Environ. Sci. Technol. 1986, 20, 187-195
Role of Coagulation in the Kinetics of Sedimentation Kevin J. Farieyt and Frangois M. M. Morel" R. M. Parsons Laboratory, 48-425, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
The combined results of analytical, numerical, and laboratory studies are used in examining the kinetic behavior of sedimentation in well-mixed systems. The rate of mass removal of solids, dC/dt, as a function of mass concentration C, can be described as the sum of three power laws, dC/dt = -BdsC2.3- B,hC1.g- BbC1.3, each term of which corresponds to a particular coagulation mechanism: differential settling, shear, and Brownian motion. Empirical relationships for the coefficients Bds,Bsh,and Bb as a function of system parameters are provided. Introduction Understanding sedimentation kinetics is important for determining the performance of treatment processes and assessing the impact of waste disposal in natural waters. Sedimentation rates are now typically described by a fixed distribution of settling velocities, corresponding to the discrete settling of individual particles. Experimental results (1-7) however have shown the importance of particle-particle interactions, most notably coagulation, resulting in the aggregation of smaller particles into larger ones. According to Stokes law, large particles settle much faster than smaller ones, and overall sedimentation rates can thus be dramatically affected by coagulation. Basic coagulation theory, which is largely attributable to the work of Smoluchowski (8,9), is presented elsewhere (5-7,lO-12). Combining coagulation theory with Stokes law for gravity settling yields the dynamic equation of sedimentation kinetics. No general analytical solution exists for this equation. Investigators have thus relied on laboratory observations (3-6, 13), simplified analytical solutions (5, 6,14), and numerical simulations (12,15) to determine sedimentation behavior. A number of problems are associated with these individual approaches. First, extrapolating experimental settling rates to field Eonditions is difficult since dispersion is not represented in laboratory studies and the appropriate scaling of the laboratory results is unknown. Second, simplified analytical solutions, namely those developed by Hunt (5, 6) and Morel and Schiff (14), describe the mass removal of solids by a second-order dependency on mass concentration. Although this second-order rate law appears to be an adequate description of laboratory data, it has only been tested over a limited range, encompassing less than an order of magnitude change in suspended mass. Further testing of this rate law is clearly needed. Finally, numerical simulations based on the dynamic equation for sedimentation have so far been performed for case-specificapplications-providing little insight into the general behavior of sedimentation. Since no systematic comparison of numerical results with laboratory or field data has been performed, the applicability of the numerical simulations remains questionable. The goal of this study is to combine results of laboratory, analytical, and numerical studies for predicting the rate of solids removal from water. Simplified descriptions of sedimentation behavior previously hypothesized by Hunt (5,6), Morel and Schiff (14), and Farley (16) are used as a starting point and tested in a series of numerical simu'Present address: Tetra Tech, Inc., 1911 N. Fort Myer Drive, S-601,Arlington, VA 22209. 0013-936X/86/0920-0187$01.50/0
lations. On this basis a new simplified equation of sedimentation kinetics is proposed and tested with laboratory data, including new data covering a 2 order of magnitude decrease in mass concentration. Sedimentation Theory. Since soagulation affects the distribution of particle settling rates, one must consider coagulation and settling simultaneously to describe sedimentation kinetics. For a vertically homogeneous water column of depth h, the dynamic equation for the particle size distribution has been given by Friedlander (10) as
where n(u) is the particle size distribution, given as the number concentration of particles as a function of particle volume (number mL-l ~ m - ~p(i, ) , j ) is the collision frequency function, which depends upon the mode of interparticle contact (see Table I), and w,(u) is the settling velocity assumed to be given by Stokes law. Kinetic expressions for the change in total particle numbers, NT, and the wet weight of suspended solids, C, are obtained from integrations of eq 1
dC
dt = -pe
Jm
u F n ( u )du
(3)
where pe is the effective particle density (floc density) and is dependent on particle density, pp, fluid density, pf, and aggregate porosity, e Pe = (1- e)p, + ep, (4) Values for the total particle number, NT, and the wet weight of suspended solids, C, are related to moments of the size distribution
C = pe
Jm
un(u) du
In this description of sedimentation, the following assumptions have been made: (1)particles before and after each aggregation are spherical; (2) particles approach one another on rectilinear paths, the path of one particle not being affected by the presence of another; (3) no force acts on the particles until they come into physical contact, after which they adhere. Thus, all possible collisions are successful in forming aggregates and the efficiency of particle collisions is unity. The effects of collision efficiencies that are less than one (and assumed constant with respect to particle size) are discussed subsequently; (4) collision functions for Brownian motion, fluid shear, and differential settling are additive; (5) the floc density is constant over the size distribution (see Appendix for limiting case); and (6) the water column is vertically homogeneous with re-
0 1986 American Chemical Society
Environ. Scl. Technol., Vol. 20, No. 2, 1986
187
Table I. Functional Relationships for Collision Frequency and Stokes Settling Rate’
Brownian Motion p(i, j ) = ~ , , [ ( i / ~ + ~( ~) /’u/ ~~) ’ / ~ +] [ u ~ ~ / ~ Kb = 2kT/3p (cm3 8-l)
ut/3]
Fluid Shear K,h = G / T
(8-l)
Stokes Settling S / h = ( 1 / 6 ~ * ) ’ / ~ @ / 3 ~- hp f)) (/ ppf ,(cm-’ sd) OK,,, Knh,&, and S / h are dimensional constants; k is the Boltzmann constant; T is the absolute temperature; p and u are the dynamic and kinematic viscosities, respectively, of the fluid; G is the shearing rate of the fluid; g is the gravitational acceleration; pe is the effective particle density (floc density); pf is the fluid density; and h is the depth of the vertically homogeneous water column.
