Journal of Geoscience and Environment Protection, 2014, 2, 13-23
Published Online December 2014 in SciRes. http://www.scirp.org/journal/gep
http://dx.doi.org/10.4236/gep.2014.25003
How to cite this paper: Zhu, Z. F. (2014). Theory on Orthokinetic Flocculation of Cohesive Sediment: A Review. Journal of
Geoscience and Environment Protection, 2, 13-23. http://dx.doi.org/10.4236/gep.2014.25003
Theory on Orthokinetic Flocculation of
Cohesive Sediment: A Review
Zhongfan Zhu
College of Water Sciences, Beijing Normal University, Beijing, China
Email: z huzh ongfan1985@ gmail.co m
Received N ovemb er 20 14
Abstract
Investigation on flocculation phenomenon of cohesive fine-grained sediment has been a important
part of sediment dynamics. During all of three dynamical factors (i.e., Brownian motion, flow shear
and differential settling) that have been verified to play important roles in promoting flocculation
of cohesive sediment, the influence of flow shear on sediment flocculation has been paid great at-
tention by many researchers (this flocculation pattern has been termed as “orthokinetic floccula-
tionin most of published literatures). Among many researches regarding orthokinetic floccula-
tion, the dynamical equation developed by Sm olu chows ki in 1917 (we called it as Smoluchowski
equation hereafter) has been widely adopted as an origin and basement for theoretically analyz-
ing sediment flocculation under a shear flow. Meanwhile, many researchers have also pointed out
the deficiencies of Smol uc h owski equation (this is because the derivation of Smolu ch owsk i equa-
tion was based on six different assumptions), and correspondingly have amended this equation
from different aspects. In this paper, we attempt to summarize these results, hopefully providing
the theoretical research of sediment orthokinetic flocculation with some references.
Keywords
Sediment, Orthokinetic Flocculation, Review
1. Introduction
Flocculation phenomenon of cohesive sediment can widely be observed in most of estuaries, reservoirs and
channels. Sediment flocculation has been confirmed to play an important role in influencing development of
estuary delta, geomorphologic variation of sand bar and dredging of mud layer channel. Mechanisms of floccu-
lation are complex, and there are many factors that have been verified to play some roles in influencing the
flocculation process, including physical and chemical factors (such as mineral composition of sediment particle,
sediment concentration, particle size, type of electrolyte in the suspension, concentration of electrolyte, temper-
ature, pH value), as well as dynamical factors (flow shear, Brownian motion and differential settling). At present,
most of researches have been carried out to demonstrate the mechanism and dynamic characteristic of sediment
flocculation (Yang & Qian, 1986; Liu, 1994; Guan & Chen, 1995; Jiang & Zhang, 1995; Guan et al., 1996;
Zhang, 1996; Jin et al., 1998; Shi, 2000; Chen et al., 2001; Jin et al., 2002; Jiang et al., 2002).
In particular, in the case of estuary and coastal area, flocculation of sediment is inseparable with a flow shear
Z. F. Zhu
14
due to inland runoff, tidal current from open sea, as well as interaction of salty water and fresh water. The kind
of flocculation due to a flow shear has been termed as orthokinetic flocculation, and it is of great interest for
many researchers. The theoretical basement of most of researches regarding orthokinetic flocculation can be
found to originate from a pioneering dynamical equation put forward by Smoluchowski in 1917 (we shall call it
Smoluchowski equation hereafter). Smoluchowski equation can simply describe concentration variation of dis-
crete particle during flocculation due to a flow shear. The term in this equation that can describe the influence of
a flow shear on flocculation is simply velocity gradient of a laminar flow. Later, Camp and Stein (1943) intro-
duced the root-mean-square (RMS) of velocity gradient representing the turbulent intensity into Smoluchowski
equation, which makes the discussion on effect of a turbulent flow on flocculation become possible. Although
Smoluchowski equation has been widely as a origin for theoretically analyzing flocculation due to a shear flow,
derivation of this equation was based on some assumptions, which inevitably simplifies the real engineering sit-
uations and thereby limiting its applicability, as many researches shown. Plenty of researches have been per-
formed to amend Smoluchowski equation from different aspects, and in this paper we attempt to summarize
these research results scatted in different literatures, hopefully providing the theoretical study regarding sedi-
ment orthokinetic flocculation with some references.
2. Theoretical A nalysis of Flocculation Dynamics
In 1917, Smoluchowski derived a dynamical equation to describe orthokinetic flocculation of discrete particle
(Smoluchowski, 1917)
1 max
33
1, 1
14 4
23 3
k
ki j ijki ik
ii jki
ndu du
nn RnnR
t dzdz
= +==
 
