A hybrid transient CFD-thermoelectric numerical model for automobile thermoelectric generator systems

doi:10.1016/j.apenergy.2022.120502

Applied Energy 应用能源

Volume 332, 15 February 2023, 120502
第 332 卷，2023 年 2 月 15 日，120502

, Yuying Yan ^c,
Yuying Yan ^c 、, , , ,

^a: College of Electrical Engineering & New Energy, China Three Gorges University, Yichang, China
中国三峡大学电气工程与新能源学院，中国宜昌

^b: School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China
中国深圳，香港中文大学科学与工程学院

^c: Faculty of Engineering, University of Nottingham, Nottingham, UK
英国诺丁汉大学工程学院

^d: School of Automobile and Traffic Engineering, Jiangsu University, Zhenjiang, China
江苏大学汽车与交通工程学院，镇江，中国

Received 4 August 2022, Revised 25 November 2022, Accepted 5 December 2022, Available online 22 December 2022, Version of Record 22 December 2022.
2022 年 8 月 4 日收到，2022 年 11 月 25 日修订，2022 年 12 月 5 日接受，2022 年 12 月 22 日在线提供，2022 年 12 月 22 日记录版本。

https://doi.org/10.1016/j.apenergy.2022.120502 Get rights and content 获取权利和内容

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Highlights 亮点

•
A hybrid transient CFD-thermoelectric numerical model is proposed for the first time.
首次提出了一种混合瞬态 CFD 热电数值模型。
•
Dynamic output power keeps in a smoother change trend than exhaust temperature.
与排气温度相比，动态输出功率的变化趋势更为平稳。
•
Steady-state analysis overestimates output power and underestimates conversion efficiency.
稳态分析会高估输出功率，低估转换效率。
•
Conversion efficiency can be enhanced by periodic exhaust waste heat.
定期排出废热可提高转换效率。

Abstract 摘要

Dynamic performance prediction of the automobile thermoelectric generator system is one of the research hotspots in the field of thermoelectric technology. In this work, a hybrid transient CFD-thermoelectric numerical model is proposed for the first time to predict the dynamic response characteristics of an automobile thermoelectric generator system. Taking the exhaust gas of a heavy truck under a highway fuel economy test driving cycle as the transient heat source, the transient numerical study on the automobile thermoelectric generator system is carried out. It is found that the dynamic output power of the automobile thermoelectric generator system changes more smoothly than exhaust temperature due to the effect of thermal inertia, while the conversion efficiency fluctuates greatly. The transient output power at t = 467 s and transient conversion efficiency at t = 635 s reach the highest values of 45.16 W and 39.68 %, respectively. Within the period of 765 s, the total power generation and average conversion efficiency of the automobile thermoelectric generator system are 26460 J and 3.29 %, respectively. Through the transient experimental validation, the average error of transient output voltage between experimental and model results is 6.43 %. This work fills the gap in the dynamic performance prediction of thermoelectric devices used for fluid waste heat recovery. The findings are helpful in better understanding the dynamic response characteristics of the automobile thermoelectric generator system.
汽车热电发电机系统的动态性能预测是热电技术领域的研究热点之一。本研究首次提出了一种混合瞬态 CFD-热电数值模型，用于预测汽车热电发电机系统的动态响应特性。以重型卡车在高速公路燃油经济性测试行驶循环下的尾气为瞬态热源，对汽车热电发电机系统进行了瞬态数值研究。研究发现，由于热惯性的影响，汽车热电发电机系统的动态输出功率变化比排气温度变化更平稳，而转换效率波动较大。t = 467 s 时的瞬态输出功率和 t = 635 s 时的瞬态转换效率分别达到最高值 45.16 W 和 39.68 %。在 765 秒内，汽车热电发电机系统的总发电量和平均转换效率分别为 26460 J 和 3.29 %。通过瞬态实验验证，实验结果与模型结果之间的瞬态输出电压平均误差为 6.43%。这项研究填补了用于流体余热回收的热电设备动态性能预测方面的空白。研究结果有助于更好地理解汽车热电发电机系统的动态响应特性。

Graphical abstract 图形摘要

Based on the proposed hybrid transient CFD-thermoelectric model, we analyzed the dynamic response characteristics of an automobile thermoelectric generator (TEG) system under a complete driving cycle.
基于所提出的混合瞬态 CFD-热电模型，我们分析了汽车热电发电机（TEG）系统在一个完整驾驶循环下的动态响应特性。

Keywords 关键词

Dynamic performance

Numerical model

Thermoelectric

Thermoelectric generator system

Exhaust gas

Dynamic response

动态性能数值模型热电热电发电机系统废气动态响应

Nomenclature 术语

Symbols 符号

area, m² 面积，米 ²

specific heat, J kg⁻¹·K⁻¹
比热，J kg ⁻¹ -K ⁻¹

\vec{E}

electric field density vector, V m⁻²
电场密度矢量，V m ⁻²

output current, A 输出电流，A

\vec{J}

current density vector, A m⁻²
电流密度矢量，A m ⁻²

turbulent kinetic energy, m² s⁻²
湍流动能，m ² s ⁻²

\dot{m}

mass flow rate, g s⁻¹
质量流量，g s ⁻¹

pressure, Pa 压力，Pa

output power, W 输出功率（瓦

heat absorption, W 吸热量，W

resistance, Ω or K/W 电阻，Ω 或 K/W

\dot{S}

source term 源词

time, s 时间，秒

temperature, K 温度，K

output voltage, V 输出电压，V

\vec{v}

velocity, m s⁻¹ 速度，m s ⁻¹

Greek symbols 希腊文符号

thermal conductivity, W m⁻¹ K⁻¹
导热系数，W m ⁻¹ K ⁻¹

density, kg m⁻³ 密度，千克米 ⁻³

dynamic viscosity, Pa s
动态粘度，Pa s

turbulent dissipation rate, m² s⁻³
湍流耗散率，m ² s ⁻³

electrical conductivity, S m⁻¹
电导率，S m ⁻¹

σ^-1

electrical resistivity, Ω m
电阻率，Ω m

electrical potential, V 电位，V

Seebeck coefficient, μV K⁻¹
塞贝克系数，μV K ⁻¹

conversion efficiency 转换效率

Subscripts 下标

cold side 冷侧

co 媾

copper 铜

ex 从

exhaust gas 废气

exi

exhaust inlet 排气口

exo 外型

exhaust outlet 排气口

hot side 热面

i = 1, 2, 3, and 4
i = 1、2、3 和 4

load resistance 负载阻力

material name 材料名称

n-type thermoelectric semiconductor
n 型热电半导体

out 向外

outlet surface 出口表面

p-type thermoelectric semiconductor
p 型热电半导体

Abbreviations 缩略语

CFD 差价合约

computational fluid dynamics
计算流体动力学

EUDC 欧盟数据中心

extra urban driving cycle
城市额外行驶周期

FEM

finite element method 有限元法

FVM

finite volume method 有限体积法

LHDC

long haulage driving cycle
长途运输驾驶周期

TEG

thermoelectric generator
热电发生器

TEM

thermoelectric module 热电模块

HWFET

highway fuel economy test
公路燃油经济性测试

1. Introduction 1.导言

For engine-powered vehicles, the waste heat contained in automobile exhaust gas accounts for about one-third of the total energy generated by burning fossil fuels and is directly discharged into the environment [1], resulting in serious energy waste and environmental pollution. Recycling this waste heat is one of the effective ways to improve engine thermal efficiency and reduce fuel consumption. Thermoelectric power generation is considered to be a promising technology, which can directly convert exhaust heat into electricity and improve the fuel economy of vehicles [2]. It was reported that the automobile thermoelectric generator (TEG) system has the potential to save 6 % of fuel consumption [3]. In recent years, a great number of automobile TEG systems have been reported in the literature, which can produce a maximum output power of tens of watts to kilowatts, as listed in Table 1. The power generation of the mentioned automobile TEG systems was obtained by experimental measurements or theoretical models. As is well known, the theoretical model is necessary to guide the design and optimization of automobile TEG systems. Therefore, the development of a reasonable model for the automobile TEG system has become the focus of researchers [4].
对于以发动机为动力的汽车而言，汽车尾气中所含的余热约占化石燃料燃烧产生的总能量的三分之一，并直接排放到环境中[1]，造成严重的能源浪费和环境污染。回收利用这些废热是提高发动机热效率和降低油耗的有效方法之一。热发电被认为是一种很有前途的技术，它可以直接将废热转化为电能，提高汽车的燃油经济性[2]。据报道，汽车热电发电机（TEG）系统有可能节省 6% 的燃油消耗[3]。近年来，文献中报道了大量汽车 TEG 系统，其最大输出功率从几十瓦到几千瓦不等，如表 1 所示。上述汽车 TEG 系统的发电量是通过实验测量或理论模型获得的。众所周知，理论模型是指导汽车 TEG 系统设计和优化的必要条件。因此，建立合理的汽车 TEG 系统模型已成为研究人员关注的焦点 [4]。

Table 1. Automobile thermoelectric generator systems in published literature.
表 1.已发表文献中的汽车热电发电机系统。

Vehicle or engine type 车辆或发动机类型	Number of TEG modules TEG 模块数量	Working conditions 工作条件	Steady-state or transient state 稳态或瞬态	Experimental test or model prediction 实验测试或模型预测	Generated power 发电量	Ref 参考文献
BMW X6 宝马 X6	–	US06 driving cycle US06 行驶周期	transient state 瞬态	experimental test 试验	from 0 to ∼ 500 W 从 0 到 ∼ 500 W	[5]
Lincoln MKT 林肯 MKT	–	highway driving cycle 公路行驶周期	transient state 瞬态	experimental test 试验	from ∼ 50 W to ∼ 300 W 从 ∼ 50 W 到 ∼ 300 W	[5]
heavy-duty diesel engine 重型柴油发动机	20	non-road transient cycle 非道路瞬时周期	transient state 瞬态	model prediction 模型预测	from 0 W to ∼ 300 W 从 0 W 到 ∼ 300 W	[6]
off-road vehicle 越野车	240	engine power: 51 kW 发动机功率： 51 千瓦	steady-state 稳恒	experimental test 试验	maximum power: 944 W 最大功率： 944 瓦	[7]
gasoline engine 汽油发动机	12	EUDC driving cycle 欧盟数据中心驱动周期	transient state 瞬态	experimental test 试验	from ∼ 2 W to ∼ 30 W 从 ∼ 2 W 到 ∼ 30 W	[8]
heavy-duty vehicle 重型车辆	240	LHDC driving cycle LHDC 驱动周期	transient state 瞬态	experimental test 试验	from ∼ 300 W to ∼ 820 W 从 ∼ 300 W 到 ∼ 820 W	[9]
gasoline engine 汽油发动机	306	engine power: 30.4 kW 发动机功率： 30.4 千瓦	steady-state 稳恒	model prediction 模型预测	maximum power: 515 W 最大功率： 515 W	[10]

* Extra Urban Driving Cycle (EUDC); ** long haulage driving cycle (LHDC).
* 额外城市驾驶循环 (EUDC)；** 长途运输驾驶循环 (LHDC)。

The existing theoretical models for predicting the performance of the automobile TEG system can be classified into two categories: the analytical model [11] and the numerical model [12]. The analytical model is deduced by analyzing the heat transfer process from the exhaust gas to the heat exchanger, thermoelectric module (TEM) arrays, heat sinks, and finally to coolant, which can quickly calculate the output performance under the given temperature and mass flow rate, but can not consider the factors such as heat loss, temperature distribution, and complex geometry. The analytical model has been proved to be unreasonable compared with the numerical model [13]. Especially in recent years, with the development of numerical analysis software, more and more researchers use numerical models based on the finite element method (FEM) [14] or finite volume method (FVM) [15] to predict the performance of TEGs and TEG systems. Furthermore, the numerical models for the automobile TEG system include: (i) computational fluid dynamics (CFD) model, which can analyze the thermal and pressure performance of the automobile TEG system without considering the thermal-electric coupling effects [16]; (ii) thermal-electric numerical model, which can analyze the thermoelectric performance of the system by simplifying the heat transfer from the exhaust gas to TEM arrays [17]; and (iii) recently reported fluid-thermal-electric numerical model, which can comprehensively evaluate the output performance of the system through some geometric simplifications [18].
现有的汽车 TEG 系统性能预测理论模型可分为两类：分析模型 [11] 和数值模型 [12]。解析模型是通过分析废气到热交换器、热电模块（TEM）阵列、散热器，最后到冷却液的传热过程推导出来的，可以快速计算出给定温度和质量流量下的输出性能，但无法考虑热损失、温度分布和复杂几何形状等因素。事实证明，与数值模型相比，解析模型是不合理的[13]。特别是近年来，随着数值分析软件的发展，越来越多的研究人员使用基于有限元法（FEM）[14] 或有限体积法（FVM）[15] 的数值模型来预测 TEG 和 TEG 系统的性能。此外，汽车 TEG 系统的数值模型包括(i) 计算流体动力学（CFD）模型，可在不考虑热电耦合效应的情况下分析汽车 TEG 系统的热性能和压力性能[16]；(ii) 热电数值模型，可通过简化废气到 TEM 阵列的热传导来分析系统的热电性能[17]；(iii) 最近报道的流体-热电数值模型，可通过一些几何简化全面评估系统的输出性能[18]。

However, most of the models reported at present are steady-state, which is not consistent with the actual vehicle conditions. Steady-state models can not meet the needs of dynamic performance prediction of the automobile TEG system under actual driving conditions. Also, it was reported that the power generation of the automobile TEG system under a complete driving cycle is lower than that expected from a steady-state analysis [8]. To obtain more accurate analysis results, it is necessary to extend the model from steady-state to transient state. There are three routes to develop the transient theoretical models: transient analytical model, transient numerical model, and hybrid transient model.
然而，目前报道的大多数模型都是稳态模型，与实际车辆状况不符。稳态模型无法满足实际驾驶条件下汽车 TEG 系统动态性能预测的需要。另外，有报道称汽车 TEG 系统在一个完整驾驶周期下的发电量低于稳态分析的预期值[8]。为了获得更精确的分析结果，有必要将模型从稳态扩展到瞬态。建立瞬态理论模型有三种途径：瞬态分析模型、瞬态数值模型和混合瞬态模型。

Firstly, the analytical model has the advantages of no consumption of computing resources and simple modeling. In the previous studies [19], [20], [21], some steady-state analytical models for automobile TEG systems have been established, which can consider the temperature dependence of thermoelectric materials and heat loss. However, the developed models can not be used to predict dynamic performance. By introducing a time variable into the equations of the steady-state analytical model, Lan et al. [6] proposed a transient analytical model to estimate the dynamic output performance of an automobile TEG system. The discrete data of exhaust temperature and mass flow rate under the New European Driving Cycle (NEDC) were imported into the dynamic model, and the output parameters at corresponding time points were calculated. Nevertheless, the transient analytical model features the major drawback that the thermal inertia can not be considered, which further increases the error of the analytical model itself.
首先，分析模型具有不消耗计算资源、建模简单等优点。在以往的研究中[19]、[20]、[21]，已经建立了一些汽车 TEG 系统的稳态分析模型，这些模型可以考虑热电材料的温度依赖性和热损失。但是，所建立的模型不能用于预测动态性能。Lan 等人[6]通过在稳态分析模型的方程中引入时间变量，提出了一种瞬态分析模型来估计汽车 TEG 系统的动态输出性能。他们将新欧洲驾驶循环（NEDC）下排气温度和质量流量的离散数据导入动态模型，并计算出相应时间点的输出参数。然而，瞬态分析模型的主要缺点是无法考虑热惯性，这进一步增加了分析模型本身的误差。

