Model of the aggregation process of fine ferromagnetic particles

. The most important result of the effect of interparticle interactions of ferromagnetic particles in suspensions is the formation of aggregates and modification of their physical properties, which allows changing the kinetics of separation processes. The interest of the authors to this problem is dictated by both its theoretical content and purely applied aspects within mineral processing. Based on the block-structural approach to technological equipment modeling within the framework of the ideal mixing model, a mathematical model of processes occurring in the working volumes of apparatuses using magnetic forces to intensify the separation of mineral suspensions was created. The use of the MATHCAD software has allowed obtaining analytical solutions of the differential equations system of the model and predictions of the kinetics of aggregation of ferromagnetic particles and physical properties of the formed aggregates, such as coarseness, density, deposition velocity in the gravity field as functions of medium parameters and external magnetic fields. The analytical solutions are necessary for developing simulation models of magnetic separation processes in strong, weak, uniform and non-uniform magnetic fields, magnetic deslammers and filters.


Introduction
In the presence of an external magnetic field, a magnetic force = 0 ∇ acts on particles of any nature, where ∇ is a gradient of the external magnetic field strength, and M and Vp are magnetization and volume of a particle [1].
The combined action of gravitational and magnetic forces results in particle acceleration. Operation of dressing equipment for magnetic separation is provided by attraction of ferromagnetic particles to the poles of magnetic systems under the action of gradient magnetic force. The magnetization of the ferromagnetic particle M depends on the strength of the external magnetic field H and the coercive force of a mineral which it consists of [2].
If the coercive force is significant, the particle magnetization does not decrease to zero when the external field disappears, i.e. the particle acquires a nonzero magnetic moment and turns into a magnetic dipole. In mineral suspensions, in addition to the forces of magnetic attraction to the poles of magnetic equipment systems, forces of interparticle interaction appear. Interparticle magnetic interactions are based on the attraction or repulsion of the magnetic dipole poles under the action of the magnetic force Fm [3].
Magnetic interparticle interactions of particles in ferromagnetic suspensions based on external or intrinsic magnetic fields lead to formation of particle aggregates. Aggregation processes modify physical and physicochemical properties of particles and make them suitable for various technological processes. In magnetic separation processes, aggregation affects the kinetics of particle separation processes of suspensions according to their magnetic properties; in industrial wastewater treatment processes the aggregates -floccules and coagules -are necessary for adsorption of pollutant impurities. In gravitational separation processes, particle aggregation changes the deposition velocities or hydraulic particle sizes. Physical and physicochemical properties of the resulting aggregates, such as density, size, shape, hydraulic coarseness, magnetic susceptibility, and adsorption capacity depend on the strength of the external magnetic field, ferromagnetic particle size distribution, magnetic susceptibility, and their volume content in the suspension.
The suspension stirring intensity has a great influence on the physical properties of the formed magnetic aggregates of particles. In [4], a mathematical model of formation of ferromagnetic aggregates in the suspension under the action of a homogeneous constant magnetic field is developed, taking into account the action of resistance forces to particle motion. In [5], a mathematical model of the convergence of ferromagnetic particles in a magnetic field was proposed. Using the Smoluchovskii model, the authors thoroughly analysed the probabilities of binary union of ferromagnetic particles in a homogeneous magnetic field for the mono-and polydisperse compositions of the suspension. The work [6] studied the formation of magnetite floccules in strong gradient fields of magnetic drum separators, which are reliable apparatuses for iron ores processing. In [6] the solutions of equations of motion of particles in the cylindrical part of magnetic hydrocyclone based on the Lagrangian approach to determine the optimal configuration of magnetic field were obtained and mathematical model of aggregation in the working volume of the apparatus was developed.

