A comprehensive analysis of the performance of thermoelectric generators with constant and variable properties

Applied Energy 241 (2019) 11–24

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A comprehensive analysis of the performance of thermoelectric generators with constant and variable properties☆

T

Wei-Hsin Chena,b, , Yi-Xian Lina, Xiao-Dong Wangc, Yu-Li Lind ⁎

a

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan Research Center for Energy Technology and Strategy, National Cheng Kung University, Tainan 701, Taiwan c State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China d Energy and Environmental Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan b

HIGHLIGHTS

GRAPHICAL ABSTRACT

of TEGs with constant/ • Performances variable materials’ properties are studied.

cold surface temperature • Decreasing with fixed ΔT may intensify or abate the performance.

hot surface temperature • Oscillating may increase or decrease the performance.

performance of a TEG and its • Practical deviation from theory depend on properties adopted.

variable properties • Considering material provides a more realistic

of

outcome.

ARTICLE INFO

ABSTRACT

Keywords: Thermoelectric generator (TEG) and generation Variable properties Self-consistency Impedance matching Temperature oscillation

Material properties of a thermoelectric generator play a pivotal role in directly converting heat into electricity. To provide a comprehensive study on power generation of thermoelectric generators, four different materials with constant and temperature-dependent properties are investigated numerically. The influences of the temperature difference across the elements, the temperature at the cold side surface, and the temperature oscillation at the hot side surface on the performance of the generators are simulated numerically and analyzed systematically. The predictions indicate that the output power, temperature-difference square rule, and impedance matching of the materials with the variable properties deviate from the theoretical results with constant properties, but their behavior always obeys the self-consistency. At a given temperature difference, a decrease in the cold surface temperature may intensify or abate the performance, depending on the material properties adopted. Oscillating the hot surface temperature may increase or decrease the output power and efficiency. In summary, the practical performance of a thermoelectric generator and its deviation from the theoretical performance depend strongly on the properties adopted. Therefore, the consideration of the variable properties of materials can provide a more realistic outcome compared to the predictions with constant properties.

The short version of the paper was presented at ICAE2018, Aug 22–25, Hong Kong. This paper is a substantial extension of the short version of the conference paper. ⁎ Corresponding author at: Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan. E-mail addresses: [email protected], [email protected] (W.-H. Chen). ☆

https://doi.org/10.1016/j.apenergy.2019.02.083 Received 23 November 2018; Received in revised form 7 February 2019; Accepted 16 February 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

Applied Energy 241 (2019) 11–24

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Nomenclature

A D

E I

J k L P Q q q RTE Re S T t V

W

width (mm)

Greek letters

surface area (mm2) depth (mm)

electric field intensity vector (V m−1) electric current (A)

e

electric current density vector (A m−2) thermal conductivity (W m−1 K−1) length (mm) output power of TEG (W) heat transfer rate (W) heat generation per unit volume (W m−3) heat flux vector (W m−2) electric resistance (Ω) external load resistance (Ω) seebeck coefficient (V K−1) temperature (°C) time (s) voltage (V)

efficiency (%) electrical resistivity (Ω m) electric scalar potential (V) amplitude (K) period (min)

Subscripts

c h n p TE IM

1. Introduction

cold side of TE element hot side of TE element n-type TE element p-type TE element thermoelectric element impedance matching

heat problem. Jiang et al. [9] developed a set of road TEG system to simultaneously harvest energy from a road and reduce the road surface temperature. This helped to alleviate the urban heat island effect and reduce pavement defects caused by high temperatures. Allouhi et al. [10] performed the theoretical analysis of a thermoelectric heating system coupled with an office room, which saved 55–64% energy compared to conventional electric heaters. Lertsatitthanakorn [11] evaluated a TEG for biomass cooking stoves and showed that it generated approximately 2.4 W at the temperature difference of 150 °C. In recent years, many researchers have focused on the development of thermoelectric materials. There are many kinds of materials such as lead telluride (PbTe) [12] and bismuth telluride (Bi2Te3) [13] have been used in TEGs. Otherwise, many synthetic routes of thermoelectric materials like mechanical alloying [14,15], hydrothermal [16,17], melt and growth [18,19], and microwave synthesis [20,21] have been well developed to improve the performance of TEGs. Furthermore, it has been found that the nanostructure could effectively reduce the thermal conductivity [22] and lead to a higher figure-of-merit (ZT) of a TEG. Numerical simulations of thermoelectric generation also play a very important role in research because they are a good way to save costs and time with high accuracy while recognizing TEG characteristics and designing TEG devices. In this aspect, Fernández-Yañez et al. [23] considered the effects of size and internal topology of TEG to maximize the electrical power output using computational fluid dynamics (CFD). Zhao et al. [24] established a mathematical model of TEG to recover the waste heat of wet flue gas. Muralidhar et al. [25] optimized the heat transfer process in a TEG system to improve energy recovery and fuel saving in a hybrid heavy duty vehicle. Fan and Gao [26] established a three-dimensional finite element model of annular TEG and studied the performance and mechanical reliability of the TEG. Bai et al. [27] constructed a TEG model which was attached to the heat exchanger and cooling tank and calculated the performance of an automotive thermoelectric generator employing metal foam. Jia and Gao. [28]

