ELECTRO-THERMAL ANALYSIS OF PELTIER COOLING USING FEM

Transcription

1 ISSN Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) ELECTRO-THERMAL ANALSIS OF PELTIER COOLING USING FEM D. ENESCU, E.O. VÎRJOGHE 2, M. IONEL, M.F. STAN 2 Electronic, Telecommunications and Energetics Department 2 Automatics, Informatics and Electrical Engineering Department Valahia University of Targoviste, Electrical Engineering Faculty 8-20 Unirii Ave., denescu@valahia.ro Abstract: Peltier technology has been the subject of major advances in recent years, due to the development of semiconductors and the incorporation of the thermoelectric devices into domestic appliances. According to the environmental problems produced by chlorofluorocarbons, the development of equipment based on the Peltier technology increased in the last years. Thermoelectric systems are used for measurement techniques and in other devices where a highprecision temperature control is essential (thermocouples and thermopiles), for Peltier cooling (Peltier elements for CPU cooling, refrigeration, temperature stabilization) and direct energy conversion of heat (thermoelectric generators, driven by waste heat, radioactive decay, combustion). In this paper an implementation of thermoelectric effects in ANSS Multiphysics is described. The authors present a computational model which simulates thermal and electric performance of a thermoelectric cooling device based on thermoelectric technology. The finite element method (FEM) is used here in order to solve the system of thermoelectric equations providing values for temperature distribution, thermal flux, temperature gradient, and voltage distribution. Keywords: thermoelectric cooler, Peltier elements, Seebeck effect, finite element method FEM, thermal gradient, thermal flux, voltage distribution.. INTRODUCTION Generally, if a thermal gradient is applied to a solid, it will always be accompanied by an electric field in the opposite direction. This process is called as the thermoelectric effect (TE). Thermoelectric material applications include refrigeration or electric power generation. The efficiency of a thermoelectric material is given by the figure of merit,, which is defined as []: α 2 σ =, k K α material's Seebeck coefficient, V/K, σ electrical conductivity of material, S/m, k thermal conductivity of material, W/(m. K). 2 The numerator α σ in equation is called the power factor. Therefore, the most useful method in order to describe and compare the quality and thermoelectric efficiency of different material systems is the dimensionless () figure of merit (T), where T is the temperature of interest. Therefore, equation () can be rewritten as: α 2 σ T T = (2) k An important point it is represented by achieving a high value of T, this being carried out by increasing the power factor (α 2 σ) and decreasing the thermal conductivity (k). One of the main applications of thermoelectrics is for refrigeration purposes. An electrical current applied across a material will cause a temperature differential which can be used for cooling. As it is known, metals are poor thermoelectric materials because they have a low Seebeck coefficient and large electron contribution to thermal conductivity k, so electrical conductivity σ and thermal conductivity k will cancel each other out. A low thermoelectric effect is carried out by insulators which have a high Seebeck coefficient and small electron contribution to thermal conductivity, so their charge density and electrical conductivity are low. The best thermoelectric materials are between metals and insulators (i.e., semiconductors) []. The thermoelectric materials of choice for the steadystate simulations illustrated in this paper on a thermoelectric element Peltier cooler are Bismuth- Tellurium (Bi-Te) and Lead-Tellurium (Pb-Te). They have a high Seebeck coefficient α, a good electric conductivity σ, and a poor thermal conductivity k. This paper presents the finite element formulation, which, in addition to Joule heating, includes Seebeck, Peltier and Thomson effects. An implementation of thermoelectric effects in ANSS Multiphysics is described in the next sections. Numerical results and their interpretation are provided and compared with other literature results. 2. THERMOELECTRIC COOLER MODELING 2. Thermoelectric Cooler The basic unit of a thermoelectric (TE) cooler is composed of two semiconductor elements connected at a copper strap as shown in Figure. It consists of an n-type and a p-type thermoelement connected electrically in series by a conducting strap. 89

