Evaluation method for assessing heat transfer enhancement effect on performance improvement of thermoelectric generator systems

Applied Energy 263 (2020) 114688

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Evaluation method for assessing heat transfer enhancement effect on performance improvement of thermoelectric generator systems

T

Yurong Yang, Shixue Wang , Yu Zhu ⁎

School of Mechanical Engineering, Tianjin University, PR China Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Ministry of Education, Tianjin University, Tianjin 300350, PR China

HIGHLIGHTS

evaluation method for thermoelectric generator systems was proposed. • An proposed index was theoretically derived and validated with adequate accuracy. • The proposed evaluation method is convenient and time-efficient for engineering application. • The • The application of the proposed index was analyzed by case studies. ARTICLE INFO

ABSTRACT

Keywords: Thermoelectric generator Heat transfer enhancement Net power Evaluation method

The total power output of a thermoelectric generator system can be increased by enhancing the heat transfer performance of its hot-side heat exchanger. However, heat transfer enhancement is usually accompanied by the consumption of additional pump power. Therefore, it is unclear whether the net power output, that is, the difference between the total power output and the pump power, increases. Developing a comprehensive evaluation method based on the net power output is necessary to determine the heat transfer enhancement effect on performance improvement of the system. In this paper, the concept of net power ratio which integrates the total power output and the pump power consumption was proposed to evaluate the heat transfer enhancement effect on the performance improvement of a thermoelectric generator system. Based on a ring-shaped thermoelectric generator, the analytical solution of the net power ratio was theoretically derived. Compared with the numerical and experimental methods, the proposed net power ratio in a thermoelectric generator system is convenient and time-saving with adequate accuracy for engineering applications. Moreover, the application of net power ratio in a real thermoelectric generator system was investigated by case studies. Results show that both the inlet fluid temperature and the mass flow rate affect the net power ratio. When using net power ratio to evaluate the performance improvement of a thermoelectric generator system, net power should also be considered to obtain optimal working conditions for a thermoelectric generator system.

1. Introduction With the rapid development of industrialization and drastic increase in population, the world is currently facing an energy crisis. Maximizing energy utilization is critical for sustainable development [1]. However, with low energy utilization and severe energy wastage, the status quo of energy utilization worldwide is poor. A latest report [2] has shown that 20–50% of energy in the transportation, power, industrial, and residential sectors is released into the atmosphere as heat. To improve energy efficiency and save energy, waste heat recovery

⁎

technologies have received increased attention in recent years [3]. Among them, thermoelectric technology based on a thermoelectric generator (TEG) can be used to directly convert thermal energy into electrical energy [4]. Compared with other waste heat recovery devices, the TEG is small, lightweight, free of moving parts or liquid medium, and is safe and reliable [5]. TEGs are all-solid environment-friendly waste heat recovery devices [6]. Sargolzaeiaval et al. [7] designed a flexible TEG that harvests thermal energy from the human body and converts it into electrical energy to power wearable electronics. Fernández-Yáñez et al. [8] tested the performance of the same TEG in a diesel and a gasoline engine. Their results show that up to ~0.6% and

Corresponding author. E-mail address: [email protected] (S. Wang).

https://doi.org/10.1016/j.apenergy.2020.114688 Received 15 September 2019; Received in revised form 13 February 2020; Accepted 14 February 2020 0306-2619/ © 2020 Published by Elsevier Ltd.

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Y. Yang, et al.

