Journal of Colloid and Interface Science 218, 488 – 499 (1999) Article ID jcis.1999.6424, available online at http://www.idealibrary.com on
Particle Deposition under Unfavorable Surface Interactions Renbi Bai 1 and Chi Tien Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore Received March 8, 1999; accepted July 12, 1999
cient, l 0, to its value in the absence of unfavorable surface interactions, (l 0) f, or
Experiments were conducted on particle deposition in granular media (deep bed filters) under the condition of unfavorable surface interactions. The data collected together with those of previous investigators constitute a sufficiently large data base on which correlations were developed to predict the extent of the reduction in particle deposition as a function of relevant operating and system variables. The new correlation equation was found to describe the experimental results reasonably well from the present experimental work or the present plus early experimental works, and appears to be much better than any other earlier correlations. The limitations of the new correlation equation, as well as those of all earlier ones, were also discussed. © 1999 Academic Press Key Words: deep-bed filtration; particle deposition; filter coefficient; unfavorable surface interactions; correlation.
a5
1
N col 5
kH , e 0ez pzg
[2]
where H is the Hamaker constant, k the reciprocal of the Debye thickness, z p and z g the particle and filter grain zeta potentials, respectively, and e 0 and e the permittivity in vacuum and relative permittivity of the suspending liquid. For water filtration, N col is strongly dependent on the ionic concentration of the water to be filtered. A correlation between a and N col for several sets of filtration data involving latex particles and soda-lime glass beads was developed (7),
a 5 0.0257~N col) 1.19.
[3]
More recently, the present authors (8) showed that the filter coefficient ratio, a, in principle, is a function of 11 dimensionless parameters,
l0 a5 5A ~ l 0! f
PN , 11
ai i
[4]
i51
where the definitions of N i (i 5 1, 2, . . . , 11) are given in Table 1, and A and a i are empirical constants.
To whom correspondence should be addressed.
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
[1]
and obtained an empirical relationship between a and the ionic strength of the suspension based on their data as well as on those of previous investigators (5, 6). In a later study, Elimelech (7) argued that reduction in deposition rate can be directly related to the magnitude of the maximum of the repulsive force between particles and filter grains, which is determined by the dimensionless parameter N col, defined as
INTRODUCTION
In deep bed filtration, the surface interaction forces between filter grains and particles to be removed include the double layer force and the London–van der Waals force. In the case when the combined force is repulsive (unfavorable), particle deposition may be retarded. Accurate prediction of the effect of unfavorable interactions is therefore important not only to the design and operation of filtration systems, but also to the formulation and establishment of guidelines making suspensions more amenable to filtration treatment. Theoretical analyses on the effect of surface interactions on particle deposition abound in the literature. A summary of these studies can be found in Tien’s monograph (1) and the more recent review article of Ryan and Elimelech (2). Generally speaking, these analyses have not yielded satisfactory results for the unfavorable surface interactions case. Theoretical predictions show a catastrophic decline in deposition rates with the onset of a repulsive force barrier between particles and collectors, which is not observed in experiments (3, 4). To provide a practical way of predicting deposition rates under unfavorable surface interactions, Vaidyanathan and Tien (3) suggested that the reduction in deposition rate in clean filters can be described by the ratio of the initial filter coeffi-
l0 , ~ l 0! f
488
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
TABLE 1 Definitions of Dimensionless Parameters Notation N1 N2 N3 N4 N5 N6 N7 N8 N9
5 5 5 5 5 5 5 5 5
N Re NR N Pe N LO NG N Fr 1/ p Nrtd N E1 N E2
N 10 5 N E3 N 11 5 N DL
489
a 5 1.0118 3 10 23 ~N LO! 0.8459 3 ~N E1! 20.2676~N E2! 3.8328 ~N DL! 1.6776.
