A MODEL OF MULTIPLE MAGNETIC SEPARATION IN THE CONTINUOUS PROCESS

In continuous enrichment processes, in industrial conditions and due to the generally required high efficiency of machines, a relatively high concentration of feed is applied. Since it is necessary to liberate the useful minerals, the feed has to be ground thoroughly and this fact results in the percentage growth of the fraction of particles measuring a few tens or several micrometers. When the particles are so fine and the feed is so much con- centrated, the magnetic particles interactions play a significant role and they lead to the phenomenon of magnetic flocculation. The flocs contain the particles of waste rock and intergrowths. In order to obtain the final product of high quality the material is subjected to multiple enrichment operations. This paper presents a mathematical model of multiple magnetic separation in which the grade ofthe magnetic concentrate was expressed as the function of multiplication ratio of separation and the value of external forces acting on a particle as well as interactions ofthe magnetic particles. The dependence of the content of magnetic component in the high- gradient separation concentrates on the multiplication ratio of separation, flow velocity of the suspension through the separator working space, the intensity of matrix vibration and the concentration of solid particles in the feed for the separation of magnetite ore was verified. The verification was carried out according to the author’s own research and the reference.


INTRODUCTION
Industrial processes of magnetic separation, due to their mass character and the necessity of enrichment of large quantities of the throughput, are carried out in a continuous way. In the course of the process the 213 214 M. BROEK feed is continuously transported to the separator and the enrichment products are also continuously collected. These are: the magnetic product and the non-magnetic product. Since it is necessary to liberate the useful minerals, the feed for the enrichment process must be first crushed to the range of a few tenths of a millimeter. The material, crushed in such a way, is subjected to the action of magnetic field in the separator working space. Under the influence of the magnetic field the magnetic particles obtain magnetic moments whose mutual interactions lead to the phenomenon of magnetic flocculation. The non-magnetic particles are bound in the internal structure of flocs by magnetic, surface and mechanical forces.
If the separation is performed in the constant magnetic field, the nonmagnetic particles are transported, together with the magnetic component, to the magnetic product, decreasing the quality of this product. Due to that, in order to obtain the final product of high quality, the material should be subjected to multiple separations (cleaning operations). When the material of high magnetic susceptibility, such as, for instance, magnetite ore, is enriched, the magnetic product is sometimes subjected, between enrichment operations, to demagnetization in order to break the floc structure. During each consecutive separation (stage) a certain portion of the non-magnetic component is removed from the enriched material.
In the process of cyclic separation the magnetic component, due to the forces of the magnetic field, is removed from working chamber of the separator while the non-magnetic component remains in the working chamber during the entire process. The amount of the separated magnetic component is the function of time and the concentration of this component in the separator working space changes in time. Transporting the magnetic component outside working space of the separator in the process of the cyclic separation is a pure death process from the point of view of Markovian stochastic processes [1]. The duration of the separation process may be arbitrarily long. Even if magnetic particle returns to the working chamber of the separator from the magnetic product, it can be brought again to the magnetic product in the next time interval. Because of that, the entire amount of the magnetic component can be separated in the cyclic process.
As opposed to the cyclic separation, in the continuous process, the chance of return of the magnetic particle to the magnetic product is MAGNETIC SEPARATION IN CONTINUOUS PROCESS 215 slight, due to the short residence time the material in the working chamber of the separator. Because of that, magnetic separation in the continuous process is a process of birth and death [1]. In this system only a part of the magnetic component can be separated. With fixed and time independent distributions of magnetic properties of feed particles and with fixed capacity and separation conditions the relative amount of the separated magnetic component does not depend on time.
A MODEL OF MULTIPLE MAGNETIC SEPARATION As it was mentioned at the very beginning, the process of multiple separation proceeds in a continuous way. A magnetic product from the separation in the kth stage constitutes a feed for separation in the (k + 1)th stage. In every stage the material, in the magnetic field zone, travels a certain distance along the separator working space, called the separation path. In the process of multiple separation the effective length of the separation path will be proportional to the number of stages The effective path is a path travelled by the material in the magnetic field zone.
