Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 479

outgoing stream-flow rates, and the lake is well mixed (i.e., as a completely-mixed tank
reactor, CSTR), we can have

V^= (QC 0 + S) - QC - v s AC

(Eq. 15.9)

where V = lake volume, L ; Co and C = NM's concentration in the incoming stream
flow, and effluent of the lake, respectively, M/L 3 ; Q = the stream inflow and outflow
rate, L 3 /T; S = the rate of addition of NM from the point source, M/T; v s = the settling
rate of NMs, L/T; and A = the surface area of the lake, L 2 . It would be easy to add other
sources, including nonpoint sources if their NMs' loading rates could be estimated. The
settling rate, v s can be determined with some equations shown in Section 15.2.2.1, but
the stability of NMs is often not available (Phenrat et al., 2007). Therefore, v s is usually
an empirically determined quantity with limited confidence.

Eq. 15.9, however, may not be always valid because most lakes are not well
mixed at all. In these lakes, the structure (e.g., temperature or density profile) of the lake
and the mixing patterns of the water determine the form of mass balance equations,
which can be very complex in some cases. For example, nonaggregated NPs in free
water columns are subject to a variety of fluid motions ranging from laminar flows to
less predictable turbulent flows; these NPs may interact with and then be removed by
phytoplankton. The mass transport of NMs in the proximity of a phytoplankton cell (or
microbial cell) is governed by the advection-diffusion equation

^ + UVC = DV 2 C

(Eq. 15.10)

where C = NM concentration, M/L 3 ; U = the velocity vector, L/T; D = the turbulent

diffusion coefficient of NMs, L 2 /T; and V = gradient operator, L" 1 . Using the technique
of inner region and outer region expansions (Hondzo and Al-Homoud, 2007), eq. 15.10
can be solved, and the following model can be used to evaluate the effects of small-scale
turbulence on the mass transport of NMs to a phytoplankton cell:
Sh = (1 + a Pe 1 / 2 + a 2p e )

(Eq. 15.11)

where Sh = the Sherwood number = Q/QD, namely the ratio of the total NM flux to a
phytoplankton cell in a turbulent fluid (Q) versus the total nutrient flux to the cell by

molecular diffusion (QD), unitless; Pe = uL c /D, the Peclet number, with u = 0.5(ev) 1/ 4 =
the Kolmogorov velocity average over the Kolmogorov length scale, L/T; e =

dissipation of turbulent kinetic energy, L 2 /T 3 ; v = the kinematic viscosity of a fluid,
L /T; and a = the order "1" shape coefficient. Eq. 15.11 indicates that mass transport of
NMs is enhanced with increased e (= increased small-scale fluid motion) in the turbulent
flow. Hondzo and Warnaars (2008) reported that the field estimates of e ranged from

m 2 /s 3 in Square Lake located at Washington County, Minnesota. The e