A model for eddy current separation

inmllnlmonNJmlllllltU mlnERnt PRO[E|SInG ELSEVIER Int. J+Miner. Process. 49 (1997) 193-200 A model for eddy current separation P.C. Rem *, P.A. Leest, A.J. van den Akker T.U. Delft, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands Received 29 April 1996; accepted 5 December 1996 Abstract We present a model for the motion of particles in an eddy current separation process. The model describes the action of the transient magnetic field of the separator by a first-order linear differential equation for the particle magnetic moment. From the model, qualitative conclusions can be drawn regarding the effects of particle size, shape and conductivity on the particle trajectory. It is also possible to solve the equations of motion for particles of simple shapes by numerical integration. The results of such simulations are compared with experimental data. The aim of the model is to improve the understanding of the behaviour of small particles in eddy current processes in order to optimize separator design and operation. Keywords: eddy current; separation; modelling; recycling 1. Intr,~luction Eddy current separation is an effective way of removing non-ferrous metals from stream,,; of industrial or municipal waste (Dalmijn, 1990). The process is used to separate aluminium and copper from car scrap and to remove metals from recycle glass. The separation technique is based on the fact that conductive materials resist being moved in a gradiLent magnetic field and, vice versa, will accelerate in a moving gradient field (Edison, 1889). In the most widely used type of process the particles are transported over a fast spinning drum, the surface of which consists of layers of magnets of alternating polarity. As the drum is rotated, the particles present in the field will tend to follow the motion of the drum (see Fig. 1). Poorly conducting materials, on the other hand, will stay on the belt and drop close to the drum. The trajectory of particles is generally determined by a combination of gravity, the frictional forces of the tranportation belt and the air, and the electromotive force. * CoFesponding author. Fax: + 31 15 278-2836; E-mail: [email protected] 0301-7516/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0301-75 16(96)00045-2 194 P. C. Rem et al. / lnt. J. Miner. Process. 49 (1997) 193-200 ~ - rnetals ~ Side View + plastics, glass, etc. drum i particle I belt ~ Top View sNNsN" It Fig. 1. An eddy current separator consists of a drum covered with magnets that are oriented alternately N-S and S-N. The fluctuating field of the spinning drum induces eddy currents in electrically conductiveparticles moving close to the drum. The particles are then expelled along the direction of rotation. The pictures show an eddy current separator as viewed from aside (top) and from above (bottom). The force that separates the conductive particles from the remaining material is the Lorentz force: the force of the magnetic field on eddy currents inside the particle. These eddy currents, in turn, are induced by fluctuations of the same magnetic field as these are picked up by the moving particle (see Fig. 2). The complex interaction between the field and the particle have troubled efforts to model the particle trajectory, although results have been obtained for simplified situations (van der Valk, 1988). Recently, a comprehensive approach has been started (Leest et al., 1995), stimulated by the fact that nowadays increasing attention is being paid to the separation of small particles, for which analysis seems more tractable than for large particles. In order to derive the model, we will first consider a particle with a given trajectory and rotation in a time-dependent field. The calculations will be limited to lowest order in eddy current ) Lorentz force particle Fig. 2. Particles carrying induced currents are expelled from the magnetic field of eddy current separators by the Lorentz force. The vectors have been drawn, assuming that the magnets below the particle are moving from left to right. P.C. Rem et al, / Int. J. Miner. Process. 49 (1997) 193-200 195 the size of the particle, i.e., we assume that the particle is sufficiently small with respect to the scale of the fluctuations of the external magnetic field that this field can be regarded as essentially constant within the volume of the particle. Next, we will compute the Lorentz force of the field on the induced currents to derive the equations of motion for the particle. Again, these calculations will be to lowest order in the particle size, in this case involving only linear variations of the field. Finally, we close the model by an approxiraate differential equation for the particle magnetic moment. The remaining part of the paper deals with the experimental validation of the model. It describes the experiments and compares results of numerical simulations of the model with particle trajectories obtained by image analysis. 