Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius

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1 ISSN(Print): X ISSN(Online): VOLUME 1, NUMBER 2, JUNE 2014 OPEN JOURNAL OF MODERN PHYSICS Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius A. M. Khaleghi 1 *, M. H. Tavakoli 1 and E. Mohammadi Manesh 2 1 Physics Department, Bu-Ali Sina University, Hamedan, I.R. Iran. 2 Physics Department, Malayer University, Malayer, I.R. Iran. *Corresponding author: mteskh@gmail.com Abstract: An induction coil with two different radiuses for oxide Czochralski crystal growth systems is considered and corresponding results of electromagnetic field and volumetric heat generation have been computed using a finite element method (FlexPDE package). The calculation results show the importance of the coil radius on the heat generation distribution in a CZ growth system. Keywords: Computer Simulation; Czochralski Method; Induction Heating 1. INTRODUCTION The principle of induction heating is applied by the Czochralski (CZ) crystal growth technique to supply the required thermal energy to the melt part, Figure 1. This power is generated by induction heater coils (inductor) that surround the metal crucible. An alternating current is passed through the coils. This azimuthal current produces an approximately orthogonal time-varying magnetic field outside the coils (Amperes law) which in turn generates (induces) an oscillating azimuthal electric field (Faraday s law). Both fields penetrate the metal crucible and other metallic parts (such as active afterheater, chamber and... ), to an extent that depends on the electrical conductivity of the metal. The electric field within the metal walls causes a parallel current flow (Ohm s law) and the product of the electric field strength with the current describes the rate of energy dissipation in the metal - the familiar I 2 R heating - in the form of temporal and spatial volumetric heating [1 4]. In order to gain a better understanding of the entire CZ process as applied to the growth of oxide crystals (e.g., YAG (Y 3 Al 5 O 12 ), sapphire (Al 2 O 3 ), GGG (Gd 3 Ga 5 O 12 ) and BGO (Bi 4 Ge 3 O 12 )), which includes various modes of heat transfer and a complex range of fluid dynamic phenomena, the simulation of the CZ growth is necessary. The simulation should be started with a proper description of the source term for crystal growth - power supplied to the melt [5 7]. In this article, at first we explain a two dimensional steady state mathematical model for induction heating process and then detailed specifications of electromagnetic field distribution and heating structure in a CZ system with different configuration of Radio frequency (RF) coil are described. 41

2 Figure 1. Schematic diagram of the inductively heated Czochralski furnace (crystal pulling). 2. MATHEMATICAL MODEL The following assumptions are valid in our numerical calculation of induction heating: (1) all materials are isotropic and non-magnetic and have no net electric charge. (2) The displacement current is neglected. (3) The heat generation is independent on the temperature of crucible and afterheater. Under these assumptions, the governing equations are [1, 2, 8]. 1 r r ( J 0 cosw t J f = s c r Y B t Y B + r z 1 r Y B = µ 0 J f (1) z driving current in the coil eddy current in the conductors Y B (r,z,t)=c (r,z)cosw t + S(r,z)sinw t (3) q(r,z)= s cw 2 2r 2 C 2 + S 2 (4) Where y B (r,z,t) is the magnetic stream function, C(r,z) and S(r,z) the in-phase and out-of-phase component, respectively; q(r,z) the volumetric power generation in the metallic parts, w frequency of the driving current in the induction coil, J the charge current density, s the electrical conductivity, µ 0 the magnetic permeability of free space and t the time. The boundary conditions are y B =0; both in the far field (r, z! ) and at the axis of symmetry (r=0). The driving current density in the induction coil is calculated by J 0 =s co V co /(2pR co N) where V co is the total voltage of the coil, R co is the mean value of the coil radius and N is the number of coil turns. (2) 42

3 Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius Table 1. Symbol Value s co s cr s ch Table 2. OPERATING PARAMETERS USED FOR CALCULATIONS Description (units) Symbol Value Crucible inner radius (mm) r c Crucible thickness (mm) l c Crucible inner height (mm) h c Afterheater inner height (mm) h af Afterheater hole (mm) r af Coil inner radius (mm) r co Coil thickness (mm) l co Height of coil turns (mm) h co Distance between coil turns (mm) d co Current frequency of RF coil (khz) f , The set of fundamental equations with boundary conditions have been solved using the finite element method (the FlexPDE package [9]). Values of electrical conductivity employed for our calculations are presented in Table 1 and operating parameters are listed in Table 2. In order to compare the results of electromagnetic field and heat generation distribution, we have assumed a driving electrical current with total voltage of 200 v and a frequency of 10 khz in the RF coil (typical values) for all cases. For the magnetic permeability (µ) we assume that it is everywhere the constant value of free space µ = µ r µ 0 = µ0 (i.e. µ r = 1) where µr is the relative magnetic permeability. The results based on this set of parameters will be presented now. Values of electrical conductivity (mho/cm) used in our calculations; the subscripts co, cr and ch denote coil (copper), crucible and afterheater (iridium), and chamber (steel), respectively [8]. 3. RESULTS AND DISCUSSION In order to find a good information of the induction heating in the CZ system, we have considered these cases: Case a: crucible, no afterheater, Case b: crucible and afterheater, Case c: crucible and afterheater, change in the radius of the RF coil, In all cases the induction coil and chamber have been taken into account. 3.1 Case a. Crucible, no afterheater In the first case, there are crucible, RF coil and chamber in the CZ system. The simplest equivalent cylinder model of the 6-turn coil was employed for the simulation, i.e. rather than model each individual coil, the total current was assumed to flow uniformly through a single right cylindrical annulus. Figure 2 shows the contours of in-phase component (right hand side) and out-of phase component (left hand side) of the magnetic stream function for the CZ system. The maximum value of in-phase component 43