The subject-matter of the considerations will be constituted by the separation of the non-magnetic component of susceptibility Xn from the stream of magnetic product (main stream). In every stage a certain amount of non-magnetic component is carried off.
Let r denote an effective length of the separation path. The value P(r > s)=f(s) (1) will denote the probability of the non-magnetic particle remaining in the main material stream to a point s or, in a different way, the probability of non-separation of the non-magnetic particle into the non-magnetic product on the separation path s when the condition is fulfilled.
A non-separation of the non-magnetic particle in two consecutive sections of the effective path s and u (s, u > 0) will be a pair of independent events. Therefore the probability of non-separating the 216 M. BROEK non-magnetic particle in the joint section of the effective path s + u will be, according to the formula of total probability, equal to [2] P(r > s + u) P(r > u)e(r > s), After differentiating expression (3b) in relation to u (assuming that there is a density f'(s + 0), s >_ 0), we can obtain Dividing both sides of Eq. (3b) by Eq. (4) we obtain f() -.
As it was said before, f(s) denotes a probability of non-separation of the non-magnetic particle of susceptibility n into the non-magnetic product. The measure of this probability will be constituted by a number of particles of susceptibility Xn, contained in the magnetic product after travelling the path s (number of non-separated particles) to the total number of particles of this susceptibility in the feed Therefore from expression (7) after considering Eq. (8) we obtain Nm(slXn N(OlXn)e -"s.
In the above equations Nm(s Xn) denotes the number of non-magnetic particles of susceptibility Xn, contained in the magnetic product after travelling the path s while N(0IXn) denotes the number of particles of susceptibility Xn in the feed. The value Nn(sIXn) N(0IXn) Nm(slXn) N(0IXn)(1 es) (0) represents the number of particles of susceptibility Xn separated to the non-magnetic product after travelling the path s by the main stream.
The total recovery of these particles in the non-magnetic product (to the point s of the effective path of the main stream) will be equal to Nn(s[Xn) e e(SlXn) N(01X ) (11) If the length of the separation path in a single stage is equal S1, then after k stages the effective separation path will be s sk. In such a situation expression (11) can be written as follows: (kiX) exp(-/slk). (2) The length of the separation path in one stage s depends on the type and dimensions of the working space of the separators applied in a given technological process.
It results from formula (12) that with the growing number of stages (cleaning separations) the recovery ofthe non-magnetic component into the non-magnetic product is growing and consequently the residue of this component in the magnetic product is decreasing.

TECHNOLOGICAL INDICES OF SEPARATION
The residue of the non-magnetic component in the magnetic product after k separations is equal to cr(klXn e(klXn) exp(-#slk). (1-an) in the feed increases, then the content of the magnetic component in the magnetic product decreases.
Analogously, for . 1 In all formulas shown above the constant / is associated with the physical properties of the mixture components and the separation conditions as well as with the value of particle interactions.
Assuming/ 1 (So specific length of the separation path), the quotient So/S will be such a number of separations after which the residue of the non-magnetic component in the magnetic product will be 0.33 or the recovery of the non-magnetic component in the non-magnetic product will be 0.67. Magnetic separation aims at arranging a set of particles into two or more subsets according to their magnetic properties. In the case of a two-component mixture it is arranged into two subsets. This arrangement requires the work against the forces ofparticle interactions to be performed. This work is performed by mechanical forces.
Paper [4] contains a derivation of the formula for the number of non-magnetic forces contained in a volume unit of the magnetic stream in a plate separator, after travelling the separation path x: where No is the number of non-magnetic particles in the feed, V is the interaction potential of a non-magnetic particle and its surrounding particles, rn is the mass of the particle and g the acceleration of gravity.
The exponent numerator of the power of expression (20) contains the force executing the work against particle interacting forces, i.e. the force contributing to the arrangement of particles (in the plate separator it is the force of gravity), while the denominator contains the potential of interactions hindering this arrangement. 220 M. BRO;EK Formula (9) has the same form as formula (20). Consequently, the expression of constant # in the above formulae can be expressed as follows: -'iFic (21) #= Vo where i F represents the sum of all competing forces aiding the exit of non-magnetic particles from the magnetic stream and performing work against the interaction forces. The speed of the arrangement process will therefore depend on the relationship of the values of forces favouring the arrangement to the interactions hindering the arrangement. In order to improve the separation efficiency we should improve this relationship. It can be done either by means of decreasing the particle interactions or by increasing the forces which support the exit of non-magnetic particles from the magnetic stream.