spect to particle concentrations, thus vertical mixing of the water column is fast relative to sedimentation. This is a useful approximation for many engineering applications where the depth scale is on the order of meters. As shown subsequently, the assumption of a vertically homogeneous water column also appears reasonable for unstirred laboratory columns. Simplified Analytical Solutions. Through different approaches, Hunt (5,6) and Morel and Schiff (14) have derived a simplified solution that describes the mass removal of solids by a second-order dependency on mass concentration dC _ - -Be2 (7) dt where B is the rate coefficient. To derive this equation, Hunt ( 5 , 6 )assumed that one coagulation mechanism was dominant for any given particle size: Brownian coagulation for the smallest sizes; shear coagulation for intermediate sizes; and differential settling for the largest particle sizes. This implies that collisions between similar size particles are most important. Following Friedlander (17,18), Hunt also assumed that the volume flux of particles through the size distribution was constant at any instant in time. (For a discrete size distribution, the volume flux is equivalent to the coagulation rate of particles out of a size interval.) Morel and Schiff (14) considered the extreme case in which coagulation is the rate-limiting step in the overall sedimentation process. In this situation, small particles are assumed to have effectively zero net settling velocity, but coagulation takes place and large particles are formed continuously. Above some fixed critical volume size the particles are considered to settle infinitely fast (the Stokes velocity being proportional to u ~ / ~ )For . particle sizes less than the critical volume size, the shape of the particle size distribution is assumed invariant while the total number of particles is decreasing due to coagulation and settling. Farley (16) derived another simplified “similarity” solution assuming that the fraction of particles in a given size range is a function of two normalizing parameters n(u, t ) = b(t)J,(u/a(t)) (8) where J, is the “self-preserving”function and a ( t ) and b(t) are normalizing parameters related to moments of the size distribution. For a continuously depleting size distribution, the parameter b ( t ) alone represents a uniform reduction of particle numbers over the entire distribution and a ( t ) represents the faster depletion of larger sized particles. Solutions using the similarity transformation are obtained by substituting the relationship given by eq 8 into eq 1-6 and solving these equations simultaneously. Solutions found in this way are possible “characteristic” solutions, which are believed to be asymptotically approached in time; they have been shown to be valid for the 188
Environ. Sci. Technol., Vol. 20, No. 2, 1986
simple case in which the collision frequency function, @(i, j ) , is constant (16).
