### An Impact of the Particle Size Distribution on CP Degree at the Mechanism of Shear-Induced Diffusion in Cross-Flow Microfiltration

#### Abstract

Model for concentration profile at the mechanism of shear- induced back migration has been proposed. The model is based on the following assumptions: (1) Flow is assumed to be isothermal, incompressible, fully developed and steady– state. The model implies the position dependence of transverse velocity and shear rate; (2) Dispersed phase consists of rigid bodies characterized by identity variables and particle- size distribution; (3) Back migration being non- diffusive in nature is characterized by certain kinetic behavior. The migration of particle is determined by shear- induced force (moving force) and drag forces (resistance force); (4) The migration of particles was assumed to occur within the zone referred to as shear-diffusion migration zone. Migration zone ranges from membrane to its threshold value at the upper boundary where the state of pseudo-equilibrium takes place; Vector sum of the forces acting the particle (shear and drag forces) ranges from its maximum value at the surface of membrane to zero at the upper boundary of migration zone where the state of pseudo-equilibrium takes place; (5) The driving force being degree of deviation from the state of equilibrium was expressed as the difference between local shear rate and its threshold value at the upper boundary of migration zone; Transport mechanisms such as Brownian diffusion, inertial lift, interaction based on Van der Waals or electrostatic forces, combined effects of particle-particle and particle–membrane interactions are outside the scope of the model. The proposed model has the following possible implications: (A) It can be used for quantitative estimation of the probability of fouling caused by different fractions at existing hydrodynamic conditions; (B) The model allows analyzing the impact of hydrodynamic conditions (such as shear stress and transverse velocity) on the probability of fouling caused by certain dispersed fraction; (C) It allows estimation of the threshold value of the shear rate specific for the certain fraction that should be exceed in order to prevent accumulation of this fraction; (D) Proposed model permits analyzing

**the influence of shear rate at the surface of membrane on the transverse concentration distribution; (E) The model can be used for quantitative analysis of CP distribution of individual fraction while considering poly-disperse systems as soon as any fraction is characterized by its individual migrating behavior.**#### Keywords

Shear-induced diffusion, concentration polarization, cross-flow microfiltration, modeling

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ISSN: 1929-6037