Page 273 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

258 Principles and Methods

medium are assumed to be spherical collectors of radius a , surrounded
c
by an imaginary outer fluid sphere of radius b with a free surface
(Figure 7.15). The imaginary fluid envelope contains the same amount
of fluid as the relative volume of fluid to the collector volume in the
1/3
entire medium [i.e., (a /b)(1 ) ] where is the porosity of the given
c
medium. Particles can be transported to the collector surface through
a combination of interception, gravitational settling, and diffusion
transport mechanisms. Interception occurs when fluid streamlines
pass sufficiently close to the collector so that contact results. Contact
through gravitational settling and diffusion results from the particles
crossing the fluid streamlines via the respective mechanisms within
a critical region around the collector and contacting the surface.
Analytical solutions for particle transport due to Brownian diffusion
have been combined with particle trajectory calculations to yield a
closed-form solution for the transport of particles to the surface of spher-
ical collectors expressed as the theoretical single collector contact effi-
ciency ( ) [10]:
0
5 1 1 G (18)
0
I
D
is the single collector contact efficiency for transport by diffusion;
where D
is the single collector contact efficiency for transport by interception; and
I

Fluid flow

a c

Figure 7.15 Illustration of the
b Happel sphere model in which the
spherical collector having a radius
a c is surrounded by a fluid envelope
having a radius b.