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Research Papers: Energy Systems Analysis

# Energy Conversion by Nanomaterial-Based Trapezoidal-Shaped Leg of Thermoelectric Generator Considering Convection Heat Transfer EffectPUBLIC ACCESS

[+] Author and Article Information
Abu Raihan Mohammad Siddique, Franziska Kratz, Bill Van Heyst

School of Engineering,
University of Guelph,

Shohel Mahmud

School of Engineering,
University of Guelph,
e-mail: smahmud@uoguelph.ca

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received November 14, 2018; final manuscript received December 24, 2018; published online February 14, 2019. Assoc. Editor: Esmail M. A. Mokheimer.

J. Energy Resour. Technol 141(8), 082001 (Feb 14, 2019) (11 pages) Paper No: JERT-18-1837; doi: 10.1115/1.4042644 History: Received November 14, 2018; Revised December 24, 2018

## Abstract

Thermoelectric generators (TEGs) can harvest energy without any negative environmental impact using low potential sources, such as waste heat, and subsequently convert that energy into electricity. Different shaped leg geometries and nanostructured thermoelectric materials have been investigated over the last decades in order to improve the thermal efficiency of the TEGs. In this paper, a numerical study on the performance analysis of a nanomaterial-based (i.e., p-type leg composed of BiSbTe nanostructured bulk alloy and n-type leg composed of Bi2Te3 with 0.1 vol % SiC nanoparticles) trapezoidal-shaped leg geometry has been investigated considering the Seebeck effect, Peltier effect, Thomson effect, Fourier heat conduction, and surface to surrounding irreversible heat transfer loss. Different surface convection heat transfer losses (h) are considered to characterize the current output, power output, and thermal efficiency at various hot surface (TH) and cold surface (TC) temperatures. Good agreement has been achieved between the numerical and analytical results. Moreover, current numerical results are compared with previous related works. The designed nanomaterial-based TEG shows better performance in terms of output current and thermal efficiency with the thermal efficiency increasing from 7.3% to 8.7% using nanomaterial instead of traditional thermoelectric materials at h = 0 W/m2K while TH is 500 K and TC is 300 K.

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## Introduction

Over the last two decades, energy conversion from waste heat to electricity using thermoelectric generators (TEGs) has gained a considerable amount of attention as an alternative to other electricity generation techniques using fossil fuels [1,2]. TEGs are able to convert thermal energy into electric current and vice versa due to the Seebeck and Peltier effects [3]. The technology has several advantages compared to others in the energy harvesting field including the robustness and durability of the system due to no moving components, their silent operation, and their low environmental impact [4,5]. These advantages are contrasted, however, by the expensive nature of the materials used for TEGs and the very low energy conversion efficiency [6,7], typically with a maximum thermal efficiency less than 10% [7]. TEGs are currently most often deployed in systems that generate electricity from waste heat in order to offset a small portion of the heat losses.

The performance of TEGs depends basically on three parameters: the temperature difference between the hot and cold surface of the thermoelectric (TE) leg, the figure of merit (Z), and design parameters such as the shape of the leg geometry and the external load [5]. While numerous researchers have been attempting to improve the figure of merit of the TE materials [8,9], another research avenue that has been explored is the effect of the shape of the TE legs on the thermal performance and power output from a numerical and experimental perspective. This has resulted in a variety of different leg geometries (i.e., rectangular, cylindrical, trapezoidal, and octagonal-shaped) being studied in order to improve the thermal efficiency of the TEG [7,10,11]. Among these geometrical configurations, the trapezoidal-shaped TE leg is most often considered in the literature.

For legs with a varying cross-sectional area, Sahin and Yilbas [7] reported that thermal efficiency was higher in the case where the leg geometry differed from a rectangular form. Subsequently, Al-Marbati et al. [12] analyzed the thermal stress on trapezoidal-shaped TE legs and reported that the thermal efficiency and temperature deviation changed with changing the leg geometry. Moreover, Ali et al. [13] varied the cross-sectional area of TE leg exponentially and examined the thermoelectric generation of the device and reported that it increased the thermal efficiency.

Erturun et al. [10] also studied the effect of different leg geometries on the performance and the thermomechanical effects. In their study, rectangular, trapezoidal, cylindrical, and octagonal prisms were investigated. The investigation included temperature distribution, power output, conversion efficiency, and thermal stress. Recently, Lamba and Kaushik [14] added Thomson effect to analyze the trapezoidal-shaped TE legs and showed that the thermal efficiency improved over that using flat rectangular shaped TE legs.

One method to optimize the thermal efficiency of the TEG is to minimize the thermal losses caused by convection at the leg's surface [1517]. This can be achieved by taking advantage of the high power factor and low thermal conductivity typical of nanomaterials. These nanomaterials can then be used for the TE materials for enhanced p-type and n-type legs that increase the figure of merit [2,8,1821]. Combining variable cross-sectional leg shapes synthesized using nanostructured TE materials could thus potentially increase the thermal efficiencies of TEGs.

One method to optimize the thermal efficiency of the TEG is to minimize the thermal losses caused by convection at the leg's surface [1517]. This can be achieved by taking advantage of the high power factor and low thermal conductivity typical of nanomaterials. These nanomaterials can then be used for the TE materials for enhanced p-type and n-type legs that increase the figure of merit [2,8,1821]. Combining variable cross-sectional leg shapes synthesized using nanostructured TE materials could thus potentially increase the thermal efficiencies of TEGs.

