HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 30. pp. 131- 135 (2002) 

THE SIMULATION OF CLUSTER FORMATION PROCESS IN FINE 
PARTICLES DISPERSION UNDER THE INFLUENCE OF EXTERNAL 

MAGNETIC FIELD 

A. PRISTOVNIK, L. C. LIPUS, A. KROPE and J. KROPE1 

(Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, SLOVENIA 
1 Faculty of Chemistry ar1d Chemical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, SLOVENIA) 

Received: January 18,2002 

The magnetic water treatment (MWT) devices for scale control can be used with great economical and ecological 
benefits. According to experimental results in well-controlled laboratory conditions it can be established that effects of 
MWT devices are highly dependent on composition of treated dispersion system and their working conditions. To 
investigate the effects of magnetic field on cluster formation process in fine particle dispersion under the influence of 
external magnetic field of the MWT device, a theoretical model on the DL VO theory and statistical Monte Carlo 
Metropolis method have been used. The "Open Source" computer programs for the simulation and graphical presentation 
of clustering under the influence of external magnetic field have been developed. Obtained results have been an.Uyzed by 
cluster analysis based on the partitioning and hierarchical methods (fuzzy, agglomerative and divisive analysis). 

Keywords: magnetic water treatment, scale prevention, Monte Carlo Metropolis method, cluster analysis 

Introduction 

The Magnetic Water Treatment (MWT) is frequently 
used non-chemical method for scale control. It is 
economically favorable and is one of the most 
controversial scale prevention methods. Scale 
prevention is achieved by passing the water through the 
magnetic field. Despite several decades of intensive 
research work done in this area, no scientifically 
confirmed theoretical explanation exists yet, which 
adequately describes how MWT devices work and what 
are the conditions under which is their operation most 
effective. 

Supplied natural waters are rich dispersion systems, 
which contain many colloids, ions, etc. Due to natural 
supersaturating of supplied water or supersaturating by 
changed operation conditions (such as pressure drop, 
temperature and pH) a hard scale precipitates on 
pipeline and equipment walls. The magnetic water 
treatment is particularly promising technique for 
manipulation and controlling the interactions between 
micrometer-sized colloidal particles dispersed in fluid 
suspensions [1}. 

Although many processes affect colloidal behaviour, 
the balance between the thermal, attractive and 
repulsive terms determines a criterion of the stability of 
colloid dispersions in natural waters. D.etailed 

consideration of this interplay is the basis of the DLVO 
[2] theory. The tendency of particles to aggregate via 
the short range van der Waals-London force is counter 
act by a charged layer on the particles. Thus the total 
interaction energy ( Et ) between two colloidal particles 
is the sum of the double layers energy of interaction 
(repulsive energy E,) and the energy of the interaction 

particles themselves due to van der Waals-London 
forces (attraction energy E. )[2]: 

(1) 

The repulsive energy ( E,) between two relatively 
large spherical particles depends on radii.IS of interacting 
spheres (a ), length between centers of particles ( R ) 
and dielectric constant ( e ) of dispersion medium: 

(2) 

where ( s ) represents the ratio between distance and 
radius { s ""R/a) between centers of these two spheres. 

The extension of the double layer { 8 ) is in the order 
of 1/K, where K is the Debye-Huckel parameter and 
bas the dimension of reciprocal length: 



132 

-- -. 
30 f----"-+-....._+-..,-+--+-~+.-=--i-=·~--~ 

-- ::..._ .. -- .. _ :-

!0 20 30 40 50 60 70 

Fig.l Random distributed spherical particles in a two 
dimensional square cell (A=70 units, d=l unit, N=300) 

1 
{j =-, 

K 

2 8nne~Z 
K =---

ckT 
(3) 

For oxides in water, the surface potential ( q;o) of 

particles is determined by the pH of suspension [2,3]. 
The energy of attraction ( Ea) between two spherical 

particles with identical radius is defined by Eq.(4): 

E =-kh ·(-2-+3._+lnsz-4l 
a 6 s2 - 4 S2 s2 J (4) 

where k11 is the Hamaker constant. 
Magnetostatic particle interactions modify the 

behavior of the fluid and can have a detrimental effect 
on the colloidal stability. When a dispersion of colloidal 
particles is placed in an external magnetic field 
additional magnetic force arises, which decreases the 
stability of the colloid particles when attractive. The 
energy of the magnetic attraction ( Em ) between two 

spherical particles separated by a distance ( Rr) depends 
' g 

>n magnetic field density ( B ) and angle of the external 
magnetic field, radius of the particles (a ) and magnetic 
properties of the particles: 

