## 1. Introduction

Due to the worldwide shortage of oil resources and environmental issues, energy saving and automotive emission reduction have attracted much attention in recent years. For the traditional internal combustion engine (ICE), 30–45% heat from fuel combustion is discharged into the surroundings by exhaust gas, which causes a great waste of energy and accelerates the deterioration of the environment [

1,

2,

3]. One of the best solutions for these problems is to recover waste heat contained in exhaust gas. Vazquez et al. [

4] indicates that if 6% of waste heat is recovered from exhaust gas, the fuel consumption rate of an ICE would decline by 10%.

Liang et al. [

5] summarized all existing waste heat recovery technologies, such as Organic Rankine Cycle (ORC), turbocharging, etc. As for ORC, in spite of its relatively high thermal efficiency, it is not currently suitable for vehicle use because of its complicated structure and large size. Due to the requirement of relatively high exhaust gas temperature, turbocharging only have potential for the fuel saving under ICE high-load condition [

6]. Liang et al. [

5] concluded that with lots of advantages such as its simple structure, no moving parts, environment friendliness, extremely low noise, and long working hours, the thermoelectric generator (TEG) is one of the most promising methods for recovering ICE waste heat in the future.

Many studies on TEG technology for ICE waste heat recovery have been conducted. Robert J. Stevens et al. [

7] designed a thermoelectric generating system on the exhaust pipe of a heavy-duty truck, and this system could recover 16% of waste heat of the exhaust with a conversion efficiency of 3–5%. Liu et al. [

8] constructed a new automotive exhaust-based thermoelectric generator called the “four-TEGs” system and assembled it into the prototype vehicle called “Warrior”. A performance of 201.7 V (open circuit voltage)/944 W was obtained with a system efficiency of 1.85%. Car manufacture Hi-Z in the US succeeded in recovering waste heat from the exhaust using thermoelectric modules (TEMs). However, experimental result showed that the conversion efficiency was lower than 5% [

9]. It can be found from these studies that thermoelectric generating technology is confined to a low conversion efficiency of approximately 5% in general. Thus, main challenge of applying TEG technology for ICE waste heat recovery is to improve the conversion efficiency of TEG system.

Currently, the endeavor to improve the conversion efficiency of the TEG system has concentrated on two aspects. Some focused on the efficiency of the thermoelectric material. However, although thermoelectric materials with high performance have been developed [

10,

11,

12,

13], commercial high-efficient thermoelectric modules are still not available. Another way is to maximize the temperature gradient across the embedded thermoelectric modules through system configuration optimization and heat transfer process intensification. Ahmet Z. Sahin et al. [

14] studied effect of the leg length and width of TEMs on power output and conversion efficiency for the generating system with Bi

_{2}Te

_{3} materials. Zhiqiang Niu et al. [

15] established a 3D model to assess impact of the size of exhaust flow channel and the angle of fins in heat exchanger on TEG system. Simon Belanger et al. [

16] optimized the internal structure of a commonly used heat exchanger and investigated the effect of TEMs quantity and circuit current on output performance of TEG system. Tongcai Wang et al. [

17] designed a new heat exchanger filled with metal foam for TEG system, which realized a high heat transfer efficiency of 83.6% between hot air and coolant and increased output voltage. Yuchao Wang et al. [

18] indicated that water-cooling was more competitive than air-cooling in a TEG system. Moreover, considering the limited size of exhaust flow channel and pressure, as well as the difficulty of enhancing heat transfer between the high-temperature exhaust and hot side of TEMs, more efforts should be put on the heat transfer enhancement between cold side of TEMs and coolant.

As to enhance heat transfer between cold side of TEMs and coolant, applying nanofluid as coolant is an attractive method. Nanofluids refer to fluids containing thermally conducting nanometer-sized solid particles [

19], and are known to possess novel properties that make them have great potential in heat transfer applications. The promising features of nanofluids are high thermal conductivity due to much higher thermal conductivity of metals and metallic oxide nanoparticles than that of liquids (based fluid) and micro motions of nanoparticles in based fluid, which will lead to promising heat transfer performance of nanofluid. Meanwhile, due to their nanometer size, nanoparticles can be dispersed into the base fluids stably and thereby without problems such as abrasion and clogging. Thus, nanofluids can be treated as an excellent coolant in thermoelectric-based automotive waste heat recovery system. Since Choi et al. [

19] proposed the conception of nanofluid with excellent heat transfer performance, many studies on the application of nanofluid for heat transfer enhancement have been conducted in recent years. Tzeng et al. [

20] added CuO and Al

_{2}O

_{3} nanoparticles into engine oil, and the results showed that two types of nanofluid enhanced cooling effect of automotive power transmission system and avoided high thermal stress of components. Kulkarni et al. [

