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Fluid mechanics and fractal aggregates
Wat. Res. Vol. 27, No. 9, pp. 1493-1496,1993 Printed in Great Britain.All rights reserved
0043-1354/93 $6.00+ 0.00 Copyright © 1993Pergamon Press Ltd
RESEARCH
FLUID MECHANICS
AND
NOTE
FRACTAL
AGGREGATES
SHANKARARAMAN CHELLAM and MARK R. WIESNERO Department of Environmental Science and Engineering, George R. Brown School of Engineering, P.O. Box 1892, Rice University, Houston, TX 77251, U.S.A. (First received August 1992; accepted in revised form February 1993)
Aima'act--The disturbances in uniform creeping flow in the presence of an isolated porous floc are investigated theoretically. Using the Carman-Kozeny equation, the floc permeability is related to its fractal dimension, D. Fluid streamlines, drag coefficient and the fluid collection et~ciency of porous aggregates are expressed in terms of D. As D increases, for a fixed packing factor and ratio of primary particle radius to floc radius, the permeability is found to decrease and the fluid mechanics resembles more closely that of an isolated impermeable sphere. As a simplification, it is suggested that a rectilinear model for flow up to an impervious sphere may be a reasonable approximation for aggregatv-aggregat¢ and particle-aggregate interactions if D ~ 2. Curvilinear models for flow up to an impervious sphere may be accurate approximations for interactions involving aggregates with higher fractal dimensions (D ;~ 2.3). Key words--fractal dimension, flocculation, sedimentation, fluid mechanics, streamlines, water treatment
INTRODUCTION A description of the fluid mechanics of porous clusters is of interest in modeling unit processes such as flocculation, sedimentation and filtration (Lawler, 1986; Li and Ganczarczyk, 1992; Mulder and Gimbel, 1991) as well as in polymer physics (Felderhof, 1975). A theoretical expression for the drag force experienced by isolated porous spheres under creeping flow conditions was first developed by Sutherland and Tan (1970) as a model for floes. Fluid flow internal to the floes was modeled using Darcy's law. Their treatment was later refined by Neale et al. (1973) by using Brinkman's equation to describe flow internal to the sphere. Experimental verification of their expression was provided by Matsumoto and Suganuma (1977) and Masliyah and Polikar (1980). Initial interest in the mechanics of isolated porous clusters was restricted to calculation of the drag coefficient. Adler (1981) investigated streamlines in such geometries under both uniform as well as simple shear flows. Floes encountered in water and wastewater treatment applications can be described using fractal geometry (Li and Ganczarczyk, 1989). By fractal we mean that the spatial variation of mass m, with radial distance r, can be described using a power law of the type, m ocr ~
(I)
where 1 < D < 3 is the fractal dimension of the floe. In this paper we use a simple model to express the floc permeability in terms of the fractal dimension. Using this relation we analyze the perturbation in uniform
creeping flow caused by an isolated porous sphere as a function of its fractal dimension. GOVERNING EQUATIONSOF FLOW The isothermal flow of an incompressible, Newtonian fluid having viscosity tJ, exterior to the floc is assumed to satisfy
~V2vffi Vp
(2)
V.v=O
(3)
where p is the pressure and v is the velocity vector. Flow internal to the sphere having permeability k is modeled either using Darcy's law - 7
K
v=
vp
(4)
or the equation proposed originally by Brinkman (1947) /t - ~ v ÷ ~ V 2 v = Vp
(5)
Neale et al. (1973) provide an excellent discussion of the appropriate boundary conditions. Boundary conditions at the surface of the sphere involve imposing the continuity of the fluid velocity vector and the pressure. At large distances from the sphere, the velocity should asymptotically approach the undisturbed velocity. A!so, at the center of the sphere the velocities should remain finite.