Similarity solutions derived by this approach for the separate cases of coagulation by Brownian motion, shear, and differential settling show that mass removal rates are expressed by power laws, the exponent of which is dependent on the mode of coagulation (16). For multiple coagulation mechanisms, similarity transformations are not successful (19). However, if only one coagulation mechanism is dominant at a given mass concentration (or if eq 1 can be treated as linear), the mass removal rate of solids can be expressed approximately as the sum of three power laws
According to this equation, differential settling should control the removal kinetics at high solids concentration, shear at intermediate concentrations, and Brownian motion at low concentrations. Materials and Methods Numerical Simulations. Numerical simulations involving the solution of the dynamic equation for the particle size distribution (eq 1) are used to examine sedimentation behavior. Here the particle size distribution is expressed as a finite number of size intervals that are numerically integrated in time. The discrete form of the governing equation for sedimentation is given by
(wk/h)nk (10) where a(i, j ) = 1 for i # j and a(i,j ) = 2 for i = j . The subscripts i, j , and k denote particles of a particular volume size, and m is the maximum number of size classes considered in each integration step. The value of m is determined internally in the numerical solution by expanding and contracting the number of size classes to include all size classes having particle numbers greater than particles mL-l. Since a wide range of particle sizes is considered in the numerical solution, it is most convenient to employ a logarithmic division of particle sizes. With standard particle sizes that are equally spaced on the basis of the logarithm of their volumes, the coagulation of particles does not produce a new particle of standard size. By following the method of Lawler et al. (15) weighted fractions of the new particle are assigned to standard particle sizes. This allows particle volume to be conserved. Unless otherwise stated, initial conditions for the size distribution were defined by considering all particles to be present in the three smallest volume sizes.
Numerical Results
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Flgure 1. Slmulation results for quiescent sedimentatlon wlth dlfferent lnlthl mass concentrations, expressed as mass concentration vs. t h e (pe = 2.0 g cm-!', p, = 1.0 g cm-!', h = 3.3 cm, G = 0 si, T = 20 O C , p = 0.01 g cm-i si).
Laboratory Experiments. To test the proposed rate laws for mass removal, experiments were performed over a wide concentration range, from a few hundred to a few milligrams per liter. Small sample volumes were used to prevent drawdown of the water column. As discussed subsequently, it was also desirable to employ high-density particles to minimize the effect of shear coagulation caused by extraneous sources of turbulence. Metallic copper particles were obtained from Alfa Products, Inc., and goethite particles were prepared in our laboratory. The copper particles have a diameter of approximately 1pm and a particle density of 8.9 g cmS; the particle density of goethite is approximately 4.5 g ~ m - ~ For . each experiment, particles were added to a mixing tank containing 20 L of artificial seawater. A 23-cm-diameter, 60-cm-high Plexiglass sedimentation column was then filled from the mixing tank, with the sedimentation column initially characterized by a vertically uniform concentration of particles. Samples were collected over time at two sampling ports using 5-mL syringes with 15-gaugeneedles. Solids concentrations were measured by digesting the sample suspension in acid and determining the metal concentration by atomic absorption. For copper particles, nitric acid was used in the digestion step; digestion of the goethite particles was performed by using hydrochloric acid and heating the samples at 70 OC. Since the experiments were conducted at a pH of approximately 8 (dissolved metal concentration less than a microgram per liter), total metal measurements represent the solids concentrations. Results and Discussion Numerical Simulations. Results of numerical simulations for quiescent sedimentation with various initial mass concentrations are presented in Figure 1 as mass concentration vs. time. For the lower initial concentrations there is a longer delay before removal of solids. As mass is reduced below its initial value, solids concentrations at any given time all approach the same value regardless of the initial concentration. To examine the reasons for this behavior, the simulation results are presented in Figure
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Mass Concentration, C (rng L-I) Flgure 2. Simulation results for quiescent sedimentation with different initial mass concentrations, expressed as the mass removal rate of solids, dCldt, vs. mass concentration (same conditions as in Figure 1).
2 as the mass removal rate of solids, dC/dt, vs. mass concentration. During the initial period of sedimentation, which corresponds to the development of a characteristic size distribution, smaller particles are aggregated into larger ones through coagulation, resulting in an increased rate of solids removal. Mass concentration remains relatively constant during this time. Since the coagulation rates are related to particle numbers, the time period is longer for the lower initial mass concentrations. A "characteristic" rate of solids removal is then achieved which is a function of the mass concentration and is not affected by initial conditions. The characteristic rate of removal for this particular simulation can be described by a second-order rate law (slope of 2 on Figure 2) over a concentration range of 1000-30 mg L-I. However, a better description for the characteristic removal rate over the entire range of mass concentration appears to be given by the summation of two power laws corresponding approximately to the differential settling term and the Brownian motion term in eq 9. The evolution of the particle size distribution over time provides a check on the assumptions of the simplified derivations. Results of the numerical simulation for the particle size distribution are presented in Figure 3a as dV/d(log u ) vs. u where
is employed since it emphasizes where the particle volume, and hence mass concentration, is located in the distribution. The results show that through coagulation the initial Envlron. Sci. Technol., Voi. 20, No. 2, 1986
189
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Flgure 4. Differential flux rate of particles, dE(v)/d(iog V), coagulating - to~ collision with particle out of size interval v = 0.68 - 1.47 ~ mdue of size V (abscissa). Particle Volume ( p m 3 )
Flgure 3. (a) Characteristicvolume distributions from simulation results with different inltiai mass concentrations (same results obtained for Co = 50 000 mg L-'; 5000 mg L-', and 500 mg L-I). (b) Coagulation rate of particles, E( v ) ,out of each particle size interval.