Air pollutants and CO2 emissions from fossil fuel combustion due to anthropogenic activity have received global concern and become important issues for environmental sustainability in recent years. Though the awareness of environmental protection has risen, fossil fuels still dominate the energy demand and supply in the next few decades. As a consequence, how to improve the fuel consumption efficiency and reduce fossil fuel usage becomes a remarkable challenge currently. Based on the statistical report, only around 30–35% combustion energy in power plants is effectively utilized, while the rest is wasted in the form of heat [1]. Waste heat recovery is an issue of great importance nowadays and has been used in many fields. Mukherjee et al. [2] investigated the feasibility of a novel low-temperature gas-to-gas heat recovery system for a food manufacturing process. Ganguly et al. [3] presented a new approach to improve the thermal efficiency of a solar pond by recovering heat lost. Chen et al. [4] recovered waste heat in a Swiss-roll reactor to enhance methanol partial oxidation for hydrogen production. Jiang et al. [5] claimed that organic Rankine cycle (ORC) was an effective device to produce power from the recovery the lowtemperature waste heat. Among all of the waste heat recovery methods, thermoelectric generators (TEGs) show their high value with many advantages such as eco-friendly and easy use, lightweight device, no moving part and noise free, etc. The applications and researches of TEGs are more and more widespread. For example, Wang et al. [6] constructed a wearable TEG for converting human body heat into electricity, which could be utilized to power a miniaturized accelerometer to detect human body motion through harvesting wrist heat. Montecucco et al. [7] presented the application of TEG to a solid-fuel stove to charge a lead-acid battery and transfer heat to circulating water for heating or household use. Su et al. [8] designed a distributed thermoelectric device to extract the waste heat of coal fire that could prevent heat buildup and solve the waste

12

13

Bi2Te3

Bismuth-telluride and constant-property material

Sb2Te3 and Bi2Te3(assume as constant properties)

Constant-property material

Constant-property material

Th = 423 K Tc = 298 K Temperature oscillations on hot and cold side surfaces ΔT = 30 K to 150 K Solar power

Surfaces that providing appropriate temperature gradients

Vehicle exhaust system

Heavy duty vehicle

SiGe and DD0.76Fe3.4Ni0.6Sb12 (p-type) Ba0.3Ni0.05Co3.95Sb12 (n-type) Bi2Te3 Th = 225 °C Tc = 25 °C Th = 157.4 °C to 223.5 °C Th = 80 °C ΔTinit = 10 K to 20 K

Wet flue gas

Constant-property material

Application Diesel engine

Conditions

Bi2Te3

Material

Table 1 A literature review of TEG numerical simulations.

The thermal resistance model (assuming as constant properties) overestimated the output power and conversion efficiency of the TEG, it was especially true at large currents and high-temperature differences. Reducing the cross-sectional area of the thermoelectric element was a better way to improve performance when the length of the component was fixed.

Optimal structural parameters of the annular thermocouple could be found for the improvement of thermoelectric performance and mechanical reliability. The increase of metal foam thickness would contribute to a higher temperature and output power of TEG, and a better sound reduction performance. A new folding scheme for the thin-film thermoelectric generators was proposed. The folding scheme enabled high packing densities without compromising the thermal contact area to the heat source and sink. The excitation of temperature fluctuation at the two surfaces, especially at the hot side surface, was an effective countermeasure to enhance the performance of a TEG.

A modest amount of energy could be recovered in the high load and high engine speed region of common driving conditions. The increasing of temperature in the flue gas led to a gradual decrease in the output power from latent heat, whereas the output power of sensible heat increased greatly. Both silicon germanium and Skutterudites create substantial fuel savings when used in the vehicle.

Outcome

[31]

[34]

[37]

[30]

[27]

[26]

[25]

[24]

[23]

Ref.

W.-H. Chen, et al.

