THE ASSESSMENT OF THE APPLICABILITY OF A MODEL OF NONINERTIAL MOTION OF A PARTICLE IN A MAGNETIC SEPARATOR

Using two models, namely the initial differential equation of motion and the reduced equation obtained from the initial one by neglecting the inertial force comparative calculations of the capture cross section of paramagnetic particles by a cylindrical ferromagnetic collector (a wire) were carried out. For various combinations of the particle diameter, the wire diameter, medium flow velocity, strength of the magnetic field, the error in the capture cross section caused by neglecting the inertial force in the equations of motion was determined. Approximate formulae for the determination of this error and of the ratio of the inertial force to the magnetic force as a function of the main separation parameters were proposed.


INTRODUCTION
When designing magnetic filters or separators it is expedient to carry out preliminary determination of the efficiency of extraction of material depending on its properties (magnetic susceptibility, density and the diameter of particles) and technological parameters (strength of the magnetic field, velocity of the slurry movement).For the working space 74 Yu.S. MOSTIKA et al.   filled with ferromagnetic elements (rods, wire mesh, steel wool) this efficiency can be approximately determined on the basis ofcalculation of the capture cross section ofparticles of the extracted material by a single wire.This problem is well described in [1][2][3][4] and in other publications.THEORETICAL

MODELS
In order to determine the capture cross section a theoretical model can be, in some cases used [1,2], which we shall call for short a model of "noninertial" motion of particles.It is a system of differential equations of motion, where the terms corresponding to the inertial force are rejected.As a criterion of validity of such a model a condition that quantity r is much smaller than unity was proposed in [2]: (It has to be noted that r is much smaller than 1 s, since r has a dimension of time.) Here pp and rp are density and radius of a particle, respectively and r/is the dynamic viscosity of the carrying medium.
In this work, in order to estimate the influence of the inertial force, two methods are used: the first one determination of the ratio of the inertial force to the magnetic and hydrodynamic drag forces, the second one determination of the capture cross sections calculated with and without the inertial force.In each theoretical model, "inertial" and "noninertial", the capture cross section was estimated for given com- binations of the determining parameters on the basis of the limiting trajectory calculations.Differential equations of motion proposed in work [6] were used.By a numerical solution of the initial equations and of the corresponding reduced equations assuming that a parametric study oferrors ofcalculation ofthe capture cross section of paramagnetic particles caused by neglecting the inertial force in the equations of motion was carried out.Here mp is the mass of a particle, 17p is the particle velocity.The results thus obtained allow to estimate the validity of the simplified model of the "noninertial" movement of particles for a given combination of parameters: the diameter of a particle dp, the wire diameter dw, the medium flow velocity V0, and the strength of the magnetic field H0.
Taking into account the estimates of the influence ofthe gravitational force Fg carried out in [6] we shall consider separately the influence of the inertial force on the capture cross section of particles assuming Fg 0. In this case the criterion ofvalidity ofthe "noninertial" equations of motion can be expressed by the following inequalities: Zm << or Z d << 1, where Zm--Fin/Fro; Zd Fin/Fd; Fin mpldp/dt[ is the inertial force; Fm is the magnetic force; Fd is the hydrodynamic drag.It must be noted that for the calculation of Fd the dependence ofthe hydrodynamic drag factor on Reynolds number as given in [5] was used.