No literature is currently available on nanomaterial-based, variable cross-sectional area TEGs that consider convectional heat transfer losses. As such, this paper presents a numerical study on the performance analysis of nanomaterial-based, trapezoidal-shaped leg TEG considering convection heat transfer effects. The effects of different convection heat transfer coefficients and different temperature values for the hot and cold surfaces have been examined numerically for their effect on the current output, power output, and thermal efficiency. These numerical results have been compared with analytical results and with the limited literature using similar configurations.

## Geometry and Material Specifications

Figure 1(a) presents a schematic representation of a typical single unit of a TEG. The TEG mainly consists of the p-type and n-type semiconductor legs (usually with a flat rectangular geometry) made of TE materials. The legs are placed vertically between two rigid substrates where these legs are connected thermally in parallel and electrically in series. Figure 1(b) gives the corresponding schematic for a TEG with trapezoidal-shaped legs.

For the current numerical modeling work, a trapezoidal-shaped leg geometry was considered where the cross-sectional area at the top (A1) of the legs was held constant but the cross-sectional area down the leg was allowed to vary with height as seen in Fig. 2(a). The dimensions of each leg were 3 mm (w1 and w3) × 3 mm (b) × 3 mm (h2), and the cross-sectional area was allowed to change using a geometric imperfection parameter (a). This produced a bottom cross-sectional area A2 that was smaller than A1 and gives the overall leg a trapezoidal shape. The trapezoidal shape also has the advantage of requiring less volume of TE materials to build compared to the flat rectangular leg. The gap between the two legs was held constant at 1.5 mm (w2), and the thickness of the top and bottom copper substrate strip was 0.12 mm (h1 and h3). Figure 2(b) illustrates a three-dimensional view of a trapezoidal-shaped leg, including the dimensional changes as a function of vertical position, y.

In the numerical modeling scenarios, two pairs of TE materials were considered. The first pair was composed of traditional materials and includes a p-type leg of 25% Bi2Te3 and 75% Sb2Te3 (1.75% Se) and an n-type leg of 75% Bi2Te3 and 25% Bi2Se3. The second pair was composed of nanomaterials that include a p-type leg of BiSbTe nanostructured bulk alloy and an n-type leg of Bi2Te3 with 0.1 vol % SiC nanoparticles [17,19,20,22]. This nanomaterial was selected for its low-cost and high-performance characteristics (dimensionless figure of merit for the p-type leg, ZTp, is around 1.2 and, for the n–type leg, ZTn, is around 1.04) [19,20]. The TE properties are, however, temperature-dependent and polynomial functions have been used to describe the temperature dependencies on the various properties as indicated in Table 1. The temperature ranges for the parameters vary as indicated with the traditional material typically having a much higher upper extreme temperature. The value of the parameters was calculated using the average temperature of the hot and the cold surface and was considered to be representative of the entire leg.

## Mathematical Modeling

The end goal of the developed mathematical model was to determine the temperature distribution and the electrical potential distribution, both of which are dependent on the leg geometry and TE materials as well as the heat convection transfer from the surface to the surroundings. The mathematical model includes Fourier heat conduction, convection at the surface, internal Joule heat, as well as the Seebeck, Peltier, and Thomson effects. The mathematical molding is divided into two-dimensional (2D) heat transfer modeling with boundary conditions and one-dimensional (1D) analytical approaches.

###### Two-Dimensional Heat Transfer Analysis and Boundary Conditions.

In setting up the 2D model for the numerical analysis, the following assumptions were employed:

• A steady-state heat transfer condition is considered.

• Both p-type and n-type leg materials are temperature-dependent homogeneous and isotropic.

• The contact resistance between the copper strips and TE legs are negligible.

• Convective heat transfer is considered to occur from the leg surface to the surroundings.

• The top and bottom surface of the TE leg is exposed to a constant hot temperature (TH) and a constant cold temperature (TC).

The governing equation of energy transport of a TEG can be written as [15] Display Formula

(1)$ρ c∂T∂t+∇⋅q=q˙gen$

where ρ, c, q, T, and $q˙gen$ are the density of the TE material, specific heat of the TE material, heat flux vector, temperature, and heat generation per unit volume, respectively. Under steady-state condition, the first term on the left-hand side of Eq. (1) disappears (i.e., $ρ c (∂T/∂t)=0$). According to the conservation law, the continuity of electric charge flow through the TE legs must satisfy the following equation: Display Formula

(2)$∇⋅(J+∂D∂t)=0$

where J is the electric current density vector, and D is the electrical displacement field. In the steady-state condition, the time-dependent partial derivative term in Eq. (2) disappears (i.e., $(∂D/∂t)=0$). The constitutive relationships for the heat flux vector (q) and current density (J) in Eqs. (1) and (2), are provided below in the presence of thermoelectric effect [23] Display Formula

(3)$J=σ (E−α ∇T)$
Display Formula
(4)$q=α T J−k ∇T$

where k is the thermal conductivity, α is Seebeck coefficient, and σ is the electrical conductivity of TE material. The variable E is the electric field intensity vector which can be expressed as the negative gradient of the electric potential (i.e., $E=−∇ϕ$, where $ϕ$ is the electric potential). The heat generation term, $q˙gen$, in Eq. (1) can be further expressed as Display Formula