Secondly, the numerical model can simulate the actual working conditions of the automobile TEG system and predict its performance with high accuracy, but it has the disadvantage of long computation time. The steady-state CFD model [22] is the most widely used model to analyze the thermal–hydraulic performance of automobile TEG systems. Kempf and Zhang [23] first optimized the parameters of the heat exchanger via the steady-state CFD model and then designed a 1-kW automobile TEG system prototype based on CFD results [24]. However, there is no report about transient CFD modeling for the automobile TEG system. Different from the CFD model, the thermal-electric numerical model has made great progress. From 2D [25] to 3D [14], from steady-state [26] to transient state [27], a large number of thermal-electric numerical models have been used to evaluate the performance of TEGs [28]. The thermal-electric numerical model suffers from the main limitation that the fluid flow of exhaust gas and cooling water can not be considered. To address this issue, Ma et al. [29] and Yan et al. [18] proposed a 3D steady-state fluid-thermal-electric multiphysics numerical model of the automobile TEG system by integrating a thermal-electric numerical model into CFD model. In their studies, the TEM was simplified into a thermoelectric couple [29], and the geometry of heat sinks was ignored [18]. The fluid-thermal-electric multiphysics numerical model takes into account the coupling effects of fluid, thermal, and electric fields. Even though it needs huge computing time and resources, extending it from the simplified geometry to the whole structure, from steady-state to transient state is the best way to predict the transient performance of automobile TEG systems. However, it still requires tremendous studies.
其次，数值模型可以模拟汽车 TEG 系统的实际工况，对其性能进行高精度预测，但存在计算时间长的缺点。稳态 CFD 模型[22]是目前应用最广泛的汽车 TEG 系统热液性能分析模型。Kempf 和 Zhang [23]首先通过稳态 CFD 模型优化了热交换器的参数，然后根据 CFD 结果设计了 1 kW 的汽车 TEG 系统原型 [24]。然而，目前还没有关于汽车 TEG 系统瞬态 CFD 建模的报道。与 CFD 模型不同，热-电数值模型取得了长足的进步。从二维[25]到三维[14]，从稳态[26]到瞬态[27]，大量的热-电数值模型已被用于评估 TEG 的性能[28]。热-电数值模型的主要局限是无法考虑废气和冷却水的流体流动。针对这一问题，Ma 等人[29]和 Yan 等人[18]通过将热-电数值模型集成到 CFD 模型中，提出了汽车 TEG 系统的三维稳态流体-热-电多物理场数值模型。在他们的研究中，TEM 被简化为热电偶[29]，散热器的几何形状被忽略[18]。流体-热-电多物理场数值模型考虑了流体场、热场和电场的耦合效应。尽管需要大量的计算时间和资源，但将其从简化几何扩展到整个结构，从稳态扩展到瞬态，是预测汽车 TEG 系统瞬态性能的最佳方法。然而，这仍然需要大量的研究。

Thirdly, another effective way to develop a transient model for automobile TEG systems is to combine different models to form a hybrid model and extend it from steady-state to transient state. Li et al. [30] used the temperature data from steady-state CFD simulation as the boundary conditions of the analytical model of TEMs, and then worked out the output performance of the automobile TEG system. The model makes full use of the advantages of the CFD model and the analytical model, but it inevitably has the limitations of the analytical model. Furthermore, Luo et al. [31] proposed a steady-state hybrid numerical model for an air-to-water TEG system by combining the thermal-electric numerical model with the CFD model. In the following work [32], the authors extended the model from a TEG system with one TEM to a TEG system with a series of TEMs. The proposed hybrid steady-state numerical model was experimentally verified, which provides a new approach to evaluating the performance of automobile TEG systems. Comparing the abovementioned methods of extending these two hybrid models from steady-state to transient state, the latter one is a more attractive one, because the analytical model in the first model can not consider the thermal inertia, which further increases its own error. The hybrid transient CFD-thermoelectric numerical model can not only maintain a high accuracy of the numerical models but also reduce the computational complexity and time, which is the key point of this work.
第三，开发汽车 TEG 系统瞬态模型的另一种有效方法是将不同模型组合成混合模型，并将其从稳态扩展到瞬态。Li 等人[30]利用稳态 CFD 模拟的温度数据作为 TEM 分析模型的边界条件，进而计算出汽车 TEG 系统的输出性能。该模型充分发挥了 CFD 模型和分析模型的优势，但也不可避免地存在分析模型的局限性。此外，Luo 等人[31]通过将热电数值模型与 CFD 模型相结合，提出了空气-水 TEG 系统的稳态混合数值模型。在随后的研究中[32]，作者将该模型从带有一个 TEM 的 TEG 系统扩展到带有一系列 TEM 的 TEG 系统。实验验证了所提出的混合稳态数值模型，为评估汽车 TEG 系统的性能提供了一种新方法。将上述两种混合模型从稳态扩展到瞬态的方法进行比较，后一种方法更具吸引力，因为前一种模型中的分析模型无法考虑热惯性，这进一步增加了其自身误差。瞬态 CFD-热电混合数值模型既能保持数值模型的高精度，又能降低计算复杂度，缩短计算时间，这正是本研究的重点。

Table 2 summarizes the latest development of theoretical models for automobile TEG systems. It can be noticed that there are few reports on the transient theoretical model. However, due to the overestimation of performance in steady-state analysis, it is necessary to study the dynamic performance of the automobile TEG system. Although the fluid-thermal-electric numerical model features ultrahigh precision and reasonability, the existing steady-state model has a large number of geometric simplifications. Besides, the fluid-thermal-electric numerical model needs plenty of computing time and resources, and it suffers from convergence problems during the numerical calculation. Therefore, the hybrid CFD-thermoelectric numerical model is an attractive choice to develop its transient model. Compared with the fluid-thermal-electric numerical model, the hybrid CFD-thermoelectric numerical model can significantly shorten the execution time, because the CFD model and the thermal-electric numerical model are solved separately at the same time. The only drawback of the hybrid CFD-thermoelectric numerical model is that it can not consider the influence of heat generated by TEMs on the CFD model. However, compared with the heat contained in the exhaust gas, the heat generated by TEMs is tiny and can be ignored. Therefore, this assumption will not seriously affect the predicted output performance. More importantly, the validity of the hybrid CFD-thermoelectric numerical model in steady-state has already been verified by experiments in previous studies [32], [33].
表 2 总结了汽车 TEG 系统理论模型的最新发展。可以看出，有关瞬态理论模型的报道很少。然而，由于稳态分析会高估性能，因此有必要研究汽车 TEG 系统的动态性能。虽然流体-热-电数值模型具有超高精度和合理性，但现有的稳态模型存在大量几何简化。此外，流体-热电数值模型需要大量的计算时间和资源，并且在数值计算过程中存在收敛问题。因此，在开发瞬态模型时，CFD-热电混合数值模型是一个有吸引力的选择。与流体-热电数值模型相比，由于 CFD 模型和热电数值模型是同时分别求解的，因此 CFD-热电混合数值模型可以大大缩短执行时间。CFD 热电混合数值模型的唯一缺点是无法考虑 TEM 产生的热量对 CFD 模型的影响。然而，与废气中所含的热量相比，TEMs 产生的热量微乎其微，可以忽略不计。因此，这一假设不会严重影响预测的输出性能。更重要的是，CFD-热电混合数值模型在稳态下的有效性已在之前的研究[32]、[33]中得到实验验证。

Table 2. Recent advances of theoretical models for automobile TEG systems.
表 2.汽车 TEG 系统理论模型的最新进展。

Model type 型号		Current states 当前状态	Advantages 优势	Disadvantages 缺点
analytical model 分析模型		steady-state: [19], [34] 稳态：[19], [34] transient state: [6] 瞬态：[6]	• rapid computation - 快速计算 • convenience for modeling - 便于建模	• rough precision - 粗精度 • low dimensional - 低维 • ignoring the thermal inertia - 忽略热惯性
numerical models 数值模型	CFD model CFD 模型	steady-state: [35], [36] 稳态：[35], [36] transient state: not reported 瞬态：未报告	• convenience for studying the thermal–hydraulic performance - 方便研究热液性能	• ignoring the thermoelectric coupling effects - 忽略热电耦合效应
	thermal-electric model 热电模型	steady state: [37], [38] 稳态：[37], [38] transient state: [25], [27] 瞬态：[25], [27]	• convenience for studying the thermoelectric performance - 方便研究热电性能	• ignoring the fluid flow and conjugate heat transfer - 忽略流体流动和共轭传热
	fluid-thermal-electric model 流体-热-电模型	steady-state: [18], [29] 稳态：[18], [29] transient state: not reported 瞬态：未报告	• ultrahigh high precision - 超高精度 • potential to fully simulate actual conditions - 完全模拟实际条件的潜力	• huge computing time and resources - 巨大的计算时间和资源 • many geometric simplifications for the reported models - 报告模型的许多几何简化
hybrid models 混合动力车型	CFD-analytical model CFD 分析模型	steady-state: [30] 稳态：[30] transient state: not reported 瞬态：未报告	• relatively short computing time - 计算时间相对较短	• ignoring the thermoelectric coupling effects - 忽略热电耦合效应
hybrid models 混合动力车型	CFD-thermoelectric model CFD 热电模型	steady state: [32], [33] 稳定状态：[32], [33] transient state: not reported 瞬态：未报告	• relatively short computing time and high precision - 计算时间相对较短，精度较高	• ignoring the effect of heat generated by TEMs on the CFD model - 忽略 TEM 产生的热量对 CFD 模型的影响

This work serves the purpose of providing a reasonable transient model to evaluate the dynamic response characteristics of the automobile TEG system under actual vehicle transient driving cycles. For this reason, a hybrid transient CFD-thermoelectric numerical model considering transient input of the exhaust heat source, temperature dependence, impedance matching, and heat loss is proposed for the first time. It features the advantages of high accuracy and short calculation time. This study fills the gap in the theoretical prediction of the transient behavior of the automobile TEG system and provides a new perspective for predicting dynamic response characteristics of the automobile TEG system. This paper is structured as follows: Section 2 introduces the hybrid transient CFD-thermoelectric numerical model in detail, including the geometry structure of the automobile TEG system, governing equations, boundary conditions, grid independence analysis, and experimental validation; Section 3 elucidates the results and gives a detailed analysis of the results; Finally, the main research findings are summarized in Section 4.
这项工作的目的是提供一个合理的瞬态模型，以评估汽车 TEG 系统在实际车辆瞬态驾驶循环下的动态响应特性。为此，首次提出了一种混合瞬态 CFD 热电数值模型，该模型考虑了排气热源的瞬态输入、温度相关性、阻抗匹配和热损失。该模型具有精度高、计算时间短等优点。该研究填补了汽车 TEG 系统瞬态行为理论预测的空白，为预测汽车 TEG 系统的动态响应特性提供了新的视角。本文的结构如下：第 2 节详细介绍了 CFD-热电混合瞬态数值模型，包括汽车 TEG 系统的几何结构、调控方程、边界条件、网格独立性分析和实验验证；第 3 节阐明了研究结果，并对结果进行了详细分析；最后，第 4 节总结了主要研究成果。

2. Hybrid transient CFD-thermoelectric numerical model
2.混合瞬态 CFD 热电数值模型

2.1. Structure of the automobile thermoelectric generator system
2.1.汽车热电发电机系统结构

In this work, an automobile thermoelectric generator (TEG) system presented in the previous study [32] is taken as the research objective, and the dynamic response characteristics of the automobile TEG system under a complete transient driving cycle are studied through the proposed hybrid transient computational fluid dynamics (CFD) -thermoelectric numerical model. The structure of the automobile TEG system is shown in Fig. 1, which includes an actual prototype (Fig. 1(a)) and the structure diagram (Fig. 1(b)). The automobile TEG system consists of a heat exchanger, sixteen thermoelectric modules (TEMs), and four heat sinks, which are clamped and fixed by stainless steel bars. Both heat sinks and the heat exchanger are made of 6063 aluminum alloys. Twenty long fins with a size of 20 mm × 2 mm × 200 mm are arranged on the upper and bottom walls of the heat exchanger. High-temperature exhaust gas enters and leaves the heat exchanger through a circular tube with a diameter of 50 mm. The diameter of the coolant channel in the heat sink is 5.5 mm. Detailed geometric features and dimensions can be found in Fig. 1.
本研究以先前研究[32]中提出的汽车热电发电机（TEG）系统为研究对象，通过提出的混合瞬态计算流体动力学（CFD）-热电数值模型，研究了汽车 TEG 系统在完整瞬态驾驶循环下的动态响应特性。汽车 TEG 系统的结构如图 1 所示，包括实际样机（图 1(a)）和结构图（图 1(b)）。汽车 TEG 系统由一个热交换器、十六个热电模块（TEM）和四个散热器组成，散热器由不锈钢条夹紧固定。散热器和热交换器均由 6063 铝合金制成。在热交换器的上下壁上布置了 20 个长翅片，尺寸为 20 毫米 × 2 毫米 × 200 毫米。高温废气通过直径为 50 毫米的圆管进出热交换器。散热器中冷却剂通道的直径为 5.5 毫米。详细的几何特征和尺寸见图 1。

Besides, the sixteen TEMs are evenly distributed on the upper and bottom sides of the heat exchanger. The TEM consists of two ceramic plates, 256 copper sheets, 128 pairs of p-type and n-type thermoelectric legs. The dimensions of the hot-side ceramic plate, copper sheets, thermoelectric legs, and cold-side ceramic plate are 44 mm × 40 mm × 0.8 mm, 3.8 mm × 1.4 mm × 0.35 mm, 1.4 mm × 1.4 mm × 1 mm, and 40 mm × 40 mm × 0.8 mm, respectively. During transient CFD numerical simulations, to shorten the calculation time, the complete TEM is simplified to a cube with an equivalent size of 40 mm × 40 mm × 3.3 mm. According to the CFD results, the average surface temperatures on both sides of TEMs are obtained, which are used as the temperature boundary conditions of the transient thermal-electric numerical model. During transient thermal-electric numerical simulations, the complete TEM structure is adopted. A load resistance is connected in series with the four TEMs. Detailed properties of the materials in the automobile TEG system are listed in Table 3. Here, the thermal conductivity of the equivalent TEM is obtained according to the reported method in [20], where the contact thermal resistance inside the TEM has already been considered in the experimental measurement process. As for the contact thermal resistances between the TEM and heat exchangers, they are ignored due to the use of clamping devices and thermal extensional structures [39], [40]. The material properties of exhaust gas and coolant are not given, because they are replaced by dry air and water, respectively. The properties of air and water are from the material library of COMSOL Multiphysics [41], and the temperature-dependent properties of the thermoelectric materials are taken into account.
此外，16 个 TEM 均匀分布在热交换器的上下两侧。TEM 由两块陶瓷板、256 片铜板、128 对 p 型和 n 型热电腿组成。热侧陶瓷板、铜片、热电腿和冷侧陶瓷板的尺寸分别为 44 mm × 40 mm × 0.8 mm、3.8 mm × 1.4 mm × 0.35 mm、1.4 mm × 1.4 mm × 1 mm 和 40 mm × 40 mm × 0.8 mm。在进行瞬态 CFD 数值模拟时，为了缩短计算时间，将整个 TEM 简化为等效尺寸为 40 mm × 40 mm × 3.3 mm 的立方体。根据 CFD 结果，可以得到 TEM 两侧的平均表面温度，并以此作为瞬态热-电数值模型的温度边界条件。在瞬态热电数值模拟中，采用了完整的 TEM 结构。在四个 TEM 上串联了一个负载电阻。表 3 列出了汽车 TEG 系统中材料的详细属性。这里，等效 TEM 的热导率是根据 [20] 中报告的方法获得的，在实验测量过程中已经考虑了 TEM 内部的接触热阻。至于 TEM 与热交换器之间的接触热阻，由于使用了夹紧装置和热延伸结构 [39]，[40]，因此忽略不计。没有给出废气和冷却剂的材料特性，因为它们分别由干燥空气和水代替。空气和水的属性来自 COMSOL Multiphysics 的材料库[41]，热电材料的属性与温度有关。

