SELECTION OF AN EXPRESSION FOR THE HYDRODYNAMIC DRAG ON A PARTICLE IN A MAGNETIC SEPARATOR

The results ofa numerical solution ofequations ofthe motion ofa paramagnetic particle in the working gap of a magnetic separator for various values of magnetic induction (from 0.2 to 2 T) and the particle diameter (from 10 tm to mm) show that along the particle trajectory various types of flow modes: e.g. laminar, transitional and turbulent can be present. It is also shown that some of the well-known formulae approximating the experimental dependence of the hydrodynamic drag coefficient on Reynolds number Re (the so-called standard hydrodynamic drag curve) as a step-smooth function, do not ensure the condition of continuity of the hydrodynamic drag on the boundaries of the range of values Re for which these formulae were obtained. One of the variants of approximation of the standard hydrodynamic drag curve for the case where along the same trajectory various types ofregimes offlow-around a particle are present, is proposed.


INTRODUCTION
In magnetic separation the main force determining the process of separation is the magnetic force acting on a particle in the working gap. 46 Yv.S. MOSTIKA et al.
However, in order to estimate parameters ofthis process it is necessary to determine the complete forces: the hydrodynamic drag and the gravitational force. In many cases we can neglect the influence of the gravitational force (small enough sizes of particles or for small difference of the densities of particles and medium flow), whereas the hydrodynamic drag force plays as important role as the magnetic force. To determine the hydrodynamic drag force of spherical particles Stokes formula is often used. It is applicable to the values of Reynolds numbers Re < Rel where Rel is assumed to be equal to 0.3; 0.5 or depending on the level of acceptable error (for example, with Re the error of the Stokes formula is about 10%). In order to extend the range of solved problems it is necessary to use the hydrodynamic drag estimation formula and the equation of the motion of particles applicable not only in the Stokes regime but also in a wider range of values of Reynolds number. Such formulae and equations of motion will be considered below.

THEORETICAL MODEL
Equations of motion of a paramagnetic particle in the working gap of a magnetic filter (separator) with cylindrical ferromagnetic collectors were obtained in [1]. The hydrodynamic drag force of a particle was determined by the Stokes formula. In a more general case when the hydrodynamic drag depends on the particle flow regime (the Stokes transitional or Newton regime) we can obtain the following equations: (d2r (d0) 2) Vm(A -\dt2 r 3 -q-cos20 Nd(gp,rgf,r) d20 dr dO) where r and 0 are, respectively, the radius and the angle of polar system of coordinates, the center of which is coincides with the center of the cross section of the cylindrical collector; is time; r/rw; rw is the AN  Nd-Cd/Cd,o; Cd, Cd,0 are the coefficients of the hydrodynamic drag of a particle in the general case and for the Stokes regime respectively; dp is the diameter of a particle; g is the acceleration of gravity;/3 is the angle between axis x (from this axis the polar angle 0 is measured) and the vector (the axis x is parallel to the vector 0); Mis the magnetization of the ferromagnetic cylinder; H0 is the strength of the magnetic field; A #w #f. #w + #f' #w and #f are the magnetic permeabilities of the cylinder and the medium, respectively, /0 is the magnetic constant (permeability of free space);/p and nf are the volume magnetic susceptibilities of the particle and the medium, respectively, pp and pf are the densities of particles and carrying medium, respectively, /is the dynamic viscosity of the carrying medium; Vp, Vf are the velocity vectors of the particle and of the carrying medium; Vp,r, Vf  The plus and minus signs before the terms on the right-hand sides of formulae (1) and (2) and also before V0 in the two last equations correspond to the origin and the direction of measurement of angles 0, c,/3, which are shown in Fig. 1.

