STRUCTURAL OPTIMIZATION OF A THERMOELECTRIC GENERATOR BY NUMERICAL SIMULATION

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1 Électrotechnique et électroénergétique STRUCTURAL OPTIMIZATION OF A THERMOELECTRIC GENERATOR BY NUMERICAL SIMULATION ALEXANDRU MIHAIL MOREGA 1,2, MIHAELA MOREGA 1, MARIUS A. PANAIT 1 Key words: Thermoelectric cells, numerical simulation, structural optimization. There is a growing interest in utilizing electrical power sources that recuperate part of the exergy destruction by conventional power systems, and thermoelectric generators (TEGs) are sound candidates. This paper reports a mathematical model based on Onsager formalism that relates gradients (temperature, electric potential) to fluxes (heat flux, electrical current) and its numerical, finite element (FEM) implementation that is used to simulate the underlying, multiphysics processes. In this study, an elemental TEG cell the simplest system under investigation is optimized for maximum efficiency. Then, it is packed into more complex, higher order ensembles that inherit its outlining features. This growing technique is derived from the constructal theory that explains shape and structure in systems of finite size (volume, resources), with internal flows (e.g., heat flux, currents), subject to specific internal and external constraints. 1. INTRODUCTION Recently, there is a growing interest in utilizing electrical power sources that recover part of the exergy lost in conventional power systems. For instance, the efficiency of a modern internal combustion engine is relatively low (37 50%), and much of the available work is lost to the environment. Thermoelectric generators (TEGs) are an alternative, zero-emission technology solution [1]; they have no moving parts and do not pollute. TEG modules are commercially available. Micro TEG (µ-teg) devices are designed following the system-on-a-chip concept to provide for a high degree of integration of thermal management, power, and electronics. Still under research [2], they are developed using integrated-type fabrication processes, electrochemical deposition, and high thermal conductivity substrates. Recent reports [3] suggest that nanoengineered TE materials could present higher values of the thermoelectric figure of 1 Faculty of Electrical Engineering, University POLITEHNICA of Bucharest, Splaiul Independeţei nr. 313, sector 6, Bucharest, , Romania, amm@iem.pub.ro 2 Gheorghe Mihoc Caius Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest, Romania. Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 55, 1, p. 3 12, Bucarest, 2010

2 4 Alexandru Mihail Morega, Mihaela Morega, Marius Alexandru Panait 2 merit, σ. A value of 2.4 for σ can be achieved in Sb 2 Te 3 / Bi 2 Te 3 thin films about 1 nm thick [4] fabrication of thin film µ-teg is cited by [5]. Numerical study on µ-teg has also gained attention, for instance the FEM analysis of thermoelectric and thermomagnetic effect reported in [6]. One of the main goals of the TEG design is to determine the optimum cell geometry, for available thermoelectric material and manufacturing technology, and meeting the given application specifications. In this paper we report a mathematical model that may be used to simulate the outlining processes in TEGs [6], based on the Onsager formalism that relates gradients (of temperature, electric potential) to fluxes (heat flux, electric current), and on specific constitutive (material) laws. First, the elemental TEG is defined and optimized for maximum thermodynamic performance (efficiency). Next, the optimized cell is packed into more complex, higher order ensembles. This growing technique is derived by the constructal theory, which explains shape and structure of systems with internal flows (e.g., heat flux, currents), of finite size (volume, resources), and under specific internal and external constraints [7]. The outcome is optimal, scalable TEG ensembles. The models and optimization technique developed in this study are helpful in reducing the time and cost of the design and development of TEGs. 2. THE ELEMENTAL SYSTEM, ITS STRUCTURAL OPTIMIZATION 2.1. STRUCTURAL OPTIMIZATION Figure 1 shows the notional schematic of the elemental TEG cell. Fig. 1 The structural optimization of the elemental TEG cell-notional representation; α and β are symbols used to distinguish the two columns.