Table 3. Material parameters of the automobile TEG system.
表 3.汽车 TEG 系统的材料参数。

Material 材料	Components 组件	Parameter name 参数名称	Value 价值	Unit 单位
p-type thermoelectric material p 型热电材料	p-type thermoelectric legs p 型热电腿	Seebeck coefficient 塞贝克系数	$\begin{matrix} - 1.80268 \times 10^{- 7} T^{4} + 3.23632 \times 10^{- 4} T^{3} \\ - 0.21537 T^{2} + 62.97444 T - 6616.56781 \end{matrix}$	W m⁻¹ K⁻¹
		electrical resistivity 电阻	$\begin{matrix} - 3.08802 \times 10^{- 9} T^{4} + 4.56531 \times 10^{- 6} T^{3} \\ - 2.58541 \times 10^{- 3} T^{2} + 0.65579 T - 60.58804 \end{matrix}$	10^-5 Ω∙m
		thermal conductivity 导热性	$\begin{matrix} - 3.05948 \times 10^{- 9} T^{4} + 4.56781 \times 10^{- 6} T^{3} \\ - 2.51621 \times 10^{- 3} T^{2} + 0.61074 T - 53.98632 \end{matrix}$	μV K⁻¹
		density 密度	6600	kg m⁻³ 千克米 ⁻³
		specific heat 比热	188	J kg⁻¹ K⁻¹
n-type thermoelectric material n 型热电材料	n-type thermoelectric legs n 型热电腿	Seebeck coefficient 塞贝克系数	$\begin{matrix} 1.80268 \times 10^{- 7} T^{4} - 3.23632 \times 10^{- 4} T^{3} \\ + 0.21537 T^{2} - 62.97444 T + 6616.56781 \end{matrix}$	W m⁻¹ K⁻¹
		electrical resistivity 电阻	$\begin{matrix} - 3.08802 \times 10^{- 9} T^{4} + 4.56531 \times 10^{- 6} T^{3} \\ - 2.58541 \times 10^{- 3} T^{2} + 0.65579 T - 60.58804 \end{matrix}$	10^-5 Ω⋅m
		thermal conductivity 导热性	$\begin{matrix} - 3.05948 \times 10^{- 9} T^{4} + 4.56781 \times 10^{- 6} T^{3} \\ - 2.51621 \times 10^{- 3} T^{2} + 0.61074 T - 53.98632 \end{matrix}$	μV K⁻¹
		density 密度	6600	kg m⁻³ 千克米 ⁻³
		specific heat 比热	188	J kg⁻¹ K⁻¹
copper 铜	copper sheets 铜板	electrical resistivity 电阻	1.75 × 10^-8	Ω⋅m
		thermal conductivity 导热性	165.64	W m⁻¹ K⁻¹
		density 密度	8978	kg m⁻³ 千克米 ⁻³
		specific heat 比热	381	J kg⁻¹ K⁻¹
–	load resistance 负载阻力	electrical resistance 电阻	17	Ω
		thermal conductivity 导热性	165.64	W m⁻¹ K⁻¹
		density 密度	8978	kg m⁻³ 千克米 ⁻³
		specific heat 比热	381	J kg⁻¹ K⁻¹
ceramic 陶瓷	ceramic plates 陶瓷盘	thermal conductivity 导热性	22	W m⁻¹ K⁻¹
		density 密度	3600	kg m⁻³ 千克米 ⁻³
		specific heat 比热	850	J kg⁻¹ K⁻¹
6063 aluminum alloys 6063 铝合金	heat exchanger and heat sinks 热交换器和散热器	thermal conductivity 导热性	201	W m⁻¹ K⁻¹
		density 密度	2719	kg m⁻³ 千克米 ⁻³
		specific heat 比热	871	J kg⁻¹ K⁻¹
–	equivalent TEM 等效 TEM	thermal conductivity 导热性	$\begin{matrix} - 2.90574 \times 10^{- 9} T^{4} + 4.33411 \times 10^{- 6} T^{3} \\ - 0.00239 T^{2} + 0.57868 T - 51.05908 \end{matrix}$	W m⁻¹ K⁻¹
		density 密度	6600	kg m⁻³ 千克米 ⁻³
		specific heat 比热	188	J kg⁻¹ K⁻¹

2.2. Governing equations of the hybrid transient CFD-thermoelectric numerical model
2.2.CFD- 热电混合瞬态数值模型的控制方程

The hybrid transient CFD-thermoelectric numerical model consists of two sub-models: a transient CFD model and a transient thermal-electric numerical model. During the complete numerical calculation process, the transient CFD model is firstly solved, and then the transient thermal-electric numerical model is solved. The average surface temperatures on both sides of TEMs obtained by the transient CFD model are used as temperature boundary conditions of the transient thermal-electric numerical model. For the transient CFD model, the transient mass, momentum, and temperature conservation equations [42] can be expressed as follows:(1)

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{v}) = 0

(2)

\frac{\partial}{\partial t} (ρ \vec{v}) + \nabla \cdot (ρ \vec{v} \vec{v}) = - \nabla p + \nabla \cdot [μ (\nabla \vec{v} + \nabla {\vec{v}}^{T})]

(3)

ρ c \frac{\partial T}{\partial t} + ρ c \vec{v} \cdot \nabla T - \nabla \cdot (λ \nabla T) = 0

where ρ, t,

\vec{v}

, p, μ, c, T, and λ represent the density, time, velocity, pressure, dynamic viscosity, specific heat at constant pressure, temperature, and thermal conductivity of fluids, respectively.
瞬态 CFD-热电混合数值模型由两个子模型组成：瞬态 CFD 模型和瞬态热电数值模型。在整个数值计算过程中，首先求解瞬态 CFD 模型，然后求解瞬态热电数值模型。瞬态 CFD 模型得到的 TEM 两侧平均表面温度用作瞬态热-电数值模型的温度边界条件。对于瞬态 CFD 模型，瞬态质量、动量和温度守恒方程 [42] 可表示如下： (1)

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{v}) = 0

(2)

\frac{\partial}{\partial t} (ρ \vec{v}) + \nabla \cdot (ρ \vec{v} \vec{v}) = - \nabla p + \nabla \cdot [μ (\nabla \vec{v} + \nabla {\vec{v}}^{T})]

(3)

ρ c \frac{\partial T}{\partial t} + ρ c \vec{v} \cdot \nabla T - \nabla \cdot (λ \nabla T) = 0

其中，ρ、t、

\vec{v}

、p、μ、c、T 和 λ 分别代表流体的密度、时间、速度、压力、动态粘度、恒压比热、温度和导热系数。

In general, the fluid flow of exhaust gas in the heat exchanger and that of coolant in heat sinks are turbulent and can be considered incompressible. The standard k - ε turbulence model [42] is adopted to compute the turbulent flow in the current research, and the transport equations for k and ε are:(4)

\frac{\partial}{\partial t} (ρ k) + ρ (\vec{v} \cdot \nabla) k = \nabla \cdot [(μ + \frac{μ_{t}}{σ_{k}}) \nabla k] + P_{k} - ρ ε

(5)

\frac{\partial}{\partial t} (ρ ε) + ρ (\vec{v} \cdot \nabla) ε = \nabla \cdot [(μ + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + C_{1 ε} \frac{ε}{k} P_{k} - C_{2 ε} ρ \frac{ε^{2}}{k}

with(6)

μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}

where k, ε, and P_k are the turbulence kinetic energy, dissipation of turbulence kinetic energy, and shear production of turbulence kinetic energy, respectively. C_1ε = 1.44, C_2ε = 1.92, C_μ = 0.09, σ_k = 1.0, and σ_ε = 1.3 [43] are model constants.
一般来说，热交换器中的废气流和散热器中的冷却剂流都是湍流，可以认为是不可压缩的。本研究采用标准 k - ε 湍流模型[42]计算湍流，k 和 ε 的传输方程为 (4)

\frac{\partial}{\partial t} (ρ k) + ρ (\vec{v} \cdot \nabla) k = \nabla \cdot [(μ + \frac{μ_{t}}{σ_{k}}) \nabla k] + P_{k} - ρ ε

(5)

\frac{\partial}{\partial t} (ρ ε) + ρ (\vec{v} \cdot \nabla) ε = \nabla \cdot [(μ + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + C_{1 ε} \frac{ε}{k} P_{k} - C_{2 ε} ρ \frac{ε^{2}}{k}

与 (6)

μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}

其中，k、ε 和 P _k 分别为湍流动能、湍流动能耗散和湍流动能的剪切产生。C _1ε = 1.44、C _2ε = 1.92、C _μ = 0.09、σ _k = 1.0 和 σ _ε = 1.3 [43] 为模型常数。

The governing equations of the fluid are given by Eqs. (1)-(6). For the solid region in the CFD model, to calculate the heat transfer of the thermal field, the transient energy conservation is defined as:(7)

{(ρ c)}_{m} \frac{\partial T}{\partial t} = \nabla \cdot (λ_{m} \nabla T)

where subscript m denotes different materials.
流体的控制方程由式 (1)-(6) 给出。对于 CFD 模型中的固体区域，为计算热场传热，瞬态能量守恒定义为 (7)

{(ρ c)}_{m} \frac{\partial T}{\partial t} = \nabla \cdot (λ_{m} \nabla T)

其中，下标 m 表示不同的材料。

The above equations constitute the governing equations of the transient CFD model. The transient CFD model can be used to analyze the transient thermodynamic performance of the automobile TEG system. However, the transient response of the electric field of the automobile TEG system can not be obtained by the transient CFD model. In this work, the transient thermal-electric numerical model is used to calculate the electrical output performance of the automobile TEG system.
上述方程构成了瞬态 CFD 模型的控制方程。瞬态 CFD 模型可用于分析汽车 TEG 系统的瞬态热力学性能。但是，瞬态 CFD 模型无法获得汽车 TEG 系统电场的瞬态响应。本研究采用瞬态热-电数值模型来计算汽车 TEG 系统的电输出性能。

The governing equations of the transient thermal-electric numerical model include the energy conservation equation, electric field conservation equation, and current continuity equation. Here, the energy conservation of the transient thermal-electric numerical model is expressed as [27]:(8)

{(ρ c)}_{m} \frac{\partial T}{\partial t} = \nabla \cdot (λ_{m} \nabla T) + {\dot{S}}_{m}

with

{\dot{S}}_{m} = \{\begin{matrix} σ_{p}^{- 1} (T) {\vec{J}}^{2} - T_{p} \vec{J} \cdot \nabla α_{p} (T) - \frac{\partial α_{p} (T)}{\partial T} T \vec{J} \cdot \nabla T_{p}; p - type thermoelectric leg (9 - 1) \\ σ_{n}^{- 1} (T) {\vec{J}}^{2} - T_{n} \vec{J} \cdot \nabla α_{n} (T) - \frac{\partial α_{n} (T)}{\partial T} T \vec{J} \cdot \nabla T_{n}; n - type thermoelectric leg (9 - 2) \\ σ_{co}^{- 1} {\vec{J}}^{2}; copper sheet (9 - 3) \\ σ_{L}^{- 1} {\vec{J}}^{2}; load resistance (9 - 4) \\ 0; ceramic (9 - 5) \end{matrix}

where

\dot{S}

_m denotes the energy source term, which is induced by the electric field. σ^-1, α, and

\vec{J}

are the electrical resistivity, Seebeck coefficient, and current density vector, respectively. Subscripts p, n, co, and L represent the p-type thermoelectric leg, n-type thermoelectric leg, copper sheet, and load resistance, respectively. The first term on the right side of Eqs (9–1) and (9–2) represent Joule heat, the second term represents Peltier heat on both ends of thermoelectric legs, and the third term represents Thomson heat along thermoelectric legs.
瞬态热-电数值模型的控制方程包括能量守恒方程、电场守恒方程和电流连续性方程。这里，瞬态热-电数值模型的能量守恒表示为[27]： (8)

{(ρ c)}_{m} \frac{\partial T}{\partial t} = \nabla \cdot (λ_{m} \nabla T) + {\dot{S}}_{m}

与

{\dot{S}}_{m} = \{\begin{matrix} σ_{p}^{- 1} (T) {\vec{J}}^{2} - T_{p} \vec{J} \cdot \nabla α_{p} (T) - \frac{\partial α_{p} (T)}{\partial T} T \vec{J} \cdot \nabla T_{p}; p - type thermoelectric leg (9 - 1) \\ σ_{n}^{- 1} (T) {\vec{J}}^{2} - T_{n} \vec{J} \cdot \nabla α_{n} (T) - \frac{\partial α_{n} (T)}{\partial T} T \vec{J} \cdot \nabla T_{n}; n - type thermoelectric leg (9 - 2) \\ σ_{co}^{- 1} {\vec{J}}^{2}; copper sheet (9 - 3) \\ σ_{L}^{- 1} {\vec{J}}^{2}; load resistance (9 - 4) \\ 0; ceramic (9 - 5) \end{matrix}

其中，

\dot{S}

_m 表示能量源项，由电场引起；σ ^-1 、α 和

\vec{J}

分别为电阻率、塞贝克系数和电流密度矢量。下标 p、n、co 和 L 分别代表 p 型热电脚、n 型热电脚、铜片和负载电阻。公式 (9-1) 和 (9-2) 右侧的第一项代表焦耳热，第二项代表热电腿两端的珀尔帖热，第三项代表热电腿上的汤姆逊热。

Compared with Eq. (8), the energy source term is absent in Eq. (7), because the electric field is ignored in the transient CFD model.
与公式 (8) 相比，公式 (7) 中没有能量源项，因为在瞬态 CFD 模型中忽略了电场。

The electric field conservation equation of the transient thermal-electric numerical model can be written as [44]:(10)

\vec{E} = - \nabla ϕ + α_{p, n} (T) \nabla T

where

\vec{E}

and ϕ are the electric field density vector and electrical potential, respectively.
瞬态热-电数值模型的电场守恒方程可写成 [44]： (10)

\vec{E} = - \nabla ϕ + α_{p, n} (T) \nabla T

其中

\vec{E}

和 ϕ 分别为电场密度矢量和电动势。

Besides, the current flows through the p-type and n-type thermoelectric legs, copper sheet, and load resistance is continuous, which can be defined as [45]:(11)

\nabla \cdot \vec{J} = 0

with(12)

\vec{J} = σ_{m} \vec{E}

此外，流经 p 型和 n 型热电脚、铜片和负载电阻的电流是连续的，可定义为 [45]： (11)