2 Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) ISSN In the application shown in this paper, the n-type and p- type elements have a length 3.8 mm, and the element width is 2.5 mm. The width of the copper strap is mm. Thermoelectric power generation results in order to provide a temperature gradient across a material. Seebeck coefficients are also central to Peltier cooling. Cooling occurs by the absorption of heat as an electrical current passes through a junction between materials with different Seebeck coefficients. { E} - electric field, V/m. The general equation of heat flow used in the thermoelectric analysis is given by: T ρ c + {} v T + {} q = &&& q (6) t ρ density, kg/m 3, c specific heat capacity, J/(kg. K), t time, s, - represents the grad operator, - represents the divergence operator, vx {} v = vy - velocity vector for mass transport of heat vz & q& - heat generation rate per unit volume, W/m 3. Fourier s law of heat transfer by conduction is used in order to relate the heat flux vector to the thermal gradients: { q} [ k] T = (7) Figure. The thermoelectric cooler (Source: [2]). 2.2 Governing Equations of Thermoelectricity The new set of ANSS coupled-field elements developed in [2] enables users to accurately and efficiently analyze thermoelectric devices. The finite element method (FEM) flexibly used here can model arbitrary shaped structures, work with complex materials, and apply various types of loading and boundary conditions. The method can easily be adapted to different sets of equations, which makes it particularly attractive for coupled-physics simulation. The coupled thermoelectric equations are [3], [5]: { q} [ Π]{ J} []{ k T} { J} = [ ]{ ( E} [ α ]{ T} ) Replacing [ Π ] with [ α] = (3) σ (4) T in eq.3, it results: { q} T[ ]{ J} []{ k T} = α (5) [ Π ] = T[ α] - Peltier coefficient matrix, V, T - absolute temperature, K, [ α ] - Seebeck coefficient matrix, V/K, k - thermal conductivity matrix, W/m. K [] { q} - heat flux vector, W/m 2, { J} - electric current density, A/m 2, { T} [ ] - thermal gradient, K/m, σ - electrical conductivity matrix, S/m, Using eq. (6) and eq. (7), it results [4]: {} T ρ c { L} T { L} T + v = ([]{} k L T ) + && q& (8) t Developing eq. (8) it results: ρc + v t + k x x + v y + k + v y z = && q& + z + k z z z (9) The velocity vector for mass transport of heat is zero, in order to obtain the general equation of thermal conduction. ρ c = &&& q + k t x + k y + k z The continuity of electric charge equation is: D {} + = 0 z z (0) J ( ) t and the equation for a dielectric medium is given by [ ] E D = ε ( 2) 90

3 ISSN Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) D - electric flux density, C/m 2, [] ε - dielectric permittivity matrix, F/m. In the absence of time-varying magnetic fields, the electric field E is irrotational ( E = 0), and can be derived from an electric scalar potential ϕ : E = ϕ (3) Substituting eqs. (3)-(3) into eqs. (6)-(), a system of coupled equations of thermoelectricity is obtained [3]: ([ Π] J ) ( [] k T ) = && q& ρ c + (4) t ϕ [] ε + ( [ σ ] [ α ] T ) + ([ σ ] ϕ) = 0 (5) t where the heat generation term && q& in eq. (4) includes the electric power J E spend on Joule heating and on work against the Seebeck field [ α ] T. The system thermoelectric finite element equations can be obtained by applying the Galerkin FEM procedure to the coupled equations derived in the previous section. This technique involves the following steps [4]: a) approximate the temperature T and the electric scalar potential ϕ over a finite element as [3]: { N} { T} e { N} ϕe T = (6) ϕ = (7) { N} - vector of element shapes functions, - vector of nodal temperatures, {} T e ϕe - vector of nodal electric potentials; b) write the system of eqs. (4) and (5) in a useful form; c) integrate the equations by parts; d) take into account the Neumann boundary conditions. The resulting system