Environ. Sci. Technol. 2005, 39, 5496-5497
Response to Comment on “Correlation Equation for Predicting Single-Collector Efficiency in Physicochemical Filtration in Saturated Porous Media” We thank Rajagopalan and Tien (1) for their interest in our paper describing the development of a new closed-form equation for the single-collector contact efficiency (η0) in physicochemical filtration (2). Here, we address the specific points raised and re-iterate the main contribution of our correlation equation. We believe that the points raised result from several misinterpretations and misunderstandings of our work. Main Contribution of the Tufenkji and Elimelech (TE; 2) Equation. We agree with Rajagopolan and Tien (1) that the primary objective of a correlation equation is to capture the essential physics of the mechanisms governing particle deposition. In effect, this was the greatest motivation for the development of our equation. The TE equation was not derived to improve the “level of accuracy” of previous approaches as suggested in ref 1 but to include all physicochemical mechanisms governing the single-collector contact efficiency. The comment by Rajagopalan and Tien (1) did not recognize this central point. In developing the correlation equation for the individual contribution by diffusion (ηD), we considered the influence of hydrodynamic interactions and van der Waals forces. The effect of these important interactions is reflected in the dimensionless parameters NR and NvdW in the first term of our correlation equation, with all parameters defined in (2): -0.081 -0.715 0.052 η0 ) 2.4A1/3 NPe NvdW + 0.55ASN1.675 N0.125 + S NR R A
0.053 0.22N-0.24 N1.11 R G NvdW (1)
In previous approaches, such as the commonly used Rajagopalan and Tien (RT) equation (3), the influence of hydrodynamic interactions and van der Waals forces on deposition by diffusion is not considered. Instead, the deposition of particles dominated by Brownian diffusion is described by the simplified Smoluchowski-Levich approximation (4, 5), which consists only of the dimensionless parameter NPe. Excluding the influence of hydrodynamic interactions and van der Waals forces on the deposition of particles that are dominated by Brownian diffusion leads to significant overestimation of η0. Removal by the diffusion mechanism is important for particles as large as a few micrometers for filtration at low flow rates (or low Peclet numbers), such as in groundwater aquifers. We illustrated these important points in Figures 2 and 3 in ref 2, where the RT equation overestimates η0 by as much as 60% for particles with a diameter of about 2 µm. Other Significant Improvements in the TE Equation. Additional disparities between the TE and RT equations are observed in the last term in eq 1 (i.e., deposition by gravity). First, we note that the porosity-dependent parameter (AS) is not included in this term, whereas it is present in the RT equation for the gravitational deposition mechanism. The porosity-dependent parameter (AS) should not be included. Rigorous numerical solution of the convective-diffusion equation for different porosities showed that deposition by gravity barely changes with porosity (2). The other significant difference is the presence of the van der Waals number (NvdW) 5496
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in our gravitational deposition mechanism, whereas it is absent in other models, such as the RT equation. Inherently, the universal van der Waals interaction should be included in all three transport mechanisms. Accuracy of Experimental Data. We agree that there are limitations in using theoretical models to predict particle transport and deposition in porous media. We disagree, however, that there is no scientific value in improving our ability to predict particle transport and deposition in these environments. It also does not make sense to present a correlation equation that is not in good agreement with rigorous numerical calculations of the complete deposition model with all physicochemical mechanisms fully incorporated. We emphasize that the purpose of our published work (2) was not only to present a correlation equation for the sake of obtaining more accurate predictions of η0; rather, our equation provides significant physical insights into the problem of particle deposition which cannot be observed from earlier models (3, 4). Tufenkji-Elimelech Results. In comparing predictions based on our correlation equation for η0 with calculations using other models, we ensured that the forms of the latter equations were defined on a similar basis as described in Chapter 12, Section 12.2.2., of Elimelech et al. (5). In Figure 4 of Tufenkji and Elimelech (2), calculations of η0 based on each model are compared to previously published experimental data. These data were obtained from well-controlled column experiments under favorable conditions for deposition, using uniform spherical particles and collectors. We used only data where it was clear that the conditions are favorable for deposition, namely, very high ionic strength and oppositely charged particles and collectors, where repulsive electrostatic interactions are absent. Rajagopolan and Tien (1) make a good observation that some of the data points in the YHO plot (Figure 4a) seem to be missing. This is due to the fact that they are off the maximum abscissa scale (i.e., the deviation between ηexp and ηYHO is considerable). In all three plots in Figure 4, the same data points were included, and the regression analyses were conducted by including every data point. Absence of Double-Layer Forces. The general approach for predicting the single-collector contact efficiency (η0) is to solve the convective-diffusion equation with all relevant interactions and forces considered simultaneously, with the exception of electrostatic double-layer forces. In fact, this is the approach taken by Rajagopolan and Tien in developing their correlation equation within the framework of trajectory analysis (3) and, earlier, by Yao et al. (4). Obviously, in most aquatic systems, repulsive double-layer interactions influence particle deposition. Within the context of filtration theory (4), the effect of double-layer interactions is accounted for by introducing the attachment (sticking) efficiency (R) as described below. In studying filtration in aqueous systems, the actual singlecollector removal efficiency is given as η ) R × η0. Here, the single-collector contact efficiency (η0) describes the rate of collisions between particles and filter grains and the attachment efficiency (R) accounts for the fraction of collisions that actually result in attachment. The parameter η0 describes the physics of the problem while R describes the effects of chemistry, since the fraction of successful collisions is controlled by solution chemistry and the chemical characteristics of the particles and grains. The reason for separating the chemistry and physics of the problem is the breakdown of current theories to predict the actual deposition rate when considering electrostatic double-layer effects (5). In this case, 10.1021/es050810g CCC: $30.25
2005 American Chemical Society Published on Web 06/07/2005
the theory predicts deposition rates that are several orders of magnitude smaller than the actual deposition rate (5).
(5) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. A. Particle Deposition and Aggregation: Measurement, Modelling, and Simulation; Butterworth-Heinemann: Oxford, England, 1995.
Literature Cited (1) Rajagopalan, R.; Tien, C. Comment on “Correlation equation for predicting single-collector efficiency in physicochemical filtration in saturated porous media”. Environ. Sci. Technol. 2005, 39, 5494-5495. (2) Tufenkji, N.; Elimelech, M. Correlation equation for predicting single-collector efficiency in physicochemical filtration in saturated porous media. Environ. Sci. Technol. 2004, 38, 529536. (3) Rajagopalan, R.; Tien, C. Trajectory analysis of deep-bed filtration with sphere-in-cell porous-media model. AIChE J. 1976, 22, 523-533. (4) Yao, K. M.; Habibian, M. T.; O’Melia, C. R. Water and wastewater filtrationsconcepts and applications. Environ. Sci. Technol. 1971, 5, 1105-1112.
Nathalie Tufenkji Department of Chemical Engineering McGill University Montreal, Quebec H3A 2B2, Canada
Menachem Elimelech* Department of Chemical Engineering Environmental Engineering Program Yale University New Haven, Connecticut 06520 ES050810G
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