3 3 Structural optimization of a thermoelectric generator 5 Thermoelectric voltage is produced when the cell is subject to a temperature gradient between the hot (top) and cold (bottom) end, the T H and T L heat sources. Basically, the cell works as a temperature-controlled voltage source. Its columns are made of materials of different Seebeck coefficients, e.g. n and p-doped InSb. We assume that the material properties are linear, homogeneous, isotropic, temperature independent (this assumption will be discarded later). The p-n junction, exposed at the hot end, is a shunt, electrically insulated with respect to the heat source. At the cold end, the columns are electrically insulated with respect to the cold source and to each other. Heat flux and electric current are assumed unidirectional (vertical) throughout the columns. The optimization reported here follows closely the analysis introduced in [8]. 2 The TE conversion efficiency of the TEG is defined by η = RI Q& H, where R is the external load, and I is the electric current produced by the TEG. The heat transfer rate Q & H at the cold end in eq. (1) may be written in terms of design quantities: geometry data, properties, and electrical load η = π 2 Aα I Lα 1 1 ( T ) I + T ( k + Xk ) ( k Xk ) α, β H α β e, α + Lα 2 Aα Here = ε = ( ε ε ) = ε ( T ) π α, β αβ β α T H αβ H RI 2 e, β. (1) is the Peltier coefficient of the junction, A (α,β) the cross-sectional area of the column, L (α,β) the column length, k (α,β) and k e(α,β) the thermal and electrical conductivities, and X = ( A L) β ( A L) α is the geometric aspect ratio. The linearized Onsager equations are dt dt dφ q = k + Tε ( T) J( T), J = εke ke, dx dx dx (2) Peltier-Joule effect where ε is Seebeck coefficient and J is the electric current density. The electric TH I = ε ε dt R+ R. Consequently, the current may be written as ( β α) ( internal ) TL efficiency depends on two parameters only: the ratio between the external resistance, R, and the cell internal resistance, Ri nt ernal = Lα ( ke, α Aα ): R R i nt ernal, and the geometric aspect ratio, X, suggesting that the TEG may be twice optimized for X η = Z 1 T L H < 1 T L TH, Z T T ( 1 T ) opt L H ηcarnot opt = ke, αkα ke, βkβ, and ( R R ) opt = Z ( 1+ ke, αkα ke, βkβ ) internal. In eq. (3), (3)

4 6 Alexandru Mihail Morega, Mihaela Morega, Marius Alexandru Panait 4 2 { ε αβ [ k k + k k ]}, Z = 1+ Tmed α e, α β e, β Z = 1+ Tmed σ, T med ( TH + TL ) 2 =. The message conveyed by eq. (3) is remarkable: the TEG efficiency may not exceed the Carnot limit. To achieve higher efficiency, the semiconductors should be weak thermal conductors, and good electric conductors; the lower conductivity column (+) should be robust, i.e. of large cross-sectional area and short length. Because the figure of merit (σ) of thermoelectric materials is a function of temperature and because TEG s columns even when optimized are sieges of longitudinal (vertical) temperature gradient, i.e. temperature varies along their length from the hot end to the cold end, they do not operate with the same efficiency along their height. The current flow in such a structure is accompanied by Seebeck, Peltier, Joule (heating) effects, and the conduction heat transfer, driven the temperature difference between the hot and the cold junctions. To investigate the analysis presented above, we formulate 2D numerical models with different geometric aspect ratios for the elemental TEG, discarding the assumption in virtue of which the material properties are temperature independent. Here we report only the results obtained for the optimized elemental TEG. Next, a first order, cascaded ensemble is proposed. By numerical simulation, we investigate the open circuit and the shortcircuit working conditions. In these experiments, the heat transfer related to the hot end cold end temperatures is responsible for the thermoelectric (Seebeck) voltage produced by the TEG, whereas Peltier effect is a menace to the TEG efficiency its minimization is a design objective STRUCTURAL OPTIMIZATION MATHEMATICAL MODEL The physical model of the thermoelectric phenomena is made of the eqs. (2) that give the heat and current fluxes [6], rewritten as Φ = ρ { J + ε T {, q = Φ { J k { T + ε { T J. (4) Ohm Seebeck Joule Fourier Peltier The partial differential equations that make the mathematical model result by specifying the heat and current sources for steady state working conditions, i.e. by taking the divergences of the heat and current fluxes, q = 0 and J = 0 { ( T ) T α( T ) TJ φj} = 0 14 k , (5) heat flux { σ( ) φ + σ( T ) α( T ) T} = 0 14 T electric flux We assume that the TEG is made of homogeneous materials, and all properties are functions of temperature only in view of [6] the temperature dependence is (6)