(5)$q˙gen=J⋅E=σ ∇ϕ⋅∇ϕ+σ α ∇T⋅∇ϕ$

Finally, the conservation equations for energy and electrical charge transports can be expressed, in terms of temperature and electric potential, as Display Formula

(6)$∇⋅ [−k ∇T−σ α T ∇ϕ]=σ ∇ϕ⋅∇ϕ+σ α ∇T⋅∇ϕ$
Display Formula
(7)$∇⋅ [−σ ∇ϕ−σ α ∇T]=0$

Equations (6) and (7) were solved using the following thermal and electrical boundary conditions:

Thermal boundary conditionsDisplay Formula

(8a)$Top wall (isothermal boundary condition): T=TH$
Display Formula
(8b)$Bottom wall (isothermal boundary condition): T=TC$
Display Formula
(8c)$Side walls (convection boundary condition):−k∂T∂n̂ |sw−h T |sw=−h T∞$

Electrical boundary conditionsDisplay Formula

(8d)$Top wall (electrical insulation): ∂ϕ∂y=0$
Display Formula
(8e)$Bottom wall (electrical insulation): ∂ϕ∂y=0$
Display Formula
(8f)$Side walls (electrical insulation): ∂ϕ∂n̂ |sw=0$

where $T∞$ is the surrounding temperature, h is the convection heat transfer coefficient, and $n̂$ refers to the direction normal to side walls. The subscript “sw” refers to the value at the sidewall. The surface of the copper between the two legs, which is not exposed to a constant temperature, is assumed to be electrically and thermally insulated. Figure 3 presents a schematic illustration of boundary conditions for a single TE leg.

###### One-Dimensional Analytical Approaches.

The heat input to the 1D TE leg, as presented in Fig. 1(a), is defined as [1] Display Formula

(9)$Q˙in=K(TH−TC)+αTCIL−12IL2Ri$

where $Q˙in$ is the heat input to the system. K, TH, TC, α, IL, and Ri are the thermal conductance of the material, hot surface temperature, cold surface temperature, load current, and internal electrical resistance, respectively. Under the scenario considered, the load current of the TEG depends on the Seebeck coefficient of p-type (αp) and n-type (αn) materials, as well as the hot and cold surface temperature, internal resistance (Ri), and load resistance (see Eq. (10)). The efficiency (η) of the TEG is given by Eq. (11) [24] Display Formula

(10)$IL=(αp−αn)(TH−TC)Ri+RL$
Display Formula
(11)$η=PQ˙in=IL2RLK(TH−TC)+αTCIL−12IL2RL$

In Eq. (11), P is the power output of TEG. As the cross-sectional area of the trapezoidal is not uniform and changes with the height of the leg, the internal thermal and electrical resistivity, therefore, needs to change with the changing cross-sectional area of the leg. Hence, the overall cross-sectional area of the designed trapezoidal leg can be written as [12] Display Formula

(12)$A=(A2−A1h2)y+A1$
Display Formula
(13)$A=A∗y+A1$

where A1 is the hot surface cross-sectional area, A2 is the cold surface cross-sectional area, A* = (A2−A1)/h2 for the condition where A2 < A1, h2 is the height of the TE leg, and y is the distance from the hot surface toward the cold surface. According to the first law of thermodynamics, the energy balance equation of the system is given by Display Formula

(14)$E˙in−E˙out=dEdt|system=0$
Display Formula
(15)$Q˙y−(Qy˙+∂Q˙∂ydy)=0$
Display Formula
(16)$∂Q˙∂ydy=0$

where $Ein˙$ and $Eout˙$ are the energy in and out of the system, and $Qy˙$ is the heat transfer rate along the y-direction. The cross-sectional area (A) and temperature (T) of the TE leg change with the distance from the top surface (i.e., A and T are dependent variables). According to Fourier's law of thermal conduction, Eq. (16) can be written as Display Formula

(17)$K∂∂y(AdTdy)dy=0$

By solving Eq. (17) based on position, the equation for T becomes Display Formula

(18)$T=C1ln(A∗y+A1)+C2$

where C1 and C2 are constants of integration. According to Fig. 2(b), the boundary conditions of Eqs. (12) and (18) are given by Display Formula

(19)$A(y)y=0=A1$
Display Formula
(20)$A(y)y=h2=A2$
Display Formula
(21)$T(y)y=0=TH$
Display Formula
(22)$T(y)y=h2=TC$

Applying the boundary conditions with Eq. (18), results in C1 and C2 having the following forms: Display Formula

(23)$C1=TH−TCln(A1)−ln(A2)$
Display Formula
(24)$C2=ln(A1)TC−ln(A2)THln(A1)−ln(A2)$

By substituting the values of C1 and C2 in Eq. (18) and applying boundary conditions in Fourier's law of thermal conduction and for the derivatives of T, $Qin˙$ becomes Display Formula

(25)$Q˙in=−KA1(TH−TC)(A2−A1)[ln(A1)−ln(A2)]h2A1$
Display Formula
(26)$Q˙in=(TH−TC)h2K(A1−A2)[ln(A1)−ln(A2)]$
Display Formula
(27)$Q˙in=(TH−TC)RiT$

where the internal thermal resistance of the trapezoidal leg is, $RiT=(h2[(ln(A1)−ln(A2)]/K(A1−A2))$, and where K is the thermal conductivity of the leg. Similarly, the internal electrical resistance can be written as $RiE=(h2[(ln(A1)−ln(A2)]/σ(A1−A2))$, where σ is the electrical conductivity.