= −
 
 
∑∑
(1)
where
i
,
j
,
k
are grades of sizes of the flocs,
max
is that of the maximum floc,
i
n
,
j
n
,
are number
concentrations of
i
,
j
,
k
-grade flocs respectively (here
kij= +
, meaning a
k
-grade floc results from col-
lision and subsequent adhesion between
i
and
j
-grade flocs),
du dz
is velocity gradient in a laminar
flow(here
u
is fluid velocity along the mentioned direction, and
z
is the direction perpendicular to the men-
tioned direction in a simple two-dimensional right-handed coordinate system),
ij i j
R rr= +
,
iki k
R rr= +
are
radii of collisions between
i
,
j
-grade flocs and between
i
,
k
-grade flocs, respectively,
i
r
,
j
r
,
k
r
are ra-
dii of
i
,
j
,
k
-grade flocs respectively.
As Thomas et al. (1999) introduced, there exists six assumptions in the derivation of Equation (1), and they
are summarized as follows: 1) flow is assumed to be laminar; 2) all particles are special-shaped, and they are
composed of solid spheres. The moving trajectory of one particle are assumed completely independent from
others, and this trajectory should be rectilinear; 3) no breakup or sedimentation during process of random colli-
sion and subsequent adhesion can happen; 4) all of inter-particle collisions can result in corresponding adhesion,
indicating the so-called collision efficiency coefficient is equal to be unity; 5) sizes of all particles are the same
at the beginning of flocculation (this state can be regarded to be in the mono-disperse condition, in contrast to
the poly-disperse condition); 6) collisions between particles only happen between two particles, and those of
three or more particles are never mentioned. These assumptions have been found to oversimplify the real engi-
neering situation. As a result, this limits the applicability of Equation (1). Here, we attempt to summarize some
results regarding amendments of Smoluchowski equation according to every assumption mentioned above.
2.1. About the First Assumption
2.1.1. Laminar Non-Uniform Shear Flow
Realizing the fact that Equation (1) can only be applied in a laminar uniform flow, Camp and Stein (1943) in-
troduced the root-mean-square (RMS) of velocity gradient,
G
, in a non-uniform flow into Equation (1) instead
of
dduz
as follows
1
2
G
µ

Φ
=

, (2)
where
Φ
is dissipate work of flow (per volume per time) due to viscous shear. For the entire system, the sta-
tistically-averaged velocity gradient,
G
, should be
Z. F. Zhu
15
1
2
G
µ

Φ
=


, (3)
where
Φ
is the averaged value of
Φ
over the system.
Kramer et al. (1997) and Kramer (1997) pointed out that there may be two problems in the study of Camp and
Stein. Tangential parts of rate of strain tensor were only mentioned in their study, whereas normal parts were
neglected, as long as Smoluchowski equation is extended from two-dimensional form to three-dimensional form.
Statisticall y-averaged velocity gradient,
G
, was simply adopted as a unique parameter to evaluate flocculation
process in the study of Camp and Stein, and this will cause some errors. Experiments have demonstrated that
flocculation process and resultant effect may be favorably different, although a same
G
can be found in dif-
ferent flocculation reactors. Furthermore, by transforming normal rate of strain tensor into a special rate of strain
tensor with only major stress rate remaining, Kramer (1997) derived the flocculation dynamical equation with
the following new form:
( )
( )
1 max
33
''
max max
1, 1
14 4
23 3
k
kij ijki ik
ii jki
na nnrrna nrr
t
ππ
= +==
=+− +
∑∑
, (4)
where all of parameters have the same meaning as introduced above, except for
'
max
a
being a absolute value of
maximum value of rate of strain tensor after coordinate system transformation. Moreover, Pedocchi and Piedra-
Cueva (2005) clarified that rotation of flow field cannot cause the inter-particle collision, and put forward the
following equation to describe the collision between particles by using some mathematical methods:
( )
3
22 2222'
00
1sin cos(1 )sin sincossin
2
ijiji j
Nbbd drrnn
ππ
θϕθϕθ θθϕλ
=+−−∗ +
∫∫
, (5)
where
ij
N
is the number of collisions between
i
,
j
-grade flocs (per volume per time);
θ
,
ϕ
are basic pa-
rameters in the spherical coordinate system (
02
ϕπ
≤≤
,
0
θπ
≤≤
);
b
is a coefficient related to three eigenva-
lues of deformation rate tensor after coordinate system transformation,
'
λ
is a “common part” of these ei
genvalues, thus the eigenvalues are
'
b
λ
,
( )
'
1b
λ
,
'
λ
. For a laminar shear flow with velocity gradient being
a constant (
0du dzC= >
), three eigenvalues become
2C
, 0,
2C
(
1b=
,
'
2C
λ
=
can also be obtained
correspondingly), and thus Equation (5) reduces to
( )
3
43
ijiji j
NrrCn n= +
, which is consistent with the expre-
ssion describing number of collisions between
i
,
j
-grade flocs in Smoluchowski equation.
2.1.2. Turbulent Flow
Turbulent flow is accompanied by a series of eddy motions. Those large eddies which contain the energy of
system have been termed as “containing-energy eddies”. Energy can be transported from those large eddies to
smaller eddies, until all of energy can be dissipated by viscous force of a particular eddy with a minimum size.
This kind of eddy with the minimum size has been termed as “Kolmogorovf micro-scale eddy” (Xia, 1992), and
its size,
η
, can be expressed to be
1
34
ν
ηε

=

, (6)
where
ν
is kinematic viscosity of flow, and
ε
is energy dissipate rate of the turbulent flow.
Camp and Stein (1943) stated that it will be reasonable to substitute velocity gradient in the turbulent flow,
G
,
with the following form for
dduz
in Equation (1)
1
2
G
ε
ν

=

. (7)
Since the system is mentioned, average velocity gradient,
G
, with the following form should be used cor-
respondingly
Z. F. Zhu
16
1
2
G
ε
ν