[5]
Definition Reynolds number Interception parameter Peclet number London number Gravitational number Froude number Retardation parameter First electrokinetic parameter Second electrokinetic parameter Third electrokinetic parameter Double-layer force parameter
u s r d g /m d p/d g d gu s/D BM 4H/(9 pm d p2 m s) (D r )d p2 g/(18 m u s) u s2/(gd g) d p/l e ee 0(z p2 1 z g2)/(3pmu sd p) 2z pz g/(z p2 1 z g2) N AId g3
kd p
As a simplification, the ratio of l 0/(l 0) f can be considered as a function of four dimensionless parameters, N LO, N E1, N E2, and N DL. The correlation developed by Bai and Tien is
Equation [5] was based on the data reported by Vaidyanathan and Tien (4), Elimelech and O’Melia (9), and Elimelech (7). The data of Elimelech and O’Melia and Elimelech were obtained using submicron particles. On the other hand, Vaidyanathan and Tien conducted their experiments using relatively large particles (11.3 mm in diameter). Because of the modest data base size, the correlation given by Eq. [5] can only be regarded as a preliminary one. A more general correlation cannot be developed unless more data become available. The purpose of the present study is twofold, to experimentally determine filter coefficients with both micron and submicron particles under various unfavorable surface interactions conditions, and to establish, based on the new data and previously reported ones, general correlations for estimating the effect of unfavorable surface interaction. These results are presented in the following sections preceded by a description of the experimental work.
FIG. 1. Schematic diagram of experimental setup.
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FIG. 2. Particle distribution in aqueous phase, (—r—) D.I. water only, (—F—) decanted water from water– glass beads mixture.
EXPERIMENTS
Experiments were performed to obtain filtration data under unfavorable surface interactions. The main component of the experimental setup included a suspension preparation tank, a peristaltic pump, an overflow constant-head tank, and two experimental filters in parallel, each equipped with a rotometer and two flow rate control valves. A schematic diagram of the experimental setup is shown in Fig. 1. The suspension tank was a plastic container (10 liter in volume), and test suspension was recycled by the pump in order to keep the suspended particles from settling and agglomerating. Each of the experimental filters was constructed from a plexiglass cylinder with an inside diameter of 25 mm and a length of 300 mm. They were placed 1.5 m below the overflow constant-head tank. A divergent funnel preceded the inlet to each filter to ensure that a flat liquid velocity profile was achieved as the suspensions flowed into the column. An 80mesh screen was placed at the bottom of each filter to retain and also support the filter media. Samples of the influent were
collected from the inlet pipe of the filter and those of the effluent from the outlet pipe after the rotometer, respectively. Two types of Ballotini glass beads were used as filter grains. Type 1, specified as lead and white-brown in color, had a specific gravity of 2.95 and diameter of 0.29 ; 0.42 mm with deviations of about 621 ; 29 mm. Type 2, specified as soda and brown in color, had specific gravity of 2.55 and diameter of 0.40 ; 0.52 mm with deviations of about 626 ; 40 mm. Before their use in experiments, the glass beads were soaked in reagent HCl for 4 h (though HCl cleaning of the glass beads did not make a notable difference in the particle deposition rate in a set of comparison experiments). Then they were transferred and kept in D.I. water for another 2 days. Following that, the glass beads were packed in a filter column where they were backwashed with tap water at 40 ; 50% bed expansion for 3 h and then with D.I. water for another 30 min. The glass beads were subsequently dried in an oven at 105°C for 2 days and then stored in a desiccator for use as filter media. To prepare a filter bed for filtration experiments, the filter column was half filled with D.I. water, then a certain amount of the dried glass beads were weighed and packed into the column. The glass beads were soaked in the filter column for at least 1 day before experiment; during this period, they were backwashed several times in order to remove air bubbles trapped in the filter. The filter media depth was 10 cm with a porosity of 0.41. Polystyrene latices of three different sizes, 3.004 mm (standard deviation of 0.04 mm), 0.802 mm (standard deviation of 0.0096 mm), and 3.063 mm (standard deviation of 0.03 mm) in diameter, obtained from Duke Scientific Corp. (Palo Alto, CA, USA), were used in the experimental work. Suspensions were prepared by adding particles of one given size into D.I. water. A small volume of latex particles was first added to 100 ml D.I. water and placed in an ultrasonic bath for 10 min to ensure particle dispersion. Then the solution was poured into a 10-liter container with 5 liter D.I. water in it and stirred for half an hour before experiments. Sodium chloride (99.8% purity) was added to the suspension to give the desired ionic concentration. A specified amount of sodium chloride was weighed and added to 1 liter D.I. water to prepare a standard solution. For each experiment, a projected volume of the standard solution was measured in a volumetric pipette and added to the suspension while the latex particle solution was poured into the 5 liter D.I. water for stirring. The suspensions have been made very dilute so as to avoid any possible aggregation problems. Particle size analysis by Coulter Counter did not show obvious change of particle size distribution in the solution. To begin an experiment, the test suspension in the suspension tank was pumped to the constant head tank and passed through one of the filter columns ready for experiment. The flow rate was controlled by adjusting the valve to a preset rotometer reading and kept constant through the entire experiment. Influent and effluent samples of about 15 c.c. were taken at the start of a run and then at 5-min intervals.
