Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 487

(3)= ZLi[Q'nh + v h .n h .A+-D h ^.A]

(Eq. 15.32)

where vj, = the settling velocity of NPs of size h within the control volume, L/T (cm/s),
which is the mass leaving rate of NPs in size h due to advection in the flow direction and
the settling out of the control volume. The item (4) in eq. 15.1 lumps the mass transport
rates due to all of the interphase transfer and transformation processes that the NMs are
involved in. For example, the rate of the particles of size h formed due to collision of
smaller particles of size i and j minus that disappeared into other size classes due to
collision with any other particles can be described with

V- [ZLiCjZi+HhaPninj) - ZLi(Zr=iapnjn h )]

(Eq. 15.33)

Note that the notation (i.e., i, j) in parentheses for (3, as in eq. 15.25, is no longer
needed. Eq. 15.33 is usually used as a general coagulation equation in summation form.
If there are no other processes acting as sinks or sources of NPs, item (4) of eq. 15.1
equals eq. 15.33. It should be pointed out that NPs may be generated or degraded
(decay) due to other mechanisms (see Sections 15.3 and 15.4). For example, the
concentration of algae (a specific NP) increases during a phytoplankton boom can be

T i= *£=»•»*

(Eq. 15.34)

where |i s = the growth rate, 1/T. Note that |i s may need to be modified for settling if eqs.
15.31 and 15.32 do not include the settling of algae into and out of the control box. In
those cases, the rate of NP mass generated (r g V) or decayed (-iaV) should also be added
into eq. 15.33. In addition, item (4) of eq. 15.1 includes many interphase transfer
processes, such as adsorption, dissolution, volatilization and absorption, which need to
be included in a mass balance equation (eq. 15.1) (see Section 15.3).

The aforementioned models (eqs. 15.24-34, also known as rectilinear models)
are valid for completely destabilized particles (e.g., clusters of NPs). These models are
based on the assumption that a spherical particle sweeps out all other particles in the
water in its path, in addition to six major assumptions (Smoluchowski, 1917; Thomas et
al., 1999). In recent decades, the rectilinear models have been modified by considering
(a) a smaller cross-sectional area used to calculate collision frequencies, (b) the water to
be squeezed out between two particles, and (c) short-range attractive and/or repulsive
forces between the particles, leading to different curvilinear collision models (Logan,
1999; Thomas et al., 1999). As shown in Fig. 15.3, the collision frequency functions
predicted by the rectilinear and curvilinear models are within the same order of
magnitude for NPs (10-100 nm) but are separated by seven orders of magnitude for
particles of 100 |im.