=


, (8)
where
ε
is the averaged value of
ε
over the system.
Saffman and Turner (1956) also analyzed aggregation phenomenon of particle in a turbulent flow. By assum-
ing that floc size is smaller than Kolmogorovf micro-scale
η
and that the turbulent flow is homogeneous, they
presented the following expression to describe inter-particle collision
( )
3
8
15
ijiji j
Nr rnn
πε
ν
= +
, (9)
whereas Dilichatsios and Probstein (1974) studied the aggregation of particles whose sizes are between Kolmo-
gorovf micro-scale
η
and size of containing-energy eddy, and put forward a new mathematical expressions as
follows:
( )
17
33
1.37
ijiji j
Nr rnn
πε
= +
. (10)
2.2. About the Second Assumption
2.2.1. Fractal Structure of Floc
The second assumption behind Smoluchowski equation has been regarded to be incompatible with some expe-
rimental observations of porous flocs, and experiments have shown that flocs have the fractal structures(Burban
et al., 1989; Li et al., 1998; Ma & Pierre, 1999; Lartiges et al., 2001; Li et al., 2004). For a single floc having the
fractal structure, there is a simple mathematical relation to represent the relation between its mass and size as
follows(Meakin, 1988):
f
D
ff
Md
, (11)
where
f
M
is mass of the floc,
f
d
characteristic length of the floc (this characteristic length can be com-
monly regarded to be equal to size of the floc), and
f
D
is mass fractal dimension of the floc, representing
compactness degree of the floc (here
13
f
D
<≤
). The expression between characteristic length of the floc and
number of single primary particle contained in the floc can be expressed as follows (Feder, 1988):
0
f
D
f
d
id

=

, (12)
where
i
is number of single primary particle contained in the fractal floc, and
0
d
is size of single primary
particle. On substituting Equation (12) into Equation (1), the dynamical equation to describe the process of
flocculation of fractal floc can be rewritten (Kramer & Clark, 1999):
11
3
3
11
1 max
33
00
1, 1
14 4
23 3
DD
ff
ff
kDD
kij ijki ik
ii jki
ndu du
nnr iinnr ii
t dzdz
= +==


 

=+− +
 
 
 
