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Mathematical Modelling and Computer Simulation of
Electrical Precipitation and Separation
(Full text in English)
Jaroslav DŽMURA1, Milan BERNÁT2, Ladislav RUDOLF3
1
Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Electric
Power Engineering;
2
University of Prešov in Prešov, Faculty of Humanities and Natural Sciences, Department of Physics,
Mathematics and Techniques;
3
University of Ostrava, Pedagogical Faculty, Department of Technical Education
Abstract
The paper deals with the application of the method for mathematical modelling and simulation at solving some
issues in the area of electrostatic technology. It focuses on the processes in electrostatic separation and
precipitation. The computer simulation is highly required in equipment design and their diagnostics in critical
operating states using theoretical calculations and experimental data evaluation. The presented computer
models may be applied both by project and design engineers using the most advanced computer-aided design
of electrostatic technologies.
Keywords: mathematical modelling, computer simulation, electrostatic force, high voltage
1. Introduction
Up to present, in the study of electric
separation and sorting process, as well as in
the development of relevant equipment, the
attention has been primarily directed
towards constructional and operational
aspects with empirical knowledge being the
main and practically the sole source of
information.
Notwithstanding that ballistics of a
particle (knowledge of the equation of
motion and its solutions) is a keystone of the
theoretical analysis of electric separation
and sorting, any comprehensive solution of
the examined subject has not hitherto been
found.
A few attempts, trying to elucidate the
issue by means of approximate methods,
achieved only limited success. They could
unravel the issue at the kinematic level,
albeit only under conditions of not taking full
account of all the factors affecting the
ballistics of motion. The use of numerical
methods at structuring the ballistics of
particles was unfeasible principally due to
the absence of appropriate possibilities for
calculation. The primary goal of the cur-rent
research is to contribute to some extent to
building a theoretical basis inevitable for a
further thorough study of electrostatic
technology processes. Our intention is not to
disclose the essence of the investigated
phenomena, but to identify the potential
application areas for our research findings.
The secondary - but not marginal - goal of
the research is to explore effects of the
direct and alternating (hereinafter DC and
AC) electric fields on the transport of
electrically charged particles.
Marton [3] [7] [2] and Džmura [8] [9] [10]
focused on a particular shape, given by the
geometry of a particle. In their calculations,
they did not take full account of all factors
affecting the motion of particles in the
electric field created by high voltage. The
most comprehensive experiments have been
conducted by [13] [14] [15] [16] [17] [18] [19]
[20] [21].
The research method of mathematical
modelling and computer simulation has
proven to become a powerful approach for
understanding the complexity of a physical,
biological, chemical system [6] [5] [4] [11]
[22] [23]. It enables to examine the system
based on virtual prototyping.
2. Computer model of electric separatorsorter of cylindrical type
2.1. Description of the physical model
The charged (e.g. by corona charging)
macroscopic particles in the separator inflow
jet area are carried by the flowing gas into
the separation space (Figure 1).
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ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
(shown in Figure 1) charged by DC voltage
takes the following form:
m
(
(
)
)
ε −1
d 2r
qU
dr
U2
− 6πηR − 4πε R 3 r
=
0
dt
ε +2
r 
dt 2  r2 
r
ln
r
ln 2  2  r 3
 r 
r 
 1
 1
(1)
m
Figure 1. Kinematic scheme
The separation space is created by a set
of coaxial cylindrical electrodes (a thin bar
or a piece of wire may serve as the internal
electrode) among which the direct
(hereinafter DC) or the alternating (herein
after AC) electric field is generated in dependency on the type of electrode charging.
The force interaction of the electric field and
the charged particles causes the separation
and sorting of these particles from the air
mass to the external electrode wall. The use
of inhomogeneous electric field separators or
sorters (as it is in our case) un-like
homogeneous electric field equipment
requires to take account of gradient force
and its co-effects (together with some other
factors) on the dynamics of a macroscopic
particle. (The choice of the coordinate
system and the kinematic scheme is shown in
Figure 1).
