance, the heat flow rate into the module at the hot side (${\dot{\mathrm{Q}}}_{\mathrm{H}}$) can be calculated using the following equation:

$$\dot{{\mathrm{Q}}_{\mathrm{H}}}=\left[\mathsf{\alpha }\mathrm{I}{\mathrm{T}}_{\mathrm{H}}-\mathrm{K}\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)-0.5\mathrm{R}{\mathrm{I}}^2\right]\mathrm{N}$$

(7)

where α is the Seebeck coefficient of thermoelectric material, I is an electric current in the circuit, $\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)$ is the difference between the hot and cold ends of the temperature module, R is the resistance of the thermoelectric module,$\mathrm{K}$ is the thermal conductivity of the module, and $\mathrm{N}$ is the number of pairs in the module. Two sets of equations are used (n-type and p-type thermoelectric Bi₂Te₃) to calculate the thermoelectric properties (α, $\mathrm{K}$, R) as a function of temperature [11]. The physical thermoelectric properties (Custom Thermoelectric Model (Bishopville, MD, USA), 00711-5A30-12CU4) are as follows: Q_max, 0.6 W; I_max, 1.2 Amp; V_max, 0.85 V; ∆T_max, 69 K; T_max, 125 °C. At ∆T less than 400 K, the thermoelectric material has a high figure of merit which is less than 1 compared with other semiconductors. The high numerical value of the figure of merit is indicated to design a high-performance thermoelectric module for use in precise engineering applications [29]. Significantly, there is an equivalence between the base surface of the thermoelectric module and the base surfaces of the hot and cold heat sinks for useful heat transfer in the system. However, the heat balance between the heat flow rate into the TE module and the total heat rate to the system can be expressed as

$$\dot{{\mathrm{Q}}_{\mathrm{H}}}={\dot{\mathrm{Q}}}_{\mathrm{i}\mathrm{n}}-\dot{{\mathrm{Q}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}$$

(8)

where $\dot{{\mathrm{Q}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}$ is the total thermal loss between the hot and cold borders, which depends on the number of legs, the cross-sectional area of a leg, and air gaps between the legs, as shown in Figure 3. There is a reasonable mechanism to calculate the total thermal losses inside the thermoelectric generator which is employed in the model [29]. The heat flow rate out of the module at the cold side (${\dot{\mathrm{Q}}}_{\mathrm{C}}$) can be calculated using [11] the following equation:

$$\dot{{\mathrm{Q}}_{\mathrm{C}}}=\left[\mathsf{\alpha }\mathrm{I}{\mathrm{T}}_{\mathrm{C}}-\mathrm{K}\left({\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{C}}\right)-0.5\mathrm{R}{\mathrm{I}}^2\right]\mathrm{N}$$

(9)

Both Equations (4) and (5) are repeated to calculate the total heat transfer rate out of the system at the cold surface heat sink. Therefore, the heat balance between the heat flow rate out of the module and the total heat rate out of the system can expressed as

$$\dot{{\mathrm{Q}}_{\mathrm{C}}}={\dot{\mathrm{Q}}}_{\mathrm{o}\mathrm{u}\mathrm{t}}+{\mathrm{P}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$

(10)

where ${\mathrm{P}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the electrical power of the thermoelectric generator. Equations (8) and (10) can be substituted into Equation (1):

$$\dot{{(\mathrm{Q}}_{\mathrm{H}}}-\dot{{\mathrm{Q}}_{\mathrm{C}})}+\dot{{\mathrm{Q}}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}}+{\mathrm{P}}_{\mathrm{o}\mathrm{u}\mathrm{t}}={∆\dot{\mathrm{Q}}}_{\mathrm{S}}$$

(11)

Equation (11) calculates the total heat balance of the micro-thermoelectric generator system. Finally, the conversion efficiency of the micro-thermoelectric generator system $\xi$ can be expressed as

$$\xi ={\mathrm{P}}_{\mathrm{o}\mathrm{u}\mathrm{t}}/{\dot{\mathrm{Q}}}_{\mathrm{i}\mathrm{n}}$$

