Specific Cake Resistance due to Filtered Aggregates: Creeping Flow Relative to a Swarm of

Solid Spheres with a Porous Shell

Albert S. Kim and Rong Yuan

Department of Civil and Environmental Engineering

University of Hawaii at Manoa, Honolulu, Hawaii 96822

Submitted to

Journal of Membrane Science

November 14, 2003

Corresponding Author: Dr. Albert S. Kim Department of Civil and Environmental Engineering University of Hawaii at Manoa Honolulu, HI 96822 808-956-3718 808-956-5014 (fax) AlbertSK@hawaii.edu

Abstract

A simple model to evaluate hydrodynamic cake resistance due to filtered aggregates

has been developed. An aggregate is treated as a hydrodynamically equivalent solid core

with a porous shell. Creeping flow past a swarm of the composite spheres has been

solved using the Stokes equation and Brinkman’s extension of Darcy’s law. The

dimensionless drag force exerted on a composite sphere in the swarm is analytically

determined by four parameters, i.e., radius of the solid core, thickness of the porous shell,

permeability of the aggregate, and occupancy fraction defined in this study. In certain

limiting cases, the final expression, represented as Ω , converges to pre-existing analytic

solutions for an isolated (i) impermeable, (ii) uniformly porous, and (iii) composite

sphere, and a swarm of (iv) impermeable, and (v) uniformly porous spheres. This

expression is then used to predict the specific resistance of aggregate cake formed on

membrane surfaces.

Keywords: Aggregation; Fractal Aggregate; Cake Resistance; Cell model; Settling

velocity

1

1. Introduction

Microfiltration (MF) and ultrafiltration (UF) are widely used to remove suspended

solid, colloidal particles, macromolecules, bacteria, and viruses from suspensions with

particular applications as alternative, substitutional, or supplementary process for

conventional water and wastewater treatment. The major limiting factor of MF/UF

application is the ratio of permeate flux (with appropriate water quality) to invested cost

since membrane fouling often gives rise to additional maintenance cost. While filtering

particulate matter, MF/UF membranes undergo three different patterns of matter-stacking

phenomena on their surfaces that constrict permeation, i.e., concentration polarization,

(followed by) colloidal cake/gel formation [1-16], and aggregate cake (i.e., cake of

retained aggregates composed of many small primary colloidal particles) formation [17-

20] with each effect bringing out its own permeate flux decline behavior. Interestingly,

particle aggregation occurring in relatively high ionic strength ( ≥ 0.01 M) remarkably

reduces the flux decline by developing more porous cake layer on the membrane surfaces

as compared to ones built with individual colloidal particles.

Waite et al. [17] observed distinct flux decline behaviors from two different

aggregation schemes in the bulk phase of stirred cell ultrafiltration membrane, i.e., rapid

aggregation and slow aggregation; and showed that specific resistances of cakes formed

from aggregates generated under the rapid aggregation are at least one order of

magnitude lower than those of cakes formed from aggregates generated under the slow

aggregation. With very high overall porosity of more than 0.99, the rapidly formed

aggregates provides significantly less flux decline with fractal dimensions of 1.8 – 2.2,

which are between the typical fractal dimensions of diffusion-limited aggregation (DLA),

2

2.5 [21], and diffusion-limited cluster aggregation (DLCA), 1.75 [22]. On the other hand,

the slowly formed aggregates cause higher flux decline and render the fractal dimension

to be 2.4, relatively higher than the typical value of reaction-limited cluster aggregation

(RLCA), 1.9 – 2.2 [23-28]. The differences in the fractal dimensions are considered to be

due to one or some of shear-induced aggregation, breakage, and reworking (re-

aggregation after breakage). In their work, the aggregation regime has been controlled by

ionic strength (salt concentration), pH, and fulvic acid concentration, which significantly

affect aggregate structure, overall cake porosity, and flux decline behavior in a correlated

manner. On the other hand, specific cake resistance are assumed to be estimated by

substituting the overall porosity of aggregates into the Carman-Kozeny equation [29] of

which its validity is questionable as the porosity increases over 0.8 (see Table 8-4.2 of ref.

[30]).

