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 IEEE
The 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 IEEE
1 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 IEEE
Fig. 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. IAS 2005 1433 0-7803-9208-6/05/$20.00 © 2005 IEEE
For h = 30 and 40 mm the variations of electric field [3] L. Dascalescu, A. Iuga, R. Morar, V. Neamtu, I. Suarasan, A. Samuila, strength at the surface of the plate are very close to that of the and D. Rafiroiu, "Corona and electrostatic electrodes for high-tension separators,". J. Electrostatics , vol. 29, pp. 211-225, 1993. wire-plate case and the metallic particles acquires practically [4] R. Morar, A. Iuga, L. Dascalescu, and A. Samuila, "Factors which the same charge in the central zone of the corona discharge. influence the insulation-metal electroseparation," J. Electrostatics, vol. These results show that only the insulating particles charging 30, pp. 403-412, 1993. depend significantly of the arrangement of the dual electrode. [5] A. Iuga, R. Morar, A. Samuila, and L. Dascalescu, “Electrostatic separation of metals and plastics from granular industrial wastes,” IEE Proc.-Sci. Meas. Technol., vol. 148, pp. 47-54, 2001. [6] L. Dascalescu, R. Tobazeon, and P. 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Moreau-Hanot, “La charge des particules Council, France. sphériques dans un champ ionisé”, J. Phis. Radium, vol. 3, pp. 590-613, 1932. [20] N. J. Félici, “Forces et charges de petits objets en contact avec une électrode affectée d’un champ électrique”, Rev. Gen. Electr., vol. 75, REFERENCES 1966, pp. 1145-1160. [21] A. Samuila, M. Mihailescu and L. Dascalescu, “Unipolar Charging of [1] A .D. Moore (Ed), Electrostatics and Its Applications, Wiley, New York, Insulating Spheres in Rectified AC Electric Fields”, IEEE Trans. Ind. 1973. Appl., vol. 34, pp. 726-731, 1998. [2] J.S Chang, A.J. Kelly and J.M. Crowley (Eds), Handbook of Electrostatic [22] K. Adamiak, A. Krupa, and A. Javorek, “Unipolar particle charging in Processes. Dekker, New York, 1995. an alternating electric field”, Electrostatics ’93, York, U.K. Inst. Of Physics, 1995, pp. 275 -278. IAS 2005 1434 0-7803-9208-6/05/$20.00 © 2005 IEEE
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