(12)

2.3. Energy Modeling—Thermal and Flow Simulations

The model processes the steady-state and transient solutions starting from the initial boundary conditions of the temperatures. It presents the temperature distributions inside the micro-thermoelectric generator system at any point in time. At the modeling stages, there are concerns surrounding the temperatures and the heat transfer in and out of the micro-thermoelectric generator system. This model allows the hot and cold sides of the thermoelectric module to change dynamically through conduction, convection, and radiation modes, depending on the temperature of the heat sink in contact with the module. The discretization scheme used in the computational model with all stages is shown in Scheme 1.

2.3.1. Material and Volumetric Properties

Professional software is used to simulate the thermoelectric micro-generator and to analyze the thermal performance. Data from this modeling are used to estimate the temperature distribution and heat flux based on the material and volume properties of the system components. These properties are listed in

Table 1. Material and volumetric properties of the integrated system components.

Component—Material	Amount	Volumetric Properties	Treated as
Heat Sink—Aluminum	2	Mass = 41.16 milligrams Volume = 15.25 mm3 Surface Area = 85.80 mm2	Solid Body
TE Plate—Ceramic	2	Mass = 10.14 milligrams Volume = 4.41 mm3 Surface Area = 39.48 mm2	Solid Body
TE Strip—Copper	17	Mass 1.16 milligrams Volume = 0.13 mm3 Surface Area = 3.10 mm2	Solid Body
TE Leg—Bi2Te3	16	Mass = 8.59 milligrams Volume = 1.12 mm3 Surface Area = 7.13 mm2	Solid Body

.

2.3.2. Mesh Generation

Professional SOLIDWORKS 2024 (Design Study: Thermal and Flow Simulation) is the software used to create 3D meshes through the functional finite element plus solver. The main characteristics of mesh processing are the accuracy and consistency of the results. Tetrahedral elements use four-node regular-stress elements. The elements provided for each node are embodied in a three-dimensional space with three degrees of independence in the X, Y, and Z directions, taking into account mesh values such as mesh type, tolerance, aspect ratio, and mesh quality. For the reliability of the multivariate discretization method, the density of the lattice plays a crucial role in achieving an accurate solution. It is recognized that any increase in density around the 3D mesh will reduce the error.

Table 2. Mesh information of the integrated micro-thermoelectric generator system.

Mesh Density	Coarse	Medium	Fine	Micro
Mesh Type	Solid	Solid	Solid	Solid
Mesh Used	Standard	Standard	Standard	Standard
Jacobian Points	16	16	16	16
Element Length	0.817 mm	0.613 mm	0.357 mm	0.104 mm
Tolerance	0.0486 mm	0.0306 mm	0.0178 mm	0.0085 mm
Mesh Quality	High	High	High	High
Total Elements	2970	4026	13,476	55,691
Total Nodes	5769	7937	23,563	88,194
Maximum Aspect Ratio	26.368	16.186	5.315	1.006
Quality of Elements	46.7%	59%	90.9%	99.7%

presents the lattice information of the thermoelectric micro-generator system. In this analysis, the boundaries of the lattice density (2970–55,691) are tetrahedral cells. We performed a mesh refinement process to verify the mesh quality and checked that the mesh tolerances for each case were 0.0486 mm, 0.0306 mm, 0.0178 mm, and 0.0085 mm, where the element lengths were 0.817 mm, 0.613 mm, 0.357 mm, and 0.104 mm. Therefore, for heat flow simulations, the choice of micro-grid spacing is shown in Figure 4.