1493
1494
Research Note
PERMEABILITY AND FRACTAL DIMENSIONS OF AGGREGATES
A simple model relating the permeability of a cluster of radius a, composed of smaller spheres of radius ap to its packing factor ~ and fractai dimension D can be derived using the Carman-Kozeny equation. Assuming that the porosity within the cluster is constant, Rogak and Flagan (1990) present the following expression for the dimensionless permeability (~-2__ k/a2). Note that their result has a typographical error and the correct expression should read, ~-2 = [ 1 - • (ap/a)3-D] ((aP/~22D - 6 2(a~ a)~-3
+ 1
spherical floc immersed in a uniform flow are shown in Fig. I for three cases; one characteristic of DLA (D -- !.78), another produced by RLA (D = 2) and another characteristic of that produced with aggregation followed by restructuring (D = 2.3). It is noted that as D decreases and the permeability increases, streamlines approach that of uniform rectilinear flow. The ratio of the flow rate passing through the sphere to that approaching it is defined as the fluid collection efficiency r/. This ratio is dependent on the equation used to model internal flow (Adler, 1981; Neale et al., 1973). 3 r / = 3 + 2~ e if Darcy's law is used
)(a4~
(6)
(7)
~1 = 1 - d / ~ - c / ~ 3 if Brinkman's law is used (8)
Low values of D correspond to very porous flocs for which the Kozeny equation is not strictly valid. Thus, equation (6) provides a relationship between the permeability and fractal dimension for D I> 2. Theoretically, the packing factor can assume values between zero and a/a v. The case ofy = 1 corresponds to a solid object if D -- 3. As 3,---*0 floc size is large compared with the size and number of the primary particles making up the floc. Using experimental data on clay-aluminium flocs (Tambo and Watanabe, 1979) we have estimated a value of y equal to 0.5 which we have used throughout this paper. Flocs encountered in water and wastewater treatment processes can have a wide range of fractal dimensions. Fractal dimensions calculated for aggregates formed in natural and engineered systems typically fall between 1.7 and 2.85 (Jiang and Logan, 1991; Li and Ganczarczyk, 1989; Wiesner, 1992). One natural limit is that of diffusion limited aggregation (DLA) which occurs when particle transport is dominated by Brownian motion and all collisions result in aggregation. Large floes formed under such conditions are expected to be very porous. Fractal dimensions obtained under DLA conditions have been reported to be approximately equal to 1.78 (Jullien and Botet, 1987). If all collisions do not result in aggregation (reaction limited aggregation, RLA) aggregates are more compact and D ~ 2. Restructuring in the floc after initial aggregation produces even more compact flocs with higher values of D. Under such conditions, fractal dimensions in the range 2 < D < 2.85 have been observed in numerical and laboratory experiments (e.g. Jullien and Botet, 1987 and references within). STREAMLINES, FLUID COI~[CrlON EFFICIENCY AND DRAG
Simultaneous solution of equations (2), (3) and (4) is possible in spherical polar coordinates by introducing the stream function (Sutherland and Tan, 1970). Streamlines for the case of a porous
Fig. I. Streamlines for creeping flow in the neighborhood of a porous sphere for three cases each with
ap/a=
0.005,
~, -- 0.5. The top figure corresponds to aggregation followed by restructuring (D =2.3, ~-2=0.0036). The middle figure represents reaction limited aggregation (D =2,
-2= 0.0882). The bottom figure depicts the DLA limit ~-'=0.9126). It should be noted that the relationship between D and ~ -2 is strictly valid for 2 ~
Research Note
Dimensionlesslloc permeability ~-2 0 t L
2
4
6
I
8
I
10
I
x
0.6
2~2
" "
'-.
',,
~o.4
o.,f 0
1
1.6
'
= - -
I
- .... ".,..