\nabla \cdot \vec{J} = 0

与 (12)

\vec{J} = σ_{m} \vec{E}

Eqs. (8)-(12) constitute the governing equations of the transient thermal-electric numerical model. To solve the partial differential conservation equations of the transient CFD model and the transient thermal-electric numerical model, the numerical calculation method is essential. The finite element method, finite volume method, and finite difference method are commonly used to solve numerical models. In the present work, the backward difference method and the finite element method are used to discretize the time and space variables, respectively, by the commercial numerical software package of COMSOL.
式 (8)-(12) 构成了瞬态热电数值模型的控制方程。要求解瞬态 CFD 模型和瞬态热电数值模型的偏微分守恒方程，数值计算方法至关重要。有限元法、有限体积法和有限差分法是求解数值模型的常用方法。在本研究中，利用 COMSOL 商业数值软件包，分别采用后向差分法和有限元法对时间变量和空间变量进行离散计算。

2.3. Boundary conditions of the hybrid transient CFD-thermoelectric numerical model
2.3.瞬态 CFD-热电混合数值模型的边界条件

Boundary conditions are essential for solving the governing equations of the hybrid transient CFD-thermoelectric numerical model. There are two kinds of boundary conditions: the boundary conditions of the transient CFD model and the boundary conditions of the transient thermal-electric numerical model. Considering the huge computing time and resources required for transient numerical calculation, this study only takes 1/4 of the automobile TEG system as the research objective due to its completely symmetrical structure.
边界条件对于求解瞬态 CFD-热电混合数值模型的控制方程至关重要。边界条件分为两种：瞬态 CFD 模型的边界条件和瞬态热电数值模型的边界条件。考虑到瞬态数值计算所需的大量计算时间和资源，本研究仅以完全对称结构的汽车 TEG 系统的 1/4 为研究目标。

In practice, exhaust parameters are time-dependent and related to vehicle driving conditions. Consequently, the transient inlet boundary conditions of exhaust gas in the transient CFD model, including the transient inlet temperature and mass flow rate of the exhaust gas, are defined. However, other boundary conditions, such as coolant inlet, exhaust outlet, coolant outlet, heat loss, and symmetrical surface, are steady-state. To obtain accurate transient exhaust temperature and mass flow rate, the operating conditions of the heavy truck were simulated by the vehicle simulation software package of ADVISOR [46]. On the platform of ADVISOR, the vehicle type was defined as a heavy truck, and the automobile TEG system was generally installed between catalytic converter and muffler; the vehicle weight and cargo weight were set at 7068 kg and 2000 kg respectively; the heavy truck was powered by a diesel engine with a displacement of 7.2 L and a maximum power of 206 kW. Besides, the vehicle was set to operate under the Highway Fuel Economy Test (HWFET) driving cycles. For the first several driving cycles, the exhaust temperature increases dramatically from room temperature, and it reaches an equilibrium state at the 10th driving cycle. At this time, the transient temperature (T_ex(t)) and mass flow rate (

\dot{m}

_ex(t)) of the exhaust gas at the corresponding position were extracted from simulation results and used as the inlet boundary conditions of the transient CFD model, as shown in Fig. 2. Here, the period is 765 s for a complete driving cycle. The exhaust mass flow rate was divided by 4 because only a quarter of the automobile TEG system was considered. It can be observed that the exhaust temperature and mass flow rate change with vehicle speed. However, the change in exhaust mass flow rate is more severe than those in exhaust temperature and vehicle speed because of the influence of thermal inertia on temperature and mechanical inertia on vehicle speed.
实际上，排气参数是随时间变化的，与车辆行驶条件有关。因此，在瞬态 CFD 模型中定义了废气的瞬态入口边界条件，包括废气的瞬态入口温度和质量流量。然而，其他边界条件，如冷却剂入口、排气口、冷却剂出口、热损失和对称面等，则是稳态的。为了获得准确的瞬态排气温度和质量流量，我们使用 ADVISOR 车辆仿真软件包[46]对重型卡车的运行条件进行了仿真。在 ADVISOR 平台上，车辆类型被定义为重型卡车，汽车 TEG 系统一般安装在催化转换器和消声器之间；车辆重量和货物重量分别设定为 7068 千克和 2000 千克；重型卡车由排量为 7.2 L、最大功率为 206 kW 的柴油发动机提供动力。此外，车辆设定在高速公路燃油经济性测试（HWFET）驾驶循环下运行。在前几个驾驶循环中，排气温度从室温急剧上升，在第 10 个驾驶循环时达到平衡状态。此时，从模拟结果中提取相应位置的废气瞬态温度（T _ex (t)）和质量流量（

\dot{m}

_ex (t)），作为瞬态 CFD 模型的入口边界条件，如图 2 所示。这里，一个完整驾驶循环的周期为 765 秒。由于只考虑了汽车 TEG 系统的四分之一，因此排气质量流量除以 4。可以看出，排气温度和质量流量随车速变化。然而，由于热惯性对温度和机械惯性对车速的影响，排气质量流量的变化比排气温度和车速的变化更为剧烈。

Regarding the steady-state boundary conditions of the transient CFD model, an inlet temperature of 363.15 K and a velocity of 1 m/s are defined as the coolant inlet boundary conditions. Compared with a coolant temperature of room temperature as the inlet boundary condition in other research, the coolant temperature of 363.15 K is more in line with the actual situation. A boundary condition with a standard atmosphere pressure was defined on the outlet surfaces of the exhaust and coolant channels. In this study, exhaust gas and coolant are replaced by dry air and water, respectively. The heat loss boundary condition was defined on the surfaces of the automobile TEG system exposed to the surroundings, of which the ambient temperature is 300 K and the natural convection coefficient is 15 W m⁻² K⁻¹. In addition, the symmetric boundary condition was defined on the symmetric surfaces.
关于瞬态 CFD 模型的稳态边界条件，将入口温度为 363.15 K 和速度为 1 m/s 定义为冷却剂入口边界条件。与其他研究中以室温作为冷却剂入口边界条件相比，363.15 K 的冷却剂温度更符合实际情况。在废气和冷却剂通道的出口表面定义了标准大气压的边界条件。在本研究中，废气和冷却剂分别由干燥空气和水代替。热损失边界条件定义在汽车 TEG 系统暴露于周围环境的表面上，环境温度为 300 K，自然对流系数为 15 W m ⁻² K ⁻¹ 。此外，还在对称表面上定义了对称边界条件。

As for the transient thermal-electric numerical model, its boundary conditions include transient temperature boundary conditions and steady-state electric field boundary conditions, in which the transient temperature boundary conditions are extracted from the transient CFD results. Based on the CFD results, the transient average temperatures on the hot and cold side surfaces of the equivalent TEMs are obtained, which are taken as the working temperature of the complete TEMs in the transient thermal-electric numerical model. In the quarter structure of the automobile TEG system, the four TEMs are marked as TEM1, TEM2, TEM3, and TEM4 respectively along the direction of the downward flow of the exhaust gas, as can be found in Fig. 1. Two adjacent TEMs are connected in series with copper electrodes, and the load resistance is connected to the remaining terminal of TEM1 and TEM4, as can be seen in Fig. 4(b). In this way, a closed electric circuit is formed. The terminal between TEM4 and load resistance is set to be grounded.
至于瞬态热-电数值模型，其边界条件包括瞬态温度边界条件和稳态电场边界条件，其中瞬态温度边界条件是从瞬态 CFD 结果中提取的。根据 CFD 结果，得到等效 TEM 冷热侧表面的瞬态平均温度，并将其作为瞬态热-电数值模型中完整 TEM 的工作温度。在汽车 TEG 系统的四分之一结构中，四个 TEM 沿废气向下流动的方向分别标记为 TEM1、TEM2、TEM3 和 TEM4，如图 1 所示。相邻的两个 TEM 用铜电极串联，负载电阻连接到 TEM1 和 TEM4 的剩余端子，如图 4（b）所示。这样就形成了一个闭合电路。TEM4 和负载电阻之间的端子接地。

Fig. 3 shows the relationship between the governing equations and boundary conditions of the hybrid transient CFD-thermoelectric numerical model. As can be seen, the transient CFD model of the equivalent automobile TEG system is solved firstly, under the transient boundary conditions of exhaust gas and other steady-state boundary conditions. Then, the transient hot-side temperatures (T_{h_TEM1}(t), T_{h_TEM2}(t), T_{h_TEM3}(t), and T_{h_TEM4}(t)) and transient cold-side temperatures (T_{c_TEM1}(t), T_{c_TEM2}(t), T_{c_TEM3}(t), and T_{c_TEM4}(t)) of four TEMs extracted from CFD results are used as transient temperature boundary conditions. Combined with the grounded boundary condition of the transient thermal-electric numerical model, the transient thermal-electric numerical model of the complete TEMs is solved, and the dynamic electric response characteristics of the automobile TEG system are obtained. According to the physical field distribution obtained by the transient CFD model and the dynamic output obtained by the transient thermal-electric numerical model, the transient performance of the automobile TEG system is comprehensively analyzed in the following sections.
图 3 显示了瞬态 CFD-热电混合数值模型的控制方程与边界条件之间的关系。可以看出，等效汽车 TEG 系统的瞬态 CFD 模型首先是在废气瞬态边界条件和其他稳态边界条件下求解的。然后，求解瞬态热端温度（T _{h_TEM1} (t)、T _{h_TEM2} (t)、T _{h_TEM3} (t)和 T _{h_TEM4} (t)）和瞬态冷端温度（T _{c_TEM1} (t)、T _{c_TEM2} (t)、T _{c_TEM3} (t)和 T _{c_TEM4} (t)）、T _{c_TEM2} (t)、T _{c_TEM3} (t) 和 T _{c_TEM4} (t)）作为瞬态温度边界条件。结合瞬态热电数值模型的接地边界条件，求解完整 TEM 的瞬态热电数值模型，得到汽车 TEG 系统的动态电响应特性。根据瞬态 CFD 模型得到的物理场分布和瞬态热-电数值模型得到的动态输出，下文将对汽车 TEG 系统的瞬态性能进行全面分析。

2.4. Parameter definitions
2.4.参数定义

Output power and conversion efficiency are the two key parameters to characterize the output performance of the automobile TEG system. Here, the output power of the automobile TEG system is expressed as:(13)

P_{out} (t) = \frac{U_{L}^{2} (t)}{R_{L}}

where U_L is the output voltage of the four TEMs obtained by the transient thermal-electric numerical model. R_L represents the load resistance. According to the maximum power point in the previous study [32], R_L is set to 17 Ω in this work.
输出功率和转换效率是表征汽车 TEG 系统输出性能的两个关键参数。这里，汽车 TEG 系统的输出功率表示为 (13)

P_{out} (t) = \frac{U_{L}^{2} (t)}{R_{L}}

其中 U _L 是通过瞬态热电数值模型获得的四个 TEM 的输出电压。R _L 表示负载电阻。根据先前研究 [32] 中的最大功率点，R _L 在本文中设定为 17 Ω。

Furthermore, the thermal-to-electric conversion efficiency of the automobile TEG system equals the ratio of output power (P_out) to heat extracted from exhaust gas (Q_h), that is:(14)

η (t) = \frac{P_{out} (t)}{Q_{h} (t)} = \frac{P_{out} (t)}{c_{ex} {\dot{m}}_{ex} (t) [T_{exi} (t) - T_{exo} (t)]}

where T_exi(t) and T_exo(t) are the inlet and outlet temperature of the exhaust gas respectively, which are obtained by the transient CFD model.
此外，汽车 TEG 系统的热电转换效率等于输出功率（P _out ）与从废气中提取的热量（Q _h ）之比，即 (14)

η (t) = \frac{P_{out} (t)}{Q_{h} (t)} = \frac{P_{out} (t)}{c_{ex} {\dot{m}}_{ex} (t) [T_{exi} (t) - T_{exo} (t)]}

其中，T _exi (t) 和 T _exo (t) 分别为废气的入口温度和出口温度，由瞬态 CFD 模型求得。

The output performance of TEMs has an important impact on the overall performance of the automobile TEG system. Also, the TEMs located at different positions produce different outputs due to the uneven temperature distribution. To study the influence of the position of the TEM on its output performance, the output power and conversion efficiency of each TEM are defined as follows:(15)

P_{TEMi} (t) = I_{L} (t) U_{TEMi} (t) = \frac{U_{L} (t)}{R_{L}} U_{TEMi} (t)

(16)

η_{TEMi} (t) = \frac{P_{TEMi} (t)}{Q_{TEMi} (t)} = \frac{P_{TEMi} (t)}{λ_{ce} \frac{A_{h_ce}}{δ_{h_ce}} [T_{h_TEMi} (t) - T_{h_cei} (t)]}

where P_TEMi(t), U_TEMi(t), η_TEMi(t), and Q_TEMi(t) are the output power, output voltage, conversion efficiency, and heat absorption of different TEMs, respectively. T_{h_cei}(t) is the average temperature of the hot-side ceramic plate of different TEMs on the top surface. The subscript i = 1, 2, 3, and 4 denotes the TEM1, TEM2, TEM3, and TEM4 respectively. I_L(t) is the output current. A_{h_ce} and δ_{h_ce} are the cross-sectional area and thickness of the hot-side ceramic plate, respectively. Here, the data of P_TEMi(t), U_TEMi(t), η_TEMi(t), Q_TEMi(t), T_{h_cei}(t), and I_L(t) are obtained from the transient thermal-electric numerical model.
TEM 的输出性能对汽车 TEG 系统的整体性能有重要影响。此外，由于温度分布不均匀，位于不同位置的 TEM 会产生不同的输出。为了研究 TEM 位置对其输出性能的影响，对每个 TEM 的输出功率和转换效率定义如下： (15)

P_{TEMi} (t) = I_{L} (t) U_{TEMi} (t) = \frac{U_{L} (t)}{R_{L}} U_{TEMi} (t)

(16)

η_{TEMi} (t) = \frac{P_{TEMi} (t)}{Q_{TEMi} (t)} = \frac{P_{TEMi} (t)}{λ_{ce} \frac{A_{h_ce}}{δ_{h_ce}} [T_{h_TEMi} (t) - T_{h_cei} (t)]}

其中，P _TEMi (t)、U _TEMi (t)、η _TEMi (t) 和 Q _TEMi (t) 分别为不同 TEM 的输出功率、输出电压、转换效率和吸热量。T _{h_cei} (t) 是不同 TEM 上表面热侧陶瓷板的平均温度。下标 i = 1、2、3 和 4 分别表示 TEM1、TEM2、TEM3 和 TEM4。I _L (t) 是输出电流。A _{h_ce} 和 δ _{h_ce} 分别是热侧陶瓷板的横截面积和厚度。这里，P _TEMi (t)、U _TEMi (t)、η _TEMi (t)、Q _TEMi (t)、T _{h_cei} (t) 和 I _L (t) 的数据来自瞬态热电数值模型。

