KEYWORDS: Manganite, modeling, dielectric heating, resistive heating, microwave heating

Transcription

1 ESTIMATING RESISTIVE AND DIELECTRIC EFFECTS DURING MICROWAVE HEATING OF Fe 0.22 Ni 0.67 Mn 2.11 O 4 Juan A. Aguilar-Garib, Felipe García, Zarel Valdez Universidad Autónoma de Nuevo León, Facultad de Ingeniería Mecánica y Eléctrica Pedro de Alba s/n, Cd. Universitaria, San Nicolás de los Garza, NL 66450, México Tel. +52 (81) x 5770, Fax. +52 (81) , juan.aguilargb@uanl.edu.mx ABSTRACT Dielectric approaches for estimating microwave heating of ceramics and polymers assuming that resistive effects are small enough to be neglected have been applied quite often. However, these approaches are not valid for semiconductors. Permittivity and resistivity are very sensitive to chemical composition; therefore it is difficult to extrapolate properties from similar materials as can be done with thermal properties. The objective of this work is to estimate the dielectric and resistive contributions during microwave heating, solving the electromagnetic and heat transfer equations by means of Finite Element Method (FEM) using the software ANSYS TM. The experiments were performed over a parallelepiped made of compacted and sintered powders of the manganite, which is a negative thermal coefficient thermistor, placed inside a waveguide and exposed to 2.45 GHz microwaves at different powers. Temperature was taken with sheathed thermocouples that also served as electrodes for monitoring electric resistance. Matching experimental data and simulation results provided values for the properties within acceptable range. An operative model for simulating heating of a sample placed in a waveguide was built, considering either resistive or dielectric effects, which in turn helps to estimate these properties applying it jointly with the experimental technique shown. KEYWORDS: Manganite, modeling, dielectric heating, resistive heating, microwave heating INTRODUCTION It has been demonstrated that microwave heating produces temperatures high enough for producing alumina magnesia spinel [1], calcium zirconate [2] and silicon carbide [3]. It is assumed that the main mechanism of energy absorption in these systems is dielectric, letting out possible resistive effects. This assumption could be valid for polymers and ceramics, but semiconductors are far of being in the extreme of conductive or insulator materials. There are reports about microwave sintering of manganites, manganese oxides, with Nickel: Mn(Ni,Fe) 3 O 4 [4, 5], which are semiconductors and are used as a test material in this research. The aim of this work was to use a simulation model built with electromagnetic field and flow heat equations upon the consideration of both, resistive and dielectric heating. In this case the analytical solution of this equation system is complicated and numeric methods are necessary; hence in this work finite element method was employed using ANSYS TM solution software.

2 RESISTIVE DIELECTRIC HEATING MODEL Description of the heating process of semiconductors requires a model that considers that energy dissipation in the sample is due to both, resistive and dielectric effects. The equations that describe these phenomena (equations 1-3) were coupled in order that thermal behavior can be described from a given set of electromagnetic and thermal properties. The model also allows estimating the parameters when heating data is available. 2 Ε = ω 2 μ ε Ε Electric field: (1) T 2 2 Heat transfer: = T + f ( x, y, z) t α (2) 1 2 Power dissipation: P = ( σ + ω ε ) E E dv V (3) Where: E: electric field (V/m); ω: frequency (rad/s); μ: permeability (H /m), ε: permittivity (F/m); x, y, z: spatial coordinates (m); T: Temperature (K); t: time (s); α 2 : thermal diffusivity (m 2 /s); P: dissipated power (W); σ: conductivity (S/m). Resistivity as function of temperature σ=f (T), temperature as function of time T=f (t) at a given power and frequency ω are known from the heating tests and can be applied to a resistivedielectric model. Permeability of the sample is assumed as (μ=1) and its thermal diffusivity (α 2 ) is taken from values of materials of the same family because they exhibit similar thermal properties. Therefore, the only unknown parameter is the complex permittivity: 1) real part of the permittivity (ε ), and 2) imaginary part of the permittivity (ε ). The real part is related to power dissipation dielectric and resistive. The magnitude of the complex electric permittivity or loss factor is related to power dissipation by dielectric effects. Experimental data and the certainty of having a correct mathematical proposal that considers the two heating phenomena allow estimating these values and reproducing the thermal steady state. EXPERIMENTAL Sample It consists of a manganite semiconductor (Fe 0.22 Ni 0.67 Mn 2.11 O 4 ) produced by sintering of compacted powders. It was a parallepiped of a 5.9 mm X 11.4 mm X 1.6 mm; density was 4.2 g/cm 3. Both ends of the sample (5.9 mm X 1.6 mm faces) were coated with a 2 microns layer of silver for having good electrical contact with the electrodes for resistivity measurements.