By using Brinkman's equation to describe internal flow, the expression for [1 becomes (Neale et al., 1973),
+ I
2
i
22
f~ =
I
2.4
'
I
2.6
Fractal dimension D
I
2.8
3
Fig. 2. Fluid collection efficiencies as a function of the fractal dimension and the dimensionless permeability for a fixed value of the packing factor (y ffi 0.5). Fluid flow within the porous cluster was modeled using Brinkman's equation. Note however that the relationship between the fractal dimension and the agg~gate l~'rmeability was derived using Darcy's law to describe internal flow and hence is strictly valid only for 2 g D ~<3. The coefficients d and c appearing in equation (8) are given below: d =.3 ~ 3 ( 1 I{
tanh~ ~ ) tanh ~ (3~5 + 6~3)}
(9)
(14)
SUMMARY AND IMPLICATIONS Using a simple model the permeability of a porous aggregate is related to its fractal dimension. It is demonstrated that the fractai dimension increases as the dimensionless permeability decreases. Fluid drag on porous spheres characterized by fractal dimensions in the range reported for flocs in water and
(10)
Dimensionless permeability E-2
0
J = 2~ 2 + 3 - 3 tanh ~
2~2(1 - tanh ~/~) 2~ 2 + 3(1 - tanh ~/~)
Equations (I 3) and (14) are presented graphically in Fig. 3 as a function of the dimensionless permeability and fractal dimension of the floe. When D increases, floes become more compact and less permeable and f~---*l. Again, considerable difference is observed in the predictions of Darcy's and Brinkman's laws. Since using Brinkman's equation to model internal flow results in greater fluid collection efficiencies compared to Darcy's law we see that it also predicts lower drag.
and
1
2
3
4
5
(11)
Figure 2 shows the fluid collection efficiency calculated by equations (7) and (8) expressed in terms of both the dimensionless permeability as well as the fractal dimension for throe different ap/a ratios. As fractal dimension decreases (more open clusters) floes can be predicted to have higher permeability (high -2). As expected, we see in Fig. 2 that for any fixed a~/a, ~--.+0 as D---~3 (~-2--~0) and ff---~l as D---q (~-2---~oo). For dense aggregates having the same fractai dimension (D > 2), fluid collection efficiency increases with ap/a. In this regime, Darcy's law is a valid description of internal flow. For highly porous and permeable floes (D < 2), Brinkman's extension of Darcy's law may be preferred and hence equation (6) may not be accurate in relating the fractal dimension and the permeability, We see qualitatively different behavior as regards the fluid collection efficiency in this regime, i.e. ~ decreases with increasing ap/a ratio for aggregates having the same fractal dimension.
WR 2T/9---G
(13)
2~2+3
... "',,
1,8
(I 2)
Sutherland and Tan (1970) used Darcy's law and derived
+ i,/+,+iii
~
The drag force experienced by the floc can be expressed as a dimensionless ratio (Neale et al., 1973). [~ = drag on the permeable sphere of radius a drag on a solid sphere of radius a
=.o.sj~\ I ~~ ~,.. \ "~
1495
a
0.8
:~,
O
\
';
,
+ ;~ "~ 0
0.4 t/
/'/-
/o .o05
,,i/, , ,. ............ Dar::equation / ' ~ k m a n ' s equation ,'.. "~'/
0.2
01,6
""'"'"'"...
1.8
2
2,2
2.4
21.6
21.8
Fractal Dimension D
Fig. 3. Comparison of the ratio of the drag experienced by a permeable sphere to that of a solid sphere of the same radius. When Brinkman's equation is used to describe internal flow (instead of Darcy's law), a smaller value of drag is predicted.