Using crossflow microfiltration (MF) membranes, Hwang and Liu [20] investigated

effects of ionic strength (IS) on the particle aggregation followed by corresponding

changes in (quasi-) steady state flux, indicating that the steady state permeate flux

remained constant while IS < 0.01 M, but quickly increased if IS > 0.01M due to

increasing cake porosity caused by the particle aggregation. Upon scrutinizing their

experimental data, however, it is found that the steady state fluxes commonly have

minimum values near IS = 0.005 M, displaying the normal decreasing behavior with

respect to the increasing ionic strength [12, 31], on which small aggregates start forming

with less repulsive interparticle interactions. Theoretically, they calculated the cake

porosity profile across the entire cake layer consisting of aggregates as well as separated

particles in void space outside the aggregates, and substituted the porosity into the

3

Carman-Kozeny relation. This approach, as also found in Waite et al.’s [17], implies

possible replacement of a swarm of aggregates, of which sizes are presumed to be similar,

by same number of smaller-sized impermeable spheres so that both the aggregates and

equivalent solid spheres can provide identical cake porosity as well as hydrodynamic

drag. It seems to be, however, less intuitive rather than replacing the aggregates by

uniformly porous spheres [32], having equal size of the aggregates and characterized by a

volume-averaged permeability of the aggregates [33, 34].

Mechanism of collapse of aggregate cake by the permeate flow velocity has been

recently studied through Small Angle Neutron Scattering (SANS) experiments by Cabane

et al. [19]. They demarcated dependence of scattering density I (of aggregates

generated in rapid coagulation, i.e., diffusion-limited regime) on scattering vector in

three typical regions, where the inverse scattering vector

q

1q− is much less than, almost

equal to, and much larger than the particle size. This method characterizes the structure

of the bulk aggregates and aggregate-collapsed cake by corresponding fractal dimensions,

i. e., the slopes of log I vs. plots. Furthermore, it is then used to investigate the

collapsing criteria by increasing trans-membrane pressure. Even without shear, however,

the fractal dimension of bulk aggregates composed of spherical latex particles is found to

be 2.45, which is somewhat higher than that of aggregates made by RLCA.

log q

Results and implications of the current research listed earlier show that the slow

aggregation through RLCA with or without ambient shear flow occurs at the ionic

strength of an order of 10 2 M− under which aggregation efficiency is too low, generating

relatively dense and compact aggregates with the fractal dimensions typically greater

than 2.0. As ionic strength increases, the coagulation rate becomes fast so that the formed

4

aggregates show sparse and loose structure with smaller fractal dimensions typically less

than 2.0. However, specific hydraulic resistance due to a swarm of aggregate, i.e.,

aggregate cake, is still approximately determined in a qualitative as well as quantitative

manner by simply substituting aggregate porosity into hydrodynamic drag expressions for

a swarm of impermeable solid spheres [17, 20], e. g., Carman-Kozeny equation [29] or

possibly Happel’s expression [35]. Therefore, it is now of great necessity to accurately

predict the permeate flux decline due to cake structures of porous aggregates deposited

on the membrane surfaces.

There has been extensive research for the creeping flow relative to porous medium

composed of mono-sized spheres. As described earlier, Happel [35] developed a

hypothetical cell model to investigate the flow through a swarm of impermeable spheres

packed in beds. So, this model converges to the Stokes’ law for infinitely dilute limit of

colloidal suspensions. Brinkman [36] showed an expression of the flow through an

isolated permeable sphere of uniform permeability without detailed derivation, and later

Neale et al. [32] located the permeable sphere within the Happel’s hypothetical cell of a

tangential stress free surface so that their expression reduces to the Happel’s if

permeability of the permeable sphere vanished. Masliyah et al. [37] developed a

mathematical expression that include the Stokes’ law and Brinkman’s [36] expression by

considering an isolated composite sphere comprised of a solid core and a porous shell,

and they used the expression to investigate the settling velocity of a solid sphere to which

various flexible threads are attached. Besides the analytic expressions [32, 35-37] listed

above, Ooms et al.[38] numerically confirmed the Brinkman’s expression [36] that has

been reported without derivations. Moreover, Veerapaneni and Wiesner [33] investigated

5

hydrodynamic properties of an isolated fractal aggregate with radially varying porosity

and hence permeability by considering many consecutive thin shells of constant

permeability within the fractal aggregate.

From a hydrodynamic point of view, the flow through a uniform porous sphere can be

categorized into two representative types: a slow interior flow driven by fluid pressure

and a fast exterior flow due to shear stress around the permeable sphere surface [39]. This

phenomenon is more apparent in a fractal aggregate of which its central region is much

denser than its edge. Therefore, in this work, a fractal aggregate, which is presumed to be

characterized by its length scale (radius) and fractal dimension followed by its radially

varying local permeability, is simplified by using instead an inner spherical core of

impermeable solidity and an outer porous shell of volume-averaged overall permeability

of the fractal aggregate. Later, we follow Neale et al.’s approach [32] by positioning the

simplified aggregate at the center of the tangential stress free cell [35], applying proper

boundary conditions, and calculating the hydrodynamic drag due to a swarm of the

aggregates with the hydrodynamically simplified structure.

