Particle shear induced diffusion

S. S. Hsiau and M. L. Hunt. "Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer," /. Fluid Mech., 251, 299-313, 1993. 

However, the shear-induced particle diffusivity Dp( ) may be described via relation (3.1.74) and a nondimensional particle diffusivity Dt as... 

It has been observed that particle flux expressions developed based on shear-induced particle diffusivity describe the observed solvent flux through a microflltration membrane much better than those based on Brownian diffusivity of a particle. For the following system properties, determine the ratio of the solvent fluxes based on shear-induced particle diffusivity and Brownian diffusivity. You should employ the particle volume fraction based solvent flux expression based on the gel polarization model used in ultrafiltration (equation (7.2.72)) ... 

The simple form in Eq. (7) can be maintained by replacing the Brownian diffusion coefficient in the expression kc = /-An /5 by the shear-induced hydrodynamic diffusion coefficient for the particles, Ds. Shear-induced hydrodynamic diffusion of particles is driven by random displacements from the streamlines in a shear flow as the particles interact with each other. For particle volume fractions between 20 and 45%, Ds has been related to... 

Leighton, D. Acrivos, A. Viscous resuspension. Chem. Eng. Sci. 1986, 41 (6), 1377-1384. Leighton, D. Acrivos, A. Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 1987, 177, 109-131. Leighton, D. Acrivos, A. The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 1987, 181, 415-439. Altobelli, S.A. Givler, R.C. Fukushima, E. Velocity and concentration measurements of... 

Breedveld, V., e. a. (1998). The measurement of the shear-induced particle and fluid tracer diffusivities in concentrated suspensions by a novel method, /. Fluid Mech. 375 297-318. 

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

Eor a given crossflow filtration, the dominant particle back transport mechanism may depend on the shear rate and the particles size [4]. Brownian diffusion is only important for particles smaller than only a few tenths of a micron in diameter with relative low shear, whereas inertial lift is important for particles larger than several tens of microns with higher shear rates. Shear-induced back transport appears to be important for intermediate particle sizes and shear rates. Li et al. [7] reported that the shear-induced mechanism was able to predict fluxes comparable with the critical fluxes identified by the DOTM. 

The positive effect of velocity on the permeate flux is a result of enhanced hydrodynamic effects at the membrane surface, since high velocities lead to high shear and turbulent flow, which results in the formation of vortices and eddies that minimize the concentration polarization effects and the development of a fouling layer. The bigger the thickness of this layer, the higher its flow resistance and the smaller the permeate flux through the membrane becomes. Under turbulent flow conditions, shear effects induce hydrodynamic diffusion of the particles from the boundary layer back into the bulk, with a positive effect on the permeate flux. 

He have presented a simple procedure whereby one can estimate the stability of a colloidal system undergoing simultaneous creaming and gravity-induced flocculation. This procedure is by no means restricted to only this case. One can easily take into account other particle loss mechanisms, such as shear-induced flocculation or Brownian flocculation. What is required in these cases are the appropriate particle/particle collision kernels, which can be computed by solving the governing convective-diffusion equation. 

A particle migration model was proposed by Gadala-Maria and Acrivos to describe experimental shear-induced migration observations. This model allows for a better understanding of the shear effects on particle diffusion for concentrated suspensions. Based on these studies, a conservation equation for the solid phase was established by Phillips, Amstrong, and Brown, which takes into account convective transport, diffusion due to particle-particle interactions, and the variation of viscosity within the suspension, namely ... 

Although this picture is remarkably generic, the mechanisms responsible for the formation of a particle-lean layer adjacent to the wall depend on the properties of the material under consideration. For the case of solid particle dispersions, wall depletion, particle migration, and solid-liquid separation are the most frequent sources of solvent layer lubrication. Wall depletion occurs whenever dispersions are brought into contact with smooth and solid surfaces because the suspended particles cannot penetrate rigid boundaries [147]. Particle migration is due to various forces arising from fluid inertia, fluid elasticity, and shear-induced diffusivity effects [165]. Solid-liquid separation, which frequently occurs in flocculated suspensions like... 

In practice, however there could be differences between the observed and estimated flux. The mass transfer coefficient is strongly dependent on diffusion coefficient and boundary layer thickness. Under turbulent flow conditions particle shear effects induce hydrodynamic diffusion of particles. Thus, for microfiltration, shear-induced difflisivity values correlate better with the observed filtration rates compared to Brownian difflisivity calculations.Further, concentration polarization effeets are more reliably predicted for MF than UF due to the fact diat macrosolutes diffusivities in gels are much lower than the Brownian difflisivity of micron-sized particles. As a result, the predicted flux for ultrafiltration is much lower than observed, whereas observed flux for microfilters may be eloser to the predicted value. 

Several types of dispersions show strain rate thinning, and a quantitative explanation is not easily given. We will briefly consider two cases. The first one concerns shear flow. As discussed (above, Factor 4), anisometric particles show rotational diffusion and thereby increase viscosity. This effect will be smaller for a higher shear rate when the shear-induced rotation is much faster than the diffusional rotation, the latter will have no effect anymore. The shear rate thinning effect is completely reversible. Something comparable happens in polymer solutions (Section 6.2.2). 

For l-/im particles, the inertial force is quite small relative to the drag force, and rapid formation of a particle or gel layer is predicted. However, the particle layer does not grow steadily until it plugs the channel but seems to reach a steady-state thickness. One explanation is that the high shear rate near the wall causes a tumbling motion of individual particles, which expands the layer and leads to migration away from the wall. This shear-induced dispersion and the particle movement toward regions of lower concentration can be modeled with a particle diffusivity, which is proportional to the shear rate and the square of the particle size. More work is needed to understand the effects of particle shape and surface characteristics. 

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