ABOUT THE EQUATIONS OF MOTION OF A MAGNETIC PARTICLE IN A MAGNETIC SEPARATOR

Equations of motion of magnetic particles in the flow medium near a cylindrical ferromagnetic collector have been developed. A vector equation, expressing the balance of the inertia force and the vector sum of three forces: the magnetic, the gravitational and the hydrodynamic drag force of a particle was taken as the initial equation. Reduced equations obtained from the initial one by rejection of terms corresponding to the inertia or the gravitational force were also considered. Examples of numerical evaluation in which the motion trajectories calculated with the use ofthe initial equation were compared with the reduced equations of motion are given.


INTRODUCTION
In the number of papers [1][2][3][4] dealing with the theory of magnetic separation, in addition to the initial differential equation ofmovement of a magnetic particle also simplified equations, obtained from the initial one by rejecting the terms conforming the inertia force were considered.
It is assumed that such a replacement is admissible for small sizes of particles or with a specific combination of sizes ofparticles, their density and viscosity of the medium flow, for example, with the.use the Parker's criterion [2].It is usually considered that if with some conditions the influence of the inertia force on the movement of a particle is negligibly small, then for these conditions it is possible to neglect the gravitational force.It is of interest to carry out numerical evaluation for two specific cases.

THEORETICAL MODEL
Let us write the equation of motion of a magnetic particle in the flow medium near the cylindrical collector: dVp mp -Fm + Fd + Fg, (1)   where mp is the mass of a particle; Vp is the velocity vector of a particle and Fm, Fd, Fg are the vectors of magnetic, hydrodynamic and grav- itational forces acting on a particle.The gravitational force is an algebraic sum of the weight and Archi- medes forces.The magnetic force is caused by joint influence of the external magnetic field H0 and the magnetization of the cylinder that is placed in this field.
Equation (1) can be expressed as its projection onto the coordinate directions r, 0 of the polar coordinate system, the center of which coincides with the center of cylinder cross section: mpWr fm,r + fd,r + fg,r; (2 where wr, Wo are the projections of particle acceleration vector onto coordinate directions r, 0.
The second indices r or 0 of the quantities Fm,r,..., Fg,O denote the components of corresponding vectors with directions r, 0.
Sometimes reduced equations of motion [1,2] are used in which the inertial force is neglected.Such equations can be obtained by sub- stituting the expressions for the components of the forces into the fol- lowing system of equations: Fm,r 4-fd,r + Fg,r 0; (4) Fm,o + Fd,O -+-Fg,O 0.
(5) Usually, although not always, for the separation conditions that allow the use of Eqs. ( 4) and (5), it is possible to accept one additional simplification: to neglect the gravitational force in comparison with magnetic and hydrodynamic ones.Then the movement of a particle is described approximately by equations equivalent to the next system of equations Fm,r + Fd,r 0;  Calculations show that the criterion: 2pprZp << 1, that was proposed in [2], does not always guarantee the validity of the replacement of Eqs. ( 2) and (3) by Eqs.(4) and (5) or by Eqs. ( 6) and (7).A more reliable admissibility criterion of such a substitution is the condition that the inertial force Fin is much smaller than the magnetic force Fm, hydrodynamic drag Fd force or than the sum of one of them with the gravitational force Fg: Zm << Zm=l/rn_+_/g 1, Zd--i/d+/g I, Fin--mp   Let us express the components of the forces on the right-hand side of Eqs.
(2) and (3).The magnetic force, acting on a particle, is determined by the formula [3,5]: where/p and f are the volume magnetic susceptibilities of a particle and the medium, respectively, Vp is the volume of a particle and H H(r, O) is the magnetic field strength in the vicinity of a ferromag- netic collector.
We shall determine the function H H(r, O) assuming a single cylinder with its axis perpendicular to the vector of the external magnetic field using formulas [3,6].Then from Eq. ( 10) we obtain following expres- sions for the Fm,r, Fm,o components of magnetic force vector Fm" where where rw is the radius of a cylindrical collector; r/rw; pp is the density of the particle material and M= 2AHo is the magnetization of the collector; #w and #f are the magnetic permeabilities of the cylinder and the medium (#f #0), respectively.The vector of the hydrodynamic drag force can be expressed as: ffd --1/2 Cd,ofl lp,fl lp,fSm, where Cd is the coefficient of the hydrodynamic drag gp,f-gp Vp, Vf are the vectors of velocity of a particle and medium flow; pf is the medium density and Sm is the cross sectional area of a particle.The hydrodynamic drag coefficient Ca depends on the shape of a particle and on the Reynolds number where r/is the dynamic viscosity of the carrying medium and dp is the diameter of a particle.
For a spherical particle a number of different formulas, approx- imating the experimental dependence Cd(Re) are known.A formula for of a particle is spherical, then Sm 7rd2p/4.
In this case we obtain from Eq. (15) the following expression of projections of the hydrodynamic drag vector onto the coordinate directions r, 0: Fd,r---mpgd(vp,r-Vf,r); The Fg, and Fg,o projections of the Fg vector of the gravitational force are determined by formulas: Fg,r Qg cos(O Fg,o -Qg sin(0 3), where Qgmpg(1 (pf/pp)); g is the acceleration of gravity and/3 is the angle between x axis and the vector g.
The x axis is parallel to the H0 vector and the positive direction of the axis x is opposite to the V0 vector of the medium flow velocity in the "longitudinal configuration" (170 0).
Let us express the w and Wo components of the vector of particle acceleration with the time derivative of coordinates r, 0 of the polar system:  2) and (3) and the Eqs.( 11)-( 13), ( 16), ( 17), ( 19) and (20) into the right- handsides of the Eqs.( 2) and (3) using a notation: d2p MHo gm #0(/p Nf) 187 rw (23) as introduced in [1].Taking into account Eq. ( 18), we obtain: ---g + cos 20 Nd(Vp,r Vf,r) + g(1-P-pf)COS(0 -/3); (24) Components of the medium flow velocity vector Vf, Vf,o in the right- hand sides of Eqs. ( 24) and (25) are considered to be known functions of coordinates r, 0. These functions have different forms for different models of flow around the cylindrical collector by fluid, e.g. the poten- tial, viscous laminar flow and the potential flow with viscous boundary layer on the .surface of the cylinder.
Taking into account the analysis of these models carried out in [3], we shall consider the model of the potential flow around the cylinder, with the vector of the unperturbered flow V0, normal to the axis of cylinder.
We shall consider additional limitations: (1) the diameter of particles and the medium flow velocity are quite small, so that when determining the hydrodynamic drag of a particle it is possible to use the Stokes formula; (2) the mass of a particle or the difference of densities of the particles and the liquid are quite small, so that the gravitational force (the algebraic sum of the gravity and Archimedes forces) can be neglected; (3) the magnetization of a cylindrical ferromagnetic element reached the value of saturation (M-Ms).Under these assumptions Eqs. ( 24) and (25) then yield equations given in [1].Analogical equa- tions given in [3] are applicable to a more general case.In contrast Eqs. ( 24) and (25) can be used for any arbitrary flow regime around the cylinder, not only for the Stokes condition.

