Fuel Cells

Thermoelectric Generator: Modeling and Analysis

Table of Contents1. Introduction.....................................................................................................................2

2. Theoretical analysis.........................................................................................................3

2.1. Thermoelectric generator technology..................................................................3

2.2. Seebek Effect......................................................................................................3

2.3. Thermoelectric cells............................................................................................4

2.4. Flux and energy conservation equations for thermoelectric materials.................5

3. COMSOL Multiphysics.....................................................................................................6

3.1. The COMSOL Multiphysics software...................................................................6

3.2 The Finite Element Method..................................................................................6

4. Design considerations......................................................................................................8

4.1. Methodology........................................................................................................8

4.2. Simulation process..............................................................................................9

4.3. Thermoelectric materials.....................................................................................9

4.4. Creating the Thermoelectric module.................................................................10

4.5. Material properties.............................................................................................10

4.6. Parameter settings for the Simulation...............................................................12

5. Results...........................................................................................................................15

................................................................................................................................. 15

1. IntroductionWith the rise in power demand, it is becoming essential to find alternative energy sources which are eco-friendly, as well as economical. Recently, researchers are showing interest towards Thermo Electric Generators (TEGs), which are devices that convert heat into electrical energy, taking advantage of the phenomenon called “Seebeck effect” [1].

The aim of the current project is to create a model of a thermoelectric generator, simulate the heat flow and obtain the voltage that it will generate for different parameters. The report presents the behavior of a TEG made of bismuth telluride (Bi2Te3), with different temperatures being applied across them and by varying the load resisitor connected across the thermoelectric cell, using the software package "COMSOL Multiphysics".

2. Theoretical analysis2.1. Thermoelectric generator technologyThermoelectric modules consist of a number of N-type and P-type semiconductors. The latter are connected electrically in series with copper electrodes and are "sandwiched" between metalized ceramic plates to connect thermally in parallel.Normally, modern thermoelectric cells use highly doped semiconductors made out of bismuth telluride (Bi2Te3), lead telluride (PbTe), calcium manganese oxide (Ca2Mn3O8) or combinations of those materials. The selection of the most suitable semiconductor can also be made based on temperature [1]. Various efficiencies can be achieved, depending on the chosen material, and the same material can achieve different efficiencies according to the temperature.The following figures show parts of a thermoelectric module, which will be discussed in the next chapters.

Figure 2.1: (a) A thermoelectric leg, (b) Isometric view of a basic sketch of a thermoelectric cell

a)b)

2.2. Seebeck EffectBy placing a conductor in a temperature gradient, an electric voltage can be generated. This phenomenon is called the Seebeck effect [2]. The Seebeck coefficient (S), also known as thermopower, constitutes a measure of the efficiency of this phenomenon, and it is defined as the ratio of the generated electric voltage to the temperature difference. It is determined by the scattering rate and the density of the conduction electrons. The effect can be exploited in thermal electric-power generators by connecting two conductors with different Seebeck coefficients. At the atomic scale, charged carriers of the semiconductor are diffused from the hot side to the cold side with the applied temperature gradient.In the open circuit case, charge carriers will accumulate in the cold region, resulting in the formation of an opposing electric field that halts the diffusive motion. The voltage drop generated by this electric field is known as the "thermo-voltage", V th, and it is proportional to the temperature differential and a material dependent constant known as the thermo power, S:

(2.1)

The thermopower of a material can be either negative or positive, depending on the dominant type of charge carriers. For an electron dominated material (n-type), we know that and for a hole dominated material (p-type), we have that .The Seebeck effect is capable of producing very small voltage values. There can be only a low voltage of only a few mV per Kelvin degree of temperature difference. Once a larger temperature difference is created, some devices are capable of producing a few volts. A number of such devices can be connected either in series or in parallel, so as to increase the output voltage or the maximum deliverable current. Therefore, in order to provide useful electrical power, large arrays of Seebeck-effect devices should be connected, with large temperature differences applied across the junctions.

Figure 2.2: A graphical representation of the Seebeck effect

2.3. Thermoelectric cellsTypical thermoelectric cells are not manufactured from a single material; instead they are constructed from two dissimilar materials, called "elements". The reason for this is the need to complete an electric circuit, for the cell to be useful. The total thermo-voltage generated by a cell is given by the difference between the thermo voltages generated by each element:

(2.2)

The typical choice for commercial cells is one of n-type and one of p-type [3]. A thermoelectric cell can operate in two possible ways: as a generator, or as a cooler. In the generator configuration, the Seebeck effect is utilized in order to generate useful electric power from an applied thermal gradient. In the cooler configuration, the Peltier effect allows for an electric current to move heat from one thermal reservoir to the other.