1496
Research Note
REFERENCES wastewater applications is significantly smaller than that on non-porous spheres. It is observed that fluid Adler P. M. (1981) Streamlines in and around porous collection efficiencies for such floes can also be signifiparticles.J. Colloid Interface Sci. 81, 531-535. Brinkman H. C. 0947) A calculation of the viscous force cantly greater than zero. extended by a fowing fluidon a dense swarm of particles. There is considerable discussion as to the appropriAppL Sciet. Rea. AI, 27-34. ateness of rectilinear and eurvilinear models in aggreFclderhof B. U. (1975) Frictionalproperties of dilutepolygation studies (e.g. Han and Lawler, 1991, 1992). For mer solutions. Physica A 80, 63-75. floes in the DLA and RLA regimes (D ~< 2), rectilin- Han M. and Lawler D. F. (1991) Interactionsof two settling spheres: settling rates and collision efficiency.J. Hyd. ear models for particle-aggregate and aggreEngng I17, 1269-1289. gate-aggregate interactions may be more accurate Han M. and Lawler D. F. (1992) The (relative)insignifithan curvilinear models that in each case, model flow cance of G in flocculation.J. Am. Wat. Wks Ass. 84, around the floc as that around an "equivalent" solid 79-91. sphere. At higher fractal dimensions, fluid collection Happcl J. and Brenner H. (1973) Low Reynolds Number Hydrodynamics. Noordhoff, Leyden, The Netherlands. efficiencies are low and the error associated with a Jiang Q. and Logan B. E. (1991) Fractal dimensions of simple rectilinear model for fluid flow is expected to aggregates determined from steady-state size distribe significant. These observations are very sensitive to butions. Envir. Sci. Technol. 25, 2031-2038. the value assigned to the packing factor. F o r a fixed Jullicn R. and Botet R. (1987) Aggregation and Fractal Aggregates. World Scientific,Singapore. D, the permeability of a cluster increases as ~, deLawler D. (1986) Removing particlesin water and wastecreases. Therefore, the error associated with computwater. Envir. Sci. Technol. 20, 856-861. ing the interactions with floes having small values of Li D. and Ganczarczyk J. (1989) Fractal geometry of using a rectilinear model may be very small. particle aggregates generated in water and wastewater treatment processes. Envir. Sci. Technol. 23, 1385-1389. Omission of the inertial terms in the Navier-Stokes equations compared to the viscous terms (creeping Li D. and Ganczarezyk J. (1992) Advective transport in activated sludge floes. Wat. envir. Res. 64, 236-240. flow) is an excellent approximation in sedimentation Masliyah J. H. and Polikar M. (1980) Terminal velocity of where the Reynolds number (based on particle diamporous spheres. Can J. Chem. Engng 58, 299-302. eter) Re, is typically less than 0.1. Even at very high Matsumoto K. and Suganuma A. (1977) Settling velocity of a permeable model floe. Chem. Engng Sci. 32, 445--447. hydraulic loading rates, the packed bed applications encountered in water and wastewater treatment (con- Mulder T. and Gimbel R. (1991) On the development of high performance filtration materials for deep bed filters. ventional filters, GAC column adsorbers, etc.) are Sep. Technol. 1, 153-165. characterized by Re < 1. Deviations from linearity Neale G., Epstein N. and Nader W. (1973) Creeping flow are small in packed beds for Re < 5 (Happel and relative to permeable spheres. Chem. Engng Sci. 2~ 1865-1874. Brenner, 1973). Hence, the assumption of creeping flow is also valid in packed beds of spheres. Calcu- Rogak S. N. and Flagan R. C. (1990) Stokes drag on self-similar clusters of spheres. J. Colloid Interface Sci. lations of the collision rate kernels in particle popu134 206-218. lation models for flocculation may require a Sutherland D. N. and Tan C. T. (1970) Sedimentation of a porous sphere. Chem. Engng Sci. 25, 1950-1951. consideration of inertial effects. In flocculation basins operating under turbulent flow conditions, the analy- Tambo N. and Watanabe Y. (1979) Physical characteristics of flocs--I. The foe density function and aluminium floe. sis presented here may only be a reasonable descripWat. Res. 13, 409--419. tion of aggregate--flow interactions occurring below Wiesner M. R. (1992) Kinetics of aggregate formation in the Kolmogorov microscale. rapid mix. Wat. Res. 26, 379-387.