distribution of small particles is developed into a characteristic shape with a relative depletion in both small and large particles. Small particles are depleted due to efficient Brownian coagulation. Large particles are removed by settling faster than they are replenished by coagulation as mass concentration (related to particle numbers) is reduced. This results in a nonuniform reduction in the size distribution. The derivation of second-order removal kinetics by Morel and Schiff (14) in which the shape of the size distribution is assumed invariant over some fixed volume size is not consistent with these numerical results. Numerical results, presented in Figure 3b, also show that at any instant in time the rate of coagulation of particles out of a size interval roughly follows the volume distribution. This rate is not constant across the size distribution in violation of Hunt's (5, 6) second assumption. The contribution of each size class to the coagulation rate of particles out of a particular size interval (e.g., u = 0.68 - 1.47 pm3 in Figure 4) is represented as the differential volume flux rate, dE(u)/d(log b), vs. 8 , where
--
- 2 . 3 ~ b , P ( b~ ), n ( ~ ) n ( dAU ) (12) d(1og b) For the higher solids concentration (C = 360 mg L-') the coagulation of 0.68 - 1.47 pm3 size particles is controlled by collisions with particles of much larger size and not with similar size particles as assumed by Hunt (5,6). Although it may be argued that collision efficiencies would be expected to favor the coagulation of similar size particles, 190
Environ. Sci. Technoi., Voi. 20, NO. 2, 1986
Valioulis (12) has reported that this possibility may be expected only for the case of shear-induced coagulation. With the differential flux rate plotted on a linear scale and particle volume on a logarithmic scale, the relative importance of each collision mechanism in controlling coagulation out of a particular size interval (u = 0.68 - 1.47 pm3, Figure 4) can also be assessed by comparing the areas under each curve. For the higher solids concentration (C = 360 mg L-l) differential settling controls the coagulation of 0.68 - 1.47 pm3 size particles. At low solids concentration (C = 2.6 mg L-l) Brownian motion controls. These results contradict the first assumption employed by Hunt (5, 6) that one coagulation mechanism is dominant at a given particle size. Numerical results showing a nonuniform reduction in the characteristic size distribution (Figure 3) and coagulation of particles out of a particular size interval being controlled by different collision mechanisms as mass concentration is reduced (Figure 4) are all consistent with the simplified description of sedimentation proposed by Farley (16) and given in eq 9. To test this equation in greater detail, simulations have been performed for the separate cases of coagulation by Brownian motion, shear, and differential settling. Results presented in Figure 5 confirm that the characteristic rates of solids removal can be described by power law dependencies on mass concentration and that the exponent is dependent on the mode of coagulation. For Brownian motion, solids removal is proportional to C1.3,for shear proportional to C1.9, and for differential settling proportional to C2.3.These exponents correspond reasonably well with those of eq 9, the greatest difference (2.3 vs. 3.0) being observed for the case of coagulation by differential settling. The explanation for differences in the rate laws derived by similarity (16) and those obtained from numerical
Mass Concentrat on ' m g ~i
Flgure 5. Simulation results for the mass removal of solids for the separate cases of coagulatlon by Brownian motion, shear, and differential settling (p, = 2.0 g ~ m - p~p ,= 1.0 g ~ m - h~ =, 3.3 cm, 0 = 3.0 s-', T = 20 OC, p = 0.01 g cm-i Si).
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Flgure 7. Simulation results for the mass removal rate for a case in which all coa ulation mechanisms (including shear) are considered (p, = 2.0 g cm-y, p, = 1 0 0111-3, h = 3.3 cm, G = 10.0 s-1, = 20 o c , p = 0.01 g Cm-l's-B,.
