Elsevier

Applied Energy

Volume 129, 15 September 2014, Pages 373-383
Applied Energy

High fidelity finite difference model for exploring multi-parameter thermoelectric generator design space

https://doi.org/10.1016/j.apenergy.2014.04.088Get rights and content

Highlights

  • A system level numerical model for thermoelectric power generation.

  • 1st and 2nd generation Mg and Mn silicide TE devices fabricated.

  • The numerical model validated experimentally for power output.

  • TEG module geometric design space parameters optimized numerically.

  • Effects of thermal and electrical contact resistance quantified.

Abstract

Thermoelectric generators (TEGs) are being studied and developed for applications in which waste heat, for example, from the exhaust of motor vehicles is converted into usable electricity. TEGs consisting of TE elements integrated with an exhaust heat exchanger require optimization to produce the maximum possible power output. Important optimization parameters include TE element leg length, fill fraction, leg area ratio between n- and p-type legs, and load resistance. A finite difference model was developed to study the interdependencies among these optimization parameters for thermoelectric elements integrated with an exhaust gas heat exchanger. The present study was carried out for TE devices made from n-type Mg2Si and p-type MnSi1.8 based silicides, which are promising TE materials for use at high temperatures associated with some exhaust heat recovery systems. The model uses specified convection boundary conditions instead of specified temperature boundary conditions to duplicate realistic operating conditions for a waste heat recovery system installed in the exhaust of a vehicle. The 1st generation, and an improved 2nd generation TEG module using Mg2Si and p-type MnSi1.8 based silicides were fabricated and tested to compare TE power generation with the numerical model. Important results include parameter values for maximum power output per unit area and the interdependencies among those parameters. Heat transfer through the void areas was neglected in the numerical model. When thermal contact resistance between the TE element and the heat exchangers is considered negligible, the numerical model predicts that any volume of TE material can produce the same power per unit area, given the parameters are accurately optimized. Incorporating the thermal contact resistance, the numerical model predicts that the peak power output is greater for longer TE elements with larger leg areas. The optimization results present strategies to improve the performance of TEG modules used for waste heat recovery systems.

Introduction

Energy conservation is becoming increasingly important because of growing energy demands. This compels us to promote the development of energy efficient systems. Exhaust gases from the tail pipes of motor vehicles are responsible for dissipating roughly one third of the energy content of the fuel to the environment as heat [1], [2], [3]. Therefore, various waste heat recovery technologies are being investigated to capture that waste heat. Such technologies include organic Rankine cycles [4], turbo-compounding [5], direct use of the waste heat as thermal energy, power absorption chillers [6], and thermoelectric devices [4], [7], [8], [9], [10]. Thermoelectric devices in particular have gathered considerable attention in the last few decades. They are solid state heat engines that use the Seebeck effect to directly convert heat to electrical energy. They operate quietly, without moving parts [11], and are compact and lightweight. However, thermoelectric devices have low efficiency (<5%) in addition to being expensive; this has limited their widespread use. Nonetheless, the potential use of relatively inexpensive materials like Mg2Si and p-type MnSi1.8 based silicides shows promise for cost reduction and improved performance. A schematic of a typical TEG (Thermoelectric Generator) module is shown in Fig. 1. System level thermoelectric heat exchangers for waste heat recovery consist of several TEG modules integrated with a compact heat exchanger that can extract maximum heat from the exhaust flow.