In the case of dry separation the possibilities in this respect are slight.
Only the effect of the force of gravity can be improved by an appropriate construction of separators. Applying separators with alternating magnetic field and selecting a proper frequency of the field, the exit of nonmagnetic particles from the magnetic stream is facilitated, without falling of the magnetic particles into the non-magnetic product. In the constant magnetic field, due to the fact that the Van der Waals interaction in the air is much larger than in water, dry magnetic separation is performed for loose materials. Much larger possibilities of control of separation conditions occur in the case of wet separation. It transpires from formula (18) that with the increase of the cleaning separations (stages) the content of the magnetic component in the magnetic concentrate increases. This dependence was verified using natural samples ofilmenite: the ore ofparticle sizes 0.2-0; 0.1-0; 0.075-0 and 0.063-0mm. The magnetite fractions were separated from the samples of the magnetite-ilmenite ore of the above particle sizes by means of the plate separator. The non-magnetic product of this separation, containing mainly ilmenite and the waste rock, constituted the material for further investigation. Samples of the ilmenite ore were subjected to a multi-stage separation in the polygradient separator. The magnetic product of the preceding stage was the feed for the next stage.
The separation conditions in each stage were identical: diameter of steel balls 10mm, intensity of external magnetic field 750kA/m, height of working space 0.1 m, velocity of the suspension flow 0.08 m/s, volume content of solid parts in the suspension 10%. After each stage of separation in the magnetic product the TiO2 content was determined.
The verification of formula (22) was performed according to the literature data. Therefore the character, resulting from formula (22), of the dependence of the content of magnetic component in the magnetic product (after a single separation) on the value of competing forces and particle interactions was verified.
The value of the sum of mechanical forces in the high-gradient separator is affected by the flow velocity ofsuspension through the separator working space and the intensity of vibrations of fibres of the separator matrix (explained further in the text). Therefore the dependence of the content of magnetic component on flow velocity and on the intensity of the current inducing the vibrations of fibres was investigated.

M. BRO,EK
Particle interactions can be divided into magnetic and surface interactions. Magnetic interactions in the separator working space (at the constant ore composition and fixed conditions of separation) depend on the content of solid parts in the suspension directed to the separator. Therefore the dependence of the content of the magnetic component in the magnetic product of the drum separator on the content of solid parts in the suspension was given.

ANALYSIS OF RESULTS
The influence of the number of stages on the results of separation  the increase of size reduction the number of separations necessary for exceeding a certain threshold of content, referring to the content/3t, increases. At a higher size reduction, as it can be seen in Table I, the content of TiO2 increases in the purest particles of the magnetic product (/3t grows) which is associated with the growth ofmagnetic susceptibility of magnetic particles. With the increase of susceptibility of magnetic particles, the length of the separation path and consequently, the number of separations increases [4]. In order to reach the content -0.95/t it is necessary, in case of the particles 0.2-0 mm, to carry out nine separations while in case of the particles smaller than 0.06 mm fourteen separations are required. where is the coefficient of dynamic viscosity of a liquid; B and C the constants given by B S mg C 6Sl 7rr/a It was simultaneously assumed that the movement of a liquid with velocity v is of a laminar character and the force of hydrodynamic resistance is expressed by Stokes' formula. Therefore the sum of the competing forces contains two components; i.e. the force of gravity and the force of hydrodynamic resistance.