2. Theoretical

2.1. Flow through a swarm of aggregates

A primary assumption employed in this study is that a solid spherical core surrounded

by a porous shell, as shown in Fig. 1, can be hydrodynamically equivalent to a fractal

aggregate with radially varying permeability. The radius of the impermeable core, the

distance between the center of the core and the outer surface of the shell (i.e., aggregate

6

radius), and the radius of the hypothetical spherical cell (on which the tangential stress

vanishes [35]) are denoted as , a ( )b a> , and ( )b>c respectively, where is the

thickness of the shell characterized by its constituent primary particles of radius ,

porosity

b a−

pa

ε , and thereby accompanying permeability κ . Relative to this composite

sphere (i.e., a core with a shell) in the hypothetical cell, the creeping flow of Newtonian

fluid with absolute fluid viscosity µ is considered to be steady and axi-symmetric. For

mathematical convenience, the center of the composite sphere is selected as the origin of

spherical coordinates [ , , ]r θ ϕ , while the fluid is approaching toward the membrane

surface as well as partially passing through the composite sphere in the z− direction with

velocity V .

Governing Equations

The governing equations of incompressible Newtonian creeping flow in void space

are

(1) 2uµ∇ = ∇p

and

, (2) 0u∇ ⋅ =

where u u[ , , ]r u uθ ϕ= denotes the fluid velocity vector, and p the fluid pressure. The

corresponding equation for describing Newtonian creeping flow through a uniformly

porous medium of permeability is known as the Brinkman’s equation (Brinkman’s

extension of Darcy’s law), i.e.,

κ

7

*

* 2 * * *u u pµµκ

∇ − = ∇ (3)

with the macroscopic incompressibility assumption [40, 41] of

, (4) * 0u∇ ⋅ =

where * denotes any macroscopically averaged quantity specifically pertaining to the

porous medium. In this light, *µ represents an effective viscosity in the porous medium,

often being assumed to be identical to µ especially when the porous medium is highly

porous and hence permeable. Therefore, with the incompressibility of the fluid in both

void space and porous medium, Eqs. (1) and (3) are valid within the void space

and the porous shell (b r c≤ ≤ ) ( )ba r≤ ≤ , respectively.

Boundary Conditions

The boundary conditions that are physically realistic and mathematically consistent

for the current problem are as follows:

(5) ( )* ,ru a θ = 0

0

),

(6) ( )* ,u aθ θ =

( ) (* ,r ru b u bθ θ= (7)

( ) (* ,u b u bθ θ ),θ θ= (8)

( ) (* ,rr rrb ),bτ θ τ θ= (9)

( ) ( )* ,r rb bθ θ ,τ θ τ θ= (10)

( ), coru c V sθ θ= − (11)

8

(12) ( ),r cθτ θ = 0

with 0 2θ π≤ < , where (*)rrτ and ( )*

rθτ denote the normal and tangential components of the

stress tensor, respectively, i.e.,

( ) ( )( )*

* *(*) 2 rrr

upr

τ µ ∂= − +

∂ (13)

and

(*) 1 rr

u uur r r r

θ θθτ µ ∂∂= + − ∂ ∂

. (14)

No-slip boundary conditions are applied on the surface of the core sphere of radius

(Eqs. (5) and (6)). The normal and tangential components of velocity and stress tensor are

considered to be continuous across the permeable interface at

a

r b= (Eqs. (7)-(10)), and

fluid velocity V exerted by the composite sphere is reflected in Eq. (11), which is also

corresponding to such a situation that the sphere moves in the opposite direction with

velocity V where the fluid is stationary. The condition of the free tangential stress, first

used in Happel’s original work [35], on the surface of the hypothetical cell is

implemented in Eq. (12) to mimic the adjacent presence of other aggregates of similar

size. Additionally, it has been proved that the continuity of the normal stress (Eq. (9)) is

equivalent to that of the fluid pressure [42, 43], i.e.,

( ) ( )* , ,p b p bθ θ= (15)

from the incompressibility of Eqs. (2) and (4).

9

Method of Solutions

Independence of the flow on the azimuthal angle, so called cylindrical or axi-

symmetry, introduces stream functions ψ and *ψ related to the velocity fields by

( )( )*

*2

1sinru

rψ

θ θ− ∂

=∂

(16)

and

( )( )*

* 1sin

ur rθ

ψθ

∂=

∂, (17)

where 0 2θ π≤ ≤ for both outside (unstarred) and inside (starred) the shell. Taking the

curl of the Stokes equation (Eq. (1)) and Brinkman equation (Eq. (3)), one obtains

(18) 4 0E b rψ = ≤ ≤ c

b

and

, (19) 4 * 1 2 * 0E E a rψ κ ψ−− = ≤ ≤

respectively, where

2

22 2

sin 1sin

Er r

θθ θ θ

∂ ∂ = + ∂ ∂ ∂ ∂

(20)

with the macroscopic assumption of

* 1µµ

= . (21)