Substantial research has been done in recent years regarding the design and optimization of thermoelectric heat exchangers. Applications of various thermoelectric materials exhibiting peak thermoelectric efficiencies at different temperature ranges have been studied in the past for installation in exhaust heat recovery systems [12], [13], [14]. Stobart and coworkers modeled and experimentally tested the TE performance of devices with exhaust temperatures up to 800 K. Thermal asymmetry between the hot-side and cold-side accounting for the difference in heat transfer due to internal electric energy conversion was not considered by Stobart et al. Their model was based on the average figure of merit (ZT) of the TE material that assumes optimal device geometry and optimal current [15], [16]. Geometric parameters that affect TE performance include n-/p-type leg area ratio, leg length, the area of individual legs and the distance between adjacent legs. Modeling efforts by Hendricks et al. [7], [10] considered temperature-dependent TE properties to determine optimal TE leg areas, lengths, and device designs. Modeling carried out by Xuan and Cheng [17], [18], [19], which dealt with thermoelectric coolers, considered the geometrical optimization of the length of TE legs only. Miller et al. studied heat transfer and heat exchanger optimization for a combined TE and organic Rankine cycle waste heat recovery system. Miller et al. calculated the TE device efficiency based on an average ZT for state of the art TE materials evaluated at typical operating temperatures [4]. A report by Wang and Dai presented an optimization study on thermoelectric generator systems for waste heat recovery in motor vehicles. They discovered that power output was more sensitive to hot side heat transfer coefficient than the cold side. They also discovered other key optimization results for TEG modules such as sensitivity of peak power to PN junction height and convection heat transfer coefficients [20]. Chen and Li et al. considered a two stage TEG module system for their study and reported optimization studies for the heat transfer surface area as well as the total number of TE elements on each TEG stage for peak power output [21]. A study by Gou, Xiao, and Yang presented a system model for low-temperature thermoelectric generators for use in waste heat recovery systems, which was verified with some experiments. An appropriate range of heat sink surface area as well as cold-side heat transfer capacity for peak power output were discussed [22]. A study by Matsubara et al. based on thermoelectric stacks composed of segmented legs projected highly efficient systems (up to 10%) [23]. Such efficiencies could produce enough power to supplement a vehicle alternator or perhaps replace it altogether [11], [24]. Hussain et al. developed a model with thermally lumped TE devices that accounted for transient behavior and thermal asymmetry. In their model, the spatial variation and temperature dependence of the TE properties in individual TE devices was accounted for. Crane and coworkers developed a system level model integrating thermoelectric devices with a heat exchanger for cross-flow and counter-flow heat exchanger configurations. They used an analytical, thermally lumped TE leg performance model that correctly accounted for thermal asymmetry. Some of this modeling work was validated with experimental results. Crane’s most recent model incorporated transient performance. The report also presented a novel high-power-density subassembly of the TEG module that has various advantages over the conventional TEG assembly [9], [25], [26]. Kumar and coworkers presented a thermal resistance based numerical model to study the electrical power output and pressure drop for various flow rates for a General Motors Co. prototype TE generator designed for a Chevrolet Suburban [27]. This study incorporated junction-averaged temperature dependent TE properties. A recent report by Espinoza and coworkers reported modeling efforts which take into account the temperature dependent properties along the heat exchanger, but not within the legs [28]. In addition, vehicle manufacturers including BMW, Ford, GM, and Ford have all been involved in studies on thermoelectric heat exchangers for automobiles in partnership with the US Department of Energy [11].

Optimization studies on TEGs in the past have assumed a constant temperature boundary condition for simplified analysis. However, for a thermoelectric waste heat recovery system installed in the exhaust of a vehicle, these temperatures will be dictated by convection heat transfer on both sides (exhaust and coolant). Therefore, a more appropriate/realistic boundary condition to govern the performance of TEGs is convection heat transfer rather than constant temperature boundary conditions for TE junctions. Gomez et al. reported a modeling study incorporating constant reservoir temperatures instead of constant TE element junction temperatures. They discussed the relationship between fill fraction, leg length, and the ratio of load resistance and internal resistance for optimal performance [29]. However, their model did not account for temperature dependent TE properties. The study was carried out for low temperature ranges (∼350 K) based on experimentally measured TE properties of a commercial TEG module.

None of the previous work discussed here used a TE model that accounted for spatial- and temperature-variant properties within the TE material of an individual TE couple. In addition, the majority of them have focused mainly on the optimization of the heat exchanger geometry only. The ones that discuss the optimization of TEG module geometry itself do not consider the interdependencies between the optimization variables like leg area ratio, area of individual TE legs, spacing between the legs, and the load resistance. A comprehensive study regarding the optimization of TEG modules that includes such analysis is lacking.