3 Experimental arrangement Experimental arrangement consisted in a 2.45 GHz magnetron connected to a waveguide and other WR284 accessories. The sample was placed inside the waveguide (figure 1) between the generator and a short for having a stationary wave, locating the maximum amplitude of the microwave over the sample. Tests (Transverse Electric, TE 10 mode) were conducted at powers between 30 Watts and 130 Watts. The sample was hold between two thermocouples that had a double purpose; taking sample temperature, and forming a circuit with it for having resistivity measurements considering that [5]: Ungrounded type K thermocouples are covered with a stainless steel sheath, hence microwave do not interfere with the thermocouple wires. The stationary wave does not provide a variable magnetic field in the thermocouple placed perpendicularly to the electric field in the waveguide. The sheaths of the thermocouples used as electrodes, were not in electrical contact with the waveguide. Temperature, electrical conductivity and applied power were recorded during the tests. The sheaths of the thermocouples are the same diameter as the thickness of the sample (1.6 mm) for avoiding discontinuities in the discretization arrangement in the computational model. A detail that must be considered is that the measured resistance was DC, no reactance, taken with the current that passes through the sample, the electrode-sample interface and the thermocouple sheath itself. Impedance requires that an alternating current is fed at a given frequency to the sample under test. In a strict sense this frequency must be the same than the microwaves that are being employed for heating. However, Joule Effect for these materials is somehow controversial [6, 7] and without leaving out that impedance is highly sensitive to frequency, there are reports [8] showing strong increments of both, resistance and reactance, at 1 MHz in La0.7Sr0.3MnO 3. It was considered in this work that using DC resistance approach gives a conservative figure for considering in the model. SIMULATION Simulation was carried out by finite element (FEM); computer aid is necessary because thousands of discrezitation elements are often required in an ordinary system, which implies solving large equation systems. In this case commercial software ANSYS TM was used for performing this task. Definition of the geometric simulation domain Symmetry along axe Z was considered for reducing calculation load considerably because the sample was located in the center of the coordinate (a) of the waveguide (figure 1), taking one thermocouple and half of the sample. Even when the wave travels 2.24 m in the waveguide, the

4 model is built only around 23.2 cm (1 wave length) around the sample, covering two wavelengths in total. Discretization of the geometric simulation domain The defined geometric domain was discretized in a three-dimensional element mesh, which includes the waveguide, the thermocouple and the sample. It is considered during generation of the mesh the finite element model that: The amount of elements should minimize the possible discrepancies between the real and the simulated geometry. The mesh must be adapted so that the model produces physically consistent results. The amount of elements and nodes must be minimized to reduce computing time. Elements in the sample were smaller than 1 mm for minimizing discrepancies among the real and the simulated geometry, however, using this size of element in the waveguide that is very large compared to the sample would give too many elements, then a transition from relatively large elements in the waveguide to smaller ones in the thermocouple-sample set was performed. The strategy of meshing consisted on the elaboration of two auxiliary volumes that involved the sample-thermocouple set (figure 2). The volume of the waveguide around the outer auxiliary volume was meshed with elements of 8 mm by side, and then at the outer auxiliary volume was refined to 4 mm, and then to less than the 1 mm at the inner auxiliary volume. The sample and the thermocouple were meshed with elements of 0.7 mm and 0.8 mm respectively (figures 3 and 4). The amount of elements dedicated to each component is presented in Table I. Due to the considerable difference of size of the elements of the mesh of the finite element model, it was decided to verify its precision and physical sense by means of a comparison of the electric field distribution calculated with this model, against the electric field distribution in the domain of the same model evaluated according to the analytical solution of the wave equation of the electric field in an empty waveguide using microwaves: 130 Watts at 2.45 GHz. The analytical solution of the wave equation was evaluated by means of a program codified in MATLAB TM giving that the electrical field distribution produced by the FEM model was practically the same than the electric field distribution of the same model according to the analytical solution of the wave equation, which is a sort of proof of the physical consistence of the FEM model and the validity of the developed meshed. Boundary conditions, materials and parameter properties of the electromagnetic model The waveguide is full of air (ε=1, μ=1), the walls are metallic; aluminum in the waveguide and stainless steel at the test section, properties of the materials are given in Table II [9]. The Neumann condition frontier [10] was applied to the symmetry region of the waveguide, including the cross section of the sample over the symmetry plane. C u n = 0 (4)