Nomenclature

Subscripts

Symbols

0 C cav cin cout f fav fin fout H L m

cross-sectional area of air pipe, m2 specific heat capacity, J kg−1 K−1 diameter of air pipe, m electromotive force, V Darcy drag coefficient cross-sectional area of a PN couple, m2 heat transfer coefficient, W m−2 K−1 current, A length of air pipe, m mass flow rate, kg s−1 number of PN couples Nusselt number dimensions of PN couple (length/width/height), mm pressure drop, Pa power output, W flow power consumption, W heat flow rate, W thermal resistance, K W−1 internal resistance, Ω load resistance, Ω temperature, K air flow rate, m s−1 volume flow, m3 s−1 figure of merit, K−1

A cp D E f F h I L m N Nu l/w/h Δp P ΔP Q R Rn Rw T u V Z

Greek symbols α η λ ρ ρf

TH

TL TH

ZTH ZTm + ZTH + 4

seebeck coefficient, V K−1 conversion efficiency, % thermal conductivity, W m−1 K−1 resistivity, Ω m air density, kg m−3

Abbreviations NPR TEM TEG PPI

1.1% of fuel savings under common driving conditions can be obtained using the TEG for diesel and gasoline engines, respectively. Ma et al. [9] used segmented TEGs to recover engine waste heat, and through segmented leg length ratio optimization, the power output in the TEG system increased by 6.8%. Tahami et al. [10] developed a TEG system that utilizes the thermal gradients between the pavement surface and the soil below the pavement to convert it to electricity. TEGs are often integrated with photovoltaic cells as a system to recover solar energy [11]. However, the low power generation efficiency of TEG systems limits their application and development [12]. A TEG ideally operates at maximum power output, in which case the thermoelectric efficiency can be expressed as follows [13]:

=

cases before heat transfer enhancement cooling water average values of cooling water inlet cooling water outlet cooling water air average value of air inlet air outlet air hot side of PN couple cold side of PN couple middle of PN couple

net power ratio thermoelectric module thermoelectric generator pores per linear inch

cold-side fluid is usually water and the hot-side fluid is exhaust gases whose heat transfer coefficient is an order of magnitude lower than that of water [16]. Heat transfer enhancement measures are typically implemented in hot-side heat exchangers with high thermal resistance. Kim et al. [17] enhanced the heat transfer performance of the heat exchanger of an automobile TEG using a fin structure. Through numerical calculation, they found that an increase in the number and thickness of fins increases the temperatures of the exhaust pipe and the hot-side of the TEM accompanied by a gradual increase in the pressure drop across the pipe. Cao et al. [18] designed and tested a-heat-pipe assisted TEG with 36 TEMs for automobile exhaust heat recovery. Their experimental results showed that the maximum open circuit voltage was 81.09 V, the corresponding power output and voltage drop were 13.08 W and 1657 Pa, respectively, and the TEG’s optimal thermoelectric power generation efficiency was 2.58%. Lesage et al. [19] enhanced the heat transfer performance of a hot-side heat exchanger by incorporating three spoiler elements in the flow path and then measured the pressure drop. Their results showed that the net power output of the TEG system with a spiral element was the lowest, whereas that of the TEG system with a convex panel was the highest. Li et al. [20] experimentally studied a TEG system using automobile exhaust and improved heat transfer efficiency using metal foam in the hot-side heat exchanger. When the inlet air temperature was 300 °C and the flow rate was 120 m3/h, the metal foam with a pore density of 20 PPI and a porosity of 75% quadrupled the convective heat transfer coefficient of the channel and doubled the power output of the system. However, the metal foam increased the pressure drop across the pipe. Under this condition, the pressure drop was 17 kPa. Wang et al. [21] also used metal foam to study the heat transfer performance of a heat exchanger used in a TEG system. To minimize the adverse effect of flow pressure drop on the system performance, a metal foam with a low pore density and a high porosity was used. Lu et al. [22] demonstrated that a TEG with metal foam gave a higher power output and efficiency, as well as a higher pressure drop, than a TEG with rectangular fins, particularly