Figure 2.3: A thermoelectric cell, acting as a generator (left) and a cooler

In order to fully describe the heat flow through a thermoelectric material, and thus describe a thermoelectric generator or cooler, the effects of Seebeck and Peltier effects on the electric current density and heat flux should be discussed.

2.4. Flux and energy conservation equations for thermoelectric materialsIn general, the electric current density (J) and the heat flux (Φ), are related to the electric field (Ε) and the thermal gradient (ΔΤ), through the following two equations:

(2.3)

(2.4)

where σ is the electrical conductivity, k is the thermal conductivity, and S is the thermopower.

The second term in the first equation, , is known as the thermocurrent, and represents the contribution to the electric current density due to the Seebeck effect. The first term in the second equation, , describes the Peltier heat flux through the material. The remaining two terms describe standard voltage-driven electric currents, and thermally-driven heat flux, respectively.

In addition to the two flux equations, there are two more equations:For the conservation of energy: (2.5)

For the conservation of charge: (2.6)

Equations 2.3 to 2.6 are sufficient for describing the linear response of thermoelectric phenomena in any material. Hence, these equations have been used throughout the following simulation process in the COMSOL Multiphysics software.

3. COMSOL Multiphysics

3.1. The COMSOL Multiphysics softwareCOMSOL Multiphysics is a general-purpose software platform, based on advanced numerical methods, for modeling and simulating physics-related problems. As its name denotes, Comsol Multiphysics can examine a problem from various physical aspects simultaneously, such as electrical, mechanical, fluid flow and chemical. The software uses dedicated interfaces for the various applications, providing a wide variety of tools for the input of data and the setup of the necessary equations, as well as a graphical depiction of the resulting simulations. The resulting plots are fully controllable, in order to provide a thorough overview of the phenomena in question and thus lead to a better understanding of the underlying processes. The main reason for that is the software's Model tree incorporated in the Model Builder, which gives the user a full overview of the model and access to all functionality – geometry, mesh, physics settings, boundary conditions, studies, solvers, postprocessing, and visualizations. Thus, individual aspects of the model can easily be adjusted. Moreover, with COMSOL Multiphysics it is possible to make use of add-on products in a seamless way, without affecting the performance of the main software platform. The functionality can be further extended by creating apps with the Application Builder tool, or take advantage of the built-in COMSOL API for use with Java that enables connecting the user-created COMSOL Multiphysics models with other, external applications. The user may also create his own sets of equations, which will in turn allow the implementation of new interfaces based on these specific models. The new interfaces can for instance contain input and output fields that the user of the app is given access to.

3.2 The Finite Element MethodThe method deployed in COMSOL Multiphysics is the Finite Element Method (FEM), or else Finite Element Analysis. This method derives approximate solutions to boundary value problems for partial differential equations. The principles of the FEM method are:

The subdivision of the problem's domain into simpler, elementary parts (called "finite elements"). Each subdomain is represented by a set of a equations for the original problem.

The sets of equations for all subdomains are recombined into a global system of equations for the final calculation. This global system of equations has known solution techniques and can be calculated from the initial values of the original problem, in order to obtain a numerical answer.

The following example will provide an idea of how the FEM method works. Let us consider a case from structural mechanics, where we want to calculate the stress tension in the following flat plate (Figure 3.1), which has a square hole cut in its center and is subjected to uniaxial tension:

Figure 3.1: A square metal plate, for calculating the structural tension

As a general rule, in such a geometry we can exploit the symmetry and perform our calculations only on one fourth of the problem's area. The reason for this is that working on a smaller part of the geometry requires fewer computational resources and less time, something that can be significant in some cases, depending on the mesh refinement we have chosen.

Figure 3.2: The actual calculation area, due to the symmetry of the plate

As we can see in Figure 3.3 at first the FEM method divides the problem's area into small subareas, where the equations will be applied. This group of basic smaller areas is called the "mesh". In simple geometries, such as a square, the mesh consists of a symmetrical set of triangles, but in more complex geometries the mesh loses this symmetry and becomes more dense (i.e. more elements are inserted ) near the areas of change (such as the lower left corner in Figure 3.3-a). That way, the solution of the problem's equations will become more accurate in areas of change, where the boundary conditions apply (and where, understandably, the error is estimated to be large). Calculating for the desired physical quantity (i.e. electric field, magnetic flux, or in this case, structural tension), we acquire a colored depiction of this quantity's behavior. In case of a relatively simple mesh (Figure 3.3-a), the resulting plot is rather "raw" and the color changes between each pair of triangles are visible and abrupt. This can be avoided by refining the mesh, as in (b) and (c) in Figure 3.3. - of course, this results to more processing time and load, but also offers a "smoother" graphical representation of the results.