 
∑∑
, (13)
where
0
r
is radius of single primary particle, and
i
i
,
j
i
,
k
i
are numbers of single primary particle contained
in
i
,
j
,
k
-grade flocs respectively.
2.2.2. Moving Trajectory of Particl e
1) Rectilinear trajectory, curvilinear trajectory and others Each of Smoluchowski equation, Camp and
Stein method, and Saffman and Turner equation neglects influences of hydrodynamic force and short-range in-
teractive force, when one particle touches and collides with another particle, and considered moving trajectory
of the particle to be rectilinear. This understanding may be not in accordance with real situations. When two
mentioned particles get into contact, surrounding fluid between these particles would be squeezed out, conse-
quently leading to the rotation of particle relative to another particle and finally resulting in deviation of moving
trajectory of particle from the rectilinear one; meanwhile, short-range interactive force between particles should
Z. F. Zhu
17
also play some role in the collision of particle. Han and Lawler (1992) pointed out that moving trajectory of par-
ticle may be curvilinear, and that frequency function of collision between particles can be expressed as follows:
curcur rec
e
ββ
=
, (14)
where
cur
β
is a frequency function of collision between particles (or collision kernel in some literatures) under
the assumption of a curvilinear moving trajectory(its mathematical expression such as frequency function,
ij
β
,
of collision between
i
,
j
-grade flocs is:
ijiji j
N nn
β
=
),
rec
β
being a frequency function of collision under
the assumption of a rectilinear trajectory (from Smoluchowski equation, expression of frequency function,
( )
,
rec
ij
β
, of collision between
i
,
j
-grade flocs can be found to be
( )
( )
3
, 43
reci j
ijGr r
β
= +
), and
cur
e
is a
correction coefficient(this coefficient can be gotten by numerical simulation). For a curvilinear trajectory model,
it can be inferred that flocculation rate is no longer proportional to velocity gradient,
G
.
Neither the rectilinear trajectory model nor the curvilinear trajectory model considers the porous and loose
properties of floc, which means that a simple usage of the rectilinear or curvilinear trajectory model may bring
some deviations from real situation. There are some researches that attempt to tackle this problem. Veerapaneni
and Wiesner (1996) and Thill et al. (2001) proposed a new formula to calculate frequency function of collision,
ij
β
, between
i
,
j
-grade flocs by incorporating the considerations of fractal structure and permeability of floc
as follows
( )
3
4
3
ijfi ifjj
G ErEr
β
= +
, (15)
where
fi
E
,
fj
E
are efficiency coefficients of collection of fluid surrounding
i
,
j
-grade flocs, respectively,
and they are closely related to permeability of floc (their values can be determined using some empirical formu-
lae). On the other hand, Kusters et al. (1997) have developed a simple shell-core efficiency model to calculate
frequency function of collision as expressed as follows
( )
3
1.294
ijijij
Gr r
βα
≅+
, (16)
where
ij
α
is a collision efficiency coefficient,
0.18
ij
Fl
α
(van de ven & Mason, 1977), and
Fl
is a ratio
of the hydrodynamic force and Van der Waals force between primary particles contained in
i
,
j
-grade flocs
(when calculating the ratio
Fl
, it needs to be noticed that core is not penetrative in the shell-core model, whe-
reas shell is penetrative). In particular,
ij
α
can be calculated by analyzing the moving trajectory of particle.
Furthermore, Li and Logan (1997) paid special attention to the collision between a fractal floc and a small par-
ticle, and put forward the following formula to calculate frequency function,
frac
β
, of collision:
fracfrac rec
e
ββ
=
, where
frac
β
is a frequency function of collision between a fractal floc and a small particle, the
meaning of
rec
β
has been explained as before, and
frac
e
is a correction coefficient. Some experiments have
shown that there is a empirical relation
1 0.33
f
D
frac
G
β
. Thus this expression will reduce to
frac G
β
when
the floc is infinitely loose(that is
0
f
D=
). In this case, collision trajectory between the floc
and the small particle should be rectilinear, and the model put forward by Li and Logan becomes to agree with
the rectilinear trajectory model; while the floc is a compact sphere (that is
3
f
D
), the expression reduces to
the relation indicating
frac
β
has little relationship with
G
. In this case, collision trajectory between the floc
and the small particle should be curvilinear, and thus Li and Logan's model presents to be in agreement with the
curvilinear trajectory model. As an addition, some calculations have shown that
frac
β
is larger than the value
from a curvilinear trajectory model by five orders of magnitude and smaller than that from the rectilinear one by
two orders of magnitude, respectively.
2) Two expression forms of frequency function of collision In Smoluchowski equation, all of particles that
can collide with the mentioned particle is regarded simply to be in a cylinder with its radius being equal to the
collision radius
collision
R
(a algebraic sum of radii of colliding particles), so this kind of expression form of
frequency function of collision has been termed as cylinder formula. However, Saffman and Turner equation
(1956) mentioned that all of particles that can collide with the mentioned particle should be on the surface of the
sphere with its centre being the centroid of the mentioned particle and its radius being equal to the collision ra-
dius
collision
R
, and in contrast this kind of expression form has been termed as spherical formula. Two kinds of