2. Theory As a particle moves through a vector field, it experiences a change of size and orientation of the field due to its translational and rotational motion as well as through the inherent time-dependence of the field. Let Ba(r,t) be the magnetic field that is generated by the rotating magnet configuration of the eddy current separator. If we neglect spatial variations of B a on the scale of the particle dimensions, we can define a magnetic field vector ~'a(t) as the applied field experienced by the particle in its own coordinate frame: a frame that is rigidly connected to the particle ~. Let U(t) be the coordinate transformation matrix between the particle frame and the lab frame (Fig. 3), the columns of which are the unit vectors along the three axes of the particle coordinate system as expressed in the lab frame. Then ,_~a(t) is related to Ba(r,t) by: U(t) "~a(t) = Ba(r(t),t) (1) and the variation of B a and U with time is given by: d 0 U . - - ~ , a = - - B a + ( u • V ) B a - ~'] X B a, dt 0t d --U=/~×U dt d Here lilt) is the position of the particle and u = - - r and ff] are its translational and dt rotational velocities. For any given particle trajectory, the equations above can be used to compute the applied field ~ a ( t ) . The distribution of eddy currents inside the particle, )~(~,t), then follows from solving Maxwell's equations. In general, analytical results can be obtained We will keep to script letters for vector quantities relative to the coordinate frame in which the particle is translationally and rotationally at rest and use the corresponding roman bold type for the same quantities relative to the laboratory frame. 196 P. C Rein et al. lint. J. Miner. Process. 49 (1997) 193-200 e~ e~ J Fig. 3. Relation between Lab frame and particle frame. The coordinate transformation matrix is U = (e¢,e n,e¢ ). only for simple particle shapes and linear constitutive equations. Therefore, we will assume that the current density is linear with the electric field: y=o-~ For a given eddy current distribution, the Lorentz force is integrated over the volume, V, of the particle to find the total force and torque on the particle: F = u- f v J x ~ ' d ~, T = U.fv~× (J×~')d~: For a sufficiently small particle, variations of the applied field within the particle are small and both the force and the torque can be expressed in terms of the magnetic moment of the particle: M= ~ffxJd, (3) In that case, V = M' T=M×B (4) ~7B a a (5) Hence, the problem of computing the forces on a particle in a time-dependent field reduces to computing the particle magnetic moment ~ ' ( t ) and relating its behaviour in time to that of the applied field ~'~(t) experienced by the particle. 2.1. Evolution equation for the magnetic moment The magnetic moment of particles of simple shapes can be computed by classical techniques if the applied field is an harmonic function of time, i.e., for .-.~a(t) -=~aei°Jt. For example, the magnetic moment of cylindrical particles of radius R and length L >> R with their axis parallel to the z-axis in an harmonic field of frequency co is given by: ..[l'(t)=.~t~oeit; ./~¢'w= ./]/'o, ' .-~ 'a (6) P.C. Rein et al. / Int. J. Miner. Process. 49 (1997) 193-200 197 with 2 R2 2,i32 0 Jr'°' tx° Jo(i3/:~tZo °J°'R2 ] 0 0/ 0 1 (7) where J0 and J2 are Bessel functions of order 0 and 2, respectively. Now, the magnetic moment for fields L~a(t) that are arbitrary functions of time can be found by transforming relation (6) from the frequency domain into an ordinary differential equation in time. For example, for the case of a cylindrical particle this transforraation results in the following first-order approximation: d 8 rrRZL d --~= ( 2 ~ _ +~'1~) (8) dt /z0 o-R :I¢" /x0 dt ~1~ and ~ _ refer to the field components parallel and perpendicular to the axis of the cylinder, respectively. The simple result (8) matches relation (7) both for the low- and high-frequency limit. Higher-order approximations can be found in principle, but this is outside the scope of the present work. 3. Comparison with experimental results Experiments were carried out in order to validate the first order model for the particle magnetic: moment. Two industrial eddy current separators of the rotating drum design of Bakker Magnetics (Bakker) were used and particles of various shapes, sizes and compositions were subjected to the field. The particles were dropped along a vertical passing at a few millimeters from the spinning drum (see Fig. 4) and the detailed trajectories of the particles were recorded. The option to drop the particles, rather than conveying the particles over the belt, was chosen to minimize the introduction of other forces which could complicate model validation. As is shown in Fig. 4, the action of the field is to push the particles away from the drum in the X - Y plane. The positions that were captured by the video were reduced to a set of (X,Y) points using image analysis software., (DIFA). In order to get a comparison between experiment and theory, the equations of motion were integrated with the help of the DAE-integrator DASSL (Petzold, 1982; Brenan et al., 1989) for the conditions of the experiments. Table 1 defines the particle and Fig. 4. In the experiment,particles are dropped close to the rotating magnetic drum and the deviationof the trajectoryfrom the verticalis measured. 198 P.C. Rem et al. lint. J. Miner. Process. 