The general solutions of Eqs. (18) and (19) are, respectively

( ) 2 4 2, sin2V A B C Dκ ,ψ ξ θ ξ ξ ξ θ β ξ γ

ξ

= + + + ≤ ≤

(22)

( )* 2 2cosh sinh, sinh cosh sin2

,

V E F G Hκ ξ ξψ ξ θ ξ ξ ξ θξ ξ ξ

α ξ β

= + + − + −

≤ ≤

(23)

10

where rξ κ= , aα κ= , bβ κ= , and cγ κ= . Having satisfying the set of

required boundary conditions listed in Eqs. (5)-(12), the constants , A B , , , , ,

, and will have the specific values as given in the Appendix.

C

VbΩ

D E F

G H

Integrating the normal and tangential stress distributions over the porous surface

yields the drag force F a experienced by the composite sphere

(24) [ ]2

02 cos sin sin 6rr r r b

b dπ

θπ τ θ τ θ θ θ πµ=

= − =∫F

where

2 3B βΩ = + , (25)

of which its physical meaning is

Hydrodynamic resistance experienced by a swarm of the composite sphere with a core of radius and a porous shell of thickness -

Hydrodynamic resistance experienced by an impermeable sphere of radiusa b a

b.

The sign of Eq. (25) refers to the direction of the force (+z) that sustains the composite

sphere in downward ambient fluid flow toward the membrane surface.

Comparison with other related works

The correctness of our expression of Ω may be confirmed, although the calculation is

quite routine, by scrutinizing five important cases of which analytic solutions are already

available: (i) as α β→ (or ) and 0κ → γ → ∞ , the Stokes law is obtained, i.e., Ω = ,

(ii) as

1

α β→ (or ), Happel’s well known formula [35] is retrieved, i.e., 0κ →

a The necessary FORTRAN subroutine to calculate F can be provided upon request to the corresponding author.

11

52

353 3

2 2

1lim1Hα β

η6η η η→

+Ω ≡ Ω =

− + −, (26)

where (b c )η β γ= = of which the cube can be interpreted as volume fraction of the

sphere-packed porous medium, (iii) as 0α → and γ → ∞ , Brinkman’s formula [38, 44]

for an isolated homogeneous sphere of radius is replicated, i.e., b

( )( )

2

20

2 1 tanhlim

2 3 1 tanhBαγ

β β ββ β β→

→∞

−Ω ≡ Ω =

+ −, (27)

(iv) as 0α → , Eq. (25) in this study becomes Eq. (30) of work of Neale, Epstein, and

Nader [32], i.e.,

( )

( )

2 2 5 5 2 2 5 543

2 2 2 5 2 6 2 5 5 60

2 2 5 2 6 2 5 5 6

tanh2 20 2 8 20lim

2 3 3 2 90 42 30 3tanh 3 15 12 90 72 30 3

NENα

ββ β η η β β η ηβ

β β η β η β η β η η ηβ β η β η β η β η η η

β

−→

−

+ + − + +Ω ≡ Ω =

− + − + + − + − − + − + + − +

,(28)

and finally (v) as γ → ∞ , Masliyah et al.’s work [37] is reproduced, but detailed formula

of is omitted in this paper due to length and complexity of the mathematical

expression. Proving the above five known expressions by inspecting certain limiting

cases of Ω calculated in this study, we confirm that our general solution of Eq. (25) is

mathematically correct, having appropriate physical meanings.

limγ →∞

Ω

2.2. Filtration Equation: Darcy’s law

Permeate flux across a membrane containing a deposited layer (i.e., aggregate cake

layer in this study) may be described by the Darcy’s law with the resistance-in-series

model:

12

( )m c

PVR Rµ∆

=+

, (29)

where V is the permeate velocity, P∆ is the pressure drop across the membrane, mR is

the membrane resistance, and cR is the cake resistance, which can be described as

c c cR r δ= , (30)

where and cr cδ are specific resistance and thickness of the cake layer, respectively. In

the current model system shown in Fig. 1, the pressure gradient across the cake layer

should be equal in magnitude to the drag force density inside the hypothetical cell of

which its radius is c , i.e.,

( )3

6 , ,4 3

c

c

bVPc

πµ α β γδ π

Ω∆= , (31)

where cP P RmVµ∆ = ∆ − . Manipulating Eq. (31) with Eq. (29) gives

( )2

, ,2 9c b

rα β γ

λΩ

= (32)

where 3λ η= is defined in this study as “occupancy fraction”, which is always greater

than the volume fraction φ of the aggregate cake layer due to the porous region

of the composite sphere. Now it is worth noting that the term (a )r b< < 22 9b λ in Eq.