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simulations is found by examining the dynamics of the particle size distribution. One may express the numerical results in terms of two normalizing parameters: the normalizing volume size 17 = u / a ( t ) and the self-preserving size distribution $$(v) (see eq 8). The resulting graph of the self-preserving size distribution vs. 71 shown in Figure 6 is analogous to Figure 3a and indicates the relative contribution of each size class to the total mass. The results show that some variation in the self-preserving function occurs as mass concentration is reduced, particularly for the case of coagulation by differential settling. Clearly a simple similarity transform is not strictly applicable. It has been shown to work when the collision frequency
function is constant (16). For more complex forms of the collision frequency function, the transform can only approximate the dynamics of the size distribution. In addition, for the hypothetical cases of coagulation by shear and differential settling, the contribution of smaller particles t~ mass concentrations does not necessarily approach zero. This results in a similarity prediction that continually creates mass at the very small particle sizes. For the case of coagulation by differential settling, these very small particle sizes make a significant contribution to the predicted m a s concentration. The characteristic removal rate given by eq 9 can thus only be viewed as a first approximation. We now propose instead a semi-empirical kinetic equation that has the form of eq 9 and includes the exponents of the three power laws obtained from the numerical simulation:
The dimensions for the rate coefficients are given as Bds (mg-1.3L1.3s-l)9 B ah (mg4.' Lo.'s-l) and Bb (mg4.3 s-l). This equation has been employed in fitting the characteristic rate of solids removal presented in Figure 2. For this case of quiescent sedimentation, the following rate coefficients were used: Bds = Bb = and Bsh = 0. As shown in Figure 2, solids removal is controlled by differential settling at high solids concentrations and by Brownian motion at low concentrations. For the intermediate concentration range, solids removal is determined by the addition of the two terms in accordance with eq 13. Numerical simulations have also been performed for cases in which shear coagulation is important. Results of Environ. Sci. Technol., Vol. 20, No. 2, 1986
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Table 11. Final Results for the Proposed Power Law Removal Relationship"
KBh, and Kb are dimensional parameters for the collision frequency functions; ab, Ct&, and adsare the collision efficiencies; (S/h)is the dimensional parameter for Stokes settling in a vertically homogeneous water column; and pe is the floc density (see Table I for details). I
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particles are present in the size distribution-a condition that does not occur at low concentrations where the Brownian motion term controls the rate of mass removal. The coefficient for removal by shear coagulation Bshis expected to depend on four parameter groupings (Slh,p e , Kb, Kb is included since its value is important in determining the coagulation of small particles present in the size distribution. With dimensions as a constraint, the relationship for Bsh (mg4.' Lo.' s-l) has one degree of freedom B~~oc (S/h)0.6(1-~)~ b0 . 4 ( 1 - a ) ~ ~ ~ a ~ ~ - 0 . 9 (15)
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The final relationship for Bsh obtained by regression of numerical data is shown in Figure 8b. A similar procedure is taken in determining a relationship for Bds (mg-1.3L1.3 s-'). For the case of quiescent sedimentation, Bdsis expected to depend on four parameter groups (Slh, p e t Kb, KdS)as follows:
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one simulation are presented in Figure 7. By use of values of Bdsand Bb obtained for quiescent sedimentation and a value of Bsh of the characteristic removal can adequately be described by the summation of power laws given in eq 13. Comparison of the mass removal rate with the rate obtained for quiescent sedimentation (dotted line on Figure 7) shows that shear coagulation primarily affects the removal rate for an intermediate range of mass concentration. Values for the sedimentation rate coefficients (Bb,Bsh, BB) obtained from all our numerical simulations were used in developing functional relationships describing the dependency of the rate coefficients (given in eq 13) on system parameters. Parameters that are involved in the dynamics of solids removal can be represented by five independent groupings: the four dimensional constants in Table I (Slh, Kb, Ksh, Kds) PIUS the floc density, pe. (Note that the parameter h is included since it is important in defining the suspension time of each particle size and also in determining the likelihood of coagulation prior to removal.) For solids removal controlled by Brownian motion, the coefficient Bb is expected to depend on three parameter groupings (S/h,pe, &). A relationship for Bb (mg-".3 s-l) is then obtained directly by dimensional analysis Bb a
(s/ h)0'6Kb0'4pe-0'3
(14)
As shown by the solid line in Figure Ba, this relationship agrees well with the rate coefficients obtained from numerical simulations. Ksh and Kdshave not been included in this relationship since coagulation by shear and differential settling would only be important when larger 192
Environ. Sci. Technol., Vol. 20, No. 2, 1986