DISCUSSION OF RESULTS
Let us consider examples of trajectories of particle motion calculated by numerical solution of the Eq. ( 24) and (25) equivalent to Eq. (1).At the same time we shall consider trajectories, which were calculated using the same initial data, from the reduced equations obtained from Eqs. ( 24) and (25) assuming that the left-hand sides are equal to zero (neglecting the inertia force).
The influence of the gravitational force on the capture cross-section of particles can be significant or negligibly small, depending on the diameter of the particle dp, the medium flow velocity V0, strength of the external magnetic field H0, densities of a particle and the medium pp and pf respectively, the volume magnetic susceptibilities of particles and the medium Np and f, the angle/3 between the vectors Fg and H0, and on the angle c between the vectors V0 and H0.
With c=0 and /3=r/2 the influence of the gravitational force becomes apparent in the displacement of the central line of the particle capture cross-section zone with respect to the axis x, which is the axis of symmetry of the magnetic field and the slurry flow.Calculations show that for a given configuration the inclusion of the Fg force in the equations of motion leads to insignificant alterations of the capture zone.It is explained by that fact that increments of the capture cross- sections for Y > 0 and for Y < 0 have opposite signs and differ insig- nificantly in absolute value.Fig.
Trajectories a and lb are calculated by neglecting the gravitational force (Fg-0); while trajectories 2a, 2b with the gravitational force included.The vector Fg is perpendicular to the vectors H0 and V0.The inertial force was taken into account in all four cases.For Fg 0 (curves a and b) width of the capture zone is h 235c 5.54 at the distance 2-3L from the axis of.the cylinder; for Fg-J= 0 (curves 2a and 2b)