2.5. Grid analysis 2.5.网格分析

The accuracy and computing time of the transient CFD model and transient thermal-electric numerical model are sensitive to the grid parameters of the finite element model. The greater the grid size is, the lower the accuracy and the less the computing time will be. Fig. 4 shows the finite element model of the automobile TEG system, including the equivalent geometry of the CFD model and four complete TEMs of the thermal-electric numerical model. In the equivalent geometry, the tetrahedral mesh is used due to the irregularity of structure, and all meshes were generated according to their specific physical fields, such as the fined mesh at corners and the boundary layer mesh for the contact surfaces between the solid region and fluid region. In the complete TEM geometry, hexahedron mesh was used because of its regular structure, and all meshes were generated by a sweeping method. It is necessary to choose a reasonable grid size to make a tradeoff between model accuracy and computing time.
瞬态 CFD 模型和瞬态热电数值模型的精度和计算时间对有限元模型的网格参数非常敏感。网格尺寸越大，精度越低，计算时间越短。图 4 显示了汽车 TEG 系统的有限元模型，包括 CFD 模型的等效几何体和热电数值模型的四个完整 TEM。在等效几何图形中，由于结构的不规则性，采用了四面体网格，所有网格都是根据其特定的物理场生成的，如边角处的细化网格和固体区与流体区接触面的边界层网格。在完整的 TEM 几何图形中，六面体网格因其规则的结构而被采用，所有网格都是通过扫频方法生成的。有必要选择合理的网格大小，以便在模型精度和计算时间之间做出权衡。

In this work, steady-state numerical calculations of the transient CFD model and the transient thermal-electric numerical model are carried out to check the grid independence. The average exhaust temperature of 615.46 K and average exhaust mass flow rate of 18.26 g/s under the complete HWFET driving cycle are used as steady-state exhaust inlet boundary conditions. Other boundary conditions were the same as those in transient numerical calculation. All numerical calculations were performed in a workstation with 8 cores and 64 G storage. The hot-side temperature of TEM1 and output voltage of four TEMs were taken as the basis for selecting a reasonable grid system of the abovementioned models. As listed in Table 4, the steady-state results of the CFD model and the thermal-electric numerical model under four grid systems are given. Taking the case with the largest number of grids as the baseline, the corresponding error was calculated. The time required for transient numerical calculation is much longer than that of steady-state numerical calculation for a complete HWFET driving cycle. Therefore, to shorten the computing time and maintain the model accuracy, a grid system with 751672 grids was selected to conduct transient numerical calculation of the transient CFD model. Taking the corresponding steady-state temperature as the boundary conditions of the thermal-electric numerical model, the output performance of TEMs under four grid systems was obtained. Compared with the CFD model, the computing time of the thermal-electric numerical model is much less. Similarly, the grid system with 50619 grids was selected to carry out the transient numerical calculation of the transient thermal-electric numerical model.
在这项工作中，对瞬态 CFD 模型和瞬态热电数值模型进行了稳态数值计算，以检验网格的独立性。以完整 HWFET 驱动循环下的平均排气温度 615.46 K 和平均排气质量流量 18.26 g/s 作为稳态排气入口边界条件。其他边界条件与瞬态数值计算中的边界条件相同。所有数值计算均在具有 8 个内核和 64 G 存储空间的工作站中进行。以 TEM1 的热侧温度和四个 TEM 的输出电压为基础，选择上述模型的合理网格系统。表 4 列出了 CFD 模型和热电数值模型在四种网格系统下的稳态结果。以网格数最多的情况为基准，计算了相应的误差。对于一个完整的 HWFET 驱动周期，瞬态数值计算所需的时间远远长于稳态数值计算所需的时间。因此，为了缩短计算时间并保持模型精度，选择了 751672 个网格的网格系统对瞬态 CFD 模型进行瞬态数值计算。以相应的稳态温度作为热-电数值模型的边界条件，得到了四种网格系统下 TEM 的输出性能。与 CFD 模型相比，热-电数值模型的计算时间更短。同样，选择 50619 网格系统进行瞬态热电数值模型的瞬态数值计算。

Table 4. Grid analysis of the hybrid transient CFD-thermoelectric numerical model.
表 4.瞬态 CFD-热电混合数值模型的网格分析。

Steady-state results of the CFD model CFD 模型的稳态结果
grid number 网格数	computing time 计算时间	hot-side temperature of TEM1 TEM1 的热端温度	error of temperature 温度误差
2,952,180	12h16min 12 时 16 分	506.52 K 506.52 K	0
1,268,538	6h48min 6 小时 48 分钟	506.79 K 506.79 K	0.053 %
751,672	4h30min 4 小时 30 分钟	506.92 K 506.92 K	0.079 %
347,888	2h16min 2小时16分钟	507.31 K 507.31 K	0.156 %

Steady-state results of the thermal-electric numerical model 热电数值模型的稳态结果
grid number 网格数	computing time 计算时间	output voltage of four TEMs 四个 TEM 的输出电压	error of output voltage 输出电压误差
99,840	34 min 34 分钟	12.651 V 12.651 V	0
57,885	20 min 20 分钟	12.654 V 12.654 V	0.024 %
50,619	17 min 17 分钟	12.655 V 12.655 V	0.032 %
46,523	12 min 12 分钟	12.659 V 12.659 V	0.063 %

2.6. Model validation 2.6.模型验证

To verify the accuracy of the hybrid transient CFD-thermoelectric numerical model, transient experiments for the automobile TEG system were performed. Fig. 5 shows the designed transient performance test bench. An air heater (F1-R1055, FTV, China) was used to provide a heat source for the automobile TEG system. The air temperature and air velocity can be adjusted instantaneously by the knob on the air heater. The maximum power of the air heater is 5 kW. The tap water with a constant temperature of 284.85 K and a constant flow rate of 21.19 g/s flows through heat sinks of the automobile TEG system to dissipate the heat of TEMs. Two K-type temperature sensors (WRNT, Huarun, China) were installed at the inlet and outlet of the automobile TEG system to measure the air temperatures. The air temperatures were read and recorded by a temperature data logger (RDXL4SD, OMEGA, US) with an accuracy of

\pm

0.4 %. To test the output performance of the automobile TEG system in the load circuit, an electronic load (IT8500+, ITECH, China) was connected in series with the four TEMs in the same row. The load resistance was set to 17 Ω. As the electronic load can not record the transient output voltage, a voltage data logger (KSF, Keshun, China), with an accuracy of ± 0.2 %, was used. To record the transient air velocity, a hot-wire anemometer (HHF-SD1, OMEGA, US) with an accuracy of

\pm

5 %, was installed on the pipe behind the automobile TEG system. The operating temperature of the hot-wire anemometer shall not exceed 50℃. Therefore, an air cooler was installed between the automobile TEG system and the hot-wire anemometer, which was powered by a DC power supply (UTP1305, UNI-T, China). The sampling time of the temperature data logger, voltage data logger, and hot-wire anemometer can be set as 1 s, 2 s, 5 s, 10 s, etc. To obtain more accurate test results, a minimum sampling time of 1 s was adopted.
为了验证 CFD-热电混合瞬态数值模型的准确性，对汽车 TEG 系统进行了瞬态实验。图 5 显示了设计的瞬态性能测试台。汽车 TEG 系统使用空气加热器（F1-R1055，FTV，中国）提供热源。空气温度和风速可通过空气加热器上的旋钮即时调节。空气加热器的最大功率为 5 千瓦。温度恒定为 284.85 K、流速恒定为 21.19 g/s 的自来水流经汽车 TEG 系统的散热器，以散发 TEM 的热量。在汽车 TEG 系统的入口和出口处安装了两个 K 型温度传感器（WRNT，中国华润），用于测量空气温度。空气温度由温度数据记录器（RDXL4SD，美国欧米茄）读取和记录，精度为

\pm

0.4 %。为了测试汽车 TEG 系统在负载电路中的输出性能，在同一排的四个 TEM 上串联了一个电子负载（IT8500+，ITECH，中国）。负载电阻设定为 17 Ω。由于电子负载无法记录瞬态输出电压，因此使用了电压数据记录器（KSF，中国科顺），精度为 ± 0.2 %。为了记录瞬态风速，在汽车 TEG 系统后面的管道上安装了一个热线风速计（HHF-SD1，美国 OMEGA 公司），精度为

\pm

5 %。热线风速计的工作温度不得超过 50℃。因此，在汽车 TEG 系统和热线风速计之间安装了一个空气冷却器，由直流电源（UTP1305，UNI-T，中国）供电。温度数据记录器、电压数据记录器和热线风速计的采样时间可设置为 1 秒、2 秒、5 秒、10 秒等。为获得更准确的测试结果，最小采样时间为 1 秒。

During transient experiments, the transient air temperature and velocity were obtained, which were taken as the transient inlet boundary conditions of the air in the transient CFD model. However, there is a large temperature drop from the inlet to the outlet of the air cooler, resulting in the error between the measured air velocity and the actual one, which is affected by the temperature dependence of air density. To eliminate this error, the inlet air velocity is modified by

v_{i n} = \frac{ρ_{air} (T_{test}) \times v_{test} \times A_{test}}{ρ_{air} (T_{in}) \times A_{in}}

. Furthermore, the transient average temperatures on both ends of the TEMs obtained from the transient CFD model were used as the transient temperature boundary conditions of the transient thermal-electric numerical model. Ultimately, the transient output performance of the automobile TEG system was predicted. Fig. 6 shows the comparison of transient output voltage between experimental and model results. The predicted output voltage enables a greater delay in response than the experimental voltage, which can be attributed to the signal delay of the sensors. When the air temperature changes instantaneously, the temperature sensors will not respond immediately due to the thermal inertia. The average error of transient output voltage between experimental and model results is 6.43 %. Due to the measurement error of instruments and the neglect of thermal grease in the numerical model, this tiny error is acceptable for transient experiments.
在瞬态实验中，获得了瞬态空气温度和速度，并将其作为瞬态 CFD 模型中空气的瞬态入口边界条件。然而，由于空气密度与温度有关，空气冷却器从入口到出口存在较大的温降，导致测量的空气速度与实际速度之间存在误差。为了消除这一误差，对入口空气速度进行了

v_{i n} = \frac{ρ_{air} (T_{test}) \times v_{test} \times A_{test}}{ρ_{air} (T_{in}) \times A_{in}}

修正。此外，瞬态 CFD 模型获得的 TEM 两端的瞬态平均温度被用作瞬态热电数值模型的瞬态温度边界条件。最终，对汽车 TEG 系统的瞬态输出性能进行了预测。图 6 显示了实验结果与模型结果之间的瞬态输出电压对比。与实验电压相比，预测输出电压的响应延迟更大，这可归因于传感器的信号延迟。当空气温度发生瞬时变化时，由于热惯性，温度传感器不会立即做出响应。实验结果与模型结果之间的瞬态输出电压平均误差为 6.43%。由于仪器的测量误差和数值模型中对热油脂的忽略，这一微小误差对于瞬态实验是可以接受的。

3. Results and discussion
3.结果和讨论

3.1. Physical field distribution characteristics of the automobile thermoelectric generator system
3.1.汽车热电发电机系统的物理场分布特征

CFD model is widely used to analyze the thermodynamic performance of automobile TEG systems. Considering the time dependence of exhaust temperature and mass flow rate, the transient CFD model is used to study the dynamic performance of the automobile TEG system under a complete HWFET driving cycle. Fig. 7 shows the physical field distribution of the automobile TEG system at t = 100 s, 300 s, 500 s, and 700 s obtained by the transient CFD model. At t = 100 s, 300 s, 500 s, and 700 s, the exhaust temperatures are 614.48 K, 598.96 K, 620.29 K, and 620.89 K, respectively, and the exhaust mass flow rates are 22.54 g/s, 33.30 g/s, 18.18 g/s, and 27.01 g/s, respectively. In general, the hot-side temperature of the heat exchanger increases with the increase of exhaust temperature and mass flow rate, and the influence of exhaust temperature is more obvious than that of exhaust mass flow rate. According to the temperature distribution of the equivalent automobile TEG system (see Fig. 7(a)), the hot-side temperature of the heat exchanger is the highest at t = 500 s, followed by those of t = 700 s, t = 100 s, and t = 300 s. Although the exhaust mass flow rate at t = 500 s is lower than that at t = 700 s, and the exhaust temperature is almost the same, the hot-side temperature of the heat exchanger at t = 500 s is higher. This can be attributed to thermal inertia, that is, the exhaust temperature drops from a higher level before the time point of t = 500 s, and rises from a lower level before t = 700 s, as can be found in Fig. 2. A similar phenomenon between t = 100 s and t = 300 s can also be explained by this reason.
CFD 模型被广泛用于分析汽车 TEG 系统的热力学性能。考虑到排气温度和质量流量的时间依赖性，瞬态 CFD 模型用于研究汽车 TEG 系统在完整 HWFET 驱动循环下的动态性能。图 7 显示了瞬态 CFD 模型得到的汽车 TEG 系统在 t = 100 s、300 s、500 s 和 700 s 时的物理场分布。在 t = 100 s、300 s、500 s 和 700 s 时，排气温度分别为 614.48 K、598.96 K、620.29 K 和 620.89 K，排气质量流量分别为 22.54 g/s、33.30 g/s、18.18 g/s 和 27.01 g/s。一般来说，热交换器的热端温度随着排气温度和质量流量的增加而增加，排气温度的影响比排气质量流量的影响更为明显。根据等效汽车 TEG 系统的温度分布（见图 7(a)），热交换器的热端温度在 t = 500 s 时最高，其次是 t = 700 s、t = 100 s 和 t = 300 s。这可以归因于热惯性，即排气温度在 t = 500 秒的时间点之前从较高水平下降，在 t = 700 秒之前从较低水平上升，如图 2 所示。在 t = 100 秒和 t = 300 秒之间的类似现象也可以用这个原因来解释。

Fig. 7(b) shows the velocity distribution on the selected cross-sections. The fluid velocity of exhaust gas is directly dependent on the exhaust mass flow rate. Therefore, the fluid velocity of exhaust gas at t = 300 s is the highest, followed by those of t = 700 s, t = 100 s, and t = 500 s. Besides, the closer to the center of the exhaust channel is, the greater the fluid velocity of exhaust gas is.
图 7(b) 显示了所选截面上的速度分布。废气流体速度直接取决于废气质量流量。因此，t = 300 s 时的废气流速最高，其次是 t = 700 s、t = 100 s 和 t = 500 s。

Fig. 7(c) shows the temperature distribution on the hot-side surface of the equivalent TEMs. Similarly, the cold-side temperature distribution can be obtained. By extracting average surface values, the transient average temperature on the hot side of the equivalent TEMs T_{h_TEMi}(t) and that on the cold side T_{c_TEMi}(t) are used as temperature boundary conditions of the transient thermal-electric numerical model with which, the output performance of TEMs can be predicted. Also, the overall hot-side temperature of the four TEMs at t = 500 s is the highest, which presents the same trend as that of Fig. 7(a). As the exhaust gas flows downward, the exhaust temperature decreases accordingly, resulting in the decrease of hot-side temperature from TEM1 to TEM4. Consequently, the outputs of TEMs located at different positions are various, with TEM1 showing the highest output performance. As the four TEMs are connected in series, the overall output current is limited by the smallest output current among the TEMs. Therefore, it is necessary to compare the output performance of different TEMs.
图 7(c) 显示了等效 TEM 热侧表面的温度分布。同样，也可以得到冷侧的温度分布。通过提取平均表面值，等效 TEM 热侧的瞬态平均温度 T _{h_TEMi} (t)和冷侧的瞬态平均温度 T _{c_TEMi} (t) 被用作瞬态热电数值模型的温度边界条件，并以此预测 TEM 的输出性能。此外，四台 TEM 在 t = 500 s 时的整体热侧温度最高，与图 7（a）的趋势相同。随着废气向下流动，废气温度相应降低，导致 TEM1 至 TEM4 的热侧温度降低。因此，位于不同位置的 TEM 的输出各不相同，其中 TEM1 的输出性能最高。由于四个 TEM 是串联连接的，整体输出电流受限于各 TEM 中最小的输出电流。因此，有必要比较不同 TEM 的输出性能。

