Capture of Small Particles by London Forces from Low-Speed Liquid Flows Lloyd A. Spielman' and Simon L. Goren Department of Chemical Engineering, University of California, Berkeley, Calif. 94720
The motion of a small particle near a much larger collector is discussed using realistic expressions for the London attraction and especially the hydrodynamic forces when the distance between particle and collector becomes comparable to the particle size. By inspection of the equation of motion, one can deduce the few dimensionless groups o n which the single collector efficiency depends and, in certain limiting cases, determine this dependency.
T
he present work grew out of a n experimental study of the coalescence of liquid-liquid dispersions by flow through fibrous media (Spielman, 1968; Spielman and Goren, 1970). Here, as in more conventional filtration, the solid and coalesced liquid matrix may capture droplets by a number of mechanisms : interception, Brownian diffusion, inertial im paction, gravity settling, and long-range attractive forces. Order-of-magnitude estimates indicated that interception should be the dominant mechanism under our experimental conditions. The existing theory of particle capture by interception from low-speed flows, largely developed for aerosol filtration, predicts the filter coefficients varies as the square of the particle size and the inverse cube of the collector size, and is independent of flow rate. I n contrast, our experiments showed more nearly a first power dependence on particle size, inverse 2.5 power dependence o n fiber size, and inverse 0.25 power dependence on flow rate. Other recent liquid filtration experiments where interception is expected to be the dominant capture mechanism also give a first power dependence of the filter coefficient o n particle size. We were therefore led to re-examine the existing theory of particle capture by interception from low-speed flows. 'The classical model for particle capture by interception assumes the center of a small particle to follow exactly a n undisturbed fluid streamline near a larger collector until the particle and collector touch, whereupon the particle is retained by molecular adhesion. Fluid motion is considered to enable collision and molecular attraction to prevent reentrainment. However, a particle in close proximity to a collector must deviate from the undisturbed streamline. The continuum description of fluid motion with "no slip" at solid boundaries becomes invalid for describing fluid movement in the gap between particle and collector when the gap becomes comparable to the mean free path of fluid mole-
cules. For such small gaps the hydrodynamic resistance to closer approach of the particle is lower than that predicted by continuum theory with no slip. The mean free path of air molecules at standard conditions is about 0.1 p ; thus, the capability of describing capture of micron and smaller particles from air by classical interception, which neglects viscous interactions between particle and collector altogether, should be promoted by the breakdown of the no slip continuum flow model. For liquids, however, the mean free path is of the order of the molecular dimensions; hence, the no slip continuum description should remain valid virtually to contact and the classical model for interception appears doubtful as a n accurate description of capture from liquids. In fact, from a no slip continuum point of view, a spherical particle cannot truly contact the collector as a result of hydrodynamic forces alone because the rate of drainage of fluid between particle and collector would become infinitesimally small as the gap narrows. However, London-van der Waals forces increase very rapidly as the particle approaches the collector and becomes strong enough to overcome the otherwise slow drainage. I n this paper a n equation is established describing the trajectory of a very small spherical particle in close proximity to a much larger cylindrical or spherical collector, and subject to hydrodynamic and London forces. Knowledge of the particle trajectory permits calculation of the efficiency of collection by individual obstacles which, in turn, is related to the filter coefficient of the macroscopic medium. Collection by simultaneous interception and Brownian diffusion is also discussed. Motion of a cery small particle in proximity to a much larger cylinder
The undisturbed low-speed flow field near a cylinder is given by the stream function \k = 2AFl%aF-'(r- ap)'sin 0
(1)
where A F is a parameter characterizing the flow model. For isolated cylinders A F is found from Lamb's solution:
For fibers in a fibrous mat, A F is a function of the fraction solids a and is given as
by Happel (1959), 1 Now at Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02138.
Volume 4, Number 2, February 1970 135
by Kuwabara (1959), and AF
=
‘/nzKi(z)/Ko(z)
+
where cy = z2[z 2zKl(z)/Ko(z)]/8by Spielman and Goren (1968). We assume inertial impaction, sedimentation, and Brownian diffusion to be negligible. I n many practical cases, the suspended particles are much smaller than the cylinder. It is then a good approximation to assume that the flow field past the cylinder is undisturbed by the particles except in the latter’s immediate vicinity, and that at separations greater than several particle diameters, particle centers move along the undisturbed fluid streamlines given by Equation 1 . At closer approaches it is necessary to take more realistic hydrodynamic interactions into account. Here we make the further approximation that the particles are so small compared to the cylinder that within separations where a particle trajectory deviates significantly from a n undisturbed fluid streamline, the cylinder can be treated as a plane wall (except insofar as its size and shape determine the flow field far from the particle). The flow field in the neighborhood of a small moving particle is governed by the incompressible creeping flow equations
A. Particle moving under applied force
8. S t a t i o n a r y p a r t i c l e
C. Freely m o v i n g p a r t i c l e .
I n addition to giving the particle trajectory at large distances from the collector, Equation 1 also determines the boundary conditions far from the particle for the solution to Equations 2 for the locally disturbed flow field. With y = r - aF and x = aF(O - e,), where O p is the angular displacement of the particle from the forward stagnation point, the undisturbed flow field far from a particle can be resolved into two flows, one a planar stagnation flow iist (arising from that velocity component at infinity along the line of centers of the cylinder and particle) and the other a shear flow (arising from the flow normal to the line of centers).
Figure 1. Resolution of induced particle motion
dyp
C R = - - -dt- -
dh
F”f;(H)
- dt - 67rpaP
f l ( H ) being a universal function of the dimensionless gap width H = h/ap. For large H , the function f;(H) becomes unity, since far from the obstacle the particle obeys Stokes’ law : f i ( H ) E 1 for H>> 1
-u,), = 4 C I P - ~ AUFsin Opy -is
(4)
These formulas for the undisturbed fluid velocity are valid for aF >> (x2 y 2 ) l i 2>> U P and are correct to second order in x and y . Denoting the minimum gap between the particle and the assumed “flat” surface by h, the particle center is located at x = Oandj- = y p = a p 12. The particle movement and accompanying flow disturbance can be obtained by considering radial and angular motions of the particle separately, since the linear Equations 2 apply and the boundary conditions at the particle surface, cylinder surface, and far from the particle are additive (Figure 1). The radial (y-directed) motion of the particle may be resolved further into two separate flows. In one flow, the particle moves under the influence of an externally applied force F,, consisting of a molecular component and a hydrodynamic component, the fluid velocity vanishing far from the particle and on the “flat” surface of the cylinder. Since the y-directed particle velocity, cR, depends only on the quantities which enter into the equation of motion and boundary conditions of the problem,
,fi(H) = H
136 Environmental Science & Technology
for H
> 1 .
is termed the adhesion number. The particle velocity c ' ~tangential to the surface of the obstacle is obtained by considering the spherical particle as freely rotating and translating under the boundary conditions that far from the particle the flow field is the uniform shear flow given by Equation 4, while o n the surfaces of the cylinder and sphere there is no slip. u8 must depend only o n the parameters entering into the equations of motion 2 and the boundary conditions; thus
- 1) S Z 4Y
( W