DISCUSSION
The results of calculation of distribution of quantity Zd (the ratio of the inertial force to the hydrodynamic drag) along the trajectory of parti- cles, in the form (Fin/Fd) {r/rw}, where r is the radial coordinate of a given point of trajectory (rw dw/2), are shown in Fig. 1 (a).The vari- ables were the diameter of a particle dp, the magnetic induction B0 of external magnetic field and the medium flow velocity V0.Other deter- mining parameters in all variants were equal and had following values: density of particles pp=4.7 x 103 kg/m3; specific magnetic susceptibility of particles Xp 1.5 x 10 -6 m3/kg; diameter of the cylindrical ferromagnetic collector dw mm.
The longitudinal configuration [3], i.e. 170 0 was considered.The medium fluid was assumed to be water at room temperature.
Magnetic permeability of the cylinder material (a steel containing 0.2% of carbon), which is used to estimate the magnetization of the cylinder, was determined using the reference tables and approximations as a function of H0.Fig. 1 (b) shows the dependencies (Fin/Fm) {r/rw}; the curve numbers correspond to the same combinations as in Fig. l(a).dependencies (Fin/Fd) {r/rw}.However the value Z m (provided Fg 0 which is assumed above) at any point of the trajectory is less than unity, whereas the value Zd can exceed unity.This usually takes place for r < r, and for larger values dp and V0 and as well for r > r,, where r, is the radial coordinate of the equilibrium point for a given trajectory.
It can be seen from Fig. l(a) and (b) that the dependencies Zd() and Zm(), where r/rw, have two characteristic sections, located on the left and right-hand sides of the Zd and Zm zero value points.Each of these points corresponds to a special point of the system of differential equations of motion of a particle (the equilibrium point).For some combinations of values of dp, V0, H0 and other parameters the points of zero value of Zd or Zm is absent; in these cases equations of motion do not have special points in the interval > 1.
It is obvious that the capture cross section is determined by the parameters of the particle motion in the section of the trajectory located upstream from the equilibrium point, i.e. in the section r > r,.The values of Fro, Fd, Fin change along the trajectory, therefore let us con- sider, for the estimation of the dependence of ratios Zd Fin/Fd and Zm Fin/Fm on parameters dp, dw, V0, B0 the average-integral values Z, Zm of these ratios: lf0L' Zi --ZddL; Z= where L1 is the length of a section of the limiting trajectory from the point r--rl to the point r=r,; dL is an element of length of the trajectory.
In preliminary calculations the rl value was varied, after that it was chosen to be equal to r =3r,.The Z and Z' m dependencies on induction B0 of the external magnetic field for various values of dp, dw, and V0 are shown in Fig. 2. Values of these parameters corresponding to the variants of curves to 9 are shown in the Table in Fig. 3.The reduction of values Z and Zn with increasing B0, keeping other parameters fixed, is caused by the increase of the distance r, from the axis of the cylinder to the equilibrium point on the limiting trajectory.
Thus the section of the limiting trajectory r, < r < rl is displaced in the area of relatively small gradients of the magnetic field and relatively weak perturbation of the flow.This results in the reduction of Yu.S. MOSTIKA et al. 0,8 FIGURE 2 The average-integral values of the ratio of the inertial force to the hydro- dynamic drag (Z) and to the magnetic force (Ztm) as functions of the background magnetic induction B. the acceleration of a particle in a given section.The difference between dependencies Z'd(BO and Z'm(Bo increases with the increase of the diameter of a particle dp and the velocity V0 and with the reduction of the wire diameter dw, when the above parameters are fixed.Parameters dp and V0 are related to the Reynolds number Re0 by: Re0 pf Vodp (1) FIGURE 3 The relative errors AYc/ Ye of the determination of the capture cross section caused by neglecting the inertial force in the equations of particle motion as functions of the relative magnetic velocity Vm/Vo; F0; longitudinal configuration.and the field strength H0 is related to the "magnetic" velocity Vm according to [1,3]: Z' m can then be expressed by following approximate formula: 0.58Redl "2   Ztrn + O.14(Vm/Vo) '5'   (3) where a 1.2dl; and dl dpldw.In (1) to (3) pf is the density of the flow medium; M is the magnetization; M 2AH0; #w #f.#w + #f' #w and #f are the magnetic permeabilities ofthe cylinder and the medium, respectively, p and /f are the volume magnetic susceptibilities of a particle and the medium, respectively.Equation (3) approximates the averaged (in the section r, < r < 3r, of the limiting trajectory) values of the ratio of the inertial force to the magnetic force for the values d o, dw, and V0 corresponding to the variants 1 to 9 (see Fig. 3).
The range of the H0 values ofthe magnetic field strength where, Eq. ( 3) is applicable is determined on the basis of inequality Vm/Vo> (Vm/Vo)min, with Vm on the left-hand side given by Eq. ( 2).The right hand side of this inequality is given by the following approximation: The dependencies of the relative error Ay/ Ye of the capture cross section on the ratio Vm/Vo caused by neglecting the inertial force in the movement equations is shown in Fig. 3.Here A y Ye,0-Ye where Yc, Yc,0 are the capture cross sections at distance x 3r, from the axis of the cylinder calculated with and without the inertial force; in both cases the gravitational force was neglected.The curves 1, 2, 7, 9 are obtained for the particles with diameter dp 50 lxm; 3, 4, 8 100 txm; 5, 6  200 lxm; the value of the medium flow velocity V0 for these curves: 3 V0 0.02 m/s; 1,4, 5, 7, 8, 9 0.1 m/s; 2 and 6 0.2 m/s; dw 1 mm for the variants 1-6; dw 0.5 for the variants 7, 8; dw 0.2 for the variant 9.For all variants pp =4.7 103 kg/m3; pf= 103 kg/m3; Xp= 1.5 x 10-6m3/kg.
The maximum point of curve (AYe/Ye)(Vm/Vo) corresponds to such a combination of parameters when the equilibrium point of the limiting trajectory is close to the surface of the cylinder.When Vrn/Vo increases in the section Vm/Vo > (Vm/V0)m where (Vm/V0)m is the value of Vm/Vo at the maximum point the value A Ye/Ye decreases at the expense ofthe growth ofthe function Y(Vm/Vo).Function A Yc(Vm/Vo) increases slowly, asymptotically approaching the constant value for high values of Vm/Vo.The value of (Vm/V0)m can be calculated from the These dependencies, in the first approximation, are similar to the 100