491
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
TABLE 2 Conditions Used in the Experimental Work Series no. I
II
III
Exp. no.
NaCl (mol/liter)
dp (mm)
dg (mm)
zP (mV)
zG (mV)
V (m/h)
L (cm)
t (°C)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
0 0.0001 0.001 0.01 0.1 0.2 0.1 0.0001 0.0001 0.0001 0.001 0.01 0.01 0.01 0.1 0 0 0 0.0001 0.0001 0.0001 0.001 0.001 0.001 0.01 0.01 0.01 0.03 0.03 0.03 0.06 0.06 0.06 0.1 0.1 0.1 0.2 0.2 0.2 0 0.001 0.01 0.1 0.1 0.1
3.004 3.004 3.004 3.004 3.004 3.004 3.004 0.802 0.802 0.802 0.802 0.802 0.802 0.802 0.802 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063
0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.4 ;0.52 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42 0.29;0.42
220.5 219.6 218.1 213.9 26 25.1 26 220.7 220.7 220.7 219.3 215.7 215.7 215.7 27 225.5 225.5 225.5 224.5 224.5 224.5 223 223 223 215 215 215 210 210 210 28 28 28 26.8 26.8 26.8 25.5 25.5 25.5 225.5 223 215 26.8 26.8 26.8
225 222.8 221.2 218.1 211.2 28 211.2 222.8 222.8 222.8 221.2 218.1 218.1 218.1 211.2 216.4 216.4 216.4 212.9 212.9 212.9 211 211 211 28 28 28 25 25 25 24 24 24 23 23 23 22 22 22 216.4 211 28 23 23 23
3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 6.1 9.8 3.7 3.7 6.1 9.8 3.7 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 3.7 3.7 3.7 3.7 3.7
10.3 10.2 10.2 10.2 10.1 10.2 10 10.3 10.3 10.3 10.2 10.3 10.3 10.3 10.3 10 10 10 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10 10 10 10 10 10 10.1 10.1 10.1 9.9 9.9 9.9 10.1 10.1 10.1 10.1 10.1 10.1
23 23 21.5 22 22.5 22.5 21.8 21 21 21 23.3 22.8 22.8 22.8 21.5 22.5 22.5 22.5 24 24 24 24 23 24 23.5 21 24 21.5 21.5 21.5 22 22 22 23.5 23.5 22.5 23.5 24 25 23 23 23 23 23 22
Samples were analyzed for particle concentration and size distribution using a Coulter Counter (Model IIe, Coulter Electronics Ltd., Luton, England) equipped with a multichannel analyzer interfaced to a Pentium computer. The particle counter was calibrated weekly using the standard particles supplied by the vender and was checked before each experiment. Zeta potentials of the latex particles were measured by a
ZetaPlus4 instrument (Brookhaven Instruments Corp., Holtsville, NY). Calibration was done before each experiment, according to the manufacturer’s instructions. The same measurements were repeated several times in order to obtain the average values. To determine the zeta potentials of filter grains (glass beads), glass beads were placed in a 100-ml vial with 50 ml D.I. water and the vial was vibrated in a sonic bath for 24 h. The liquid in
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BAI AND TIEN
potentials of the glass beads (0.29 ; 0.42 mm) were found to be positive (not shown in the figure). As mentioned earlier, the effect of unfavorable surface interactions on deposition can be presented by the ratio of the filter coefficient under unfavorable surface interactions to the corresponding filter coefficient under favorable interactions, a. To determine the value of l 0, the equation
S D
l 0 5 lim 2 t30
1 C eff ln , L C inf
[6]
was used, where C eff and C inf denote the effluent and influent particle concentrations, respectively, and L is the filter bed thickness. This equation is based on assumption that the conditions throughout the filtration are uniform. A typical set of filtration experimental results is shown in Fig. 4. For a given series, the value of l 0 at the highest NaCl concentration was taken as the initial filter coefficient under favorable interactions, (l 0) f, and used to calculate the filter coefficient ratio, a. Table 3 summarizes the results of the present study. The results of previous investigations are given in Table 4.