(1)

where TL and TH are the hot- and cold-side temperatures of the thermoelectric module (TEM), respectively; ZTH is the figure of merit at temperature TH ; and ZTm is the figure of merit at the average temperature Tm (Tm = (TH + TL ) 2 ). According to this equation, the thermoelectric conversion efficiency of a TEG can be improved by increasing the figure of merit ZT or the temperature difference between the hot and cold sides of the TEM. Poudel et al. [14] improved the bismuth antimony telluride (BiSbTe) bulk alloy to a nanocrystalline BiSbTe bulk alloy and obtained high-thermoelectric performance. Compared with the BiSbTe bulk alloy, the dimensionless ZT increased from 1.0 to 1.4. Perumal et al. [15] achieved a high ZT of ~2.1 in GeTe at 723 K through the complementary co-doping of Bi and In. The thermoelectric efficiency reached 12.3% for a single-leg thermoelectric device of In and Bi co-doped GeTe at a temperature difference of ~445 K. In addition to ZT, the temperature difference between the hot and cold ends of the module also significantly affects the overall performance of TEG. For automobile exhaust heat recovery applications, the 2

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at high mass flow rates. Massaguer et al. [23] investigated the performance of an automotive thermoelectric generator and revealed that the total power generation of the TEG can be significantly improved by maximizing the heat transfer through TEMs using a finned geometry. In addition, they demonstrated that the pressure drop plays a key role in the design of automotive TEGs for practical applications. Although the power output of a TEG system can be improved by implementing heat transfer enhancement measures on the heat exchanger, this method is accompanied by an increase in flow pressure drop, resulting in an increase in pumping power. It is necessary to establish a comprehensive evaluation method considering the total power generation and power consumption of the TEG to evaluate the heat transfer enhancement effect on TEG performance improvement. Wang et al. [24] proposed a performance evaluation index (P/P0)/ [(Δp/Δp0)(R/R0)] for TEG systems. In this index, P represents the power output, Δp represents the pressure drop, R represents the total thermal resistance, and parameters with the subscript “0” correspond to the case before heat transfer enhancement. Wang et al. [24] concluded that the greater the evaluation parameter, the more the system performance improved by heat transfer enhancement. However, the evaluation parameters cannot provide a critical point at which the increase in power output owing to heat transfer enhancement is offset by the flow power consumption. Borcuch et al. [25] observed that a fin structure generates a high system power consumption; therefore, they used the net power, removing non-negligible power consumption, to evaluate the effect of heat transfer enhancement of different heat transfer enhancement structures on TEG performance under different operating conditions. Ma et al. [26] studied the effect of longitudinal vortex generators on TEG performance using net power through numerical simulation. Their results showed that the net power of longitudinal vortex generator channels is 59–150% greater than that of a smooth channel. Choi et al. [27] conducted experiments on the heat transfer enhancement of a hot-side heat exchanger in a TEG with perforated plates. A maximum back pressure of 3 kPa was chosen as the standard limit to evaluate the feasibility of the heat transfer enhancement measures. Herein, the concept of net power ratio (NPR), which integrates the total power output and flow power consumption, was used to evaluate the heat transfer enhancement effect on the performance improvement of a TEG system was proposed. Based on a ring-shaped TEG, an analytical solution of NPR was derived and validated by numerical and experimental methods. Compared with numerical and experimental methods, the proposed NPR is convenient and time-saving with adequate accuracy for application. This study also investigated the applications of NPR in real TEG systems through case studies.

NPR is the ratio of the net power of the TEG system after heat transfer enhancement of the hot-side heat exchanger to the net power of the TEG system before heat transfer enhancement, as shown in Eq. (2). When the NPR is greater than 1, the net power of the system after heat transfer enhancement is greater than that before. When the NPR is less than 1, the net power of the system after heat transfer enhancement is smaller than that before.