(32) is the permeability of solid spheres of radius in a dilute limit while the porous

shell is disappearing, i.e.,

b

λ φ→ . For the case of a swarm of solid spheres, the

corresponding is then obtained by substituting Happel’s expression (Eq. (26)) into

Eq. (32), i.e.,

2cb r

5

3

513 3

22 3

, 23 32 2

192 1c Hb r φφ

φ φ φ + = − + −

, (33)

13

or often by Carman-Kozeny equation instead

2,

92c CK KCb r φ = Ω

, (34)

where

( )3

21CK

kφφ

Ω =−

(35)

with found empirically. The dimensionless specific cake resistance b r of Eqs.

(33) or (34) are commonly used to estimate the resistance

5k ≈ 2c

cR with the volume fraction of

the colloidal cake layer [9, 12, 45-47], so that they will be compared to the b r of the

aggregate cake layer in the following section.

2c

3. Results and discussion

3.1. Dimensionless Specific Cake Resistance, b r 2c

Fig. 2 shows the dimensionless specific cake resistance b r defined in Eq. (32) as a

function of the occupancy fraction

2c

λ with given β values, where 0 α β≤ ≤ and

3γ β λ= , in addition to Eqs. (33) and (34). The corresponding analysis of Fig. 2 is as

follows.

Converging to Happel’s model. Figs. 2(a)-(c) show using selected 2cb r α ’s and

β ’s, and commonly expose that reaches , as proved in Eq. (26) and shown in

Fig. 2(d), while the porous shell is disappearing (

2cb r 2

,c Hb r

α β→ , so consequently λ φ→ ).

14

Aggregate as a uniform porous sphere. On the other hand, b r with 2c 0α =

decreases as ( b )β κ≡

2b r

decreases due to the increasing constant permeability of the

uniformly porous sphere of radius b . Therefore, it is obvious that the dimensionless

specific cake resistance will vanish when both

κ

c α and ( )β α≥ reach zero with fixed

λ , which physically implies the absence of any material retained on the membrane

surface.

Effect of porous shell thickness. Comparing ( )2 , ,cb r α β γ of α β= and 0.95α β=

with 0.5λ = gives quite interesting result that the thin porous shell, of which its

thickness is only 5% of the aggregate radius , significantly reduces b r down to 60%

compared to

b 2c

(2 , ,cb r )β β γ . This result means that aggregates generated in RLCA regime

still provide significantly less drag compared to solid spheres of same size, and further

implies that a cake layer composed of relatively big colloidal particles with only small

roughness can drastically reduce the cake resistance, avoiding no-slip boundary

conditions on outer surfaces of the rough particles.

Effect of overlap and/or breakage. Another interesting behavior of in Fig.

2(a)-(c) is that, unlike Eqs. (33) and (34), does not diverge as long as

2cb r

2cb r α is less than

(but not equal to) β , even if the occupancy fraction λ reaches a unity. This limit of

is corresponding to cases (i) when the aggregates are closely packed so that

portions of their sparse edges are somehow overlapped and so partially broken to a

( )1λ →

15

certain extent, leaving almost no pure void space out of the aggregates by producing

detached primary particles that can be re-aggregated by themselves or re-attached to

nearby aggregates in a way, and (ii) when big solid colloidal particles, adsorbing polymer

chains on their surfaces and being inter-linked by the polymers, are transported (from

bulk phase) to and then retained on membrane surfaces with relatively high volume

fraction. Obviously, this limiting value of as2cb r λ is converging to unity decreases as

α and/or β decrease, which is physically inferring that the solid core is of smaller radius

and/or the porous shell is of lower volume fraction (accordingly reflected as higher

permeability ), respectively.

a

cr

κ

2c

2kc CKΩ ≈

3.2. Happel’s Cell Model and Carman-Kozeny Equation

Invalidity of Carman-Kozeny equation for highly porous aggregates. When

compared as b r , the Happel’s (Eq. (33)) and Carman-Kozeny’s (Eq. (34)) expressions

are almost indistinguishable, as shown in Fig. 2(d). However, if permeability as inverse

of is considered, the Carman-Kozeny equation shows quite different asymptotic

behavior from the Happel’s expression in an infinitely dilute limit. For solid spheres of

radius b in the dilute limit, the Stokes’ law brings out 1Ω = and leaves the

corresponding permeability as 22 9b φ . In this circumstance, Eq. (35) becomes

,0limφ

φ→

, (36)

and implies the invalidity of Carman-Kozeny equation where 0.1φ ≤ , bolstering the

necessary condition to use Eq. (34), i.e., 0.3φ ≥ with 5k ≈ (see Table 8-4.2 of ref. [30]).

16

Therefore, it is quite apparent that the Carman-Kozeny equation highly underestimates

the dimensionless drag force , and thereby significantly overestimates the

corresponding permeability (see Fig. 1 of ref. [34]) of aggregates found to have porosity

( ) in natural and engineered systems [48-52].

