ap-v2.dvi


Acta Polytechnica Vol. 50 No. 2/2010

Flock Growth Kinetics for Flocculation in an Agitated Tank

R. Šulc, O. Svačina

Abstract

Flock growth kinetics was investigated in baffled tank agitated by a Rushton turbine at mixing intensity 40 W/m3 and
kaolin concentration 0.44 g/l. The tests were carried out with a model wastewater (a suspension of tap water and kaolin).
The model wastewater was flocculated with organic Sokoflok 56 A flocculant (solution 0.1 % wt.). The flock size and
flock shapewere investigated by image analysis. A simple semiempirical generalized correlation for flock growth kinetics was
proposed, andwasused for data treatment. Theflock shapewas characterized by fractal dimension Df2. Using the statistical
hypothesis test, the fractal dimension was found to be independent of flocculation time, and the value Df2 =1.442 ±0.125
was determined as the average value for the given conditions.

Keywords: flocculation, flock growth, flock size, mixing, Rushton turbine, kaolin slurry.

1 Introduction

Flocculation is one of the most important operations
in solid – liquid separation processes in water supply
and wastewater treatment. The purpose of floccula-
tion is to transform fine particles into coarse aggre-
gates – flocks that will eventually settle – in order to
achieve efficient separation.

The properties of separated particles have a major
effect on the separation process and on separation ef-
ficiency in a solid – liquid system. The solid particles
in a common solid – liquid system are compact and
are regular in shape, and their size does not change
during the process. The size of these particles is usu-
ally sufficiently described by the diameter or by some
equivalent diameter.

The flocks that are generated are often porous and
are irregular in shape, which complicates the floccu-
lation process design. Flock properties such as size,
density and porosity affect the separation process and
its efficiency.

The aim of this work is to propose a simple semiem-
pirical generalized correlation for flock growth kinetics
and to verify the proposed model experimentally.

2 Generalized correlation for
flock growth kinetics

Turbidity measurement has been used and recom-
mended for flocculation performance assessment in
routine control in industry. Flocculation efficiency has
frequently been expressed as the rate of turbidity re-
moval:

Z∗e (tF ) =
Ze(tF )

Z0
=

Z0 − Zr(tF )
Z0

= 1 − Z∗r (tF ) (1)

where Z∗e is turbidity removal degree, Z
∗
r is residual

turbidity degree, Z0 is turbidity of a suspension before

the beginning of flocculation, Ze is eliminated turbid-
ity due to flocculation, Zr is residual turbidity of clari-
fied water after flock separation, and tF is flocculation
time.

Šulc [1] found that flocculation kinetics expressed
as the dependence of the residual turbidity rate on the
flocculation time can be expressed by a simple formula,
taking into account flock breaking:

Z∗r = AZr∗ · log
2(t∗F ) + BZr∗ · log(t

∗
F ) + CZr∗ (2)

where Z∗r is residual turbidity degree, t
∗
F is dimension-

less flocculation time, and AZr∗ , BZr∗ , CZr∗ are the
model parameters. For flocculation taking place in an
agitated tank, Šulc and Ditl [2] recommend dimension-
less flocculation time t∗F given by:

t∗F = n · tF (3)

where tF is flocculation time, and n is impeller rota-
tional speed. The proposed definition of dimensionless
flocculation takes this into account due to the charac-
teristic time choice. The chosen characteristic time,
tchar ∝ 1/n, is proportional to the number of liquid
passages through an impeller.

Based on Eq. (2) Šulc [1], Šulc and Ditl [3] have
proposed a generalized correlation for flocculation ki-
netics in an agitated tank that takes into account flock
breaking, as follows:

ΔZ∗r = A
∗
Zr∗ ·

(
Δ[ntF ]

∗
log

)2
, (4)

where

ΔZ∗r =
Z∗r − Z∗rmin

Z∗rmin
(5)

Δ[ntF ]
∗
log =

log(ntF ) − log([ntF ]min)
log([ntF ]min)

, (6)

where Z∗r min is minimal residual turbidity degree
achieved at time [ntF ]min, [ntF ]min is the dimension-
less flocculation time in which Z∗r min can be achieved,

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Acta Polytechnica Vol. 50 No. 2/2010

A∗ is the residual turbidity shift coefficient, tF is the
flocculation time, and n is impeller rotational speed.

