A new heat-treatment system has been developed using an arc driven by an alternating magnetic field. The arc motion was theoretically investigated by the method of non-dimensionalanalysis. After the definition of the pertinent characteristic length and time, the momentum equation was converted into the dimensionless form. This approach gave us not only a short cut to simulate the arc motion but also clear understanding on the nature of the magnetically driven arc

Akiho, R.

Takeda, K.

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

doi:10.14311/ppt.2018.2.49Plasma Physics and Technology 5(2):49–54, 2018 © Department of Physics, FEE CTU in Prague, 2018SIMULATION OF ARC MOTION IN ALTERNATING MAGNETIC FIELDUSING DIMENSIONLESS MOMENTUM EQUATIONK.Takeda∗, R.AkihoFaculty of Systems Science and Technology, Akita Prefectural University, Yuri-Honjyo, Akita, 015-0055, Japan∗ ktkd@ruby.ocn.ne.jpAbstract. A new heat-treatment system has been developed using an arc driven by an alternatingmagnetic field. The arc motion was theoretically investigated by the method of non-dimensionalanalysis. After the definition of the pertinent characteristic length and time, the momentum equationwas converted into the dimensionless form. This approach gave us not only a short cut to simulate thearc motion but also clear understanding on the nature of the magnetically driven arc.Keywords: heat treatment, arc motion, magnetic field, momentum equation, non-dimensional analysis.1. IntroductionWhen an alternating magnetic field is applied per-pendicularly to an arc, the arc swings back and forthlike a pendulum [1–3]. The authors refer to such anarc as a magnetically driven arc. The magnitude ofthe arc power and the amplitude of its oscillatorymotion can be adjusted by changing the arc currentand the applied field strength, respectively. The heatflux distribution can be controlled by changing thewave form of the alternating field. As shown in Figure1, the distribution of the heat flow is rather flat forthe sinusoidal wave form. While for the rectangularwave form, heat flows are concentrated at the bothends of the oscillation amplitude.Using the magnetically driven arc, a new heat treat-ment system has been developed. As shown in Figure2, it is composed of several sections. A transferred dcarc burns between a plasma torch and a work piecewhich serves as an anode. Various types of alternatingmagnetic field could be produced by an ac currentsupply. The work piece travells at any arbitrary veloc-ity in horizontal direction by a driving mechanism ofan anode platform. After the heating, it is quenchedby cooling water sprayed from its back side. Thesequences of the treatment are proceeded automati-cally by a programmable logic controller. To checkthe performance of the treatment, Vickers hardnesstests were carried out. The test for the carbon steelafter the heat treatment revealed that the hardness in-creased more than three times compared to the initialone [4].To expand the application of this heat treatment,it is important to know the exact behaviours of thearc under various operating conditions. Few investiga-tions were carried out for various alternating magneticfields [5, 6]. As the arc motion depends on many op-erating parameters, it is not easy to understand thenature of the arc. The aim of the present work is toopen a simple way to simulate the arc motion using adimensionless analysis. In the following section, thegoverning equation will be prepared together with nec-Figure 1. Control of the heat flow distribution bychanging the wave form of the magnetic field.Figure 2. Arrangement of the developed heat treatmentsystem.essary initial conditions. In Section 3, a new approachusing non-dimensional analysis will be discussed tosolve the governing equation. Numerical calculationswill be carried out in Sections 4 and 5. To confirmthe theoretical