Correspondence/Rebuttal pubs.acs.org/est
Response to Comment on “Extending Applicability of Correlation Equations to Predict Colloidal Retention in Porous Media at Low Fluid Velocity” relatively high fluid velocities (e.g., NPe > 70), to the authors’ knowledge no favorable-condition experiments have been published for very low fluid velocities (e.g., NPe < 70). Furthermore, the only low-velocity reference cited in Nelson et al.1 was the work by Nagasaki et al.,3 who studied colloid transport behaviors for three particle sizes but one velocity case (NPe < 10). Trends of η with respect to velocity and particle size at the low velocity regime under favorable conditions have not been experimentally established. The mass transfer analyses in Ma et al.2 showed that the Sherwood number (NSh) became independent of NPe at low velocity regime (e.g., NPe < 70), indicating that Brownian motion was predominantly responsible for colloid mass transfer to collector surfaces under these conditions, and that the unity asymptote represents a reasonable approximation of η under very low fluid velocity conditions for small particle sizes. Nelson et al. correctly point out that our modified correlation equation provides no explicit transition to the asymptote of η = 1 for particles < ∼40 nm, and also under-predicts η for these nanoparticles under conditions involving both very low fluid velocities and high porosities. As we acknowledged (Figure 4b and the last paragraph of section 3 in Ma et al.2), the underpredictions/discontinuities to the asymptotes were solely an artifact of lack of transition from the correlation equation to the asymptote, not from our mechanistic simulations. As we stated, the asymptote rather than the correlation equation should be used for particle sizes less than 40 nm under very low fluid velocity conditions. An assumption that the observed inverse relationship between η and velocity (and particle size) at high velocity applies also to very low velocity regime underlies Nelson et al.’s1 proposed eq 1. While we agree that this equation provides a clearly defined transition from the correlation equation to the asymptote, the assumed trend is not proven experimentally. As shown in Nelson et al.’s1 Figure 1b, the erf-based expression provides incremental improvement over the so-called MFJ, but both the MFJ and the erf-based function perform significantly better than the non-erf-corrected NG expression that was the subject of our comment. Nelson et al.1 strongly disagreed with our statement that existing correlation equations for favorable conditions basically agree with one another and with available experimental data from uniform granular media (within roughly a factor of 2), excepting the very low velocity condition where a “constraint” is needed on the correlation equations to avoid superunity η values. Figure 1c of Nelson et al.1 intends to illustrate the degree of agreement between example correlation equation predictions and 112 favorable-condition experiments gathered from the literature (see Table 3 and Figure 5 in Nelson and Ginn4). However, as we stressed in our paper,2 our goal was to
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e appreciate the opportunity to clarify issues raised by the comment of Nelson et al.1 on our paper2 describing the causes of and solutions to the nonphysical collector efficiency (η) values at low fluid velocities under favorable conditions. We address these issues in order from broadest to most specific. First, we stress that our paper specifically concerned favorable attachment conditions (i.e., absence of repulsive colloid-collector interactions), which is the condition under which all existing correlation equations for η have been developed, as stated in the first sentence of our paper.2 It is possible for colloid−colloid interactions to be either favorable or unfavorable under the condition of favorable colloid− collector interactions, and we made reference to possible blocking or ripening effects resulting from such. However, the issues explored in our paper corresponded to favorable colloid− collector interactions. Nelson et al.1 state agreement with our comment that application of existing η correlation equations at low fluid velocities under favorable conditions may be a largely hypothetical exercise because of expected violation of clean bed conditions. They go on to make the point that it is possible to match observed colloid retention under unfavorable low velocity conditions using calibrated α values combined with near-unity η values. However, while our comments addressed the application of η under favorable (not unfavorable) conditions, it should be noted that at the time of these comments, α remains an empirical parameter, since no mechanistically based correlation equation exists for prediction of colloid retention under unfavorable conditions. In fact, the “calibrated α values” of Nelson et al.1 were obtained via dividing the experimental η values (Nagasaki et al.3) by their theoretical η values (averaged over for different particle sizes),4 which of course would be expected to yield the observed retention under unfavorable conditions. Nelson and Ginn4 considered this a “successful” modeling of “accurate near-unity η prediction”. Had Nelson et al.1 not averaged the α value across