Meshing is an important step in the design analysis. The mesh equations depend on the finite element method for engineering analysis. The finite element method estimates the solution rather than the equation. The essential idea is to substitute the continuous function u(x) with the finite dimensional estimation $ú\left(x\right)=\sum _{k=1}^n{a}_k{\mathrm{\varnothing }}_k\left(x\right)$, where ${\mathrm{\varnothing }}_k$ is the basis functions with area support. These basis functions are typically low-order polynomials, so that the action of the differential operator ζ${\mathrm{\varnothing }}_k$ is uncomplicated to compute. Because the term $ú\left(x\right)$is defined universally on the domain, the analysis of convergence can be created in a continuous manner instead of point-wise as in the finite difference method. The finite element method achieves a discrete estimation by requiring that the differential equations be satisfied for some set of test functions ${\mathsf{\psi }}_k\left(x\right)$, ${\int }_{\Omega }^n\left(\zeta ú\right){\mathsf{\psi }}_i={\int }_{\Omega }^n{f}^{\prime }{\mathsf{\psi }}_i$; an implementation of the finite element method includes the computation of coefficients ${\mathrm{A}}_{ik}={\int }_{\Omega }^n\left(\zeta ú\right){\mathsf{\psi }}_i$, where ${\mathrm{A}}_{ik}$ means the set of elements for which both the basis function ${\mathrm{\varnothing }}_k$ and the test function ${\mathsf{\psi }}_i$ are nonzero. These functions depend on the following factors: geometry, contact conditions, and tolerance.

2.3.3. Thermal Conditions

Computer modeling software analyzes thermal performance and heat transfer patterns in a micro-thermoelectric generator system over time. Under steady-state and transient conditions, the thermal balance of solid and liquid is divided into two categories:

Heat transfer mode A—Solid thermal simulation 2024: This software is used for the detailed simulation of conduction heat transfer in solid components.

Heat transfer mode B—Fluid flow simulation 2024: This software is used to simulate in detail the natural convection and thermal radiation of solid components exposed to laminar fluid flow. According to theses analyses, the details of the thermal conditions of the system can be summarized in

Table 3. The thermal conditions of the integrated micro-thermoelectric generator system.

Component	Conditions	Condition Image	Conditions Details
Hot Surface Heat Sink	1—Initial Temperature 2—Solid Conduction 3—Natural Convection 4—Heat Radiation		Entities = 22 T initial = 20 °C h air = 10 W/(m2.K) Emissivity: 0.95 K Al = 237 W/(m.K) State = Transient to Steady State
Thermoelectric Module	1—Initial Temperature 2—Solid Conduction 3—Natural Convection 4—Heat Radiation		Entities = 178 T initial = 20 °C U overall = 1.72 W/K Emissivity: 0.95 K overall = 0.01 W/(m.K) State = Transient to Steady State
Cold Surface Heat Sink	1—Initial Temperature 2—Solid Conduction 3—Natural Convection 4—Heat Radiation		Entities = 22 T initial = 20 °C h air = 10 W/(m2.K) Emissivity: 0.95 K Al = 237 W/(m.K) State = Transient to Steady State

. With the details in Table 2, the spatial discretization scheme of the micro-thermoelectric generator is shown in Figure 5.

According to the above section, this study has several strengths:

• The model was improved to deal with the components of the micro-thermoelectric module (ceramic, semiconductor and copper strip) separately.
• The model was improved to calculate the conversion efficiency of the system at maximum temperature difference of the thermoelectric module.
• The model was improved to analyze the transfer of useful heat to electrical power and the thermal losses at maximum temperature difference of the module.
• The model can optimize the power output of the integrated micro-thermoelectric generator system.
• The model was improved to estimate the total energy consumed to produce the integrated micro-thermoelectric generator system.
• The model was improved to be employed for medical and electronic micro-applications.

The limitations of the integrated micro-thermoelectric generator system are that it is relatively hard to manufacture depending on the application and it requires a constant heat source.

