Simulation of corona discharge in point–plane configuration☆
Introduction
Despite widespread use of the electric corona discharge, there is no reliable and accurate numerical model for the computer simulation of this phenomenon. Existing numerical techniques are sufficient for rough estimate of the basic characteristics of the discharge (V–I curve, electric field, current density), when the corona electrode is not too sharp. However, they fail if high accuracy of results is required or in the case of electrodes with very small radii of curvature.
The main reason for the lack of a more sophisticated corona simulation algorithm is complexity of this phenomenon. The full model in two-dimensional space has been attempted by very few authors. Morrow [1] was the first who tried to analyze all essential processes (ionization, attachment of electrons, recombination, etc.) in the time domain. He was apparently successful, although the results were not validated experimentally, but the algorithm must have led to very time consuming computations due to a very fine spatial discretization needed, especially in the ionization layer, and very irregular dynamics of the problem, with rapid time variation of some parameters followed by much longer intervals of relatively slow changes. Therefore, such an approach can be useful for detailed study of some physical processes, but not as a design or optimization tool in engineering applications.
Much better documented are one-dimensional models, which do not require so demanding computer resources [2]. Pontiga and others have concentrated their interest on the numerous processes and chemical reactions occurring in the corona discharge [3]. However, this was done for steady-state conditions.
Nearly all the authors interested in engineering applications of the corona discharge completely ignore the processes in the ionization layer, what is justified by its small thickness, generally on the order of the corona electrode radius of curvature [4]. The ionization of gas is, by definition, limited to this layer, but attachment is not for the negative corona; nobody yet has tried to investigate these effects outside the ionization layer. The most common approach is to consider the two-dimensional DC monopolar model, where one species of ionic charges is injected from the discharge electrode and the ions trajectories are predicted by solving coupled equations for the electric field and the charge transport. Many versions of this basic monopolar corona model differ by using various numerical techniques. Iterative algorithms are predominant, where both problems are solved subsequently, until convergence is reached. A different philosophy has been presented only by Feng [5], who combined both equations into a one non-linear set of partial differential equations.
Differential techniques are preferred for the Poisson equation governing the electric field distribution. Earlier attempts with the finite difference method could not handle the sharp geometry of the discharge electrode very well [6]. The finite element method (FEM) has removed this limitation and is commonly accepted [7], although a proper discretization may lead to quite large algebraic systems. The boundary techniques (charge simulation method and boundary element method) are better suited to strong field gradients, but inclusion of the space charge is much troublesome [8].
Choice of the numerical technique for the charge transport equation is much less obvious. Three basic groups of methods can be identified:
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the method of characteristics (MOC), which predicts the space charge density along some lines, called characteristics, which are the field lines and ions trajectories here [9].
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the finite volume method, especially in the version called the donor-cell method, using the charge conservation equation in the integral form [10].
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the FEM, which may be prone to the numerical diffusion, predicting some space charge density in the whole space while in reality it is limited to some area between both electrodes [5].
Interfacing both problems (electric field and space charge) may be problematic in some combination of the techniques mentioned above. The differential techniques for the electric field require values of the space charge density at the model nodes, while the MOC calculates them at its own set of points. From the most common suggestions, one proposes to regenerate the mesh at each iteration step [11] and another one uses a special interpolation of charges [12]. Both can significantly extend the computing time.
This paper presents a hybrid approach to the monopolar corona modeling: the electric field is solved by combined BEM-FEM technique. The first one is used for the Laplace equation, and the second one for the Poisson equation. The MOC handles the charge transport equation and a simple interpolation scheme provides an interface between both steps. The Kaptzov assumption (restricted to the zone of the needle electrode where the corona discharge sustains) is approximately satisfied by introducing an injection law, which gives the density of injected charge carriers as a function of the local field. This leads to delineate the part of the needle where the corona discharge takes place and, hence, to determine the inter-electrode zone with non zero charge density. In particular the radius at which the current density on the ground plate abruptly decreases down to zero is well predicted for different values of the tip radius and the needle-plate distance.
A close examination of computed and measured current–voltage characteristics I(V) shows that their shape is well predicted and that a good approximation is given by the expression I=aKεV(V−V0)/d where K is the ions mobility, ε the gas permittivity, V the applied voltage and V0 the corona onset level. The constant a takes a value close to 1.58. This makes it possible to estimate the mobility of ions from the I(V) measurement.
Section snippets
Numerical model
A classical electric corona discharge configuration is investigated in this paper with two electrodes of distinctly different radii of curvature separated by some air gap. The sharp electrode is supplied with a positive high voltage and one species of positive ions is injected into air gap. These ions move with a constant mobility towards the grounded plate. As the space charge is present, the distribution of electric scalar potential V is governed by the Poisson equationwhere ρ is the
Computational algorithm
A hybrid FEM-BEM-MOC technique has been used to solve , and to find the magnitude and distribution of the injected charge. All steps are performed in a sequence forming two iterative loops.
Geometry of the problem
The presented algorithm has been applied to the point–plane configuration. The point electrode was a long cylindrical wire of in diameter with a conical ending of length and capped with a hemispherical tip [21]. Two different needles were tested with an average radius of curvature equal to 95 and . The needles were mounted perpendicularly to the aluminum plate of at the distance of and . The high voltage supplied to the needle varied between the corona onset
Conclusions
The results of the numerical simulation of the electric corona discharge have been compared with the experimental data for the point–plane geometry. Assuming that the mobility of the positive ions is equal to 2.3×10−4, the numerical algorithm predicts reasonably well V–I characteristics of the discharge. The I/V versus V curve is almost perfectly linear and follows a simple analytical formula, with the proportionality coefficient equal to 1.58.
Prediction of the current density distribution on
Acknowledgements
This work was supported in part by Joseph Fourier University and CNRS (through invitation of K. Adamiak to the LEMD) and the Natural Sciences and Engineering Research Council (NSERC) of Canada.
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Presented at the 2003 Joint Meeting of the Electrostatics Society of America and the IEEE Industry Applications Society Electrostatic Processes Committee, June 24–27, 2003, Little Rock, AR, USA.