predictions, experimental observationwill be reported also in Section 5. Summary talk willbe presented in Section 6.Main parameters and their symbols used in thisarticle are listed in Table 1 .49K.Takeda, R.Akiho Plasma Physics and Technologysymbol nomenclatureB magnetic flux density (vector)B absolute value of BB* critical value of magnetic fieldI arc currentL z-position of the anodeQ mass flow rate of plasma gasR dimensionless position, R = r/ζT dimensionless time, T = t/τX dimensionless x-position, X = x/ζY dimensionless y-position, Y = y/ζZ dimensionless z-position, Z = z/ζb unit vector, b = B/Bj current density of arc (vector)r position of plasma gas (vector)t timev velocity of plasma gas (vector)vo initial velocity of plasma gasx x-component of ry y-component of rz z-component of rα proportional constantλ operating variable, λ = IB/Qθ intersection angle of B to z-axisρ density of plasma gasζ characteristic length, ζ = −vo/λτ characteristic time, τ = 1/λTable 1. List of main parameters used in the modelling2. Momentum equation for the plasmagasThe arrangement of the theoretical model is illustratedin Figure 3. A transferred arc is generated between theanode and the cathode in the torch. Plasma forminggas is ionized in the torch and ejected from the exit ofthe torch nozzle at the velocity vo. It is assumed thatthe magnetic field B is uniform in space and it hasno y-component (By = 0). In an alternating magneticfield, the intersection angle between the field vectorand z-axis changes from θ for a certain half period to(θ + pi) for another half period, since B reverses itsdirection at the time interval of the half period. Thefrequency (f ) of the alternating field is also assumedto be so small that the ionized gas may travel as ifthe field was in a steady state during the flight fromFigure 3. Schematic illustration of the the model.the torch to the anode. The flight time of the plasmagas is estimated to be L/vo in order of magnitude.Typically, L/vo ∼ 10−3 s as the distance (L) and thevelocity (vo) are 10−1m and 102ms−1 respectively.Therefore, if f is restricted in the range of 100Hz orless, the assumption of the steady field can be satisfied[5].The key idea in the present modelling is that thetrajectory of the plasma gas motion represents thearc profile, since the arc current has to pass alongthe stream line of the plasma gas which is electricallyconductive. The momentum equation of the plasmagas is expressed asρdvdt = j×B. (1)Using a unit vector b, the magnetic field is written asB = Bb. (2)When intersection angle equals θ,b = (bx, by, bz) = (sin θ, 0, cos θ) . (3)Based on the fundamental idea used in the modelling,j is expressed asj = −αv (4)where α is a positive proportional constant. A minussign results from the anti-parallel relation between theplasma gas motion and the arc current. Consideringthe continuity of the gas flow and that of the arccurrent, the following relation is obtainedI = αQρ. (5)Then the equation (1) is rewritten asdvdt = −λ (v× b) , (6)whereλ = IBQ. (7)Sincev = drdt , (8)50vol. 5 no. 2/2018 Non-dimensional analysis of arc motionthe momentum equation is expressed asd2rdt2 = −λ(drdt × b). (9)The plasma gas starts from the torch exit at t = 0,with initial velocity vo directed downward along z-axis. The trajectory of the plasma gas is obtained bysolving the momentum equation using a Lagrangianmethod. Initial conditions at t = 0 are given as(r)t=0 = (x, y, z)t=0 = (0, 0, 0) (10)(drdt)t=0=(dxdt ,dydt ,dzdt)t=0= (0, 0, vo) , (11)and (d2xdt2 ,d2ydt2 ,d2zdt2)t=0= (0,−λvosinθ, 0) . (12)When the applied magnetic field is alternating, λin equation (9) and initial condition (12) varies itsmagnitude depending on the field strength B. Andthe vector b changes its direction from θ for a halfperiod to (θ + pi) for another half period. Therefore,for another half period, equation (3) and equation(12) should