Progress In Electromagnetics Research Letters, Vol. 11, , 2009

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1 Pogess In Electomagnetics Reseach Lettes, Vol. 11, , 2009 COMPUTATIONAL MODELING OF INDUCTION HEATING PROCESS M. H. Tavakoli, H. Kabaschi, and F. Samavat Physics Depatment Bu-Ali Sina Univesity Hamedan 65174, Ian Abstact An accuate 2D steady state mathematical model fo induction heating pocess is descibed and additional esults of electomagnetic field, eddy cuents distibution and volumetic heat geneation have been computed fo a sample setup using a finite element method. Fo the calculations, the input voltage of induction coil is set to be 200 V with a fequency of 10 khz. It was shown that fo the case consideed hee, the distibution of eddy cuents density along the adius/thickness of the wokpiece has a damped sinusoidal wave-shaped fom. 1. INTRODUCTION Radio fequency induction heating is a non-contact heating pocess and has the geatest application in mateial pocessing such as heat teating, joining, welding, bazing, soldeing, melting and testing. Induction heating is the pocess of heating an electically conducting mateial (usually a metal) by electomagnetic induction, whee eddy cuents ae geneated within the wokpiece and esistance leads to Joulean heating (I 2 R) of the mateial in the fom of tempoal and spatial volumetic heating. Induction heating povides a numbe of advantages such as: quick heating, high poduction ates, ease of automation and contol, and safe and clean woking [1 3]. An induction heating installation has thee impotant pats: a souce of high-fequency altenating cuent, an induction coil (RFcoil) and the wokpiece (metallic mateial) to be heated. The RFcoil establishes an altenating electomagnetic field in the setup which induces temendous cuents to flow though the wokpiece. These ae known as eddy cuents. In this aspect, the amount of induced Coesponding autho: M. H. Tavakoli (mht@basu.ac.i).

2 94 Tavakoli, Kabaschi, and Samavat eddy cuents and powe as well as thei spatial distibution in the wokpiece ae the majo paametes to be detemined. Undestanding the physics of these popeties is quite cucial and impotant when designing induction heating systems. In this aticle, at fist we explain a two dimensional steady state mathematical model fo induction heating pocess and then detailed specifications of electomagnetic field distibution, eddy cuents patten and heating stuctue in a system ae descibed. 2. MATHEMATICAL MODEL Fo calculating an electomagnetic field it is necessay to solve Maxwell s equations [1 3]. In ode to solve these equations fo ou pupose, we have to make seveal assumptions: (1) The system is otationally symmetic about the z-axis, i.e., all quantities ae independent of the azimuthal coodinate φ, (2) all mateials ae isotopic, non-magnetic and have no net electic chage, (3) the displacement cuent is neglected, (4) the distibution of electical cuent (also voltage) in the RF-coil is unifom, (5) the self-inductance effect in the RF-coil is taken into account, (6) the cuents (impessed and induced) have a steady state quality and as a esult the electomagnetic field quantities ae hamonically oscillating functions with a fixed single fequency. Unde these assumptions, Maxwell s equations in diffeential fom and in the mks units (mete-kilogam-second-coulomb) can be witten as: E = 0 (fom Gauss s law) (1) B = 0 (fom Gauss s law) (2) B = µj (fom Ampee s law) (3) E = B t (fom Faaday s law) (4) J = σe (5) whee E is the electic field intensity, B is the magnetic flux density, J is the fee chage cuent density, µ is the magnetic pemeability and σ is the electical conductivity of the medium, which is a non-zeo value only in metallic pats. Intoducing the vecto potential A as B = A (6) and assuming axi-symmetic condition, we can tansfom Eqs. (1) (5)