5 This indicates that the net flow, both electrical and magnetic field through the symmetry region is zero. The characteristics of the microwave were specified at the sample zone. In the rest of the faces of the sample the frontier condition of Dirichlet [10] was applied: C u n 0 (5) Meaning that in this case a net flow of electrical and magnetic field is taking place. Electromagnetic properties of the thermocouple wires are not necessary because its sheath is not being penetrated by the microwaves. In the case of the sample of manganite the resistivity was taken experimentally, as explained, while the dielectric properties were estimated with the proposed model adjusted to temperature data. Boundary conditions and material properties in the thermal model The thermal model considered the geometric domain of the sample and the thermocouple only, temperature is calculated by the general diffusion equation of heat, including heat generation in the sample by the action of the microwave according to equation (2). Heat from these components to the surrounding air inside the waveguide is given by convection: T n = h k [ T T ] (6) Where: h: convection coefficient (W/m 2 ); k: heat transfer coefficient (W/m K). The temperature of the air (T = 23 C) was taken from the average of the two thermocouples at the beginning of the experiments (sample at room temperature) and the value of the convection coefficient (h=5 W/m²) corresponds to a typical value for the stationary air. In the case of the heat transfer coefficient (k) of the manganite was taken from manganese oxide, which is the majority component according to the NASA s Materials Properties Database [11]. The thermocouple was considered entirely built of stainless steel; thermal properties of these materials are shown in Table III. Assumption of thermocouple sheaths taken as solid rods suppose a larger mass of thermocouples, which is not important once that the steady state has been achieved, but it could be very important regarding heat transfer from the sample because of the high thermal conductivity of metals and the effective area compared to the sheath, actually the cross section of a pipe. This difference, as other issues in models where as many as possible considerations are included, becomes one of the factors that must be evaluated.

6 RESULTS AND DISCUSSION Temperature and resistivity tests of the manganite Power inputs were chosen in order to heat the sample to less than 100 C for avoiding aging [12] so that the same sample can be used for all the tests. The microwaves were maintained long enough to let the sample achieving a steady temperature state, specifically when the temperature did not change in more than 1 C in 5 minutes. Temperature of the sample evolved in a similar way in all the tests, being increased relatively rapid in the first five minutes, before reaching a steady thermal state (figure 5). Resistivity exhibited an inverse behavior to temperature, confirming that indeed this material operates as a negative temperature coefficient thermistor (figure 6). Table IV summarizes the temperature and resistivity data at the beginning and at the end of each test. The resistivity values at final heating stages when temperature was steady are also shown in figure 7, following the curves that were taken during heating at different powers and follows the same behavior at different powers, which indicates that indeed there was no appreciable aging of the material at this temperature range. Uncertainties at low temperatures are normal, because they are close to the room temperature and it is difficult to establish the thermal profile given by the microwaves. Estimating the electric permittivity and the loss factor of manganite For estimating the magnitude orders of the two unknown parameters of the resistive-dielectric heating, it is necessary to consider one of the unknowns at a time. The resistive part can be solved taking the resistivity data from the heating experiments, being the permittivity the only unknown to be determined. This parameter is adjusted until the differences among the temperatures given by the model against the experimental ones are minimal. Comparisons of actual temperatures and those produced by the modeled are taken from a node that is at the center of the thermocouple where it is in contact with the sample. Once the permittivity has been estimated, the thermal model is ran again, but now considering only the dielectric part, hence the only unknown parameter is the loss factor, which is adjusted until, once again, the difference among experimental and modeled temperatures are minimized. The permittivity values that explain a purely resistive heating in steady state are presented in table V. Figures 8-11 show examples of the electric field and temperature distributions of the manganite and thermocouple calculated with the model implemented in ANSYS TM during the process for estimating permittivity. Estimated temperature-permittivity relationship is shown in figure 12; found values are in agree with the magnitude orders reported in literature for this kind of materials [13]. Differences around room temperature were already explained. It was observed during the analysis of the permittivity that as this value was increased, the electric field and the power absorption were decreasing, and consequently, the temperatures calculated by the model were lower than the experimental ones (equation 7-9). This difference can be compensated in the model with higher dielectric losses, which means that values of electric permittivity obtained with the supposition of purely resistive heating, corresponds to the minimal values of permittivity of the manganite. The assumption of solid rod thermocouples could be a source of error; however trials reducing conductivity by a factor given by the cross section of the sheath taken as a pipe to solid rod did not affect the found values of electric permittivity and loss factor of the manganite when temperature differences were minimized.