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Table 2 Properties of materials. Material Formula A

B

C

D

Bi2Te3 (assume as constant)

(Bi0.2Sb0.8)2Te3

Cu11Ni1Sb4S13

AgxSb2−xTe3−x x = 0.93

Parameter

Property ρp = ρn = 1.447 × 10

ρ (Ω m) −1

k (W m K S (V K−1)

−1

)

k (W m K S (V K−1)

[39]

kp = kn = 1.52 Sp = − Sn = 226.8 × 10−6 ρp = ρn = (0.1449T3 − 19331T2 + 22342063T − 3502255974) × 10−11

ρ (Ω m) −1

Ref. −5

−1

)

3

2

kp = kn = (0.2538T + 956.77T − 163847T + 77278578) × 10 Sp = − Sn = (0.249T3 − 3559.86T2 + 2522962T − 219937413) × 10−12

ρ (Ω m)

ρp = ρn = (−0.2378T3 + 1060.8T2 − 1021250T + 3584129731) × 10−7

k (W m−1 K−1) S (V K−1)

kp = kn = (0.2716T3 − 1659.74T2 + 2113123T − 90916832) × 10−9 Sp = −Sn = (0.2762T3 − 660.31T2 + 586427T + 40952813) × 10−12

ρ (Ω m)

ρp = ρn = (0.083267T3 − 135.92T2 + 68631T − 52234) × 10−11

−1

k (W m K S (V K−1)

−1

)

[40]

−8

3

2

[41]

[42] −8

kp = kn = (−0.5798T + 1144.34T − 718227T + 221203518) × 10 Sp = − Sn = (−0.2048T3 + 237.67T2 − 81592T + 33905793) × 10−11

developed a unique linear-shape TEG and discussed its performance under small and large temperature differences. The relevant literature is tabulated in Table 1. The use of simulation software to analyze the thermoelectric power has been widely employed. However, it is often limited by the thermoelectric power output formula in the analysis process. In many studies, constant thermoelectric material properties were assumed for simplifying the physical problem and easy calculation [29,30], but the material’s properties such as electrical resistivity, thermal conductivity, and Seebeck coefficient are actually temperature-dependent. Past studies found that assuming the properties of thermoelectric materials as constant may induce a numerical deviation at high temperatures [31]. A few studies used constant and variable properties to investigate the difference in TEC or TEG behavior. Wang et al. [32] developed a threedimensional TEC model coupling the temperature field and electric potential field, and found that constant property model underestimated the electric potential and overestimated the cooling capacity of TEC compared to variable property model. Meng et al. [33] examined the effect of material properties on the dynamic characteristics of the TEC at small, medium, and large currents, and pointed out that the variable property model was more appropriate to predict TEC transient performance accurately, especially at medium and high currents. They [34] also showed that the thermal resistance model with constant properties overestimated the output power and efficiency of a TEG, resulting from the underestimation of electrical resistance and overestimation of thermal conductance and Seebeck coefficient. The literature reviewed above indicates that the constant-property material may lead to prediction error in the simulation of the TEG in performance, however, some important information remains unclear. For this reason, the purpose of this research is to provide a comprehensive analysis and comparison on the effects of constant and variable material properties upon the TEG performance such as output power and efficiency using numerical simulations. To achieve this target, four different materials with constant or variable properties are considered.

The conventional phenomena discussed in theoretical analysis such as ΔT2 law, heat transfer mechansims, self-consistency, and impedance matching are examined deeply. Moreover, the impacts of (1) the cold surface temperature with a fixed temperature difference between the hot and cold surfaces and (2) the oscillated hot surface temperature under specified mean temperature upon TEG performance will also be explored. On account of the significant progress in industrial applications of TEG, it is believed that the obtained results are able to give a more practical approach for TEG simulations and are conducive to TEG system design using numerical methods with better accuracy. 2. Mathematical formulation 2.1. Physical model A thermoelectric generator with 127 pairs of thermoelectric couples, which is the common design in commercial TEGs, serves as the basis of the present study. Each thermoelectric couple consists of a pair of p-type and n-type elements. The length and width of the thermoelectric module (TEM) are 4.0 cm and 4.0 cm, respectively. The copper connection thickness on both sides of the p-type and n-type electrodes is 0.2 cm. The geometries of p-type and n-type elements are equivalent, and the length, width, and depth each element is 1.6 mm, 1.4 mm and 1.4 mm, respectively. Only a pair of thermoelectric elements is taken into account in the simulations. Thermal, electrical, and thermoelectric effects are involved in TEG. The principle of energy conservation is given by:

·q = q

(1)

where q and q stand for the heat flow vector and heat generation, respectively. Meanwhile, the steady-state principle of current is expressed as:

·J = 0

14

(2)

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Thermal conductivity (Wm-1K-1)

1.8

(a) Thermal conductivity

where J represents the current density vector. The heat flux vector q and current density vector J are coupled by [32]:

1.6

q = STJ

1.4

Material

1

(

+ S T)

e

(4)

element, and E is the electric field intensity vector. The electric field

and

shown as E = [31]. Substituting Eqs. (3) and (4) into Eqs. (1) and (2) yields the coupled governing equations of temperature and electric potential as:

0.4 0.2 300

350

400

Temperature (K)

450

·(STJ )

500

·

(b) Seebeck coefficient

1 e

·(k T ) = q +

·

S

T =0

e

(5) (6)

For the boundary conditions, the hot surface, cold surface, and lateral surfaces are regarded. In the calculations, the temperatures of the hot and cold surfaces are given, and the lateral surfaces are assumed to be adiabatic. In regard to the electric boundary conditions, the temperature difference across the TEG under a given external load resistance will drive current from the inlet which is the side surface of the electrode. The zero electric potential of the outlet is assumed, that is = 0 . Meanwhile, it is postulated that there is no current flowing

0.00028 -1

Seebeck coefficient (VK )

1

vector E in Eq. (4) is derived from an electric scalar potential

0

0.00026 0.00024 0.00022

through the lateral surfaces so that J · n = 0 [35].