Figure 3.3: (a) A simple mesh of the problem area, (b) A more refined mesh ,(c) A thoroughly refined mesh, (d) Graphical results for mesh c

Conclusively, the FEM method is based on the same concept as in the notion that we can always represent a circle by using a set of tiny straight lines. It is easily understood that if we have only a few lines, the produced circle will be rather crude and unsightly; however, the more lines we use, the more satisfactory the appearance of the circle is going to be, and after we use a satisfactory amount of lines, the resulting figure will be a perfect circle.

3.3 The governing equations The governing equations determining the analysis of the Seeback and Peltier effects are:

Electric current balance: (3.1)

Heat energy balance: (3.2)

where: σ: electric conductivity (S/m) V: electric potential (V) ρ: density (kg/m2) Cp: heat capacity (J/(kg.K)) q: heat flux (W/m2) P: Peltier coefficient (V) J: current density (A/m2) Q: (= Joule heating(W/m3)

Thomson’s second relation between the Seebeck (S) and Peltier (P) coefficients provides:

(3.3)

For the transfer of the energy balance to a weak form, we multiply each side of the energy balance by a test function, Ttest, and integrate over the computational domain Ω:

(3.4)

Using the vector identity: (3.5)

Then the previous equation becomes:

(3.6)

By using the Gauss theorem:

(3.7)

where n is unit normal to the boundary of domain , then equation 3.7 becomes:

(3.8)

The energy flux is given by:

(3.9)

Hence equation 3.8 becomes:

(3.10)

The Peltier weak contribution is then given by:

(3.11)

where COMSOL notation for the Ttest = test(T)and partial derivatives.

= , = and = .

4. Design considerationsIn the simulation process, hot surface temperature has varied to understand the behavior of electric generation and the load resistance changed during the simulation for understand the loading effect certain hot surface temperature values were selected based on temperature ranges where each material properties measured in COMSOL. The range of the load resistor values is selected by doing several simulations and the effective range for the demonstration was selected.

4.1. Simulation processIn the modeling process, a system of two thermoelectric legs was considered and bismuth telluride was chosen as the semiconductor material. In addition, copper (Cu) was used as the electrical conductor between the thermoelectric legs.

Figure 4.1: A schematic diagram of the 2 thermoelectric legs used in our modelling process

4.2. Thermoelectric materialsThe following 3 parameters have to be considered when selecting thermoelectric materials for the thermoelectric generator: (a) electrical conductivity σ, (b) thermal conductivity λ, and (c) the Seebeck coefficient S. For the chosen material to be suitable, it should be characterized of high electrical conductivity, low thermal conductivity and a high Seebeck coefficient. The above three properties depend on the charge carrier concentration. For a given material, a combination of these 3 parameters is summarized in the form of the material's efficiency, which in turn is given by its Figure of Merit (FOM): , with T being the average temperature of the material. Thermoelectric devices have been further classified with respect to the temperature ranges over which they can be usefully employed. To facilitate optimization, a quantity called electrical power factor α2σ, has been introduced. The main three semiconductors known to combine the desired low thermal conductivity and high power factor were: bismuth telluride (Bi2Te3), lead telluride (PbTe), and silicon germanium (SiGe). These materials contain very rare elements which make them very expensive compounds.

Table 1.1. Temperature ranges for 3 thermoelectric materials

Temperature range

Materials

Low (300 - 450 K) Bi2Te3,BiSe3

Intermediate (300 - 850 K) PbTeHigh (300 -1300K) SiGe

In this project, the chosen material is Bismuth Telluride (Bi2Te3), already available in the thermoelectric material library of the COMSOL software. Bismuth Telluride is a promising material, as far as low temperature applications are concerned.

4.3. Creating the Thermoelectric module A prototype of a thermoelectric module has been created, which consists of 64 legs at a dimension of 5x5x5 mm, and a 3 mm gap between the legs. Each leg pair is made of p-type Bi2Te3 and n-type Bi2Te3. The two adjacent legs in every pair are connected to each other with 0.5 mm copper strips.