21] adopted Al

_{2}O

_{3}-EG nanofluid as a coolant for water jackets of a diesel generator, which obviously strengthened the cooling effect. S.M. Peyghambarzadeh et al. [

22] studied the cooling effects of pure water, pure EG, Al

_{2}O

_{3}-water nanofluid, and Al

_{2}O

_{3}-EG nanofluid for a vehicle radiator and results showed that nanofluid had a 40% increase on cooling effect than base fluid. Although there has been previous research on the heat transfer performance of nanofluid, to the best of our knowledge, the influence of nanofluid as coolant on the performance of TEG has not been previously investigated.

At present, the coolant used in engines is EG-W. In order to compare with traditional coolant, we choose EG as the base fluid. Cu, CuO, Al, and Al_{2}O_{3} are commonly used nanoparticles dispersed into EG-based fluid. Due to the much higher thermal conductivity of Cu-EG naofluid than the others, Cu-EG nanofluid was chosen as the new coolant in this study.

The present work studied the performance of a thermoelectric-based automotive waste heat recovery system with nanofluid coolant. Temperature distribution of thermoelectric modules, power output, and conversion efficiency for TEG system were obtained through simulating calculation. Affecting factors on the system performance, such as inlet temperature of exhaust gas, concentration of nanofluid, and total area of TEMs, are discussed in detail. Comparative analysis has also been carried out between Cu-EG nanofluid and traditional EG-W.

## 2. Mathematical Model

Waste heat contained in exhaust gas is transferred from the hot side to the cold side of TEMs and is finally taken away by the coolant. This heat transfer process is vital for the performance of the TEG system. Due to the complexity of this process, some hypotheses are made for simplifying the mathematical model as follows:

- (1)
The heat transfer process is steady;

- (2)
All the TEMs in the system are in series. Geometric configurations properties of P-type and N-type materials are identical, and physical properties of each P-type legs are identical, as well as the N-type legs;

- (3)
Air between TEMs and heat exchangers are omitted because it is quite small. Thermal radiation is not taken into consideration. Contact resistance between TEMs and heat exchangers, thermal resistance perpendicular to flow direction of fluid, and Thomson effect are also omitted;

- (4)
The external load resistance is equal to the internal resistance;

- (5)
Thermoelectric material used in this study is Bi

_{2}Te

_{3}, its thermoelectric parameters are constants, and

Table 1 presents the basic calculation parameters.

According to the different flow directions of the hot and cold fluid, there are two typical modes exist in the TEG system, namely, parallel flow mode and counter flow mode. These two modes are completely symmetrical in structure and perform under the same mechanism. Thus, only a parallel flow heat exchanger, namely, flow directions of exhaust and coolant are the same, is adopted for the TEG system in this study. The schematic of the TEG system is presented in

Figure 1a, and the mathematical model for the TEG system is shown in

Figure 1b. All the TEMs are divided into

n_{x} ×

n_{y} computation units, and each unit is a P-N junction. The direction of fluid is defined as

x direction, and along this direction there are

n_{x} rows, which are marked by

i (

i = 1, 2, …,

n_{x}). Similarly, the perpendicular direction is defined as

y, and there are

n_{y} columns that are marked by

j (

i = 1, 2, …,

n_{y}). A coordinate plane is formed based on the row and column number and the coordinate of computing unit can be represented as (

i,

j).

A P-N junction marked (i, j) along the y direction is chosen for calculation. The inlet temperature of exhaust gas is ${T}_{f}^{i,j}$ and the outlet temperature is ${T}_{f}^{i+1,j}$. The inlet temperature of coolant (Cu-EG nanofluid or EG-W) is ${T}_{c}^{i,j}$ and the outlet temperature is ${T}_{c}^{i+1,j}$. Due to the small cross-sectional area of a junction, the average value of the inlet temperature and outlet temperature is used to represent fluid (exhaust gas/coolant) temperature in this junction, namely, ${T}_{fav}^{i,j}=\left({T}_{f}^{i,j}+{T}_{f}^{i+1,j}\right)/2$ and ${T}_{cav}^{i,j}=\left({T}_{c}^{i,j}+{T}_{c}^{i+1,j}\right)/2$. Temperature of the hot side and cold side is ${T}_{H}^{i,j}$ and ${T}_{L}^{i,j}$, and heat flux across the hot side and cold side is ${q}_{H}^{i}$ and ${q}_{L}^{i}$, respectively. It should be pointed out that the outlet temperature of fluid in the former junction is equal to the inlet temperature of fluid in the latter junction. Apparently, along the y direction, all the P-N junctions in the same column have identical temperature distribution. Thus, each column along the y direction is considered as a computing unit.