The system of forces affecting the motion
of particles (according to Figure 1) is
presented in Figure 2.
d2y
dy
= mg − 6πηR
2
dt
dt
(2)
If the separator is charged by AC voltage,
the equations take the following forms:
m
d 2 r qU m sin (ωt + φ 0 )
dr
=
− 6πηR −
2
dt
dt
 r2 
 ln  r
 r1 
− 4πε 0 R 3
m
(ε r − 1) U m
(ε r + 2)
2
sin (ωt + φ 0 )
(3)
2
r
ln 2  2
 r1
 3
 r

d2y
dy
= mg − 6πηR
dt
dt2
(4)
Cauchy initial conditions for the solution
of the system of differential equations
(constructed by us) take for the chosen
coordinate system the following form:
r(0) = r0; vr(0) = vr0; y(0) = y0; vy(0) = vy0
(5)
He following relation (6) was taken from
[1] where it had the form FgradGVr
(ε r − 1)
FgradGVr = −4πε 0 R
(ε r + 2)
3
U2
r
ln 2 ( 2 )r 3
r1
(6)
Note: The derivation of the relation
results from the fact that the particle in the
electric field is also charged by q0. The
particle is affected by the vector sum of
forces - the gradient force (affecting the dipole moment according to (7) and the force
affecting the charge generated in the
charging process (8).
r
r
r
∂E ( x )
F = kE (x )
∂x
Figure 2. System of forces
2.2 Synthesis of the Mathematical model
Newton’s kinematic equation (the
mathematical model) for the motion of a
macroscopic particle in the separator space
r
r
FE = q 0 .E
(7)
(8)
To simplify it, we consider that the vector
E has only the direction x.
Then the resulting force is as follows:
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
r
r
r
r
r
r
∂E ( x )
∂E ( x )
F = kE ( x )
+ q 0 E ( x ) = E ( x )[q 0 + k
]
∂x
∂x
(9)
The research method of mathematical
modelling and computer simulation has
proven to become a powerful approach for
understanding the complexity of a physical
(biological, chemical) system. It enables to
examine the systems based on virtual
prototyping.
An original physical system is replaced by
its computer-based virtual prototype to be
used for various computer experiments
whose results are then applied to the original
physical system.
53
the computational modelling of all the
relations between the state values of the
modelled system. In the present research, it
assists to solve the following tasks:
kinematics of the particle motion in the
modelled physical system,
dynamics of the particle motion in the
modelled physical system, and,
values characterizing the electric field.
These values are presented in the
following graphs (see Figure 3 to Figure 10).
2.3 Synthesis of the computer model and
simulation
Based on the model of the separator
(synthetized above), we simulated the
separation and sorting of the particles with
varying size and weight, but identical
material parameters. In order to reflect the
real-world technological conditions we
selected the parameters of the particle 1 and
the particle 2 in accordance with the
measurements carried out in [1] [12].
The constructed here systems of
differential equations are systems of nonlinear differential equations (due to the nonlinear differential equations (1) and (3)).
Since the analytical solution of such
equations has not been possible, the
approximation of the methods are the only
way of solving the given set of equations. We
used the Runge-Kutta 3 method in the Matlab
software to carry out the execution of the
tasks.
electric field DC: U=15 000 V; r1= 0.001 m;
r2= 1 m; ε0=8,854.10-12 F/m;
electric field AC: Um=15000 V; ω=314 rad/s;
φ0=0; r1=0.001 m; r2=1 m;
dynamic viscosity of the environment:
η=18,1.10-6 kg/ms;
initial conditions: r(0)=0,001 m;
vr(0)=y(0)=vy (0)=0 m/s.
