Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 480

defines the thickness of the diffusive sub-layer around individual algal cells and,
consequently, determines NMs or nutrient uptake in a moving fluid (Maltis et al., 2004;
Hondzo and Al-Homoud, 2007).

15.2.1.3 Mass Balance Equations in Porous Media

For NMs present in porous media, the mass balance relationship, that is
independent of the details of the phases that are present, is as follows:

M T = M v + M N + M w + M s + MO = V V C V + M N + V W C W + sp b V Soi l + V O C O (Eq. 15.12)
where MT = total mass of NMs, M; My to Mo = mass of NMs in the soil gas phase,
NAPL (nonaqueous phase liquid) phase, the moisture (water phase), adsorbed in the soil,
and the other phase (e.g., grease), respectively, M; Vy, Vw, Vo = the respective volume
of gas, water, and the other phase in the soil, L 3 of gas, water, and other phase,
respectively; Cy, Cw, Co = the respective NMs' concentration in gas, water, and the
other phase, M/L 3 of gas, water, and the other phase, respectively; s = mass of NMs
sorbed per unit dry mass of soil, unitless; pb = the dry bulk density of the soil, M/L 3 of
the soil; and Vsoii = the volume of the soil, L 3 . Eq. 15.12 is useful to determine the mass
of the NMs distributed in different phases.

The generalized one-dimensional transport equation describing NMs transport in
porous media at non-steady-state conditions can be

+ u|=»LS + S^ (Eq. 15.13)

where c = NMs concentration, M/L 3 ; u = the actual average pore velocity of the water,

L/T; DL = a longitudinal dispersion coefficient, L 2 /T; and Rj = the 1 t h transfer/
transformation process that the NMs are involved in. Notice that the Darcy velocity, U
(also called a superficial velocity or empty-bed velocity) is related to u by soil porosity

U = u6 (Eq. 15.14)

The complexity of eq. 15.13 depends on the definition of the term Rj and transport
conditions that the NMs are involved in. For example, if deposition is the only transfer
process involved by NPs, and the deposition rate rdeposMon = - kc (with k = the l st -order
deposition rate coefficient, 1/T), eq. 15.13 becomes the same as eq. 15.5; the only
differences are that c, v and DL may be defined with respect to the porous media.
Consequently, under the BCs of eq. 15.6, the solution of NPs transport in the porous
media (eq. 15.13) is the same as eqs. 15.7 and 15.8.