5 5 Structural optimization of a thermoelectric generator 7 moderate. Hence, the space derivatives are negligibly small. This hypothesis may not provide for accurate results when higher temperature differences are considered. The simulations were performed for: ρ = 1/k e = 8.0 T Ωm; ε = ±( (T 273)) 10 4 VK 1 ; k = T 1.65 WK 1 m 1. The boundary conditions (BCs) that close the model areas follows: for the heat transfer part of the problem, adiabatic conditions for all sides (Neumann, homogeneous BCs), except for the thermal contacts with the heat sources where temperature is set (Dirichlet BCs, T H and T L, respectively); for the electric field part of the problem, insulation (Neumann, homogeneous BCs) for all sides, except for the terminals where voltage and normal current density conditions are set. Special attention was devoted to implement the resistive load working condition, where the current and voltage at the electrical terminals are related through Ohm s law. To model this restriction, we set one terminal (A) to the ground (V A = 0), and utilized Comsol s integral coupling (boundary) variables technique [9] to transfer (couple) the current in that column to the terminal, B, whose the BC was set as a voltage condition, considering Ohm s law, V B = RI B STRUCTURAL OPTIMIZATION NUMERICAL MODEL Figure 2 shows the computational domain and the FEM mesh used for numerical simulation. a Fig. 2 The elemental TEG ( T = 200ºC, T C = 20ºC): a) open circuit the computational domain; b) shortcircuit the FEM mesh made of 22,000 Lagrange elements. The heat flux (thick arrows) and the current (thin arrows) flows are in opposite directions, in the n-type column, and in the same direction, in the p-type column. To implement the diffusion-type PDEs for heat and current flow, we used b

6 8 Alexandru Mihail Morega, Mihaela Morega, Marius Alexandru Panait 6 the general diffusion-convection PDE model, in coefficient form [5, 8] diffusion convection source c u + αu γ conservative flux au absorbtion convection + β u = { f source in Ω, (7) ( c u u ) qu g h T, on n +α γ + = µ Ω, (8) hu = r, on Ω where Ω is the computational domain and Ω is its boundary; c(t), α(t), γ(t), a(t), β(t) are coefficients; g, f are sources; the conservative flux is either the heat flux, or the electric current density (4), and u is either temperature, T, or electric potential, V; h T is the transpose of h (as h is scalar, h T =h); µ is Lagrange multiplier. The temperature difference between the TEG s hot and cold ends, T, was set to 30, 80, 120, and 280ºC, successively. In the iterative solution procedure, it was assumed that the initial temperature is 300 K for T = 20 ºC. The solutions for higher T s were obtained starting from the solutions obtained for lower T s. To solve the non-linear set of algebraic equations, consistent with the FEM technique, we used the direct SPOOLES [9] solver. Mesh independent solutions were obtained for roughly 22,000 elements. Figure 3 depicts the thermal and electrical fields for open circuit and shortcircuit working conditions, for T = 80ºC. Apparently, the electric field (Fig. 3a) and the temperature field (Fig. 3b) are stratified. The results are for a TEG cell that is 2 mm wide, 16 mm tall, and 2 mm thick. a Fig. 3 The elemental TEG ( T = 80ºC, T L = 20ºC): a) open circuit electrical field (voltage), contour lines; heat flux, arrows; b) shortcircuit temperature, contour lines; current density (arrows, streamlines). The non-dimensional open circuit voltage vs. temperature, and shortcircuit current vs. temperature curves are presented in Fig. 4. b

7 7 Structural optimization of a thermoelectric generator 9 Recalling the analysis presented in Section 2.1, and assuming that the properties are temperature independent, the efficiency of the optimized TEG depends on its apparent internal resistance, R internal, the load, R, and the TEG geometric aspect ratio, X. Since R internal is invariant with respect to T, the maximum power (higher for higher T s) is expected to occur for the same load, R. As seen in Fig. 5 numerical simulations confirm this behavior. The bell-shaped output power curves are consistent with finite-power sources characteristics. By increasing T, the output power increases less and less, until a limiting curve is attained (here, ~280ºC). Beyond this margin, the output power decreases. a b Fig. 4 The elemental TEG characteristics non-dimensional ordinates: a) shortcircuit, current [A]; b) open circuit, voltage [V]. Fig. 5 The output power for the elemental TEG.