of thermoelectric finite element equations is [2], [5]: C 0 TT 0 T& e k + ϕϕ C & ϕe k TT ϕt T P e 0 e Q+ Q + Q = ϕϕ k ϕ e I (8) where the element matrices and load vectors are obtained by numerical integration over the element volume V. The corresponding expressions are: - element diffusion conductivity matrix: k TT = N λ N { } [ ] { }dv - element electrical conductivity coefficient matrix: ϕϕ k = N σ N { } [ ] { }dv - element Seebeck coefficient coupling matrix ϕ { N} [ σ ] [ ] { N}dV k T = α - element specific heat matrix C TT = ρ C N { }{ N}dV - element dielectric permittivity coefficient matrix ϕϕ C = N ε N { } [] { }dv - vector Q of combined heat generation loads - Peltier heat load vector: Q P = N Π J { } [ ] {}dv - electric power load vector: Q e = N E { }{ } { J}dV - electric current load vector I. Thermal loads (Q) can be in the form of imposed temperature, point heat flow rate, surface heat flux, convection, or radiation, as well as body heat generation rate for causes other than electric power dissipation (accounted for in Q e ). Electrical loads (I) can be in the form of imposed electric potential and point electric current. Linear electric circuit components (resistors, capacitors, and voltage or current sources) can be connected to the finite element model to simulate passive and active electrical loads [2], [5]. The ANSS input of material matrices [k], [σ], [α], [ε] is in the form of their diagonal terms, i.e., material coefficients along the x, y, z axes. This input can be combined with an arbitrarily oriented element coordinate system to account for an alternative material orientation. Electrical properties are input as resistivity and internally converted into conductivity [σ], which is the conductivity evaluated at zero temperature gradient. The input [λ] is the thermal conductivity evaluated at zero electric current ( J = 0). All material properties can be temperature dependent. In particular, Thomson effect is taken into account by specifying temperature dependent Seebeck coefficients [α] [3]. The global matrix equation is assembled from the individual finite element equations, and is nonsymmetric like eq. (8). Since the thermal load vector depends on the electric solution, the analysis is nonlinear and requires at least two iterations to converge. The solution yields temperatures (T e ) and electric potentials (φ e ) at unconstrained nodes, or reactions in the form of heat flow rate and electric current at nodes with imposed temperature and electric potential respectively. The temperature gradient and electric field are calculated as [3]: { N} T e T =, (9) { E} { N} ϕe =, (20) and then substituted into Eqs. (3)-(2) to obtain the values of J, q, D fields, and Joule heat generation density for each element. 9

4 Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) ISSN STEAD-STATE ANALSIS OF A THERMOELECTRIC ELEMENT Numerical simulation is carried out by using a finite element package ANSS. This package operates with three stages: preprocessor, solver and postprocessor. The procedure for doing a static thermoelectricity analysis consists of following main steps: create the physics environment, build and mesh the model and assign physics attributes to each region within the model, apply boundary conditions and loads (excitation), obtain the solution, review the results. In order to define the physics environment for an analysis, it is necessary to use the ANSS preprocessor (PREP7) and to establish a mathematical simulation model of the physical problem [5]. In order to do this, the following steps are presented below: set GUI Preferences, define the analysis title, define element types and options, define element coordinate systems, set real constants and define a system of units, define material properties. ANSS includes three elements which can be used in modeling the thermoelectricity phenomenon [5]. Element types establish the physics of the problem domain. Depending on the nature of the problem, it is necessary to define several element types to model the different physics regions in the model. In the present application, for modeling the electric and thermal fields the SOLID227 element was chosen. SOLID227 has the following capabilities: structuralthermal, piezoresistive, electroelastic, piezoelectric, thermal-electric, structural-thermoelectric, thermalpiezoelectric. The element has ten nodes with up to five degrees of freedom per node. Thermoelectric capabilities include Seebeck, Peltier, and Thomson effects, as well as Joule heating. ELEMENTS APR :2:55 values for Bismuth-Telluride, Lead-Telluride and copper were taken from [6]. Table. Numerical material properties from [6], [0] Material properties from Seebeck Coefficient Electric resistivity Thermal conductivity Density Heat capacity Units measure α [V/K ] ρ [S/m ] λ [W/m/K ] δ [V/K ] C [V/K ] Bismuth- Tellurium (Bi-Te) Lead- Tellurium (Pb-Te) Cooper p:200e -6 p:75e e -6 n:-200e -6 n:-75e e e e Usually, those material properties depend on the temperature and may be anisotropic. Here only isotropic material properties are used at constant material parameters. Thermoelectric (TE) materials based on (Bi,Sb) 2 (Te,Se) 3 are the best and, in fact, the only materials used for cooling. These include bismuthtellurium (Bi-Te) and antimony-tellurium (Sb-Te) compounds. More recently, nanostructured materials have been investigated as candidates to increase the performance of thermoelectric devices. PbTe nanocomposites have been prepared from PbTe nanocrystals, synthesized via chemical route, by compaction under high pressure and temperature. The thermoelectric (TE) properties are found to vary with the shape and size of the composites nanostructures. Transport properties of PbTe nanocomposites have been evaluated through temperature-dependent electrical conductivity, Seebeck coefficient, room temperature, and thermal conductivity measurements [7], [8]. STEP= SUB = TIME= TEMP (AVG) RSS=0 S =273.5 SM =34.5 APR :30:4 M Figure 2. Meshing the model with triangular elements Next step in the preprocessor phase is mesh generation and load application on the elements. It was used a mesh with 6366 nodes and 3249 triangular elements. The finite element mesh of the thermoelectric element is shown in Figure 2. The following examples show results of calculations for typical thermoelectric applications. The material properties for the calculations with temperatureindependent values are shown in Table. Here typical Figure 3. Distribution of temperature for Bi 2 Te 3 material A calculation model that simulates the thermal and electric performance of whole cooling based on thermoelectric technology has been implemented. The model inputs are: semiconductor materials and geometry, Peltier pellet components type, electrical voltage supplied to the Peltier pellet components and the hot and cold side temperatures. The distribution of temperature for Bi 2 Te 3 material is shown in Figure 3 and the distribution of voltage for Bi- Te material is shown in Figure 4. 92

5 ISSN Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) When an electric current is running from the cold end to the hot end, the Joule heating is generated uniformly inside the element. The dissipated heat must reach both ends equally by conduction. STEP= SUB = TIME= TFSUM (AVG) RSS=0 S =.858E-06 SM =60888 APR :36:05 M STEP= SUB = TIME= VOLT (AVG) RSS=0 SM =.0924 APR :30:58 M.858E Figure 6. Distribution of thermal flux for Bi 2 Te 3 material The distribution of temperature for PbTe material is shown in Figure Figure 4. Distribution of voltage for Bi 2 Te 3 material The cooler with Bi 2 Te 3 material is designed to maintain the cold junction at a temperature T c =273.5 K and to dissipate heat from the hot junction T h =34.5 K on the passage of an electric current of magnitude I=0.535 A. The positive direction of the current is from the n-type material to the p-type material. STEP= SUB = TIME= TEMP (AVG) RSS=0 S =273.5 SM =336.5 APR :8:07 STEP= SUB = TIME= TGSUM (AVG) RSS=0 S =.53E-08 SM =30456 APR :52: Figure 7. Distribution of temperature for PbTe material M M The distribution of voltage for PbTe material is shown in Figure 8. STEP= SUB = TIME= VOLT (AVG) RSS=0 SM = APR :8:59.53E M Figure 5. Distribution of thermal gradient for Bi 2 Te 3 material After the simulation, the model-returned outputs are: temperatures, heat flows, thermal gradient, thermal flux, and voltage distribution. The distribution of thermal gradient for Bi 2 Te 3 material is shown in Figure 5 and the distribution of thermal flux for Bi 2 Te 3 material is shown in Figure 6. The cold junction is at a temperature of T c =273.5 K and dissipates heat from the hot junction T h =336.5 K for PbTe material Figure 8. Distribution of voltage for PbTe material The temperature dependence of Seebeck coefficient, electrical conductivity and power factor of the PbTe material lie within the temperature range K. Thermal conductivity reduction has played a central role in improving the thermoelectric figure-of merit, T, of materials that already have a good power factor. 