Figure 3. Graphical representation of the motion
trajectory (y = f(r) of particle 1 and
particle 2 moving in the space of the
separator charged by DC voltage
Figure 4. Graphical representation of comparison of
forces affecting particle 1 in the relation
F = f(r), (DC voltage), Fgrad, FSr, FE, FG
We solved the system of the differential
equations of motion (the mathematical
model)
numerically
in
the
Matlab
computational
environment)
for
the
following values:
particle 1: q=0,5.10-8C; m=1.10-3 kg; εr=3,5;
R=2,5.10-3 m;
particle 2: q=0,4232.10-8C; m=0,807.10-3 kg;
εr=3,5; R=2,3.10-3 m
Moreover, Matlab offers an opportunity of
Figure 5. Graphical representation of motion
trajectory (y = f(r)) of particle 1 and
particle 2 moving in the space of the
separator charged by AC voltage
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ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
Figure 6. Graphical representation of comparison of
forces affecting particle 1 in the relation
F = f(r), (AC voltage), Fgrad, FSr, FE, FG
Figure 10. Graphical representation of the intensity
course of the electric field (dependency
Er = f(r) – r) in the inter-electrode space
(b curve: the course if non-existence of
space charge, g curve: “deformation
contribution”)
2.4 Analysis and discussion on the
simulation results
Figure 7. Dependence Er = f(r) on the trajectory of
the particle motion in the space of the
separator charged by DC voltage
Figure 8. Graphical representation of velocity of
particle 1 motion in the space of the
separator (shown in Figure 1) charged by
AC voltage; the relation vr = f(t)) for b
curve (φ0 = 0°), for r curve (φ0 = 180°)
Figure 9. Graphical representation of the motion
trajectory of a particle moving in the
inter-electrode space (dependency
y=f(r)). (r curve: trajectory if motion in
the field of space charge, b curve: nonexistence of space charge)
A proper control over the separationsorting technological process is guaranteed
by achieving high degree of control over
kinematics and dynamics of the particles
motion in the separator space. This is not
workable without recognising and identifying
the effects of the individual quality and
quantity factors on this motion. This is
exactly what the current research is aimed
at.
The key finding that resulted from the
present computer simulation may be drawn
as follows: when analysing dynamics of the
charged macroscopic particle motion in the
separator-sorter it is necessary that also
gradient force be taken into account because
gradient force is comparable with the other
force interactions (see Figure 4 and
Figure 6).
Another important accomplishment is the
revealing of possible occurrence of critical
operating states of the separator-sorter
provided, which is charged by AC voltage
(see Figure 8.).
The b curve in Figure 8, illustrates the
relation vr(t) for t=<0; 0.01 s> provided that
φ0=0°.
The r curve in Figure 8 illustrates the same
relation provided that φ0=180°.
Our investigation showed that under the
condition that φ0=180° the operation of the
separator-sorter might become critical. If
this is the case, the velocity of the motion
vr(t) takes the opposite direction in
comparison to that stated as the positive
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
one. If the velocity of the motion reached a
negative value during the time t close to
zero, the particles would adhere to the wall
of the separator-sorter. It would then
depend upon nothing but the value vr(0) and
the adhesive conditions. It is essential that
this observation be taken into consideration
in the process of designing and constructing
an AC voltage separator-sorter.
The most important research contributions are: improvement of the existing
research methods in the technology of
separation and sorting (the emphasis has
been so far put on constructional and
operational
aspects
with
empirical
knowledge being the main and practically
the sole source of information), parallel
solution of the aspects of kinematics,
dynamics, and the electric field of a moving
macroscopic
particles
(in
the
real
environment of electrostatic technologies).
Unlike the past, the method proposed
offers a more complex solution to the issue.
It solves the issue not only from the
viewpoint of kinematics but also from the
viewpoint of dynamics and of the theory of
the electromagnetic field taking into account
all the major factors affecting the ballistics
of a particle. If we realise that the then used
empirical methods were capable of solving
only the kinematics of a particle but not its
dynamics, the method proposed by us seems
to be at present the only complex solution.
The overall result of the present computer
simulation is the knowledge that the
operation of the examined type of a
separator is also possible in the condition,
when it is charged by AC voltage (it is
necessary to consider
the possible
occurrence of critical operating states).