be replaced by the following equationsb = (sin(θ + pi), 0, cos(θ + pi)) = (− sin θ, 0,− cos θ)(13)and (d2xdt2 ,d2ydt2 ,d2zdt2)t=0= (0, λvosinθ, 0) . (14)3. Non-dimensional analysis3.1. Definition of characteristic length andtimeIn an alternating magnetic field, the arc changes itsprofile with B. To know the variation of arc, it isrequired to solve the differential equation (9) manytimes for various λ. To avoid such troublesome, non-dimensional analysis was proposed in this work. Here,dimensionless time (T) and dimensionless position(R) were considered using characteristic time (τ) andlength (ζ). The time was normalized by the character-istic time and converted into the dimensionless timeasT = t/τ. (15)The position was normalized by the characteristiclength and converted into the dimension position asR = r/ζ (16)orX = x/ζ, Y = y/ζ, Z = z/ζ. (17)Where the characteristic time and length were definedrespectively asτ = 1/λ = Q/ (IB) (18)andζ = −voλ = − (voQ) / (IB) . (19)3.2. Dimensionless momentum equationUsing the dimensionless time and the dimensionlessposition, the momentum equation (9) was transformedinto the following dimensionless equation asd2RdT 2 =(dRdT × b). (20)And the initial conditions presented in equations (10-12) were transformed into(R)T=0 = (X,Y, Z)T=0 = (0, 0, 0) , (21)(dRdT)T=0=(dXdT ,dYdT ,dZdT)T=0= (0, 0,−1)(22)and (d2XdT 2 ,d2YdT 2 ,d2ZdT 2)T=0= (0, sin θ, 0) . (23)To calculate the trajectory of the plasma gas mo-tion, the dimensionless momentum equation (20) wasdecomposed into three scalar equations. For the X-component,d2XdT 2 = −dYdT cos θ. (24)For the Y-component and the Z-component were re-spectively rewritten asd2YdT 2 = −dZdT sin θ +dXdT cos θ, (25)andd2ZdT 2 = −dYdT sin θ. (26)While scalar product of the vector b to the vectorequation (20) yielded the following relationddT(dXdT sin θ +dZdT cos θ)= 0. (27)Considering the initial condition at T = 0 , equation(27) lead todXdT sin θ +dZdT cos θ = − cos θ. (28)Taking the above relation into account, the simul-taneous equations (24-26) were rearranged into thefollowing three third-order linear ordinary differentialequationsd3XdT 3 +dXdT + sin θ cos θ = 0, (29)d3YdT 3 +dYdT = 0, (30)andd3ZdT 3 +dZdT + cos2 θ = 0. (31)As you notice, all the experimental parameters exceptθ disappear in these differential equations and initialconditions. Therefore, the trajectory of the plasma gasin the dimensionless Cartesian coordinate system canbe calculated indifferently to the operating conditionssuch as λ and vo.51K.Takeda, R.Akiho Plasma Physics and Technology4. Solution in dimensionlesscoordinates4.1. Trajectory in the perpendicular fieldTo solve the differential equations (29-31) with theinitial conditions (21-23), the commercial softwareMathematica[7] was used. In a perpendicular mag-netic field (θ = pi/2), differential equations becamequite simple asd3XdT 3 +dXdT = 0, (32)d3YdT 3 +dYdT = 0 (33)andd3XdT 3 +dXdT = 0. (34)Initial conditions were(X,Y, Z)T=0 = (0, 0, 0) , (35)(dXdT ,dYdT ,dZdT)T=0= (0, 0,−1) (36)and (d2XdT 2 ,d2YdT 2 ,d2ZdT 2)T=0= (0, 1, 0) . (37)The trajectory obtained in dimensionless coordinatesystem was represented in Figure 4. At T = 0, theplasma gas started from the origin(0, 0, 0) of the di-mensionless Cartesian coordinate system (X,Y, Z)and it moved along a circle on a Y-Z plane in theclock-wise direction with the increase of T.While for θ = 3pi/2 , equations (32-34) and initialconditions (35-36) were same but the third initialcondition was different from equation (37). It wasdescribed as(d2XdT 2 ,d2YdT 2 ,d2ZdT 2)T=0= (0,−1, 0) . (38)The resulting trajectory was illustrated in Figure 5.Both trajectories shown in Figure 4 and Figure 5 aremirror symmetrical with respect to a X-Z plane atY=0.4.2. Trajectory in the oblique fieldIn case of an oblique field (θ 6= pi/2), differential equa-tions (29-31) were solved under the initial