Ω

0.99≥

9 1

p p

kaρ

−

pρ

5.0

Equivalent sphere. Some researchers have replaced a fractal aggregate by an

equivalent, but smaller, solid sphere and applied the Carman-Kozeny equation to estimate

the cake resistance [17, 20]. Although Waite et al. [17] calculated quite reasonable

values of cake porosity by considering several cases of fractal dimension and ratios of

aggregate to primary particle size (i.e., pb a ), the estimation of the specific cake

resistanceb through

( )2 3

εε

, (37)

where is the mass density of the primary particles, can be underestimated due to the

coefficient of which value is about 10 times greater than the empirically obtained value

of if the porosity reaches 0.99 (see Table 3 of ref. [17] and Table 8-4.2 of ref. [30]).

On the other hand, Hwang and Liu [20] used a porosity-dependent for flow

perpendicular to the cylinders (Eq. (8-4.33) of ref. [30]) to evaluate the specific cake

resistance defined as Eq. (37) although it seems more reasonable to use for flow

through random orientation of cylinders. However, they used a different interpretation of

Happel’s original cell model [35] by considering additional void space among the

hypothetical cells of free tangential stress [53]. Nevertheless, the measured and estimated

k

k

k

b The specific cake resistance of Waite at el.’s work [17] is defined as cake resistance divided by deposited mass per unit membrane area.

17

porosity at high ionic strength ( 0.01M≥ ), i.e., fast aggregation (DLCA) regime, are well

coincided between 0.67 and 0.73 although faster axial velocity seems to affect

configuration of aggregates on the membrane surface or partially break the aggregates to

a certain extent.

An alternative way to link the porosity and the specific cake resistance of an

aggregate can be to treat the aggregate as a uniform porous sphere [36] if the porosity is

very high nearing 1.0. For a relatively smaller or denser aggregate generated in slow

aggregation (i.e., RLCA) regime, however, negligible flow near central region of the

aggregate should be implemented by possibly locating a solid core at the center of the

uniform porous sphere, which is to pragmatically consider deflecting flow around the

edge of the aggregate [34, 39] and thereby accurately estimating the specific cake

resistance defined as Eq. (32). In this light, the constant permeability κ of the porous

shell in the current model can be replaced by an average or edge-value of the radially

varying local permeability of the fractal aggregate and the parameter ( a )α κ≡ can be

obtained from the according porosity profile due to fractal nature of the aggregate.

3.3. Interpretation of Settling Velocity

As a reverse concept of the hydrodynamic drag force, settling velocity of spherical

objects can be easily addressed by analyzing Ω defined as Eq. (25), of which the inverse

can be interpreted as

( )1

Settling velocity of a swarm of the composite spheres with a core of radius and a porous shell of thickness -, ,

Settling velocity of an impermeable sphere of radius a b

bα β γ−Ω =

a , (38)

18

where the composite and impermeable spheres have same mass. For a single isolated

composite sphere, i.e., as γ → ∞

)

, Eq. (38) represents the settling velocity ratio of the

composite spheres to the same-sized (by radius ) solid sphere, which is denoted here as

. Then, the ratio of

b

(1 , ,α β γ−∞Ω = Ω → ∞ 1−Ω to 1−

∞Ω provides the dimensionless settling

velocity defined as

( )( )

1

1

, ,, ,sv

α β γα β γ

−

−∞

Ω → ∞Ω≡ =

Ω Ω, (39)

of which the physical meaning is expressed in a specific form:

settling velocity of a swarm of composite spheres settling velocity of an isolated composite sphere

.

Therefore, sv for solid spheres and uniformly porous spheres can be assessed by taking

limits of α β→ [35] and 0α → [32], respectively.

Fig. 3(a) and (b) then show the dimensionless settling velocity sv defined as Eq. (39)

as a function of the occupancy fraction λ with β values of 1 and 8, where 0 α β≤ ≤

and 3γ β λ= . The corresponding analyses are as follows: (i) similar to what

previously explained in Fig. 2, the dimensionless settling velocities for solid spheres (i.e.,

α β= ) are identical in Fig. 3(a) and (b) with different β values, (ii) as α reduces, sv

increases at a fixed value of λ since the sold core shrinks accordingly, leaving less

surface area of no-slip fluid flow, (iii) interestingly, unlike the specific cake resistance

shown in Fig. 2, the difference between settling velocities of solid spheres 2cb r ( )α β=

and rough spheres ( )0.95α β= is not remarkable as illustrated in Fig. 3, and (iv) for

relatively large β due to small permeability, the presence of the solid core does not

19

significantly affect the settling velocity since the flow penetration into the composite

sphere is negligible. It is now worth noting that quantitative comparison of any two

curves in Fig. 3(a) and (b) with a same α (except α β= ) may not provide appropriate

physical meaning since the reference velocities (i.e., 1−∞Ω ) are not identical to each other

due to the different β values.