FIG. 3. Zeta potentials of glass beads and test particles.
the vial with small fragments from the glass beads in it was then decanted for measurements. Figure 2 gives the size distributions of particle fines in the decanted liquids. Compared with blank test results (i.e., D.I. water only) the difference clearly demonstrated that a certain amount of fragments had come off the glass beads’ surface. The zeta potential of these fragments was taken to be the zeta potential of the glass beads. RESULTS
The experimental conditions used in this study were summarized in Table 2. Some of the measurements were repeated in order to ensure data reproducibility. There were three series of experiments, based on the sizes of the particles and glass beads. In each series, measurements were made with test suspensions of different NaCl concentrations. Figure 3 gives the zeta potentials of test particles and glass beads as a function of NaCl concentration. The zeta potential values shown in this figure represented the average values based on several measurements. Generally, the zeta potentials of both the latex particles and the glass beads were negative and their magnitude decreased with the increase in NaCl concentration. In some cases when the NaCl concentrations exceeded 0.1 M, the zeta
FIG. 4. A typical set of filtration experiment results (d g 5 0.29 ; 0.42 mm, d p 5 0.802 mm, V 5 3.7 m/h).
493
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
TABLE 3 Summary of Experimental Data Collected in the Present Study Exp. no.
NaCl (mol/liter)
dp (mm)
dg (mm)
zP (mV)
zG (mV)
V (m/h)
t (°C)
l0 (m 21)
a 5 l 0/(l 0) f
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
0 0.0001 0.001 0.01 0.1 0.2 0.1 0.0001 0.0001 0.0001 0.001 0.01 0.01 0.01 0.1 0 0 0 0.0001 0.0001 0.0001 0.001 0.001 0.001 0.01 0.01 0.01 0.03 0.03 0.03 0.06 0.06 0.06 0.1 0.1 0.1 0.2 0.2 0.2 0 0.001 0.01 0.1 0.1 0.1
3.004 3.004 3.004 3.004 3.004 3.004 3.004 0.802 0.802 0.802 0.802 0.802 0.802 0.802 0.802 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063 3.063
0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.46 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35
220.5 219.6 218.1 213.9 26 25.1 26 220.7 220.7 220.7 219.3 215.7 215.7 215.7 27 225.5 225.5 225.5 224.5 224.5 224.5 223 223 223 215 215 215 210 210 210 28 28 28 26.8 26.8 26.8 25.5 25.5 25.5 225.5 223 215 26.8 26.8 26.8
225 222.8 221.2 218.1 211.2 28 211.2 222.8 222.8 222.8 221.2 218.1 218.1 218.1 211.2 216.4 216.4 216.4 212.9 212.9 212.9 211 211 211 28 28 28 25 25 25 24 24 24 23 23 23 22 22 22 216.4 211 28 23 23 23
3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.7 6.1 9.8 3.7 3.7 6.1 9.8 3.7 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 6.1 9.8 3.7 3.7 3.7 3.7 3.7 3.7
23 23 21.5 22 22.5 22.5 21.8 21 21 21 23.3 22.8 22.8 22.8 21.5 22.5 22.5 22.5 24 24 24 24 23 24 23.5 21 24 21.5 21.5 21.5 22 22 22 23.5 23.5 22.5 23.5 24 25 23 23 23 23 23 22
0.063 0.0813 0.4578 1.7633 8.0452 8.294 8.0726 0.0217 0.0161 0.0133 0.2517 0.9469 0.868 0.8369 3.8956 0.0404 0.0451 0.035 0.0701 0.0559 0.0381 0.1865 0.1499 0.1199 1.122 1.0798 0.8999 2.5732 1.9685 1.6999 5.3048 4.1637 3.5438 7.894 6.0877 4.8404 8.25 6.352 5.145 0.0448 0.2098 1.5417 7.821 7.9521 8.168