7 E Y With: β π 2 E sin x sin 10 t a ( β z) sin( ω ) = o (7) π = ω με 2 (8) a And the absorbed power: 1 P = 2 σ E E dv V Where β 10 : phase constant (rad / m) (9) Once that the minimal values of permittivity of the manganite were estimated considering resistive heating only, an analogous procedure was performed for estimating the values of the loss factor of this material. In this case the dielectric part was the only parameter considered in the resistive-dielectric heating model, obtaining the values of the loss factor and tan (δ) (Table VI). Found values are plotted in figures 13 and 14, as for the temperature permittivity case, the estimated temperature loss tangent and temperature loss factor curves show similar behavior and values reported for the same kind of materials [13]. The results show agreement is the same sense that has been presented by Ehlers [14]. CONCLUSION An operative FEM model for heating simulation of a sample placed in a waveguide, considering either resistive or dielectric effects has been achieved. This model in combination with experimental data provides a way for estimating the range of the permittivity values. Taking resistance instead of impedance is a very important issue; therefore this model is prepared for allocating impedance values, when available, and elucidating contribution of several variables to the microwave heating. Although the calculations provided values within the traditionally reported data for this kind of materials, the results given by the model are a starting set of values for prediction issues because the model is based on steady state. Uniform heating can be accepted if the sample is small, in terms of wave penetration depth and the properties, in the case of dielectric heating can be proposed that large low loss materials can be heated as uniformly as small ones with higher values. Resistive heating can be also predict because induced current and skin effect can be estimated also, thus as in the dielectric case it is also possible to say how a material will be heated. Dielectric and resistive heating depend on temperature as at the same time temperature depends on energy input.

8 Materials of the same type often exhibit very different permittivity, temperature relationships and impedances among them, and then a model that provides estimated values and narrows uncertainties is always useful. REFERENCES [1] AGUILAR J, GONZALEZ M, GOMEZ I., Microwaves as an energy source for producing magnesia-alumina spinel. Journal of the Microwave Power and Electromagnetic Energy 1997; 32(2): [2] AGUILAR J, GOMEZ I., Microwave processing of calcium zirconate from CaO and ZrO 2. Advances in Technology and Materials Processing Journal 2003; 5(2): [3] AGUILAR J, RODRÍGUEZ J, HINOJOSA M. Production of β-sic with microwaves as an energy source. Journal of the Microwave Power and Electromagnetic Energy, IMPI 2001; 36(3): [4] VALDEZ NAVA Z. Sinterización de manganitas Ni-Fe empleando microondas como fuente energía. Universidad Autónoma de Nuevo León. Doctoral Thesis, [5] VALDEZ Z, GUILLEMET S, AGUILAR J, DURAND B, HINOJOSA M. Conductivity measurements of a spinel manganite ceramic in a microwave field. 9th International Conference on Microwave and High Frequency Heating [6] GHATAK S.K, KAVIRAJ B, DEY T.K., Giant magnetoimpedance in Ag-doped La 0.7 Sr 0.3 MnO 3. Journal of Applied Physics, 2007, Volume 101, Issue 2, pp [7] Nurgaliev T. Modeling of the Surface Impedance of Conductive Ferromagnetic Films on Different Substrates Proceedings of the CSMAG'07 Conference, Kosice, July 9-12, 2007, Published on ACTA PHYSICA POLONICA A (2008) Vol. 113 No [8] MERCONE S, FRESARD R, CAIGNAERT V, MARTIN C, SAUREL D, SIMON C, ANDRE G, MONOD P, FAUTH F. Nonlinear effects and Joule heating in I-V curves in manganites Journal of applied physics. 2005, vol.98, n o 2, pp [9] CARTER R.G. Electromagnetism for Electronic Engineers. Kluwer Academic Publisher, [10] STANLEY J.F. Partial Differential Equations for Scientifics and Engineers. New York: Dover Publications, [11] National Aeronautics Space Administration. Material Properties Database: Web Edition Version 4.2, Last visited: August [12] DMITRY A. KUKURUZNYAK, JEROME G. MOYER, FUMIO S. OHUCHI. Improved Aging Characteristics of NTC Thermistor Thin Films Fabricated by a Hybrid Sol-Gel- MOD. Process Journal of the American Ceramic Society, 2006, 89 (1): [13] VON HIPPEL A. Dielectrics and Waves. Artech House Publishers, [14] EHLERS R. METAXAS R. An investigation on the effect of varying the load, mesh and simulation parameters in microwave heating applications. Journal of the Microwave Power and Electromagnetic Energy 2007; 40(4):

