Particle Charging in Combined Corona-Electrostatic Fields
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Particle Charging in Combined
Corona-Electrostatic Fields
Laurentiu-Marius Dumitran1, Octavian Blejan1, Petru Notingher1, Adrian Samuila2, Lucian Dascalescu2
1
Laboratory of Electrotechnical Materials,
Politehnica University of Bucharest, Splaiul Independentei nr. 313, 060042 sector 6, Bucharest, Romania
2
Electronics and Electrostatics Research Unit, LAII-ESIP, UPRES-EA 1219
University Institute of Technology, 4 avenue de Varsovie, 16021 Angoulême Cedex, France
Abstract – The association of several ionizing and non- The physical model of the corona field is rather well
ionizing electrodes generate combined corona-electrostatic established [8] and can be easily solved for simple two-
fields, characterized by space charge zones of well-defined electrode configurations like wire-plane or point-plane, using
extensions. In a previous paper, the authors presented an various techniques, such as finite-differences [9], finite
effective numerical method for the computation of such fields. element [10-13], the combined method of finite element and
The aim of the present work is to show how these results can be
the method of characteristics [14], and the charge simulation
employed for estimating the charge acquired by insulating and
method [15]. The formulation of the mathematical model is
conducting particles when passing through the space charge
zones generated by various corona-electrostatic electrode less easy to handle for the combined corona-electrostatic
geometries. The study is done under several assumptions that fields, generated by ionising electrodes associated with non-
authorize the use of Pauthenier’s formula: diffusion charging ionising electrodes at the same or a different potential. Such
can be neglected, the applied electric field is quasi-uniform in configurations, designated as “dual electrodes” [5, 7], are
the vicinity of particles, and particle speed is low compared to characterized by the existence of singular points where the
that of air ions. The charging model takes into account the electric field is zero. Very few algorithms are able to provide
computed spatial distribution of the electric field and charge a solution to this problem [16, 17].
density. The computations were performed for various values of In a recent paper [18], conformal mapping was employed
the geometrical parameters of the electrode system and of the to transform the geometrical domain into one easily tractable
particle transit time through the corona discharge zone. The by classical numerical methods. The aim of the present work
results can be used for the design of the electrode system of any is to show how these results can be employed for estimating
electrostatic process employing corona discharge fields.
the charge acquired by insulating and conducting particles
Index Terms : Computational electrostatics, Particle charging, when passing through the space charge zones generated by
Corona discharge various corona-electrostatic electrode geometries.
I. INTRODUCTION II. CORONA FIELD COMPUTATION
Corona charging of particulate matter is a physical In drum-type electrostatic separators, a “dual electrode”
mechanism frequently employed in electrostatic processes connected to a negative high-voltage supply is facing a roll
such as dust precipitation, electrostatic painting, powder electrode connected to the ground (Fig 1, a). The roll radius
coating, and separation of granular mixtures [1,2]. In many being significantly larger than the inter-electrode spacing and
situations, non-ionizing electrodes are associated with the the characteristic dimensions of the other electrodes, the most
corona (ionizing) electrode, in order to increase the efficiency simple electrode geometry that could model this situation of
of the charging or enhance the electric forces exerted on the practical interest is the three-electrode system consisting of a
particles. The electric field generated by any such electrode wire, a cylinder, and a plate, shown in Fig. 1,b. The wire and
arrangement has been referred to as corona-electrostatic [3], the cylinder are parallel to each other and connected to the
and has been the object of several experimental studies [4, 5]. same high voltage potential compose a “wire-type dual
The physical phenomena associated with corona charging electrode”. The small radius wire is the ionizing electrode.
and particle motion in electric fields affected by the presence The ionic charge injected at the wire surface is repelled by
of space charge have been thoroughly studied [6, 7]. the large radius cylinder (the non-ionizing electrode) and
However, none of the mathematical models elaborated for collected by the grounded plate electrode, normal to the plane
this purpose are capable of simulating the distortion of the defined by the axes of the other two electrodes.
electric field due to the presence of the ionic space charge.
IAS 2005 1429 0-7803-9208-6/05/$20.00 © 2005 IEEEThe electric field E affected by the ionic space charge is
governed by the following equations:
- the Poisson equation :
ρ
∆Φ = − , (1)
ε0
where ρ is the ionic space charge density and Φ is the
electric potential related to the electric field through :
G
E = − gradΦ ; (2)
- the charge conservation law :
∂ρ G
+ div j = 0 , (3)
∂t
a)
G G
where j = ρ K i E is the corona current density when the
diffusion current is neglected. Ki = 2ּ10-4 m2/Vs being the ion
mobility [18].