NPR =

Pnet Pnet0

(2)

where the parameter with subscript 0 corresponds to the case before heat transfer enhancement, and the parameter without any subscript corresponds to the case after heat transfer enhancement. The following parameters use the same denotation. The flow power consumption induced by the flow pressure drop is , and the total power output of the TEG system is P . The net power P. output Pnet of the TEG system can be expressed as Pnet = P Detailed calculation of NPR for a TEG system will be described in the following sections. 3. Mathematical model and analytical solution 3.1. Model description and assumptions Based on a cylindrical heat source pipeline of a TEG system, we selected ring-shaped TEMs for the analysis conducted in this study. Fig. 1 shows a schematic of the TEG model. Air, which is the heat source of the TEG system, flows through the inner tube. The error associated with the use of air instead of exhaust gas is usually less than 2% [28]. The cold source of the system is cooling water, which flows along the outermost annular casing. The hot and cold fluids flow in the same direction. A ring module group is situated between the cold and heat source channels. The calculations performed in this study are based on the following assumptions. (1) The influence of copper sheets and ceramic substrates used in an actual TEG is not considered. (2) The temperature change of the module in the flow direction is not considered. (3) The fluid temperature change in the flow direction is linear. (4) The load resistance of the TEG is equal to its internal resistance. (5) The inlet temperature, mass flow rate, and fluid properties of the hot and cold fluids before heat transfer enhancement are the same as those after. (6) The parameters of the TEM do not change with temperature. (7) All the PN junctions are connected in series. (8) The influence of the Thomson effect is ignored.

2. Concept of net power ratio In this paper, the concept of net power ratio (NPR) is proposed to evaluate the heat transfer enhancement effect on the performance improvement of a TEG system. For automobile exhaust heat recovery applications, the fluid in the cold-side heat exchanger is usually water, and the fluid in the hot-side heat exchanger is exhaust gas whose heat transfer coefficient is an order of magnitude lower than that of water. Herein, we focus on hot-side heat exchangers with poor heat transfer.

HZ-20-type thermoelectric material (Hi-Z Company) [29] was used in the study. This thermoelectric material is a commonly used and commercially available Bi-Te based material. Table 1 lists the basic calculation parameters of the TEM. Fig. 1. Thermoelectric generator (TEG) model and P/N element structure.

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Table 1 Basic calculation parameters of the TEM.

Table 3 Basic calculation parameters of fluids.

Parameters

P-type semiconductor leg

N-type semiconductor leg

Seebeck coefficient α (V K−1) Resistivity ρ (Ω m) Thermal conductivity λ (W m−1 K−1) Dimensions of PN couple (length/width/height) l/w/h (mm)

2.037 × 10−4 1.314 × 10−5 1.265

−1.721 × 10−4 1.119 × 10−5 1.011

5/5/5

5/5/5

Parameters

Values

Inlet cooling water temperature Tcin (K) Mass flow rate of cooling water mc (kg s−1) Heat transfer coefficient of the cold-side heat exchanger hc (W m−2 K−1) Specific heat capacity of air cp,f (J kg−1 K−1) Specific heat capacity of cooling water cp,c (J kg−1 K−1)

298.15 0.025 1000 1010 4200

3.2. Governing equations A mathematical model was established for the heat transfer and power generation process of the TEG system. The heat flow rate at the hot and cold sides of the TEM can be expressed as the sum of the heat flow rates due to the Peltier, Fourier, and Joule effects, as follows [30]:

QH = N [ QL = N [

pn ITH pn ITL

+ Kpn (TH + Kpn (TH

TL )

0.5I 2Rpn]

(3)

TL ) +

0.5I 2Rpn]

(4)

where QH is the heat flow rate at the hot side of the module; QL is the heat flow rate at the cold side of the module; N is the number of PN couples in the TEG; pn is the Seebeck coefficient of the PN couple, which is equal to the difference between the Seebeck coefficient of the P- and N-type semiconductor legs; TH is the hot-side temperature of the module; TL is the cold-side temperature of the module; I is the current; Kpn is the thermal conductivity of a PN couple; and Rpn is the resistance of a PN couple. The amount of heat released or absorbed by the heat exchanger can be expressed by Newton’s law of cooling, as follows:

Fig. 2. Comparison of the present and numerical results of power output.