The present study focuses primarily on modeling the performance and optimization of TEG modules integrated with a system level heat exchanger and using Mg2Si and p-type MnSi1.8 based silicides as the TE materials. A numerical model was developed using a finite difference technique which accounts for spatial- and temperature- dependent thermoelectric properties. The model also couples hot-side and cold-side convection heat flux, thus accounting for the thermal asymmetry using a numerical root finding algorithm. The model was developed to understand and analyze the interdependencies among various parameters such as the height of the TEG modules, volume of TE material, the area ratio between n-type and p-type legs, and the load resistance. The set of parameters that provide the greatest power conversion for a given heat exchanger and operating conditions is determined. In addition, two TE leg pair devices, each with a different approach to fabrication, were assembled, and an experimental setup was constructed to validate the numerical model. Electrical contact resistances have been experimentally measured and incorporated into the numerical model for validation purposes. One unique feature of this study is that it is focused on Mg and Mn silicides. These materials have potential cost advantages with comparable ZT values at higher temperatures [30], suitable for integration into heat exchangers employed in diesel and gasoline vehicles as opposed to conventional TE materials, e.g. bismuth telluride, lead telluride, and silicon germanium alloys.

Section snippets

Thermoelectric device model

To account for the temperature dependence of the TE properties in the direction of heat conduction and the thermal asymmetry, a TE device finite difference model was developed. The TE leg pairs were modeled at the device level using an iterative finite difference algorithm described by Hogan and Shih [31]. A schematic of the finite difference numerical scheme is shown in Fig. 2. The equations that govern the heat transfer and temperature distribution are:ΔTiΔx=1ki-1JTi-1αi-1-qi-1ΔqiΔx=ρi-1J21+Zi

Device fabrication

Two different TE devices – the 1st and 2nd generation, each composed of a single TE leg pair were fabricated for the purpose of experimental model validation. The compositions of the TE elements used in the 1st and 2nd generation TE devices are shown in Table 1. The dimensions of the TE elements are listed in Table 2. The 1st generation TE device was assembled by bonding copper interconnects directly to the TE elements using a commercial silver paste with high thermal and electrical

Model results

As mentioned above, four parameters, leg length, fill fraction, leg area ratio, and load resistance were considered for optimization of a TEG. Boundary conditions used for this modeling study are listed in Table 4. To simulate a typical finned heat exchanger geometry, the overall heat transfer coefficients averaged in the streamwise direction for both the hot and cold sides were used, as previously reported by Baker et al. [33]. These numbers were reported for a heat exchanger installed in the

Conclusions

This paper introduces a numerical model for predicting power output of thermoelectric heat exchangers using specified convection and temperature boundary conditions. The study was carried out for a heat exchanger installed in the exhaust of a Cummins 6.7 L diesel engine designed for TEG modules that are to use Mg2Si and p-type MnSi1.8 based silicides as TE materials. The model was used to optimize TEG module fill fraction, leg length of TE elements, leg area ratio between n- and p- type legs,

Acknowledgements

This project was carried out under a grant from the NSF/DOE Joint Thermoelectric Partnership, award number CBET-1048767. Libin Zhang and Xi Chen are thanked for their help in preparing the silicide thermoelectric materials used in the experiments.

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      Nonetheless, device research focusing on maximizing power output from human body heat remains scarce. There have been a few studies on utilizing materials (including thermoelectric material itself, contact materials, electrodes, and substrate materials) with better thermal properties,13,21–23 tuning the fill factor and the geometry of the thermoelectric pillars,24–33 and adoption of novel configurations of the thermoelectric pillars and heat exchangers.34–39 Despite theoretical studies that have suggested further improvement based on device-level optimization or heat transfer analysis,30,32 there is still a lack of effective and economical design that would generally work for wearable thermoelectric devices.

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