32m6xaBa 
E =- (5) 

m 9f,t.,R~ 

Thus the total energy of interaction ( E, ). of colloid 
particles in the external magnetic field is: 

(6) 

Statistical numerical methods known as Monte Carlo 
methods ate methods that utilize sequences of random 
numbers to perform the simulation. The presented 
model is based on the Monte Carlo Metropolis [4,13] 
method and bas been used to investigate the properties 
of colloid particle dispersion in water under the 
influence of magnetic field [:2,3,5,6,7]. 

The model is based on a two-dimensional iquare ceU 
with side length (A) containing (N) random distributed 
spherical particles (Fig. I). AU the particles are identical. 

" a , ~ ~'*"" 
y a 

Jl~ 

\If\ ~ 
... (Xj.Yj) 

!>' 

~~ 

l v 
LA_ ~ 

X 

Fig.2 A pair of spherical particles (i, j) in a two dimensional 
square cell under the magnetic field applied at angle \jl [8] 

The position of any particle is known and can be 
specified with coordinates (x, y ,8), where (8) is the 
angle between the magnetic moment of the particle and 
the applied magnetic field [6] (Fig.2). 

The base energy of the system ( E8 ) is determined as 

the sum of the total energy of interaction of all colloid 
particles in that square cell: 

N 

Es = LEt(i) 
i=l 

(7) 

The technique consists of calculating the energy 
change ( AE) when the coordinates of one particle of 
the representative ensemble are changed at random by a 
small amount. 

If the new total energy of the system is less than 
previous, the particle stays in new position, otherwise 
the factor ( P ) is calculated and compared with random 
number(x; xE [0,1]): 

-liE 

P=ekT (8) 

If factor P is greater than random number the 
particle retains its new position, otherwise it is returned 
on its original position. This procedure is applied for of 
N particles in a square celL 

Experimental 

A computer program for the simulation of cluster 
formation 

Well-known theories from the field of colloid, statistic 
and mathematics science have been taken into the 
consideration and been applied in the magnetic water 
treatment research. An "Open Source" computer 
program (MCM) [14] for-two-dimensional simulation 
of the cluster formation process in fine particles 
dispersion under the influence of external magnetic field 
which according to some authorst9.lll1, take place in 



y 
25.00 

18.75 

12.5G 

0.00 

0.00 6.25 

Fig.Ja Position of the particles after 10 moves [8] 

MWT device, was developed. For the graphical 
presentation (MCM View) [8] and cluster analysis 
(Fanny&Twins) [11J the set of computer programs were 
used. 

The data are entered with two input files. In the first 
input file the size of the cell, number and physical 
properties of the particles are determined and structure 
for the intermediate output files is defined. In the second 
input file the origin position of the particles is 
determined. 

The results of simulation are written in various 
formats and can presented with the programs such as 
Microsoft Excel [14] or Microcal Origin [14] as a front 
end. For the graphical presentation the "MCM View" 
program has been developed on the basis of Compaq 
Array Visualiser [14]. 

The results have been further analysed with the 
partitional and hierarchical methods for cluster analysis 
of authors Kaufman and Rousseauw [11] (fuzzy, 
agglomerative and divisive analysis). For that purposes 
programs ''FANNY" any "TWINS" have been used. 

Results and Discussion 

As the reference data for the primary numerical 
calculations, the physical properties of the hematite 
particles ( Fez03 ) in aqueous dispersion have been used. 

On this basis, the numerical calculations for some of the 
scale forming minerals in the water such as: 
diamagnetic calcium carbonate { CaC03 ), calcium 

sulfate ( CaS04 ), silicon dioxide ( Si02 ), paramagnetic 

hematite ( Fe20 3 ), goethite ( FeOOH) and 

ferrimagnetic magnetite ( Fe,04 ) have been carried out. 