Fig.10. P-type legs (blue color) Fig.11. N-type legs (blue color)

Fig.11. Copper plate legs (blue color)

4.5. Material properties

Bismuth Telluride and Copper, already in the material library of COMSOL, have the following properties:

Name Value Unit

Heat capacity at constant pressure 154 J/(kg.K)Density 7700 kg/m3

Seebeck coefficient S(T) V/KElectrical conductivity sigma(T) S/mThermal conductivity k(T) W/(m.K)Relative permittivity 1 1Heat capacity at constant pressure 154 J/(kg.K)Density 7700 kg/m3

Table 2. Properties of Bi2Te3

For the P –type material Seebeck coefficient is positive and for the N-type materials it is considered to be negative. Other parameters remain same for both types.

Fig.12. Bi2Te3 electrical conductivity vs temperature

Fig.13. Bi2Te3 Seebeck coefficient vs temperature

Fig.14. Bi2Te3 thermal conductivity vs temperature

Name

Table 3. Properties of Copper

4.6. Parameter settings for the Simulation

The COMSOL thermoelectric library uses two basic multiphysics libraries. The first one is the "Heat transfer in solids" library for defining the boundary conditions for heat transfer and other temperature-related parameters. The second library is the "Electric current" library, used for setting up the boundary conditions and the other conditions for the electric energy being generated. Low temperature is applied to the bottom copper plates and kept at 293.15 K while top surface temperature is applied to the top copper plates and varies from 300 K to 450 K. One simulation is carried out for the whole module (64 legs) and results are obtained for the variation of the open circuit voltage between two free ends, with respect to temperature. As this simulation was too demaning on terms of time needed and available computational resources, the rest of the simulation was completed using two pairs of thermo electric legs, as in Fig.15:

Value Unit

Electrical conductivity 5.998.107 S/mHeat capacity at constant pressure 385 J/(kg.K)Relative permittivity 1 1Density 8700 kg/m3

Thermal conductivity 400 W/(m.K)Young's modulus 110.109 PaPoisson's ratio 0.35 -Reference resistivity 1.72.10-8 ohm.mResistivity temperature coefficient 0.0039 1/KReference temperature 298 K

Fig.15. A 2-leg thermoelectric module

Figure 16 shows that all the boundaries are thermally insulated (blue), except for the top and bottom surfaces (grey), where the temperature is applied.

Fig.16. Thermal insulation of the 2-leg thermoelectric module

Electrical contacts have been added between the legs. These contacts, made of copper (blue lines in Fig.17), have the default properties shown in Table 4.

Fig.17. Electrical contacts added to the module

Description

Value

Constriction conductance correlation Cooper - Mikic - Yovanovich correlationSurface roughness, asperities average height

1[um]

Surface roughness, asperities average slope

0.4

Hardness definition MicrohardnessMicrohardness 3[GPa]Contact pressure 100[kPa]

Table 4. Properties of the electrical contacts

All the surfaces are electrically insulated (blue color surfaces), except for the ground terminal (right hand corner) and the positive terminal (left hand corner):

Fig.18. Voltage terminals and electrically insulated surfaces

During the simulation process, certain parameters of the mesh had to be determined. As already mentioned in the FEM section, understandably these parameters have a firect impact on the quality of the acquired results.

Fig.19. The mesh used in the model

NameValue

Maximum element size 3.2Minimum element size 0.576Curvature factor 0.6Resolution of narrow regions 0.5Maximum element growth rate 1.5

Table 5. Properties of the mesh

Using the Comsol AC/DC library, an external resistance has been added between the two output terminals. Finally, a load voltage at different load resistances was obtained by changing the hot side temperature. The value of the simulation resistance has been changed from 0 to 1 Ω in steps of 0.05 Ω, and the hot side temperature changes from 300 K to 450 K in steps of 10.

5. Results5.1 64-leg thermo electric module

To get an idea of the possibilities of FE modelling some exemplary plots (figures 5.17 and 5.18) are presented. The open circuit voltage distribution of the 64-leg model for two different top surface temperatures are shown. It can be seen that open circuit voltage is increased with increasing the Top surface voltage.