0.0002

2.2. Numerical method

0.00018 0.00016 300

Electrical resistivity (ohm m)

S T) =

In the preceding equations, S is the Seebeck coefficient, T is temperature, k is thermal conductivity, e is the electrical resistivity of the

0.6

0.00014

(E

e

0.8

0.0003

1

J =

A B C D

1.2

(3)

k T

350

400

Temperature (K)

450

500

450

500

The physical phenomena are predicted by the commercial software ANSYS 18.0. The governing equations are discretized using a finite element scheme in which the Galerkin method is used [36]. In the present study, the geometry of the thermoelectric generator is the same as that in a previous study [37] in which strict numerical validation and grid independent test have been performed. For the numerical validation, the predicted values of power per area agreed well with experimental measurements. For the grid independence test, three different orthogonal grid systems of 2912, 5080, and 14,340 grids have also been tested [37] and compared with the analytical analysis of Chen et al. [38], for the purpose of verifying the validity and a sufficient number of grids in the numerical simulation. The comparison suggested that the grid system with 5080 grids satisfied the requirement of grid independence, and the developed numerical method was consistent with the analytical analysis (the maximum error in voltage was 0.8%), thereby achieving the numerical validation and grid independence. In the predictions, after the temperatures at the cold and hot surfaces are given, which induces the heat flow entering the hot surface (Qh), the generated voltage (V), output power (P), and efficiency (η = P/Qh) could be acquired.

(c) Electrical resistivity

0.00012 0.0001 8E-05 6E-05 4E-05 2E-05 0 300

2.3. Materials, operating conditions, and physical qualities 350

400

Temperature (K)

In this study, four different TEG materials are considered and their performances are compared with each other. The materials include Bi2Te3 (Material A) [39], (Bi0.2Sb0.8)2Te3 (Material B) [40], Cu11Ni1Sb4S13 (Material C) [41], and AgxSb2−xTe3−x x = 0.93

Fig. 1. Profiles of (a) thermal conductivity, (b) Seebeck coefficient, and (c) electrical resistivity of four different materials.

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1.6

1.3

1.5

1.1 1 0.9 0.8 0.7 0.6 300

0.8

350

400

Temperature(K)

450

0.4 0.3 0.2 0.1 350

1.4 1.3 1.2 1.1 1 0.9

400

Temperature(K)

450

350

450

500

Experiment Curve fitting

0.9 0.8 0.7 0.6 0.5 0.4 300

500

400

Temperature(K)

(d) Material D

1

0.5

0 300

Experiment Curve fitting

1.1

Experiment Curve fitting

0.6

(b) Material B

0.8 300

500

(c) Material C

0.7

Figure of merit (ZT)

Figure of merit (ZT)

Experiment Curve fitting

1.2

Figure of merit (ZT)

Figure of merit (ZT)

(a) Material A 1.4

350

400

Temperature(K)

450

500

Fig. 2. Figure-of-merit (ZT) profiles of materials (a) A (Bi2Te3), (b) B ((Bi0.2Sb0.8)2Te3), (c) C (Cu11Ni1Sb4S13), and (d) D (AgxSb2−xTe3−x, x = 0.93).

(Material D) [42]. The properties of Bi2Te3 such as the electrical resistivity, thermal conductivity, and Seebeck coefficient are assumed to be constant, whereas those of the other three materials are temperaturedependent. The three properties of the four materials are tabulated in Table 2. In the first section of this study, the cold surface temperature (Tc) is fixed at 25 °C, while the hot surface temperature (Th) is in the range of 50–200 °C. A number of crucial phenomena such as ΔT2 law, heat transfer mechanisms, self-consistency, and impedance matching will be examined. In the second section, the temperature difference is fixed at 100 °C, and the five different cold surface temperatures of 5, 10, 15, 20, and 25 °C are regarded. In the last section, Tc is fixed at 25 °C, and Th with temperature oscillation following the sinusoidal function is adopted. Three mean temperatures of Th (50, 100, and 150 °C) along with the amplitude of the temperature oscillation of 20 °C are taken into account. The period of the sinusoidal temperature function is fixed at 30 min. A temperature difference will induce current I. For the thermoelectric couple with constant properties, the induced current I can be expressed as

I=

S T RTE + R e

(7)

where S, RTE, and Re are the Seebeck coefficient (S = Sp Sn ), electrical resistance of the TEG, and the external load resistance, respectively. RTE is defined as [43]:

RTE =

e , p Lp

Ap

+

e, n Ln

(8)

An

where L and A are the length and cross-sectional area (=1.4 mm × 1.4 mm) of the elements, respectively, and the subscripts p and n designate the p-type and n-type elements, respectively. The theoretical output power of the TEG is given by

P=

S 2 T 2R e (RTE + Re )2

T2

(9)

When the impedance matching [38,44] is invoked, the electrical resistance of the TEG is equivalent to the external load resistance, that is

RTE = R e

16

(10)

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W.-H. Chen, et al.