Z. F. Zhu
18
expression forms can be shown as follows:
( )
2
2
ijcollisioni j
Rurru
βπ π
== +
, (17)
( )
2
2
22
sph
ijcollision rijr
Rurr u
βπ π
== +
, (18)
where
sph
ij
β
is frequency function of collision bet ween
i
,
j
-grade flocs corresponding to the spherical mod-
el,
u
is the statistically-averaged value of the relative velocity of two arbitrary points in the flow filed, whe-
reas
r
u
is counterpart along the radial direction. Comparison of Equation (17) and Equation (18) can
find that Equation (17) was larger than Equation (18) by more than twenty-five percents when the mentioned
flow is homogeneously turbulent and by twenty percents when the flow is a laminar one, and that these equa-
tions were the same when a random motion or a simple gravity sedimentation is mentioned (Wang et al., 1998).
2.3. About the Third Assumption
2.3.1. Breakup of Fl oc
The porous and loose floc is prone to undergo breakup in a shear flow. It may be understood that breakup of the
floc will happen as long as a shear stress,
τ
, exerted by the fluid on the floc exceeds strength of the floc,
T
σ
.
Two kinds of models to describe breakup of the floc have been widely accepted (Muhle, 1993; Thomas et al.,
1999).When size of the floc exceeds Kolmogorovf micro-scale,
η
, the floc should be in the inertial sub-range
in a turbulent flow. In this case, the floc will undertake the action of fluctuating pressure of the turbulent flow,
which may cause a large-scale fragmentation of the floc, and this kind of breakup has been termed as large-scale
fragmentation. While for those flocs with their sizes being smaller than Kolmogorovf micro-scale,
η
, they
should be in the viscous sub-range of the turbulent flow. In this case, the flocs bear the action of surface disrup-
tion of the turbulent flow, consequently making some small particles in the floc be easy to separate them from
main part of the floc, and in contrast this kind of floc breakup has been termed as surface erosion.
2.3.2. Maximum Size of Floc (or Critical Size)
When size of the floc reaches the maximum,
max
d
, the floc should be on the verge of its breakup. Many works
have been carried out to discuss this critical condition corresponding to breakup of the floc, and three kinds of
viewpoints can simply be summarized. (1)When a hydrodynamic stress,
τ
, exerted by fluid on the floc is equal
to strength of the floc,
T
σ
, the mentioned floc is in the critical condition of its breakup; (2)When the force due
to the flow shear,
F
τ
, is equal to comprehensive interactive force between primary particles in the floc,
F
, the
floc will be in the critical condition of breakup; (3)When the external kinetic energy is equal to bonding energy
of the floc, the floc is in the critical condition. On the first viewpoint, Muhle and Domasch (1990) and Lu et al.
(1998) presented different formulae for calculating size of the maximum floc depending on different sub-ranges
where the floc as expressed as follows:
Viscous ra nge :
( )
11
1
22
4
max
2
w
dF
ρ εν
=
;
Transition range:
111 1
-
1
222 2
max 0
1.5 w
d dF
ρ εν
=
;
Inertial range:
3611 93
5520205
max 0
0.68 w
d dF
ρ εν
−−−
=
(19)
where
F
is the comprehensive interactive force between primary particles contained in the floc as explained
above, and
w
ρ
is density of fluid. From the second viewpoint, Coufort et al. (2005; 2008) concluded that there
was a simple expression representing the relation between size of the maximum floc and the turbulent intensity
as follows
1
4
max
d
ε
, (20)
regardless of the fact that the floc may be in different sub-ranges. Based on the third viewpoint, Bache (2004)
presented a more general mathematical expression between size of the maximum floc and the turbulent intensity
Z. F. Zhu
19
as follows
-2
max
d
m
c
ε
=
, (21)
where
c
,
m
are two positive constants. Similar works can also be found in Bouyer and Line (2004). On subs
tituting Equation (7) into Equation (21), we can have
'-
max
d
m
cG=
or
'
max
loglog log
dcm G= −
, where
c
is
a constant (
2
m
cc
ν
=
). In this logarithmic mathematical expression, we can infer that the larger
m
coupled
with the same
G
corresponds to the small
max
d
. This means
m
may be a parameter to represent floc strength.
Additionally,
'
c
is also a parameter related to floc strength, and it may be closely related to the method for
measuring size of the floc.
2.4. About the Fourth Assumption
Smoluchowski equation assumed that each collision between two particles can result in a subsequent adhesion
between particles. This assumption means the so-called collision efficiency coefficient is equal to unity. The va-
lidity of this assumption has been questioned by many researchers (e.g., Elimelech & O’Melia, 1990; Kim &
Kramer, 2007). They stated that whether the collision between particles can lead to subsequent adhesion or not
depended on many factors (such as the repulsive force of double-electrical layer on the surface of particle, spa-
tial hindering interaction or hydrodynamic interaction between one particle and another particle, as well as
viscous effect of fluid). These factors can be characterized by the collision efficiency coefficient, but at present
the determination of this coefficient seems a difficult work.
It should be pointed out that hydrodynamic interaction between colliding particles may play a important role
in influencing adhesion between particles, in particular when particles are in the turbulent flow. In fact, the in-
fluence of hydrodynamic interaction on the collision efficiency coefficient can be analyzed in terms of the mov-
ing trajectory of particle, and a detailed summary can be found in Section 2.2.2.
2.5. About the Fifth and Sixth Assumptions