(S/h)0.2(3-W b0~. 2 ( 2 - a ) ~ ~ ~ a ~ ~ - 1 . 3
(16)
With one degree of freedom, a final relationship for Bds is obtained by regression (Figure 8c). For cases in which shear coagulation was also considered, any possible effects of Kshon the rate coefficient Bds could not be detected in the simulation results. The relationship for Ads given in Figure 8c is therefore assumed to hold. As shown in Figure 8, similar relationships can be obtained for cases where collision efficiencies for Brownian motion, fluid shear, and differential settling ( a b , a s h , ad?) are less than unity. General expressions for the sedimentation rate coefficients, including the effects of collision efficiencies, are presented in Table 11. In fitting the numerical results for mass removal kinetics (with collision efficiencies of l),we included an initial lag time prior to the establishment of characteristic removal (Figure 1). This lag time, which corresponds to the time required by the coagulation mechanisms to transform the initial size distribution into its characteristic shape, was found to be dependent on the initial mass concentration (C, = 50000 mg L-l, to = 15 s; Co = 5000 mg L-l, to= 180 s; C, = 500 mg L-l, to = 2300 s). Since the initial size distributions included only small particles (ca. lo4 pm3), the values presented are indicative of maximum lag times. A comparison of the characteristic removal, assuming no initial lag time, and simulation results for various initial size distributions are presented in Figure 9. For several initial distributions, the assumption that characteristic removal is established almost instantaneously is a good approximation. Laboratory Experiments. Since an idealized description of sedimentation behavior is considered in the numerical simulations, laboratory data were analyzed to test the applicability of the proposed power law removal equation and rate coefficient relationships given in Table 11. (Note that all laboratory results for mass concentrations and sedimentation rate coefficients are based on dry weight
Particle Volume (brn3) 3
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Corresponding to the Characteristic Removal Equation with no Time Lag
v
Time (seconds)
Figure 11. Laboratory results for quiescent sedimentation of goethite particles in seawater (p, N 4.5 g p, = 1.025 g ~ m - h,,, ~ , = 44 cm, T = 20 "C).
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Flgure 10. Laboratory results for the quiescent sedimentation of metallic copper particles in seawater (pp = 8.9 g cmm3,pf = 1.025 h,, = 44 cm, = 20 OC). g
r
measurements.) Results for the quiescent sedimentation ~ ) presented of metallic copper particles ( p , = 8.9 g ~ m - are in Figure 10. Although there exists a vertical gradient in the solids concentration during part of the experiment, the water column is considered vertically homogeneous for data analysis. The assumption that characteristic removal is established almost instantaneously is also employed. A good agreement of the experimental data with the proposed power law equation is obtained by using values of and 10-6.afor the Sedimentation rate coefficients Bds and Bb, respectively. As shown in Figure 10, the differential settling term controls the removal kinetics for the
higher concentration range. At low solids concentration (less than 10 mg L-l), solids removal begins to be influenced by the Brownian motion term. In conducting "quiescent" sedimentation experiments, it is nearly impossible to eliminate all extraneous sources of turbulence (due to convection,sampling, etc.). Although some low level of turbulence may exist in the sedimentation column, from analysis of numerical results we have concluded that for high-density particles such as copper a relatively high shearing rate of 0.2 s-l would be necessary for shear coagulation to have any effect on the removal rate. Low turbulence levels however may inhibit the settling of very small particles. This would result in a decreasing rate of solids removal at very low concentrations and is offered as a possible explanation for the last few data points shown in Figure 10. Results of a similar experiment for goethite particles (p, N 4.5 g ~ m - are ~ ) presented in Figure 11. The power law removal equation was fitted to the data by considering only the differential settling term to be important (Bds= lo4.%). As shown, the removal equation provides a good description of the data after approximately 2 h of sedimentation. Prior to this time, solids removal is probably affected by initial conditions. For instance, if the initial size distribution contains large particles we would expect solids removal to be initially fast due to settling and coagulation of the large particles. It should be noted that these experimental data, as well as the copper data (Figure lo), cannot be described adequately by a second-order rate law. To examine the effect of shear coagulation on mass removal kinetics, we have analyzed the laboratory data of Hunt and Pandya (13). Comparison of their data with the power law removal equation is presented in Figure 12. In fitting the data, the Brownian motion term was considered not to be important in determining solids removal over the observed range of concentration. For all shearing rates, the removal equation provides a good description of the data. Comparisons of sedimentation rate coefficients obtained from analysis of laboratory data (based on dry weight) with the rate coefficient relationships determined from numerical results (based on wet weight) are performed for B s h and Bds with the conversion factor relating dry weight to wet weight given as
