- Email: [email protected]
Comment on “Permeability of fractal aggregates”
Water Research 36 (2002) 3416–3417
Letters to the editor
Comment on ‘‘Permeability of fractal aggregates’’ L. Gmachowski Polish Academy of Sciences, Institute of Physical Chemistry, 01-224 Warsaw, Poland
The authors (X.Y. Li and B.E. Logan, Water Res. 35(2001)3373) analyzed the permeability of fractal aggregates using two different fractal scaling approaches. The single-particle-fractal model supposed the possibility to calculate the permeability assuming uniform distribution of primary particles. In the clusterfractal model aggregates were composed of primary particles separated into individual clusters, uniformly distributed inside aggregates. It was shown that the second one was more reliable in predicting the hydrodynamic properties of aggregates. If an aggregate has a self-similar structure, the fractal dimension can be determined by covering the object with the sets of spheres of increasing size [1]. The number of spheres will decrease as a negative power of their size. In this way fractals may be produced of lower and lower numbers of constituent clusters, contained in covering spheres. Aggregates composed of small number of constituents are called embryonic [2]. Since the number of clusters diminishes as a negative power of its diameter, the corresponding volume fraction f of clusters in an aggregate is the following function of the number of clusters: fpn13=D :
ð1Þ
The lower value of the number of clusters, the higher value of f and hence the narrower the range of its variability due to the radial dependence. This is the reason why the standard permeability models, applicable for systems with uniform distribution of porosity, are more reliable when applied in the cluster-fractal model comparing with that of single-particle-fractal model. In this comment I would like to controvert the method of verification of the cluster-fractal model. The authors compared the results with their experimental data, previously obtained [3]. The values of the ratio G of the settling velocity of an individual aggregate to that of an isolated sphere of the same radius and bulk density are thought to be much higher than expected [4]. This is in agreement with my calculations. The ratio G is the E-mail address: [email protected] (L. Gmachowski).
reciprocal normalized hydrodynamic radius of aggregate, derived previously in the following form [5]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r D 2 0:228; ð2Þ ¼ 1:56 1:728 R 2 which, apart from the previous verification by the aggregate structure and aggregation kinetics [5,6], has been recently used to describe the structure of bead models for macromolecules and bioparticles [7]. According to this equation the ratio G is 1.41 for D ¼ 1:81 and 1.13 for D ¼ 2:33; whereas the experimental data taken into analysis are 2.84 and 2.39, respectively. The permeability of aggregates of reduced number of constituents can be calculated using the standard permeability models after the replacement of the clusters by the corresponding spheres of hydrodynamic radii to exclude the necessity to take into account their internal permeability. In this way the problem is reduced to calculate the permeability of embryonic aggregates composed of solid impermeable particles. The Stokes regime steady-state falling speed of an aggregate of hydrodynamic radius r; composed of n spheres of radius a; normalized by the falling speed of the primary particle u1 ; is described as un r2 ð4=3Þpa3 n n r n ) ¼ ; ¼ ¼ ðr=aÞ R ðR=aÞðun =u1 Þ u1 a2 ð4=3Þpr3
ð3Þ
where the aggregate radius is the radius of circumscribed sphere. It is also possible to calculate the normalized hydrodynamic radius by the Brinkman equation for individual permeable sphere [8]: r 2x2 ðx tanh xÞ ¼ 3 ; R 2x þ 3ðx tanh xÞ
ð4Þ
using a reliable dependence of the reciprocal square root of dimensionless internal permeability of an aggregate x on the packing fraction f; calculated as follows: ð4=3Þpa3 n n ¼ : ð5Þ ð4=3ÞpR3 ðR=aÞ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The values of ðun =u1 Þ are available [9,10] for some embryonic aggregates of different structure, depicted by Kaye [2], including linear chains up to n ¼ 8; as well as
f¼
0043-1354/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 3 - 1 3 5 4 ( 0 2 ) 0 0 0 0 4 - 0
1.0
1.0
0.9
0.9
(r/R)experimental
(r/R)experimental
L. Gmachowski / Water Research 36 (2002) 3416–3417
0.8 0.7 0.6
0.8 0.7 0.6
0.5
0.5
0.4
0.4 0.4
0.5
0.6
0.7
0.8
0.9
3417
1.0
0.4
(r/R)calculated
0.5
0.6
0.7
0.8
0.9
1.0
(r/R)calculated
Fig. 1. Comparison of normalized hydrodynamic radius of aggregate determined from steady-state falling speed (experimental) with that calculated using the Happel equation (6), performed for different embryonic aggregates including: JFlinear aggregates; KFmore compact aggregates of one sphere surrounded by others.
Fig. 2. Comparison of normalized hydrodynamic radius of aggregate determined from steady-state falling speed (experimental) with that calculated using Eq. (7), performed for different embryonic aggregates.