Although the establishment of Smoluchowski equation is strictly based on the assumption of a mono-disperse
flocculation system, it is also applicable to a poly-disperse flocculation system after introducing the size group
met ho d (Chang et al., 1992).
Regarding the sixth assumption behind Smoluchowski equation (i.e., the mutual collision is only between two
particles), if number concentration of single particle in a flocculation system is low, the error caused by this as-
sumption may be negligible; whereas for those flocculation systems in which number concentration of single
particle is so high that collisions among three or more particles are not easily neglected, collision patterns of
particles will be complex, and a simple usage of Smoluchowski equation will lead to some unexpected errors. At
present, the research regarding this aspect have been rarely found (Chang et al., 1992).
As a summary, a complete dynamical equation describing the flocculation of particle incorporating breakup of
the floc can be expressed as follows (Kim & Kramer, 2007):
1 maxmax
1, 11
1
2
k
kijij ijkikikik kiki i
ii jkiik
dn nnnnS nSn
dt
αβ αβγ
= +===+
=− −+
∑∑ ∑
, (22)
where
ik
α
,
ik
β
are collision efficiency coefficient and collision frequency function of
i
,
k
-grade flocs re-
spectively,
k
S
is a function of breakup of
k
-grade floc,
ik
γ
is distribution percent of
k
-grade floc in all of
the flocs due to breakup of
i
-grade floc with its size being larger than
k
-grade floc. On the right-handed side
of Equation (22), the first term represents the rate of increase of number concentration of
k
-grade floc due to
flocculation of small flocs with their sizes being smaller than
k
-grade floc, the second term denotes the rate of
decrease of number concentration of
k
-grade floc due to flocculation of
k
-grade floc with other flocs with
different sizes, the third term showing the rate of decrease of number concentration of
k
-grade floc due to its
breakup, and the last term presents the rate of increase of number concentration of
k
-grade floc due to breakup
of those big flocs with their sizes being larger than
k
-grade floc. There are many research results regarding
mathematical expressions of
,
ik
γ
, and two summaries can be found in studies of Han et al. (2003) and Kim
and Kramer (2007). In fact, it will be difficult to obtain the analytical solution to Equation (22). Thus, some nu-
Z. F. Zhu
20
merical methods have commonly been adopted to deal with this equation, and some physical insights regarding
flocculation of particle have been demonstrated in many researches(Higashitani & Iimura, 1998; Kramer &
Clark, 1999; Runkana, 2003; Selomulya et al., 2003; Zhang & Li, 2003; Li et al., 2004; Prat & Ducoste, 2006;
Coufort et al., 2007; Kim & Kramer, 2007).
3. Some Remarks
Most of researches on orthokinetic flocculation of particle have been carried out in the fields of colloidal chemi-
stry, marine engineering, water treatment and the study of water environment. In contrast, some theoretical stu-
dies regarding orthokinetic flocculation of sediment particle have not been commonly found. Whether Smolu-
chowski equation can be directly applied to investigate flocculation phenomenon of sediment particle or not is
worthy being studied. Here we attempt to put forward some viewpoints, hopefully providing the theoretical
study of sediment orthokinetic flocculation with a fe w re ferences.
1) Particularity of flocculation of sediment Smoluchowski equation dealt with the particle with single min-
eral components, a spherical structure, a single charge distribution on its surface and a uniform size. However, in
real situations, properties of sediment particle are indeed complex, and sediment particle has diverse mineral
components (for example, Illite, Kaoline, Montmorillonite), an irregular shape, and a complex distribution of
surface charges (Charges distributed on the surface and edge of sediment particle are opposite). It will be ques-
tionable to apply Smoluchowski equation into the research field of sediment orthokinetic flocculation without
any modifications (Yang et al., 2003).
2) Difficulty of method for determining maximum size of the floc It is widely recognized that size of the
floc reaches the maximum only if the floc is in the critical condition of its breakup. However, it may be difficult
to determine maximum size of the floc because of a lack of elaborate knowledge on the interactions among ed-
dies in the turbulent flow and the flocs, which needs to be paid much attention in further research.
3) Complexity of structure of floc Introduction of fractal theory has tackled the problem that it is difficult to
describe complex structure of the floc mathematically to some degree. But as Maggi et al. (2007) and Son and
Hsu (2009) have pointed out by experiment and numerical simulation, fractal properties of sediment floc were
not in a good agreement with fractal theory(for example, experiments shown that the fractal dimension of sedi-
ment floc was variable, which is contradictory to the conclusion from fractal theory that the fractal dimension
should be constant for a given type of sediment). More detailed analysis on floc structure may be helpful to un-
derstand the process of orthokinetic flocculation of sediment particle.
4) Difficulty of determination of collision efficiency coefficient Determination of collision efficiency coef-
ficient is of theoretical and practical importance in the discipline of particle flocculation. In the context of col-
loidal chemistry, this coefficient has been regarded as the reciprocal of stability ratio of coagulation, and hence it