According to Lambert’s law, the turbidity depends
on the cross-sectional area of the flock σ and flock con-
centration Np, as follows:

Zr ∝ Np · σ (7)

Assuming that the cross-sectional area of flock σ is
proportional to flock size df

σ ∝ d2f (8)

and particle/flock mass conservation must be fulfilled
at any time, flock concentration Np can be expressed
as follows:

Np · d3f ∝ const. (9)
Using Eqs. (7), (8) and (9), the dependence of flock
size on flocculation time can be given by a simple for-
mula taking into account flock breaking:

(1/df ) = Af · log2(ntF ) + Bf · log(ntF ) + Cf (10)

where (1/df ) is reciprocal flock size, ntF is dimension-
less flocculation time, and Af , Bf , Cf are the model
parameters. Assuming that the minimum residual tur-
bidity degree corresponds to the maximum flock size,
the dimensionless model variable ΔZ∗r can be rewrit-
ten as follows:

ΔZ∗r =
Z∗r − Z∗rmin

Z∗rmin
=

Zr − Zrmin
Zrmin

∝

Nf
Nfmax

· d2f − d
2
fmax

d2fmax
=

(1/df ) − (1/dfmax)
(1/dfmax)

(11)

where df is flock size and df max is maximal flock size.
Then the generalized correlation for flock growth

kinetics that takes into account flock breaking can be
derived as follows:

Δ(1/df )
∗ = A∗f · (Δ[ntF ]

∗
log)

2 (12)

rewritten

dfmax
df

= 1 + A∗f · (Δ[ntF ]
∗
log)

2 (13)

where

Δ(1/df )
∗ =

(1/df ) − (1/dfmax)
(1/dfmax)

, (14)

A∗f =
B2f

4 · Af · Cf − B2f
, (15)

where dfmax is the maximum flock size reached at time
[ntF ]max, [ntF ]max is the dimensionless flocculation
time in which dfmax can be achieved, Δ[ntF ]

∗
log is the

variable defined by Eq. (6), and Af , Bf , Cf are pa-
rameters of Eq. (10).

The generalized correlation parameters dfmax,
[ntF ]max and A

∗
f depend generally on the flocculation

process conditions, e.g. mixing intensity and floccula-
tion dosage.

3 Model evaluation

The proposed generalized correlation was tested on the
published experimental data by Kilander et al. [4]. Ki-
lander et al. [4] investigated the local flock size distri-
butions in square tanks of different sizes (5, 7.3, 28 and
560 l). The areas investigated in the 7.3 and 28 l tanks
were the upper left corner (noted UC), the lower left
corner (noted LC), directly over the impeller (noted
OI) and directly under the impeller (noted UI). In the
5 and 560 l tanks, the areas LC, OI and UI were inves-
tigated. A suspension of buffered water and kaolinite
clay was used as the model flocculation system. The
tanks were agitated by a Lightnin A 310 hydro foil im-
peller. The 5, 7.3 and 28 l tanks were operated at two
specific energy inputs, 1.72 W/m3 and 2.55 W/m3,
corresponding to average gradient velocity 41.2 and
50.4 s−1, respectively. The 560 l tank was operated
only at specific energy input 2.55 W/m3. No model of
flock growth kinetics was applied for data interpreta-
tion.

The data in the UC area for the 7.3 l and 28 l tanks
at specific power input 1.72 W/m3 were analyzed. A
comparison of the experimental data and a generalized
correlation are depicted in Fig. 1. The generalized cor-
relation parameters fitted for the measured data are
presented in Table 1.