## Numerical Study

The numerical solution of the model system was carried out with flexpde software, a finite element model builder for partial differential equations. The system was solved for the two-dimensional case and under steady-state conditions.

Copper was used as the top and bottom strip for the TEG with a Seebeck coefficient of 1.9 × 10−6 (V K−1), a thermal conductivity of 400 W m−1 K−1, and an electrical resistivity of 1.721.9 × 10−8 (Ω m) [25]. Equation (28) was used to calculate the power output of the TEG for the numerical and analytical analysis. Moreover, Eq. (11) was used to determine the efficiency of the TEG. The power output (P) from the TEG, was estimated using Display Formula

(28)$P=IL2RL$

where IL is the load current (defined in Eq. (10) which is used for analytical study), and RL is the external load resistance which is equal to the internal resistance at the optimum operation of the TEG. The internal resistance (Ri) was calculated using the properties of the TE materials and also the dimensions of the TE leg. The area of the TE leg was varied linearly with the height of the leg, and the properties of the TE materials were varied with the average temperature.

## Results and Discussion

Based on the solutions to the numerical model described above, the performance of various trapezoidal-shaped TE leg was investigated for both traditional materials and nanomaterials. Table 2 shows the effect of geometric imperfection parameter (a, see Fig. 1(b)) on the internal electrical resistance, heat input, heat output, load current, power, and efficiency of the TEG based on the TE material's properties with a TH of 500 K and a TC of 300 K. The convection heat transfer coefficient (h) was kept at zero in this case.

It can be clearly seen from Table 2 that the electrical resistance increases with increasing geometric imperfection parameter, a, which is obvious according to the law of electrical resistance. Therefore, the load current and the power output decreases as a increases. However, the efficiency of the TEG increases up to a certain range of “a” because the heat input decreases less compared to the power output which results in a higher efficiency. The efficiency is at a maximum when a =1.25 (mm) for both materials. It is also noticeable that the efficiency of the TEG with nanomaterial is higher by approximately 1.4% than the traditional TE materials.

Increasing the geometric imperfection parameter means decreasing the bottom cross-sectional area which becomes a thermal stress handling problem. Lamba and Kaushik [14] and Al-Merbati et al. [12] have demonstrated that a TEG can easily handle the thermal stress generated under a geometry of A1/A2 = 2. Therefore, considering the thermal stress handling capability from this previous research [12], the geometric imperfection parameter was fixed at 0.75 mm, resulting in A1/A2 = 3/(2 × 0.75) = 2, throughout the rest of the numerical simulations and for comparison purposes to the limited literature.

Table 3 presents the effect of convection heat transfer coefficient (h) on the efficiency when a is 0.75 mm with 0.153 Ω for traditional TE materials and 0.01 Ω for nanomaterial-based TEG. The thermal efficiency decreases with increasing h. The difference between $Q˙in$ and $Q˙out$ indicates that the more heat coming to the hot surface of the leg, the less heat is coming out of the cold surface due to increasing surface heat surface losses via h. The thermal efficiency varies between 7.9% and 7.4% for the leg composed of traditional material and between 9.3% and 8.2% for nanomaterial legs with $0≤h≤60$ W m−2 K−1.

###### Temperature and Electrical Potential Distribution.

The temperature distribution and heat flow profile of single cell of the TEG, as simulated by flexpde, are presented in Fig. 4. Figures 4(a) and 4(b) illustrate the temperature and heat flux flow profiles for traditional TE material-based TEG at h =0 W m−2 K−1 and h =60 W m−2 K−1, respectively, and Figs. 4(c) and 4(d) illustrate the profiles for nanomaterial-based TEG at h =0 W m−2 K−1 and h =60 W m−2 K−1, respectively. The heat flow is plotted by the vertical arrow lines, whereas the temperature distribution is presented by multicolored background with isothermal lines which vary from 500 K to 300 K. The temperature profiles are constant for different values of h irrespective of the composition of the p-type and n-type TE legs. The heat flow lines, however, are changing due to the change of h for both materials. The heat entering into the hot surface goes straight to the bottom cold surface with little loss when h =0 W m−2 K−1 (see Figs. 4(a) and 4(c)). However, when for h = 60 W m−2 K−1, the heat flow lines tend slightly toward the side walls (i.e., leaving the TEG instead of going straight toward the cold surface) because of losing heat through the surface to the surrounding from convection heat flow (see Figs. 4(b) and 4(d)). A higher heat transfer coefficient would mean there will be higher irreversible convection losses through the surface of the TE legs and lower conversion rate of thermal energy into electrical energy. The surface convection heat transfer is more visible for nanomaterial-based TE legs than traditional one.

Figures 5(a) and 5(b) show the current flow and electric potential contours for traditional and nanomaterial based TEGs, respectively. The electric potential is represented by a multicolored background with isopotential blue lines, whereas the current flow is indicated by the black arrow lines. There is no significant difference in the current flow and electric potential due to irreversible convection losses. However, the magnitude of the electric potential contour is different and more segmented for the nanomaterial-based TE legs than for the traditional one.

