Numerical modeling of a nonmonotonic separation hydrocyclone curve

Abstract

In the context of the mechanics of interpenetrating continua, numerical modeling of separation of a polydis- perse suspension in a hydrocyclone is carried out. The so-called "mixture model" valid for a low volume frac- tion of particles and low Stokes numbers is used for description of the suspension and particle motion. It is shown that account taken of the interaction between large and small particles can explain the nonmonotonic behavior of the separation curve. Introduction. To characterize the efficiency of separation of small and large suspension particles in hydrocy- clones one uses the separation curve T(d) which shows the share of particles of each fraction of the suspension intro- duced into the apparatuses that are removed through the lower nozzle of a classification apparatus (Fig. 1). It follows from classical representations of the dependence of the rate of sedimentation of a particle on its size d p (1-3) that the separation curve T(d) is monotonically increasing from a certain value T(0) < 1 at d p → 0 to T = 1 at d p → ∞. This representation is based on the fact that as the size of particles increases, their sedimentation rate rises; therefore, the larger the particles, the more they can be drawn to the wall of the apparatus and taken out through the lower outlet. Indeed, it has been repeatedly noted, e.g., in (3-12), that the separation curve has a minimum in the range of particle size from 10 μm and lower. This behavior of the curve is called the "fish-hook" effect. It has not been given an unambiguous explanation to date. There are several versions of interpretation of this phenomenon in the literature: insufficiently accurate granulometric analysis; dependence of the materials' density on the particle size; flocculation and agglomeration of the dispersed material; hydrodynamic interaction of particles of various size. The analysis of the causes of the "fish-hook" effect in (13) leads to the conclusion that it is only hydrody- namic interaction of particles that can serve as a source of nonmonotony of the separation function. Hydrodynamic mechanisms of the manifestation of a "fish-hook" effect are discussed in the literature (6, 8-12) but basically at the qualitative level, which only results, as a rule, in the formulation of some empirical formulas. This helps characterize the classification process but it does not explain the phenomenon. Investigations (14) have been carried out in recent times which formulate the concepts that suggest that the abnormal motion of small particles in the apparatus is caused by the stochastic motion of particles under con- ditions of turbulence. Accordingly, stochastic models of the phenomena are constructed, whose implementation re- quires the involvement of numerical methods. At the same time, the interaction of particles of various fractions between each other is either not taken account of (14) or hypotheses are adopted that are not corroborated in ex- periments. The importance of a correct account of the said interaction is obvious if only because the parameters of the "fish-hook" effect depend on both the concentration of the solid phase in a suspension and on the granulomet- ric composition of a material. Using the below kinematic model of acceleration of sedimentation of small particles at the expense of large ones (15, 16), not only is explanation given to the anomalous behavior of the separation curve at small values of the particle size (this explanation has already been given in simplified form (11-13)) but a calculation is also provided of the separation curve for a hydrocyclone. At the same time, the manifestation of a "fish-hook" effect is illustrated at