Fig. 1: Generalized correlation Δ(1/df)
∗ = f(Δ[ntF ]

∗
log) –

Kilander et al. [4]

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Acta Polytechnica Vol. 50 No. 2/2010

Table 1: Generalized correlation parameters fitted – data Kilander et al. [4]

V εV n [ntF ]max tFmax dfmax A
∗
f Iyx δrmax/δrave

∗1

(L) (W/m3) (rev/min) (–) (min) (mm) (–) (–) (%)

7.3 1.72 199 6 429 32.3 0.231 4 39.108 0.998 5 1.4/3.2 ∗2

28 1.72 148 5 217 35.3 0.169 9 18.998 0.997 8 1.6/4 ∗2

∗1 Relative error of flock size df : maximum/average absolute value.
∗2 Data for tF = 2.22 min and tF = 3.89 min were excluded.

4 Experimental

The flock growth kinetics was investigated in a baf-
fled tank agitated by a Rushton turbine at mixing in-
tensity 40 W/m3 and kaolin concentration 0.44 g/l.
The tests were carried out on the kaolin slurry model
wastewater. The model wastewater was flocculated
with organic Sokoflok 56A flocculant (solution 0.1 %
wt.). The flock size and its shape were investigated by
image analysis. The proposed simple semiempirical
generalized correlation for flock growth kinetics was
used for data treatment. The fractal dimension of the
flocks was also determined.

4.1 Experimental apparatus

The flocculation experiments were conducted in a fully
baffled cylindrical vessel of diameter D = 150 mm with
a flat bottom and 4 baffles per 90◦, filled to height
H = D by a kaolin slurry model wastewater (tap wa-
ter + kaolin particles). The vessel was agitated by a
Rushton turbine (RT) of diameter d = 60 mm that
was placed at an off-bottom clearance of H2/d = 0.85.
Baffle width B/D was 0.1. The impeller motor and the
Cole Parmer Servodyne model 50000-25 speed control
unit were used in our experiments. The impeller speed
was set up and the impeller power input value was cal-
culated using the impeller power characteristics. The
agitated vessel dimensions are shown in Fig. 2.

Image analysis technique

The flock size was determined using a non-intrusive
optical method. The method is based on an analysis
of the images obtained by a digital camera in a plane
illuminated by a laser light. The method consists of
three steps:

1. illumination of a plane in the tank with a laser
light sheet (sometimes called a laser knife) in or-
der to visualize the flocks,

2. a record of the images of the flocks, using a cam-
era,

3. processing the images captured by image analy-
sis software.

The illuminated plane is usually vertical and a
camera is placed horizontally (e.g. Kilander et al. [4],

Kysela and Ditl [5, 6]). Kysela and Ditl [5, 6] used this
technique for flocculation kinetics observation. They
found that the application of this method is limited
by the optical properties of the system. The required
transparency limits the maximum particle concentra-
tion in the system. Therefore we do not observe the
flock size during flocculation but we observe it during
sedimentation, thus the limitation should be overcome.
Therefore the laser illuminated plane is horizontal and
perpendicular to the impeller axis. The scheme of the
experimental apparatus for image analysis is shown in
Fig. 2.

Fig. 2: Schema of the experimental apparatus for image
analysis

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Acta Polytechnica Vol. 50 No. 2/2010

Table 2: Technical parameters

Item Specification

Laser diode: NT 57113, 30 mW, wave length 635 nm (red light), Edmund Optics, Germany

Diode optics: optical projection head NT54-186, Projection Head Line, Edmund Optics, Germany

Camera: colour CMOS camera SILICON VIDEO R©SI-SV9T001C, EPIX Inc., USA
Camera optics: objective 12VM1040ASIR 10–40 mm, TAMRON Inc., Japan

Image processing card
(so-called grabbing card):

PIXCI SI PCI Image Capture Board, EPIX Inc., USA

Camera control software: XCAP R©, EPIX Inc., USA
Operation software: Linux CentOS version 5.2, Linux kernel 2.6

Software for image analysis: SigmaScan Pro 5.0

The agitated vessel was placed in an optical box (a
water-filled rectangular box). The optical box reduces
laser beam dispersion, and thus it improves the opti-
cal properties of the measuring system. The camera
with the objective and laser diode are placed on the
laboratory support stand. The technical parameters
are presented in Table 2.

4.2 Experimental procedure

The maximum flock sizes formed during flocculation
were measured for various flocculation times at mix-
ing intensity ε = 40 W/m3, constant flocculant dosage
DF = 2 ml/l and initial kaolin concentration 440 mg/l.
The flock sizes were measured during sedimentation.
The experimental parameters are summarized in Ta-
ble 3.

