Fate and Transport of Nanomaterials in Aquatic Environments - Nanotechnologies for Water Environment Applications - page 494

covering the entire range of microbial particles of environmental relevance (Tufenkji
and Elimelech, 2004). Tufenkji and Elimelech (2004) obtained a correlation equation for
predicting the overall collector efficiency (T|) by regressing the dimensionless
parameters governing particle deposition against the theoretical value of the single-
collector efficiency over a broad range of parameter values and incorporating rigorous
numerical solution of the convective-diffusion equation with hydrodynamic interactions
and universal van der Waals attractive forces. The predictions of T| based on the new
correlation equation are in much better agreement with the experimental data than those
based on the Yao, YH and RT equations. However, none of these models were
developed for NMs, whose stability is governed, at least, by (a) the balance of various
interactions (e.g., van der Waals attraction, double layer repulsion, solvation forces, and
steric interaction) and (b) hydrodynamic factors (e.g., flow velocity, particle size,
collector size, and grain geometry).
Second, these models are all derived under the steady-state conditions,
neglecting (a) diffusion into and out of the control volume, (b) changes in A, as a function
of filtration time and the location in the filter, and (c) changes in hydrodynamic
variability, grain dimensions (or bed porosity) as particles accumulate. To address these
issues, more complicated models (e.g., phenomenological depth filtration models, filter
ripening models) or models at non-steady-state conditions need to be evaluated. For
example, Wang et al. (2008a) and Li et al. (2008) developed a mathematical model by
incorporating non-equilibrium attachment kinetics [i.e., SSLjRij = (p b /6)(3S/3t) = k^C,
see eq. 15.13, where, C = the concentration of nCeo in aqueous phase, S = the
concentration of attached or solid-phase nCeo, t = time, pb = the soil bulk density, 6 = the
volumetric water content, ki = a first-order rate constant of particle attachment, \\i = a
blocking function] with a particle maximum retention capacity S ma x [4> = (S ma x - S)/S max ]
to describe the deposition process of nanoscale fullerene aggregate (nCeo) in water-
saturated porous media (Ottawa sand). The value of \\i decreases as more nCeo
aggregates are deposited on the grain surface, resulting in a reduction of the effective
attachment rate (ki\|/, which approaches a value of zero as S approaches S max ).
Essentially, ki controls the time required for nCeo to appear in the column effluent, while
Smax controls the shape and location of the retention profile (Li et al., 2008). The work
(Li et al., 2008) has improved the clean-bed filtration model, but further research is
needed because S ma x is a function of flow velocity, NP size, and mean grain size of the
Despite aforementioned limitations, the clean-bed filter theory has been widely
used currently for evaluating NMs attachment efficiency and mobility in porous media.
As shown in Table 15.5, the sticking coefficient was estimated by different researchers
by interpreting C/Co data for natural and engineered NPs. Table 15.5 shows that the
index of NP mobility (distance for a 3-log reduction in NP number concentration) is
different for different NPs. The indices of n-Ceo (0.1 m) and oxides are much smaller