Analogically, it is possible to synthetize a
computer model of the separator-sorter with
the electric field of spatial charge
distribution (the model is even more
identical to the real-world model) and to use
it for the simulation of particle separation
and
sorting
(including
programming
equipment as well as programming
technology). This problem is dealt with in
details in [1], from where we select and
discuss the following issue: the motion of a
particle of macroscopic size in the electric
field of a space charge (the density of the
space charge is p = k/r) as shown in the
picture 1.
55
Solution of the electric field:
∆V = −
k
ρ(r)
=−
ε0
ε0
(10)
1 d  dV 
ρ(r )
k
. r  = −
=−
r dr  dr 
ε0
ε0 r
(11)
For boundary conditions:
r = r1 ; V ( r1 ) = U ; r = r 2 , V ( r 2 ) = 0
V=−
(12)
k
r + C1 ln r + C2
ε0
(13)
∂V k C1
= −
∂x ε 0 r
(14)
E r (r ) = −
Equations
model):
of
motion
(mathematical
m
d 2r
dr
= q.E r (r ) − 6πηR
2
dt
dt
(15)
m
d2y
dy
= mg − 6 πηR
2
dt
dt
(16)
The solution is for the values:
p.p: r(0)=0.005 m; vr (0)=15 m/s; y(0)=0 m;
vy(0) particle: q=1.10-8C; m=5,5.10-8 kg;
R=0,5.10-3 m electric field in the direction r:
U=15000 V; r1=0,005 m; r2=0,5 m,
k=6.10-5; C1=1.105; s0=8,854.10-12 F/m
dynamic viscosity of the environment:
n=18,1.10-6 kg/ms
3. Precipitator with linear coronary
electrodes supplied by alternating high
voltage
Macroscopic
particles
diffused
in
surroundings of coronary electrode, which is
connected to alternating high voltage, are
not charged by hypothesis. For this reason, it
is necessary to create electro-physical
condition for creation of mono-polar electric
charges in area of feeder embouchure. One
possibility is to utilise the physical
phenomena
in
metal-dielectric-gas
boundary.
During our experiments, a cascade coaxial
precipitator with four sections was
constructed (Figure 11).
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ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
Insulating barrier must not to change their
mechanical and dielectric attributes under
temperature influence.
For finding the efficiency of precipitator
model, three types of dust with different
values of resistance were used. The model of
precipitator with length 30 cm has efficiency
in the range of 60 % to 97 % for each of the
dust type.
Table 1. Average values of efficiency for each of
sections of cascade tube-type precipitator
supplied by alternating voltage
Efficiency [%]
1.
2.
3.
4.
section section section section
Powder
with ρe= 3 Ω.m
Powder
with ρe= 4 kΩ.m
Powder
with ρe= 4,8 MΩ.m
Figure 11. Sections configuration of cascade tubular
precipitator supplied by alternating high
voltage
The PVC tube serving as collecting
electrode is the basic part of precipitator
section. Thin aluminium electrode is
attached on the outside of the tube and it
creates the ground electrode.
Inside of the tube, there is fixed a thin
cooper wire serving as coronary electrode.
Any sections of cascade contain different
number of coronary electrodes and they are
placed in different distances from collecting
electrode. The function of coronary
electrode is the creation of a strong
inhomogeneous electric field. This field can
be created with electrodes with small radius
of bend. If coronary electrode is positive, the
new electrons are created as consequence of
bomb of point by electrons. If coronary
electrode is negative, it is analogical, but the
difference is that the avalanches emitted
from negative point are moved to more
homogeneous field, consequently, of it the
mobility and ability to ionise decreases.
In electric precipitator supplied by
alternating voltage by the use of the metal
coronary electrode there is used the
insulating barrier as a collecting electrode.
For maximal efficiency, it is necessary to
choose insulating materials with great value
of resistance and permittivity. The choice of
materials depends on another nonelectric
quantity, as temperature of flying gas,
enough mechanical solidity and other too.