0043-1354/93 $6.00+ 0.00 Copyright © 1993Pergamon Press Ltd
RESEARCH
FLUID MECHANICS
AND
NOTE
FRACTAL
AGGREGATES
SHANKARARAMAN CHELLAM and MARK R. WIESNERO Department of Environmental Science and Engineering, George R. Brown School of Engineering, P.O. Box 1892, Rice University, Houston, TX 77251, U.S.A. (First received August 1992; accepted in revised form February 1993)
Aima'act--The disturbances in uniform creeping flow in the presence of an isolated porous floc are investigated theoretically. Using the Carman-Kozeny equation, the floc permeability is related to its fractal dimension, D. Fluid streamlines, drag coefficient and the fluid collection et~ciency of porous aggregates are expressed in terms of D. As D increases, for a fixed packing factor and ratio of primary particle radius to floc radius, the permeability is found to decrease and the fluid mechanics resembles more closely that of an isolated impermeable sphere. As a simplification, it is suggested that a rectilinear model for flow up to an impervious sphere may be a reasonable approximation for aggregatv-aggregat¢ and particle-aggregate interactions if D ~ 2. Curvilinear models for flow up to an impervious sphere may be accurate approximations for interactions involving aggregates with higher fractal dimensions (D ;~ 2.3). Key words--fractal dimension, flocculation, sedimentation, fluid mechanics, streamlines, water treatment
INTRODUCTION A description of the fluid mechanics of porous clusters is of interest in modeling unit processes such as flocculation, sedimentation and filtration (Lawler, 1986; Li and Ganczarczyk, 1992; Mulder and Gimbel, 1991) as well as in polymer physics (Felderhof, 1975). A theoretical expression for the drag force experienced by isolated porous spheres under creeping flow conditions was first developed by Sutherland and Tan (1970) as a model for floes. Fluid flow internal to the floes was modeled using Darcy's law. Their treatment was later refined by Neale et al. (1973) by using Brinkman's equation to describe flow internal to the sphere. Experimental verification of their expression was provided by Matsumoto and Suganuma (1977) and Masliyah and Polikar (1980). Initial interest in the mechanics of isolated porous clusters was restricted to calculation of the drag coefficient. Adler (1981) investigated streamlines in such geometries under both uniform as well as simple shear flows. Floes encountered in water and wastewater treatment applications can be described using fractal geometry (Li and Ganczarczyk, 1989). By fractal we mean that the spatial variation of mass m, with radial distance r, can be described using a power law of the type, m ocr ~
(I)
where 1 < D < 3 is the fractal dimension of the floe. In this paper we use a simple model to express the floc permeability in terms of the fractal dimension. Using this relation we analyze the perturbation in uniform
creeping flow caused by an isolated porous sphere as a function of its fractal dimension. GOVERNING EQUATIONSOF FLOW The isothermal flow of an incompressible, Newtonian fluid having viscosity tJ, exterior to the floc is assumed to satisfy
~V2vffi Vp
(2)
V.v=O
(3)
where p is the pressure and v is the velocity vector. Flow internal to the sphere having permeability k is modeled either using Darcy's law - 7
K
v=
vp
(4)
or the equation proposed originally by Brinkman (1947) /t - ~ v ÷ ~ V 2 v = Vp
(5)
Neale et al. (1973) provide an excellent discussion of the appropriate boundary conditions. Boundary conditions at the surface of the sphere involve imposing the continuity of the fluid velocity vector and the pressure. At large distances from the sphere, the velocity should asymptotically approach the undisturbed velocity. A!so, at the center of the sphere the velocities should remain finite.