TH)

(5)

Tcav )

(6)

QH = Nh f FH (Tfav QL = Nhc FL (TL

where hf and hc are the heat transfer coefficients of the hot and cold sides of the heat exchanger, respectively; FH and FL are the areas of the hot and cold sides of a PN couple, respectively; Tfav is the average temperature of the air in the pipe, which is equal to the arithmetic mean of the inlet air temperature Tfin and the outlet temperature Tfout (Tfav=(Tfin + Tfout ) 2 ); and Tcav is the average temperature of the cooling water in the pipe, which is equal to the arithmetic mean of the cooling water inlet temperature Tcin and the outlet temperature Tcout (Tcav=(Tcin + Tcout ) 2 ). According to the law of conservation of energy, the amount of heat released or absorbed by the heat exchanger can also be expressed in terms of the difference of enthalpy before and after the fluid flows through the pipe. Fig. 3. Comparison of the present and experimental results of power output. Table 2 Structural parameters of the hot-side heat exchanger. Parameters

Case 1

Case 2

Case 3

Case 4

D (m) y/w θ (°) n e/D P/e e/H P/H

0.064 4.0 0.9 – – – – –

0.068 2.0 – 1 – – – –

0.060 – – – 0.04 20 – –

0.055 – – – – – 0.1 2

QH = m f cp,f (Tfin

Tfout )

(7)

QL = mc cp,c (Tcout

Tcin )

(8)

where m f and mc represent the mass flow rates of air and cooling water, respectively, and cp,f and cp,c denote the specific heat capacities of air and water, respectively. According to the Seebeck effect, the electromotive force E generated in the loop can be expressed as follows [31].

E=N

pn (TH

TL )

(9)

According to the Joule effect, the electromotive force E is the product of the current I and the sum of internal resistance Rn and load resistance Rw in the circuit. Rn is the total resistance of N pairs of PN couples (Rn = N ·Rpn ). The power output P is the product of the square of the current I and load resistance Rw .

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Table 4 Correlations of Nu and the Darcy resistance coefficient f. Case No.

Before heat transfer enhancement

Case 1

Nu =

(f 8)(Re 1000) Pr 1 + 12.7 f 8 (Pr 2 3 1)

Re=2300~106 f = (0.79 ln Re

Case 2

1+

()

D 2 3 L

After heat transfer enhancement

Nu = 0.076Re 0.75Pr 0.4 (y w )

,

1.64) 2 , Re = 3000~5 × 106

0.49 (y

f = 16.559Re

Nu = 0. 023Re 0.8Pr 0.4, Re = 10 4~3.5 × 10 4

0.39 (1

0.51 (1

w)

+ )

+ )

0.53 ,

0.1,

Re = 6000~2 × 10 4

Re = 6000~2 × 10 4

Nu = 0.09653Re 0.7834n0.0551 exp(0.0256 ln n2)(y w )0.2127exp( 0.1696 ln TR2), Re=6000~2.3 × 10 4

f = 0.31Re

0.25,

Re = 10 4~3.5 × 10 4

f = 1.073Re

0.0786n0.0352 exp(0.1094 ln n2)(y

w ) 0.4696exp( 0.3760 ln TR2),

Re=6000~2.3 × 10 4

Case 3

Nu =

(f 8)(Re 1000) Pr 1 + 12.7 f 8 (Pr 2 3 1)

Re=2300~106 f = (1.82 lg Re Case 4

1+

()

D 2 3 L

Nu =

,

e+ 1.64)

2,

Re = 3000~5 ×

106

0.25,

= (e D) Re f 2 ,

e+

0.95(p e )0.53]

,

> 35, Re = 6000~105

f = 2[2.5ln(D/2e ) + 0.95(P /e )0.53 3.75] 2 , Re = 6000~105

Nu = 0. 023Re 0.8Pr 0.4, Re = 3000~2 × 10 4

f = 0.316Re

(f 2) RePr 1 + f 2 [4.5(e+)0.28Pr 0.57

Nu = 0.290236Re 0.669891Pr 0.4 (e H + 1) 4.854282 (P H + 1)

Re = 3000~2 × 10 4

0.0726 (e

f = 0.237223Re

H + 1)9.859262 (P H + 1)

0.86932 ,

0.47904 ,

Re = 4400~2.04 × 10 4

Re = 4400~2.04 × 10 4

Fig. 4. Net power ratio (NPR) of the TEG system versus inlet air temperature: (a) Case 1, mfin = 20 g/s; (b) Case 2; mfin = 25 g/s; (c) Case 3, mfin = 20 g/s; and (d) Case 4, mfin = 40 g/s.