Environ. Sci. Technol., Voi. 20, No. 2, 1986
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Sewage Sludge in seawater
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Sedimentation Experiments
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Laboratory rate coefficients are then expressed as equivalent wet weight values B,h(wet weight) = f-O.gB,h(dryweight)
(18)
&,(wet weight) = f-1.3Bd,(dryweight) (19) For the comparisons presented in Figure 13, a value of the vertically homogeneous water column depth was taken over a range corresponding to the sample depth and the total water column height. The floc porosity was estimated to be 0.8 (see Appendix). Since the particle density for the sludge experiments was not reported, a value of 1.5 g cm-3 was assumed based on the range 1.37-1.69 g cm-3 previously reported by Faisst (4). As shown in Figure 13, laboratory coefficients for the copper, goethite, and sludge data previously presented and for fly ash data (7) correspond reasonably well with our rate coefficient relationships for collision efficiencies ranging from 0.2 to 1.0. A similar range of collision efficiencies has been reported by Gibbs (20) for coagulation in seawater solutions. With the exception of the copper data (Figure lo), it was not possible to determine a laboratory value for the rate coefficient Bb The value of Bb obtained from the copper experiment is approximately an order of magnitude lower than what would be expected based on numerical results. Since metallic particles typically exhibit a relatively high surface charge, it is possible that even in seawater, double-layer effects could be important in reducing the efficiency of particle collision. Although further laboratory study is needed, we have concluded from simulation resulta that, for field applications where mixing typically occurs over depths much larger than those in a laboratory column, the Brownian motion term in the power law removal equation would not be significant for mass concentrations greater than 1 mg L-l.
Conclusions The resulta of the analytical, numerical, and laboratory studies presented in this paper appear to be consistent and their combination provides insight into the kinetic behavior of sedimentation. A semiempirical approximate solution to the dynamic equation for sedimentation describes the mass removal of solids as a simple summation of power laws (Table 11). Relationships for sedimentation rate Environ. Sci. Technol., Vol. 20,No. 2, 1986
/+-
Sedimentation Experiment
Flgure 12. Analysis of laboratory results for the sedimentation of sludge for various fluid shearing rates. Data are from Hunt and Pandya ( 13).
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Flgure 13. Comparison of laboratory rate coefficients (in equivalent wet weight values) with the rate coefficient relationships determined from numerical results for Bshand Bds.
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Flgure 14. Laboratory results (from transmission-electron micrographs) for two-dimensional projections of gold collold aggregates. Colloids are unlformly sized, spherical particles approximately 14.5 nm in diameter. Data are from Weitz and Oliveria (27).
coefficients (Bb, Bsh,Eds) have been determined as functions of system parameters. Laboratory rate coefficients for seawater experiments compare reasonably well with our rate coefficient relationships for collision efficiencies ranging from 0.2 to 1.0. Additional work on floc porosity and collision efficiencies however is needed. In applying the proposed power law removal equation to field conditions, only the differential settling term and the shear term are expected to be important. Removal rates could therefore be approximated reasonably well by a second-order expression. However, it should be emphasized that (whether by direct calculations from system
Environ. Sci. Technol. 1986, 20, 195-200
California Institute of Technology). Hunt, J. R. Environ. Sei. Technol. 1982, 16, 303-309. Farley, K. J.; Morel, F. M. M.; Harleman, D. R. F. “Ponding of Effluents from Fossil Fuel Steam Electric Power Plants”; R. M. Parsons Laboratory, Massachusetts Institute of Technology: Cambridge, MA, 1984. Smoluchowski, M. 2. Phys. Chem. (Munich) 1916, 17, 537-585. Smoluchowski, M. 2. Phys. Chem. (Munich) 1917, 92, 129-168. Friedlander, S. K. “Smoke, Dust and Haze: Fundamentals of Aerosol Behavior”; Wiley-Interscience: New York, 1977. Stumm, W.; Morgan, J. J. “Aquatic Chemistry”, 2nd ed.; Wiley-Interscience: New York, 1981. Valioulis, I. A. W. M. Keck Laboratory of Environmental Engineering Science, Report KH-R-44, California Institute of Technology: Pasadena, CA, 1983. Hunt, J. R.; Pandya, J. D. Environ. Sci. Technol. 1984,18, 119-121. Morel, F. M. M.; Schiff, S. L. R. M. Parsons Laboratory, Report 259, Massachusetts Institute of Technology: Cambridge, MA, 1980. Lawler, D. F.; O’Melia, C. R.; Tobiason, J. E. In “Particulates in Water”; Kavanaugh, M. C., Leckie, J. O., Eds.; American Chemical Society: Washington, DC, 1980, pp 353-390. Farley, K. J. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1984. Friedlander, S. K. J . Meterol. 1960, 17, 373-374. Friedlander, S. K. J . Meterol. 1960, 17, 479-483. Wang, C. S.; Friedlander, S. K. J . Colloid Interface Sci. 1967,24, 170-179. Gibbs, R. J. Environ. Sci. Technol. 1983, 17, 237-240. Weitz, D. A,; Oliveria, M. Phys. Rev. Lett. 1984, 52, 1433-1436.
