Spherical collector efficiencies

N. Single Spherical Collector Efficiencies. Four collection mechanisms are considered in the present analysis inertial impaction, interception. Brownian movement and Coulombic forces. Although in our previous analysis the electrical forces were considered to be of the induced nature (13), there is evidence that it is the Coulombic forces which dominate the electrical interactions between the particle and collector (, ], 22). Taking the net effect as the simple summation of each collection mechanism results in the single spherical collector efficiency equation. 

It is conventional to define a dimensionless collection efficiency by comparing the actual diffusional mass flow rate to the mass flow rate of particles to the collector for straight particle trajectories, that is, the mass swept out by the projected cross-sectional area of the collector. From this definition the spherical collector efficiency is... 

For all capture mechanisms where the suspended particles are independent of one another and capture results from individual encounters with the collectors making up the porous medium, the filter coefficient A is independent of the suspended particle number density n. Examples of this behavior are seen in the single spherical collector efficiencies given by Eqs. (8.3.15), (8.3.23), and (8.4.26). If the incoming suspended particle number density is q and the filter medium is uniform and unclogged, Eq. (8.5.13) integrates to... 

The modification is seen to parallel the incorporation of the Stokes-Oseen function (Eq. 8.3.24b) into the solution for the collector efficiency of a single cylindrical collector. A similar change would be made for an assemblage of cylindrical collectors, though the volume fraction function that would replace the Stokes-Oseen function would differ from the one for spherical collectors. 

Single-collector efficiency for monolayer filtration vreis estimated with the expression developed by Rajagopalan and Tien [128], obtained by the combination of the trajectory analysis of a spherical particle in the vicinity of a spherical collector with the contribution of the Brownian diffusion. For fine-fine capture step, filtration becomes driven by the fine-fine interaction forces yielding a multilayer deposit for which the filter coefficient no longer remains constant in time. The change of the filter coefficient as a function of the specific deposit was estimated using the correlation developed by Tien et al. [129]. Extra information about trickle-bed deep-bed filtration model is given in Iliuta and Larachi [130] and Iliuta et al. [119]. 

When particles experience a mean curvilinear motion and also have Brownian agitation, they are deposited on obstacles by both mechanisms. For very small particles of radii less than 0.1 /xm, Brownian motion dominates particle collection on surfaces. For larger particles, inertial forces dominate. An example of the difference in collection efficiency for spherical collectors of different size is shown in Fig. 3 for different particle diameters and aerosol flow velocity. 

The effectiveness of deep-bed filters in removing suspended particles is measured by die value of die filter coefficient which in turn is related to the capture efficiency of a single characteristic grain of the bed. Capture efficiencies are evaluated in the present paper for nil cases of practical importance in which London forces and convective-diffusion serve to transport particles to the surface of a spherical collector immersed in a creeping How field. Gravitational forces are considered in some cases, but the general results apply mainly to submicron or neutrally buoyant particles suspended in a viscous fluid such as water. Results obtained by linearly superimposing the in-... 

As with diffusion, the same behavior with particle radius is found for the spherical and cylindrical collection efficiencies. This is again a consequence of the similarity of the velocity fields near the collector surfaces. However, in contrast to the diffusion collection efficiency, which decreases with increasing particle radius as the interception collection efficiency increases with... 

As an illustration of the modification, the Brownian collection efficiency for a porous medium made up of an assemblage of spherical collectors would become, from Eqs. (8.3.13) to (8.3.15),... 

In Fig. 10.7 the capture efficiency of particles in the water-suspension flow at normal temperature over spherical collector is shown [36]. The capture vs. particle-size dependencies for the two above-considered mechanisms of capture are shown, and in the intermediate area the dependence obtained by adding the efficiencies Edif + Ecoii due to both mechanisms of capture is depicted. 

As the capture coefficient (3) was found to be proportional to the flow rate for a given sandstone (Berea sandstone was used in this work), one is encouraged to further substantiate the capture rate equation using a simple concept of deposit of particles on spherical collectors. This approach has been successfully used in waste water filtration (9). The efficiency (ri) of a single collector is defined as the ratio of two rates—the rate at which particles strike the collector and the rate at which particles flow toward the collector. Rate of flow of particles toward a spherical collector is given by (UC) 7rd /4 where U is the undisturbed flow velocity, C is the concentration of particles and 7rd /4 is the projected area of a spherical collector of diameter (d). Therefore the rate of capture of particles for a single collector is equal to u(er 7rd /4)C, where e is the retention efficiency. 

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