4 Figure 2. Components of the magnetic stream function (y B ) calculated for Case a. The right hand side shows the in-phase component (C) and the left hand side shows the out-of-phase component (S). Table 3. HEAT GENERATED IN THE DIFFERENT PARTS Part Heat generated (Watt) Percentage (%) Crucible bottom Crucible wall Chamber 72 3 Total (C max = Weber) is located in the middle part of the coil and its value rapidly decreases toward the crucible wall, while the maximum value of the out-of-phase component (S max = Weber) is located on the middle part of crucible wall and its value rapidly reduces toward the centerline. Although C max is about 8.3 times greater than S max, the magnitudes of C and S are similar within the crucible and afterheater and also chamber wherein the phase shift occurs and the required heat generation is produced. It means that within the conductors both components (C and S) contribute to the heat generation about equally. In the chamber the maximum values of in-phase and out-of-phase components are C max chamber = Weber and S max chamber = Weber, respectively and in the middle part of its side wall. The volumetric heat generation rate (q) in the crucible wall and bottom has been shown in Figure 3. The maximum value of energy deposition in the crucible is q max crucible =44 Watt/cm 3 and it is found at the outer surface of crucible wall. Figure 4 shows the profiles of the generated heat along the outer surfaces of the crucible. Figs. 3 and 4 indicate that the heat is mostly generated in the crucible wall and less in the bottom. The total energy deposition rate in the system (crucible and chamber) is Q totall system =2.11 kwatt by using integral over their volume. More details of heat generated in the different parts have been shown in the Table 3. Detail information about the heat generated in the different parts of the CZ system, calculated for the Case a. 44

5 Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius Figure 3. Contours of the volumetric power distribution (q) in the bottom and side wall of crucible, computed for Case a (for a better demonstration the wall and bottom part are separately magnified). (a) crucible bottom (b) crucible wall Figure 4. Profiles of the heat generated along the outer surface of crucible bottom and side wall calculated for Case a 3.2 Case b. Crucible and afterheater In this case, the configuration contains crucible and active afterheater without any gap between them. Also the RF coil has an additional turn for afterheater. In this case we can not assume the simplest equivalent cylinder model for whole RF coil including this additional turn because the distance between the main coil and this single turn is D coil = 5.5 cm and it is so larger than the distance between the turns in the main coil d co = 3 mm, i.e. we have considered an equivalent cylinder plus a single turn. Figure 5 shows the contours of in-phase component (right hand side) and out-of-phase component (left hand side) of the magnetic stream function for this configuration. The maximum value of in-phase component is C max = Weber located in the middle part of the main coil and the maximum value of the out-of-phase component is S max = Weber located in the middle part of crucible wall. The magnitude of C max is about 7.9 times greater thans max. Also, there is a local maximum of 45

6 Figure 5. Components of the magnetic stream function (y B ) calculated for Case b. The right hand side shows the in-phase component (C) and the left hand side shows the out-of-phase component (S). Table 4. HEAT GENERATED IN THE DIFFERENT PARTS Part Heat generated (Watt) Percentage (%) Crucible bottom Crucible wall Afterheater wall Afterheater top cover 27 1 Chamber 88 3 Total in-phase-component (Csecond coil max = Weber) in the second coil which comes from the driving current in it. The volumetric heat generation rate (q) in the crucible and afterheater has been shown in Figure 6. The maximum value of energy deposition in the crucible and afterheater is q crucible max = 44.6 Watt/cm 3 and it is placed at the outer surface of crucible side wall. Figure 7 shows the profiles of the generated heat along the outer surfaces of the crucible (bottom and side wall) and afterheater (side wall and top cover). It indicates that the heat is mostly produced in the crucible wall and less in the afterheater. The total energy deposition rate in the crucible, afterheater and chamber is Q system total = 2.64 kwatt. More details of the heat generation in the different parts have been shown in the Table 4. Detail information about the heat generated in the different parts of the CZ system, calculated for the A. It should be mentioned that although the values of C max is the same in both Cases a and b, but the value of S max has been increased about 6% in Case b and as a result q crucible max has been increased about 2%. Consequently, qtotal crucible has an increase of 2.2% (45 Watt). Because by placing the active afterheater, the domain of the out-of-phase component (S) is extended to include this active afterheater too. Also noteworthy is the position of the q crucible max does not change. 46

