Elsevier

Applied Energy

Volume 239, 1 April 2019, Pages 1003-1013
Applied Energy

Design of segmented thermoelectric Peltier coolers by topology optimization

https://doi.org/10.1016/j.apenergy.2019.01.247Get rights and content

Highlights

  • A numerical material distribution method is presented and used to optimize TECs.

  • Important model parameters of TECs are identified and discussed.

  • Heat flux and conversion efficiency are improved by up to 48% and 11%, respectively.

  • Insight in segmentation of TECs are provided and an analytic approach is suggested.

Abstract

A density-based topology optimization approach is used to optimize the cooling power and efficiency (coefficient of performance) of thermoelectric coolers by spatially distributing two different thermoelectric materials in a two dimensional design space. With basis in three numerical examples we identify important model parameters, such as the choice of objective function, the temperatures of the thermal reservoirs, the heat transfer rates and the available electrical energy. By using the topology optimization approach, we demonstrate that the cooling power and efficiency of thermoelectric coolers can be improved by 48.7% and 11.4%, respectively, compared to optimization results from in the literature.

Introduction

Thermoelectricity is a physical phenomenon which concerns the interaction between electric and thermal energy in semi-conducting materials. Thermoelectricity can be characterized by two separately identified effects, the Seebeck effect which concerns the conversion of thermal energy into electric energy, and the Peltier effect which concerns the conversion of electric energy into thermal energy [1]. With reference to the sketch in Fig. 1, a thermoelectric cooler is a solid-state heat pump which uses the Peltier effect to convert electrical energy into a thermal energy flux and hereby providing cooling power at a specified surface.

Compared to vapor-compression refrigeration systems, thermoelectric coolers offer reliable and silent operation due to the simple system designs without moving parts and circulating fluids. Due to the miniature scales of the systems and their flexibility in packaging and integration, thermoelectric coolers are often seen in applications, such as cooling of electronics [2]. As thermoelectric coolers are fabricated and operate without using chlorofluorocarbons or other chemicals that may be harmful to the environment, thermoelectric coolers are often considered as an environmentally friendly alternative to conventional refrigeration systems. Thermoelectric coolers are therefore by many researchers in science and industry often predicted to be an important entrant in the green energy changeover [3].

A thermoelectric cooler consists of a number of modules which are connected electrically in series and thermally in parallel, see Fig. 1. The modules are build up by three main components: conductors, legs and substrates. The conductors connect the legs electrically and the substrates constitute the interface between the ambient and the compartment. Thermoelectric coolers can be enhanced by so-called segmentation where two dissimilar thermoelectric materials are connected thermally and electrically in simple one dimensional interfaces. However, it is still an open question if the segmentation approaches can be improved further by allowing two and three dimensional geometries of the segmented materials. In the present study we seek to answer this question by using a density-based topology optimization approach to spatially distribute two different thermoelectric materials in a thermoelectric leg and hereby optimize the cooling power and efficiency of thermoelectric coolers. The paper is therefore concerned with energy conversion, optimal use of energy resources and analysis and optimization of energy processes.

Compared to vapor-compression refrigeration systems, thermoelectric coolers are so far limited to niche applications due to their relatively low operational cooling power and efficiency (coefficient of performance). Despite a considerable amount of scientific efforts, performance improvements of thermoelectric coolers are still required to increase the range of applications [4], [3]. The main efforts to increase the performance of thermoelectric coolers have so far been a broad search for identification and development of advanced thermoelectric materials [5], [6], however, in this paper we address a purely mathematical optimization approach aiming at finding the best spatial distributions of available materials in order to optimize a specified performance measure.

In the literature, the performance of thermoelectric coolers has been characterized in different ways, see e.g. Seifert, Müller, and Walczak [7] or Bian, Wang, Zhou, and Shakouri [8]. As we see it, the performance measures can be divided into four categories: (A) the temperature at the compartment surface, fT. (B) the heat flux at the compartment surface, fQ. (C) the coefficient of performance at the compartment surface, fμ. (D) and the dimensionless figure-of-merit of the device, fZT=ασ/κ, where α is the Seebeck coefficient, σ is the electric conductivity, κ is the thermal conductivity and T is the temperature. In this study we address objectives (A), (B) and (C).

Only a minor part of the scientific efforts concerned with improving thermoelectric energy conversion takes basis in mathematical optimization approaches. The available approaches can generally be sorted in three categories: (a) functionally graded material studies, (b) compatibility and segmentation approaches, and (c) geometrical optimization approaches. The topology optimization approach proposed in present thesis is a sub-class of (b).

Functionally graded material studies are aiming at identifying spatial profiles of relevant material parameters which optimize a prescribed performance measure of thermoelectric coolers and generators. The design solutions of functionally graded material studies are characterized by macroscopic gradients in the material parameters, which may be linked to the composition (including doping) or micro structures of the functional properties of the material [9].