1493
1494
Research Note
PERMEABILITY AND FRACTAL DIMENSIONS OF AGGREGATES
A simple model relating the permeability of a cluster of radius a, composed of smaller spheres of radius ap to its packing factor ~ and fractai dimension D can be derived using the Carman-Kozeny equation. Assuming that the porosity within the cluster is constant, Rogak and Flagan (1990) present the following expression for the dimensionless permeability (~-2__ k/a2). Note that their result has a typographical error and the correct expression should read, ~-2 = [ 1 - • (ap/a)3-D] ((aP/~22D - 6 2(a~ a)~-3
+ 1
spherical floc immersed in a uniform flow are shown in Fig. I for three cases; one characteristic of DLA (D -- !.78), another produced by RLA (D = 2) and another characteristic of that produced with aggregation followed by restructuring (D = 2.3). It is noted that as D decreases and the permeability increases, streamlines approach that of uniform rectilinear flow. The ratio of the flow rate passing through the sphere to that approaching it is defined as the fluid collection efficiency r/. This ratio is dependent on the equation used to model internal flow (Adler, 1981; Neale et al., 1973). 3 r / = 3 + 2~ e if Darcy's law is used
)(a4~
(6)
(7)
~1 = 1 - d / ~ - c / ~ 3 if Brinkman's law is used (8)
Low values of D correspond to very porous flocs for which the Kozeny equation is not strictly valid. Thus, equation (6) provides a relationship between the permeability and fractal dimension for D I> 2. Theoretically, the packing factor can assume values between zero and a/a v. The case ofy = 1 corresponds to a solid object if D -- 3. As 3,---*0 floc size is large compared with the size and number of the primary particles making up the floc. Using experimental data on clay-aluminium flocs (Tambo and Watanabe, 1979) we have estimated a value of y equal to 0.5 which we have used throughout this paper. Flocs encountered in water and wastewater treatment processes can have a wide range of fractal dimensions. Fractal dimensions calculated for aggregates formed in natural and engineered systems typically fall between 1.7 and 2.85 (Jiang and Logan, 1991; Li and Ganczarczyk, 1989; Wiesner, 1992). One natural limit is that of diffusion limited aggregation (DLA) which occurs when particle transport is dominated by Brownian motion and all collisions result in aggregation. Large floes formed under such conditions are expected to be very porous. Fractal dimensions obtained under DLA conditions have been reported to be approximately equal to 1.78 (Jullien and Botet, 1987). If all collisions do not result in aggregation (reaction limited aggregation, RLA) aggregates are more compact and D ~ 2. Restructuring in the floc after initial aggregation produces even more compact flocs with higher values of D. Under such conditions, fractal dimensions in the range 2 < D < 2.85 have been observed in numerical and laboratory experiments (e.g. Jullien and Botet, 1987 and references within). STREAMLINES, FLUID COI~[CrlON EFFICIENCY AND DRAG
Simultaneous solution of equations (2), (3) and (4) is possible in spherical polar coordinates by introducing the stream function (Sutherland and Tan, 1970). Streamlines for the case of a porous
Fig. I. Streamlines for creeping flow in the neighborhood of a porous sphere for three cases each with
ap/a=
0.005,
~, -- 0.5. The top figure corresponds to aggregation followed by restructuring (D =2.3, ~-2=0.0036). The middle figure represents reaction limited aggregation (D =2,
-2= 0.0882). The bottom figure depicts the DLA limit ~-'=0.9126). It should be noted that the relationship between D and ~ -2 is strictly valid for 2 ~
Research Note
Dimensionlesslloc permeability ~-2 0 t L
2
4
6
I
8
I
10
I
x
0.6
2~2
" "
'-.
',,
~o.4
o.,f 0
1
1.6
'
= - -
I
- .... ".,..