conditions(21-23) for various values of θ. The plasma gas de-scribed a helical line as its trajectory. The projectionof the helix in the plane normal to b was a circle withradius equal to sin θ. The center of the circle movedin the direction of −b (anti-parallel to b) at the con-stant velocity equal to cos θ. For 0 < θ < pi/2 , thetrajectories in the magnetic field with the intersectionangle θ and that with θ+ pi were illustrated in Figure6 and in Figure 7 respectively. These two trajectorieswere mirror symmetrical to each other with respectto a X-Z plane at Y=0.Figure 4. The movement of the plasma gas underthe perpendicular field (θ = pi/2) in the dimensionlessspace.Figure 5. The movement of the plasma gas under theperpendicular field (θ = 3pi/2) in the dimensionlessspaceThe trajectory in the magnetic field with θ + pi/2and that with θ + 3pi/2 were also calculated. It wasknown that the trajectories in the field with θ andwith θ+ pi/2 were mirror symmetrical with respect toa Y-Z plane at X=0. The trajectory in the field withθ + pi and that in the field with θ + 3pi/2 were also insimilar symmetry.5. Arc motion in real coordinatesThe trajectories of the plasma gas in a real spaceare obtained by the inverse transformation of thedimensionless coordinates (X,Y, Z) to the dimensionalones (x, y, z) using equations (17-19). Arc profile isrepresented by the trajectory of the plasma gas fromthe torch exit to the anode. Numerical calculationson the deformation of arc profiles with the change ofB were carried out under the operating conditionsQ = 3.0 × 10−4 kg s−1, I = 90A , and vo = 80m−1.It was assumed that the anode plane located at theposition of z = −L = −5.0× 10−2m.5.1. Arc motion in the perpendicular fieldIn the perpendicular magnetic field (θ = pi/2), the arcprofiles for various magnitude of B are obtained bythe inverse transformation along the circle shown inFigure 4. Deformations of the arc profiles with thechange of B are illustrated in Figure 8. For B = 0,the position of the arc root on the anode locates atA0 and it moves along the y-axis through A1 and A252vol. 5 no. 2/2018 Non-dimensional analysis of arc motionFigure 6. The movement of the plasma gas underthe oblique field (0 < θ < pi/2) in the dimensionlessspaceFigure 7. The movement of the plasma gas underthe oblique field (pi < θ < 3pi/2) in the dimensionlessspaceto A∗ with the increase of B. If the magnitude of B islarger than this critical strength B∗, the arc goes upwithout contact with the anode. In such situation, thearc becomes unstable as the circuit of the arc currentdestroys. The critical magnetic field is expressed asB∗ = voQIL(39)The maximum displacement of the anode root A∗equals L.In the perpendicular field for θ = 3pi/2, the arcprofiles for various magnitude of B are shown in Figure9. The whole view of the arc oscillation driven byalternating magnetic field is shown in Figure 10. Thearc oscillates in a flat y-z plane.5.2. Arc motion in the oblique fieldThe arc profiles in the oblique magnetic field withthe intersection angle of θ = pi/6 are shown in Figure11, where the helical motion in Figur 6 is replacedby the inverse transformations into different helicalmotions with various values of B. The plasma gastravels on the three dimensional curved plane. TheFigure 8. The movement of the plasma gas under theperpendicular field of θ = pi/2 in the real spaceFigure 9. The movement of the plasma gas under theperpendicular field of θ = 3pi/2 in the real spacearc root on the anode plate moves in the counter-clockwise direction with the increase of B. When theintersection angle of the field is θ = 7pi/6, the arcprofiles are mirror symmetric with respect of a x-zplane at y = 0. The whole view of the oscillatory arcin the oblique alternating magnetic field is illustratedin Figure 12.5.3. Experimental confirmationExperimental observation was performed under theoperating