Particle diffusion boundary layer

The velocity of liquid flow around suspended solid particles is reduced by frictional resistance and results in a region characterized by a velocity gradient between the surface of the solid particle and the bulk fluid. This region is termed the hydrodynamic boundary layer and the stagnant layer within it that is diffusion-controlled is often known as the effective diffusion boundary layer. The thickness of this stagnant layer has been suggested to be about 10 times smaller than the thickness of the hydrodynamic boundary layer [13]. 

Reduction of particle size increases the total specific surface area exposed to the solvent, allowing a greater number of particles to dissolve more rapidly. Furthermore, smaller particles have a small diffusion boundary layer, allowing faster transport of dissolved material from the particle surface [58]. These effects become extremely important when dealing with poorly water-soluble drugs, where dissolution is the rate-limiting step in absorption. There are numerous examples where reduction of particle size in such drugs leads to a faster dissolution rate [59-61], In some cases, these in vitro results have been shown to correlate with improved absorption in vivo [62-64]. 

Since the diffusion coefficient of the dust particles is very small, the thickness of the diffusion boundary layer is small compared to the radius R of the collector. Therefore, the concentration distribution and the rate of deposition can be calculated by substituting for the velocity and electric fields the expressions valid for y/R 1. In that region, Eqs. (182) and (183) become... 

Consider a sphere of radius a held fixed in a creeping flow field with approach velocity U. The fluid contains Brownian particles having a diffusion coefficient D. Should the Peclet number 2aUiDUj have a value much greater than unity, the diffusion boundary layer will be sufficiently thin so that curvature effects and tangential diffusion are negligible. Under these conditions the convective-diffusion equation assumes the following form ... 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

To determine the net rale of adsorption of particles suspended in a fluid that is flowing over the collector, one may then solve the usual convective-diffusion equation subject to a reversible first-order reaction as the boundary condition, provided the diffusion boundary layer is much thicker than the interaction boundary layer. 

Fig. 3- Cose 1. Sherwood numbers computed for the convective-diffusion of particles of finite sine to the surface of a spherical collector by neglecting interaction forces. The dashed line is the Levich-LighthilJ equation (19) which is valid when a diffusion boundary-layer exists and the particles are infinitesimal.

Use of Ficks law to describe the diffusion process requires the solute particle to be small compared with the diffusion boundary layer. The analysis presented above suggests that, for Peelet numbers greater than 100, the ratio 8o/0p is proportional to (Pe)ua/R. The solid curves in Figure 3 are truncated at the value of the Peelet number corresponding to Pe/R3 10 "2, where an inspection of the radial concentration profile revealed that the ratio 8d/Op is about ten. 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

This corresponds to mass-transfer limitation of the apparent surface reaction. Thus the combination of Eqs. p] and 2] is expected to give reasonable estimates of the rate of deposition for all particle-collector interaction profiles, provided the interactions are confined to a region which is thin compared to the diffusion boundary layer. 

Hull and Kitchener (2) measured the rate of deposition of 0.3- an-diameter polystyrene latex particles onto a rotating disk coated with a film of polyvinyl formaldehyde. In electrolytes of high ionic strength (where the double-layer repulsion is negligible), they found close agreement between experiments and the prediction of Levich s boundary-layer analysis (Eq. 3]), indicating that a diffusion boundary layer exists and that its thickness is large compared to the domain of van der Waais and hydrodynamic interactions. These are neces-... 

Compared to small molecules the description of convective diffusion of particles of finite size in a fluid near a solid boundary has to account for both the interaction forces between particles and collector (such as van der Waals and double-layer forces) and for the hydrodynamic interactions between particles and fluid. The effect of the London-van der Waals forces and doublelayer attractive forces is important if the range over which they act is comparable to the thickness over which the convective diffusion affects the transport of the particles. If, however, because of the competition between the double-layer repulsive forces and London attractive forces, a potential barrier is generated, then the effect of the interaction forces is important even when they act over distances much shorter than the thickness of the diffusion boundary layer. For... 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

This is the usual boundary condition for molecular diffusion to surfaces in gases and liquids for a perfectly ab.sorbing surface. Hence the results of experiment and theory for molecular diffusion in the absence of a force field can often be directly applied to particle diffusion. However, the effect of finite particle size is very imporiaiU when diffusion boundary layers are present as discussed in the next chapter. 

So long as this ratio is small, deposition is controlled by diffusion. For particles large compared with the diffusion boundary layer thickness, interception controls. 

Diffusion Boundary Layer Near the Surface of a Particle... 

As Pe increases, a diffusion boundary layer is formed near the surface of the sphere. The ratio of the thickness of this layer to the radius of the particle is of the order of Pe 1//3. In this region, the radial component of molecular diffusion to the surface of the particle is essential, and tangential diffusion may be neglected. Convective mass transfer due to the motion of the fluid must also be taken into account. 

For a viscous flow past drops (bubbles) and for an ideal fluid flow past solid particles, we have m = 1. For a laminar viscous flow past smooth solid particles, one usually has m = 2 there also exist some examples in which m = 3 [166]. It follows from the preceding that in the leading approximation, the tangential velocity vv (4.6.19) in the diffusion boundary layer near the drop surface is constant and is equal to the fluid velocity on the interface, whereas in the diffusion boundary layer close to the surface of a solid particle, the tangential velocity in the leading approximation depends on the distance to the surface linearly (sometimes, even quadratically) and is zero on the surface. 

A method for solving three-dimensional problems on the diffusion boundary layer based on a three-dimensional analog of the stream function, was proposed in [348, 350]. In [27, 166, 353], this method was used for studying mass exchange between spherical particles, drops, and bubbles and three-dimensional shear flow. 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

The diffusion boundary layer in problem (4.12.1), (4.12.2) is first adjacent to the particle surface and then rapidly spreads over the flow region with the subsequent exponential relaxation to a steady state. The characteristic relaxation time rr is of the order of Pe-2 3 for a solid particle and of the order of Pe-1 for bubbles and drops of moderate viscosity. 

Table 4.8 presents a comparison of the mean Sherwood numbers calculated according to Eq. (4.12.3) with available data for various flows past spherical drops, bubbles, and solid particles at high Peclet numbers (in this table, we use the abbreviation DBLA for diffusion boundary layer approximation ). 

At high Peclet numbers, the diffusion flux to the surface of the first solid particle can be found by solving the conventional diffusion boundary layer equation the presence of the second solid particle influences only by changing the velocity... 

Mass exchange of the second solid particle is more complicated. The main role is played by the interaction of the diffusion boundary layer of the second particle with the diffusion wake of the first particle. 

In Table 5.1, the maximum error of formulas (5.3.8) and (5.3.9) are shown in the entire range of the parameter ky for six different kinds of spherical particles, drops, or bubbles. All these estimates were found by comparison with the closed-form solution of problem (5.3.1), (5.3.2) obtained in the diffusion boundary layer approximation [363]. 

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