7 Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius Figure 6. Contours of the volumetric power distribution (q) in the crucible and afterheater, computed for Case b. (a) crucible bottom (b) crucible wall (c) afterheater top cover Figure 7. Profiles of the heat generated along the outer surfaces of crucible and afterheater calculated for Case b 47

8 Figure 8. Components of the magnetic stream function (y B ) calculated for Case c. The right hand side shows the in-phase component (C) and the left hand side shows the out-of-phase component (S). 3.3 Case c. Crucible and afterheater, change in the radius of the RF coil In this case the inner radius of the RF coil is increased from r co = 8.5 cm (Case b) to r co = 9.5 cm (i.e. Dr coil = 12%), on the other hand the distance between crucible and coil is increased about 30%. Figure 8 shows the contours of in-phase component (right hand side) and out-of-phase component (left hand side) of the magnetic stream function for this configuration. The maximum value of in-phase component (C max = Weber) is located in the middle part of the main coil and the maximum value of the out-of-phase component (S max = Weber) is placed on the middle part of crucible wall. The magnitude of C max is about 10 times greater than S max. By comparison with the Case b, we can find that the in-phase component is less compressed between crucible and the RF coil because the distance between them has been increased. Also C max has an increase of 19% but S max has been reduced about 7%. The reason is that by increasing the distance between conductors (crucible and afterheater) and induction coil, the influence of their interaction becomes less effective and as a result there is an increase in the component C and a decrease in the component S. The volumetric heat generation rate (q) in the crucible and afterheater has been shown in Figure 9. The maximum value of energy deposition in the crucible and afterheater is q crucible max = 38.3 Watt/cm 3 and at the outer surface of crucible wall. Figure 10 shows the profiles of the generated heat at the outer surface of the crucible and afterheater. The total energy deposition rate in the crucible, afterheater and chamber is Q system total = 2.04 kwatt. More details of the energy generation in the different parts have been presented in the Table 5. The decrease in total power generation of the crucible is 13% (278 Watt) and in contrast, the power dissipated in the chamber is markedly increased 58% (51 Watt) and so the total heat generation in the system is reduced about 9.2% (244 Watt). Also noteworthy is that the change in the coil diameter has not any effective influence on the spatial distribution of heat generation in the crucible and afterheater. Detail information about the heat generated in the different parts of the CZ system, calculated for the Case c. 48

9 Numerical Modeling of Induction Heating in Crystal Pulling Method Change in the Coil Radius Figure 9. Contours of the volumetric power distribution (q) in the crucible and afterheater, computed for Case c. (a) crucible bottom (b) crucible wall (c) afterheater top cover Figure 10. Profiles of the heat generated along the outer surfaces of crucible and afterheater calculated for Case c 49

10 Table 5. HEAT GENERATED IN THE DIFFERENT PARTS Part Heat generated (Watt) Percentage (%) Crucible bottom Crucible wall Afterheater wall Afterheater top cover 24 1 Chamber 70 4 Total CONCLUSION We have presented and demonstrated some results of induction heating for an oxide Czochralski crystal growth system using a finite element method. The following conclusions were obtained: 1. The maximum value of in-phase component (C max ) is located in the induction coil and the maximum value of the out-of-phase component (S max ) is placed on the crucible side wall. 2. Distribution of the in-phase component depends on the coil geometry and for the out-of-phase component depends on the crucible-afterheater orientation. 3. The major portion of the heat generation is in the crucible side wall. 4. By including an active afterheater, the total heat generation in the crucible will be increased. 5. Change in the coil diameter has not any effective influence on the spatial distribution of heat generation in the crucible and afterheater. References 50 [1] P. Gresho and J. Derby, A finite element model for induction heating of a metal crucible, Journal of Crystal Growth, vol. 85, no. 1, pp , [2] M. H. Tavakoli, Modeling of induction heating in oxide czochralski systems advantages and problems, Crystal Growth and Design, vol. 8, no. 2, pp , [3] A. Leatherman and D. Stutz, Induction heating advances. National Aeronautics and Space Administration. [4] R. Valery, L. Don, C. Raymond, and B. Micah, Handbook of induction heating, Manufacturing Engineering and Materials Processing, vol. 61, [5] J. Derby, L. Atherton, and P. Gresho, An integrated process model for the growth of oxide crystals by the czochralski method, Journal of crystal growth, vol. 97, no. 3, pp , [6] M. Tavakoli and H. Wilke, Numerical investigation of heat transport and fluid flow during the seeding process of oxide czochralski crystal growth part 1: non-rotating seed, Crystal Research and Technology, vol. 42, no. 6, pp , [7] M. H. Tavakoli and H. Wilke, Numerical study of heat transport and fluid flow of melt and gas during the seeding process of sapphire czochralski crystal growth, Crystal growth & design, vol. 7, no. 4, pp , [8] M. H. Tavakoli, H. Karbaschi, and F. Samavat, Computational modeling of induction heating process, Progress In Electromagnetics Research Letters, vol. 11, pp , [9]

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