0.0076 0.0098 0.0552 0.2126 0.970 1 0.9733 0.0039 0.0029 0.0024 0.0453 0.1704 0.1562 0.1506 0.701 0.0049 0.0071 0.0068 0.0085 0.0088 0.0074 0.0226 0.0236 0.0233 0.1360 0.1700 0.1749 0.3119 0.3099 0.3304 0.6430 0.6555 0.6888 0.9569 0.9584 0.9408 1 1 1 0.0054 0.0253 0.1859 0.9431 0.9589 0.99
1. a versus Ionic Concentration It was suggested previously that a may be related to the ionic concentration of the suspension (3, 4). Figure 5 gives the results of a versus I, the ionic concentration in the test suspension. Generally, a is found to increase with the ionic concentration when I is less than 100 mol/m 3. As shown in Fig. 5, the linear relation obtained from regression is obviously less
satisfactory than the parabolic expression in data fitting. Both expressions overpredict a, giving values which may even be greater than unity at high ionic concentration, which, of course, is physically wrong. A similar analysis was done for all the data (present plus earlier data), and the results were shown in Fig. 6. Though the overall tendency indicates an increase in a with the increase of
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BAI AND TIEN
TABLE 4 Summary of Experimental Data from Previous Studies
Reference Vaidyanathan and Tien (1989)
Elimelech and O’Melia (1989)
Elimelech (1992)
Exp. no.
NaCl (mol/liter)
dp (mm)
dg (mm)
zP (mV)
zG (mV)
V (m/h)
t (°C)
a
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.181 0.181 0.181 0.181 0.096 0.01 0.181 0.181 0.181 0.181 0.096 0.01 0.00316 0.00316 0.01 0.01 0.01778 0.03162 0.03162 0.05623 0.1 0.1 0.00316 0.00316 0.01 0.01 0.01778 0.03162 0.03162 0.05623 0.1 0.1 0.00316 0.01 0.01778 0.03162 0.05623 0.001 0.00316 0.01 0.01778 0.03162 0.05623 0.001 0.00316 0.01 0.01778 0.03162 0.05623
11.4 11.4 11.4 6.1 11.4 11.4 11.4 11.4 11.4 6.1 11.4 11.4 0.378 0.378 0.378 0.378 0.378 0.378 0.378 0.378 0.378 0.378 0.753 0.753 0.753 0.753 0.753 0.753 0.753 0.753 0.753 0.753 0.060 0.060 0.060 0.060 0.060 0.189 0.189 0.189 0.189 0.189 0.189 0.376 0.376 0.376 0.376 0.376 0.376
0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.345 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
21 21 21 21 21 211 21 21 21 21 21 211 295 295 284 284 270 254 254 242 232 232 288 288 279 279 272 260 260 250 240 240 276 261 251 237 233 286 296 286 270 255 243 289 287 281 271 262 250
23 23 23 23 23 213 23 23 23 23 23 213 256 256 246 246 242 238 238 230 228 228 256 256 246 246 242 238 238 230 228 228 257 247 243 239 233 261 257 247 243 239 233 261 257 247 243 239 233
7.2 10.8 14.4 7.2 7.2 7.2 7.2 10.8 14.4 7.2 7.2 7.2 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9 4.9
20 20 20 20 20 20 20 20 20 20 20 20 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
1 0.7937 0.6343 0.9086 0.6599 0.2892 1 0.5341 0.5341 0.7503 0.4685 0.2130 0.0102 0.0115 0.0234 0.0263 0.0490 0.0933 0.1000 0.2089 0.3548 0.4467 0.0195 0.0115 0.0407 0.0324 0.0676 0.1585 0.1413 0.3162 0.5754 0.4467 0.0107 0.0324 0.0724 0.1585 0.3020 0.0028 0.0110 0.0251 0.0490 0.0977 0.2042 0.0089 0.0155 0.0372 0.0676 0.1514 0.3162
Log a 5 a~Log I! 2 1 b Log I 1 c.