93

6 Scientific Bulletin of the Electrical Engineering Faculty ear 0 No. (2) ISSN CONCLUSIONS STEP= SUB = TIME= TGSUM (AVG) RSS=0 S =.53E-08 SM = E M APR :20:06 Figure 9. Distribution of thermal gradient for PbTe material The distribution of the thermal gradient for PbTe material is shown in Figure 9 and the distribution of thermal flux for Bi-Te material is shown in Figure 0. STEP= SUB = TIME= TFSUM (AVG) RSS=0 S =.858E-06 SM =60935 M.858E APR :20: Figure 0. Distribution of thermal flux for PbTe material Materials and device characterization play a key role in thermoelectric research. Materials composition and parameters affect the achieved thermoelectric (TE) performance (for example, functional properties and figure-of-merit of materials, efficiency, coefficient of performance, or sensitivity of devices). Applications of the two materials Bi 2 Te 3 and PbTe demonstrate the Peltier effect for thermoelectric cooling. It can be seen that a smaller thermal conductivity will decrease the heat transfer between the two ends, a smaller electrical resistivity will reduce the Joule heating, and a larger Seebeck or Peltier coefficient will enhance the heat removal. For most metals, the thermal conductivity is too high and the Seebeck coefficient is too small for refrigeration applications. Some insulators can have a large Seebeck coefficient but their electrical resistivity is too high for them to be used in thermoelectric devices. When lead telluride PbTe is compared with bismuth telluride Bi 2 Te 3, a temperature difference of nearly 63 K for cooler with PbTe and temperature difference of nearly 68 K for cooler with Bi 2 Te 3 is noted. The voltage at the upper electrode is 46.2 mv for cooler with Bi 2 Te 3 and 43.2 mv for cooler with PbTe. This means that, although the value of the figure of merit of PbTe is lower than for Bi 2 Te 3, the latter material is used. In fact, lead telluridebased materials have been used for a range of purposes in the hot-junction temperature range 600 to 900 K [9]. Conversely, PbTe has been considered more as a material for thermoelectric generation at moderately high temperatures rather than for refrigeration at room temperature and below [0]. PbTe thermoelectric generators have been widely used by the US army, in space crafts to provide onboard power, and in pacemaker batteries. The application shown in this paper can be useful to represent the characteristics of the Peltier cooling through numerical models. 4. REFERENCES [] Kimmel, J. Thermoelectric Materials, Physics 52, Special Topics Paper, March 2, 999 [2] Angrist, S. W., Direct Energy Conversion, 3rd Edition, Allyn and Bacon (Boston, 976), pp ; [3] Antonova E.E, Looman D.C. Finite Elements for Thermoelectric Device Analysis in ANSS, International Conference on Thermoelectrics 2005, /05 IEEE; [4] Silvester P. P. and Ferrari, R. L., Finite Elements for Electrical Engineers, 3rd Edition, University Press (Cambridge, 996); [5] ANSS Release.0 Documentation; [6] Jaegle M. - Multiphysics Simulation of Thermoelectric Systems - Modeling of Peltier - Cooling and Thermoelectric Generation, Proceedings of the COMSOL Conference 2008 Hannover; [7] Biplab Paul and Pallab Banerji - Grain Structure Induced Thermoelectric Properties in PbTe Nanocomposites, Nanoscience and Nanotechnology Letters, Vol., , 2009; [8] huomin M. hang, Nano/Microscale Heat Transfer, Mc Grow Hill, 2007; [9] D.M. Rowe, Ph.D., D.Sc., Thermoelectric Handbook Macro to Nano, 2006 by Taylor & Francis Group, LLC; [0] H. Julian Goldsmid, Introduction to Thermoelectricity, Springer Series in Materials Science,