For all of the enumerated minerals, the simulation of 
cluster formation has been carried following out 
physical and chemical conditions: 

• Different number of particles and their radius in the 
square cell; 

133 

y 
25.DJ 

1&.75 

12.50 

~-25 

0.00 

13.75 25.00 

FigS& Position of the particles after 600 moves [8] 

- Different angle(1Jf) of the applied magnetic field (0, 
30, 60 and 90 degree); 
Different retardation potential (<p), which determine 
the strength of the repulsion force (10, 20 and 30 
mV); 
Different density (B) of the applied magnetic field 
(from 0 to IT); 

- Different pH values of the aqueous dispersions and 
others181 • 

The total interaction energy for the paramagnetic 
hematite particles ( N = 40 + 300 ) with the diameter 
( d = 1 + 10 pm ), volume magnetic susceptibility 

(X= 0.02) and Hamaker constant ( kh = 5-10-20 J ) in 

square unit cell with side length (A == 25 + 70Jlm) has 

been computed. The absolute temperature and pH of the 
dispersion, have been set to 300 K and 7.0, respectively. 

According to the previous research [ 6] the 
recommended rate of convergence is up to 600 moves 
per particle. The origin position of the particles has been 
set up as 5x8 matrix. Fig.3 presents position of the 
particles after 10 and 600 moves under magnetic field of 

density 0.5 T, applied at 30° angle. 
Fig.3b it is quite easy to perceive the formation of 

one major and two minor clusters, which are well 
arranged in the direction of applied magnetic field. 
From the same figure, the intensity or even the size of 
the cluster cannot be obtained. thus the results have to 
be analyzed with the adequate mathematical method 
suitable for cluster analysis. The Fuzzy clustering 
method, as a generalization of partitioning, was used in 
computer program Fanny. The intensity of the cluster 
formation process had been measured with the !alue of 

normalized version of the partition coefficient ( Sk~s ). 
Fig.4 shows the intensity of the cluster formation 

process of the hematite particles at the different 
retardation potentials and under the magnetic field 
applied at different angles. 

According to experimental results (12], the effects of 
the magnetic water treatment devices are high 
dependent on composition of treated dispersion system. 



134 

0,80 

0,70 

~ 
./"" ---

0,40 ~ 

0,50 0,75 

'¥ =30",<p = 20mV 

1,00 
B(l) 

0,80 ,.......----------·----, 

0,70 .J------------------1 

s 0,6U ~=::::::==::;=--======t 
~ 0,50 -1-------------l 

0,40 +----------------; 
(),30 +------------------1 
0,20 +----~----.-------1 

o.zs 0,50 0,75 B(l} 1,00 

'¥ = 60" ,<p= 20mV 

0,80-.--------------. 
/ 0,70 -1----------~----:r'--l 

s 0,60 +"'" ... ..............,=--------------::?""c__---l 
~ 0,50 +-----=""---""""'--------l 

0,40-!---------------J 

0,30-t---------------J 

0,20 +---------------~ 
0,2S 0,50 0,75 B(l) 1,00 

'¥ = 90" ,<p = 20mV 

0,80 

0,70 

s 0,60 
~ 0,50 

0,40 

0,30 

0,20 

~ 

0,25 

0,30 

0,70 

s o.ro 

~ 0,50 

0~40 

0~30 

0.20 

0,25 

0,50 0,75 

'¥ =30" ,<p=30mV 

0,50 0,15 

'¥ = 60",q>=30mV 

~ 
l 

1,00 
B(I') 

~ -
I 

1,00 
B(f) 

0,80,------------, 

0,70f-------------:J 
s o.ro-l-,_-=....--"'=~:::::::::~~----1 

' ~ I ~ O,S<>.J-----------""'~~------jj 
0,40.J--------------"'ooo;;:-j 

·"! 0,30f-------------:! 
0,20-t--------~-----i 

0,25 0,15 1.00 0,5<> 
B(f) 

'¥ = 90" ,<p = 30mV 

-
Fig A The intensity of the cluster formation process (normalized version of the Dunn's partition coefficient, s k~s ) under the 

magnetic field of density 0.5 T applied at different angles and different retardation potentials [8] 

!
' 0,411 •• • • y. •• 

o,:lli " 
I o.30c " I.-

I 
I 

0~-~---~~--~4-~~---~-~~~~+-~--l 

1 2 3 4 5 
, .... ,.._..,.(~! 

G 7 a ~ w u a a M 
pll(~ 

FigS The intensity of the cluster formation process with pH 
change and graph approximation with the polynomial of 

second order [8] 

and working conditions. The pH value of the system is 
on~ of the most influential conditions for the duster 
formation process. For the successful clustering of 
hematite particles the optimal pH range of treated 
dispersion (FigS) is from5.6 and up to !0.3. 