In the works of Müller, Walczak, and Seifert [10], Bian and Shakouri [11], Bian and Shakouri [12] and Bian, Wang, Zhou, and Shakouri [8], the coefficients in arbitrary interpolation functions for the spatial profiles of Seebeck coefficient, α(x), and the electric conductivity, σ(x), were optimized with basis in parameter studies and non-gradient algorithms. Various types of genetic algorithms have also been applied to optimize the properties of thermoelectric devices in the work of Heghmann and co-workers [13], Chen and co-workers [14] and Wang and co-workers [15]. Such algorithms are inadequate for design problems with many design variables such as the density-based topology optimization approach [16] used in the present study.

Later, a gradient-based optimization approach for functionally graded materials was introduced in Gerstenmaier and Wachutka [17] and later extended to physically realistic boundary conditions in Gerstenmaier and Wachutka [18]. The topology optimization methodology [19], [20] used in this study is also gradient-based and supports the same type of boundary conditions as Gerstenmaier and Wachutka, however, the two methodologies take completely different offset and modeling approaches and hereby result in completely different design solutions.

Compatibility approaches were originally suggested for thermoelectric generators in the work of Ursell and Snyder [21] and have later been developed in a series of studies in e.g. Seifert, Müller, and Walczak [22], Snyder, Toberer, Khanna, and Seifert [23] and Seifert, Pluschke, and Hinsche [24]. By identification of compatible materials, it has been shown that the performance of thermoelectic generators and thermoelectic coolers can be considerably improved by segmentation. Compatible materials operate optimally under the same external electrical resistance and are therefore suited for being segmented, i.e. connected thermally and electrically in series. The design solutions of the compatibility approach are generally characterized by one dimensional (1D) line interfaces between the materials phases, where the design solutions of the topology optimization approach support arbitrary two dimensional (2D) features. The compatibility approach is, as the functionally graded material approach, related to the topology optimization methodology, however the approaches take very different offsets and converge to different design solutions.

By studying the volume fraction between two materials connected thermally and electrically in series, Yang, Xie, Ma, and Lei [25] presented a mathematical optimization approach aiming at increasing the effective figure-of-merit of two segmented materials. It was shown that the figure-of-merit of the composite medium could exceed the figure-of-merit of the constitutive materials, if the electric potential difference was chosen sufficiently large when evaluating the electric conductivity. A related approach was utilized to optimize the conversion efficiency in Yang, Ma, Lei, and Liu [26].

System configurations where a vertically directed heat flux is converted into a horizontally directed electric current are often referred to as off-diagonal problems. These problems were addressed in Sakai, Kanno, Takahashi, Tamaki, Kusada, Yamada, and Abe [27], who studied the tilting angle and volume fraction between two segmented materials in order to optimize the device figure-of-merit. The approach, which has been theoretically improved and discussed in Lundgaard and Sigmund [28], was limited to fixed temperature boundary conditions, simple topological design solutions and constant material parameters.

In the work of Schilz, Müuller, Helmers, Kang, Noda, and Niino [9] and Müuller, Walczak, and Seifert [10], the cooling power of thermoelectric coolers was optimized by maximizing the local figure-of-merit with respect to the local temperature conditions of the device during operation. The interaction between figure-of-merit and electric power output for thermoelectric generators was addressed in Lundgaard and Sigmund [28], and we do therefore not consider this measure in the present paper.

The topology optimization approach used in this study uses a completely different offset and modeling approach and converges to different design solutions compared to the functionally graded material, compatibility and homogenization approaches. The topology optimized design solutions are characterized by two separately identified material phases and two dimensional features, and if the design problems are solved for physical material parameters, the design solutions can straight-forwardly be interpreted and manufactured without any consideration of the local functional properties of the materials.

Section snippets

The design problem

The topology optimization methodology [19], [20] used in this study is based on a finite element formulation of the generalized Ohm’s and Fourier’s law [29], the method of moving asymptotes [30], adjoint sensitivity analysis [31], and various filter operations [32], [33]. The framework supports advanced physical modeling concepts such as temperature dependent material parameters, complex geometries and advanced boundary conditions. A detailed description and implementation details of the

Results

Five numerical examples are presented to demonstrate that the topology optimization methodology is suitable for optimizing thermoelectric coolers. By identifying and discussing important model parameters such as the objective functions, Section 3.1; the temperatures of thermal reservoirs, Section 3.2; the electric power supply, Section 3.3; the heat transfer rates, 3.4; and the features of the design solutions, Section 3.5.

The design solutions presented throughout the paper are dependent on the

Discussion

The topology optimization approach for thermoelectric coolers presented in this paper is related to well-accepted work in the literature such as functionally graded materials [37], the compatibility approach [21], the thermoelectric homogenization approach [25], [27] and sizing approaches [13], however the methodology takes a completely different offset and modeling approach and therefore opens a complete new branch for optimization of thermoelectric coolers.

Conclusion

A density-based topology optimization approach is used to optimize the spatial distribution of two materials in order to optimize the performance of thermoelectric coolers. The design problems are solved for physically realistic boundary conditions and model dimensions, however, the design problems are purposely limited to temperature independent materials in order to ease the interpretation of the design features. The physical modeling is based on a fully coupled non-linear finite element

Acknowledgements

The authors acknowledge the financial support received from the TopTen project sponsored by the Danish Council for Independent Research (DFF-4005-00320).

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