Environ. Sci. Technol. 1999, 33, 1318-1319
Response to ‘Comment on “Method for Calculating Bacterial Deposition Coefficients Using the Fraction of Bacteria Recovered from Laboratory Columns” ’ SIR: We appreciate the comment by Logan et al. (1) on our recent paper (2) and regret the omission of several important references noted by these authors. Logan et al. (1) essentially raise three concerns with our work: (1) the importance of dispersion in laboratory columns, (2) the validity of the analytical solution used in our paper, and (3) the usefulness of an assay that only measures the fraction of bacteria recovered from laboratory columns. Our response to each of these issues follows. (1) Logan et al. (1) state that the effects of dispersion in laboratory columns are negligible and therefore can be neglected. This was first asserted by Qi (3), and we agree that in most cases the effect of dispersion will be minimal. Logan et al. (1) suggest negligible effects for Pe . 1 since L/Pe1/2 will be ,1 but give no further quantitative analysis. A thorough analysis of the effects of the Peclet number on fractional recoveries, however, clearly shows that for low Peclet numbers and low fractional recoveries the effects of dispersion are not negligible (see eqs 10 and 11 as well as Figure 1 from our original work (2)). While Peclet numbers of greater than 100 are often observed for laboratory column experiments (2, 4, 5), in some instances the effect of dispersion on fractional recovery may be significant enough to require its inclusion when analyzing data from bacterial transport experiments. An example is the analyses of effluent or attached bacteria from extremely short columns. The coefficient of hydrodynamic dispersion (D) used to determine the Peclet number (Pe ) vL/D) is defined as
D ) RLv + D*
(1)
where RL is dispersivity and D* is the molecular diffusion coefficient. For a uniform grain size distribution dispersivity will mainly depend on grain size (6); therefore, in cases where molecular diffusion is negligible the Peclet number can be estimated by
Pe ≈
L dc
(2)
where L is the length of the column and dc is the grain diameter. For short columns and millimeter sized sand grains, Peclet numbers may be small enough to affect bacterial deposition. Simoni et al. (7) report values for Pe ranging from 8 to 15 in short columns varying in length from 0.3 to 6.4 cm. Studies that look at bacterial recovery or retention over short column lengths or in heterogeneous sediments may need to account for the Peclet number when calculating deposition coefficients. In fact Logan et al. (8) show that for cases with high enough dispersion, apparent sticking efficiencies, as measured with models that neglect dispersion, may decrease with column length, giving the appearance of a distribution of sticking efficiencies of the influent bacteria. Bolster et al. (9) report Peclet numbers of 8 and 45 for column lengths of 15 and 45 cm in conjunction with fractional recoveries of less than 0.001. It should be pointed out, however, that the interpretation of sticking efficiencies obtained in heteroge1318
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FIGURE 1. Effect of Peclet number (Pe) on the relationship between the dimensionless deposition coefficient and the fraction of bacteria recovered. Symbols with lines represent the relationship as calculated by eq 10 in Bolster et al. (2). Symbols alone are those calculated by eqs 5 and 6 in Logan et al. (1) (transformed to fluxaveraged concentrations) for unit values of M, x, v, A, and θ where fractional recovery was determined by numerically integrating under the calculated breakthrough curves. Time steps were chosen so that subsequent decreases in time steps did not yield different values of fractional recovery. Differences between infinite and semiinfinite solutions are noticeable for Peclet number of less than 50, but effects of dispersion are evident for both. neous sediments should be done with caution because collision efficiencies are calculated assuming homogeneous conditions. (2) Logan et al. (1) correctly point out that for cases where dispersion is high the proper choice of boundary conditions is essential. The authors imply that inappropriate boundary conditions were used in the solution of Parlange et al. (10) and therefore our findings that deposition, and hence recovery, is affected by dispersion for low Peclet numbers is invalid. The boundary conditions for the solution of Parlange et al. (10) are
c(0,t) ) c0
(3)
∂c (∞,t) ) 0 ∂x
(4)
Parker and van Genuchten (11) have shown that to conserve mass the proper inlet boundary condition for solving the advection-dispersion equation for a semi-infinite column is (for nonzero velocities)
c(0,t) ) c0(t) +
D ∂c(0,t) v ∂x
(5)
The advection-dispersion equation was derived from an Eulerian approach (12), and as a result the concentration, c, in the above equations is the volume-averaged, or resident, concentration, defined as the mass of solute (or colloids) per unit volume of fluid at a given instant in time. The fluxaveraged concentration is defined as the ratio of the solute (or colloid) flux to the volumetric fluid flux. Column effluent data should be treated as flux-averaged concentrations (11, 13). The flux and resident concentrations are related by
cf ) cr 10.1021/es992004d CCC: $18.00
D ∂cr v ∂x
(6)
1999 American Chemical Society Published on Web 03/10/1999
It is easily shown that the governing equation for fluxaveraged concentrations is the same as that for resident concentrations (11, 13). Insertion of eq 6 into eq 5 yields an inlet boundary condition of
c(0,t) ) c0
(7)
The solution of Parlange et al. (10) is for flux-averaged concentrations, appropriate for column effluent studies. We believe that the literature (10, 11, 14-21) clearly supports the use of solutions for semi-infinite columns when analyzing effluent data from laboratory columns, provided appropriate boundary conditions are used. The solution of Logan et al. (1) is for volume-averaged or resident concentrations for an infinite column (13). Even using the solution presented by Logan et al. (1) one cannot conclude that effects of dispersion are negligible for low Peclet numbers and low fractional recoveries. Solving eqs 5 and 6 of Logan et al. (1), with eq 6 (this paper) to obtain flux-averaged concentrations, reveals that fractional recovery is dependent on dispersion (Figure 1). The bottom boundary condition that we use does not affect our solution significantly. Calculations using a solution by Kreft and Zuber (13) with a concentration of zero at x ) ∞ (rather than eq 4) as a bottom boundary condition give results essentially indistinguishable from those we presented (data not shown). For Peclet numbers less than 4 the outlet boundary condition is not as well defined (10, 15), and Logan et al. rightly point out that the result for Pe ) 1 in our original figure may not be strictly valid. (3) Finally, our paper should not be read as a criticism of the MARK assay or column dissections in general. We believe column dissection is an excellent approach for obtaining insights into the processes controlling deposition. In fact we have recently reported the results of dissections of cores containing intact aquifer sediments (9) and strongly argue that column dissection is necessary to further our understanding of bacterial deposition. Logan et al. (1) correctly point out that measures of fractional recoveries contribute limited information on the processes involved in bacterial deposition. Nevertheless, important inferences can be obtained from experiments measuring fractional recoveries. For instance, fractional recoveries have been used to determine the role of bacterial population heterogeneities on bacterial transport through porous media (7). We do disagree with the statement of Logan et al. (1) concerning the ease to which column dissections can be performed. In collecting both effluent data and dissecting cores and enumerating deposited bacteria we find that analyzing effluent data is much less time consuming. It is true that the use of radiolabeled bacteria greatly reduces the time and effort needed to collect both effluent and sedimentassociated bacteria. However, those who may be most interested in such an assay (e.g., local water protection agencies or environmental consulting firms) may not have access to radiolabeling techniques. In these cases it is clear that the collection of effluent data will be much less time consuming. Additionally, in some cases quantifying the concentrations of sediment-associated bacteria may be less reliable than the enumeration of bacteria in the effluent. A drawback to column dissections is that all of the deposited bacteria on aquifer sediments are rarely recovered. Camesano and Logan (22) report recoveries of less than 100% of the deposited bacteria on an Arizona soil and cite the findings of Rogers (23) who could recover only ∼44% of the radiolabel from these soils. We have observed recoveries of between 80 and 95% on unconsolidated aquifer sediments (9). Because calculations of sticking efficiencies are dependent on the amount of bacteria removed from previous (upstream)
samples, incomplete recoveries may be a serious source of error, especially if cell removal efficiency is a function of the deposited bacteria concentrations. We conclude by stating that our original formulation was theoretically sound and therefore our findings valid; that is dispersion will affect bacterial recovery under the proper conditions (low Peclet number and/or low fractional recoveries). We agree with Logan et al. (1) that these conditions may not often be realized in the laboratory. We also believe that measuring fractional recoveries is an efficient way to collect a large amount of data on bacterial deposition that would be particularly useful in characterizing the spatial variability of bacterial deposition in the field.
Literature Cited (1) Logan, B. E.; Camesano, T. A.; DeSantis, A. A.; Unice, K. M.; Baygents, J. C. Environ. Sci. Technol. 1999, 33, 1316-1317. (2) Bolster, C. H.; Hornberger, G. M.; Mills, A. L.; Wilson, J. L. Environ. Sci. Technol. 1998, 32, 1329-1332. (3) Qi, S. J. Environ. Eng. 1997, 123, 729-730. (4) Hornberger, G. M.; Mills, A. L.; Herman, J. S. Water Resour. Res. 1992, 28, 915-938. (5) McCaulou, D. R.; Bales, R. C.; Arnold, R. G. Water Resour. Res. 1995, 31, 271-280. (6) Xu, M.; Eckstein, Y. Hydrogeol. J. 1997, 5, 4-20. (7) Simoni, S. F.; Harms, H.; Bosma, T. N. P.; Zehnder, A. J. B. Environ. Sci. Technol. 1998, 32, 2100-2105. (8) Logan, B. E.; Jewett, D. G.; Arnold, R. G.; Bouwer, E. J.; O’Melia, C. R. J. Environ. Eng. 1997, 123, 730-731. (9) Bolster, C. H.; Mills, A. L.; Hornberger, G. M.; Herman, J. S. Wate Resour. Res. in press. (10) Parlange, J.-Y.; Starr, J. L.; van Genuchten, M. T.; Barry, D. A.; Parker, J. C. Soil Sci. 1992, 153, 165-171. (11) Parker, J. C.; van Genuchten, M. T. Water Resour. Res. 1984, 20, 866-872. (12) Bear, J. Dynamics of fluids in porous media; American Elsevier: New York, 1972. (13) Kreft, A.; Zuber, A. Chem. Eng. Sci. 1978, 33, 1471-1480. (14) van Genuchten, M. T.; Parker, J. C. Soil Sci. Soc. Am. J. 1984, 48, 703-708. (15) van Genuchten, M. T.; Parker, J. C. Soil Sci. Soc. Am. J. 1985, 49, 1325-1326. (16) van Genuchten, M. T.; Parker, J. C. Soil Sci. Soc. Am. J. 1994, 58, 991-992. (17) Toride, N.; Leij, F. J.; van Genuchten, M. T. Water Resour. Res. 1993, 29, 2167-2182. (18) van Genuchten, M. T. J. Hydrol. 1981, 49, 213-233. (19) Jury, W. A.; Roth, K. Transfer functions and solute movement through soil; Birkhauser Verlag Basel: Basel, 1990. (20) Toride, N.; Leij, F. J.; van Genuchten, M. T. Report U.S. Salinity Laboratory, 1995. (21) Parker, J. C.; Valocchi, A. J. Water Resour. Res. 1986, 22, 399407. (22) Camesano, T. A.; Logan, B. E. Environ. Sci. Technol. 1998, 32, 1699-1708. (23) Rogers, B. M.S. Thesis, The University of Arizona, 1997.
Carl H. Bolster, George M. Hornberger, and Aaron L. Mills* Program of Interdisciplinary Research in Contaminant Hydrogeology, Department of Environmental Sciences University of Virginia Charlottesville, Virginia 22903
John L. Wilson Department of Earth and Environmental Science New Mexico Institute of Mining and Technology Socorro, New Mexico 87801 ES992004D VOL. 33, NO. 8, 1999 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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