[7]
the ionic concentration, a plot of a versus I does not lead to a clear relationship between the two quantities. Both the linear and parabolic expressions fail to give a good fit with the data. In an earlier study, Vaidyanathan and Tien (4) used the following format for correlating a with I:
The results from using the above format to fit the present data are given in Fig. 7. As shown in this figure, this format gives a good data fit at low ionic concentration but not higher
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
FIG. 5. The relationship between a and I (data from this work), linear equation, a 5 0.0974 1 0.005954 I, parabolic equation, a 5 0.01513 1 0.01324 I 2 4.15 3 10 25 I 2 .
concentrations. Further, when all the data were considered, the result became considerably worse (see Fig. 8). 2. a versus the Dimensionless Number, N col As mentioned before, Elimelech proposed a power-law relationship between a and the dimensionless parameter, N col. A plot of a versus N col (on logarithmic coordinates) of the present data is given in Fig. 9. The results show that the dependence of a on N col is less than that found before. When all the data (the present ones plus earlier ones from Vaidyanathan and Tien and
FIG. 6. The relationship between a and I (all data), linear equation, a 5 0.0566 1 0.004932 I, parabolic equation, a 5 20.005351 1 0.009288 I 2 2.44 3 10 25 I 2 .
495
FIG. 7. Correlation of a according to Eq. [7] (data from this work), correlation established, Log a 5 20.02643 (Log I) 2 1 0.751 Log I 2 1.491.
Elimelech et al.) were included, the results were unsatisfactory (see Fig. 10). 3. a as a Function of N Lo, N E1, N E2, and N DL Earlier, Bai and Tien (9) concluded that as a practical matter, a may be considered as a function of four dimensionless parameters, N Lo, N E1, N E2, and N DL, which determine the magnitude of the repulsive force. Based on the data obtained in this work, from regression analysis, the following correlation was obtained: log a 5 8.5783 3 10 23 N 0.5261 N E20.2346 N E2.8987 N 1.3497 Lo DL . 1 2
[8a]
FIG. 8. Correlation of a according to Eq. [7] (all data), correlation established, Log a 5 0.1242 (Log I) 2 1 0.5365 Log I 2 1.8341.
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BAI AND TIEN
FIG. 9. Correlation of a as a function of N col (data of this work), corre0.7223 lation established, a 5 8.5467 3 10 23 N col .
When all the data were included, the correlation obtained was log a 5 2.527 3 10 23 N 0.7031 N E20.3121 N E3.5111 N 1.352 Lo DL . 1 2
[8b]
FIG. 10. Correlation of a as a function of N col (all data), correlation 0.4808 established, a 5 2.59 3 10 22 N col .
sionless quantities, the number of the relevant dimensionless parameters can still be substantial. For the present study, with considerable simplifications, a is shown to be a function of
Comparisons between the predictions based on the above correlations with experiments are shown in Figs. 11 and 12. Although these two correlations are different, the predicted dependence on three of the four parameters is similar, only in the case of N Lo, the difference is significant (0.5261 versus 0.7013 as exponent of N Lo). A further comment about the correlation obtained in this work and the one established earlier (i.e., Eq. [5]), the difference between Eq. [8b] and Eq. [5] is less than that between Eq. [8a] and Eq. [5]. This is expected since the data on which Eq. [5] is based form part of the data base used to develop Eq. [8b]. DISCUSSIONS
The major problem encountered in developing correlations from deep bed filtration data is that filtration is controlled by a very large number of variables. Even with the use of dimen-
FIG. 11. Comparison of experiments with correlation of Eq. [8a].
497
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
FIG. 12. Comparison of experiments with correlation of Eq. [8b].
four parameters. The problem is further complicated by the fact that some of the physical variables cannot be arbitrarily specified. Consequently, data collection over sufficiently wide ranges of variables is not an easy undertaking and may even be impractical. In water filtration, for the overwhelming majority of cases, both particles and filter grains are negatively charged. The degree of unfavorable surface interactions is determined largely by the double layer force between particles and filter grains. Thus one may expect that the ionic concentration of the suspension involved is one of the more important factors in determining the reduction of particle deposition rates through its effect on the double layer thickness and to a lesser degree the surface potentials. This was the rationale for developing correlations between a and I. The results shown in Figs. 7 and 8 indicate that the ionic concentration has great influence on a. But ionic concentration alone does not determine a. The rather poor agreement between experiments and predictions based on a –I correlations perhaps is not surprising since particle and filter grain surface potentials which are also important in determining the interaction force are not functions of the ionic concentration only. On this account, the correlation developed by Elimelech, which relates a with N col, was an improvement. N col is defined by the ionic concentration and zeta potential, as well as by other variables. The earlier correlation by Elimelech yields very good predictions of his own data. However, when this type of correlation (a vs N col) was sought for the present data, the result was only fair (see Fig. 9). Further, when all the data were included (namely, the present ones plus those of Elimelech and O’Melia, Elimelech, and Vaidyanathan and Tien), the results became considerably worse (see Fig. 10), demonstrating rather conclusively that a cannot be a function of N col only.