###### Heat Input.

Figure 6 defines the behavior of heat input ($Q˙in$) to the TE legs as a function of TH at different convection heat transfer coefficients under a constant TC (300 K). The TE legs absorb more heat as TH increases under different h because of a large temperature difference between TH and TC for both materials. However, the hot surface of the nanomaterial-based TEG absorbs around 2.5 times less heat than the TEG composed of traditional materials. The effect of the temperature difference on the heat input is more visible and dominant for nanomaterial-based TEG, while it is quite close for traditional materials-based TEG at various values of h.

###### Current and Power Output.

The output or load current of the trapezoidal-shaped TEG is given in Fig. 7(a) as a function of TH while TC was varied between 270 K and 300 K in steps of 10 K. The nanomaterial-based TEG generates more current than the traditional material TEG due to the different properties of the TE materials (see Table 2). In addition, a comparison was made between the numerical and analytical results (based on Eq. (10)) which validate the numerical results (see Fig. 7(b)) at different TH while TC is held constant at 300 K. The analytical calculation of the output current is based on Eq. (10) which gives a linear relationship between output current and the electrical resistance, Seebeck coefficient, and temperature difference which is reflected in Fig. 7(b).

Similarly, the power output of TEG is plotted as a function of TH for various values of TC in Fig. 8(a). The power output for both materials increases with increasing TH (i.e., temperature plays a vital role in power output). The power output is higher for a traditional material TEG than for a nanomaterial TEG due to the high electrical load resistance (see Table 2). A comparison is also made between numerical and analytical results and plotted in Fig. 8(b) at different TH, while TC remains constant at 300 K with the numerical results closely following the analytical results (calculated using Eq. (28)).

The open circuit voltage and power output, as a function of current for both materials at a temperature difference of 200 K, are illustrated in Fig. 9. The open circuit voltage decreases linearly for both materials with increasing current. The power output, however, gradually increases with increasing current until it reaches a maximum peak, after which it decreases with further increases in current. The peak represents the optimum output power at a particular value of the current.

###### Thermal Efficiency.

The thermal efficiency as a function of temperature gradient at different convection heat transfer coefficients for traditional and nanomaterial-based TEG is presented in Fig. 10. Thermal efficiency increases as the temperature difference increases for both materials. Moreover, the more the irreversible heat losses through the surface of the TE legs as convection heat transfer, the more the thermal efficiency is observed due to the high heat input. The thermal efficiency of nanomaterial-based TEG increases faster than traditional material TEG ranging between 4% and 12% at different values of h. As an example, the thermal efficiency is higher for nanomaterial-based.

###### Comparison With Previous Study.

A list of similar work available in the literature is presented in Table 4. The results listed in Table 4 are for a temperature difference of 200 K where TH is fixed at 500 K and TC is fixed at 300 K with h =0 W m−2 K−1. The current results based on a trapezoidal-shaped TEG show better performance than other works by Rabari et al. [15] because of different dimensions and configuration of the TE legs used in this study. The numerical results show that the thermal efficiency of the nanomaterial-based TEG is higher than the traditional material-based TEG.

## Conclusion

In this paper, a 2D numerical analysis of the performance of trapezoidal-shaped leg geometry of a TEG was investigated. Surface to surrounding convection heat transfer was considered for a more realistic understanding of the performance of TEG at different heat transfer coefficients. In addition, p-type and n-type legs composed of traditional materials and nanomaterials were considered. The nanostructured TEG showed better performance, in terms of load current and thermal efficiency than the TEG composed of traditional material due to the high power factor and low thermal conductivity of the nanomaterials.

Different thermoelectric effects, such as the Seebeck effect, the Peltier effect, Fourier conduction heat transfer, and irreversible convection heat transfer losses were considered for the numerical study. In the numerical analysis, the heat transfer, temperature distribution, and the electric potential of the trapezoidal-shaped TEG were studied at different temperatures and at different convection heat transfer coefficients. Moreover, analytical calculation of load current and power output of the TEG indicated a good match with the numerical results. Furthermore, the designed trapezoidal-shaped leg required less amount of materials to synthesis compared to flat rectangular one which will play an important role to cut down the price of the TEG.

The thermal efficiency of TEG with both materials decreased due to the irreversible convection heat losses through the surface of the TE legs. However, the TEG composed of nanomaterials had a higher thermal efficiency with 8.74%, while for traditional materials it was 7.3% at h =0 W m−2 K−1 and when TH = 500 K and TC = 300 K for trapezoidal-shaped leg geometry.

## Nomenclature

• a =

geometric imperfection parameter (m)

• A =

cross-sectional area of trapezoidal leg (m2)

• A1 =

cross-sectional area of top surface of trapezoidal leg (m2)

• A2 =

cross-sectional area of bottom surface of trapezoidal leg (m2)

• b =

thickness of the leg (m)

• c =

specific heat capacity (J kg−1K−1)

• D =

electrical displacement field vector (C m−2)

• E =

electric field intensity vector (N C−1)

• h =

convection heat transfer coefficient (W m−2 K−1)

• h2 =

height or length of the leg (m)

• h1, h3 =

thickness of the copper strip (m)

• J =

electric current density vector (A m−2)

• k =

thermal conductivity (W m−1 K−1)

• K =

thermal conductance (W K−1)

• P =

power (W)

• q =

heat generation vector per unit volume (W m−3)

• Q =

heat (W)

• $q˙n̂,cond$ =

conduction heat flux normal to the surface (W m−2)

• $q˙conv$ =

convection heat flux (W m−2)

• R =

electrical resistance (Ω)

• t =

time (s)

• T =

temperature (K)

