JIRI GALAS Czech Technical University, Engineering, Prague.

Transcription

1 Magnetic Separatin News, Vl. 2, pp Reprints available directly frm the publisher Phtcpying permitted by license nly 1988 Grdn and Breach, Science Publishers, Inc. Printed in the United Kingdm MAGNETIC FIELD EFFECT IN THE PROCESS OF RINSING A MAGNETIC SEPARATOR MATRIX JIRI GALAS Czech Technical University, Engineering, Prague. Faculty f Mechanical Abstract This paper surveys fundamental aspects f the prblem f rinsing matrices in high gradient magnetic separatrs. This is dne, fr the first time, in terms f the magnetic circuit design. Equatins have been cnstructed t describe the effects f spurius remanent magnetic fields n the rinsing prcess. I. INTRODUCTION Streams f water are used t rinse the matrix in magnetic separatrs. Water velcity, particularly in larger matrices, is usually limited and a suitable mixture f water and cmpressed air is nt always available. This raises prblems f specifying cnditins fr sufficient and reliable rinsing f a matrix. Als, the demands n (i) the mechanism cntrlling matrix mtin as well as n (ii) screening cver, r n (iii) the whle cncept f the magnetic circuit f a separatr, shuld be taken int cnsideratin. The paper cncentrates n specifying the magnetic cnditins. A single-wire mdel has been emplyed fr an exact calculatin and parameter values have been used which are apprpriate t the highgradient magnetic separatin f (say) Kalin. A discrete mdel has been used fr the descriptin f separated particles. 119

2 120 J. GALAS 2. MAGNETIC FIELD AND MATRIX RINSE Matrix rinse means the remval f all separated, (mstly magnetic) particles frm the surface f ferrmagnetic matrix material which is frmed by a randm assembly f wires f apprximately 0.05 mm in diameter, ccupying abut 5% f the canister vlume. Rinsing is carried ut by the hydraulic frce f water pumped int the matrix at a velcity exceeding that f the nrmal separatin prcess. During the rinsing prcess, sme separated particles are held n t the matrix wires, despite the influence f inertial, gravitatinal r inter-particle frces, by magnetic frces induced by spurius magnetic fields. These magnetic frces are f a similar character t the magnetic frces in nrmal magnetic separatin. The spurius magnetic field in rinsing cnsists f a reduced (r leakage) magnetic field f the separatr magnetic circuit and the residual magnetic field f the wire, ferrmagnetic structural parts f the separatr. and pssibly als by A residual magnetic field due t the ferrmagnetic prtin f the separated particles als exists in thery but can be neglected in practice. Frm the pint f view f the presence f the spurius magnetic field, a rinse can take place under the fllwing cnditins: a) The field current f the magnetic circuit cil is full r partly reduced; the matrix is fixed. b) The field current is zer but residual magnetic fields are present; the matrix is fixed. c) The field current is full, magnetic leakage and residual fields are present; the matrix is f a withdrawal type, (i.e. it can be drawn utside the magnetic circuit wrk space). d) The spurius magnetic field is suppressed, the field current is full; the matrix is f a withdrawal type and in the

3 RINSING A MAGNETIC MATRIX 121 rinsing area, it is surrunded by auxiliary screening cvers cntaining a de-magnetizing field prduced either by cils r permanent magnets. When evaluating the rinse quality in the abve-mentined examples, it can be said that, in case (a), (paramagnetic) particles f small susceptibility will be rinsed effectively while particles f much higher susceptibility (the ferrmagnetic prtin) will gradually after several separatin cycles fill the "active" zne n the wires until a cmplete lss f separatin capability f the matrix is reached. This slutin may be applicable when using high rinsing water velcities tgether with cmpressed air. In case (b), mainly attractive frces f a nn-magnetic character will act, thugh it wuld be wise t cnsider the presence f magnetic frces caused by residual magnetic fields that may bring abut a gradual filling f the zne. In case (c), the rinse is f the same character as in case (a), as the magnetic leakage, accrding t the design, can have magnetic inductin field values cmparable with thse present under nrmal perating cnditins. Rinsing, accrding t (d) ccurs with minimum disturbance by magnetic frces, but is mre cmplicated in design. The requirements n the chice f magnetic system fr separatr, can be specified nly by quantitative design. This design shuld include the spurius magnetic field effect rinsing prcess. Precise determinatin f the value f the the in the admissible magnetic inductin f the spurius field, fr a given rinsing velcity, will prvide an imprtant parameter fr designing the separatr magnetic circuit. The separatin system, including the feed slurry, the rinsing fluid and the randm assembly f fibers in the matrix can be described in terms f varius vectr quantities:

4 122 J. GALAS (i) the wire axes can be represented by d a unit vectr whse directin is given by wire axis and whse sense can be chsen arbitrarily; (ii) a feed velcity, v s (iii) a rinsing fluid velcity,, v; (iv) the magnetic inductin field, f the separatr; (v) the magnetic inductin f the spurius magnetic field, in the rinsing zne; (vi) the gravitatinal vectr, g; and [vii) the particle separatin zne, represented by z a vectr whse directin is determined by a straight line drawn perpendicular t the wire axis and lying in the symmetry plane f the zne; the sense f this vectr pints nrmal t the wire surface and its size is prprtinal t the distance frm the tp f the separatin zne t the wire surface., Depending n the directins f the feed stream velcity s and n the directin f znes f varius directins are z frmed n the wires. Hwever, frm the pint f view f the rinsing prcess, their prjectins, z n t the directin f the spurius field, will be decisive. The directin f the spurius field is cnsidered t be cngruent with the directin, f the separatr field, which will be decisive fr carrying ut the rinse. Different directins r senses, and, may have a limited, favrable effect upn the rinsing prcess and can ccur in certain parts f the matrix. A randm wire arrangement enables arbitrary angles between a plane determined by axis d and zne but the z velcity cmpnent f the rinsing fluid velcity in the perpendicular directin t the wire within this plane is decisive fr the rinse. Even in the case f parallelism f v and the plane determined by d and a certain transient z cmpnent f the rinsing fluid velcity,

5 RINSING A MAGNETIC MATRIX 123 v, perpendicular t this plane may be cnsidered (caused by, a bubble f, fr example, cmpressed air, frced int the matrix during the rinse, passing the wire). A particular matrix design usually cntains spaces with varius senses f gravitatin f vectr g and rinsing v. T carry ut the rinse, the parallelism and the ppsite sense f g and v must be ensured (see Fig. I.) The cnditin f rinse feasibility is [ xv] 5 defined by the nn-zer triple scalar prduct: z d if the zne vectr z is f the directin the separatin vectr is f the same directin and sense as and the gravitatinal vectr g is f the same directin but ppsite sense t v. The basic rinsing system is defined by the mutually perpendicular vectrs v and when their triple scalar z prduct d reaches its. maximum value and the mst effective rinse can be expected; i.e. the cmpnent v which tgether with and z d frms the basic system, is decisive fr quantitative evaluatin. When designing the matrix frm the pint f view f the mst effective rinse is t be expected in the s-called the rinse radial arrangement where the rinsing water is frced in the radial directin f the cylindrical matrix, while the separatin is in the directin f the axis f the matrix cylinder. 3. DISCRETE DESCRIPTION OF THE ZONE In the fllwing, the term zne indicates the space arund the wire where entrapped particles ccur. By a discrete descriptin f the frce relatins, we mean that the particles r particle clusters within the zne are cnsidered separately and withut interactin. This assumptin is justified in the case where nly magnetic frces are present.

6 124 J. GALAS 3.1 Frce relatins Frces prduced by a magnetic field gradient arund a separatin wire have been derived previusly fr the case when the wire has been magnetized up t saturatin and magnetic field mntnically drps while the field current decreases r the matrix mves ut. In this case the relatins fr magnetic frces in cylindrical crdinates can be used I. The frce cmpnent in the radial directin is: Fr -2u V x A rt r 2 I (A + H s 2) (l) and in the azimuthal directin is F =-2.V.X.A (s H sin 2) (2) r A I.a s (3) 2 g where I (T) is the saturated magnetic plarizatin f the wire, s a, the wire radius, and where the ther symbls have their usual meaning. Accrding t Stall, the relatin between the utside field intensity H and the intensity H 2 inside a cylindrical wire f relative permeability, is H 2 H ( If >>I, eqn. (4) can be expressed as H H 2 H (5) Thus fr the magnetic inductin inside the wire, (B<I.4T) B % 2B (6) Magnetic plarizatin can be thus expressed (where we linearize the hysteresis curve in the bserved interval) as: j Br + 2(I 1 BI B B r () Br + KjB