Considering the corona discharge constant and uniform
all along the ionizing wire and because L >> r, d or h
(Fig. 1, a, b), the electric field and space charge problem has
a 2-D symmetry. The 2-D computation domain (Fig. 1, b) has
a symmetry axis Oy; this reduces the domain under
investigation to the quadrant x ≥ 0, y ≥ 0, denoted by Dxy.
The boundary conditions are detailed in [18]. For the
electrical potential (Poisson equation (1)), the boundary
conditions are of Dirichlet type for the wire and the non-
ionizing electrode (Φ = Φ0 where Φ0 is the applied potential)
as well as for the grounded collector plate (Φ = 0). On the
symmetry axis Oy, a Neumann type condition is imposed
(∂Φ/∂x = 0).
For the charge conservation equation, the boundary
b)
condition consists in imposing a uniform charge density
value all around the wire surface ρ0 [18]. To establish the
Figure 1. a) Schematic view of the electrodes system used in an
electrostatic separation; b) simplified 2-D geometry investigated in the value of ρ0, the Kaptzov hypothesis and Peek formula are
present study. used [18].
The computation method used to find the solution of the
system (1) – (3) is detailed in [18]. A conformal mapping
The physical model is that of the electric field affected by transforms the computation domain into another one easier to
a permanent flow of ions generated at the surface of a smooth solve. In this way the difficulties related to the singularity
cylindrical wire and can be simplified as follows: point located on the symmetry axis Oy between the wire and
• the corona discharge is regarded as only a generator high voltage cylinder and also the construction of a suitable
of monopolar (negative) ions, all phenomena related mesh are overcome. In a second step, classic numerical
to ion generation being neglected; techniques are used to solve the Poisson and charge
G G G G conservation equations. The complete field-charge solution is
• the medium is air such that D = ε 0 E , D and E obtained using the successive approximations technique [18].
being the electric displacement and field strength Fig. 2 shows the spatial distribution of the electrical potential
respectively, and ε0 the vacuum permittivity; and ionic space charge. The electrostatic cylinder create a
zone free of ionic space charge limited by the separating line
• the corona discharge is assumed to be steady and which starts from singularity point E = 0. The electric field
uniformly distributed over the wire surface and strength and the ionic space charge density at the plate
length.
surface depend of the dual cylinder position.
IAS 2005 1430 0-7803-9208-6/05/$20.00 © 2005 IEEE(a)
Fig. 3. Variation of dimensionless space charge density along of the
grounded plate for several distances wire – cylinder. Φ0 = 25 kV,
d = 45 mm, R = 12.5 mm and r = 0.15 mm).
(b)
Fig. 2. a) Equipotential lines in a Oxy plane; b) Lines of equal charge
density in the Oxy plane. Φ0 = 25 kV, d = 45 mm, h = 40 mm, cylinder
radius R = 12.5 mm and wire radius r = 0.15.
When the cylinder is close to the wire the electric field at
the surface of the ionizing wire is lower and, by consequence,
the charge injection is lower too. For small values of h, the
region with ionic charge is very confined (Fig. 3). However, Fig. 4. Variation of dimensionless electric field strength along of the
for the large distance x, there are not an important influence grounded plate for several distances wire – cylinder. Φ0 = 25 kV,
d = 45 mm, R = 12.5 mm and r = 0.15 mm).
of the cylinder position on the electric field strength (Fig. 4).
The computations are done for particle-free electric field
domains. The results of such computations remain valid as
long as the size of a particle introduced in the field domain is The physical model that will be employed for the study of
small when compared to the inter-electrode spacing. the charging process is based on the several assumptions:
Therefore, they can be used in the study of particle charging • the particles are insulating spheres of diameter dp, and
phenomena, as shown in the next section of the paper. dielectric permittivity εr; they move in the Ox
direction with an uniform velocity Vp, equal to the
magnitude of the linear velocity of a point on the
III. CORONA FIELD CHARGING OF INSULATING PARTICLES surface of the roll electrode rotating at the speed
imposed by a given -electrostatic separation process;
The outcome of several electrostatic processes, including
• the particles are located at the surface of the
separation, painting or precipitation, strongly depends on the
grounded electrode; if they are insulating, their charge
efficiency of corona charging processes. This justifies the
is not affected by the contact with the metallic
need for the development of a computational method for the
electrode; if they are conducting, they will acquire a
evaluation of the charge acquired by insulating and
charge by electrostatic induction, as shown in section
conducting particles while passing through the corona field
IV.
zone generated by electrode systems as the one in Fig. 1.
IAS 2005 1431 0-7803-9208-6/05/$20.00 © 2005 IEEE1 3
• the influence of the particles on the spatial Qpc = π ε0dp2E . (7)
distributions of the electric field and ionic space 6
charge is neglected;
By examining the equation (7) it appears that evaluating
• a small spherical particle (insulating or conductive) the charge acquired by a conductive particle in contact with
placed on the surface of the grounded electrode is the plate electrode (Fig. 1, b) involves knowing the field
considered subjected to both a constant electric field strength at the surface of the plate for each particle position
and a constant space charge density, equal to the along the Ox axis.
values of these physical parameters computed in its
center. Moreover, as the present study concerns only
large particle sizes (dp > 1 mm), the contribution of V. NUMERICAL RESULTS AND DISCUSSION
diffusing charging mechanism is completely
neglected.