Condusions 

Despite the large volume of research work in past 
decades. the theoretical understanding of MWT 

mechanism is still incomplete. This is the main problem 
in design of efficient MWT devices. The MWT 
mechanism is very complex and directly depends on 
chemical composition of water as solution/dispersion 
system and working conditions. · 

The theoretical model. for simulation of duster 
formation with hematite particles as the reference data 
has been supplement and widens to the region of 
magnetic water treatment. The model is based on the 
DL VO theory, Monte Carlo Metropolis method and 
cluster analysis theory. With the computer programs 
based on presented model, the numerical calculations 
for most of the scale forming minerals, have been done. 
The obtained results have show that the model well 
predict the essential operational conditions for the 
effective use of MWT devices, if the chemical 
comPosition of supplied water is known. 

SYMBOLS 

a radius of interacting spheres 
B magnetic field density 
e0 electron chargel.6·IO·''l As 
Ea attraction energy 

repulsive energy 

(m) 
(Vslm2) 

(As) 
(J) 

(J) 



E, total interaction energy 

kh Hamaker constant 

N, n number of particles 
R length between centers of particles 
s ratio between distance and radius 
T absolute temperature 
Z1 ion valence 
8 extension of the double layer 
£ dielectric constant 
8 angle between the magnetic moment of the 

(J) 

(/) 

(I) 
(m) 
(I) 

(K) 
(I) 

(m) 
(f) 

particle and the applied magnetic field (rad) 
Cflo surface potential (J/As=V) 

\jf angle of the external magnetic field (rad) 

REFERENCES 

1. GRIER D. G. and BEHRENS S. H.: Interactions in the 
colloidal suspensions, The James Franck Institute, 
The University of Chicago, 2001 

2. Svoboda J.: International Journal of Mineral 
Processing, 1981,8,377-390 

3. SVOBODA J.: IEEE Transactions on Magnetics, 
1982, 18(2), 796-800 

4. METROPOLIS N., ROSENBLUTH A. W., ROSENBLUTH 
M. N., TELLER A. H. and TELLER E.: J.Chem. 
Phys., 1953,21, 1087-1092 

5. CHANTRELL R. W., BRADBURY A., POPPLEWELL J. 
and CHARLES S. W.: J. Phys. D: Appl. Phys., 1980, 
13(3), 119-122 

6. CHANTRELL R. W., BRADBURY A., POPPLEWELL J. 
and CHARLES S. W.: Joqrnal Appl. Phys., 1982, 
53(3), 2742-2744 

135 

7. MILES J. J., CHANTRELL R. W. and PARKER M. R.: 
Journal Appl. Phys., 1985, 57(1), 4271-4273 

8. PRJSTOVNIK A. J.: Configuration and formation of 
clusters from dispersion particles during the 
magnetic water treatment, Draft of Doctoral 
Dissertation, University of Maribor, Faculty of 
Chemistry and Chemical Engineering, unpublished 
work, 2001 

9. IOVCHEV M.: The Effect of Magnetic Field on an 
Iron-Oxygen Water System, Khim. Tekhnol. Inst. 
19:73-84, 1966 

10. Herzog R. E., SHI 0., PATIL J. and KATZ J. L.: 
American Chemical Society, Langmuir, 1989, 5(3), 
861 

11. KAUFMAN L. and ROUSSEEUW P. J.: Finding 
Groups in Data-An Introduction to Cluster 
Analysis, A Wiley-Interscience Publication, New 
York, Toronto, 1990 

12. KULSKII L. A., KOCHMARSKY V. Z. and KRIVTSOV 
V. V.: Khimiya i Tekhnologiya Vody, 1983, 5(4), 
296-301 

13. CSEP: Introduction to Monte Carlo Methods, 
electronic book, http://csepl.phy.ornl.gov/CSEP/ 
MC/MC.html, Copyright (C) 1991, 1992, 1993, 
1994, 1995 by the Computational Science 
Education Project 

14. COMPUTER PROGRAMS: 
- MCM, MCMr_z_a in MCM view, Copyright© 

1997-2001 Andrej Pristovnik 
- COMPAQ ARRAY VISUALIZER 1.1 

Copyright© 1998-1999 Digital Equipment 
Corporation 

- Microsoft®Excel, version 97, Copyright© 
1985-1996 Microsoft Corporation 

- Microcal TM Origin, version 6.0, Copyright© 
1991-1990 Microcal Software, Inc. 


	Page 130 
	Page 131 
	Page 132 
	Page 133 
	Page 134