• TE =

thermoelectric

• TEG =

thermoelectric generator

• w =

width of the leg (m)

• x =

distance from the hot surface (m)

• X, Y =

coordinate (m)

• Z =

figure of merit (K−1)

Greek Symbols
• α =

Seebeck coefficient (V K−1)

• η =

thermal efficiency

• ρ =

density (kg m−3)

• σ =

electrical conductivity (S m−1)

• $ϕ$ =

electric potential (V)

• ϱ =

electrical resistivity (Ω m)

Subscripts
• avg =

average

• C =

cold surface

• E =

electrical

• gen =

generation

• H =

hot surface

• i =

internal

• in =

input

• L =

• n =

n-type material

• out =

output

• p =

p-type material

• sw =

sidewall of the TE leg

• T =

thermal

• x =

surrounding

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Sahin, A. Z. , and Yilbas, B. S. , 2013, “ The Thermoelement as Thermoelectric Power Generator: Effect of Leg Geometry on the Efficiency and Power Generation,” Energy Convers. Manage., 65, pp. 26–32.
Khan, A. U. , Kobayashi, K. , Tang, D. M. , Yamauchi, Y. , Hasegawa, K. , Mitome, M. , Xue, Y. , Jiang, B. , Tsuchiya, K. , Golberg, D. , Bando, Y. , and Mori, T. , 2017, “ Nano-Micro-Porous Skutterudites With 100% Enhancement in ZT for High Performance Thermoelectricity,” Nano Energy, 31, pp. 152–159.
Tang, H. , Sun, F. H. , Dong, J. F. , Asfandiyar, Zhuang, H. L. , Pan, Y. , and Li, J. F. , 2018, “ Graphene Network in Copper Sulfide Leading to Enhanced Thermoelectric Properties and Thermal Stability,” Nano Energy, 49, pp. 267–273.
Erturun, U. , Erermis, K. , and Mossi, K. , 2014, “ Effect of Various Leg Geometries on Thermo-Mechanical and Power Generation Performance of Thermoelectric Devices,” Appl. Therm. Eng., 73(1), pp. 128–141.
Crane, D. T. , and Bell, L. E. , 2006, “ Progress Towards Maximizing the Performance of a Thermoelectric Power Generator,” 26th International Conference on Thermoelectrics (ICT), Vienna, Austria, Aug. 6–10, pp. 11–16.
Al-Merbati, A. S. , Yilbas, B. S. , and Sahin, A. Z. , 2013, “ Thermodynamics and Thermal Stress Analysis of Thermoelectric Power Generator: Influence of Pin Geometry on Device Performance,” Appl. Therm. Eng., 50(1), pp. 683–692.
Ali, H. , Sahin, A. Z. , and Yilbas, B. S. , 2014, “ Thermodynamic Analysis of a Thermoelectric Power Generator in Relation to Geometric Configuration Device Pins,” Energy Convers. Manage., 78, pp. 634–640.
Lamba, R. , and Kaushik, S. C. , 2017, “ Thermodynamic Analysis of Thermoelectric Generator Including Influence of Thomson Effect and Leg Geometry Configuration,” Energy Convers. Manage., 144, pp. 388–398.
Rabari, R. , Mahmud, S. , and Dutta, A. , 2014, “ Numerical Simulation of Nano-Structured Thermoelectric Generator Considering Surface to Surrounding Convection,” Int. Commun. Heat Mass Transfer, 56, pp. 146–151.
Xiao, H. , Gou, X. , and Yang, S. , 2011, “ Detailed Modeling and Irreversible Transfer Process Analysis of a Multi-Element Thermoelectric Generator System,” J. Electron. Mater., 40(5), pp. 1195–1201.
Reddy, B. V. K. , Barry, M. , Li, J. , and Chyu, M. K. , 2013, “ Thermoelectric Performance of Novel Composite and Integrated Devices Applied to Waste Heat Recovery,” ASME J. Heat Transfer, 135(3), p. 031706.
Ma, Y. , Heijl, R. , and Palmqvist, A. E. C. , 2013, “ Composite Thermoelectric Materials With Embedded Nanoparticles,” J. Mater. Sci., 48(7), pp. 2767–2778.
Poudel, B. , Hao, Q. , Ma, Y. , Lan, Y. , Minnich, A. , Yu, B. , Yan, X. , Wang, D. , Muto, A. , Vashaee, D. , Chen, X. , Liu, J. , Mildred, S. , Dresselhaus, S. M. , Chen, G. , and Ren, Z. , 2008, “ High-Thermoelectric Performance of Nano-Structured Bismuth Antimony Telluride Bulk Alloys,” Science, 320(5876), pp. 634–638. [PubMed]
Zhao, L. D. , Zhang, B. P. , Li, J. F. , Zhou, M. , Liu, W. S. , and Liu, J. , 2008, “ Thermoelectric and Mechanical Properties of Nano-SiC-Dispersed Bi2Te3 Fabricated by Mechanical Alloying and Spark Plasma Sintering,” J. Alloys Compd., 455(1–2), pp. 259–264.
Li, H. , Tang, X. , Zhang, Q. , and Uher, C. , 2009, “ High Performance InxCeyCo4Sb12 Thermoelectric Materials With in Situ Forming Nano-Structured InSb Phase,” Appl. Phys. Lett., 94, p. 102114.
Angrist, S. W. , 1982, Direct Energy Conversion, 4th ed., Allyn and Bacon, Boston, MA.
Pereez-Aparcio, J. L. , Palma, R. , and Taylor, R. L. , 2012, “ Finite Element Analysis and Material Sensitivity of Peltier Thermoelectric Cells Coolers,” Int. J. Heat Mass Tranfer, 55, pp. 1363–1374.
Yilbas, B. S. , and Sahin, A. Z. , 2010, “ Thermoelectric Device and Optimum External Load Parameter and Slenderness Ratio,” Energy, 35(12), pp. 5380–5384.
Bentley, R., 1998, “ Theory and Practice of Thermoelectric Thermometry,” Handbook of Temperature Measurement, Vol. 3, Springer, Cham, Switzerland.
Rabari, R. , Mahmud, S. , Dutta, A. , and Biglarbegian, M. , 2015, “ Effect of Convection Heat Transfer on Performance of Waste Heat Thermoelectric Generator,” Heat Transfer Eng., 36(17), pp. 1458–1471.
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## References