Design of a mathematical model of particle aggregation in suspension
Technological equipment which uses aggregation of ferromagnetic particles has a complex design, in separate parts of which there are processes differing in the mass forces action. Structurally there are separate zones, such as a zone of initial suspension receipt, a zone of aggregation and separation of different particle fractions, and a zone of thickening and output of technological products. In all these zones, combinations of hydrodynamic resistance forces, magnetic and electric forces of different nature, forces of inertia, including centrifugal forces, act.
A mathematical model of a technological device, which has a complex functional structure, is a combination of blocks connected by material flows, and the output flow of one block is input for the next [7]. The mathematical device of each functional block describes the change in concentrations of individual fractions in its volume (Fig. 1). The zone of aggregation and separation of different fractions of particles according to physical and physical-chemical properties is the most important in a block-structural model of a technological device functioning. In many chemical-engineering processes, reagent flocculation and coagulation processes have been implemented. For example, in liquid purification processes, particle aggregates -floccule and coagulates -are used for adsorption and absorption of contaminants.
In gravitational separation processes, particle aggregation changes the deposition velocities or hydraulic particle sizes. In equipment with magnetic separation principle, the action of magnetic forces of different nature causes induction of magnetic moments of ferromagnetic particles and their interaction. Magnetic inter-particle interactions of particles based on external or own magnetic fields lead to formation of particle aggregates. Physical and physicochemical properties of the formed aggregates, such as density, size, shape, hydraulic size, magnetic susceptibility, and adsorption capacity, depend on the strength of the external magnetic field, the ferromagnetic particle size distribution, magnetic susceptibility, and their volume content in the suspension. A change in physical properties affects the kinetics of particle separation processes of suspensions according to their magnetic properties.
The system of differential equations of "ideal mixing" flow structure model was used for mathematical description of non-stationary modes of particles concentration changes in the volume of aggregation zone and particles separation. The change in concentration of particles of the initial suspension in the device's volume depends on the flow rate of its feed, velocity of gravitational deposition and flocculation of particles of the initial suspension.
The equations of the component-by-component balances of the ideal mixing model in general form are written as follows: In the case given, the volume includes a flux of thin ferromagnetic particles possessing intrinsic or induced magnetic moments. Induced magnetic moment appears in ferromagnets in the external magnetic field of a permanent magnet or electromagnetic solenoid. The The total intensity of sources (sinks) of mass, substance, due to various physical and chemical processes in the considered area, including the additional supply (withdrawal) of these flows from (to) external sources per unit volume = − + magnetic moment remains in the particles even after the external magnetic field is removed. The interaction of magnetic moments leads to interparticle interaction, which appears as flocculation or aggregation of particles. The change in concentration of flocculated particles in the volume of the device depends on Where: _ velocity of gravitational deposition of magnetite floccules in liquid particle aggregation model Integration of differential equations of the ideal mixing flow structure model [7] was carried out using forward and inverse Laplace transformations in the Mathcad software package. For this purpose, the Laplace image of the right and left parts of the derived equations (3) was obtained [8,9].
As a result of Laplace transformations and inverse assembly, the resulting system of equations with notations (0) → _ 0.
The resulting system of equations (4) is integrated with the built-in block for solving systems of equations in Mathcad complex "Given-Find". Given As a result, the vector of solutions of the equations system with respect to Laplace images has been found. By performing the inverse Laplace transformation, the solutions of the original system of equations (5) have been obtained Thus, the explicit form of the kinetic equations of the magnetite particles flocculation is as follows: Where _0 and _0 are initial concentrations of nonflocculated and flocculated particles.

Conclusion
Deterioration of quality of mineral raw materials leads to the necessity of involvement in processing of fine disseminated ores of complex material composition. Use of fine mineral crushing is accompanied by decrease of contrast properties of mineral particles in separation and slurring operations. Technologies of any separating operations are based on physical and physicochemical properties of minerals and are implemented in certain ranges of mineral particle size. The operating principles of separating devices are based on the laws of free and constrained movement of mineral particles under the action of mass or surface forces. Obviously, as the size of the particles decreases, the intensity of those or other forces responsible for the result of beneficiation changes, which can lead to losses of the useful component, which is in a fine-dispersed state. Reducing the size of mineral particles reduces the contrast of physical properties and reduces the efficiency of beneficiation processes. The most important task of intensification of technological processes in suspensions can be solved with the use of inter-particle electric and magnetic forces, the action of which leads to the formation of aggregates of particles that change their physical, chemical and physicochemical properties. For example, deposition velocity of ferromagnetic mineral particles with diameters less than 5*10 -5 m is less than 5*10 -3 m/sec. The influence of magnetic field leads to formation of ferromagnetic aggregates of various shapes and densities with sedimentation velocity more than ten times higher than the velocity of original particles. The obtained solutions are suitable for building a particle aggregation unit when creating mathematical models of technological devices with magnetic flocculation (drum magnetic separators, MGS, magnetic sludge thickeners) and with reagent flocculation and coagulation.