3 Pogess In Electomagnetics Reseach Lettes, Vol. 11, into a simple scala equation [4, 5] ( ) 1 Ψ B + z ( 1 ) Ψ B = µj φ (7) z in which Ψ B is the magnetic steam function defined by Ψ B (, z, t) A φ (, z, t), whee A φ is the azimuthal component of A and (, φ, z) is the cylindical coodinates. In othe wods, we assume that all cuents (diving and induced) flows only in the azimuthal diection both in the coil and the wokpiece. If we include the self-inductance effect in the induction coil as eddy cuents epesented by J e = (σ co /)( Ψ B / t), then { Jd +J J φ = e =J d σco Ψ B t diving and eddy cuents in the coil J e = σ w Ψ B (8) t eddy cuents in the wokpiece whee σ co and σ w ae the electical conductivity of the RF-coil and wokpiece, espectively. Setting J d = J 0 cos ωt as the diving cuent in the coil we can find a solution of the fom Ψ B (, z, t) = C (, z) cos ωt + S (, z) sin ωt (9) whee C(, z) is the in-phase component and S(, z) is the out-of-phase component of the solution. Now the coupled set of elliptic PDE s fo C (, z) and S (, z) is: ( 1 ( 1 ) C + z ) S + z ( 1 ( 1 ) C = z ) S = z ( µ co J0 σ coω S ) coil µ wσ wω S wokpiece (10) 0 elsewhee µ co σ coω C coil µwσwω C wokpiece 0 elsewhee (11) Afte solving (10) and (11) fo C (, z) and S(, z), the eddy cuents distibution and the enegy dissipation ate can be computed via J e = σ wω [C (, z) sin ωt S (, z) cos ωt] = J S sin ωt + J C cos ωt (12) and P (, z, t)= J φ 2 σ ( ) σ co ω [C 2 2 sin 2 ωt+ J0 2cos ( ] = 2 σ S 2 coω ωt+c J0 σ S coω )sin 2ωt coil ( σ w ω 2 C 2 sin 2 ωt + S 2 cos 2 ωt CS sin 2ωt ) (13) wokpiece 2

4 96 Tavakoli, Kabaschi, and Samavat Thus, the powe is geneated in all metallic pats (including RFcoil) as a function of 2ω. The peiod fo this time-dependence is τ = 2π/ω = 10 4 s fo a 10 khz induction system. Since the timehamonic function is so shot, epesentation of the heat geneation by the time aveaged quantity is moe useful, which we aveage ove one peiod to obtain the volumetic heat geneation ate, q (, z) = ω 2π = 2π/ω 0 σ co ω [C 2 + P (, z, t) dt ( J0 σ co ω S ) 2 ] coil σ w ω ( C 2 + S 2) wokpiece (14) The bounday conditions ae Ψ B = 0; both in the fa field (, z ) and at the axis of symmety ( = 0) The Calculation Conditions Values of electical conductivity employed fo ou calculations ae pesented in Table 1 and opeating paametes ae listed in Table 2. The system unde consideation includes a ight cylindical conducto Table 1. Values of electical conductivity (mho/cm) used in ou calculations; the subscipts co and w denote coil (coppe) and wokpiece (steel), espectively. Symbol Value σ co σ w Table 2. Opeating paametes used fo calculations. Desciption (units) Symbol Value Wokpiece adius (mm) w 55 Wokpiece height (mm) h w 93 Coil inne adius (mm) co 85 Coil width (mm) L co 10 Coil wall thickness (mm) l co 1 Height of coil tuns (mm) h co 13 Distance between coil tuns (mm) d co 3