Table 3: Experimental conditions

Kaolin concentration

Parameter cC0 = 0.44 g/l

εV (W/m
3) 40

n (rev/min) 180

tF (min) 4; 6.66; 9.33; 13.33; 20

ntF (–) 720, 1 200, 1 680, 2 400, 3 600

DF (ml/l) 2

No. of date 5

The experimental procedure was as follows:

1. Calibration and experimental apparatus
setting

Before each flocculation experiment the calibra-
tion grid was placed in an illuminated plane and
the camera was focused manually onto this cal-
ibration grid. Then the image of the calibra-
tion grid was recorded. For camera resolution
800 × 800 pixels, the scale 1 pixel ∝ 45μm was
found for our images. This corresponds to a

scanned area of 35 mm × 35 mm (approx. 6 % of
the cross-section area of the tank). The scanned
area was located in the middle of one quarter of
the vessel, between the vessel wall and the im-
peller.

2. Model wastewater preparation
Kaolin slurry (a suspension of water and kaolin
particles (18 672 Kaolin powder finest Riedel-de
Haen)) was used as a model system. The solid
fraction of kaolin was 440 mg/l.

3. Flocculation
The model wastewater was flocculated by the or-
ganic Sokoflok 56A polymer flocculant (medium
anionity, 0.1 % wt. aqueous solution; flocculant
weight per flocculant solution volume mF /VF =
1 mg/ml; Sokoflok Ltd., Czech Republic). The
experimental conditions are specified in Table 3.
The flocculation was initiated by adding floccu-
lant into the agitated vessel, and the flocculation
time measurement was started.

4. Image acquisition
After impeller shutdown, the flocks began to set-
tle. During sedimentation, the images of flocks
passing through the illuminated plane and hav-
ing 10-bit depth were captured at frame rate
10 s−1, exposure = 5 ms, and gain 35 dB. Image
capturing started 20 s after impeller shutdown
and took 120 s. Finally 1 200 images were ob-
tained for the flocculation experiment and some
were stored in a hard disk in 24-bit jpg format.

5. Image analysis
The images were analyzed using SigmaScan soft-
ware and its pre-defined filters (filter No. 8 for
removing one-pixel points, filter No. 10 for re-
moval of objects touching on an image border,
and filter No. 11 for filling an empty space in
identified objects caused by capture error) and
our macros (Svačina [7] in detail).

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Acta Polytechnica Vol. 50 No. 2/2010

Table 4: Generalized correlation Δ(1/df eq)
∗ = f(Δ[ntF ]

∗
log): parameters fitted

εV n [ntF ]max tFmax df eqmax A
∗
f Iyx δrave /δrmax ∗1

(W/m3) (ot/min) (–) (min) (mm) (–) (–) (%)

40 180 1 440 8 1.406 2 16.373 0.895 2 2.5/4.4 ∗2

Note: ∗1 Relative error of equivalent flock size df eq : average/maximum absolute value.

Note: ∗2 Flock size for ntF = 3 600 was excluded from the evaluation.

Fig. 3: Experimental data – Maximum flock size vs. floc-
culation time – df eq = f(tF)

Fig. 4: Generalized correlationΔ(1/df eq)
∗ = f(Δ[ntF ]

∗
log)

4.3 Experimental data evaluation

From the images that were captured, the largest flock
was identified and its projected area was determined
for the given flocculation time. The equivalent diame-
ter calculated according to the flock area plotted in de-
pendence on the flocculation time for a given flocculant
dosage and mixing intensity is shown in Fig. 3. When
flocculation time increases, the flock size increases up
to the maximum value, due to primary aggregation,
and then decreases due to flock breaking.

Generalized correlation
Δ(1/df eq )

∗ = f (Δ[ntF ]
∗
log )

The dependence of calculated equivalent diameter df eq
on flocculation time was fitted according to the gen-
eralized correlation (12). The generalized correlation
parameters are presented in Table 4. A comparison of
the experimental data and the generalized correlation
is depicted in Fig. 4.