particle sizes, and instead preserved the values for each particle size, the predicted retention from all of the examined correlation equations would have collapsed onto the experimental η trend. The variations among the different retention predictions were a direct consequence of the method used by Nelson and Ginn4 to determine α and cannot be expected to provide objective or useful information regarding the accuracy of the correlation equations examined by their method. Nelson et al.1 repeatedly raise the criticism that our modified η correlation equation2 violated an assumed inverse relationship of η with velocity for η < 1, as well as an assumed inverse relationship of η with particle size for low fluid velocity under favorable conditions. While it has been extensively demonstrated that these two inverse relationships may hold at © XXXX American Chemical Society
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dx.doi.org/10.1021/es4024942 | Environ. Sci. Technol. XXXX, XXX, XXX−XXX
Environmental Science & Technology
Correspondence/Rebuttal
provide insight regarding the source of superunity η predictions in existing correlation equations, as derived from either the correlation equations themselves versus the underlying numerical models. We did so in the context of a modified hemisphere-in-cell flow field, to produce what Nelson et al.1 referred to as the “MHJ” equation. We were explicit in NOT promoting our equation as a general improvement because (1) this was not our goal, as stated above, and (2) we anticipated the need to modify the flow field mesh and boundary conditions to address both high and low fluid velocities. Notably, overpredictions by the MHJ model correspond to the highest pore fluid velocities (on the order of 240 m/day or greater). Our analysis successfully described the contributions of correlation equations and underlying numerical models to the superunity predictions; and moreover we explicitly demonstrated that the assumed inverse power law relationship of η ∝ NPe−2/3 has no physical basis for low velocity conditions (e.g., NPe < 70). We are perplexed by Nelson et al.’s insistence on such a relationship (eq 1 in Nelson et al.1) without experimental validation. A few words should be said about the comparisons made in Nelson et al.1 Because the relationship between η and kf differs for different collector geometries, and the conversion used for the so-called MHJ equation was not described, it is likely that an incorrect η−kf conversion was used by Nelson et al.1 to represent the MHJ predictions (notably, this conversion would not be equivalent to that for the MPFJ equation because of the modified flow field). Likewise, experimental data such as that used in Figure 1c of Nelson et al.1 that were reported as η, must be converted back to kf with the η−kf relationship originally used for conversion of kf to η. Instead all such conversions were performed using a single assumed Happel relationship,4 thereby introducing additional error in the experimental kf values. Finally, our comment regarding the general agreement of correlation equations under favorable conditions reflected the fact that experiments carry their own errors that are at least on the order of a factor of 2 (e.g., Rajagopalan and Tien5). Our point was that predictions are generally reasonable under favorable conditions in uniform granular media, whereas, under unfavorable conditions and in nonuniform media,6 the predictions often miss the experimental values by orders of magnitude. On this basis we stand by our statement that predictions are generally reasonable under favorable conditions in uniform granular media.
(2) Ma, H.; Hradisky, M.; Johnson, W. P. Extending Applicability of Correlation Equations to Predict Colloidal Retention in Porous Media at Low Fluid Velocity. Environ. Sci. Technol. 2013, 47 (5), 2272−2278. (3) Nagasaki, S.; Tanaka, S.; Suzuki, A. Fast Transport of Colloidal Particles through Quartz-Packed Columns. J. Nucl. Sci. Technol. 1993, 30 (11), 1136−1144. (4) Nelson, K. E.; Ginn, T. R. New Collector Efficiency Equation for Colloid Filtration in Both Natural and Engineered Flow Conditions. Water Resour. Res. 2011, 47 (5), W05543 DOI: 10.1029/ 2010WR009587. (5) Rajagopalan, R.; Tien, C.; Tufenkji, N.; Elimelech, M. Comment on “Correlation Equation for Predicting Single-Collector Efficiency in Physicochemical Filtration in Saturated Porous Media” [1] (multiple letters). Environ. Sci. Technol. 2005, 39 (14), 5494−5497. (6) Pazmino, E. F.; Ma, H.; Johnson, W. P. Applicability of Colloid Filtration Theory in Size-Distributed, Reduced Porosity, Granular Media in the Absence of Energy Barriers. Environ. Sci. Technol. 2011, 45 (24), 10401−10407, DOI: 10.1021/es202203m.
Huilian Ma† Michal Hradisky‡ William P. Johnson*,†
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† Geology and Geophysics Department and ‡Chemical Engineering Department, University of Utah, Salt Lake City, Utah 84112, United States
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Nelson, K. E.; Ginn, T. R.; Kamai, T. Comment on “Extending Applicability of Correlation Equations to Predict Colloidal Retention in Porous Media at Low Fluid Velocity. Environ. Sci. Technol. 2013, DOI: 10.1021/es401944q. B
dx.doi.org/10.1021/es4024942 | Environ. Sci. Technol. XXXX, XXX, XXX−XXX