Fig.23. Voltage distribution of the 64-leg model for a 350 K top surface temperature

Property ValueMinimum element quality 0.1318Average element quality 0.6665Tetrahedral elements 5131Triangular elements 2610Edge elements 478Vertex elements 56

Fig.27. Voltage distribution of the 64-leg model for a 400 K top surface temperature

Figure X shows the Open circuit voltage of 64-leg thermo electric module over the Top surface temperature from 300 K to 450 K while the bottom surface keeps at 293 K.The open circuit voltage shows a near linear relationship with temperature.

Figure X: Output voltage vs. top surface temperature

5.2 Four Leg thermo electric module

Following 3D surface plots are to illustrate the capability of the COMSOL metaphysics software and much detailed plots were generated and presented later on this section which can be used to get an idea of the thermo electric module.

Figure XX shows the temperature distribution of the model for the top surface temperature of 400 K.

Fig.26. Temperature distribution for a 400o K top surface temperature and 1Ω resistance

The open circuit voltage distribution of the 64-leg model for two different Top surface temperatures are shown below. It can be seen that load voltage is increased with increasing the Top surface voltage as well as increasing the resistor value. Also the load voltage is increases when increasing the load resistance under the same temperature.

Fig.21. Voltage distribution for a 350o K top surface temperature and 0.1Ω resistance

Fig.22. Voltage distribution for a 350o K top surface temperature and 1Ω resistance

.

Fig.24. Voltage distribution for a 400o K top surface temperature and 0.1Ω resistance

Fig.25. Voltage distribution for a 400o K top surface temperature and 1Ω resistance

Behavior of load current

First set of plots illustrate the behavior of load current with other parameters.

The behavior of the load current vs. load voltage with selected resister values were plotted in the below figure XXX when increasing the temperature in the specified range. Also the Figure shows the behavior between those two parameters for different temperatures when changing load resistance value.

Figure : load current vs. load voltage

Figure : load current vs. load voltage

Figure: Load current vs. load Resistance

Figure: Load Current vs Hot Surface Temperature

Behavior Load voltage

Results of the Output voltage is a main consideration of the thermo electric module. Variation of the output voltage was simulated with load resistance voltage as well as with the top surface temperature.

Figure : Load vs Load Resistance

Figure : Output Power vs Load Resistance

Power output and Efficiency

Output power and the efficiency are the most significant for this task. Electric power output of the thermo electric is calculated using the power output of the load resistor.

The following figures show the output power variation with load resistance, load voltage and load current.

Figure : Output Power vs Load Resistance

According to the maximum power transfer theorem, the maximum power is given when the load resistance is equal to the internal resistance of the thermoelectric generator. It can be seen from the plot that the Maximum power is given at the resistor value of 0.1 Ω.

According to the graph, it can be clearly seen that the power output is increased dramatically with the resistance and gives the maximum at resistor value of 0.1 Ω. Then it gradually goes down with the increase of resistance. But from the figure below it is clear that the output can be increased with a higher top surface temperature of the thermoelectric generator.

Figure : Output Power vs Top Surface Temperature

Below in figure XXXX the variation of the output power with the terminal voltage when increasing the load resistance value, is drawn with various top surface temperatures.

Figure : Output Power vs Load Voltage

Below in figure XXXX the variation of the output power with the current through load when increasing the load resistance value is drawn with various top surface temperatures.

Figure : Output Power vs Load current

The conversion efficiency is the ratio of electrical power delivered to the load of resistance relative to thermal power input to the module. The net heat from the top surface temperature source can be found in the simulated data in COMSOL and the output power of the resistor can be calculated using its voltage and current. To understand the effect of resistance and hot surface temperature on efficiency, plots of efficiency vs load resistance and efficiency vs hot surface temperature has drawn in figure and figure respectively.

Figure : Efficiency vs Top Surface Temperature

Figure : Efficiency vs Load Resistance

Figure : Efficiency vs Load Voltage

References

[1]. https://en.wikipedia.org/wiki/Thermoelectric_generator [2]. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, E. Saitoh,” Observation of the spin Seebeck effect”, Nature 455, 778-781 (9 October 2008) | doi:10.1038/nature07321; Received 6 May 2008; Accepted 4 August 2008

[3]. Robert R. Heikes, Roland S. Ure. Thermoelectricity: Science and Engineering. Interscience Publishers, 1961

[4] https://www.comsol.no/paper/download/84033/crompton_paper.pdf

Proceedings of the 2011 comsol conference in Boston

Multiphysics Analysis of Thermoelectric Phenomena S.P. Yushanov, L.T. Gritter, J.S. Crompton*and K.C Koppenhoefer AltaSim Technologies, Columbus, OH

this is where we took equations