E = I (Rn + Rw )

(10)

P = I 2Rw

(11)

p=f

According to Eqs. (9)–(11), assuming Rn = Rw , the power output of the TEG system can be expressed as follows:

P=

N

pn

2 (T H

4Rpn

L · D

f uf

2

2

(13)

P = Vf × p

(14)

where f is the Darcy drag coefficient; L is the length of the pipe; D is the diameter of the pipe; f is the air density; and Vf is the air volume flow, which is a product of the cross-sectional area Af of the pipe section and the air flow rate u f .

- TL )2 (12)

For the heat exchanger of the TEG system, the pressure drop p and frictional power consumption P in the pipeline can be expressed as follows [26]: 5

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Fig. 8. NPR variation with air mass flow rate in Case 3.

Fig. 5. Comparison of NPR and (P/P0)/[(Δp/Δp0)(R/R0)] in Case 1.

Fig. 9. Net power variation with air mass flow rate in Case 3.

Fig. 6. NPR variation with air mass flow rate in Case 1.

Fig. 7. NPR variation with air mass flow rate in Case 2.

Fig. 10. NPR variation with air mass flow rate in Case 4.

3.3. Expression of net power ratio

the heat transfer enhancement are the same as those after. According to Eqs. (2) and (12)–(14), the analytical solution of NPR can be expressed as follows:

The object on which heat transfer enhancement was to be investigated is the hot-side heat exchanger of the TEG system; the coldside heat exchanger remained unchanged. The mass flow rate and physical properties of the heat source fluid, pipe parameters, and number of PN couples together with their installation methods before

NPR =

6

M (T f0 H M (T f0 H0

TL) 2 TL0) 2

f f0

1

(15)

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Fig. 3 shows a slight deviation between the analytical and experimental results. This may mainly result from the assumptions of constant property parameters of the TEM and the linear temperature change of the exhaust gas along the flow direction in the analytical method, which do not agree with practical scenarios. However, the maximum deviation between the analytical and experimental results is 5.7%, which is relatively small. Thus, the analytical method was deemed reliable. 4. Case studies, results, and discussion 4.1. Case description Four cases were studied using the proposed NPR to evaluate the performance improvement of a TEG system by enhancing heat transfer of a hot-side heat exchanger. The models and correlations of Cases 1–4 were adopted from [33–36], respectively. The length of the air pipe is set to 2 m [37]. Table 2 lists other structural parameters of the hot-side heat exchanger of the TEG system. The hot and cold sources of the system are air and water, respectively. Table 3 lists the basic calculation parameters for hot and cold fluids. Table 4 lists the correlations of Nu and the Darcy resistance coefficient f before and after heat transfer enhancement.

Fig. 11. Net power variation with air mass flow rate in Case 4. N pn2D

where M =

2Rpn LAf f uf 3

.

A simplified expression of the temperature difference between the hot and cold sides of the module can be derived from Eqs. (3)–(8).