References more compact aggregates of one sphere surrounded by the others up to n ¼ 13: This makes it possible to calculate the normalized hydrodynamic radius by Eq. (3). These values (experimental) are compared in Fig. 1 with those calculated by Eq. (4) employing the rearranged Happel formula [11] R x¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3fð3 þ 2f5=3 Þ 2 3f1=3 þ 3f5=3 þ 2f2
:
ð6Þ
The experimental values of normalized hydrodynamic radius are close to the values calculated using the Happel equation. This is a confirmation of the idea developed by the authors. However, there is some observable deviation from the straight line in Fig. 1. This deviation is considerably reduced by employing another formula [12], previously verified over a very wide concentration interval vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u " 4=3 Ru 9 25f x¼ t f 1þ : a 2 3ð1 fÞ3
ð7Þ
The results are depicted in Fig. 2. The agreement is achieved despite the non-homogeneous porosity distribution in aggregates, especially occurring in linear aggregates. The packing fraction of constituent particles changes in a very wide interval from 1/64 for the longest linear aggregate to 13/27 for the most compact one.
[1] Feder J. Fractals. New York: Plenum Press, 1988. p. 14–5. [2] Kaye BH. A random walk through fractal dimensions. Weinheim: VCH, p. 310–1. [3] Li XY, Logan BE. Collision frequencies of fractal aggregates with small particles by differential sedimentation. Environ Sci Technol 1997;31:1229–36. [4] Woodfield D, Bickert G. An improved permeability model for fractal aggregates settling in creeping flow. Water Res 2001;35:3801–6. [5] Gmachowski L. Estimation of the dynamic size of fractal aggregates. Colloids Surfaces A: Physicochem Eng Aspects 2000;170:209–16. [6] Gmachowski L. A method of maximum entropy modeling the aggregation kinetics. Colloids Surfaces A: Physicochem Eng Aspects 2001;176:151–9. [7] Gmachowski L. Intrinsic viscosity of bead models for macromolecules, bioparticles. Eur Biophys J Biophys 2001;30:453–6. [8] Brinkman HC.Acalculationoftheviscosity, the sedimentation velocity for solutions of large chain molecules taking into account the hampered flow of the solvent through each chain molecule. Proc R Dutch Acad Sci 1947;50:618– 24(821). . [9] Stober W, Berner A, Blaschke R. The aerodynamic diameter of aggregates of uniform spheres. J Colloid Interface Sci 1969;29:710–9. . [10] Stober W, Flachsbart H. Size-separating precipitation of aerosols in a spinning spiral duct. Environ Sci Technol 1969;3:1280–96. [11] Happel J. Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J 1958;4:197–201. [12] Gmachowski L. Flow drag in heterogeneous systems over wide intervals of porosity and Reynolds number. J Chem Eng Jpn 1996;29:897–900.
Letters to the editor
Comment on ‘‘Permeability of fractal aggregates’’ L. Gmachowski Polish Academy of Sciences, Institute of Physical Chemistry, 01-224 Warsaw, Poland
The authors (X.Y. Li and B.E. Logan, Water Res. 35(2001)3373) analyzed the permeability of fractal aggregates using two different fractal scaling approaches. The single-particle-fractal model supposed the possibility to calculate the permeability assuming uniform distribution of primary particles. In the clusterfractal model aggregates were composed of primary particles separated into individual clusters, uniformly distributed inside aggregates. It was shown that the second one was more reliable in predicting the hydrodynamic properties of aggregates. If an aggregate has a self-similar structure, the fractal dimension can be determined by covering the object with the sets of spheres of increasing size [1]. The number of spheres will decrease as a negative power of their size. In this way fractals may be produced of lower and lower numbers of constituent clusters, contained in covering spheres. Aggregates composed of small number of constituents are called embryonic [2]. Since the number of clusters diminishes as a negative power of its diameter, the corresponding volume fraction f of clusters in an aggregate is the following function of the number of clusters: fpn13=D :
ð1Þ
The lower value of the number of clusters, the higher value of f and hence the narrower the range of its variability due to the radial dependence. This is the reason why the standard permeability models, applicable for systems with uniform distribution of porosity, are more reliable when applied in the cluster-fractal model comparing with that of single-particle-fractal model. In this comment I would like to controvert the method of verification of the cluster-fractal model. The authors compared the results with their experimental data, previously obtained [3]. The values of the ratio G of the settling velocity of an individual aggregate to that of an isolated sphere of the same radius and bulk density are thought to be much higher than expected [4]. This is in agreement with my calculations. The ratio G is the E-mail address: [email protected] (L. Gmachowski).