Examination of Elimelech’s data revealed that they were obtained under rather severe unfavorable interaction conditions (i.e., high zeta potentials and low ionic concentrations) and that the values of a were low. The corresponding values of N col extend from 10 21 to 10. In contrast, the N col values of the present study ranged from 10 0 to 10 3, and those of Vaidyanathan and Tien were approximately 10 4. The correlation of Eq. [3] appears applicable only at low values of N col, probably in the range 10 21 to 10 0. The results shown in Figs. 4 through 12 demonstrate rather convincingly that among the three kinds of correlations considered (a vs I, a vs N col, and a as a function of N Lo, N E1, N E2, and N DL), the last one yielded the best and acceptable results. A summary of the regression statistics of these three kinds of correlation is given in Table 5. The difference between those of Eq. [8a] and those of Eq. [8b] is slight although Eq. [8b] is based on a much larger body of data. Accordingly, in predicting the effect of unfavorable surface interactions, the use of Eq. [8b] is preferred. In comparing Eq. [3] with Eq. [8b] (or Eq. [8a]), both correlations include the filter grain and particle surface potentials as variables, but in different ways. According to Eq. (3), a is dependent on the product z pz g. On the other hand, based on Eq. [8a] (or [8b]), a is a function of the product z pz g and (z p2 1 z g2). Consider two arbitrary cases, Case (a), z p 5 210 mV and z g 5 210 mV, and Case (b), z p 5 2100 mV and z g 5 21 mV, with all other variables being the same. Equation [3] will give the same results for these two cases, but Eq. [8b] may give different values of a for these two cases. The use of the four dimensionless parameters, N Lo , N E1 , N E2 , and N DL in correlating a is a simplification of the general conclusion that a is a function of the eleven parameters listed in Table 1. This simplification was validated in an early study (8) through a partial regression of the data of Vaidyanathan and Tien, Elimelech and O’Melia, and Elimelech. A similar analysis was made by adding the present data to the earlier ones. The partial regression results are
TABLE 5 Summary of Regression Statistics of Different Correlations
Reference
a 5 f(I) a 5 f(I 2 , I) Log a 5 f((log I) 2 , log I)
a 5 f(N col) a 5 f(N Lo, N E1, N E2, N DL)
Data resource
Multiple determination
Standard deviation
Present All Present All Present All Present All Present All
0.918 0.885 0.993 0.915 0.987 0.920 0.951 0.857 0.990 0.982
0.161 0.163 0.05 0.142 0.135 0.297 0.263 0.389 0.124 0.145
498
BAI AND TIEN
TABLE 6 Summary of Regression Coefficients and Errors: Effect of Parameter Inclusion and/or Deletion Series
A
a1
a2
a3
a4
a5
a6
a7
a8
a9
a 10
a 11
R
S
1 2 3 4 5 6 7 8 9 10 11 12
10 22.5974 10 23.9771 10 23.3735 10 22.6021 10 20.02164 10 22.9537 10 22.5718 10 22.9264 10 22.8996 10 22.6986 10 22.5344 10 22.7193
/ / / / / / 0.1260 / / / / /
/ / / / / / / 20.0809 / / / /
/ / / / / / / / 0.0661 / / /
0.7013 / 0.4579 0.6450 0.6721 0.6722 0.7030 0.6609 0.7848 0.6885 0.7025 0.6771
/ / / / / / / / / 20.0134 / /
/ / / / / / / / / / 0.0190 /
/ / / / / / / / / / / 20.0496
20.3121 0.2489 / 20.1934 20.8132 20.2963 20.2886 20.2963 20.2989 20.3091 20.3123 20.3095
3.5111 2.3536 2.4630 / 0.5656 3.4522 3.4594 3.4653 3.4826 3.5742 3.6114 3.5748
/ / / / / 0.0288 / / / / / /
1.3520 1.3321 1.5030 1.2561 / 1.3038 1.3708 1.3621 1.3604 1.3550 1.3540 1.3546
0.9821 0.8989 0.9666 0.9631 0.7262 0.9821 0.9822 0.9821 0.9821 0.9821 0.9821 0.9821