TH

a f X1 X2 + a f X3

TL

where X1 = ac (Tfin

af =

2cp,f mf hf FH 2cp,f mf + Nhf FH

4.2. Net power ratio characters with different inlet air temperature

(16)

Tcin ),

, and ac =

X2 = ac Kpn + 2cp,c m c h c FL 2cp,c mc + Nh c FL

2 pn Tcin 2Rpn

, X3 = ac + Kpn +

2 pn Tfin

2Rpn

An exhaust gas temperature range of 100–400 °C and a mass flow rate range of 7–40 g/s [38] were considered in this study to reflect the exhaust gas conditions. Fig. 4 shows that NPR is affected by the inlet air temperature. As shown in Fig. 4(a) and (b), the net power may increase (NPR > 1) or decrease (NPR < 1) after heat transfer enhancement when the air temperature is varied for the same heat transfer enhanced TEG structure. In Case 1, the net power will increase when air inlet temperature is above 107 °C, and in Case 2, the net power will increase when the air temperature is greater than 131 °C. Fig. 4(c) and (d) exhibit a relatively good heat transfer enhancement effect on the improvement of TEG net power as NPR is always greater than 1. Considering Case 1 as an example, Fig. 5 compares the NPR and the index (P/P0)/[(Δp/Δp0)(R/R0)] proposed in [24] to evaluate the heat transfer enhancement effect on the performance improvement of a TEG system. As shown in Fig. 5, compared with the index proposed in [24], the NPR is advantageous in determining whether the net power of the TEG system is enhanced after heat transfer enhancement at a certain intake temperature.

,

. See Appendix A for a detailed

derivation. Furthermore, Eq. (15) can be rewritten as follows:

( NPR = M( M

) )

2 af X1 X2 + af X3

af0 X1 X2 + af0 X3

2

f f0

(17)

3.4. Model verification 3.4.1. Numerical verification The analytical solution method was verified by comparing the power output with the numerical data of He et al. [32]. An automotive thermoelectric generator with a hot-side exchanger recovering heat from exhaust gas and a cold-side exchanger releasing heat to the ambient air was investigated in their study. In the verification performed herein, the same parameters as those used in the numerical study were adopted for calculation. Fig. 2 shows the analytical and numerical results in terms of maximum power output at different temperatures of inlet exhaust gas. As shown in Fig. 2, the maximum deviation between the present and the numerical results is 0.85%. The small discrepancies may be attributed to the assumption of the linear temperature change of the exhaust gas along the flow direction and the simplification of the electric current term in the proposed calculation method. These results demonstrate relatively good agreement with an acceptable extent of deviation.

4.3. Net power ratio with different air mass flow rates Fig. 6 shows the results of the NPR as a function of air mass flow rate in Case 1. It indicates that when the inlet air temperature is 125 °C or higher, or the air flow rate is less than 17.4 g/s, the net power increases through enhanced heat exchange. Similarly, Fig. 7 shows that when the inlet air temperature reaches or exceeds 200 °C or the air flow rate is less than 18.0 g/s, the net power increases after heat transfer is enhanced. In a TEG system, NPR should be used considering the temperature and mass flow rate variation of the hot fluid. For example, in Case 2, the net power increases after the heat transfer is enhanced when the inlet air temperature reaches or exceeds 100 °C and the air flow rate is less than 18.0 g/s; the net power also increases when the inlet air temperature reaches or exceeds 150 °C and the air flow rate is less than 28.9 g/s. By combining the NPR and the temperature and mass flow rate of the hot fluid, working conditions can be obtained wherein the net power of the TEG system will increase after heat transfer enhancement. Fig. 8 shows NPR variation with the inlet air flow rate in Case 3. All the NPRs are greater than 1 under the conducted operating conditions, indicating that the net power increases after heat transfer enhancement

3.4.2. Experimental verification The proposed analytical method was also verified using the experimental data of Choi et al. [27]. In their study, the performance of a TEG system was investigated before and after enhancing the heat transfer of the hot-side heat exchanger using perforated plates at engine speeds of 1000, 1200, and 1400 rpm. To verify the proposed analytical method, the same parameters as those of the experimental apparatus were used for calculation. Fig. 3 compares the maximum power outputs of the analytical and experimental results, including the results without a heat transfer enhancement plate and those with a perforated plate of type A at different engine rotation speeds. 7