49 (1997) 193-200 Table 1 Field, particle and operating parameters used in the experiments Parameter Experiment l Experiment 2 Rdr,m (ram) O)drum (rad/s) (Xo,Y o) (mm) ~r ( 1 / O h m m) Shapes R, L or ~ (mm) 145 - 1001r (145 + R, 250) 23 X 106 (aluminum) cyl. 2.5, 15 cyl. 5, 30 149 - 100"rr (149 + R, 250) 57 x 106 (copper) disk 5, 2 sphere 5 operating parameters used in the experiments and simulations. Below, we define the approximation used for the calculation of the magnetic field of the drum and compare the results of the simulations with data from the experiments. The field of the magnet drums was both measured with a Gauss meter (Group 3) and computed with the ANSYS simulation software (ANSYS). The results were found to differ by about 7% for the machine of the first set of experiments and by about 20% for the machine used in the second set of experiments. Because of the large differences between the two sets of results, the field measurements were adopted as the basis for the 0.3 0.2 0.I 0 y [m] -0.i -0.2 -0.3 -0.4 -0.5 0.1 0.3 0.12 0.14 I I 9 0.2 0.16 0.18 [ I 0.2 0.22 x [m] I I ' 0.24 0.26 0.28 I I I 0.3 experiment 0 theory - - 0.1 0 y [m] -0.1 -0.2 -0.3 -0.4 -0.5 0.1 I i I I 0.12 0.14 0.16 0.18 I I 0.2 0.22 x [m] I % 0.24 I I 0.26 0.28 0.3 Fig. 5. Comparison of experimental results and simulations for aluminium cylinders of sizes R × L = 2.5 X 15 mm (top) and R × L = 5 × 30 m m (bottom). P.C. Rein et al. / lnt. J. Miner. Process. 49 (1997) 193-200 199 remaining calculations. The measured field profiles were fit to a series expansion in cylindrical coordinates (r, t~, z) relative to the drum axis: 2 1)k(th - Br = E bn( r/gdrum)-(2n+l)k-lsin( 2n + tOdrumt), n=0 2 BO, = E - bn( r/Rdrum)-(2n+l)k-lcos( 2rt + 1)k(~b- tOdrumt), n=O B z = 0. This field satisfies Maxwell's equations V. B = 0 and V x B = 0 for all values of the coefficients. Actually, the approximation corresponds to an infinitely long drum with k sets of symmetric N-S magnets, rotating with angular velocity tOdrum. In the first set of experiments, cylindrical particles of various sizes and compositions were te,;ted. Fig. 5 shows some of the results as compared with simulations. In the second set of experiments, particles of roughly the same size but various shapes were tested. Fig. 6 shows the results of these experiments compared with simulations. 0.3 I l I I l I I experiment theory 0.2 <> -- 0.1 0 y [m] -0,1 -0.2 -0.3 -0.4 -0.5 0.1 I I 0.12 0.3 0.14 0.16 0.18 % I I 0.2 0.1 I O 0.2 0.22 0.24 0.26 0.28 I I I } I x [m] 0.3 experiment theory n y [m] -0.1 -0.2 -0.3 -0.4 -0.5 0.1 I I 0.12 0.14 .- 0.16 0.18 0.2 0.22 x [~1 0.24 . I I 0.26 0.28 Fig. 6. Comparison of experimental results and simulations for copper disks of size and spheres of size R = 5 mm (bottom). 0.3 R x • = 5X2 nun (top) 200 P.C. Rem et a l . / lnt. J. Miner. Process. 49 (1997) 193-200 4. Discussion The comparison of results from simulations and experiments shows a fair correspondence between the data for aluminium cylinders of the two sizes used. Stray points in the graph are due to particles accidentally hitting and jumping off the rubber transportation belt. The data on copper disks and spheres show as a trend that the simulation trajectories fall short of the measured trajectories. For the spheres the value o f / % oJo-R2 in Eq. (7) is about 4. Detailed analysis shows that for this value the tangential force is modelled accurately but the radial force is underestimated by the first-order approximation used in the simulations by about 25%. This means that one would expect the simulated trajectories to fall short by about 25% too. This is in fact very close to what is seen in the data of Fig. 6. Data on small aluminium disks, not presented here, show a similar deviation for the same reason. Considering all the experimental data, we conclude that the model captures the essential part of the action of transient magnetic fields, such as those generated by eddy current separators, on conducting particles. It appears that the largest modelling error is introduced in the approximation of the transfer function between the field and the magnetization of the particle. This suggests that better results may be obtained with a second-order differential equation for the magnetic moment. However, the model as presented allows a great deal of insight to be obtained, precisely because of its relatively simple and elegant nature. This may prove its strength in the design of specialised eddy current separators. Acknowledgements The authors are grateful to Mr. E. Schokker for computing the field of the drum magnets with the ANSYS simulation software (ANSYS). References ANSYS. ANSYS simulation software. ANSYS Inc., Houston, PA, USA. Bakker. The two eddy current separators used for the two experiments are models BM 29.701/12 and BM 29.701/18, respectively. Bakker Magnetics, Son, The Netherlands. Dalmijn, W.L., 1990. Practical applications of eddy currents in the scrap recycling. In: Proc. Second Int. Symposium Recycling of Metals and Engineering materials, the Minerals, Metals and Materials Society (1990). DIFA. Graphical software TimWin. Difa Measuring Systems B.V., Breda, The Netherlands. Edison, T.A., 1889. U.S. Patent 400,317 (1889) Group 3. Digital Teslameter. Group 3 Technology Ltd, Avondale, Auckland, New Zealand. Leest, P.A., Rem, P.C. and Dalmijn, W.L., 1995. Analytical approach for custom designing of eddy current separators. In: Proc. XLVI. Berg- und Hiittenm~nnischer Tag (1995) Freiberg. Petzold, L.R., 1982. A Description of DASSL: Differential Algebraic System Solver. SAND82-8637, Sandia National Laboratories, Sept. (1982). Brenan, K.E., Campbell, S.L. and Petzold, L.R., 1989. Numerical solution of initial value problems. In: Differential-Algebraic Equations. Elsevier, New York. Valk, H.J.L. van der, Dalmijn W.L. and Duyvesteyn, W.P.C., 1988. Erzmetall, 41(1988): 266.