By using Brinkman's equation to describe internal flow, the expression for [1 becomes (Neale et al., 1973),
+ I
2
i
22
f~ =
I
2.4
'
I
2.6
Fractal dimension D
I
2.8
3
Fig. 2. Fluid collection efficiencies as a function of the fractal dimension and the dimensionless permeability for a fixed value of the packing factor (y ffi 0.5). Fluid flow within the porous cluster was modeled using Brinkman's equation. Note however that the relationship between the fractal dimension and the agg~gate l~'rmeability was derived using Darcy's law to describe internal flow and hence is strictly valid only for 2 g D ~<3. The coefficients d and c appearing in equation (8) are given below: d =.3 ~ 3 ( 1 I{
tanh~ ~ ) tanh ~ (3~5 + 6~3)}
(9)
(14)
SUMMARY AND IMPLICATIONS Using a simple model the permeability of a porous aggregate is related to its fractal dimension. It is demonstrated that the fractai dimension increases as the dimensionless permeability decreases. Fluid drag on porous spheres characterized by fractal dimensions in the range reported for flocs in water and
(10)
Dimensionless permeability E-2
0
J = 2~ 2 + 3 - 3 tanh ~
2~2(1 - tanh ~/~) 2~ 2 + 3(1 - tanh ~/~)
Equations (I 3) and (14) are presented graphically in Fig. 3 as a function of the dimensionless permeability and fractal dimension of the floe. When D increases, floes become more compact and less permeable and f~---*l. Again, considerable difference is observed in the predictions of Darcy's and Brinkman's laws. Since using Brinkman's equation to model internal flow results in greater fluid collection efficiencies compared to Darcy's law we see that it also predicts lower drag.
and
1
2
3
4
5
(11)
Figure 2 shows the fluid collection efficiency calculated by equations (7) and (8) expressed in terms of both the dimensionless permeability as well as the fractal dimension for throe different ap/a ratios. As fractal dimension decreases (more open clusters) floes can be predicted to have higher permeability (high -2). As expected, we see in Fig. 2 that for any fixed a~/a, ~--.+0 as D---~3 (~-2--~0) and ff---~l as D---q (~-2---~oo). For dense aggregates having the same fractai dimension (D > 2), fluid collection efficiency increases with ap/a. In this regime, Darcy's law is a valid description of internal flow. For highly porous and permeable floes (D < 2), Brinkman's extension of Darcy's law may be preferred and hence equation (6) may not be accurate in relating the fractal dimension and the permeability, We see qualitatively different behavior as regards the fluid collection efficiency in this regime, i.e. ~ decreases with increasing ap/a ratio for aggregates having the same fractal dimension.
WR 2T/9---G
(13)
2~2+3
... "',,
1,8
(I 2)
Sutherland and Tan (1970) used Darcy's law and derived
+ i,/+,+iii
~
The drag force experienced by the floc can be expressed as a dimensionless ratio (Neale et al., 1973). [~ = drag on the permeable sphere of radius a drag on a solid sphere of radius a
=.o.sj~\ I ~~ ~,.. \ "~
1495
a
0.8
:~,
O
\
';
,
+ ;~ "~ 0
0.4 t/
/'/-
/o .o05
,,i/, , ,. ............ Dar::equation / ' ~ k m a n ' s equation ,'.. "~'/
0.2
01,6
""'"'"'"...
1.8
2
2,2
2.4
21.6
21.8
Fractal Dimension D
Fig. 3. Comparison of the ratio of the drag experienced by a permeable sphere to that of a solid sphere of the same radius. When Brinkman's equation is used to describe internal flow (instead of Darcy's law), a smaller value of drag is predicted.