(a) 10

Power (W)

8

Fig. 2. Overall, the ZT values of Materials A, C, and D increase with rising temperature. Material A is a typical Be-Te alloy. Material C is cubic tetrahedrite whose Cu atoms are substituted by Ni and Zn. Material D is recognized as the solid solution between Ag2Te and Sb2Te3. In contrast, the curve of Material B, a material of low-temperature TEG featured by nano-structure of BiSbTe [40], has a maximum ZT value of 1.46 at a temperature of approximately 350 K.

Material A B C D

6

3.2. Effect of temperature difference

4

3.2.1. Output power The distributions of the output power of the TEGs based on the four materials versus temperature difference (ΔT) between the hot and cold surfaces are shown in Fig. 3a. It is clear that the curves of Materials A, C, and D have the same trend. Specifically, the curves exponentially rise with increasing ΔT. This can be explained by the ZT values, the dominant factor in determining the performance of a TEG [45], of Materials A, C, and D increase monotonically as the temperature is raised (Fig. 2a, c, and d). Unlike the three materials, the ZT value of Material B has a trend to decrease with increasing temperature (Fig. 2b). This is the reason why the increase in the output power of Material B tends to slow down when ΔT increases. Overall, the output power of the four materials is characterized by the order of A > B > C > D.

2

0 25

50

75

100 o

∆T ( C)

125

150

175

(b) 10

Power (W)

8

3.2.2. ΔT2 law For a TEG material with constant properties, Eq. (9) reveals that the output power of a TEG is linearly proportional to the temperature-difference square [37], namely, the ΔT2 law. Material A has constant properties so that its output power linearly increases with ΔT2, as shown in Fig. 3b. For the materials with variable properties, thermal conductivity, Seebeck coefficient, and electrical resistivity in Eq. (9) change with the temperature (Fig. 1), so the output power deviating from the ΔT2 law is exhibited. To figure out the deviation degree of Materials B, C, and D due to the variable properties adopted, their linear regression is carried out. The results suggest that the coefficients of determination (R2 ) of Materials B, C, and D are 0.9676, 0.9898, and 0.9998, respectively. This implies, in turn, that Materials C and D roughly obey the the ΔT2 law, whereas Material B departs from the law to a certain extent.

6

4

2

0

0

10000

20000

∆T2 (oC2)

30000

Fig. 3. Distributions of output power versus (a) temperature difference and (b) temperature difference square.

and the maximum output power can be attained from Eq. (9). In the present study, the impedance matching is utilized in the numerical simulations.

3.2.3. Heat transfer mechanism The purpose of a TEG is to convert heat into electricity via the Seebeck effect [46]. The heat flux in the elements from the hot surface to the cold surface with constant material properties is expressed as:

3. Results and discussion

q=

3.1. Material properties and figure-of-merit

SI k Th + T A LTE

0.5I 2 RTE A

(11)

where L and RTE are the length and internal resistance of the elements, respectively. The three terms at the right side of Eq. (11) are the Peltier heat flux, Fourier heat flux, and Joule heat flux, respectively [44,47]. To provide the comparison of three different heat fluxes, their distributions with respect to ΔT are sketched in Fig. 4 where the distributions are calculated based on the constant material properties (i.e., S and k ) at 300 K. The figure depicts that both the Peltier and Joule heat fluxes are not linear with rising ΔT, whereas the Fourier heat flux linearly increases. These phenomena are in line with the observations of

The curve fittings of the four materials’ properties (i.e., electrical resistivity, thermal conductivity, and Seebeck coefficient) at temperatures between 300 K and 500 K based on the experimental data are shown in Fig. 1, while the obtained polynomials describing the variations of the properties with temperature are given in Table 2. It can be seen that the fitted curves agree well with the experimental data, and are thus employed in the simulations. Meanwhile, the curve-fitting profiles of the figure-of-merit (ZT) of the four materials are displayed in

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(a) Peltier 0.25

200

Material

A B C D

0.15

Heat flux (Wmm-2)

Heat flux (Wmm-2 )

0.2

0.1

150

100

50

0.05

0 25

50

75

100 o

∆T ( C)

125

150

0 25

175

(c) Joule

50

75

100

125

150

175

100

125

150

175

∆T (oC)

(d) Total

0.025

200

Total heat flux (Wmm-2)

0.02

0.015

0.01

0.005

0 25

50

75

100 o

∆T ( C)

125

150

150

100

50

0 25

175

50

75

o

∆T ( C)

Fig. 4. Distributions of (a) Peltier heat flux, (b) Fourier heat flux, (c) Joule heat flux, and (d) total heat flux.