8 10 Alexandru Mihail Morega, Mihaela Morega, Marius Alexandru Panait 8 3. CASCADED TEGs Figure 6 shows a schematic single stage TEG ensemble made of elemental TEGs sandwiched between two ceramic plates, and connected electrically in series and thermally in parallel, through metallic connectors. Fig. 6 Single stage, conventional TEG ensemble made of a number of elemental cells. The hot (upper) plate is partially represented notional representation. To further increase the overall efficiency TEG stages may be cascaded (Fig. 7a): the heat received from the hot end is transferred through the first stage layer, whose cold end is the hot end of a second stage layer. The cold head of the second stage layer contacts the cold heat source. This design avoids the difficulties presented by a segmented TEG ensemble (Fig. 6) that works between the same hot and cold heat sources. The two stages are electrically connected in series, thus avoiding the wiring exposure to high temperature that would lead to supplementary losses through the electrical resistivity increase with temperature. The heat received from the hot source by the upper (hotter) stage is transferred to the lower (colder) stage. a

9 9 Structural optimization of a thermoelectric generator 11 b c Fig. 7 First order cascaded TEG ensemble: a) the computational domain for the cascaded TEG ensemble; b) open circuit (voltage is in V); c) shortcircuit (temperature is in K). Figures 7b,c present the thermal and electrical fields for the cascaded TEG, under open circuit and shortcircuit working conditions, for T = 100 ºC. The columns in each stage have the same size, respectively. The usage of optimized elemental cells in constructing a cascaded TEG ensemble makes the object of future research. 4. CONCLUSIONS The main conclusions drawn in this study are as follow: The constructal principle utilized in the structural optimization of the elemental TEG cell is deterministic, and relies on the outlying physical laws that govern the thermoelectric phenomena in this study, Maxwell equations for the electromagnetic field in kinetic regime, and the heat transfer law, assembled both in the linearized Onsager relations. The TEG architecture is the outcome of constructal growth, from optimized elemental cell to higher order ensembles following a time arrow from small to large, from simple to complex [10]. The first construct is the elemental TEG cell in the limit, it is the smallest discernable, continuum (medium) system that exhibits the same properties, and obeys the same laws as larger scale systems. Its thermodynamic optimization performed here analytically, and assessed by numerical simulation has as objective and as high as possible efficiency, corresponding to a finite external load; as with all finite-power sources, the optimal TEG cell is adapted to a specific load. The efficiency of the optimized TEG depends on its internal resistance, R internal, the external load, R, and the aspect ratio, X. The bell-shaped output power curves are consistent with finite-power sources characteristics. By increasing DT, the output power increases less and less, until a limiting curve is attained (here, ~280 ºC). Beyond this margin, the power decreases.

10 12 Alexandru Mihail Morega, Mihaela Morega, Marius Alexandru Panait 10 Higher order ensembles result by cascading layers of elemental TEGs. They may be made of lower order TEG ensembles, and result in a tree structure. In this study we presented a first order ensemble that utilizes conventional TEGs. ACKNOWLEDGMENTS The research was conducted in the Laboratory for Multiphysics Models. The authors acknowledge the support offered by the CNCSIS grant A, no. 358/ Panait M.A. acknowledges the CNCSIS grant TD no. 315/2007. Received on 3 February 2009 REFERENCES 1. J. Vázquez, M.A. Sanz-Bobi, R. Palacios, A. Arenas, State of the Art of Thermoelectric Generators Based on Heat Recovered from the Exhaust Gases of Automobiles, (accessed in May, 2008). 2. J.P. Fleurial, G.J. Snyder, M.A. Ryan, C.-K. Huang, Solid-state power generation and cooling micro/nanodevices for distributed system architectures, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 20 th International Conference on Thermoelectrics, Beijing, P.R. China, June 8-11, L.D. Hicks, M.S. Dresselhaus, Thermoelectric figure of merit of a one-dimensional conductor, Phys. Rev., 47, B, pp , R. Venkatasubramanian, E. Siivola, T. Colpitts, B. O Quinn, Thin-film thermoelectric devices with high room-temperature figures of merit, Nature, 413, pp , S.W. Han, M.D.A. Hasan, J.Y. Kim, H.W. Lee, K.H. Lee, O.J. Kim, Multi-physics analysis for the design and development of micro-thermoelectric coolers, ICCAS2005, June 2-5, KINTEX, Gyeonggi-Do, Korea. 6. A.M. Morega, M. Morega, A FEM model for thermoelectric and thermomagnetic effects, Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 48, 2-3, pp , A. Bejan, Shape and Structure, from Engineering to Nature, United Kingdom, Cambridge U. Press, Cambridge, A. Bejan, Heat Transfer, New York, Wiley, * * *, Comsol A.B., v a, Sweden, A.M. Morega, J.C. Ordonez, M. Morega, A constructal approach to power distribution networks design, International Conference on Renewable Energy and Power Quality, ICREPQ 08, March, Santander, Spain, 2008.