7 RINSING A MAGNETIC MATRIX 125 II(T) is the magnetic plarizatin crrespnding t BI; Be(T) is the magnetic inductin f the field in which the wire has been inserted, and K. is a cnstant determined by I i characteristic wire magnetizatin curve. B I, Br frm the Kquatin (7) is applied t express the magnetic plarizatin in eqn. (3) as (B +K.B) a A= r 3 (8) 2U O We are cnscius f a certain inaccuracy here, as the magnetic plarizatin was suppsed t be cnstant, but, n the ther hand we relate the dependence f that quantity, as given by Sestak and Rieger t the drp f the external magnetic field. The particles are f hetergeneus, irregular shape and due t their mutual chesin they frm a certain vlume. The suitability f vrius substitute bdies fr particle clusters, has been cnsidered and cylinders f different radii b t b with differing distances f their axes, t the pint in questin r O have been chsen (see Fig. i). rlv, It is pssible t express the frce relatins fr cylindrical particle vlumes V at different distances r cylinder radii b, thus: I IV and f different V b2l (9) where b(m) is the radius f the particle cluster f cylindrical shapes and L(m) is the cylinder length. The resistance frce, F f the envirnment is determined by r the velcity vectr f rinsing medium v which is perpendicular t vectr and t the wire axis (see Fig. i) O F p v L b C (i0) V where Fv(N) is the frce f envirnment resistance acting n the cylinder, p (kgm is the envirnment density, v(ms 1) is the velcity f envirnment mtin, L(m) is the cylinder length,

8 126 J. GALAS FIGURE 1 general vectr directins

9 RINSING A MAGNETIC MATRIX 127 b(m) is the cylinder radius, and C is a cefficient f envirnment resistance. Fr the range f assumed velcities (v 0.08 _i t 0.3 ms and fr wire dimensins (including the zne f apprx ) C The gravitatinal frce acting n the cylinder f particles is mm), then, accrding t Sestak and Rieger entrapped F V( 0- P )g, (ii) g i where Pi(kgm-) is the particle density in the zne and where g =9.81 ms Critical velcity cnditin Overcming magnetic frces, which als hld the particles within the zne, may cause that particle t be released and swept away by rinsin water. As has been already indicated, we are cncerned with a simplified situatin where the particle cluster f cylindrical shape is acted upn nly by magnetic and gravitatinal frces as well as by the resistance frces f the envirnment. Their time variatins cnsidered are slw s that inertial frces d nt act in this quasistatinary state. Until the particles are released frm the zne, the zne will maintain a steady vlume and it will be subject t defrmatin by the envirnment resistance frces. A particle within the zne is assumed t travel alng a circular path with regard t the cylindrical wire and the directin f frce actin. If the particle buildup cylinder is screened behind the wire, the reductin f the area is expressed by a cefficient whse values range frm 1 t O. If the particle cluster in a certain area f the zne is t be released, then it is necessary that

10 128 J. AI.S F F > F (12) v g mv where F mv is the cmpnent f magnetic frces in the velcity directin. Fmv -F r sin cs (13) F#. The frce Fmv and cefficient are functins f Thus it fllws that..( Fv -... Fg > 3Fmv 14 The critical velcity, at which magnetic and gravitatinal frces are vercme, is determined frm equality in expressin (12) abve and by fulfilling the cnditin (14). It fllws that the cefficient is 5 (rc a+b (is) s*2 in the range << Fr <_ i and fr _> O. In the interval where l >>0, fter substituting relatins (1) and (2) int (13) we get: 2 a [Br r sin + B sin 3 () Frm eqns. (i), (2) and (16), the partial derivatives in (14) becme: 8(Fv.Fg). -Fv r sin % and 3 2b = a Br+ KjB B r + KjB.b..L.X 3 r 2 B 3Fmv. 2 cs + 3B cs 3# (18)

11 RINSING A MAGNETIC MATRIX 129 If the lcal extreme-maximum f functin (16), t which k crrespnds, is determined by means f eqn. (18), and if the k speed value is calculated frm the equality f relatin (12) fr k the velcity v will be the critical k velcity as lng as < The derivative value in the bserved interval is cnstantly negative. This situatin is shwn in Fig. 2, pint i. ( Fv- Fg) 3 FIGURE 2 critical retentin cnditins If the frce plts intersect in pint 2, it can be assumed that the excess f frces between the pints 1 and 2 (hatched area) will shift the particle t larger radius, F will decrease and mv the particles will n lnger be kept in the zne. Plts 3 and 4

12 130 J. GALAS are dt-and-dash marked where the cnditins f release are nt fulfilled. (Fig. 2) Critical velcity is calculated frm the equality accrding t (12) after substitutin frm (I0) and (ii) Vk..b X a Br+3 Kj_ ) ) sin + B sin 3 0r C 2 + b (i-)g (19) x a + b r fr Ok when k<. Fr the value after re-arranging equatin (15) gives cs, 4. MAGNETIC PARAMETERS AND CRITICAL RINSE VELOCITY (20) Amng the magnetic parameters we include here: (i) the magnetic inductin field (B) in which the matrix is situated. (ii) the susceptibility f the separated prtin in the zne (X) and the particle size (b). (iii) the relative permeability (), remanence (B) and wire r diameter (2a).The critical rinsing velcity, v k, is the critical velcity f rinsing water in the directin perpendicular t wire axis and directin B at which the particles, held near t the wire by magnetic frces, start t be released. Values quted in table 1 were substituted int equatin (19). First f all, values, #k have been determined by means f equatin (18), they were then cmpared with # accrding t equatin (20) and finally substituted int equatin (19) fr I. The results are shwn in Fig. 3,4 and 5, in which plts start in the pints where reaches the value i. Thugh the exact descriptin f the rinse (fr <i) cmes t an end in the directin f decreasing the