5 Pogess In Electomagnetics Reseach Lettes, Vol. 11, load (i.e., steel wokpiece) in the diection of Oz (i.e., Oz is the centeline) which is suounded by a multitun cylindical induction coil with 6 hollow ectangula-shaped coppe tuns, Figue 1. In a eal induction system this coil is usually cooled vey efficiently by wate flowing inside the coil tuns. Theefoe it is ealistic to assume that the coil is always at oom tempeatue. We assume a total voltage of the coil V coil = 200 V with a fequency of 10 khz. The diving cuent density in the induction coil is calculated by J 0 = σ co V coil /(2πR co N), whee R co is the mean value of the coil adius and N is the numbe of coil tuns. Fo the magnetic pemeability (µ) we assume that it is eveywhee the constant value of fee space µ = µ µ 0 µ 0 (i.e., µ 1) whee µ is the elative magnetic pemeability. The quantity δ w = (2/µ w σ w ω) 1/2 has the dimension of a length and is called the skin depth (o penetation depth). It is a measue of the field penetation depth into the conductos. In ou calculation the coesponding skin depth of the steel wokpiece is δ w = 2.5 mm. It should be mentioned hee that since induction heating is a vey complicated pocess, we can not expect accuate simulation of the whole chain of coupled phenomena electomagnetic, themal, mechanical, hydodynamic and metallugical duing a heating pocess. The most impotant and contollable pocess is electomagnetic which is analyzed hee. Figue 1. Sketch of the induction heating setup including a ight cylindical wokpiece and a multitun RF-coil. Figue 2. The finite element mesh of the calculation domain. Special fine gid is placed in the aeas whee esolution of the solutions is expected to be difficult.

6 98 Tavakoli, Kabaschi, and Samavat 2.2. Numeical Method The involved patial diffeential equations equie using a numeical discetization method to solve them. The calculation of the fundamental equations with bounday conditions have been made by 2D finite element method, Figue 2. Afte solving the set of equations we can obtain the electomagnetic field and eddy cuents distibution as well as the powe densities in all pats of the studied system. The esults based on this set of paametes will be pesented now. 3. RESULTS AND DISCUSSION Figue 3 shows the distibution of in-phase component (ight hand side) and out-of-phase component (left hand side) of the magnetic steam function. The maximum of in-phase component (C) is located at the lowest and top edges of the RF-coil (C max = webe) while the minimum (C min = webe) is located on the uppe and lowe cones of the wokpiece. Fo the out-of-phase component (S), the maximum is located at the oute sufaces of the induction coil tuns (S max = webe) and its distibution has a linea gadient in the space between the coil and the wokpiece. The wokpiece tends to foce the electomagnetic field components out of its body and makes Figue 3. Components of the magnetic steam function (Ψ B ) calculated fo the case consideed hee. The ight hand side shows the in-phase component (C) with C max = webe on the lowest and top edges of the RF-coil and C min = webe on the edges of the wokpiece. The left hand side shows the out-of-phase component (S) with S max = webe on the oute sufaces of the induction coil tuns.

7 Pogess In Electomagnetics Reseach Lettes, Vol. 11, them moe intense at the space between the wokpiece and the coil compaed to othe places. Defomation and distotion of the C-component in the aea close to the top and bottom edges of the wokpiece and RF-coil is also paticulaly evident in Figue 3 (edge effect). It is woth to mention that the distibution of the in-phase component (C) depends on the geomety of both induction coil and wokpiece, while the out-of-phase component (S) mainly depends on the configuation of RF-coil whee the diving cuent is poduced. Figue 4 shows the adial pofile of both components in the middle of the wokpiece. It is clea that both components have a damped oscillating behavio. Consequently, the induced electomagnetic field within the wokpiece, concentated nea the suface and decay with distance fom the suface. It is inteesting to note that both components ae quite small and also S/C 1 in the wokpiece whee the equied heat geneation occus. As a esult, both components take pat to the equied heat geneation equally, accoding to Eq. (14). Figue 5 shows the distibution of in-phase component and outof-phase component of the eddy cuents (i.e., J C and J S ) in the wokpiece. It is seen that the distibution of both components have a damped sinusoidal wave shape which is a esult of popagation of the electomagnetic wave in the conducto wokpiece [6], Figue 6. In othe wods, the damped sinusoidal eddy cuents flow is poduced by damped oscillatoy electomagnetic field in the conducto wokpiece. Both components ae always in the opposite diection to the diving Figue 4. The adial pofile of the C and S components along A-B line of the wokpiece thickness.