4. Concluding Remarks

The specific cake resistance of aggregate cake has been investigated by developing a

simple hydrodynamic model in which fractal aggregates, typically characterized by

radially decreasing mass density with power law, was replaced by a hydrodynamically

equivalent composite sphere comprised of a solid core and a porous shell located in the

center of Happel’s hypothetical cell. The equivalent sphere provides a general expression

of dimensionless drag force Ω including Stokes law and other four pre-developed

expressions [32, 35-37]. This simple model is then able to quantify the specific cake

resistances of the cake layer composed of similar-sized aggregates possibly characterized

by representative permeability and size of central region of vanishing porosity, which are

related to fractal dimension in further studies. Even if aggregates generated in slow

aggregation (RLCA) regime have relatively small and dense structure, their permeable

edges can drastically reduce hydrodynamic resistance compared to solid spherical

particles of same size. Additionally, the presented solution is applied to hindered settling

of fractal aggregates and colloidal particles on which long polymer chains are uniformly

attached.

20

Acknowledgements

The authors acknowledge support from the Water Resource Research Center and the

University Research Council at the University of Hawaii, Manoa.

21

Nomenclature

a

a

b

c

k

radius of impermeable core

p radius of primary particle

radius of outer porous shell

radius of hypothetical cell

F drag force experienced by a composite sphere

coefficient of Carman-Kozeny equation defined in Eq. (35)

p fluid pressure

r

r

radial coordinate

c specific resistance of cake layer

cR cake resistance

mR membrane resistance

fluid velocity vector in spherical coordinate, i.e., [ , , ]ru u uθ ϕ u

V approaching fluid velocity partially passing through composite sphere toward

the membrane surface, i.e., permeate velocity

sv dimensionless settling velocity defined in Eq. (39)

Greek letters

α dimensionless solid sphere radius, defined as a κ

β dimensionless outer shell radius, defined as b κ

cδ thickness of cake layer

P∆ pressure drop across membrane

22

ε porosity of porous shell

φ volume fraction

γ dimensionless hypothetical cell radius, defined as c κ

η dimensionless parameter, defined as ( )b c β γ=

ϕ azimuthal coordinate

κ constant permeability of porous shell

λ occupancy fraction, defined as 3η

µ fluid viscosity

θ polar coordinate

pρ mass density of primary particle

stress tensor τ

Ω dimensionless drag force defined in Eq. (24)

∞Ω dimensionless drag force in an infinitely dilute limit

ψ stream function

ξ dimensionless radial coordinate, defined as r κ

Subscripts

B Brinkman

c cake

H Happel

CK Carman-Kozeny

NEN Neale, Epstein, and Nader

23

r radial direction

θ angular direction

Superscripts

* porous medium

24

Equation Chapter (Next) Section 1

Appendix

Detailed expression of the coefficients A H− of the stream functions (Eqs. (22) and (23))

are written as follows, satisfying the boundary conditions of Eqs. (5)-(12). Those are:

116 5

05 01 002DD

J D D Dγ

γ γ γ=

− + + +, (A1)

where

( ) ( )2 3 33 1 sinh 3 2 3 cosh 6J α α β β α= − ∆ + + + + ∆ − α

sh

osh

sinh ∆

osh

sinh ∆

, (A2)

, (A3) ( )

( )

3 2 4 2 3 211

2 3 2 3

6 2 12 3 9 co

3 2 4 3 sinh

D αβ β β β αβ α β α

β β α β β α α

= − + + + + − ∆

− − + + + ∆

, (A4) ( )( )

2 4 305

2 3 3

3 3 2 3 cosh

3 3 3 2 sinh

D α β α β αβ

α β α β α

= − + + + ∆

+ − − − ∆

, (A5)

( )

2 5 33

01 3 2 3 2

4 23

2 2 3 2

3 2

3 2 28 603 c

60 20 15

15 60 10 483

3 5 60

36 10 3

Dαβ β β α

βα β β α α β

αβ β ββ

α β α β α

αβ β

+ + += − ∆ + + + −

+ + ++ − + −

+ +

, (A6)

3 6 2 2 4 3 33

00 3 2

2 5 33

3 2 2 2 3 3

6

3 2 18 302 c

45 135 135

45 6 4 106

2 45 15

120

Dαβ β α β β α β

βα β αβ α

α αβ β ββ

α β α β α β α

αβ

+ − + += ∆ + + −

+ + +− + − − + −

and β α∆ = − , (A7)