FIGURE
The influence of gravitational force Fg on the particle trajectories and the capture cross section when the flow velocity vector V0 is normal to the vector Fg; dp= 100 btm; dw= mm (dw--2rw); V0=0.1 m/s; B0=0.5T; la, 2a upper limits of the capture cross-section; b, 2b lower limits; a, b Fg 0; 2a, 2b Fg -0./7= (3c)2a +l(3c)2b[-5.47,i.e. ].3% less than for Fg=0.Here h/rw; c yc/rw; rw is the radius of the cylinder; indices 2a and 2b designate the capture cross-section yo, corresponding to the trajec- tories 2a and 2b in Fig. 1.For comparison we shall note that if the inertia force is neglected (Fin 0; Fg 0), then the capture cross-section will be 4% higher than in the case when the inertia force is included (Fin k 0; Fg 0).
With the decreasing diameter of particle dp and also with the increasing velocity V0 the difference between the results obtained with Fg 0 and Fg-0 will be reduced.
In the case when vectors Fg, H0 and V0 are parallel, the influence ofthe gravitational force on the capture section has a different character.In particular, there could be a case when the above mentioned influence will be significant and, at the same time, the influence of the inertial force will be negligibly small.Our calculations show that for configuration Y " -' " " ' x , , , , FIGURE 2 The influence of the gravitational force Fg and the intertial force on the particle trajectories and the capture cross-section when the flow velocity vector V0 is parallel to the vector fig; dp 50 lam, dw mm (dw 2rw); B0 0.25 T; 1,2, 3, Vo 0.01 m/s; without consideration of the inertial force; 2, 3 with consideration of the inertial force; 1,2 Fg 0; 3 Fg 0, vertical ascending flow; 4, 5, 6, 7 V0 0.02 m/s; 4 without consideration of the inertial force; 5, 6, 7 with consideration of the inertial force; 4, 5 Fg 0; 6, 7 Fg-0; 6 vertical ascending flow; 7 vertical descending flow.
It II (the vertical slurry flow) with the value of V0 approaching the value of"soaring" velocity ofa particle in the rising flow the difference of the capture sections, calculated with and without inclusion of the gravitational force, becomes more significant.This situation is illustrated in Fig. 2 where the trajectories for V0--0.01 m/s and 0.02 m/s, are compared, assuming that other parameters in both cases are the same: dp 50 tm, Bo 0.25 T; the values dw, pp, pf, p are equal to the values in the example considered above.
With V0 =0.02 m/s the trajectories of particles, estimated with and without the inclusion of the inertia force with Fg 0 (curves 4 and 5 respectively), practically coincide.The small difference between them is visible in the vicinity of the cylinder.When the gravitational force is included the trajectories and the capture section are appreciably chan- ged.For the vertical ascending flow (curve 6) the capture section is increased by 6.5%, for the descending one (curve 7) is decreased by 5.1%.Let us note that the capture cross-section is changed with the change of the distance from the initial point to the axis of the cylinder; here this distance is fixed and assumed equal to x0 3r,, where r, is the radial coordinate of the equilibrium point on the trajectory of a particle.
The analogous calculation for variant V0=0.01 m/s gives the fol- lowing results: the trajectories estimated with and without inclusion of the inertial force with Fg=0 (curves and 2 in Fig. 2) practically coincide; with Fg -0 (the gravitational force is included) for the vertical ascending flow (curve 3) the capture cross-section increases by 19.6%.

CONCLUSION
The effect of the gravitational and inertial forces, for various config- urations, on the particle capture cross-section has been investigated.
It transpires from the results that if, for a certain combination of parameters dp, Vo, Ho and others, one of the above forces can be neglected, it does not necessarily indicate the other force can also be neglected.It was also observed that with a decreasing size of a particle, the influence of both forces tends to decrease.
/Cd,o; Cd,0 and Cd are the hydrodynamic drag coeffici- ents based the Stokes formula and on a more general formula corre- sponding to the given Reynolds number Re; ppdp (18) -= 18---Vp,r, Vf, are the radial components of Vp and Vf vectors and Vp,o and Vf,o are the azimuthal components of these vectors.