pubs.acs.org/Langmuir © 2009 American Chemical Society
Comment on Particle Tracking Model for Colloid Transport near Planar Surfaces Covered with Spherical Asperities
The approach presented by Kemps and Bhattacharjee (2009) for considering the deposition of colloidal particles onto spherical asperities covering a planar surface in the presence of shear flow indicates some important aspects about how surface roughness may impact particle hydrodynamics and deposition. However, clarification is required regarding their claim that omitting the effect of hydrodynamic retardation on the diffusion coefficient would serve to decrease deposition. This is incorrect. A positive correlation between diffusion and deposition is well established. Our points of clarification below stem from both a priori theoretical considerations and recent work on our particle tracking model that shares some similarities with that of Kemps and Bhattacharjee. In the Validation of Approach section, the authors present results obtained from their method applied to the determination of the single-collector collection efficiency (η) of colloid filtration theory, with comparison to the simulation results of Nelson and Ginn,1 as well as the correlation equations of Rajagopalan and Tien2 and Tufenkji and Elimelech.3 Figure 3 in Kemps and Bhattacharjee shows markedly lower collection efficiencies for Brownian particles computed in ref 1 than those calculated by all others. Kemps and Bhattacharjee provide the explanation that the reason for the difference between the results in ref 1 and their own results is that the effects of hydrodynamic retardation on particle diffusivity are not considered in ref 1. The authors suggest that the hydrodynamic retardation of diffusivity would lead to more frequent collisions and larger η by stating the following complement: “This absence of hydrodynamic interactions would make it more difficult for each particle to contact the collector’s surface since the particle’s Brownian displacement at each time step would not be reduced as the particle approaches the collector surface.” However, this explanation is contrary to the actual effects of diffusion on particle collision with surfaces in both physical and mathematical contexts. Physically, a larger diffusivity leads to a larger effective mean free path and on average to more, not less, frequent collisions with a fixed boundary. Mathematically, it is established that greater diffusivity results in greater deposition as evidenced by the negative exponent on the Peclet number in all common correlations (and analytical solutions) for the collector efficiency, including those in refs 2 and 3. The incorporation of hydrodynamic retardation in the diffusion coefficient would then be expected to decrease diffusivity and thus deposition. There are a number of other reasons that could explain the discrepancy shown in Kemps and Bhattacharjee’s Figure 3. First, the computational domain of Kemps and Bhattacharjee is the unit cell surrounding a spherical asperity protruding from a planar surface, which is significantly different from the Happel model computational domain (used in all of the computational results used for comparison) in which the spherical collector is surrounded by a concentric liquid envelope. Did the authors remove the planar surface for this set of simulations and revise the (1) Nelson, K. E.; Ginn, T. R. Langmuir 2005, 21, 2173–2184. (2) Rajagopalan, R.; Tien, C. AIChE J. 1976, 22, 523–533. (3) Tufenkji, N.; Elimelech, M. Environ. Sci. Technol. 2004, 38, 529–536.
Langmuir 2009, 25(21), 12835–12836
particle tracking for strictly Happel sphere velocity conditions? If not, it would seem that the presence of the planar surface, through its effect of reducing the velocities from the original Stokes’ flow around the Happel sphere, would serve to increase depositions and, thus, the difference in results with those in ref 1. Also, if the authors’ planar surface remains, it is unclear how uniform particle start locations are established over the whole spherical collector. Second, we note that the authors cite a definition of the collector efficiency as the ratio of the flux of particles onto the collector to the flux of particles in the projected area of the collector upstream; this can be referred to as the isolated sphere definition. This is only one of two possible definitions, and in ref 1 it is noted that for appropriate comparison to their results it is necessary to use the definition that is correct for the Happel model (i.e., the ratio of flux onto the collector to flux in the projected area of the entire Happel sphere (collector plus liquid envelope)). By using the first definition for the results of Kemps and Bhattacharjee, as well as those of refs 2 and 3, the difference with the results of ref 1 are exaggerated. It is also worthwhile to note that the Happel model definition of η is used in the original paper2 developing the Rajagopalan and Tien equation for η. However, this equation is commonly applied in the form based on the incorrect isolated sphere definition; see also refs 4 and 5. Using the isolated sphere definition for deposition calculations based on the Happel flow field results in a nonphysical definition that overestimates η, including values greater than unity under highdeposition conditions. Thus, use of the isolated sphere definition in Kemps and Bhattacharjee’s Figure 3 for all results except those in ref 1 clearly accounts for some of the observed discrepancies. Third, we have recently updated our own Lagrangian method6 and in the process have discovered that our approximations for hydrodynamic retardation overestimated the effect at large separation distances. After correcting this, the discrepancy with refs 2 and 3 for Brownian particles diminished. Additionally, we have incorporated the effect of hydrodynamic retardation into the diffusion coefficient, and as expected, this results in lowered deposition and smaller values of η.6 Figure 1 shows the results of our simulations with the corrected hydrodynamic retardation factors on both the deterministic motion alone and deterministic plus Brownian motion, along with the results in refs 1-3 (all plotted using the Happel definition of η). This clearly shows that some of the observed discrepancies in Kemps and Bhattacharjee’s Figure 3 are due to the errors in the hydrodynamic retardation factors (used on the deterministic component of motion) in ref 1. We also address Kemps and Bhattacharjee’s neglect of gravity, which was noted to be a possible means of increasing deposition when the gravitational force acts in the opposite direction to forces leading to deposition. For the particular case of downward flow that is adopted in all of the studies1-3 used by Kemps and Bhattacharjee for comparison, the inclusion of gravity will result in more deposition on the upper half of the collector and less (4) Logan, B. E.; Jewett, D. G.; Arnold, R. G.; Bouwer, E. J.; O’Melia, C. R. J. Environ. Eng. 1995, 121, 869–873. (5) Rajagopalan, R.; Tien, C.; Pfeffer, R.; Tardos, G. AIChE J. 1982, 28, 871– 872. (6) Nelson, , K. E.; Ginn, , T. R. Environ. Sci. Technol., submitted for publication. (7) Nelson, K. E.; Massoudieh, A; Ginn, T. R. Adv. Water Resour. 2007, 30, 1492–1504.
Published on Web 08/27/2009
DOI: 10.1021/la902013w
12835
Comment
Nelson and Ginn
Figure 1
deposition on the lower half. On the basis of Figure 14 in ref 7, which gives the attachment distribution of Brownian colloids on the Happel collector, we do not believe that the decreased deposition on the lower half would be greater than the increased deposition on the upper half. Thus, we do not believe that Kemps and Bhattacharjee’s neglect of gravity accounts for any of the observed discrepancy with ref 1. In summary, this comment points out that the explanation given by Kemps and Bhattacharjee to account for the discrepancies in their Figure 3 can be ruled out a priori because increased diffusion is always known to increase deposition on a fixed boundary. Two other explanations that certainly account for some of the observed discrepancies are the following: (1) use of the isolated sphere definition for three of the η results with the Happel definition used for the fourth and (2) an error in the hydrodynamic retardation factors used on the deterministic velocities in
12836 DOI: 10.1021/la902013w
ref 1. It is also possible that inconsistencies in how the Happel sphere is used in conjunction with a planar surface by Kemps and Bhattacharjee result in additional discrepancies. A final note is that Kemps and Bhattacharjee state that all data used for comparison with their results were from correlations that smoothed out the scatter observed in their Lagrangian simulations. This is true for ref 2 and 3 but not ref 1, which also reported the results of Lagrangian simulations. Kirk E. Nelson*,†,‡ and Timothy R. Ginn† Department of Civil & Environmental Engineering, University of California, Davis, California, and ‡U.S. Bureau of Reclamation, Sacramento, California *Corresponding author. E-mail:
[email protected]. †
Received June 4, 2009
Langmuir 2009, 25(21), 12835–12836
