Similitude Law for Impaction Stokesian Particles

When an aerosol flows over an object in its path, the gas velocity decreases as it approaches the surface. Both tangential and normal components of the gas velocity vanish at the surface of a fixed solid body. The particles, however, are unable to follow the gas motion because of their inertia if they come within one particle radius of the surface, they can adhere depending on the interaction between the attraction and rebound energies discussed in previous sections. An idealized version of the situation is shown in Fig, 4,5. At large 

When the particles are much. smaller than the collector, and in sufliciently low concentration, the How Helds for the particle and collector can be uncoupled. For the gas flow Held around the collector, (he velocity distribution is detenntned by the Reynolds number based on collector diameter, independent of the presence of the particles. The particle is assumed to be located in a How with a velocity at intinity equal to the local velocity for the undisturbed gas How uroimd the collector the drag on the particle is determined by the local relative velocity between particle and gas. 

When the Reynold. number for the particle motion is small, a force balance can be written on the particle assuming Stokes law holds for the drag. For the jt direction we have 

Here u is the particle velocity, U/ i.s the local fluid velocity, and / is the Stokes friction coefficient. We call particles that obey this equation of motion Stokesian particles. The use of (4.2S) is equivalent to employing (4.19), neglecting the acceleration terms containing the gas density. Because (4.19) was derived for rectilinear motion, the extension to flows with velocity gradients and curved streamlines adds further uncertainty to this approximate method. 

For the mechanical behavior of two particle-fluid systems to be simitar, it is necessary to have geometric, hydrodynamic, and particle trajectory similarity. Hydrodynamic similarity is achieved by fixing the Reynolds number for the flow around the collector. By (4.26), similarity of the particle trajectories depends on the Stokes number. Trajectory similarity also requires that the particle come within one radius of the surface at the same relative location. This means that the interception parameter, R = dp/L, must also be preserved. 

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