Unsuitability of traditional methods for describing particle transport

Unsuitability of Traditional Methods for Describing Particle Transport 

THROUGH Hydrodynamic Boundary Layer of almost Completely Retarded 

Since water can often be strongly contaminated by organic matter and bubble surfaces can become almost completely immobilised by an adsorption layer, a development of the theory of particle transport to a bubble surface through a hydrodynamic boimdary layer is very important. 

In a paper by Nguen Van Kmet (1992), a difficult numerical integration of the Navier-Stokes equation was carried out in order to obtain information necessary for calculating collision efficiency. As usual, it is very difficult to evaluate the reliability of numerical calculations. Usually some judgement can be drawn when comparison with results tfom analytical formulas for known limiting cases or with numerical calculations based on substantially different methods are performed. Unfortunately such data are not available. 

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

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