DISCUSSION OF THE RESULTS
In order to analyse the effect of technological parameters (H0, V0) of separation and of the size of particles on the hydrodynamic flow regime numerical integration of the equation system (1) and (2) was carried out for a number of variants of the initial data assuming that particles have a spherical shape.
Values of the magnetic field strength H0, of the unpertubered medium flow velocity V0, diameter ofthe ferromagnetic collector dw 2rw and the diameter of a particle dp for these variants are shown in the Table I  Values of other determining parameters in all the variants were kept constant: specific magnetic susceptibility of particles ;p-l.5 10 -6 mS/kg (p pDp); density of particles pp 4.7 x 10 s kg/mS; density of carrying medium pr-l0 s kg/mS; dynamic viscosity of medium r/-10-3N x s/m2; saturation magnetic induction of a ferromagnetic collector Bs 2.15 T.
Magnetic permeability #w was determined for given values of H0 using known experimental dependencies. The so-called longitudinal configuration of a collector [2], in which vector V0 is parallel with the direction of vector H0 (c--0) was considered. Initial conditions for numerical integration of Eqs. (1) and (2) Table I  where Ca Ca(Re); Re is the Reynolds number; Re Pf117P'fldP (3) and Sm is the cross-sectional area of a particle.
Velocity Vv,f can exceed many times velocity V0 of the unpertubered medium flow. For example, for variants 2, 4 and 7 the ratio Vp,d Vo near collector is 4.9, 19.1 and 9.0 respectively.
In the initial stage of the trajectory the flow condition of a particle can be laminar, while when approaching the collector, as a result of the increasing velocity Vp,f, the transitional or turbulent condition can be reached. In order to assess the flow condition the value of Reynolds number is usually used [7].
Results of calculations of the Reynolds number using Eq. (3), along the trajectory of particles (i.e. depending on the relative distance r/rw)  Table I. It can be seen from these graphs that the Reynolds number Re can vary substantially for different trajectories, and for motion along a fixed trajectory. Point A on lines 2 and 6 corresponds to the equilibrium point on the trajectory y(x) (Fig. 1). And when the trajectory is passing through point A it can curve sharply.
The results show that the velocity Vp,f of the particle movement relative to the medium practically does not depend on velocity V0. It can be seen, for example, comparing curves and 9 in Fig. 3, taking into account that the dependence Re(?) differs from the dependence Vp,f(?) by a factor (ppdp)/. For variants and 9 the values of velocity V0 differ by a factor of five with other conditions constant. However, dependencies Re(?), and therefore dependencies Vp,r(?), for variants and 9, practically coincide. We can make similar conclusions when comparising curves 4 and 5 and also 7 and 8 in Fig. 3. It is usually assumed [3][4][5][6][7] that for calculations of the hydrodynamic drag of a spherical particle the Stokes formula (expressed in terms of factor Cd): Cd 24/Re (4) is applicable in the range of Reynolds numbers Re < 1, although with Re the error of the Stokes formula is about 10%. It is possible to use the generalised experimental dependence Cd(Re), the so-called standard hydrodynamic drag curve [5][6][7]10]. However, approximation of this curve by a single formula with a small error in a wide range of Reynolds factor will be a complicated and bulky expression. In [10] various approximations of the curve Cd(Re) were considered and in particular the following expressions [11] were assumed: 24/Re, Re _< 2; Cd 18.5/Re '6, 2 < Re <_ 500; (5) 0.44, Re > 500. (6) The same expressions are given in [9], but in contrast to [11] the transition from Eq. (4) to Eq. (5) is supposed to occur at Re 0.2 rather than at Re--2.
In [8] for evaluation of the hydrodynamic drag of spherical particles various approximations of the experimental dependence Cd(Re) for different intervals 10 < Re < 10 + (n 1,0, 1,2) are used. The Stokes formula is considered applicable in the range Re < 0.1, while for 0.1 < Re < the following expression is suggested: Cd Cd,0(0.947 + 0.1538Re + 0.003763/Re), where Cd,0 24/Re. This equation becomes equal to the Stokes formula if the expression in the parentheses is assumed to be equal to unity.
Apparently, the value of Re 0.2 in [9] was given by mistake, since the value of Ca calculated under Eq. (5) at Re 0.2, is approximately 60% lower than the value under the standard drag curve, whereas with Re= 2, Eqs. (4) and (5) give almost equal values (17% and 15 % below the standard curve respectively).
Equation (7) is known as the Klatchko formula (see for example [3,10]); Eq. (8) is given in [4,10]. Transition from the Stokes formula to (7) and (8) at Re the function Cd(Re) changes in a step. That is why the results of calculation using such step-smooth dependencies have singularities of artificial character which are absent in reality. For example, the dependence of the velocity of particle motion relative to the carrying medium on parameter r/rw considered above (see Fig. 2.) has "a plateau", i.e. a section of almost constant value for those values of r, which correspond to a section of trajectory with Reynolds number close to unity. Curve 4a in Fig. 2 where Cd,1 is the value of Cd, calculated using Eq. (8).
When these formulae are used the discontinuity between two different dependencies of Cd(Re) at point Re 0.3 is eliminated. Also the error in Cd(Re) decreases in comparison with Eq. (8). The upper limit of the Re range in which Eq. (9) was used is determined from the condition that Cd(Re) reaches the value corresponding to the Newton's regime: CdCOnSt=0.44. Re---920 corresponds to this condition. Figure 4 shows the relative error A Cd as a function of the Re number for different formulae Cd(Re) in relation to experimental data Cd,e(Re): Afd (Cd Cd,e)/fd,e.
The experimental data were taken from [8] where they are given in a tabular form. Lines 1, 2, 3, 4 and 5 in Fig. 4 show errors of Eqs. (4), (5), (7), (8) and (9) respectively. When using Eqs. (4) and (9) (9), respectively. and 7% for 0.1 < Re < 1000. We should note that the results of calculation expressed in Figs. 1-3 (except for curve 4a in Fig. 2), were obtained using the dependencies (4) and (9) connecting at the point Re =0.3. The calculation shows that for given values of the fluid viscosity, magnetic susceptibility of particles and other determining parameters the turbulent flow condition takes place for dp mm and magnetic induction B0= 2 T (variant 11), and in the case, when B0 < 2T, for dp> mm. With the increase of the magnetic susceptibility the limiting values of the diameter of a particle and of the magnetic field strength at which the turbulent condition develops decrease.

CONCLUSIONS
An analysis of several formulae of the coefficient of the hydrodynamic drag for a spherical particle has been carried out and a modification of AN EXPRESSION FOR HYDRODYNAMIC DRAG 55 one of the expressions, in the form of Eq. (9), was proposed. The error of this expression, relative to the experimental data in the range 0.3 < Re < 1000 does not exceed 7%, while for Re < 0.3 the Stokes formula is used. The distribution of the velocity of motion of particles relative to the carrying medium has been calculated and the corresponding Reynolds numbers along the trajectory of motion, for a number of characteristic combinations of the separation conditions, have been determined.