TEMs convert the waste heat of exhaust into electricity, and the power output depends on the circuit current. According to the thermodynamic equilibrium theory, considering Joule effect, Peltier effect, and heat conduction loss of TEMs, governing equations of heat transfer process are as follows [

23,

24]:

In Equations (1) and (2), three terms on the right side represent Joule effect, Peltier effect, and heat conduction loss, respectively.

${\alpha}_{pn}$ is Seebeck coefficient of a P-N junction,

${K}_{pn}$ is the thermal resistance of a P-N junction, and

${R}_{pn}$ is electrical resistance of a P-N junction. These three parameters can be calculated as follows:

Because the thermal resistance of the heat exchanger is omitted, heat flux across the hot side and cold side of TEMs can be described by Newton’s cooling law as Formula (6) and (7), in which A denotes the cross section area of a P-N junction and ${h}_{f}$ and ${h}_{c}$ are heat transfer coefficient across the hot side and cold side of TEMs.

From the perspective of energy transport, it can be found that, due to the assumption of steady state, heat absorbed by the hot side of TEMs is equal to the heat loss of the exhaust. Likewise, heat loss of the cold side of TEMs is equal to the heat absorbed by the coolant. These relationships can be written as follows:

All the P-N junctions in the system are in series. Thus, current in each junction is identical to the total circuit current. Along the

y direction, the total output voltage provided by junctions in a column is calculated as follows:

The total open voltage for the system can be calculated as follow:

Total resistance in the circuit is,

As a consequence, circuit current can be calculated as follows according to Ohm’s Law:

In order to solve the governing equations, initial and boundary conditions should be prescribed. In this study, the initial inlet temperatures of exhaust and coolant (Cu-EG nanofluid or EG-W) are known, and boundary conditions are given as follows:

Note that in Formulas (6) and (7), the heat transfer coefficients across the hot side and cold side of TEMs, namely,

${h}_{f}$ and

${h}_{c}$ should be prescribed. In this study, an empirical value 80 W m

^{2} K

^{−1} is given for

${h}_{f}$ [

24]. The heat transfer coefficient between cold side of TEMs and coolants (Cu-EG nanofluid and EG-W) can be calculated according to following the heat transfer correlations proposed for Cu-EG nanofluid in laminar and turbulent conditions [

25]:

in which

$N{u}_{nf}$ denotes the Nusselt number of nanofluid, heat transfer coefficient

${h}_{c}$ can be calculated through the formula

$N{u}_{nf}=\frac{hD}{{k}_{nf}}$,

$P{e}_{d}$ denotes the Peclet number of nanoparticle

$P{e}_{d}=\frac{{u}_{m}{d}_{p}}{{\alpha}_{nf}}$, Reynolds number of nanofluid

$R{e}_{nf}=\frac{{u}_{m}D}{{\nu}_{nf}}$ and Prandtl number of nanofluid

${\mathrm{Pr}}_{nf}=\frac{{\nu}_{nf}}{{\alpha}_{nf}}$, thermal diffusivity of nanofluid

${\alpha}_{nf}=\frac{{k}_{nf}}{{\rho}_{nf}\cdot {c}_{p,nf}}=\frac{{k}_{nf}}{\left(1-\varphi \right){\rho}_{f}\cdot {c}_{p,f}+\varphi {\rho}_{d}\cdot {c}_{p,d}}$. Thermophysical properties of nanofluid are defined as follows [

26,

27]:

As can be seen in Equation (13), the current I was determined by the whole temperature distribution on the TEG system. Meanwhile, the temperature distribution was also affected by the current I. As can be seen in Equations (1) and (2), the current I, cold side temperature of TEMs and hot side temperature of TEMs coexist. Therefore, there is a coupled relationship between the temperature and the electric field. Since the temperature distribution of TEMs coupled with the circuit current, the iterative method presented in

Figure 2 is applied to solve the governing equations, at the beginning, the number of rows along

x direction

n_{x} and number of columns along

y direction

n_{y} are prescribed. Then, initial current value

I_{0} is given to solve the temperature distribution of the TEG system. According to the calculated temperature field, a new current

I can be obtained and this new current

I will be compared with the initial current

I_{0}. If these two value are not consistent, the new current

I is set as the initial current

I_{0} for the iterative calculation until the two values are consistent.

Power output and thermoelectric conversion efficiency are the two most important parameters to evaluate the performance of a thermoelectric-based waste heat recovery system. Thermoelectric conversion efficiency is defined as the ratio of total power output and heat absorbed from exhaust. These two parameters can be calculated by following formulas: