Environ. Sci. Technol. 2005, 39, 6337-6342
Integral Water Treatment Plant Modeling: Improvements for Particle Processes† DESMOND F. LAWLER* AND JEFFREY A. NASON Environmental and Water Resources Engineering, Department of Civil Engineering, The University of Texas at Austin, 1 University Station, Mail Stop C1786, Austin, Texas 78712
An update of research on particle behavior in water treatment plants first performed 25 years ago under the direction of Charles O’Melia is provided. The earlier work involved mathematical modeling of the changes in particle size distributions in the flocculation and sedimentation processes in water treatment plants. The current model includes corrections for short-range interactions between particles as they approach one another. These corrections severely reduce the expected collision frequency between particles that are very different in size and, therefore, substantially change the model predictions. Both experimental and field measurements of particle size distributions are provided; such measurements were unavailable in the earlier work and represent a touchstone to reality for the modeling efforts. The short-range model successfully fits experimental results for flocculation when the mechanism of particle destabilization is charge neutralization. However, the model does not account for the creation of new solids by precipitation either when hydrolyzing salts of aluminum or iron are added for particle destabilization by “sweep floc” destabilization or lime is added to remove calcium and magnesium as calcium carbonate and magnesium hydroxide in softening. The flocculent sedimentation model yields results that are in strong qualitative agreement with typical field measurements.
Introduction Surface water treatment plants are designed to accomplish multiple objectives: the removal of particles, including microorganisms, originally present in the source water; the oxidation of some reduced species; the disinfection of microorganisms; and, in recent years, the removal (usually by adsorption onto precipitates of metal hydroxides or calcium carbonate) of natural organic matter (NOM). Although membrane processes are often considered for new water treatment plants, most plants maintain a traditional layout, consisting of a rapid mix tank (in which several chemicals might be added and mixed with the water), flocculation, sedimentation, and granular media filtration. Even when membranes are used, pretreatment with a traditional layout is often needed. Despite the multiple objectives, the overall design (i.e., sizing of units, flow patterns through the units) of such traditional plants is primarily dictated by the requirements of particle removal. †
This paper is part of the Charles O’Melia tribute issue. * Corresponding author phone: (512)471-4595; fax: (512)471-5870; e-mail:
[email protected]. 10.1021/es050089e CCC: $30.25 Published on Web 05/06/2005
2005 American Chemical Society
Twenty-five years ago, one author of this paper participated with the honoree, Professor Charles O’Melia, in an investigation of the design of particle processes in water treatment plants (1). That research involved mathematical modeling of the three main particle treatment processes (flocculation, sedimentation, and granular media filtration); the expected overall behavior of a water treatment plant was investigated by using those models in series. The flocculent sedimentation model was subsequently applied by O’Melia and co-workers (2-4) to investigate particle behavior in natural systems. In the interim, much research has been done to improve on that original work; the objective of this paper is to present the progress of the modeling of particle behavior in water treatment plants. Stated another way, the objective is the same as in the original paper. The focus in this paper is on flocculation and sedimentation.
Methods The mathematical model of flocculation is a numerical integration of the Smoluchowski equation (5), which can be described in modern terms as follows:
1 rk ) Remp (Rijβij)Totninj - Rempnk (Rikβik)Totni 2 all i and j all i
∑
∑
such that Vp,i + Vp,j ) Vp,k
(1)
where rk is the rate of change with time of the number concentration of particles of size k (with dimensions L-3 T-1); i, j, and k are size classes of particles; ni is the number concentration of particles of size i (L-3); Vp,i is the volume of particles of size i (L3); βij is the collision frequency function for each of three long-range transport processes (Brownian motion, fluid shear, and differential sedimentation) (L3 T-1); Rij is a dimensionless function accounting for short-range particle-particle interactions and is cast as a correction factor for the long-range interactions; and Remp is the empirical, dimensionless correction factor used to match experimental data to the equation. The expression (Rijβij)Tot is the sum of the Rijβij terms for each of the three collision mechanisms. The stipulation Vp,i + Vp,j ) Vp,k for the summation in the first term on the right in eq 1 reflects the assumption that volume is conserved during collisions. In water treatment plants, sedimentation usually follows flocculation; the particles continue to flocculate as they settle, so the mathematical description of the sedimentation process includes flocculation:
1 rk,m ) Remp (Rijβij)Totni,mnj,m 2 all i and j
∑
such that Vp,i + Vp,j ) Vp,k
Rempnk,m
∑(R all i
ikβik)Totni,m
+ wk
nk,m-1 zm
- wk
nk,m zm
(2)
where the vertical dimension is divided into several boxes or compartments and the subscript m refers to a particular vertical compartment, z is the depth of a box, and w is the settling velocity of a particle. Both the original work and that reported herein considered a “standard” plant and influent water characteristics. In all cases reported herein, the water temperature was assumed to be 25 °C; colder temperatures would lead to changes in VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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viscosity and density that would alter the results, but not drastically. The influent particle size distribution was characterized by a power law distribution (ni/∆dp,i ) Adp,i-β), where dp,i is the diameter of particles in class i, and ∆dp,i is the associated size increment. With the exponent β ) 4 over the size range 0.3 < dp < 30 µm, A chosen such that the total particle volume concentration was 50 ppmv, and a particle density of 2.65 g/cm3, the suspended solids concentration was 132 mg/L. This suspension might be realistic for the raw water from a turbid river source or for a softening plant (after the precipitation of CaCO3); however, the density and the concentration are both rather high for most water treatment plants using cleaner sources and alum or iron coagulation. Modeling a suspension with lower particle density and lower concentration would lead to similar but less dramatic changes in the size distributions than shown in this paper, primarily because flocculation is a second-order reaction in particle number concentration as shown in eq 1. The standard flocculation tank had plug flow, a detention time of 1 h, and a velocity gradient of 10 s-1. The sedimentation facilities had a detention time of 2 h and a depth of 2 m, yielding an overflow rate of 1 m/h; the flow was again considered plug flow. With the assumption of plug flow in both flocculation and sedimentation, the modeling work consisted of a numerical integration of eqs 1 and 2, with rk ) dnk/dt. The primary difference in the mathematical modeling between the earlier work (referred to herein as the longrange model) and the new research (referred to as the shortrange model) is the inclusion of the correction factors (Rij) for short-range interactions. These correction factors (6, 7) account for the hydrodynamic interactions and London van der Waals attraction (between perfectly destabilized or uncharged particles). The corrections for Brownian motion are relatively minor, but those for fluid shear and differential sedimentation drastically reduce the predicted frequency of collisions between large and small particles in the suspension. A second important difference between the original and current models is that only two collision mechanisms were considered in flocculation (Brownian motion and fluid shear) and in sedimentation (Brownian motion and differential sedimentation) in the earlier work, whereas all three mechanisms are considered in both flocculation and sedimentation now. A minimal velocity gradient of 1 s-1 was assumed within the sedimentation tank in the current research. As noted above, the particle volume is conserved when flocs are formed in the models used here. In reality, water is incorporated into flocs when they form, the floc density is less than the original particle density, and the flocs take on a fractal nature. Although others (8, 9) have used fractal concepts in modeling flocculation quite similarly to our work reported here, we have chosen not to incorporate those concepts at this time because, as shown below, our models tend to overpredict collisions between large and small particles, and the fractal models exacerbate that problem even more. Also, the experimental measurements made in our laboratory are performed with a Coulter Counter, which responds (theoretically) to the solid volume of a floc, so that our modeling and measurements are consistent with one another. Nevertheless, the simplification of the conservation of volume is a recognized limitation of the approach taken in this research. A template for research design that characterizes our current work is shown in Figure 1. Research questions can be formulated into mathematical terms, and the mathematical solution constitutes an answer to the question. Alternatively, the question can be translated into an experimental investigation, and the results also constitute an answer to the question. If both the mathematical and the experimental pathways are taken, the two answers can be compared. If 6338
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FIGURE 1. Research design.
FIGURE 2. Modeling flocculation and sedimentation with the longrange model. The methodology and input data are adapted from Lawler et al. (1). In that work, differential sedimentation was not included in flocculation, and fluid shear was omitted from flocculent sedimentation. these two answers are in agreement, it is likely that one has answered the research question correctly. Lack of agreement is a sign that complete understanding is still elusive, but it is common for that disagreement to lead to further insight (and often further research). This paper delineates a few “trips” through this research diagram. At the time of the original work, no detailed size distribution measurements were available in the literature, nor were any obtained at that time in our own work. Hence, judgments about the reasonableness of the modeling effort could only be made from gross parameters calculated from the more detailed predictions of the changes in the size distribution. We now have a large body of detailed particle size distribution measurements from a variety of laboratory studies and a few water treatment plants. In all cases, very detailed measurements of size distributions have been obtained using a Coulter Counter; details of the measurement procedures have been documented elsewhere (10). The existence of particle size distribution data from experiments and operating water treatment plants affords a check on the reasonableness of the mathematical modeling efforts that was not available at the time of the original paper.
Results and Discussion In the current work, the first step was to reconstitute the model as it existed in the original work. Results are shown in Figure 2 for the changes in the particle size distribution through flocculation and sedimentation in the standard plant. These results are essentially identical to those obtained and shown in the original work; as reported then (1, 11), these results gave a reasonable value for the weight fraction lost through sedimentation (approximately 85%). This overall result was quite fortuitous and, in fact, reflects some canceling errors; with our current knowledge, decisions made at the time of original modeling work seem erroneous. For example,
FIGURE 3. Flocculation modeling using the long-range model and including differential sedimentation and extra particle size classes. Notes: flocculation 1 is the flocculation effluent with inclusion of differential sedimentation; flocculation 2 is the flocculation effluent with differential sedimentation and five additional particle size classes. the omission of differential sedimentation as a possible collision mechanism in flocculation led to the small decrease in particle number concentration through most of the distribution; its subsequent inclusion in the sedimentation model caused the dramatic reduction in particle number concentration throughout the size range shown in the figure. Because of the high particle density chosen for this work, differential sedimentation causes a high collision frequency. Also obvious in Figure 2 is the fact that the largest particle size allowed in the model accumulates particles during flocculation; these particles are then removed easily in sedimentation; such an accumulation appears unrealistic. To correct the problems of that earlier work, we investigated the effects of including differential sedimentation as a flocculation mechanism and also included extra size classes to allow the particle sizes to grow gradually into larger sizes than those originally present in the distribution. As shown in Figure 3, the inclusion of differential sedimentation as a mechanism in flocculation led to a substantial reduction in the number concentration of particles throughout the first half of the distribution, as one might expect in flocculation. However, the existence of the extra bins made the predictions even less reasonable than the original work; as shown in Figure 3, the existence of extra bins led to far greater flocculation being predicted, and the amount of flocculation depended largely on the number of those extra bins for larger particle sizes. Such predictions are unrealistic; particles would be expected to grow gradually into larger sizes. In this modeling, however, as soon as a few particles exist in the larger bins, they scavenge an increasing number of small particles. These results suggest that, in the long-range force model, the collisions between large and small particles are substantially overpredicted. This conclusion was also reached on the basis of comparison (not shown) with experimental results in one of the first tests of the model with field results (12). The first comparisons of model predictions with both laboratory (13) and field measurements (12) made it quite clear that the model overpredicted collisions between large and small particles. While batch flocculation experiments incorporating destabilization by charge neutralization showed that particles grew gradually through the distribution over time, the model suggested that small particles “leap-frogged” the middle size range and were incorporated into the largest flocs directly. The field results, using measurements from the largest Austin, TX, water treatment plant, also led to the same conclusion: the model dramatically overestimated collisions between large and small particles. As stated then (12): “under different conditions, the model predictions tend
FIGURE 4. Conceptual view of the short-range and long-range flocculation models. to fit the measured results for the small or large particles, but not both”. Most previous careful experimental tests of the Smoluchowski equation for flocculation had been done with originally monodisperse suspensions (14, 15), and such tests were intrinsically incapable of discovering this problem. Many experiments had also been performed with heterodisperse suspensions (16-18), but the resulting measurements of water quality (e.g., turbidity after settling) were of insufficient sensitivity to see difficulties with the modeling. The disparity between the model predictions and experimental results led to the realization that the conceptual view of collisions that was built into the Smoluchowski equation was too simplified. The collision frequency functions (βij in eq 1) were developed with assumptions that neglected any hydrodynamic effects and short-range interactions between particles as they approached one another. The view of collisions in the original (long-range, Smoluchowski) model is depicted for collisions by differential sedimentation in the left part of Figure 4. The drawing is a Lagrangian view from the perspective of the center of the larger particle as it settles in the vicinity of two smaller particles. All three particles are settling downward, but the large particle has a greater settling velocity than small ones (assuming they all have the same density), so it appears (from the center of the large particle) that the smaller ones are moving upward. In the long-range view of collisions, a collision will occur as long as the center of a small particle, when it is well below the large particle, is within a circle defined by the sum of the radii of the large and small particles. However, just as a large ship pushes a lot of water out of its way (and will likely push a canoe in a harbor out of its way as well), a large particle will cause hydrodynamic effects that alter the trajectories of small particles. Also, when particles are in near proximity to one another, van der Waals attraction and electrostatic repulsion (if the particles are charged) influence the pathways (and therefore the collisions) of particles. These ideas are depicted in the right side of Figure 4. The figure suggests that, for a collision to occur, the circle in which the center of the small particle must reside when well below the large particle is far smaller when these short-range interactions are accounted for. The short-range corrections factors (Rij) in eq 1 are the ratio of the areas of the circles of the right and left parts of Figure 4 (i.e., Rij ) Xc2/(ai + aj)2). Following the work of Jeffrey and Onishi (19), Han and Lawler (6, 7) developed the shortrange correction factors for differential sedimentation; they also calculated values for Brownian motion that were consistent with those of Spielman (20) and Valioulis and List (21) and incorporated those by Adler (22) for fluid shear. For all collision mechanisms, the particles were assumed to have been destabilized by charge neutralization, so that the VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 5. Modeling experimental flocculation data with the longrange and short-range models: experimental particle size distribution at t ) 60 min (O); long-range model at t ) 60 min with remp,long-range ) 0.4 (- -); short-range model at t ) 60 min with remp,short-range ) 0.9 (s). correction factors account for hydrodynamic interactions and London van der Waals attraction simultaneously and explicitly. Any double-layer repulsion due to incomplete surface charge neutralization must be accounted for by Remp. The primary result of these corrections is to drastically reduce the predicted collision frequency by both fluid shear and differential sedimentation between particles that are very different in size; that is, the short-range corrections apparently accounted directly for the overprediction of the frequency of collisions between large and small particles that had been the downfall of the long-range force model. Because the short-range model appeared to correct the earlier difficulties, further experimental work was undertaken to provide a new test of the model (i.e., a new comparison of experimental and modeling results). For an adequate test, a couette flow reactor (a reactor in the annular space between a rotating and a stationary cylinder) was constructed, because these reactors give a uniform velocity gradient in laminar flow. Results from experiments in this reactor eliminate questions about the role of mixing in the turbulent conditions of standard jar tests. Example results from that work are shown in Figure 5. These results were obtained using a 50 mg/L suspension of PVC particles with a density of 1.40 g/cm3. Destabilization was accomplished by charge neutralization with 3 mg/L of alum at pH ) 6.0, and the experiment was performed at a velocity gradient of 50 s-1. Some experimental results (not shown) were used to find the values of Remp that gave the best fit of the model to measured particle size distributions for these chemical conditions; these values were found to be 0.9 for the shortrange model and 0.4 for the long-range model. The longrange model yields a lower value because Remp, long-range must account for the phenomena that are otherwise ignored. In terms of eq 1, Rij ) 1 for the long-range model for all particle collisions, whereas Rij < 1 for the short-range model; since the same experimental data are being fit with both models, Remp, long-range is necessarily less than Remp, short-range. The comparison of the experimental results with the predictions from both models, shown in Figure 5, confirms that, under conditions of destabilization by charge neutralization, the short-range model for flocculation is far superior to the long-range model. Note that the long-range model (dotted line) predicts a greater loss of small particles and a greater creation of large particles than were observed experimentally (data points), whereas the short-range model (solid line) fits the experimental results well throughout the entire size range. Since the original predictions about the behavior of particles in water treatment plants were done with the long-range model, these results suggest that the 6340
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FIGURE 6. Modeling flocculation and sedimentation with the shortrange model.
FIGURE 7. Modeled volume average diameter of remaining particles during flocculation (prior to 1 h) and sedimentation (1-3 h) and weight fraction remaining during sedimentation: short-range model (- -); long-range model as modeled by Lawler et al. (1) (s). water treatment modeling should be revisited using the shortrange model. Using the short-range model and all of the characteristics of the standard plant and influent suspension from the original work leads to the predicted size distributions shown in Figure 6. Note that these predictions include approximately a 1.5 order of magnitude (97%) reduction in the number concentration of the smallest particles through flocculation and a further reduction in those particles in the sedimentation facilties, results that appear reasonable. However, the loss of the largest particles in sedimentation is not substantial, and the entire midrange of particle sizes (0.1 < log dp < 1.1, or 1.3 < dp < 13 µm) barely changed from the influent through both flocculation and sedimentation. The differences between the original long-range model (which excluded differential sedimentation in flocculation and fluid shear in sedimentation) and the current shortrange model are shown more dramatically in Figure 7. For this figure, the results from the full particle size distribution for both models are summarized by the volume average diameter as it changes through the flocculation and sedimentation basins. As expected, this average diameter increased monotonically in flocculation in both models, and this net particle growth continued in the early part of the sedimentation reactor (i.e., in this plug flow model, the distance into the reactor is proportional to the time). However, in the long-range model, the loss of large particles by sedimentation eventually resulted in a decrease in this parameter. In the short-range model, the volume-average diameter increased through both processes; the effects of flocculation within the sedimentation tank (to increase this
FIGURE 9. Modeling sedimentation with a flocculation effluent typical of precipitative coagulation processes. FIGURE 8. Experimental particle size distributions from Georgetown Water Treatment Plant, Georgetown, TX: raw water (O); flocculation effluent (0). parameter) outweighed those of sedimentation (to reduce it) throughout the entire time. Coincidentally, the two models yield the same value at the end of the sedimentation tank. However, the weight fraction (i.e., particle volume) removal in the sedimentation was only 20% for the short-range model, far less than the 95% removal predicted by the long-range model and far less than that usually achieved in real operating water plants. So, while the short-range model clearly was superior to the long-range model in fitting experimental results for flocculation under conditions of destabilization by charge neutralization, it yields unreasonable predictions for overall water treatment plant performance. To gain some insight into the differences between the experiments that showed the excellence of the short-range model and the modeling that exhibited its weakness with respect to adequately capturing the performance of water treatment plants, we obtained measurements of particle size distributions from an operating plant in Central Texas. The results are shown in Figure 8. The influent water is from a lake, and remarkably, the particle size distribution of the influent is nearly a straight line with a slope of -4 on this figure; that is, the influent follows the power law distribution originally posited in the early work without benefit of any measurements. The particle size distribution after flocculation is dramatically different; note the substantial rise in particle concentration of the largest particles (0.8 < log dp < 1.3, or 6 < dp < 20 µm) in the suspension in comparison to the influent. This increase in particle number concentration of the large particles is due to precipitation of aluminum hydroxide. (Note that the Coulter Counter responds to the solid volume, so that aluminum hydroxide flocs that measured this size would appear to the eye to be much larger, since they incorporate a substantial amount of water; as noted above, the modeling in this paper does not account for this additional water volume in the flocs.) Because of the difficulty in sampling from a tank that has rotating paddles (that required taking the sample from a relatively dead corner of the tank) and because the velocity gradient in this tank was exceedingly low (G ) 4 s-1), it is reasonable to think that the average influent to the sedimentation tank had even more large particles than the results in Figure 8 suggest. Similar measurements at softening plants (not shown) also show a substantial hump in the distribution in the large size range (12). In the vast majority of drinking water treatment plants, new solids are formed by the precipitation of aluminum or iron hydroxides or calcium carbonate in softening. In the mathematical modeling to date, the creation of these new
solids has been ignored; work is currently being designed in our laboratory to be able to predict what those changes are. However, with the current absence of that ability, a reasonable test of the sedimentation model is to use an input size distribution that is consistent with measured results after flocculation (in lieu of predicted results that do not account for the creation of new solids as done to date). Such modeling predictions are shown in Figure 9 and indicate a substantial loss of the large particles (log dp > 1.1, or dp > 13 µm) and a small loss (due to continued flocculation) of the small particles. Because this influent distribution has a high fraction of the particle volume concentration accounted for by the large particles (due to the “hump” in the distribution created by the precipitation), these results do predict a high loss of the suspended solids concentration (87%). While perhaps a bit lower than what is normally experienced in treatment plants, the results (like those for flocculation when no new solids are formed) are a good indication that the modeling of flocculent sedimentation is reasonable. The original work was the idea of Charlie O’Melia and represented a gigantic step in modeling behavior of particles in water treatment plants; overall, it captured that behavior quite well. In the intervening 25 years, we have made substantial progress in understanding and accounting for short-range interactions between particles when they approach in close proximity. Also, the availability of detailed measurements of particle size distributions from both laboratory experiments and field measurements has increased our understanding of particle behavior. Other researchers have also contributed substantially to increased knowledge of flocculation, particularly with respect to the role of turbulence and associated floc breakup (23-25) and the fractal nature of flocs (8, 9). Despite this progress, some mysteries remain. Our current work is directed toward accounting for precipitation effects on particle size distributions, in hopes that it solves many of those mysteries.
Acknowledgments The authors thank Charles O’Melia for the original suggestion to undertake this work many years ago and for his continual encouragement to pursue it in the meanwhile; his insights have been a beacon guiding this work. The work reported herein spans many years and multiple grants, but most directly those by the NSF, the U.S. EPA, and the State of Texas Advanced Technology Program. The opinions expressed in this paper are those of the authors alone and do not reflect those of the sponsoring agencies.
Literature Cited (1) Lawler, D. F.; O’Melia, C. R.; Tobiason, J. E. Integral water treatment plant design: From particle size to plant performance. In Particulates in Water; Kavanaugh, M. C., Leckie, J. O., Eds.; VOL. 39, NO. 17, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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(2) (3) (4)
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(10) (11) (12) (13)
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Received for review January 14, 2005. Revised manuscript received April 8, 2005. Accepted April 11, 2005. ES050089E