parameters or by extrapolation from laboratory observations) the proposed power law relationship provides a better estimation of removal rates. Finally, in applying these results, particularly for the case of ocean disposal, questions concerning initial sedimentation behavior, mixing depths, and vertical mass concentration gradients should be addressed. Acknowledgments We extend our thanks to K. Stolzenbach, R. Mercier, J. Hunt, J. Connolly, D. Dzombak, and B. Hanrahan for their various contributions. Appendix In our description of sedimentation, the effective particle density (floc density) is considered constant over the size distribution. It is realized that this assumption is not appropriate for the initial aggregation of primary particles. However, as larger aggregates are formed, the porosity of the flocs is expected to reach a limiting values (resulting in a constant floc density). Such behavior is demonstrated in Figure 14 where a slope of 2 is indicative of constant porosity. From the experimental data, a value in the range 0.7-0.8 is obtained for the aggregate porosity. Literature Cited Krone, R. B. Committee on Tidal Hydraulics, Technical Bulletin 7, U.S. Army Corps of Engineers: Vicksburg, TN, 1972. Edzwald, J. K.; Upchurch, J. B.; O’Melia, C. R. Environ. Sci. Technol. 1974, 8, 58-63. Morel, F. M. M.; Westall, J. C.; O’Melia, C. R.; Morgan, J. J. Environ. Sci. Technol. 1975, 9, 756-761. Faisst, W. K. In ”Particulates in Water”; Kavanaugh, M. C., Leckie, J. O., Eds.; American Chemical Society: Washington, DC, 1980; pp 259-282. Hunt, J. R. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1980 (published as Report AC-5-80, W. M. Keck Laboratory of Environmental Engineering Science,
Received for review July 9,1984. Revised manuscript received May 23, 1985. Accepted September 23, 1985. This work was supported i n part by NOAA Grant NA79AA-D-00077 and EPA Grant CR-811181-01-01 and a combined grant from American Electric Power Service Corp., Duke Power Co., Northeast Utilities Service Co., and PS&G Research Corp.
Airborne Dioxins and Dibenzofurans: Sources and Fates Jean M. Czuczwa and Ronald A. Hltes” School of Public and Environmental Affairs and Department of Chemistry, Indiana University, Bloomington, Indiana 47405
Polychlorinated dibenzo-p-dioxins (PCDD) and dibenzofurans (PCDF) were found in urban air particulates and Great Lakes sediments. In all samples, octachlorodibenzo-p-dioxinpredominated. Combustion of municipal and chemical wastes was the most likely source of these compounds. When these data were compared with dioxins and dibenzofurans found in combustion sources, we found evidence for the loss of the less chlorinated dioxins and dibenzofurans. Using principal components analysis, we found that the most likely source of PCDD and PCDF to a western Lake Ontario site was pentachlorophenol production. Historical fluxes of dioxins and dibenzofurans to sediment cores from Lake Erie and Siskiwit Lake (Isle Royale) suggest that the incineration of chloro aromatics has been an important source of dioxins and dibenzofurans. 0013-936X/86/0920-0195$01.50/0
Introduction Polychlorinated dibenzo-p-dioxins (PCDD) have become an issue of national importance. This has probably resulted from two facts: First, 2,3,7,8-tetrachlorodibenzop-dioxin (2378-TCDD) kills guinea pigs at a dose of 0.6 Mglkg, thus making it the most toxic anthropogenic compound yet known (1). Second, Agent Orange, which was used as a defoliant in the Vietnam War, was contaminated with an average of 2 ppm of 2378-TCDD (2). Other incidents such as the explosion of a trichlorophenol reactor near Seveso, Italy (3), and the contamination of Times Beach, MO ( 4 ) ,have reinforced the public’s perception that dioxins are threats to its health. More recently, it has been learned that the incineration of municipal and chemical wastes produces dioxins, some of which are released into the atmosphere. This infor-
0 1986 American Chemical Society
Environ. Sci. Technol., Vol. 20, No. 2, 1986
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