3. Results and Discussions

3.1. Temperature Difference

Figure 6 shows the temperature difference between the hot side and the cold side of the thermoelectric micro-module after operating the system for 90 min at different (Δ$\dot{\mathrm{Q}}$s). The quasi-steady-state behavior of micro-thermoelectric generator systems has been well studied. With invariable (Δ$\dot{\mathrm{Q}}$s), the system can be observed to reach a steady state after about 70 min. In thermodynamic science, a quasi-steady-state process is a rather slow reversible process. A quasi-steady-state process can be approximated by taking the total heat transfer rate of the hot-side system and the total heat transfer rate outside the system by making them extremely slow. Each state is defined as a qualitative description and an intricate part of the thermal analysis and understanding of system characteristics and overall functionality. The Seebeck effect is thermodynamically reversible, with tiny temperature differences in accordance with the second law of thermodynamics. The increase in the total heat transfer rate to the system (from 0.03 W to 0.11 W) caused an increase in the temperature difference because of the increasing heat source temperature. At the maximum storage heat transfer rate, Figure 7 shows the steady-state behavior of the micro-thermoelectric generator system (a) and the temperature profiles of the thermoelectric legs at the two edges (b and c). It can be seen that heat flows from the 16 hot-side junctions to the 16 cold-side junctions as a result of the heat transfer rate in the thermoelectric legs. The heat transferred in both the thermoelectric legs varies along the y-axis, being red on the hot side of the heat sink and blue on the cold side of the heat sink. The reason for this appeared to be due to the relationship between the Seebeck coefficient and the temperature, as shown in Figure 7b,c, where the conversion started from 18 K to 58 K for n-type and p-type Bi₂Te₃ semiconductors. The maximum electrical power of the system (Equation (11)) can be achieved at the maximum saved heat transfer rate (0.11 W) and maximum temperature difference (58 K).

3.2. Total Heat Transfer Rate and Power Optimization

Figure 8 shows the total heat transfer rate and electrical power of the system to represent the temperature difference of the TE module. The heat flow in the module is proportional to the temperature difference. The thermal losses were considered for the model to predict realistic results. The thermal losses increased with an increase in the temperature difference between the hot and cold sides of the module. The thermal losses were significant due to the processes of convection and radiation energy. At a temperature difference of 58 K, the maximum electrical power can be reached, and the total heat losses were about 0.8 W and 0.1 W, respectively. The total heat transfer rate R² and the advanced model electrical power become a reasonable match for the theoretical equation. The total heat transfer rate and the model’s heat flow simulations are confirmed. As shown in Figure 8, the R² value (above 99%) indicates that the model accepts an accurate prediction of the thermal performance of the micro-thermal power generation system. The overall heat transfer rate at the hot and cold ends of the thermoelectric leg can be expressed as the sum of conduction through the solid and Joule heating, as shown in Figure 9. From Figure 9a, it can be seen that the half-joule heat (mainly converted from the total heat saved in the thermoelectric module) moved steadily to each junction. Therefore, when the total heat losses were considered, the difference between the heat flow into the hot-side module and that out of the cold-side module was almost equal to the electrical power produced by the thermoelectric module. A major highlight of this model was the combination of the reversible effects of inhomogeneous materials and the Joule effect in the solid-state conduction source term.

The overall heat transfer rate in the simulation results represented the solid-state conduction with the flow of heat. It was directly related to the temperature range and can be considered separately from the reversible Joule heating. The model displayed that the temperature gradient in the thermoelectric leg caused similar but opposed Joule heating intensity along the y-axis. At the maximum power point, the maximum Joule heating intensity is reached. This functionality was available in the current computational modeling software where reversible and irreversible processes were involved in calculating the overall heat transfer rate.

One main piece of equipment for satisfying the key requests of the principle of measurement was a potentiostat (Auto lab PGSTAT302N) provided by Metrohm Auto lab (Utrecht, The Netherlands), which can be precisely connected to the outputs of a thermoelectric generator and finds full I–V curves at a manageable scan rate in a range of time. This is the key to allowing the I–V measurement to be presented at a constant temperature difference.

Figure 10 shows the power output to represent the temperature difference of the integrated micro-thermoelectric generator system. The optimized power output in the integrated system progressively increased with the temperature difference. The maximum power output was achieved for a maximum temperature difference of about 80 K, whereas the minimum temperature difference was less than 20 K. As a result, there was an agreement in the power output values between the optimized values and the heat balance equations (Equations (7) and (9)).

