Correspondence/Rebuttal pubs.acs.org/est
Comment on “Extending Applicability of Correlation Equations to Predict Colloidal Retention in Porous Media at Low Fluid Velocity” ecently, Ma et al.1 sought “to extend the correction offered by Nelson and Ginn”2 for the nonphysical collector efficiency (η) values given by correlation equations (including theirs, the MPFJ equation3,4) of the colloid filtration theory (CFT) at low fluid velocities. Ma et al.1 demonstrate that the NG equation2 still yields η > 1 when small values of fluid velocity (∼10−7 m/s), porosity (∼0.25), and colloid diameter (dp ≤ 100 nm) exist concurrently, and address this by adopting the NG equation with a modified diffusion term (referred to herein as the MHJ equation) but stating it is “based on regression to mechanistic simulation results”. In this comment, we discuss important issues regarding equation comparisons, address the claim that low fluid velocity applications of CFT are “largely hypothetical”, and present a mathematical constraint that preserves the physics lost by the MHJ equation. The MHJ equation is presented as “an improvement for predicting η under a wider variety of fluid velocities” than prior correlation equations including the NG equation. However, Ma et al. recognize that their approach results in an incorrect dependence on velocity as η approaches unity and suggest correcting this via “asymptotes at the diffusion limit”; the transition point is never defined, though, leaving the suggested approach prone to ambiguous interpretation. Moreover, since the asymptote is η = 1 and the error of the MHJ equation is that it misrepresents the inverse relationship of η with velocity for η < 1, use of the asymptote will yield even larger errors. Not acknowledged by the authors is that the MHJ equation can also fail to capture the inverse relationship with colloid size (Figure 1a). Thus, the strategy employed to constrain the MHJ equation sacrifices key aspects of the physics of colloid deposition at low fluid velocities. The benefit of never exceeding unity must be weighed against these deficiencies to evaluate the claim that this is an “improvement”. Regarding inapplicability of the NG equation to “certain parametric conditions”, we note that η equations are commonly used to elucidate trends with respect to input parameters. Before we addressed the application of CFT at low fluid velocities,2 prior equations employed a power law dependence on the Peclet number (NPE) without any limiting factors. This results in high sensitivity with respect to NPE that gives extremely large errors for nanoparticles at low fluid velocities, that is, prior equations yield η values dramatically above unity with errors increasing as NPE decreases. In contrast, when the NG equation exceeds unity, the errors are relatively minor as the limiting factor moderates the error. Even when unity is exceeded slightly, the physical trends are maintained and, thus, the equation is qualitatively correct (whereas the MHJ equation is qualitatively incorrect in its reversing the trends with respect to velocity and colloid size). The benefits of constraining the MHJ equation to stay below unity are not worth the cost of losing salient aspects of the physics. Moreover, this trade-off is unnecessary as mathematical means exist to achieve the desired constraint without compromising the physics (as shown below).
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Figure 1. (a) NG equation, NG constrained equation, MHJ equation, and MHJ asymptote (dashed red line) applied to data of Nagasaki et al.6(MPFJ equation values range from 1.9 to 9.4 and are not viewable). Note that the MHJ equation reverses the dependence on colloid size and use of the MHJ asymptote increases error magnitudes of subunity η values (b) Comparison of NG equation, NG constrained equation, MPFJ equation, and MHJ equation for low Darcy velocity (U = 0.04 m/d) and low porosity (0.25) (c) NG equation, NG constrained equation, MPFJ equation, and MHJ equation compared to data set2 of 112 experiments. The first-order deposition rate coefficient (kf) is computed based on each equation’s η value. Experiments 1−15 are nanoparticles (dp ≤ 100 nm); experiments 16−48 are larger submicrometer particles (100 nm < dp < 1 μm); experiments 49− 112 are large colloids (1 μm < dp ≤ 10.1 μm). The factor-of-two level agreement is bracketed by the dashed black lines at kf model/kf experiment = 0.5 and kf model/kf experiment = 2; perfect agreement (ratio = 1) is denoted by the solid black line.
Regarding Ma et al.’s statements implying that all available η equations agree well with each other and with available data, this is demonstrated to be false in our prior comparison of available equations with 112 experiments,2 and disparities are much greater with respect to the new MHJ equation (Figure 1c). The MPFJ equation exceeded the factor-of-two level agreement 39 out of 112 times (versus 10 for the LH equation, 17 for both NG and TE, and 34 for RT). The MHJ equation exceeds this threshold 43 times. The MHJ equation performs particularly poorly for nanoparticles (dp ≤ 100 nm). The factorof-two threshold is exceeded for all nanoparticles in the data set2 with an average difference of a factor of 4.3 (compared to 1.6 for NG) greater than the experimental value and a maximum of 6.7 (compared to 2.3 for NG). For other submicrometer particles (100 nm < dp ≤ 1 μm), the MHJ equation also fares worse than all prior equations (average
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dx.doi.org/10.1021/es401944q | Environ. Sci. Technol. XXXX, XXX, XXX−XXX
Environmental Science & Technology
Correspondence/Rebuttal
(3) Ma, H.; Pedel, J.; Fife, P.; Johnson, W. P. Hemispheres-in-Cell Geometry to Predict Colloid Deposition in Porous Media. Environ. Sci. Technol. 2009, 43 (22), 8573−8579, DOI: 10.1021/es901242b. (4) Ma, H.; Pedel, J.; Fife, P.; Johnson, W. P. Hemispheres-in-Cell Geometry to Predict Colloid Deposition in Porous Media (Correction). Environ. Sci. Technol. 2010, 44 (11), 4383 DOI: 10.1021/es1009373. (5) Tufenkji, N.; Elimelech, M. Correlation Equation for Predicting Single-Collector Efficiency in Physicochemical Filtration in Saturated Porous Media. Environ. Sci. Technol. 2004, 38 (2), 529−536, DOI: 10.1021/es034049r. (6) Nagasaki, S.; Tanaka, S.; Suzuki, A. Fast Transport of Colloidal Particles through Quartz-Packed Columns. J. Nucl. Sci. Technol. 1993, 30 (11), 1136−1144, DOI: 10.3327/jnst.30.1136.
difference = factor of 2.5, maximum = 5.5). The poor performance of the MHJ equation for small colloids appears to be attributable to both the modified flow field and the reduced power on NPE in the denominator of (S7). However, by adopting our sedimentation term,2 the MHJ equation performs better relative to MPFJ for larger colloids. Even though Ma et al.1 call their constraint an improvement, they assert that application of η equations at low fluid velocities is “largely a hypothetical application due to expected violation of clean bed conditions”. While this may be the case for favorable conditions, this is not so for unfavorable conditions. Ma et al.1 state that under unfavorable conditions CFT will be inapplicable “possibly due to blocking”. However, not all electrostatically repulsive (unfavorable) conditions possess a secondary minimum. Absent the secondary minimum, we know of no mechanism that would lead to blocking. In general, biocolloids possess lower Hamaker constants than inorganic colloids,5 so Ma et al.’s supposition that near-unity η predictions are of no consequence is particularly dubious for biocolloids. It is not demonstrated for inorganic colloids either: column experiments under unfavorable conditions and low fluid velocity have exhibited low retention 6 and were successfully modeled by accurate near-unity η predictions and calibrated α values.2 Finally, the following constraint achieves the desired objective without sacrificing any physics ηconstrained = [erf(η29)]1/29
(1)
where erf is the error function and η is the η value to be corrected. The exponents are chosen such that discrepancies with the original η value remain