3.2. Physical field distribution characteristics of thermoelectric modules
3.2.热电模块的物理场分布特征

Combined with the temperature boundary conditions obtained by the transient CFD model, the physical field distribution of the complete TEMs is obtained by the transient thermal-electric numerical model, as shown in Fig. 8. Fig. 8(a) shows the temperature distributions of the complete TEMs. There is a large temperature drop in thermoelectric legs because the thermal conductivity of thermoelectric materials is lower than that of copper and ceramic. The lower the thermal conductivity of thermoelectric materials and the greater the temperature difference on both ends of thermoelectric legs are, the better the output performance is. Besides, the load resistance exhibits the highest temperature due to the effect of Joule heat. The temperature profile over time coincides well with the results presented in Fig. 7.
结合瞬态 CFD 模型得到的温度边界条件，瞬态热电数值模型得到了完整 TEM 的物理场分布，如图 8 所示。图 8(a) 显示了完整 TEM 的温度分布。由于热电材料的热导率低于铜和陶瓷，因此热电腿的温度下降幅度较大。热电材料的热导率越低，热电腿两端的温差越大，输出性能就越好。此外，由于焦耳热的影响，负载电阻的温度最高。随时间变化的温度曲线与图 7 中的结果非常吻合。

Fig. 8(b) shows the voltage distributions of the complete TEMs. The output voltages of the four TEMs at t = 100 s, 300 s, 500 s, and 700 s are 11.03 V, 10.30 V, 13.45 V, and 13.08 V, respectively, which is consistent with the hot-side temperature of the TEMs. This is because the output voltage is proportional to the temperature difference on both sides of the TEMs. Here, the load resistance is 17 Ω and connected with the four TEMs in series. The output power of the automobile TEG system, which is the total output power of the four TEMs, at t = 100 s, 300 s, 500 s, and 700 s are 7.15 W, 6.24 W, 10.64 W, and 10.06 W, respectively. The electrical potential increases from negative to positive with the increase of the number of thermoelectric legs in series. It shows that the more thermoelectric legs or TEMs are, the greater the output voltage of the automobile TEG system will be.
图 8(b) 显示了完整 TEM 的电压分布。在 t = 100 s、300 s、500 s 和 700 s 时，四个 TEM 的输出电压分别为 11.03 V、10.30 V、13.45 V 和 13.08 V，这与 TEM 的热侧温度一致。这是因为输出电压与 TEM 两侧的温差成正比。这里，负载电阻为 17 Ω，与四个 TEM 串联。在 t = 100 秒、300 秒、500 秒和 700 秒时，汽车 TEG 系统的输出功率（即四个 TEM 的总输出功率）分别为 7.15 W、6.24 W、10.64 W 和 10.06 W。随着串联热电腿数量的增加，电动势由负转正。这表明，热电腿或 TEM 越多，汽车 TEG 系统的输出电压就越大。

Fig. 8(c) shows the current density distributions of the complete TEMs. It can be observed that the highest current density occurs in the copper sheets due to the lowest electrical resistivity of copper. The absolute value of the current density of the two adjacent rows of copper sheets is equal, but the direction of the current is opposite. The overall output current can be obtained by multiplying the current density of a selected section by its cross-sectional area.
图 8(c) 显示了完整 TEM 的电流密度分布。可以看出，由于铜的电阻率最低，所以铜片上的电流密度最高。相邻两排铜片的电流密度绝对值相等，但电流方向相反。将所选部分的电流密度乘以其横截面积，即可得到总输出电流。

Driven by the temperature difference, the carriers (electrons in n-type thermoelectric legs and holes in p-type thermoelectric legs) move from the hot side to the cold side. Due to the Peltier effect, carrier dissipation on the hot side will absorb heat, while carrier accumulation on the cold side will release heat. The hot and cold side Peltier heat can be estimated by α(T_h)I_LT_h and α(T_c)I_LT_c, respectively. Fig. 8(d) shows the distribution of αI_LT in thermoelectric legs. Obviously, Peltier heat on the hot side is larger than that on the cold side. This is because the Peltier heat mainly depends on the absolute temperature for the TEMs with the same output current and similar Seebeck coefficient of thermoelectric materials.
在温差的驱动下，载流子（n 型热电脚中的电子和 p 型热电脚中的空穴）从热侧移动到冷侧。由于珀尔帖效应，热侧的载流子耗散将吸收热量，而冷侧的载流子积聚将释放热量。热侧和冷侧的珀尔帖热量可分别通过 α(T _h )I _L T _h 和 α(T _c )I _L T _c 估算。图 8(d) 显示了热电脚中 αI _L T 的分布。很明显，热侧的珀尔帖热大于冷侧的珀尔帖热。这是因为在输出电流相同、热电材料塞贝克系数相似的 TEM 中，珀尔帖热主要取决于绝对温度。

3.3. Dynamic response characteristics of the automobile thermoelectric generator system
3.3.汽车热电发电机系统的动态响应特性

With the variation of exhaust temperature and exhaust mass flow rate, the output power of the automobile TEG system fluctuates, as shown in Fig. 9. When the vehicle is under acceleration or deceleration conditions, the fuel consumption will increase or decrease correspondingly. Therefore, the exhaust mass flow rate changes consistently with the vehicle speed and fluctuates greatly. Compared with the rapid changing of exhaust mass flow rate, the transient variation of exhaust temperature is smoother, which is caused by the thermal inertia of the exhaust system. Furthermore, the change of output power of the automobile TEG system is more stable than that of the exhaust temperature. When 260 s ≤ t ≤ 296 s, the exhaust temperature decreases from 647.43 K to 593.16 K and increases from 593.16 K to 623.22 K when 296 s ≤ t ≤ 350 s. The corresponding output power decreases firstly from 8.58 W to 5.72 W and then increases to 10.11 W. In addition, the output power does not respond to the short-term fluctuation of exhaust temperature, as indicated by the profile near t = 400 s. Due to the rapid changes in exhaust temperature from one level to another, the output power does not have sufficient time to respond before the exhaust temperature returns to the original level. In the process of heat transfer from the exhaust gas to TEMs, there is a hysteresis in response due to thermal inertia.
随着排气温度和排气质量流量的变化，汽车 TEG 系统的输出功率也随之波动，如图 9 所示。当车辆处于加速或减速状态时，油耗会相应增加或减少。因此，排气质量流量随车速变化而变化，波动很大。与排气质量流量的快速变化相比，排气温度的瞬态变化较为平缓，这是排气系统的热惯性造成的。此外，汽车 TEG 系统输出功率的变化比排气温度的变化更稳定。当 260 s ≤ t ≤ 296 s 时，排气温度从 647.43 K 降至 593.16 K，当 296 s ≤ t ≤ 350 s 时，排气温度从 593.16 K 升至 623.22 K。此外，输出功率对排气温度的短期波动没有反应，如 t = 400 s 附近的曲线所示。由于排气温度从一个水平快速变化到另一个水平，输出功率在排气温度恢复到原始水平之前没有足够的时间做出反应。在废气向 TEM 传热的过程中，由于热惯性，会出现响应滞后现象。

Under the complete HWFET driving cycle, the maximum output power of 11.29 W is achieved at t = 467 s. It is worth noting that 1/4 structure of the automobile TEG system is used in this paper. Therefore, the maximum output power of the whole automobile TEG system is 45.16 W. The total electrical energy generated within the period of 765 s is 26460 J. However, the electrical energy predicted through the steady-state performance analysis within the period of 765 s is 28830 J. The energy generation predicted by the transient model is 8.96 % lower than that expected in the steady-state analysis, which indicates that the steady-state performance analysis may overestimate the output performance of the automobile TEG system under the actual driving conditions. Therefore, it is essential to accurately predict the dynamic performance via a reasonable transient model.
在完整的 HWFET 驱动周期下，t = 467 s 时的最大输出功率为 11.29 W。瞬态模型预测的发电量比稳态分析预测的发电量低 8.96%，这表明稳态性能分析可能会高估汽车 TEG 系统在实际驾驶条件下的输出性能。因此，通过合理的瞬态模型准确预测动态性能至关重要。

Fig. 10 shows the dynamic conversion efficiency of the automobile TEG system under a complete HWFET driving cycle. Different from the dynamic output power, the dynamic conversion efficiency fluctuates dramatically. According to Eq. (14), the conversion efficiency is inversely proportional to the exhaust mass flow rate. Therefore, the conversion efficiency increases with the decrease of exhaust mass flow rate, and vice versa. When the exhaust mass flow rate is close to 0, the conversion efficiency reaches the highest value. At t = 635 s, a maximum conversion efficiency of 39.68 % is achieved, which is obviously higher than the reported values (about 2 %) in previous studies. The reason for this is that even though the exhaust mass flow rate is reduced to 0, the hot-side temperature of TEMs remains near the previous state under the effect of thermal inertia. However, the ultrahigh conversion efficiency is only a transient value, which can not reflect the overall conversion efficiency of the automobile TEG system under a complete driving cycle. Here, the average conversion efficiency predicted by the hybrid transient CFD-thermoelectric numerical model is 3.29 % within the period of 765 s, while that predicted by the steady-state model is 1.66 %. It seems that the transient fluctuation of the exhaust heat source may amplify the conversion efficiency of the automobile TEG system.
图 10 显示了汽车 TEG 系统在完整的 HWFET 驱动周期下的动态转换效率。与动态输出功率不同，动态转换效率波动剧烈。根据公式（14），转换效率与排气质量流量成反比。因此，转换效率随排气质量流量的减少而增加，反之亦然。当排气质量流量接近 0 时，转换效率达到最高值。在 t = 635 秒时，转换效率达到最高值 39.68%，明显高于之前研究的报告值（约 2%）。其原因是，即使排气质量流量降至 0，但在热惯性的作用下，TEM 的热侧温度仍接近之前的状态。然而，超高转换效率只是一个瞬时值，并不能反映汽车 TEG 系统在完整驾驶循环下的整体转换效率。在这里，CFD-热电混合瞬态数值模型预测的 765 秒内的平均转换效率为 3.29%，而稳态模型预测的转换效率为 1.66%。由此看来，排气热源的瞬态波动可能会放大汽车 TEG 系统的转换效率。

Fig. 11 shows the dynamic heat absorption of the automobile TEG system extracted from the exhaust gas under a complete HWFET driving cycle. The changing trend of absorbed heat is completely consistent with that of exhaust mass flow rate because the absorbed heat is directly proportional to the exhaust mass flow rate. However, the amplitude of heat absorbed may be greater or less than that of the exhaust mass flow rate. This is because the temperature drop from the inlet to the outlet of the exhaust gas also affects the heat absorption, and there is a delay in the response of the outlet temperature of the exhaust gas.
图 11 显示了汽车 TEG 系统在完整的 HWFET 驱动循环下从废气中提取的动态吸热量。吸收热量的变化趋势与排气质量流量的变化趋势完全一致，因为吸收热量与排气质量流量成正比。不过，吸收热量的幅度可能大于或小于排气质量流量。这是因为废气从入口到出口的温度下降也会影响吸热，而且废气出口温度的反应存在延迟。

3.4. Dynamic response characteristics of thermoelectric modules
3.4.热电模块的动态响应特性

According to the CFD results, the dynamic hot-side temperatures of four TEMs are obtained, as shown in Fig. 12. It can be observed that the hot-side temperature of TEM1 is the highest, followed by those of TEM2, TEM4, and TEM3. In general, the hot-side temperature of the TEM decreases as the exhaust gas flows downward. However, the hot-side temperature of TEM4 is higher than that of TEM3, and the hot-side temperature of TEM1 is much higher than that of TEM2. This is because TEM1 and TEM4 extract heat from exhaust gas not only through the heat exchanger area covered by the TEM but also through the area between the inlet (or outlet) and TEM1 (or TEM4). Besides, the hot-side temperature curve of TEMs is flatter than that of exhaust temperature because of the thermal inertia in the process of heat transfer from the exhaust gas to TEMs. The transient hot-side temperature of TEMs in Fig. 12 is taken as the hot-side temperature boundary condition. Combined with the transient cold-side temperature of TEMs (not shown in the figure) and grounded boundary conditions, the dynamic output performance of TEMs is predicted by the transient thermal-electric numerical model.
根据 CFD 结果，得到了四个 TEM 的动态热侧温度，如图 12 所示。可以看出，TEM1 的热侧温度最高，其次是 TEM2、TEM4 和 TEM3。一般来说，TEM 的热侧温度会随着废气向下流动而降低。不过，TEM4 的热侧温度高于 TEM3，而 TEM1 的热侧温度远高于 TEM2。这是因为 TEM1 和 TEM4 不仅通过 TEM 所覆盖的热交换器区域，还通过进气口（或出气口）与 TEM1（或 TEM4）之间的区域从废气中提取热量。此外，由于废气向 TEM 传热过程中的热惯性，TEM 的热侧温度曲线要比废气温度曲线平缓。图 12 中 TEMs 的瞬态热侧温度作为热侧温度边界条件。结合 TEM 的瞬态冷侧温度（图中未显示）和接地边界条件，瞬态热电数值模型可预测 TEM 的动态输出性能。

Fig. 13 shows the dynamic output power of thermoelectric modules under a complete HWFET driving cycle. Combined with Fig. 10, The conversion efficiency increases when the power generated decreases. The reason is that the heat absorption of the automobile TEG system is significantly decreased when the vehicle speed is close to 0, yet the hot-side temperature will not respond immediately, and the output power only drops slowly. Combined with Fig. 12, it can be considered that the output power curve is obtained by amplifying the hot-side temperature curve. This is because the output power is proportional to the square of the temperature difference of TEMs, which mainly depends on the hot-side temperature of the TEMs as the cold-side temperature of the TEMs is almost constant. The output power of TEM1 is remarkably higher than that of the other TEMs. When all the TEMs are connected electrically in series or parallel, the parasitic power loss caused by the uneven output of TEMs plays a negative role in the overall output performance of the automobile TEG system. This parasitic power loss can not be neglected when there are a great number of TEMs in the automobile TEG system. In this case, the topological optimization of the TEMs is suggested. Besides, the thermal inertia in the heat transfer process from the hot side to the cold side of the TEM is smaller than that from the exhaust gas to TEMs, and there is no response delay in the electrical parameters. Therefore, the changing trend of output power is consistent with that of hot-side temperature.
图 13 显示了热电模块在一个完整的 HWFET 驱动周期下的动态输出功率。结合图 10，当发电功率减小时，转换效率会增加。原因是当车速接近 0 时，汽车 TEG 系统的吸热能力显著下降，但热端温度不会立即做出反应，输出功率只是缓慢下降。结合图 12，可以认为输出功率曲线是通过放大热端温度曲线得到的。这是因为输出功率与 TEM 的温差平方成正比，而 TEM 的温差主要取决于 TEM 的热端温度，因为 TEM 的冷端温度几乎是恒定的。TEM1 的输出功率明显高于其他 TEM。当所有 TEM 都串联或并联电气连接时，TEM 输出不均匀造成的寄生功率损耗会对汽车 TEG 系统的整体输出性能产生负面影响。当汽车 TEG 系统中的 TEM 数量较多时，这种寄生功率损耗不容忽视。在这种情况下，建议对 TEM 进行拓扑优化。此外，从 TEM 的热侧到冷侧的传热过程中的热惯性比从废气到 TEM 的传热过程中的热惯性要小，电参数没有响应延迟。因此，输出功率的变化趋势与热侧温度的变化趋势是一致的。