3.3. Influence of Compressibility inside the Integrated System

The computational modeling software has a qualitative tool which provides insight into air flow inside the system. The quantitative result of the air velocity is illustrated with two sides in Figure 11. The type of flow was subsonic with a Mach number less than 1. According to the result, the influence of the compressibility was insignificant inside the micro-thermoelectric generator system at constant atmospheric pressure. The model offers a promising route to utilize nano fluids for medical applications. It can also provide a stable energy system according to the compressibility cycles.

3.4. Conversion Efficiency of the Integrated System

One of the purposes of this work is to analyze the solid-state thermoelectric power generation (conversion efficiency) in the system, which can be designed for a specific temperature difference range of the thermoelectric modules. Note that the efficiency of the system can be determined using the overall heat transfer rate and power output of the system (Equation (12)). At steady state, the efficiency of the micro-thermal power generation system increases by 50% at five temperature differences, as shown in Figure 12. It is found that the maximum conversion efficiency of the system can reach 0.83% when the temperature difference is 58 K. However, the efficiency of the system is highly dependent on the temperature difference between the hot and cold sides of the TE strands. In order to achieve high system efficiency, the useful heat must be compared with the heat input to the system. When using thermoelectric modules in power generation mode in specialized applications, the conversion efficiency is limited by the reduction in heat losses.

Figure 13 shows the available and saved heat and heat loss (percentage) of the micro-thermoelectric generator system. With a maximum heat loss of 6%, 11% of the heat is temporarily retained in the thermoelectric module. Heat loss increases with temperature difference and natural convection (proportional to the fluid temperature) and thermal radiation (proportional to the fourth power of the fluid temperature). However, the tiny size of the TE modules improves the efficiency of the system due to the fine pitch of the thermoelectric legs leading to a reduction in natural convection and thermal radiation.

3.5. Environmental Impacts

In this research, we used a sustainability assessment method to assess the environmental impacts of the integrated system. As shown in Figure 14, when modeling environmental impact over a 10-year period, a life cycle assessment captures what happens in a system, from use to the final dismantling step. This includes the effects of transfers that occur between steps. Estimates of solids used, manufacturing, and other aspects may result in varying environmental impacts. Figure 15 shows the environmental impact of the micro-thermoelectric generator system. However, the system is environmentally friendly when using any type of fluid or nano fluid through the heat transfer processes in the system components.

Total energy consumption is measured in megajoules (MJ) of non-renewable energy associated with a component’s life cycle. These impacts include not only the energy or fuel consumed during the system’s life but also the energy required to reach and process these energies and the energy loss of component materials released during combustion. The efficiency of energy conversion (e.g., electricity, heat, steam, etc.) is considered. The total energy consumed to fabricate the micro-thermoelectric generator system is 0.015 MJ, which is a very small value compared to other electronic devices. Therefore, the micro-thermoelectric power generation system has little impact on the environment.

3.6. Validation of the Model with Experimental Works

Figure 16 and Figure 17 show the validation of the model with the experimental works [11,24,25]. An agreement can be noticed between the model prediction and the experimental data at the specific range of the temperature difference of the module. Functionally, validating the model’s assumptions and theory contributes an understanding of the model within the context of a real system.

3.7. Comparison between Different Modeling Strategies

More importantly, one of the goals of the present work is to create a general model that combines the overall heat transfer rate with the electrical power in a micro-thermoelectric generator system for efficient technology. This will provide advanced simulations that will allow the model to separately treat the different materials (ceramics, semiconductors, copper strips) of the micro-thermoelectric module and consider physically correct paths.

Table 4. Comparison between different modeling strategies.

Modeling Strategy	[11]	[19]	[23]	[24]	[25]	[30]	[31]	[32]	Present Work
Design a thermoelectric module with all Parts	Yes	No	No	Yes	Yes	Yes	No	No	Yes
Model Including Three-Dimensional Solution	Yes	No	Yes	Yes	Yes	Yes	Yes	No	Yes
Model Including Heat Transfer Rate in and out	Yes	Yes	No	Yes	Yes	Yes	Yes	Yes	Yes
Model based on the Finite Element Method	Yes	Yes	No	Yes	Yes	Yes	Yes	Yes	Yes
Model Treated from Transient to Steady State	Yes	No	Yes	No	No	No	Yes	No	Yes
Variable Temperatures as Boundary Conditions	No	Yes	Yes	No	Yes	No	Yes	Yes	Yes
Solid Conduction and Joule Heating Effects	Yes	No	Yes	Yes	Yes	Yes	No	No	Yes
Natural Convection and Radiation Effects	Yes	No	No	No	No	No	No	No	Yes
Energy Balance between Heat and Electricity	Yes	Yes	Yes	Yes	No	No	Yes	Yes	Yes
Analysis of Useful Heat Flow and Thermal Losses	No	No	No	No	No	Yes	No	No	Yes
Study the Influence of Compressibility Flow	No	No	No	No	No	No	No	No	Yes
Apply Sustainability Assessment Methodology	No	No	No	No	No	No	No	No	Yes

summarizes the comparison between the different modeling strategies discussed in the Introduction section and the rest of the current work. Modeling strategies for different thermoelectric generators are well considered. Therefore, the comparison confirms that the current work is entirely correct for the thermal performance evaluation of micro-thermoelectric generator systems.

4. Conclusions and Future Works

4.1. Conclusions

An integrated micro-thermoelectric generator system is modeled theoretically and simulated numerically for advanced energy harvesting. The system is used to transform thermal energy into electrical power in micro-scale applications according to the Seebeck and Joule effects associated with the heat transfer modes. The main conclusions are as follows:

• The model is developed to deal with the components of the micro-thermoelectric module (ceramic, semiconductor, and copper strip) separately.
• We improved the conversion efficiency of the system by about 50% when we increased the temperature difference of the thermoelectric module from 18 K to 58 K.
• We achieved a transfer of useful heat to electrical power of 83%, with 11% saved heat and thermal losses of 6% W at the maximum temperature difference of the module.
• We optimized the power output in the integrated micro-thermoelectric generator system at diverse temperature differences.
• We compared the present work with different modeling strategies to confirm that the computational modeling is accurate and efficient.
• The total energy consumed to produce the integrated micro-thermoelectric generator system was 0.015 MJ.
• The model can be employed for medical and electronic micro-applications.
• The model offers a promising route to utilize nano fluids for medical applications.

4.2. Future Works

4.2.1. Medical Application

The micro-thermoelectric generator system has several medical advantages, such as environmental safety, quiet function, light weight, long life, and an ideal nature for compact applications. Many people have to work outside with a medical face mask even when it is hot or cold or there is a sand storm. The system can be employed with a micro solar cell which is mounted on the front of a medical face mask, as shown in Figure 18, since ambient temperature is usually higher than skin temperature in summer and lower at other times of the year. The nose and mouth have to be well ventilated because they are most sensitive to thermal comfort. There are two ways to obtain thermal comfort:

A.

In the winter season, pumping heat from inside the face mask to the outside of the face mask (cooling mode);

B.

In the summer season, absorbing heat from inside the face mask to the outside of the face mask (heating mode).

According to these cases, the temperature difference of the system will range from 16 K to 32 K depending on ambient and exhaled breath temperatures.

4.2.2. Electronic Application

The advantages of the system make it very suitable for a wide range of micro applications in the thermal management of advanced electronic devices. The system can be employed to accurately control and maintain the temperature of devices within desired conditions. The system can be powered a wristwatch by converting body heat into electrical power. The temperature difference between the human body (hot side) and ambient air (cold side) is approximately 25 °C; the human body produces about 6 m W/cm² of thermal energy. At minimum hot-side temperature, the system can produce 20 μW of power and has an open-circuit potential of 200 mV with an efficiency of about 0.4%, as shown in Figure 19.

The future developments of the integrated micro-thermoelectric generator system are as follows:

• Reducing the thermal losses because of the convection heat transfer in the thermoelectric legs through computational software;
• Using different thermoelectric semiconductor materials, such as lead telluride and silicon germanium;
• Comparing the performance of the integrated micro-thermoelectric generator system with micro chemical batteries and mobile electronics.