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in Case 3. However, this does not mean that larger net power of a TEG system can be obtained using a larger NPR. This can be explained by Fig. 9, which shows the results of net power at the air inlet temperatures of 120 °C and 140 °C. Therefore, the parameter of net power should also be considered in the performance improvement of a TEG system. Fig. 10 shows NPR variation with the inlet air flow rate in Case 4. The NPR is greater than 1 under all operating conditions, implying that the net power increases after heat transfer enhancement in Case 4. At the same mass flow rate, the NPR is small when the inlet air temperature is high. However, in terms of the amount of net power, hightemperature conditions are more favorable, as shown in Fig. 11. These results also indicate that the use of the NPR in practical applications should be combined with the net power to determine optimal working conditions.

is demonstrated to be a convenient and time-saving technique with adequate accuracy. Four cases were studied herein to illustrate the application of the net power ratio in a real thermoelectric generator system. Results show that the net power ratio is affected by the inlet air temperature and air mass flow rate. Compared with other published evaluation indexes, the net power ratio has advantages in that it can determine whether the net power of a thermoelectric generator system increases after heat transfer enhancement at a certain inlet temperature. The results also show that the net power ratio should be used in combination with net power to obtain optimal working conditions for a thermoelectric generator system. CRediT authorship contribution statement Yurong Yang: Investigation, Methodology, Validation, Writing original draft. Shixue Wang: Conceptualization, Data curation, Writing - review & editing. Yu Zhu: Data curation, Writing - review & editing.

5. Conclusion The performance of a thermoelectric generator system can be improved by enhancing the heat transfer performance of its hot-side heat exchanger. However, this method is accompanied by an increase in flow pressure drop, resulting in an increase in pumping power. In this study, we proposed the concept of net power ratio, which integrates the total power output and flow power consumption of a thermoelectric generator system to evaluate the heat transfer enhancement effect on the improvement in system performance. Based on a ring-shaped thermoelectric generator, the analytical solution of the net power ratio was theoretically derived and validated by comparing the analytical results with numerical and experimental data. Comparing the numerical and experimental methods, using the net power ratio to define the change in the net power of a thermoelectric generator system

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The authors are grateful for the financial support from the International Cooperation Research Program of MOST (2017YFE0198000) and the National Natural Science Foundation of China (51876136).

Appendix A The following correlation can be obtained from Eqs. (3)–(8).

TH - TL =

pn Rpn I

3

af ) Rpn I 2

+ 0.5(ac pn

2I 2

+

pn (ac

pn (a f Tfin

+ ac Tcin ) I + af ac (Tfin

Tcin ) (A.1)

af ) I + Kpn ac + a f ac + a f Kpn

The polynomial form of Eq. (A.1) is as follows:

TH - TL =

aI 3 + bI 2 + cI + d eI 2 + fI + g

(A.2)

2 af ) Rpn , c = Tcin ), e = af ), and g = Kpn ac + af ac + a f Kpn. where a = pn , f = pn (ac pn Rpn , b = 0.5(ac pn (a f Tfin + a c Tcin ), d = a f a c (Tfin Considering Tfin = 400 °C, mfin = 20 g/s in Case 1 as an example, the order of the coefficient terms of the polynomial was analyzed. Table A.1 lists the order of the magnitude of each coefficient term of the polynomial. To simplify the calculation, Eq. (A.2) can be transformed to Eq. (A.3) by ignoring the terms with relatively low values.

TH

TL

c d af X1 I+ = g g X2 + a f X3

where X1 = ac (Tfin

Tcin ), X2 = ac Kpn +

(A.3) 2 pn Tcin

2Rpn

, X3 = ac + Kpn +

2 pn Tfin

2Rpn

.

Table A1 Polynomial coefficient terms in Case 1. Coefficient terms

Variables in Eq. (A2)

a b c d e f g

1.832 × 10−6 3.909 × 10−5 −2.166 × 10−3 6.458 × 10−3 −1.412 × 10−7 6.028 × 10−6 2.224 × 10−4

8

Applied Energy 263 (2020) 114688

Y. Yang, et al.

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