Different from the conversion efficiency of the automobile TEG system, the conversion efficiency of the TEMs varies more smoothly, as shown in Fig. 14. Here, the conversion efficiency of the TEMs is computed by Eq. (16), and it will not change with the variation of exhaust mass flow rate. Even though the exhaust mass flow rate decreases to 0 at a certain point during the transient state, the TEMs will continue extracting heat from the heat exchanger and converting it into electrical energy. This is because the internal energy in solid regions does not change instantaneously with the exhaust mass flow rate as a result of the thermal inertia effect. The average conversion efficiencies of TEM1, TEM2, TEM3, and TEM4 within the period of 765 s are 1.78 %, 1.63 %, 1.57 %, and 1.61 %, respectively. They are lower than that of the automobile TEG system (3.29 %), which can be explained by the above reasons. The conversion efficiency of TEM4 gradually increases, surpassing the conversion efficiency of TEM3 and reaching close to that of TEM2. Similar trends of hot-side temperature and output power can be observed in Fig. 12 and Fig. 13, respectively. The reason for this is that the exhaust mass flow rate reaches a relatively higher level when t > 300 s, and this larger exhaust mass flow rate leads to a more intense reverse flow of exhaust gas at the outlet of the heat exchanger, thus, enabling TEM4 to absorb more heat.
与汽车 TEG 系统的转换效率不同，TEM 的转换效率变化更为平稳，如图 14 所示。在这里，TEM 的转换效率由公式（16）计算得出，它不会随排气质量流量的变化而变化。即使排气质量流量在瞬态期间的某一时刻降至 0，TEM 仍会继续从热交换器中提取热量并将其转化为电能。这是因为在热惯性效应的作用下，固体区域的内能不会随着排气质量流量的变化而瞬间改变。在 765 秒内，TEM1、TEM2、TEM3 和 TEM4 的平均转换效率分别为 1.78 %、1.63 %、1.57 % 和 1.61 %。它们低于汽车 TEG 系统的转换效率（3.29%），原因如上。TEM4 的转换效率逐渐提高，超过了 TEM3 的转换效率，接近 TEM2 的转换效率。图 12 和图 13 分别显示了类似的热端温度和输出功率趋势。原因是当 t > 300 秒时，废气质量流量达到相对较高的水平，较大的废气质量流量导致热交换器出口处的废气反向流动更加剧烈，从而使 TEM4 能够吸收更多的热量。

Fig. 15 shows the transient heat of the TEMs under a complete HWFET driving cycle. In addition to heat absorbed from the heat exchanger, other forms of heat also occur in the working process of the TEMs, which include Fourier heat along the thermoelectric legs, Peltier heat on both hot and cold sides of thermoelectric legs, and Joule heat within the thermoelectric legs. Fig. 15 (a) shows the amount of these different heat generated in TEM1. In general, the absorbed heat is approximately equal to the hot-side Peltier heat plus Fourier heat and minus half the Joule heat. Fourier heat dominates the heat absorption, while Joule heat only accounts for a small part. For this reason, the changing trend of Fourier heat (see Fig. 15(c)) is consistent with that of absorbed heat (see Fig. 15(b)). Also, the variation of Fourier heat is slightly smoother than that of absorbed heat because of the thermal inertia. Fig. 15(d) and Fig. 15(e) show the hot-side Peltier heat and cold-side Peltier heat of thermoelectric legs for different TEMs, respectively. The difference of hot-side Peltier heat among different TEMs is greater than that of cold-side Peltier heat because the hot-side temperature of different TEMs is not uniform while the cold-side temperature is almost the same. Fig. 15(f) shows the Joule heat of thermoelectric legs for different TEMs. Here, the four TEMs are connected in series, and their internal resistances are almost the same. Therefore, there is no difference between Joule heat curves of different TEMs.
图 15 显示了 TEM 在一个完整的 HWFET 驱动循环下的瞬态热量。除了从热交换器吸收的热量外，TEM 工作过程中还会产生其他形式的热量，包括沿热电腿的傅里叶热、热电腿冷热两侧的珀尔帖热、热电腿内部的焦耳热。图 15 (a) 显示了 TEM1 中产生的这些不同热量。一般来说，吸收的热量约等于热侧珀尔帖热量加上傅立叶热量，再减去焦耳热量的一半。傅立叶热在吸热中占主导地位，而焦耳热只占一小部分。因此，傅立叶热量的变化趋势（见图 15(c)）与吸收热量的变化趋势（见图 15(b)）是一致的。此外，由于热惯性，傅立叶热量的变化比吸收热量的变化略微平滑一些。图 15(d)和图 15(e)分别显示了不同 TEM 热电腿的热侧珀尔帖热量和冷侧珀尔帖热量。不同 TEM 的热侧珀尔帖热量差异大于冷侧珀尔帖热量，这是因为不同 TEM 的热侧温度不均匀，而冷侧温度几乎相同。图 15(f) 显示了不同 TEM 的热电腿焦耳热。在这里，四个 TEM 串联在一起，其内部电阻几乎相同。因此，不同 TEM 的焦耳热曲线之间没有差异。

4. Conclusions 4.结论

In this paper, a hybrid transient CFD-thermoelectric numerical model is proposed, which considers the transient input of the heat source, temperature dependence, impedance matching, and heat loss. The model is used to predict the transient performance of the automobile TEG system under the actual driving conditions. Firstly, a heavy truck is set to run under HWFET driving conditions, and the obtained transient exhaust temperature and mass flow rate are used as the transient heat source input of the model. Then, the hybrid transient CFD-thermoelectric numerical model of the automobile TEG system is solved by using the finite element simulation of COMSOL. The dynamic output performance of the automobile TEG system, such as dynamic output power and conversion efficiency, is obtained. Finally, a transient test is carried out on the designed test bench to verify the accuracy of the hybrid transient CFD-thermoelectric numerical model. Based on the numerical results, the following conclusions are obtained:
本文提出了一种混合瞬态 CFD 热电数值模型，该模型考虑了热源的瞬态输入、温度相关性、阻抗匹配和热损失。该模型用于预测汽车 TEG 系统在实际驾驶条件下的瞬态性能。首先，设定一辆重型卡车在 HWFET 驾驶条件下运行，并将获得的瞬态排气温度和质量流量作为模型的瞬态热源输入。然后，利用 COMSOL 的有限元模拟对汽车 TEG 系统的瞬态 CFD- 热电混合数值模型进行求解。获得了汽车 TEG 系统的动态输出功率和转换效率等动态输出性能。最后，在设计的试验台上进行了瞬态试验，以验证 CFD-热电混合瞬态数值模型的准确性。根据数值结果，得出以下结论：

(1)
The hybrid transient CFD-thermoelectric numerical model can accurately predict the physical field distribution characteristics of the automobile TEG system under any transient exhaust heat source input, which provides a new tool to evaluate the dynamic performance of automobile TEG systems. The average deviation of transient output voltage between the model and experimental results is 6.43 %, which is within the acceptable range.
CFD-热电混合瞬态数值模型可以准确预测汽车TEG系统在任何瞬态排气热源输入下的物理场分布特性，为评估汽车TEG系统的动态性能提供了新的工具。模型与实验结果的瞬态输出电压平均偏差为 6.43%，在可接受范围内。
(2)
Under a complete HWFET driving cycle, the output power of the automobile TEG system changes more smoothly than that of exhaust temperature due to the thermal inertia during the heat transfer process from the exhaust gas to TEMs. For the complete geometry of the automobile TEG system, the maximum output power of 45.16 W is achieved, and the total energy generation is 26460 J within the period of 765 s, which is 8.96 % lower than that expected in the steady-state performance analysis.
在一个完整的 HWFET 驱动循环中，由于废气与 TEMs 传热过程中的热惯性，汽车 TEG 系统的输出功率变化比废气温度变化更为平稳。在汽车 TEG 系统的完整几何形状下，最大输出功率为 45.16 W，在 765 秒内产生的总能量为 26460 J，比稳态性能分析的预期值低 8.96 %。
(3)
Different from the output power, the conversion efficiency of the automobile TEG system fluctuates greatly because it is inversely proportional to the exhaust mass flow rate. The ultra-high transient conversion efficiency of 39.68 % is achieved at t = 635 s and the average conversion efficiency is 3.29 % within the period of 765 s, which is significantly higher than the average conversion efficiency of 1.66 % observed from steady-state performance analysis.
与输出功率不同，汽车 TEG 系统的转换效率波动很大，因为它与排气质量流量成反比。在 t = 635 秒时实现了 39.68 % 的超高瞬态转换效率，765 秒内的平均转换效率为 3.29 %，明显高于稳态性能分析观察到的 1.66 % 的平均转换效率。
(4)
The dynamic conversion efficiency of the TEM is quite different from that of the automobile TEG system because the TEM does not extract heat directly from the exhaust gas. The output performance of the TEM decreases with the downward flow of exhaust gas, except for the TEM in the last row. Besides, the topological relationship among TEMs plays a negative role in the overall output performance of the automobile TEG system, especially when the automobile system contains a great number of TEMs.
TEM 的动态转换效率与汽车 TEG 系统有很大不同，因为 TEM 并不直接从废气中提取热量。除最后一行的 TEM 外，TEM 的输出性能随着废气流量的下降而降低。此外，TEM 之间的拓扑关系对汽车 TEG 系统的整体输出性能起着负面作用，尤其是当汽车系统包含大量 TEM 时。
(5)
In future works, the hybrid transient CFD-thermoelectric numerical model will be further improved by returning the output of the thermal-electric numerical model to the CFD model until convergence is reached for each step. Also, the influence of the neglected heat on the model accuracy will be further studied.
在今后的工作中，将进一步改进 CFD-热电混合瞬态数值模型，将热电数值模型的输出返回到 CFD 模型，直到每一步达到收敛。此外，还将进一步研究被忽略的热量对模型精度的影响。

CRediT authorship contribution statement
CRediT 作者贡献声明

Ding Luo: Conceptualization, Methodology, Software, Writing – original draft. Yuying Yan: . Ying Li: . Ruochen Wang: Software. Shan Cheng: Investigation. Xuelin Yang: Funding acquisition. Dongxu Ji: Supervision.
丁洛构思、方法论、软件、写作 - 原稿。严玉英：.李颖： .王若晨软件Shan Cheng：调查杨学林：：获取资金。Dongxu Ji：监督。

Declaration of Competing Interest
竞争利益声明

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.
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。

Acknowledgements 致谢

This work was supported by the National Key R&D Program of China (Grant No. 2022YFB3807700), the National Natural Science Foundation of China (52072217 and 22179071), the Hubei Natural Science Foundation Innovation Group Project (2022CFA020), as well as Ningbo Science and Technology Bureau’s Technology under Grant No. 2019B10042.
这项工作得到了国家重点研发计划（批准号：2022YFB3807700）、国家自然科学基金（52072217和22179071）、湖北省自然科学基金创新群体项目（2022CFA020）以及宁波市科技局科技资助（批准号：2019B10042）的支持。

Data availability 数据可用性

Data will be made available on request.
数据将应要求提供。

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Geometry optimization of solar thermoelectric generator under different operating conditions via Taguchi method
Energ Conver Manage, 238 (2021), Article 114158
View PDF View article View in Scopus Google Scholar
[35]
K. Nithyanandam, R.L. Mahajan
Evaluation of metal foam based thermoelectric generators for automobile waste heat recovery
Int J Heat Mass Transf, 122 (2018), pp. 877-883
View PDF View article View in Scopus Google Scholar
[36]
Z. Miao, X. Meng, L. Liu
Analyzing and optimizing the power generation performance of thermoelectric generators based on an industrial environment
J Power Sources, 541 (2022), Article 231699
View PDF View article View in Scopus Google Scholar
[37]
W.-H. Chen, C.-Y. Liao, C.-I. Hung
A numerical study on the performance of miniature thermoelectric cooler affected by Thomson effect
Appl Energy, 89 (2012), pp. 464-473
View PDF View article View in Scopus Google Scholar
[38]
D. Luo, R. Wang, W. Yu, W. Zhou
Parametric study of a thermoelectric module used for both power generation and cooling
Renew Energy, 154 (2020), pp. 542-552
View PDF View article View in Scopus Google Scholar
[39]
Y. Shi, X. Chen, Y. Deng, H. Gao, Z. Zhu, G. Ma, et al.
Design and performance of compact thermoelectric generators based on the extended three-dimensional thermal contact interface
Energ Conver Manage, 106 (2015), pp. 110-117
View PDF View article View in Scopus Google Scholar
[40]
A. Massaguer, T. Pujol, M. Comamala, E. Massaguer
Feasibility study on a vehicular thermoelectric generator coupled to an exhaust gas heater to improve aftertreatment’s efficiency in cold-starts
Appl Therm Eng, 167 (2020), Article 114702
View PDF View article View in Scopus Google Scholar
[41]
E. Sandoz-Rosado, R. Stevens
Robust finite element model for the design of thermoelectric modules
J Electron Mater, 39 (2010), pp. 1848-1855
Crossref View in Scopus Google Scholar
[42]
J.D. Anderson, J. Wendt
Computational fluid dynamics
Springer (1995)
Google Scholar
[43]
Fluent A. Fluent 14.0 user’s guide2011.
Google Scholar
[44]
H. Lv, X.-D. Wang, J.-H. Meng, T.-H. Wang, W.-M. Yan
Enhancement of maximum temperature drop across thermoelectric cooler through two-stage design and transient supercooling effect
Appl Energy, 175 (2016), pp. 285-292
View PDF View article View in Scopus Google Scholar
[45]
S. Shittu, G. Li, Q. Xuan, X. Zhao, X. Ma, Y. Cui
Electrical and mechanical analysis of a segmented solar thermoelectric generator under non-uniform heat flux
Energy, 199 (2020), Article 117433
View PDF View article View in Scopus Google Scholar
[46]
K.B. Wipke, M.R. Cuddy, S.D. Burch
ADVISOR 2.1: a user-friendly advanced powertrain simulation using a combined backward/forward approach
IEEE Trans Veh Technol, 48 (1999), pp. 1751-1761
View in Scopus Google Scholar

Cited by (22)