should closely be related to physicochemical characteristics of particle surface(Chang et al., 1992). In some pub-
lished literatures, collision efficiency coefficient has been simply dealt with as a empirical constant, may due to
a lack of effective methods to determine this coefficient. Furthermore, when DLVO theory in the discipline of
colloidal chemistry (a detailed introduction can be found in Chang et al. (1992) is adopted to analyze the colli-
sion efficiency coefficient in the sediment flocculation, applicability of this theory for a micro-sized sediment
particle needs to be taken into account.
Acknowledgements
This research is supported by the Fundamental Research Funds for the Central Universities in China (Number:
2013NT50).
References
Bache, D. H. (2004). Floc Rupture and Turbulence: A Framework for Analysis. Chemical Engineering Science, 59, 25 21-
2534. http://dx.doi.org/10.1016/j.ces.2004.01.055
Bouyer, D., & Line, A. (2004). Experimental Analysis of Floc Size Distribution under Different Hydrodynamics in a Mixing
Tank. AIChE Journal, 50, 2064-2081. http://dx.doi.org/10.1002/aic.10242
Burban, P. Y., Lick, W, & Lick, J. (1989). The Flocculation of Fine-Grained Sediments in Estuarine Waters. Journal of
Geophysical Research, 94, 8323-8330. http://dx.doi.org/10.1029/JC094iC06p08323
Z. F. Zhu
21
Camp, T. R., & Stein, P. C. (1943). Velocity Gradients and Internal Work in Fluid Motion. Journal of Boston Society of Civil
Engineering, 30, 219-237.
Chang, Q., Fu, J., & Li, Z.(1992). The Principal of Flocculation. Lanzhou: Lanzhou University Press. (In Chinese)
Chen, H., Shao, M., & Li, Z. (2001). Preliminary Study on the Effect of NaCl on Fine Sediment Flocculation and Settling in
Still Water. Acta Pedologica Sinica, 1, 131-134. (In Chinese)
Coufort, C., Bouyer, D., & Line, A. (2005). Flocculation Related to Local Hydrodynamics in a Taylor-Couette Reactor and
in a Jar. Chemical Engineering Science, 60, 2179-21 92. http://dx.doi.org/10.1016/j.ces.2004.10.038
Coufort, C., Bouyer, D., Line, A., & Haut, B. (2007). Modelling of Flocculation Using a Population Balance Equation.
Chemical Engineering and Processing, 46, 1264 -12 73. http://dx.doi.org/10.1016/j.cep.2006.10.012
Coufort, C., Dumas, C., Bouyer, D., & Line, A. (2008). Analysis of Floc Size Distribution in a Mixing Tank. Chemical En-
gineering and Processing: Process Intensification, 47, 287 -294. http://dx.doi.org/10.1016/j.cep.2007.01.009
Delichatsios, M. A., & Probstein, R. F. (1974). Coagulation in Turbulent Flow: Theory and Experiment. Journal of Colloid
and Interface Science, 51, 394-405 . http://dx.doi.org/10.1016/0021-9797(75)90135-6
Elimelech, M., & O’Melia, C. R. (1990). Effect of Partilce Size on Collision Efficiency in the Deposition of Brownian
Particles with Electrostaic Energy Barriers. Langmuir, 6, 1153-1163 . http://dx.doi.org/10.1021/la00096a023
Feder, J. (1988). Fractals. New York: P le n um. http://dx.doi.org/10.1007/978-1-4899-2124-6
Guan, X., & Chen, Y. (1995). Experimental Study on Dynamic Formula of Sand Coagulation Sinking in Stationary Water in
Yangtze Estuary. Ocean Engineering, 1, 46-50. (In Chinese)
Guan, X., Chen, Y., & Du, X. (1996). Experimental Study on Mechanism of Flocculation in Yangtze Estuary. Journal of
Hydraulic Engineering, 6, 70-80. (In Chines e)
Han, B., Akeprathumchai, S., Wickramasinghe, S. R., & Qian, S. (2003). Flocculation of Piological Cells: Experiment vs
Theory. AIChE Journal, 49, 16 87-1701. http://dx.doi.org/10.1002/aic.690490709
Han, M. Y., & Lawler, D. F. (1992). The Relative Insignificance of G in Flocculation. American Water Works Association
Journal, 84, 79-91.
Higashitani, K., & Iimura, K. (1998). Two-Dimensional Simulation of Breakup Process of Aggregates in Shear and Elonga-
tional Flows. Journal of Colloid and Interface Science, 204, 320-327 . http://dx.doi.org/10.1006/jcis.1998.5561
Jiang, G., & Zhang, Z. (1995). Flocculation Deposition of Fin-Grained Sediment and the Concentration of Cation in Yangtze
Estuary. Acta Oceanologica Sinica, 17, 76-82. (In Chinese)
Jiang, G., Yao, Y., & Tang, Z. (2002). The Analysis of Factors of Flocculation of Fine-Grained Sediment in Yangtze Estuary.
Acta Oceanologica Sinica, 24, 51-56. (In Chinese)
Jin, D., Wu, H., & Shi, F. (1998). Analysis of Sediment Flocculation. Water Conservancy and Hydropower in Northeast
China, 11, 25-37 . (in Chinese)
Jin, Y., Wang, Y., & Li, Y. (2002). Experimental Study on Flocculation of Cohesive Fine Grain Sediment in Yangtze River
Estuary. Journal of Hohai University (Natural Sciences), 30, 61-63. (In Chinese)
Kim, J., & Kramer, T. A. (2007). Adjustable Discretized Population Balance Equations: Numerical Simulation and Parame-
ters Estimation for Fractal Aggregation and Breakup. Colloids and Surfaces A: Physicochemical and Engineering Aspects,
27, 173-188. http://dx.doi.org/10.1016/j.colsurfa.2006.06.020
Kramer, T. A. (1997). The Modeling of Coagulation Kinetics in Complex Laminar Flow. Ph.D. Thesis, Illinois: University of
Illinois at Urbana-Champaign.
Kramer, T. A., & Clark, M. A. (1997). Influence of Strain Rate on Coagulation Kinetics. Journal of Environmental Engi-
neering, 123, 444-452. http://dx.doi.org/10.1061/(ASCE)0733-9372(1997)123:5(444)
Kramer, T. A., & Clark, M. M. (1999). Incorporation of Aggregate Breakup in the Simulation of Orthokinetic Coagulation.
Journal of Colloid and Interface Science, 216, 116-12 6. http://dx.doi.org/10.1006/jcis.1999 .63 05
Kusters, K. A., Wijers, J. G., & Thoenes, D. (1997). Aggregation Kinetics of Small Particles in Agitated Vessels. Chemical
Engineering Science, 5, 107 -121. http://dx.doi.org/10.1016/S0009-2509(96)00375-2