Yuan, R. , Deng, Y. , Hu, T. , Su, C. , and Liu, X. , 2018, “ Energy Efficient Thermoelectric Generator-Powered Localized Air-Conditioning System Applied in a Heavy-Duty Vehicle,” ASME J. Energy Resour. Technol., 140(7), p. 072007.
Schock, H. , Brereton, G. , Case, E. , D'Angelo, J. , Hogan, T. , Lyle, M. , Maloney, R. , Moran, K. , Novak, J. , Nelson, C. , Panayi, A. , Ruckle, T. , Sakamoto, J. , Shih, T. , Timm, E. , Zhang, L. , and Zhu, G. , 2013, “ Prospects for Implementation of Thermoelectric Generators as Waste Heat Recovery Systems in Class 8 Truck Applications,” ASME J. Energy Resour. Technol., 135(2), p. 022001.
Wan, C. , Tian, R. , Azizi, A. B. , Huang, Y. , Wei, Q. , Sasai, R. , Wasusate, S. , Ishida, T. , and Koumoto, K. , 2016, “ Flexible Thermoelectric Foil for Wearable Energy Harvesting,” Nano Energy, 30, pp. 840–845.
Kim, S. L. , Hsu, J. , and Yu, C. , 2018, “ Intercalated Graphene Oxide for Flexible and Practically Large Thermoelectric Voltage Generation and Simultaneous Energy Storage,” Nano Energy, 48, pp. 582–589.
Liu, T. , and Yang, Z. , 2018, “ Performance Assessment and Optimization of a Thermophotovoltaic Converter–Thermoelectric Generator Combined System,” ASME J. Energy Resour. Technol., 140(7), p. 072010.
Fuqiang, C. , Yanji, H. , and Chao, Z. , 2014, “ A Physical Model for Thermoelectric Generators With and Without Thomson Heat,” ASME J. Energy Resour. Technol., 136(1), p. 011201.
Sahin, A. Z. , and Yilbas, B. S. , 2013, “ The Thermoelement as Thermoelectric Power Generator: Effect of Leg Geometry on the Efficiency and Power Generation,” Energy Convers. Manage., 65, pp. 26–32.
Khan, A. U. , Kobayashi, K. , Tang, D. M. , Yamauchi, Y. , Hasegawa, K. , Mitome, M. , Xue, Y. , Jiang, B. , Tsuchiya, K. , Golberg, D. , Bando, Y. , and Mori, T. , 2017, “ Nano-Micro-Porous Skutterudites With 100% Enhancement in ZT for High Performance Thermoelectricity,” Nano Energy, 31, pp. 152–159.
Tang, H. , Sun, F. H. , Dong, J. F. , Asfandiyar, Zhuang, H. L. , Pan, Y. , and Li, J. F. , 2018, “ Graphene Network in Copper Sulfide Leading to Enhanced Thermoelectric Properties and Thermal Stability,” Nano Energy, 49, pp. 267–273.
Erturun, U. , Erermis, K. , and Mossi, K. , 2014, “ Effect of Various Leg Geometries on Thermo-Mechanical and Power Generation Performance of Thermoelectric Devices,” Appl. Therm. Eng., 73(1), pp. 128–141.
Crane, D. T. , and Bell, L. E. , 2006, “ Progress Towards Maximizing the Performance of a Thermoelectric Power Generator,” 26th International Conference on Thermoelectrics (ICT), Vienna, Austria, Aug. 6–10, pp. 11–16.
Al-Merbati, A. S. , Yilbas, B. S. , and Sahin, A. Z. , 2013, “ Thermodynamics and Thermal Stress Analysis of Thermoelectric Power Generator: Influence of Pin Geometry on Device Performance,” Appl. Therm. Eng., 50(1), pp. 683–692.
Ali, H. , Sahin, A. Z. , and Yilbas, B. S. , 2014, “ Thermodynamic Analysis of a Thermoelectric Power Generator in Relation to Geometric Configuration Device Pins,” Energy Convers. Manage., 78, pp. 634–640.
Lamba, R. , and Kaushik, S. C. , 2017, “ Thermodynamic Analysis of Thermoelectric Generator Including Influence of Thomson Effect and Leg Geometry Configuration,” Energy Convers. Manage., 144, pp. 388–398.
Rabari, R. , Mahmud, S. , and Dutta, A. , 2014, “ Numerical Simulation of Nano-Structured Thermoelectric Generator Considering Surface to Surrounding Convection,” Int. Commun. Heat Mass Transfer, 56, pp. 146–151.
Xiao, H. , Gou, X. , and Yang, S. , 2011, “ Detailed Modeling and Irreversible Transfer Process Analysis of a Multi-Element Thermoelectric Generator System,” J. Electron. Mater., 40(5), pp. 1195–1201.
Reddy, B. V. K. , Barry, M. , Li, J. , and Chyu, M. K. , 2013, “ Thermoelectric Performance of Novel Composite and Integrated Devices Applied to Waste Heat Recovery,” ASME J. Heat Transfer, 135(3), p. 031706.
Ma, Y. , Heijl, R. , and Palmqvist, A. E. C. , 2013, “ Composite Thermoelectric Materials With Embedded Nanoparticles,” J. Mater. Sci., 48(7), pp. 2767–2778.
Poudel, B. , Hao, Q. , Ma, Y. , Lan, Y. , Minnich, A. , Yu, B. , Yan, X. , Wang, D. , Muto, A. , Vashaee, D. , Chen, X. , Liu, J. , Mildred, S. , Dresselhaus, S. M. , Chen, G. , and Ren, Z. , 2008, “ High-Thermoelectric Performance of Nano-Structured Bismuth Antimony Telluride Bulk Alloys,” Science, 320(5876), pp. 634–638. [PubMed]
Zhao, L. D. , Zhang, B. P. , Li, J. F. , Zhou, M. , Liu, W. S. , and Liu, J. , 2008, “ Thermoelectric and Mechanical Properties of Nano-SiC-Dispersed Bi2Te3 Fabricated by Mechanical Alloying and Spark Plasma Sintering,” J. Alloys Compd., 455(1–2), pp. 259–264.
Li, H. , Tang, X. , Zhang, Q. , and Uher, C. , 2009, “ High Performance InxCeyCo4Sb12 Thermoelectric Materials With in Situ Forming Nano-Structured InSb Phase,” Appl. Phys. Lett., 94, p. 102114.
Angrist, S. W. , 1982, Direct Energy Conversion, 4th ed., Allyn and Bacon, Boston, MA.
Pereez-Aparcio, J. L. , Palma, R. , and Taylor, R. L. , 2012, “ Finite Element Analysis and Material Sensitivity of Peltier Thermoelectric Cells Coolers,” Int. J. Heat Mass Tranfer, 55, pp. 1363–1374.
Yilbas, B. S. , and Sahin, A. Z. , 2010, “ Thermoelectric Device and Optimum External Load Parameter and Slenderness Ratio,” Energy, 35(12), pp. 5380–5384.
Bentley, R., 1998, “ Theory and Practice of Thermoelectric Thermometry,” Handbook of Temperature Measurement, Vol. 3, Springer, Cham, Switzerland.
Rabari, R. , Mahmud, S. , Dutta, A. , and Biglarbegian, M. , 2015, “ Effect of Convection Heat Transfer on Performance of Waste Heat Thermoelectric Generator,” Heat Transfer Eng., 36(17), pp. 1458–1471.