Fractal dimension

The flocks generated are often porous and have an ir-
regular shape, which complicates the design of the floc-

culation process. Fractal geometry is a method that
can be used for describing the properties of irregular
objects. Some flock properties such as shape, density,
sedimentation velocity can be described on the basis of
fractal geometry. A fractal dimension of the 3rd order
Df3 evaluated from the flock mass was usually deter-
mined. Since in our case we measured the projected
area of the flocks, the fractal dimension of the 2nd or-
der Df2 was used for flock shape characterization. The
relation among projected area A, characteristic length
scale Lchar and fractal dimension Df2 is given by:

A = C · LDf2char . (16)

The largest flocks determined in the images were used
for fractal dimension estimation. The maximum flock
size was used as a characteristic length scale. The
fractal dimension Df2 was determined for each floc-
culation time. For illustration, the dependence of the
projected area on the maximum flock size is shown in
Fig. 5 for dimensionless flocculation time ntF = 1 680.
The fractal dimension Df2 plotted in dependence on
flocculation time for a given flocculant dosage and mix-
ing intensity is shown in Fig. 6.

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Acta Polytechnica Vol. 50 No. 2/2010

Fig. 5: Fractal dimension determination – example for
ntF =1680

Fig. 6: Fractal dimension vs. dimensionless flocculation
time – Df2 = f(ntF)

Table 5: Fractal dimension – hypothesis testing

εV

(W/m3)

m

(–)

t-distribution

t(m−2),α=0.05

Relation

Df2 = B · (ntF )β

βcalc

t-characteristics |t|
Hypothesis

Df2 = B · (ntF )0

βpred = 0

40 5 3.182 5 0.045 0.97 (Yes)

Note: The absolute value of parameter t is presented in brackets.

Note: The t-distribution for (m − 2) degrees of freedom and significance level α = 0.05.

The effect of flocculation time on fractal dimension
Df2 was tested by hypothesis testing. The statisti-
cal method of hypothesis testing can estimate whether
the differences between the parameter values predicted
(e.g. by some proposed theory) and the parameter val-
ues evaluated from the measured data are negligible.
In this case, we assumed the dependence of fractal
dimension Df2 on dimensionless flocculation time, de-
scribed by the simple power law Df2 = B ·(ntF )β , and
the difference between predicted exponent βpred and
evaluated exponent βcalc was tested. The hypothesis
test characteristics are given as t = (βcalc − βpred)/sβ
where sβ is standard error of parameter βcalc. If the
calculated |t| value is less than the critical value of
the t-distribution for (m − 2) degrees of freedom and
significance level α, the difference between βcalc and
βcalc is statistically negligible (statisticians state: “the
hypothesis cannot be rejected”). The hypothesis test
result and parameter β evaluated from the data is pre-
sented in Table 5. The fractal dimension was found
to be independent of the flocculation time, and the
value Df2 = 1.442 ± 0.12 was determined as an aver-
age value.

5 Conclusions

The following results have been obtained:

• A simple semiempirical generalized correlation
for flock growth kinetics has been proposed.

• A generalized correlation has been successfully
tested and verified, using published data from
Kilander et al. (2007). The generalized correla-
tion parameters are presented in Table 1. A com-
parison of the experimental data and the gener-
alized correlation is depicted in Fig. 1.

• The flock growth kinetics was investigated in a
baffled tank agitated by a Rushton turbine at
mixing intensity 40 W/m3 and kaolin concentra-
tion 0.44 g/l. The flock size and flock shape were
investigated by image analysis.

• The tests were carried out on the kaolin slurry
model wastewater. The model wastewater was
flocculated with organic Sokoflok 56A flocculant
(solution 0.1 % wt.).

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Acta Polytechnica Vol. 50 No. 2/2010

• The largest flock was identified from the im-
ages, and its projected area was determined for
a given flocculation time. The calculated equiv-
alent diameter plotted in dependence on floccu-
lation time for the given flocculant dosage and
mixing intensity is shown in Fig. 3.

• The flock size increases with increasing floccula-
tion time up to a maximum value due to pri-
mary aggregation, and then decreases due to
flock breaking.