Using the charging models above, the insulating and
With the assumptions above, the equation of particle conductive particles charge has been computed for various
charging in an electric field E is the one established by values of the geometrical parameters of the electrode system.
Pauthenier [19]: Fig. 5 presents the variation of the insulating particle
charge and saturation charge as a function of particle
2 position. The charging process starts when the particle enters
dQp 1 Qp
= Qps 1 − s • (4) the ionic space charge zone. The maximum value of the
dt τ Qp charge is attained in a short distance approximately equal
with d. Due to the symmetry of the space charge and field
distributions, the insulating particle acquires the charge only
where the characteristic charge time constant is: in the first side of the domain Dxy (left side in Fig. 1, b). The
maximum charge is attained when the particle crosses the
4ε 0 symmetry axis Oy.
τ = . (5)
ρK i The spatial extension of region in which particle charging
takes place is due to the presence of the non-ionizing high-
and Qs represents the maximum charge (saturation charge): voltage electrode which repels the ionic charge and imposes
the limits of the space charge zone (see Fig. 4) [18].
The distance between the wire and dual cylinder has a
εr
Qps = 3πε 0 d p 2 E (6) crucial influence on the particle charge(5) (Fig. 6). For small
εr + 2 values of h, charging is not efficient. The proximity of the
non-ionizing electrode diminishes the electric field strength
at the surface of the wire and hence the density of the space
To compute the particle charge Qp the equation (4) must charge generated by corona discharge. This effect becomes
be integrated. For that the charging process is divided in very negligible for h > 30 mm; the space charge density has a
short time steps. The particles are injected at the surface of maximum value.
the grounded plate (x = -4d, y = dp/2) and leave the ionized Fig. 7 shows the charging process for different values of
field zone at (x = 4d, y = dp/2) – see Fig. 1, a. In each particle particle velocity. When the velocity increases, the residence
position the electric field strength and space charge density time in the corona region decreases and the acquired particle
are computed and the characteristic charging time τ and charge is lower.
saturation charge Qs are evaluated. The wire-plate spacing d strongly influences the particle
While Qp is smaller than saturation charge at a given point charging process (Fig. 8). For small values of d the ionic
(x, y), the particle continues to accumulate the electric charge; current at the plate surface is very strong and the particle
if Qp ≥ Qs the charging process is stopped. charge is higher. The charging process starts at different
moments according to the ionic space charge extension.
The increase of the applied voltage intensifies not only
the electric field strength at the surface of the grounded
IV. INDUCTION CHARGING OF CONDUCTING PARTICLES electrode, but also the charge injection. As a consequence,
the acquired particle charge is higher as shown in Fig. 9.
For a given conducting particle in contact with the
A spherical conducting particle in contact with a metallic
grounded plate the acquired charge depends exclusively of
electrode affected by an electric field E will acquire, by
the electric field strength. The variation of the metallic
electrostatic induction, an electric charge Qpc given by the
particle charge as function of particle position, for various
following formula [20]:
wire – cylinder distances is represented in Fig. 10.
IAS 2005 1432 0-7803-9208-6/05/$20.00 © 2005 IEEEFig. 5. Particle charge Qp and saturation charge Qs as a function of the Fig. 8. Charge of insulating particles Qp as a function of the particle position
particle position for dp = 3 mm, εr = 3, Φ0 = 24 kV, Vp = 0.5 m/s, d = 45 mm for dp = 3 mm, εr = 3, Φ0 = 25 kV, Vp = 0.8 m/s, h = 20 mm, and several
and h = 40 mm. wire plate distances d.
Fig. 6. Charge of insulating particles Qp as a function of the particle position Fig. 9. Charge of insulating particles Qp as a function of the particle position
for dp = 3 mm, εr = 3, Φ0 = 25 kV, Vp = 0.8 m/s, d = 45 mm and several wire for dp = 3 mm, εr = 3, Vp = 0.8 m/s, d = 45 mm, h = 30 mm and several
cylinder distance h. values of the applied voltage.
Fig. 7. Particle charge Qp as a function of the particle time for several values Fig. 10. Charge of conducting particles Qpc as a function of the particle
of particle velocity for dp = 3 mm, εr = 3, Φ0 = 25 kV, d = 45 mm and position for dp = 3 mm, εr = 3, Vp = 0.8 m/s, d = 45 mm, Φ0 = 25 kV and
h = 40 mm. several wire cylinder distance.
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