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Vg = the gas velocity in the x direction

Vp = the migration velocity in the y direction

The initial conditions chosen are C(0,y) = C0, and C(x,y) * 0 as y * 0, where C0 is the particle concentration at the inlet. The first condition implies a constant concentration across the inlet, while the second condition states that an infinitely long precipitator has zero penetration.

The boundary conditions are D2 3C/3y + VpC = 0 at the center plane, and D2 3C/3y — fVpC = 0 at the boundary layer. The first condition states that the diffusional flux balances the flux due to migration velocity, and no net particle flux occurs across the center plane. The second condition contains a reentrainment factor f. This factor is an empirical parameter indicating what fraction of the particles that enter the boundary layer are later reentrained back into the gas stream. The complete analytic solution is fairly complicated, and therefore only the first term of the solution is used; but it yields sufficient accuracy as follows:

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