1496
Research Note
REFERENCES wastewater applications is significantly smaller than that on non-porous spheres. It is observed that fluid Adler P. M. (1981) Streamlines in and around porous collection efficiencies for such floes can also be signifiparticles.J. Colloid Interface Sci. 81, 531-535. Brinkman H. C. 0947) A calculation of the viscous force cantly greater than zero. extended by a fowing fluidon a dense swarm of particles. There is considerable discussion as to the appropriAppL Sciet. Rea. AI, 27-34. ateness of rectilinear and eurvilinear models in aggreFclderhof B. U. (1975) Frictionalproperties of dilutepolygation studies (e.g. Han and Lawler, 1991, 1992). For mer solutions. Physica A 80, 63-75. floes in the DLA and RLA regimes (D ~< 2), rectilin- Han M. and Lawler D. F. (1991) Interactionsof two settling spheres: settling rates and collision efficiency.J. Hyd. ear models for particle-aggregate and aggreEngng I17, 1269-1289. gate-aggregate interactions may be more accurate Han M. and Lawler D. F. (1992) The (relative)insignifithan curvilinear models that in each case, model flow cance of G in flocculation.J. Am. Wat. Wks Ass. 84, around the floc as that around an "equivalent" solid 79-91. sphere. At higher fractal dimensions, fluid collection Happcl J. and Brenner H. (1973) Low Reynolds Number Hydrodynamics. Noordhoff, Leyden, The Netherlands. efficiencies are low and the error associated with a Jiang Q. and Logan B. E. (1991) Fractal dimensions of simple rectilinear model for fluid flow is expected to aggregates determined from steady-state size distribe significant. These observations are very sensitive to butions. Envir. Sci. Technol. 25, 2031-2038. the value assigned to the packing factor. F o r a fixed Jullicn R. and Botet R. (1987) Aggregation and Fractal Aggregates. World Scientific,Singapore. D, the permeability of a cluster increases as ~, deLawler D. (1986) Removing particlesin water and wastecreases. Therefore, the error associated with computwater. Envir. Sci. Technol. 20, 856-861. ing the interactions with floes having small values of Li D. and Ganczarczyk J. (1989) Fractal geometry of using a rectilinear model may be very small. particle aggregates generated in water and wastewater treatment processes. Envir. Sci. Technol. 23, 1385-1389. Omission of the inertial terms in the Navier-Stokes equations compared to the viscous terms (creeping Li D. and Ganczarezyk J. (1992) Advective transport in activated sludge floes. Wat. envir. Res. 64, 236-240. flow) is an excellent approximation in sedimentation Masliyah J. H. and Polikar M. (1980) Terminal velocity of where the Reynolds number (based on particle diamporous spheres. Can J. Chem. Engng 58, 299-302. eter) Re, is typically less than 0.1. Even at very high Matsumoto K. and Suganuma A. (1977) Settling velocity of a permeable model floe. Chem. Engng Sci. 32, 445--447. hydraulic loading rates, the packed bed applications encountered in water and wastewater treatment (con- Mulder T. and Gimbel R. (1991) On the development of high performance filtration materials for deep bed filters. ventional filters, GAC column adsorbers, etc.) are Sep. Technol. 1, 153-165. characterized by Re < 1. Deviations from linearity Neale G., Epstein N. and Nader W. (1973) Creeping flow are small in packed beds for Re < 5 (Happel and relative to permeable spheres. Chem. Engng Sci. 2~ 1865-1874. Brenner, 1973). Hence, the assumption of creeping flow is also valid in packed beds of spheres. Calcu- Rogak S. N. and Flagan R. C. (1990) Stokes drag on self-similar clusters of spheres. J. Colloid Interface Sci. lations of the collision rate kernels in particle popu134 206-218. lation models for flocculation may require a Sutherland D. N. and Tan C. T. (1970) Sedimentation of a porous sphere. Chem. Engng Sci. 25, 1950-1951. consideration of inertial effects. In flocculation basins operating under turbulent flow conditions, the analy- Tambo N. and Watanabe Y. (1979) Physical characteristics of flocs--I. The foe density function and aluminium floe. sis presented here may only be a reasonable descripWat. Res. 13, 409--419. tion of aggregate--flow interactions occurring below Wiesner M. R. (1992) Kinetics of aggregate formation in the Kolmogorov microscale. rapid mix. Wat. Res. 26, 379-387.