8 100 Tavakoli, Kabaschi, and Samavat Figue 5. Components of the eddy cuents distibution in the wokpiece. The ight hand side shows the in-phase component (J C ) with JC min = A/m 2 and JC max = A/m 2 and the left hand side shows the out-of-phase component (J S ) with JS min = A/m 2 and JS max = A/m 2. JC min and JS min ae located on both edges while JC max and JS max ae positioned on the aea close to both cones. Figue 6. The damped waveshaped pofiles of the eddy cuents components along A-B line of the wokpiece thickness. Figue 7. Volumetic powe distibution (q) in the wokpiece and the RF-coil with qmax wokpiece = W/m 3 ) on the both cones. cuent J 0 (because J C < 0 and J S < 0) on the oute suface of the wokpiece side wall. Both JC min and JS min ae located on the both edges of the wokpiece and the position of JS max as well as JC max is in the

9 Pogess In Electomagnetics Reseach Lettes, Vol. 11, wokpiece body and close to the ends because of intense edge effect. JS min shows the elated wave cest which is not tue fo JC min. All effective eddy cuents ae located in a distance of 6δ w fom the side wall and as a esult no cuent in the inne potion. The volumetic heat geneation ate (q) in the system has been shown in Figue 7. The powe intensity is at its maximum value at the both edges of the wokpiece (whee the JC min and JS min ae located) and falls off about exponentially inwad. These ae hot spots (highly heated aeas) which ae a diect esult of the edge effect and distotion of electomagnetic filed in those aeas, Figue 8. The oute suface to coe heating diffeence is a esult of the skin effect (the phenomena of nonunifom induced altenative cuent distibution within the conducto coss section) [1 3]. The appeaance of vaies heating pattens is caused diectly by diffeence in the eddy cuents distibution in the wokpiece. The spatial distibution of heat geneation in the induction coil is mostly unifom with local hot spots at the lowest and uppe edges. The skin effect and poximity effect (the effect on cuent distibution in a conducto when anothe conducto is placed neaby) ae esponsible fo these hot spots. Since the cuents (diving and induced) have the Figue 8. The W-shaped pofiles of the heat geneated along the oute suface of the wokpiece side wall. Table 3. Detail infomation about the heat geneated in the heating setup. Pat Heat geneated (Watt) Pecentage (%) wokpiece RF-coil Total

10 102 Tavakoli, Kabaschi, and Samavat same diection in the coil tuns, the eddy cuents will be concentated on opposite side of the induction coil (i.e., the top and lowest sufaces). Detail infomation about the heat geneated in the system has been shown in Table CONCLUSION Detailed specifications of an accuate 2D steady state mathematical model fo induction heating pocess is built to study the electomagnetic field distibution, eddy cuents density and heating patten in the wokpiece. Fom the obtained esults we can conclude: The finite element (FE) modeling of the induction heating is still a challenging task. The spatial stuctue of electomagnetic field is a complex function of the wokpiece and the RF-coil geomety. Fo the case consideed hee, the cuents density distibution along the adius/thickness has a damped sinusoidal wave-shaped fom, which is significantly diffeent than the classical exponential distibution. Because the cuent densities ae concentated in the extenal aeas of the wokpiece and specially in both edges, the esulting heating density is mainly poduced in those aeas (extenal heating) and the intenal heating (coe heating) will be consideably weak fo the opeating fequency consideed hee. REFERENCES 1. Leatheman, A. F. and D. E. Stutz, Induction Heating Advances, National Aeonautics and Space Administation, Rudnev, V., D. Loveles, R. Cook, and M. Black, Handbook of Induction Heating, New Yok, NY, Zinn, S. and S. L. Semiatin, Elements of Induction Heating, ASM Intenational, Gesho, P. M. and J. J. Deby, A finite element model fo induction heating of a metal cucible, J. Cystal Gowth, Vol. 85, No. 1 2, 40 48, Tavakoli, M. H., Modeling of induction heating in oxide Czochalski systems Advantages and poblems, Cyst. Gowth Des., Vol. 8, No. 2, , Reitz, J. R., F. J. Milfod, and R. W. Chisty, Foundations of Electomagnetic Theoy, Addison-Wesley, 1993.