9 ACKNOWLEGMENTS Authors express their gratitude to the Mexican Council for Science and Technology (CONACYT) and the Universidad Autónoma de Nuevo León for their grant U38672 and PAICYT CA respectively.

10 Figure 1. Location of the sample and the thermocouples inside the waveguide.

11 Figure 2. Auxiliary volumes around the sample and the thermocouple zone.

12 Figure 3. Mesh of the sample and of the thermocouple.

13 Figure 4. Location of the sample and the thermocouple in the final mesh of the model according to the experimental arrangement.

14 Figure 5. Evolution of the temperature of the sample during the heating tests at different microwave powers.

15 Figure 6. Evolution of the resistivity of the sample during the heating tests at different microwave power.

16 Figure 7. Temperature resistivity relationship in the sample during heating tests at different microwave power. Dots correspond to steady thermal states reached.

17 Figure 8. Electric field distribution of the sample and thermocouple at 30 W.

18 Figure 9. Temperature distribution of the sample and thermocouple at 30 W.

19 Figure 10. Electric field distribution of the sample and thermocouple at 130 W.

20 Figure 11. Temperature distribution of the sample and thermocouple at 130 W.

21 Figure 12. Temperature permittivity relationship estimated from the model.

22 Figure 13. Estimated temperature tan (δ) of the manganite.

23 Figure 14. Estimated temperature loss factor of the manganite.

24 Table I. Amount of elements in each component of the model. Component Amount of elements Waveguide 10,616 Outer auxiliary volume 5,723 Inner auxiliary volume 5,696 Thermocouple 1,244 Sample 1,102 Total of elements 24,381 Total of nodes 35,745 Table II. Electromagnetic properties of materials [9]. Conductivity (S/m) Relative permeability Aluminum (in the waveguide) 3.5 x Stainless steel (in the test 0.57 x section of the waveguide and in the sheath of the thermocouple) Silver coating 6.1 x Table III. Thermal properties of the sample of manganite and of the thermocouple [11]. Component Conductivity Specific heat Density (ρ) (k) (J/s-m- C) (Cp) (J/kg- C) (g/m³) Manganite * Thermocouple * Manganite was produced for this research and its density was measured.

25 Final resistivity (Ω-m) Table IV Temperature and resistivity at the beginning and at the end of each heating test. Test Power (W) Initial temperature Initial resistivity (Ω- Final temperature ( C) m) ( C) Table V. Results of the estimated permittivity of the manganite. Experimental data Data from the resistive model Power (W) Resistivity (Ω-m) Temperature ( C) Permittivity (ε) Temperature ( C)

26 Table VI. Results for the loss factor of the sample Experimental data Dielectric model data Power (W) Temperature ( C) Permittivity (ε ) tan(δ ) Loss factor (ε ) Temperature ( C )