Gou et al. [48]. It can be seen that the maximum values of the Peltier, Fourier, and Joule heat fluxes (at ΔT = 175 °C) are 0.155 W mm−2 (Fig. 4a), 166.250 W mm−2 (Fig. 4b), and 0.017 W mm−2 (Fig. 4c), respectively, implying that their orders of magnitudes are O(-1), O(2), and O(-2), respectively. It follows that the Fourier heat flux dominates the entire heat flux in the elements, followed by the Peltier heat flux and the Joule heat flux.

8 7

Material

6

Efficiency (%)

Heat flux (Wmm-2)

(b) Fourier

5

A B C D

4

3.2.4. Efficiency The efficiency distributions of the four materials are demonstrated in Fig. 5. It is obvious that the efficiency linearly increases when ΔT goes up, except for Material B resulting from its decline in ZT (Fig. 2b). For a constant-property material, its theoretical maximum efficiency in terms of the temperatures of the hot and cold surfaces and ZT is given by [49]

3 2 1 0 25

50

75

100

125

150

175

=

o

∆T ( C) Fig. 5. Distributions of the efficiency of four different materials.

P T TC = H qA TH

1 + ZT 1 + ZT +

1 TC TH

=

T TH

1 + ZT 1 + ZT +

1 TC TH

(12)

It reveals that an increase in ΔT or ZT intensifies the efficiency. For

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(a) Material B V P1 P2

4 3

0.015

2

0.01

1

Power (W)

Voltage (V)

0.02

Material B with temperatures smaller than 150 °C, its efficiency keeps growing when ΔT rises, even though its ZT diminishes. However, once ΔT is greater than 150 °C, the effect of increasing ΔT cannot keep up with that of vanishing ZT, this gives rise to the maximum efficiency of Material B exhibited at 150 °C. Material B has a higher ZT value than the other materials at lower temperatures (Fig. 2b); this is responsible for its higher efficiency than others.

5

0.025

3.2.5. Self-consistency and impedance matching In order to know the accuracy of the model calculation, the examination of the self-consistency is important [32]. There are two ways to acquire the output power of a TEG. The first route is calculating the output power based on conservation of energy along with the assumption of adiabatic lateral surfaces. Accordingly, the output power is given by P1 = Qh Qc [50]. The second route is using the formula P2 = I × V . The distributions of electric potential differences (V ) and electric powers (P1 & P2 ) of Materials B, C, and D are presented in Fig. 6. Overall, the curves of P1 closely overlap those of P2 , and the maximum relative difference between P1 and P2 is less than 0.4%. This self-consistency ascertains the accuracy of the simulations. For a TEG with constant properties, the maximum output power is obtained when the electrical resistance of the TEG is equivalent to the external load resistance, so-called impedance matching [38]. The output power curves of the four materials versus current at various temperature differences (ΔT = 25, 50, 75, 100, 125, 150, and 175 °C) are shown in Fig. 7 in which the current is varied. In the figure, the dash lines connect the maximum values of curves, while the black spots (PIM ) represent the power obtained according to the impedance matching. On account of the constant properties adopted in Material A, the dash line in Fig. 7a exactly matches the spots. Once the variable properties, namely, Materials B, C, and D, are employed, the spots depart from the dash lines, especially for Material B. It is thus underlined that the maximum power does not obey the impedance matching when variable properties are adopted. The figure depicts that the bias degree between the dash curves and black spots is ranked by B > C > D > A. A comparison to the R2 of the linear regression (Fig. 3a) suggests that the greater the coefficient of determination, the less the deviation of the impedance matching from the theory.

0.005 0 0

0

50

100

150

200

o

∆T ( C) 0.05

(b) Material C

0.8 0.7 0.6 0.5

0.03

Power (W)

Voltage (V)

0.04

0.4 0.02

0.3 0.2

0.01

0.1 0 0

50

100

150

0 200

o

∆T ( C) 0.05

3.3. Effect of cold surface temperature

(c) Material D

3.5

For a TEG with constant properties, its theoretical output power depends on the temperature difference between the hot surface and the cold surface, but it is independent of the hot or cold surface temperature, as expressed in Eq. (9). However, the experimental study of Chen et al. [51] using a commercial TEG found that a lower cold surface temperature resulted in higher output power, even though the temperature difference between the hot surface and cold one was fixed. To examine this characteristic using different materials, Fig. 8a presents the distributions of output power at different cold surface temperatures (i.e., 5, 10, 15, 20, and 25 °C with ΔT fixed at 100 °C. The figure depicts that the output power of Material A does not vary with the cold surface temperature which is in line with the theoretical result. When the temperature at the cold surface is lowered, the power output of Materials B and D increases which are consistent with the results of Chen et al. [51], whereas an opposite trend develops for Material C. To proceed farther into the recognition of the preceding behaviors, the distributions of heat flow at the hot (QH ) and cold (QC ) surfaces as well as output power (P = QH Qc ) of Materials B and C are shown in Fig. 9.