13 RINSING A MAGNETIC MATRIX 131 a= 5 10"5m B 0,4 T X i0 " X i0 "5 X=10 "6 -.I IIIII I. b= "Sm r 1.7a II. b= m r 1.9a III. b= "Sm r 1.3a IV. b= "5m r l.sa FIGURE 3 particle rinsing cnditins (a=50

14 132. l I 11 III,,.,4 3 B IT]---- FIGbqE 4 particle rinsing cnditins (a=25 m)

15 RINSING A MAGNETIC MATRIX 133 a=510"sm B r 0.ST X I0-4 X= 10-5 X= I0 "6. ii. 1. b= 5 10 "5 m r= 1.Ta b= 0.5 lo-5m r= l.ga _1 t Iii. b= 0.5 i0"5 r= 1.3a FIGURE 5 particle rinsing at higher B r (0.8T)

16 134 J. G B r =0, 4 T, Kj I. 046 variann Iml b Iml c Imi C /-/ V k /ms "1 FiE 5-5 I0 1, ,7 0,1 0,1 0,028 0,024 I- 2,5-5 I0 0,75-5 l0 l,ta 2,5 4 i 4.!... 0,1 0,i 0,i 0,1 0,029 0,021 0,020 0,015 II 3,7 0,i 0,013 II II III.: 2, ,25 0,5 5,25 l,ga 1,3a 3,7 0,i 0,I 1 0,011 0,005 _..0, ,085 0,030 0,090 II II II_. IIl III III III IV. 17) 5 l 5 0, ,5a 2,5 3,7 2, ,i 0,i 0,i 0,I 0,I 0,1 0,029 0,020 0,014 0,013 0,016 0,006 0,004 III III IV IV IV IV IV IV TABLE I

17 RINSING A MAGNETIC MATRIX 135 B, it may be assumed that, even in the range f small values f B, the rinse will take place if the velcity is higher than the velcity f the extreme left-hand pint f the diagram. Critical velcities are bserved fr wire diameters 50i00 B m, with particle susceptibility in the range frm t and i0- i0- particle size in the range 2.5 t 15 m, which is in accrdance with Gauss distributin f the size ccurrence f Kalin particles. This als crrespnds t the Czechslvak Standard Specificatin Plts in Fig. 3 and 4 marked II, III and IV, shw the situatin fr rinsing medium particles. Plts in Fig. 5 are similar but fr the value B r 0.8 T (whereas B r 0.4 T in plts shwn in Fig. 3). The influence f the susceptibility f entrapped particles is apparent frm the difference in plts drawn in a slid line ( i0 ), dash line (H 10-5 ) and dt-and-dash line ( i0 ) The influence f wire diameter 2a is evident frm cmparing the plts in Fig. 3 and CONCLUSION The paper gives a survey f the ptimum arrangement f a rinsed matrix frm the pint f view f the magnetic circuit design. Under certain simplified cnditins, frce relatins in the entrapped particle zne and the effects f spurius magnetic fields have been described. Quantitative descriptin f rinsing prcesses has been slved fr individual particles f larger dimensins variant I, smaller dimensins variants II, III, IV. The values f critical rinsing velcities, v k, f water in the matrix were btained by means f determining the cnditins fr vercming magnetic frces which are nly a part f the ttal frce hlding the entrapped particles n the wire within

18 136 J. GLA the zne. It is necessary t chse the minimum rinsing water velcity in the matrix with regard t critical velcity v k fr a certain value f magnetic inductin B f the spurius magnetic field in which the matrix during rinse is situated. Fr example, fr a magnetic inductin field B =0.05 T, fr -4 a magnetic vlume susceptibility X i0 and by cmparing the variants I IV, the minimum rinsing water velcity/perpendicular -i t B and wire axis/shuld be higher than v 0.05 ms k REFERENCES i. J.H.P. Watsn, Magnetic Filtratin J. App. Phys., Vl. 44, N. 9, P. 209, (September 1973). 2. M. Stafl, Elektrdyn..a..!ka...ve Stavbe ElektrickYch Strj.u NCAV, Praha,1962/El.ecrP.dynamics in. Cnstructin Electri c Machines. 3. J. Sestak, F. Rieger, Prensve jevy I, skriptum CVUT, Transfer Phenmena I, mimegraphed text f the Czech Technical University.