25

5 212 10

3 202 00

4 23 3

23

J C C D JC

J C C

3γ γ γ

γ γ

− + + + =

+ + (A8)

where

( )( )

2 3 2 312

2 4 2 3 2

5 2 4 3 sinh

5 2 3 12 9 cosh 103

C β β α β β α α

3β β αβ β α β α αβ

= − − + + + ∆

+ + + + − ∆ −, (A9)

( )( )

( )

3 4 3 2 2 2 210

3 5 3 2 2 2 3 3

3 2

2 10 5 48 60 3 15 60 sinh

2 2 60 60 3 15 20 28 cosh

24 10 3

C β β α β β α α β αβ

β β α β β α αβ α β α β

αβ β

= + + − − + + ∆

− + + + + − + +

+ +

∆ , (A10)

( ) ( )2 3 3 3 2 402 3 3 2 sinh 3 3 2 coC α β α α β α β α αβ β= − + + + ∆ + − + − − ∆sh , (A11)

and

( )( )

2 2 3 300

2 3 2 4 2

2 4 3 2 sinh

2 3 9 2 12 cosh3

C β α β α β α β

34β αβ α β α β β αβ

= − − + + + + ∆

+ + − + + ∆ −, (A12)

( )

012 sinh cosh D

BB C B DJ α β α

−= + −

, (A13)

where

( ) ( )

( )( )

( ) ( )

4 3 2 20

3 3 2

3 3 2

3 2 3 cosh 9 cosh sinh sinh

3cosh 2 3 sinh cosh cosh 3 sinh

3sinh 2 3 cosh 3 sinh cosh sinh

B α αβ α α α β β β α α

α β α αβ β α β β α α β α

α β α α α αβ β α β α

= + + + − −

+ ∆ + + − − + + ∆ + + + − −

,(A14)

and

( )

( ) ]

2 4 2 3 2

3 2 3

5 2 12 3 9 cosh23 3 4 2 sinh 6

DB β β β αβ α β α

α β β α β α αβ

= − + + + −

+ + + − + ∆ +

∆, (A15)

26

C DH C H DHJ+

= , (A16)

where ( ) ( )3 33 2 cosh 9 cosh sinh cosh sinhCH α β α α α α α β β = − + + − − + β (A17)

and

2 3 3 5

2 3 2 2 2

3 cosh 3 sinh 6 cosh 2 cosh15

3 cosh cosh 3 sinhDHαβ α αβ β β α β

αβ β α β α α β α

− + += −

− + −

α, (A18)

( )23 15 cosh sinhsinh cosh

C DG

α αβ α β αα β α

− − − −=

−H

, (A19)

cosh sinh3

G HF α αα+

= , (A20)

320 2 2E D B 3Fβ β= + + , (A21)

and

. (A22) ( ) (

2 3 5 3

5

cosh sinh sinh coshA E B C D F

G

D

β β β ββ β β β β β

γ

= − − − +

+ − + −

= −

) H

Eq. (A13) is then finally used to calculate Ω of Eq. (25) and to plot Fig. 2 and 3.

27

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33

θ

ac

b

ru

uθ

V

free surface z porous

shell primary particles

solid sphere

membrane

Fig. 1. The cell model of tangential-stress free surface near membrane surface, which is

applied to an impermeable solid sphere with a porous shell consisting of primary particles

of radius pa

34

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

20

40

60

80

100

λ

(d)

Happel(α = β )

Carman-Kozeny

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20

40

60

80

100

b2rc

(a) β =1

α =1

α =0.75

α =0.50α =0.25

α =0.0

α =0.95

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20

40

60

80

100

(b) β =4

α =4

α = 3

α = 2

α = 1

α = 0

α =3.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

20

40

60

80

100

b2rc

λ

(c) β =8

α = 8

α = 6 α = 4

α = 2

α = 0

α =7.6

Fig. 2. Dimensionless specific cake resistance ( )2 , ,cb r α β γ as a function of occupancy

fraction λ : (a) 1β = , (b) 4β = , and (c) 8β = , and (d) Happel (Eq. (33)) and Carman-

Kozeny (Eq. (34)) expressions. For each case of (a)-(c), α values are 0.95, 0.75, 0.50,

0.25, and 0.0 of each β , and 3γ β λ= .

35

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

vs

λ

α = 0.0 α = 2.0 α = 4.0 α = 6.0 α = 7.6 α = 8.0

(a) β = 1

α = 0.0

α = 0.25

α = 0.50

α = 0.75

α = 0.95α = 1

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

λ

(b) β = 8

36

ig. 3. Dimensionless settling velocity as a function of occupancy fraction λ : (a) 1β = ,

8β = . For each case, α values are 0.95, 0.75, 0.50, 0.25, and 0.0 of each β , and

3β λ . =

F

and (b)

γ