Design and optimal coolant flow rate analysis of an annular water-cooled automotive exhaust-based thermoelectric generator
环形水冷式汽车尾气热电发电机的设计和最佳冷却剂流速分析
2024, Case Studies in Thermal Engineering
2024，热能工程案例研究
This paper introduces a design of an annular water-cooled automotive exhaust-based thermoelectric generator (TEG). A comprehensive numerical model of TEG system is established to investigate the relationship between automotive exhaust temperature and coolant flow rate. Through numerical simulations, it is demonstrated that the implementation of this device can effectively enhance the temperature uniformity at the hot side of the heat exchanger, consequently increasing the TEG's power output. After introducing the annular water-cooled radiator, the temperature uniformity of the thermoelectric module increased by 18.8%. By utilizing the numerical simulation outcomes alongside empirical formulas, calculations for the pump power consumption, maximum output power, and maximum net output power of the thermoelectric module (TEM) are conducted. The maximum net output power can be increased by 3.30%–53.24% by controlling the coolant flow rate. As a result, a relationship curve depicting the average temperature difference between the hot and cold ends of the TEM, in relation to the optimum coolant flow rate, is obtained. This relationship curve serves as a valuable tool for adjusting the radiator's coolant flow rate and augmenting the TEG's net output power, ultimately achieving the objective of heightening the power generation efficiency of the TEG.
Energy and exergy analysis of a thermoelectric generator system for automotive exhaust waste heat recovery
汽车尾气余热回收热电发电机系统的能量和放能分析
2024, Applied Thermal Engineering
2024, 应用热能工程
Thermoelectric generators are a viable solution for recovering and producing energy utilizing wasted energy in automotive exhaust. Exergy analysis can help identify irreversible losses that occur during the transfer of exhaust energy, which is essential for comprehending and developing exhaust thermoelectric generator systems. In this study, we proposed a model based on thermodynamic of the exhaust thermoelectric generator and conducted energy and exergy analyses. The results indicate that convective heat transfer exergy loss and PN junction thermal conductivity exergy loss are the primary exergy losses of the exhaust thermoelectric generator, accounting for more than 52.0 % of the total. Increasing the temperature of exhaust can boost the performance of systems, but it may also reduce the proportion of available energy obtained by the cooling water. When the exhaust heat transfer coefficient is increased to 300, the convective exergy losses of the exhaust can be reduced to 17.6 %, resulting in an overall exergy efficiency of 38.8 %. Furthermore, by increasing the exhaust flow rate from 10 g/s to 50 g/s, the power generation capacity has increased by a factor of 6.87, with little change in exergy losses. These findings are valuable for developing high-efficiency exhaust power generators.
Performance investigation and design optimization of a battery thermal management system with thermoelectric coolers and phase change materials
使用热电冷却器和相变材料的电池热管理系统的性能调查和优化设计
2024, Journal of Cleaner Production
2024 年，《清洁生产杂志》
In this work, a novel battery thermal management system (BTMS) integrated with thermoelectric coolers (TECs) and phase change materials (PCMs) is developed to ensure the temperature working environment of batteries, where a fin framework is adopted to enhance the heat transfer. By establishing a transient thermal-electric-fluid multi-physics field numerical model, the thermal performance of the BTMS is thoroughly examined in two cases. The findings demonstrate that increasing the TEC input current, fin length, and thickness is beneficial for reducing the maximum temperature and PCM liquid fraction. Nevertheless, although the increase in fin length can lower the temperature difference, the influence of fin thickness and TEC input current on the temperature difference is tiny. Based on the numerical findings, the optimal fin length and thickness of 7 mm and 3 mm are obtained. In this situation, when the TEC input current is 3 A, the maximum temperature, temperature difference, and PCM liquid fraction in Case 1 are 315.10 K, 2.39 K, and 0.002, respectively, and those are respectively 318.24 K, 3.60 K, and 0.181 in Case 2. The configuration of Case 1 outperforms that of Case 2, due to the fewer TECs and greater distance from the battery pack to the TEC within Case 2. When experiencing a higher battery discharge rate, the TEC input current should also be correspondingly increased to ensure the temperature performance of the battery. The relative findings contribute to new insights into battery thermal management.
A comprehensive hybrid transient CFD-thermal resistance model for automobile thermoelectric generators
用于汽车热电发电机的混合瞬态 CFD 热阻模型
2023, International Journal of Heat and Mass Transfer
2023 年，《国际传热与传质杂志》
This paper proposes a comprehensive hybrid transient CFD-thermal resistance model to predict the dynamic behaviour of an automobile thermoelectric generator (ATEG) system. The model takes into account the temperature dependences, the topological connection of thermoelectric modules, and the dynamic characteristics, which has the merits of high accuracy and short computational time. The dynamic behaviour of the ATEG system is determined and thoroughly examined using the transient exhaust heat as the heat source input. According to the transient model results, the dynamic output power of the ATEG system keeps the same variation trend with the exhaust temperature, but the variation of output power is more stable. Under the whole driving cycle, the mean power and efficiency of the 1/4 ATEG system are 8.91 W and 3.39% respectively, which are 3.39% lower and 47.52% higher than those expected by steady-state analysis. Beside, the model is validated experimentally, and the mean deviations of the output voltage and outlet air temperature are 7.70% and 1.12% respectively. This model is convenient to evaluate the behaviour of the ATEG system under different topological connections and gives a fresh tool for assessing the dynamic behaviour of ATEG systems.
A hybrid battery thermal management system coupling with PCM and optimized thermoelectric cooling for high-rate discharge condition
针对高倍率放电条件的 PCM 和优化热电冷却耦合混合电池热管理系统
2023, Case Studies in Thermal Engineering
2023，热能工程案例研究
The lithium-ion battery is now abundantly available in the market due to its excellent performance. However, battery overheating is increasingly prominent when the battery discharges at high speed. In this study, an active-passive BTMS (Battery Thermal Management System) combining PCM (Phase Change Material) and TEC (Thermoelectric Cooler) was proposed. The effects of different TEC currents and delayed cooling at different PCM melting rates were analyzed under the 4C discharge condition to enhance the thermal performance. The results demonstrate that battery temperature gets effectively controlled as the TEC current increases, but the temperature difference and PCM utilization are poor. However, applying a delayed current after the PCM melting rate reaches 80% reduces energy consumption and provides better temperature uniformity. Considering the large battery heat under 4C discharge, combined with the transient supercooling effect of TEC, a continuous pulse current was used to provide additional cooling power. Results show that the battery with TEC pulsed current operates 57.8% longer at 40 °C with a temperature difference maintained at 2.5 K under 4C discharge conditions compared with the PCM model.
Optimal design of a heat exchanger for automotive thermoelectric generator systems applied to a passenger car
应用于乘用车的汽车热电发电机系统热交换器的优化设计
2023, Applied Thermal Engineering
2023，应用热能工程
The heat exchanger determines the overall performance of automotive thermoelectric generator (ATEG) systems. To obtain the accurate performance of the ATEG system under actual driving conditions, a fluid-thermal-electric multiphysics numerical model is established. Considering the backpressure loss, weight loss, and pumping power loss, a net power model of the ATEG system is established. The performance of the two heat exchangers with and without fins is investigated and compared. Through optimizations, the optimal parameters for the heat exchanger with fins are N_W = 2 rows, H = 30 mm, and N_L = 5 columns, and those of the heat exchanger without fins are N_W = 2 rows, H = 10 mm, and N_L = 4 columns. The output power, net power, conversion efficiency, and net efficiency of the optimal ATEG system with fins are 35.49 W, 22.93 W, 1.89%, and 1.22%, respectively, and those of the optimal ATEG system without fins are 16.49 W, 8.67 W, 1.51%, and 0.79%, respectively. Through the use of fins, the net power and net efficiency of the ATEG system can be increased by 164.48% and 54.43% respectively. The results are helpful to guide the optimization and design of ATEG systems.

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Evaluation of metal foam based thermoelectric generators for automobile waste heat recovery
Int J Heat Mass Transf, 122 (2018), pp. 877-883
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[36] [36]
Z. Miao, X. Meng, L. Liu
Analyzing and optimizing the power generation performance of thermoelectric generators based on an industrial environment
J Power Sources, 541 (2022), Article 231699
View PDF View article View in Scopus Google Scholar

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W.-H. Chen, C.-Y. Liao, C.-I. Hung
A numerical study on the performance of miniature thermoelectric cooler affected by Thomson effect
Appl Energy, 89 (2012), pp. 464-473
View PDF View article View in Scopus Google Scholar

[38] [38]
D. Luo, R. Wang, W. Yu, W. Zhou
Parametric study of a thermoelectric module used for both power generation and cooling
Renew Energy, 154 (2020), pp. 542-552
View PDF View article View in Scopus Google Scholar

[39] [39]
Y. Shi, X. Chen, Y. Deng, H. Gao, Z. Zhu, G. Ma, et al.
Design and performance of compact thermoelectric generators based on the extended three-dimensional thermal contact interface
Energ Conver Manage, 106 (2015), pp. 110-117
View PDF View article View in Scopus Google Scholar

[40] [40]
A. Massaguer, T. Pujol, M. Comamala, E. Massaguer
Feasibility study on a vehicular thermoelectric generator coupled to an exhaust gas heater to improve aftertreatment’s efficiency in cold-starts
Appl Therm Eng, 167 (2020), Article 114702
View PDF View article View in Scopus Google Scholar

[41] [41]
E. Sandoz-Rosado, R. Stevens
Robust finite element model for the design of thermoelectric modules
J Electron Mater, 39 (2010), pp. 1848-1855
Crossref View in Scopus Google Scholar

[42] [42]
J.D. Anderson, J. Wendt
Computational fluid dynamics
Springer (1995)
Google Scholar

[43] [43]
Fluent A. Fluent 14.0 user’s guide2011.
Google Scholar

[44] [44]
H. Lv, X.-D. Wang, J.-H. Meng, T.-H. Wang, W.-M. Yan
Enhancement of maximum temperature drop across thermoelectric cooler through two-stage design and transient supercooling effect
Appl Energy, 175 (2016), pp. 285-292
View PDF View article View in Scopus Google Scholar

[45] [45]
S. Shittu, G. Li, Q. Xuan, X. Zhao, X. Ma, Y. Cui
Electrical and mechanical analysis of a segmented solar thermoelectric generator under non-uniform heat flux
Energy, 199 (2020), Article 117433
View PDF View article View in Scopus Google Scholar

[46] [46]
K.B. Wipke, M.R. Cuddy, S.D. Burch
ADVISOR 2.1: a user-friendly advanced powertrain simulation using a combined backward/forward approach
IEEE Trans Veh Technol, 48 (1999), pp. 1751-1761
View in Scopus Google Scholar

Outline 概要

Cited by (22) 引用 (22)

Figures (16) 数字 (16)

Tables (4)

Applied Energy 应用能源

Highlights 亮点

Abstract 摘要

Graphical abstract 图形摘要

Keywords 关键词

Nomenclature 术语

Symbols 符号

Greek symbols 希腊文符号

Subscripts 下标

Abbreviations 缩略语

1. Introduction 1.导言

2. Hybrid transient CFD-thermoelectric numerical model
2.混合瞬态 CFD 热电数值模型

2.1. Structure of the automobile thermoelectric generator system
2.1.汽车热电发电机系统结构

2.2. Governing equations of the hybrid transient CFD-thermoelectric numerical model
2.2.CFD- 热电混合瞬态数值模型的控制方程

2.3. Boundary conditions of the hybrid transient CFD-thermoelectric numerical model
2.3.瞬态 CFD-热电混合数值模型的边界条件

2.4. Parameter definitions
2.4.参数定义

2.5. Grid analysis 2.5.网格分析

2.6. Model validation 2.6.模型验证

3. Results and discussion
3.结果和讨论

3.1. Physical field distribution characteristics of the automobile thermoelectric generator system
3.1.汽车热电发电机系统的物理场分布特征

3.2. Physical field distribution characteristics of thermoelectric modules
3.2.热电模块的物理场分布特征

3.3. Dynamic response characteristics of the automobile thermoelectric generator system
3.3.汽车热电发电机系统的动态响应特性

3.4. Dynamic response characteristics of thermoelectric modules
3.4.热电模块的动态响应特性

4. Conclusions 4.结论

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of Competing Interest
竞争利益声明

Acknowledgements 致谢

Data availability 数据可用性

References

Cited by (22)

Design and optimal coolant flow rate analysis of an annular water-cooled automotive exhaust-based thermoelectric generator
环形水冷式汽车尾气热电发电机的设计和最佳冷却剂流速分析

Energy and exergy analysis of a thermoelectric generator system for automotive exhaust waste heat recovery
汽车尾气余热回收热电发电机系统的能量和放能分析

Performance investigation and design optimization of a battery thermal management system with thermoelectric coolers and phase change materials
使用热电冷却器和相变材料的电池热管理系统的性能调查和优化设计

A comprehensive hybrid transient CFD-thermal resistance model for automobile thermoelectric generators
用于汽车热电发电机的混合瞬态 CFD 热阻模型

A hybrid battery thermal management system coupling with PCM and optimized thermoelectric cooling for high-rate discharge condition
针对高倍率放电条件的 PCM 和优化热电冷却耦合混合电池热管理系统

Optimal design of a heat exchanger for automotive thermoelectric generator systems applied to a passenger car
应用于乘用车的汽车热电发电机系统热交换器的优化设计

A hybrid transient CFD-thermoelectric numerical model for automobile thermoelectric generator systems汽车热电发电机系统的混合瞬态 CFD 热电数值模型

Highlights 亮点

Abstract 摘要

Graphical abstract 图形摘要

Keywords 关键词

Nomenclature 术语

Symbols 符号

Greek symbols 希腊文符号

Subscripts 下标

Abbreviations 缩略语

1. Introduction 1.导言

2. Hybrid transient CFD-thermoelectric numerical model2.混合瞬态 CFD 热电数值模型

2.1. Structure of the automobile thermoelectric generator system2.1.汽车热电发电机系统结构

2.2. Governing equations of the hybrid transient CFD-thermoelectric numerical model2.2.CFD- 热电混合瞬态数值模型的控制方程

2.3. Boundary conditions of the hybrid transient CFD-thermoelectric numerical model2.3.瞬态 CFD-热电混合数值模型的边界条件

2.4. Parameter definitions2.4.参数定义

2.5. Grid analysis 2.5.网格分析

2.6. Model validation 2.6.模型验证

3. Results and discussion3.结果和讨论

3.1. Physical field distribution characteristics of the automobile thermoelectric generator system3.1.汽车热电发电机系统的物理场分布特征

3.2. Physical field distribution characteristics of thermoelectric modules3.2.热电模块的物理场分布特征

3.3. Dynamic response characteristics of the automobile thermoelectric generator system3.3.汽车热电发电机系统的动态响应特性

3.4. Dynamic response characteristics of thermoelectric modules3.4.热电模块的动态响应特性

4. Conclusions 4.结论

CRediT authorship contribution statementCRediT 作者贡献声明

Declaration of Competing Interest竞争利益声明

Acknowledgements 致谢

Data availability 数据可用性

References

A hybrid transient CFD-thermoelectric numerical model for automobile thermoelectric generator systems
汽车热电发电机系统的混合瞬态 CFD 热电数值模型

2. Hybrid transient CFD-thermoelectric numerical model
2.混合瞬态 CFD 热电数值模型

2.1. Structure of the automobile thermoelectric generator system
2.1.汽车热电发电机系统结构

2.2. Governing equations of the hybrid transient CFD-thermoelectric numerical model
2.2.CFD- 热电混合瞬态数值模型的控制方程

2.3. Boundary conditions of the hybrid transient CFD-thermoelectric numerical model
2.3.瞬态 CFD-热电混合数值模型的边界条件

2.4. Parameter definitions
2.4.参数定义

3. Results and discussion
3.结果和讨论

3.1. Physical field distribution characteristics of the automobile thermoelectric generator system
3.1.汽车热电发电机系统的物理场分布特征

3.2. Physical field distribution characteristics of thermoelectric modules
3.2.热电模块的物理场分布特征

3.3. Dynamic response characteristics of the automobile thermoelectric generator system
3.3.汽车热电发电机系统的动态响应特性

3.4. Dynamic response characteristics of thermoelectric modules
3.4.热电模块的动态响应特性

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of Competing Interest
竞争利益声明