• The proposed simple semiempirical generalized
correlation for flock growth kinetics was used for
data treatment. The model parameters are pre-
sented in Table 4.

• The fractal dimension Df2 was determined for
each flocculation time on the basis of the ex-
perimental data. Using the statistical hypoth-
esis test, the fractal dimension was found in-
dependent of flocculation time, and the value
Df2 = 1.442 ± 0.125 was determined as the av-
erage value.

Acknowledgement

This research has been supported by The Grant
Agency of Czech Republic project No. 101/07/P456
“Intensification of Flocculation in Wastewater Treat-
ment”.

References

[1] Šulc, R.: Flocculation in a Turbulent Stirred Ves-
sel. PhD. thesis. Czech Technical University, Fac-
ulty of Mechanical Engineering, 2003 (in Czech).

[2] Šulc, R., Ditl, P.: Effect of Mixing on Floccu-
lation Kinetics. In: Proceedings of 14th Interna-

tionalCongress ofChemical andProcess Engineer-
ing CHISA 2000, (CD ROM), Prague, 2000.

[3] Šulc, R., Ditl, P.: The Effect of Flocculant Dosage
on the Flocculation Kinetics of Kaolin Slurry in
a Vessel Agitated by Rushton Turbine at Mix-
ing Intensity 168 W/m3 and Kaolin Concentra-
tion 0.58 g/l. Czasopismo Techniczne – Seria:
MECHANIKA, Vol. 105 (2008), pp. 341–349.

[4] Kilander, J., Blomström, S., Rasmuson, A.:
Scale-up Behaviour in Stirred Square Floccula-
tion Tanks. Chem. Eng. Sci., Vol. 62 (2007),
pp. 1 606–1 618.

[5] Kysela, B., Ditl, P.: The Measurement of Floc
Size and Flocculation Kinetics by the Laser Knife
and Digital Camera. In: Proceedings of 16th In-
ternational Congress of Chemical and Process En-
gineering CHISA 2004, (CD-ROM), Prague, 2004,
ISBN 80-86059-40-5.

[6] Kysela, B., Ditl, P.: Floc Size Measurement by
the Laser Knife and Digital Camera in an Agitated
Vessel. Inzynieria Chemiczna i Procesowa, Vol. 26
(2005), pp. 461–468.

[7] Svačina, O.: Application of Image Analysis
for Flocculation Process Monitoring. Diploma
Project. Czech Technical University, Faculty of
Mechanical Engineering, 2009 (in Czech).

Ing. Radek Šulc, Ph.D.
Phone: 420 224 352 558, Fax: +420 224 310 292
E-mail: Radek.Sulc@fs.cvut.cz
Department of Process Engineering,
Faculty of Mechanical Engineering,
Czech Technical University in Prague,
Technická 4, 166 07 Prague, Czech Republic

Notation

A flock area projected mm2

A∗f coefficient; model parameter (12) –

A∗Zr∗ residual turbidity shift; model parameter (4) –

df eq equivalent flock diameter according to flock area mm

df flock size mm

dfmax maximum flock size; model parameter (12) mm

D tank diameter m

DF flocculant dosage ml/l

Df2 fractal dimension of 2
nd order –

Iyx correlation index –

Lchar characteristic length scale mm

n impeller rotational speed rpm

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[ntF ]min model parameter (4) –

[ntF ]max model parameter (12) –

t hypothesis test characteristics –

tF flocculation time minute

t(m−2),α critical value of t-distribution for (m − 2) degrees of freedom and significance level α –
tsed sedimentation time minute

Z0 turbidity before flocculation FAU

Zr residual turbidity after flocculation FAU

Z∗e turbidity removal degree –

Z∗r residual turbidity degree –

Z∗rmin model parameter (4) –

Greek letters

δr relative error; %

δr = 100
∗ (experimental value – regression value)/regression value

Δ(1/df )
∗ variable –

Δ[ntF ]
∗
log variable –

ΔZ∗r variable –

εV specific impeller power input (per volume unit) W/m
3

Indices

∗ dimensionless
∗ · x link
ave average

max maximum

min minimum

29