conditions Q = 3.0× 10−4 kg s−1, I = 90Aand vo = 80m−1. The sinusoidal alternating magneticfield was imposed at the frequency f = 50Hz, and itsamplitude was B = 2.0× 10−3 T. The oscillatory arcin the perpendicular field and that in the oblique fieldwere shown in Figure 13. Although many simplifiedassumptions were used in the modelling, experimen-tal results confirmed the validity of the theoreticalconsideration.6. ConclusionsThe authors have developed a heat treatment systemwith the magnetically driven arc. To improve the sys-tem, arc behaviours in various operating conditionswere investigated theoretically. In the simulation ofthe arc motion, non-dimensional analysis was pro-posed, where the momentum equation for the plasmagas was converted into the dimensionless equationusing the characteristic time and length. When these53K.Takeda, R.Akiho Plasma Physics and TechnologyFigure 10. Oscillatory motion of the arc under analternating field imposed perpendicularlyFigure 11. The movement of the plasma gas under theoblique field (θ = pi/6) in the real spacecharacteristic time and length were defined pertinently,the dimensionless momentum equation became inde-pendent of the operating conditions B, Q, I and vo.Therefore, the trajectory of the plasma gas motioncalculated in this dimensionless coordinate system wasalso independent of the operating parameters. Thearc motions in the real space were obtained by theinverse transformations of the trajectory in dimension-less coordinates system.Main results obtained from the numerical calcu-lations are as follows: In an alternating magneticfield applied perpendicularly, the arc oscillates on aflat plane. The amplitude of its oscillatory motion in-creases with the increase of the magnetic field strength.But the arc becomes unstable when B exceeds thecritical strength equal to Qvo/IL. The arc describesa part of a circle, the radius of which depends on B.While the magnetic field is applied obliquely to thearc, the arc oscillates on a curved plane. The arcdescribes a helical line, the curvature of which alsodepends on B. Theoretical predictions were confirmedby the experimental observations.References[1] J.E.Harry and D.Goodwin. Surface heattreatmentusing a plasma torch with a magnetically traversed arc.In 4th Int. Conf. on Advance in Welding Processes,pages 181–184, 1978.Figure 12. Oscillatory motion of the arc under analternating magnetic field imposed obliquelyFigure 13. Oscillatory motion of the arc in the per-pendicular field and that in the oblique field[2] K.Takeda. Oscillating plasma arc by external acmagnetic field. In O. Solonenko, editor, Proceedings Int.Workshop on Plasam Jet in the Development of NewMaterials Technology, pages 486–492, 1990.[3] Y.Maruki T.Yamamoto T.Toh, J.Tanaka andK.Takeda. Magnetohydrodynamic simulation of dc arcplasma under ac magnetic field. ISIJ International,45(7):947–953, 2005.[4] K.Takeda Y.Noguchi R.Akiho, M.Sugimoto andT.Miura. The development of a heat treatment systemusing a magnetically driven arc. Transactions of theJSME, 79(806):3979–3992, 2013.[5] K.Takeda H.Okubo and M.Sugimoto. Time responseof arc driven by alternating magnetic field. In 13thHigh-Tech Plasma Processes Conference, editor, Journalof Physics: conference Series 505, pages 1–10, 2014.[6] K.Takeda R.Akiho and M.Sugimoto. Arc motion in anobliquely imposed alternating magnetic field. In 12thHigh-Tech Plasma Processes Conference, editor, Journalof Physics: conference Series 406, pages 1–10, 2012.[7] S. Wolfram. The Mathmatica book. 3rd edition.Wolfram Research Inc. Champaign, 1998.54

Simulation of Arc Motion in Alternating Mahnetic Field Using Dimensionless Momentum Equation

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Simulation of Arc Motion in Alternating Mahnetic Field Using Dimensionless Momentum Equation

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