3

0.04

0.03

2

1.5

0.02

1 0.01

0

0

Power (W)

Voltage (V)

2.5

0.5

50

100

150

0 200

o

∆T ( C) Fig. 6. Profiles of electric potential differences and output powers of Materials (a) B, (b) C, and (c) D.

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W.-H. Chen, et al.

(a) Material A

PIM

8

Power (W)

7

5

150 oC

6

o

25 C

o

125 C

4

o

100 C 2 0

o

75 C 50 oC 0

0.5

1

(b) Material B

6

∆T = 175 oC

Power (W)

10

4 3 2 1

1.5

2

2.5

3

0

3.5

0

0.5

1

Current (A)

0.8

(c) Material C

3.5

0.7

2.5

(d) Material D

2.5

0.5

Power (W)

Power (W)

2

3

0.6

0.4 0.3 0.2

2 1.5 1 0.5

0.1 0

1.5

Current (A)

0

0.1

0.2

0.3

0

0.4

0

0.2

0.4

Current (A)

0.6

0.8

1

Current (A)

Fig. 7. Output power profiles of Materials (a) A, (b) B, (c) C, and (d) D.

Though both QH and QC of the two materials rise with increasing Tc, the profiles of output power have different trends. This arises from the fact that the four materials have different profiles of thermal conductivity, Seebeck coefficient, and electrical resistivity (Fig. 1). The efficiency profiles of the four materials are displayed in Fig. 8b. The efficiencies of Materials A, B, and D increase when the cold surface temperature is lowered, whereas Material C has the opposite consequence. For Material B, decreasing the cold surface temperature increases P but decreases QH (Fig. 9a), this results in a higher efficiency ( = P / QH ). In contrast, both P and QH of Material C go down with decreasing the cold surface temperature. However, the decreasing extent in P is slightly higher than in QH , rendering the decline of a bit. Unlike the output power profiles in Fig. 8a, it is of interest that Material B in Fig. 8b has a higher efficiency than Material A. This is a consequence of lower QH of Material B when compared to that of Material A (Fig. 4d).

intensified when compared to that with a fixed temperature. This characteristic is also examined in this study. For the four materials, their hot surface temperature with sinusoidal function is given as

Th = Th,

=0

+

h

× sin

2 t h

(13)

where , t , and denote the temperature amplitude, time, and period, respectively. Three average temperatures of 50, 100, and 150 °C along with the amplitude of 20 °C and period of 30 min at the hot surface are considered, while the cold surface temperature is fixed at 25 °C. The output power profiles under the oscillated temperatures are shown in Fig. 10. Similar to the results without temperature oscillation (Fig. 3a), the output power profiles follow the order of A > B > D > C, regardless of the mean temperature at the hot surface. Meanwhile, Fig. 11 depicts that the efficiency profiles are ranked by B > A > D > C at the mean temperatures of 50 and 100 °C. This resembles the behavior observed in Fig. 5. However, when the mean temperature is as high as 150 °C, the efficiency of Material B is lower than that of Material A at the peaks of the temperature oscillation, ascribing to the ZT value of Material B being down to a low level (Fig. 2b). Based on the results in Fig. 11, the distributions of mean power and

3.4. Effect of temperature oscillation Past studies with constant properties [37,38] pointed out that the output power of a TEG with temperature oscillations could be

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(a) Material B

(a) 7 6

A B C D

3 2 1

2.3

10

15

20

2.2 2.15

0.31

25 0.3 275

o

280

285

290

295

2.05 300

TC (K)

(b) 7 0.12

(b) Material C

5 0.118

Heat flow rate (W)

4 3 2 1 5

10

15 TC (oC)

20

0.2

QH QC Power

0.19 0.116 0.18

0.114

0.112

25

0.11 275

Fig. 8. Distributions of (a) output power and (b) efficiency under altered cold surface temperature.