83.16
70.72
73.94
62.02
96.82
81.72
83.7
72.12
93.45
93.56
97.82
95.14
From measured values, it is possible to
estimate that the precipitation solid
admixture form flying air at alternating
voltage is comparable with precipitation at
direct voltage and at specific conditions
reaches better values of efficiency.
4. Conclusions
Computer simulation is becoming a useful
tool in the implementation of the research
findings into a manufacturing process. It can
assist in the process of de-signing and
constructing a new type of an AC separatorsorter. It can also help to get it into practice
and thus move research findings into the
real-world conditions.
The presented computer models may be
applied both by project, and by design
engineers using more advanced computeraided design of electrostatic technologies,
i.e. in the areas where computer simulation
is highly required such as prognozing and
diagnosing critical operating states.
In conclusion, we find important to draw
attention to the following: in the case of
existence of a great number of the particles
(the model more identical to the real-world
model) and their interactions (due to which
they influence each other) we recommend to
use a statistical approach and stochastic
computer modelling of the above-mentioned
processes (the creation of the stochastic
computer model is described in the
dissertation thesis [1] [12]).
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
5. Acknowledgment
We support research
activities in Slovakia. Project is co-financed from
EU funds. This paper was developed within the
Project “Centrum excelentnosti integrovaného
výskumu a vyuzitia progresívnych materiálov a
technológií v oblasti automobilovej elektroniky”
[Centre of Excellence of the Integrated Research
& Exploitation of the Advanced Materials and
Technologies in the Automotive Electronics],
ITMS 26220120055.
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Issue 29 pp.9098-9105, ISSN 1520-6106.
7. Biography
Jaroslav DŽMURA was born in
Bardejov (Slovakia), on April 28,
1972.
He graduated the Technical
University of Košice, Faculty of
Electrical Engineering and
Informatics in Košice (Slovakia), in 1995.
He received the PhD degree in electrical
engineering from the Technical University of
Košice (Slovakia), in 2002.
He is Assistant Professor at the Technical
University of Košice, in Košice (Slovakia).
His research interests concern: high voltage
engineering, electrostatics and smart electric
installation.
Correspondence address: Technical University
of Košice, Faculty of Electrical Engineering and
Informatics, Department of Electric Power
Engineering, Mäsiarska 74, 04120 Košice,
Slovakia, e-mail: [email protected]
MILAN BERNÁT was born in Kosice
(Slovakia), on June 12, 1959.
He graduated the Technical
University of Košice, Faculty of
Electrical
Engineering
and
Informatics in Košice (Slovakia),
in 1983. He received the PhD degree in
electrical engineering from the Technical
University of Košice (Slovakia), in 2001.
He is Assistant Professor at the Department of
Physics, Mathematics and Techniques at the
University of Prešov in Prešov (Slovakia).
His research interests concern: high voltage
engineering, electrostatics and theory of
technical vocational subjects teaching.
Correspondence address: University of Prešov in
Prešov, Faculty of Humanities and Natural
Sciences,Department of Physics, Mathematics
and Techniques, 17. novembra 2, 08001 Prešov,
Slovakia, e-mail: [email protected]
Ladislav RUDOLF was born in
Olomouc (Czech republic), on
May 2, 1960.
He graduated the VŠB Technical
University
of
Ostrava, Faculty of
Electrical Engineering and Computer Science
in Ostrava (Czech republic), in 1994.
He received the Ph.D degree in electrical
engineering from the VŠB - Technical
University of Ostrava (Czech republic), in
2003.
He is Assistant Professor at the Department
of Technical Education at the University of
Ostrava in Ostrava (Czech republic).
His research interests concern: high voltage
engineering, technical losses and theory of
technical vocational subjects teaching.
Correspondence address: University of
Ostrava, Pedagogical Faculty, Department of
Technical Education, Fráni Šrámka 3, 70900
Ostrava – Mariánské Hory, Czech Republic,
[email protected]