0.1455 0.3366 0.1968 0.2067 0.5281 0.1463 0.1456 0.1463 0.1463 0.1464 0.1463 0.1464
given in Table 6. The results show that a deletion of any one of the four parameters results in larger error. On the other hand, adding additional parameters to the correlation does not yield any significant improvement. Finally, it should be mentioned that, as do any correlations, Eq. [8a] and Eq. [8b] have their limitations. They are not valid for extremely high or low ionic concentrations. This limitation arises from the fact that the double layer cannot be compressed or expanded indefinitely by increasing or decreasing the ionic concentration. A prediction of a exceeding unity implies that the use of the correlation is not appropriate. In this regard, it is easy to discern that neither Eq. [8a] nor Eq. [8b] converges to unity as the repulsive force vanishes, or those correlations can be applied when the surface interactions undergo a transition from unfavorable to favorable. Equations [8a] and [8b] give physically meaningless results if either z p or z g vanishes or if the product z pz g is negative. Many other factors, such as surface roughness, chemical heterogeneity, and impurities of the filter media, may also have influence on the success of a correlation equation which did not take these factors into account. APPENDIX: NOMENCLATURE
A ai D BM dg dp H I NA Ni
Coefficient of Eq. [4] ith exponent of Eq. [4] Brownian diffusivity (kT/3 pm d p) Filter grain diameter Particle diameter Hamaker constant Ionic concentration Avogadro’s number The ith type dimensionless parameter
N DL N E1 N E2 N E3 N Fr NG N LO N Pe NR N Re N Rtd us k T
Double layer force parameter (see Table 1 for definition) First electrokinetic parameter (see Table 1 for definition) Second electrokinetic parameter (see Table 1 for definition) Third electrokinetic parameter (see Table 1 for definition) Froude number (see Table 1 for definition) Gravitational parameter (see Table 1 for definition) London force parameter (see Table 1 for definition) Peclet number (see Table 1 for definition) Interception parameter (see Table 1 for definition) Reynolds number (see Table 1 for definition) Retardation parameter (see Table 1 for definition) Superficial velocity Boltzmann’s constant Absolute temperature
Greek Letters
a Dr e0 e z g, z p k l0 (l 0) f le m r
Filter coefficient ratio Density difference Permittivity in vacuum Relative permittivity of fluid media Zeta potentials of filter grains and particles Reciprocal of double layer thickness Initial filter coefficient Initial filter coefficient under favorable surface interaction Wavelength of electron oscillation Fluid viscosity Fluid density
PARTICLE DEPOSITION UNDER UNFAVORABLE INTERACTIONS
ACKNOWLEDGMENT This study was performed with a grant from the National Science and Technology Board, Republic of Singapore.
REFERENCES 1. Tien, C., “Granular Filtration of Aerosols and Hydrosols.” Butterworth, Boston, 1989.
2. 3. 4. 5. 6. 7. 8. 9.
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Ryan, G. W., and Elimelech, M., Colloids Surf. A 107, 1 (1996). Vaidyanathan, R., and Tien, C., Chem. Eng. Sci. 43, 289 (1988). Vaidyanathan, R., and Tien, C., Chem. Eng. Commun. 81, 123 (1989). Fitzpatrick, J. A., Ph.D. Dissertation, Harvard University, Cambridge, MA, 1978. Yoshimara, Y., Ph.D. Dissertation, Kyoto University, Japan, 1980. Elimelech, M., Water Res. 26, 1 (1992). Bai, R., and Tien, C., J. Colloid Interface Sci. 179, 631 (1996). Elimelech, M., and O’Melia, C. R., Langmuir 6, 1153 (1990).