The dependence of the content of the useful component in the magnetic concentrate on the flow velocity of suspension will thus be as follows: tim(V) (1 + bexp(-cv))-t  Figure 5 shows the dependence of the Fe content in the magnetic concentrate of the oxidized iron ore (taconite) on the flow velocity of suspension through the working space of the separator [5,6]. An investigation was carried out in the high-gradient separator at the magnetic field intensity 1600 kA/m. The working space of the separator (matrix) was filled with steel wool of the packing density of 5%. The content of Fe in the feed was aFe 35% while the particle size distribution ofthe ore was below 20 gm. The maximum content ofiron in the magnetic concentrate which was reached at the enrichment ofthis ore was 65% [5]. It can be therefore assumed that/t 65%. Taking into consideration the fact that tim(0)= a 35%, the following dependence was obtained on the basis ofthe data of As can be seen. from Fig. 5 the compatibility of the experimental data with the model dependence is satisfactory.
The influence of the matrix vibrations on the separation results In order to intensify the process removal of non-magnetic particles from the working space of high-gradient separators filled with steel wool, the steel wool fibres are made to vibrate with appropriate frequency, perpendicular to the direction of the external magnetic field [7]. Vibrations of frequency contained in the range of sound frequencies are induced by an alternating magnetic field induced in the coil wound on the steel wool vessel (Fig. 6). These vibrations on the one hand keep the magnetic floc loosened while, on the other, as a result of inertia, they pass the momentum of a vibrating fibre to non-magnetic particles, facilitating them to leave the fl.oc structure. The force of the fibre vibrations is proportional to the force with which the magnetic field of the coil acts on the fibre, i.e. F,, H 2 [8]. Since the intensity of the magnetic field of the coil is proportional to the intensity of the current in the coil winding, then F 12. A floc is connected with a vibrating fibre and therefore the volume will be affected by the force of inertia, proportional to 12 and parallel to the flow direction.
In this situation the sum of competing forces occurring in expression (21) will be increased by another component, proportional to 12 Assuming that the remaining elements in the expression (21) are constant, the dependence of the content of the magnetic component in the magnetic concentrate will be expressed due to Eq. (22) by a formula similar to formula (25): /3re(l) (1 + bexp(-c12))-l/3t.
The investigation of the effect of the vibrations of the matrix fibres on the results of magnetic separation were performed on mixtures of wolframite and quartz of the concentration 0.5% WO3, particle size distribution ofwolframite 0-5 gm and quartz 0-76 gm [7]. The intensity of the external (constant) magnetic field was 1200 kA/m while the flow velocity of suspension through the working space of the separator was The continuous line of Fig. 7 was drawn according to expression (28).
Therefore the compatibility of the model with the experiments is very good. By means of a proper choice of the vibration intensity and the flow velocity of suspension it is possible to reach a very high selectivity of separation, confirmed in industrial conditions by the separation of cassiterite ore [7]. where Xm is the volume magnetic susceptibility, H the intensity of the external magnetic field, S the area of the transverse cross-section of a particle, N the demagnetization factor, r the distance between particles, and kl a constant depending on the assumed system of units.
It transpired from formula (29), that the potential of magnetic interactions between two magnetic particles in the Coulomb-type approximation is proportional to r -1 (r distance between particles). Next, the average distance between magnetic particles is proportional to the reciprocal value of the third root from the volume concentration of magnetic particles in suspension [10]. Thus, formula (29) can be written as follows: The investigation of the effect of the content of solids in the suspension were carried out using the iron ore of Krivoy Rdg [11] of particle size distribution 0.1-1 mm and Fe content ave 22%. The pure magnetic concentrate contained about 65% Fe. This value can be assumed as equal to/3t.
The magnetic separation tests were carried out using a drum seperator of drum dimensions 400 x 400 mm with a three-pole magnetic system. The magnetic field intensity on the surface was 112 kA/m and Pk being the average density of the dry feed, and p0 the density of water. Figure 8 presents the dependence of the Fe content in the magnetic concentrate as a function of the volume concentration of solids in the suspension fed into the separator. The experimental points have been taken from work [11]. The solid line represents the following dependence: /Fe(Z)--65(1 + 1.94exp(-1.2Z-1/3))-1.
The dependence/3Fe(Z) was calculated for the Coulomb-type interaction between the magnetic particles. The experimental data are in good agreement with the model for the Coulomb-type approximation of interactions while they differ completely from the dependence drawn for the dipole-dipole interaction.