reciprocal normalized hydrodynamic radius of aggregate, derived previously in the following form [5]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r D 2 0:228; ð2Þ ¼ 1:56 1:728 R 2 which, apart from the previous verification by the aggregate structure and aggregation kinetics [5,6], has been recently used to describe the structure of bead models for macromolecules and bioparticles [7]. According to this equation the ratio G is 1.41 for D ¼ 1:81 and 1.13 for D ¼ 2:33; whereas the experimental data taken into analysis are 2.84 and 2.39, respectively. The permeability of aggregates of reduced number of constituents can be calculated using the standard permeability models after the replacement of the clusters by the corresponding spheres of hydrodynamic radii to exclude the necessity to take into account their internal permeability. In this way the problem is reduced to calculate the permeability of embryonic aggregates composed of solid impermeable particles. The Stokes regime steady-state falling speed of an aggregate of hydrodynamic radius r; composed of n spheres of radius a; normalized by the falling speed of the primary particle u1 ; is described as un r2 ð4=3Þpa3 n n r n ) ¼ ; ¼ ¼ ðr=aÞ R ðR=aÞðun =u1 Þ u1 a2 ð4=3Þpr3
ð3Þ
where the aggregate radius is the radius of circumscribed sphere. It is also possible to calculate the normalized hydrodynamic radius by the Brinkman equation for individual permeable sphere [8]: r 2x2 ðx tanh xÞ ¼ 3 ; R 2x þ 3ðx tanh xÞ
ð4Þ
using a reliable dependence of the reciprocal square root of dimensionless internal permeability of an aggregate x on the packing fraction f; calculated as follows: ð4=3Þpa3 n n ¼ : ð5Þ ð4=3ÞpR3 ðR=aÞ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The values of ðun =u1 Þ are available [9,10] for some embryonic aggregates of different structure, depicted by Kaye [2], including linear chains up to n ¼ 8; as well as
f¼
0043-1354/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 3 - 1 3 5 4 ( 0 2 ) 0 0 0 0 4 - 0
1.0
1.0
0.9
0.9
(r/R)experimental
(r/R)experimental
L. Gmachowski / Water Research 36 (2002) 3416–3417
0.8 0.7 0.6
0.8 0.7 0.6
0.5
0.5
0.4
0.4 0.4
0.5
0.6
0.7
0.8
0.9
3417
1.0
0.4
(r/R)calculated
0.5
0.6
0.7
0.8
0.9
1.0
(r/R)calculated
Fig. 1. Comparison of normalized hydrodynamic radius of aggregate determined from steady-state falling speed (experimental) with that calculated using the Happel equation (6), performed for different embryonic aggregates including: JFlinear aggregates; KFmore compact aggregates of one sphere surrounded by others.
Fig. 2. Comparison of normalized hydrodynamic radius of aggregate determined from steady-state falling speed (experimental) with that calculated using Eq. (7), performed for different embryonic aggregates.
References more compact aggregates of one sphere surrounded by the others up to n ¼ 13: This makes it possible to calculate the normalized hydrodynamic radius by Eq. (3). These values (experimental) are compared in Fig. 1 with those calculated by Eq. (4) employing the rearranged Happel formula [11] R x¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3fð3 þ 2f5=3 Þ 2 3f1=3 þ 3f5=3 þ 2f2
:
ð6Þ
The experimental values of normalized hydrodynamic radius are close to the values calculated using the Happel equation. This is a confirmation of the idea developed by the authors. However, there is some observable deviation from the straight line in Fig. 1. This deviation is considerably reduced by employing another formula [12], previously verified over a very wide concentration interval vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u " 4=3 Ru 9 25f x¼ t f 1þ : a 2 3ð1 fÞ3
ð7Þ
The results are depicted in Fig. 2. The agreement is achieved despite the non-homogeneous porosity distribution in aggregates, especially occurring in linear aggregates. The packing fraction of constituent particles changes in a very wide interval from 1/64 for the longest linear aggregate to 13/27 for the most compact one.
[1] Feder J. Fractals. New York: Plenum Press, 1988. p. 14–5. [2] Kaye BH. A random walk through fractal dimensions. Weinheim: VCH, p. 310–1. [3] Li XY, Logan BE. Collision frequencies of fractal aggregates with small particles by differential sedimentation. Environ Sci Technol 1997;31:1229–36. [4] Woodfield D, Bickert G. An improved permeability model for fractal aggregates settling in creeping flow. Water Res 2001;35:3801–6. [5] Gmachowski L. Estimation of the dynamic size of fractal aggregates. Colloids Surfaces A: Physicochem Eng Aspects 2000;170:209–16. [6] Gmachowski L. A method of maximum entropy modeling the aggregation kinetics. Colloids Surfaces A: Physicochem Eng Aspects 2001;176:151–9. [7] Gmachowski L. Intrinsic viscosity of bead models for macromolecules, bioparticles. Eur Biophys J Biophys 2001;30:453–6. [8] Brinkman HC.Acalculationoftheviscosity, the sedimentation velocity for solutions of large chain molecules taking into account the hampered flow of the solvent through each chain molecule. Proc R Dutch Acad Sci 1947;50:618– 24(821). . [9] Stober W, Berner A, Blaschke R. The aerodynamic diameter of aggregates of uniform spheres. J Colloid Interface Sci 1969;29:710–9. . [10] Stober W, Flachsbart H. Size-separating precipitation of aerosols in a spinning spiral duct. Environ Sci Technol 1969;3:1280–96. [11] Happel J. Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J 1958;4:197–201. [12] Gmachowski L. Flow drag in heterogeneous systems over wide intervals of porosity and Reynolds number. J Chem Eng Jpn 1996;29:897–900.