## Figures

Fig. 2

(a) Geometry changes from the flat rectangular legs (dashed black lines) to trapezoidal-shaped legs (red lines) and (b) three-dimensional schematic views of the geometric configuration of a trapezoidal-shaped TEG leg

Fig. 1

Schematic presentation of a single TEG cell with (a) typical flat rectangular geometry legs, and (b) trapezoidal-shaped legs

Fig. 3

Schematic diagram to illustrate the boundary conditions for conduction and convection heat transfer from a TE leg

Fig. 4

Numerical results for temperature distribution profile and heat transfer contours of a trapezoidal-shaped TEG. Panels (a) and (b) are for traditional TE material based TEG with h =0 W m−2 K−1 and h =60 W m−2 K−1, respectively, and panels (c) and (d) are for nanomaterial-based TEG with h =0 W m−2 K−1 and h =60 W m−2 K−1.

Fig. 5

Numerically simulated results of electric potential contours for (a) traditional TE material-based TEG and (b) nanomaterial-based TEG with h =0 W m−2 K−1

Fig. 6

Effect of different hot side temperatures on heat input at different convection heat transfer coefficients when TC remains constant at 300 K (numerical results)

Fig. 7

(a) Output current behaviors as a function TH for traditional and nanomaterial-based TEG at different TC (numerical results) and (b) comparison between an analytical solution and the numerical simulation results

Fig. 8

(a) Power output behaviors as a function TH for traditional and nanomaterial based TEG at different TC (numerical results) and (b) comparison between numerically simulated power and analytically calculated power at different temperature gradients

Fig. 9

Analytical analysis of open circuit voltage and power output with respect to current for both materials at 200 K temperature difference

Fig. 10

Thermal efficiency with respect to temperature difference at various convection heat transfer coefficients where TH is increasing starting from 400 K and TC is kept constant at 300 K

## Tables

Table 1 Temperature-dependent polynomial function for TE material properties [17,19,20,22]
Table 2 The performance of TEG for different values of the geometric imperfection parameter (a) using TH = 500 K and TC = 300 K
Table 3 The effect of the convection heat transfer coefficient on thermal efficiency at TH = 500 K and TC = 300 K
Table 4 Comparison between previous work with current work at h = 0 W m−2 K−1, TH = 500 K and TC = 300 K

## Errata

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