Lartiges, B. S., Deneux-Mustin, S., Villemin, G., Mustin, C., Barrès, O., Chamerois, M., et al. (2 00 1). Composition Structure
and Size Distribution of Suspended Particulates from Rhine River. Water Research, 135, 808-816.
http://dx.doi.org/10.1016/S0043-1354(00)00293-1
Li, D., Tan, W., & Huang, M. (2004). Study on Fractal Properties of Flocs. Water and Wastewater Engineering, 30, 5-9. (In
Chinese)
Li, X. Y., Passow, U., & Logan, B. E. (1998). Fractal Dimensions of Small (15-200 µm) Particles in Eastern Pacific Coastal
Waters. De ep-Sea Research, 45, 115-131. http://dx.doi.org/10.1016/S0967-0637(97 )00 05 8-7
Z. F. Zhu
22
Li, X., & Logan, B. (1997). Collision Frequencies between Fractal Aggregates and Small Particles in a Turbulently Sheared
Fluid. Environmental Science and Technology, 31, 1237-1242. http://dx.doi.org/10.1021/es960772o
Li, X., Zhang, J., & Joseph Lee, H. W. (2004). Modelling Particle Size Distribution Dynamics in Marine Waters. Water Re-
search, 38, 1305-1317. http://dx.doi.org/10.1016/j.watres.2003.11.010
Liu, Y. (1994). The Influence of Temperature on Setting Velocity and Siltation of Cohesive Sediment. Express Water Re-
sources & Hydropower Information, 13, 21-24 . (In Chinese)
Lu, S., Ding, Y., & Guo, J. (1998). Kinetics of Fine Particle Aggregation in Turbulence. Advances in Colloid and Interface
Science, 78, 19 7-235. http://dx.doi.org/10.1016/S0001-8686(98)00062-1
Ma, K. S., & Pierre, A. C. (1995). Colloidal Behavior of Montmorillonite in the Presence of Fe3+ Ions. Colloids and Surfaces
A: Physicochemical and Engineering Aspects, 155, 359-372 . http://dx.doi.org/10.1016/S0927-7757 (99 )00 032 -1
Maggi, F., Mietta, F., & Winterwerp, J. C. (2007). Effect of Variable Fractal Dimension on the Floc Size Distribution of
Suspended Cohesive Sediment. Journal of Hydrology, 343, 43-55. http://dx.doi.org/10.1016/j.jhydrol.2007.05.035
Meakin, P. (1988). Fractal Aggregation. Advances in the Colloid Interface, 28, 249 -331 .
http://dx.doi.org/10.1016/0001-8686(87)80016-7
Muhle, E. K. (1993). Floc Stability in Laminar and Turbulent Floc. In B. Dobias (Ed.), Coagulation and Flocculation (pp.
355-390). New York: Marcel Dekker.
Muhle, E. K., & Domasch, K. (1990). Floc Strength in Bridging Flocculation. In H. H. Hahn, & R. Klute (Eds.), Chemical
water and Wastewater Treatment (pp.106-115). Heidelberg: Springer-Verlag Berlin Heidelberg.
http://dx.doi.org/10.1007/978-3-642 -760 93-8_ 8
Pedocchi, F., & Piedra-Cueva, I. (2005). Camp and Stein’s Velocity Gradient Formalization. Journal of Environmental En-
gineering, 131, 1369-1376. http://dx.doi.org/10.1061/(ASCE)0733-9372(2005)131:10(1369)
Prat, O. P., & Ducoste, J. J. (2006). Modeling Spatial Distribution of Floc Size in Turbulent Processes Using the Quadrature
Method of Moment and Computational Fluid Dynamics. Chemical Engineering Science, 61, 75-86.
http://dx.doi.org/10.1016/j.ces.2004.11.070
Runkana, V. (2003). Mathematical modeling of flocculation and dispersion of colloidal suspensions. Ph.D. Thesis, New
York: Columbia University.
Saffman, P. G., & Turner, J. S. (1956). On the Collisions of Drops in Turbulent Clouds. Journal of Fluid Mechanics, 1, 16-
30. http://dx.doi.org/10.1017/S0022112056000020
Selomulya, C., Bushell, G., Amal, R., & Waite, T. D. (2003). Understanding the Role of Restructuring in Flocculation: The
Application of a Population Balance Model. Chemical Engineering Science, 58, 327-338 .
http://dx.doi.org/10.1016/S0009-2509(02)00523-7
Shi, Z. (2000). Fine Sediment Processes in Yangtze River Estuary. Journal of Sediment Research, 6, 72-80. (in Chinese)
Smoluchowski, M. V. (1917). Versuch Einer Mathematischen Theorie der Koagulationskinetik kolloider Losungen. Zeit-
schrift f. Physik. Chemie. XCII, 92, 129-168.
Son, M., & Hsu, T. J. (2009). The Effect of Variable Yield Strength and Variable Fractal Dimension on Flocculation of Co-
hesive Sediment. Water Research, 43, 3582 -3592 . http://dx.doi.org/10.1016/j.watres.2009.05.016
Thill, A., Moustier, S., Aziz, J., Wiesner, M. R., & Bottero, J. Y. (2001). Flocs Restructuring during Aggregation: Exp eri-
mental Evidence and Numerical Simulation. Journal of Colloid and Interface Science, 243, 171 -182.
http://dx.doi.org/10.1006/jcis.2001.7801
Thomas, D. N., Judd, S. J., & Fawcett, N. (1999). Flocculation Modeling: A Review. Water Research, 33, 1579-1592.
http://dx.doi.org/10.1016/S0043-1354(98)00392-3
van de Ven, T. G. M., & Maso, S. G. (1977). The Microrheology of Colloidal Dispersions. VII. Orthokinetic Doublet For-
mation of Spheres. Colloid and Polymer Science, 255, 468-47 9. http://dx.doi.org/10.1007/BF01536463
Veerapaneni, S., & Wiesner, M. R. (1996). Hydrodynamics of Fractal Aggregates with Radially Varying Permeability.
Journal of Colloid and Interface Science, 177, 45-57. http://dx.doi.org/10.1006/jcis.1996.0005
Wang, L., Wexler, A. S., & Zhou, Y. (1998). Statistical Mechanical Descriptions of Turbulent Coagulation. Physic of Fluids,
10, 2647-2651. htt p:// dx.d o i.o rg/10 .10 63 /1. 8697 77
Xia, Z. (1992). Modern Hydraulics (Volume 3). Beijing: High Education Press. (In Chinese)
Yang, M., & Qian, N. (1986). The Effect of Turbulence on the Flocculation Structure of the Slurry of Fine-Grained Sediment.
Journal of Hydraulic Engineering, 8, 21-30. (In Chinese)
Yang, T., Xiong X., Zhan X., & Yang, M. (2003). On Flocculation of Cohesive Fine Sediment. Hydro-Science and Engi-
neering, 2, 65-77. (In Chinese)
Z. F. Zhu
23
Zhang, J., & Li, X. (2003). Modelling Particle-Size Distribution Dynamics in a Flooculation System. AIChE Journal, 49,
1870-1882. http://dx.doi.org/10.1002/aic.690490723
Zhang, Z. (1996). Studies on Basic Characteristics of Fine Sediment in Yangtze Estuary. Journal of Sediment Research, 1,
67-73. (In Chinese)