Power (W)

6

Efficiency (%)

2.25

0.32

2.1

5

TC ( C)

0

2.35

Power (W)

Heat flow rate (W)

Power (W)

Material

4

0

QH QC Power

0.33

5

2.4

0.17

280

285

290

295

300

TC (K) Fig. 9. Profiles of heat flow rates at the hot surface and cold surface as well as the power of Materials (a) B and (b) C.

efficiency versus mean temperature are displayed in Fig. 12. Overall, the rank in the power and efficiency with the oscillating temperature is the same as those without oscillation (Figs. 3a and 5), that is, B > A > D > C. Furthermore, the comparison in the performances of TEGs with and without temperature oscillation at the three mean temperatures are tabulated in Table 3. For the four materials with temperature oscillation, their powers are always higher than those without oscillation. These phenomena are in line with the past study [37]. However, Material B at the mean temperature of 150 °C gives an exception. This can be explained by the pronounced decrease in ZT at high temperatures (Fig. 2b). On the other hand, the temperature oscillation results in lower efficiency, except for Material C. These results are also consistent with the past observations [37]. However, for the thermal conductivity of Material C shown in Fig. 1c, it drops obviously with increasing temperature, which further lowers the heat flux at the hot surface. As a result, the lowered Qh arises the efficiency (=P /Qh ) of Material C.

4. Conclusions The performances of thermoelectric generators with constant and variable properties of materials have been comprehensively analyzed, while four different materials of Bi2Te3 (Material A), (Bi0.2Sb0.8)2Te3 (Material B), Cu11Ni1Sb4S13 (Material C), and Ag0.93Sb1.07Te2.07 (Material D) have been considered. Overall, Bi2Te3 renders the largest power, whereas (Bi0.2Sb0.8)2Te3 shows the best efficiency except at higher temperature differences (ΔT > 150 °C). In the thermoelectric elements, the Fourier heat flux dominates the entire heat flux, followed by the Peltier heat flux and the Joule heat flux. For Material A with constant properties, its performance matches the theoretical results, abiding by the ΔT2 law, self-consistency, and impedance matching. For

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o

o

4

Material

0.6

0.4

Material

3.5

A B C D

0.5

Power (W)

(a) Mean temperature 50 C

(a) Mean temperature 50 C

A B C D

3

Efficiency (%)

0.7

0.3 0.2

2.5 2 1.5 1

0.1 0

0.5 0

10

20

30

Time (min)

40

50

0

60

o

3

(b) Mean temperature 100 C 6

Efficiency (%)

Power (W)

20

30

Time (min)

40

50

60

50

60

50

60

(b) Mean temperature 100 C

5

2 1.5 1 0.5

4 3 2 1

0

10

20

30

Time (min)

40

50

0

60

o

7

10

o

2.5

0

0

(c) Mean temperature 150 C

0

10

20

30

Time (min)

40 o

(c) Mean temperature 150 C 7

6

6

Efficiency (%)

Power (W)

5 4 3 2 1 0

5 4 3 2

0

10

20

30

Time (min)

40

50

60

1

Fig. 10. Distributions of output power at mean temperatures of (a) 50 °C, (b) 100 °C, and (c) 150 °C.

0

10

20

30

Time (min)

40

Fig. 11. Distributions of efficiency at mean temperatures of (a) 50 °C, (b) 100 °C, and (c) 150 °C.

Materials B, C, and D, they also obey the self-consistency. However, their performances depart from the ΔT2 law and impedance matching, and the deviation extent depends on the adopted materials. It is found that the smaller the R2 value in the power versus ΔT2 plot, the greater the deviation between the predictions and theoretical results from the impedance matching. The output power of Material A is independent of the hot or cold surface temperature at a given ΔT, which matches the

theoretical formula. The power output of Materials B and D increases when their cold surface temperatures are lowered. This is consistent with the experimental observations. However, an opposite trend develops for Material C. Depending on the material’s properties, the oscillation in the hot surface temperature will increase or decrease the output power and efficiency of the generator. Accordingly, it should be 22

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(a)

practical insights into the applications of thermoelectric generators in industry.

5

Mean power (W)

4

Acknowledgments

Material

A B C D

3

The authors acknowledge financial support from the Ministry of Science and Technology under the grant number MOST 107-3113-E006-010- and the Bureau of Energy, Ministry of Economic Affairs, Taiwan, ROC, for this research.

2

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150

7

Mean efficiency (%)

6 5 4 3 2 1 0

50

100

Mean TH (oC)

150

Fig. 12. Profiles of (a) mean power and (b) mean efficiency. Table 3 Performances of TEG with and without temperature oscillation at three mean temperatures. Mean temperatures

Power (W)

Efficiency (%)

Without oscillation

With oscillation

Without oscillation

With oscillation

Material A 50 °C 100 °C 150 °C

0.172 1.550 4.305

0.226 1.604 4.359

1.063 3.093 5.000

1.059 3.088 4.995

Material B 50 °C 100 °C 150 °C

0.161 1.283 2.958

0.206 1.305 2.946

1.616 4.080 5.411

1.577 4.035 5.361

Material C 50 °C 100 °C 150 °C

0.010 0.101 0.310

0.013 0.105 0.315

0.289 0.920 1.585

0.292 0.921 1.585

Material D 50 °C 100 °C 150 °C

0.059 0.511 1.400

0.077 0.527 1.417

0.704 2.016 3.310

0.698 2.015 3.310

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