Numerical Analysis of Electric Wind in Corona Field

Two-dimensional digital model for analysis of direct current corona field and induced electrohydrodynamic air flow field in wire-to-plane electrode system is presented. The model based on the finite-difference method in polar coordinate system for corona field computation is suitable to use in usual personal computers. Results of computation of current-voltage discharge characteristic are compared with experimental one. The digital model of electrohydrodynamic air flow field consists of finitedifference approximation of the Navier-Stokes equation and the continuity equation in Cartesian coordinates. Digital velocity values are checked experimentally. Movement of charges in outer zone of the discharge is complicated due to drag of neutral air molecules. This is the reason of electrohydrodynamic flow rise named as electric wind. Electric wind phenomenon can be used for heat and mass enhancement, in ionic air cleaners etc. Many results of experimental and theoretical research are published [1, 2]. Because of high voltage conditions in the discharge gap authors use for experimental research laserDoppler anemometers, particle image velocimetry systems or hot-wire anemometers [3–5]. Theoretical research is related to simultaneous analysis of electric field in the electrode system with direct-current corona discharge and air flow field in the system. Analytical study of this complex problem can be performed only for onedimensional field of coaxial electrode system. Analysis of the complex field in all practical electrode systems, such as wire-to-plate, pin-to-plate etc, is available only by use of numerical methods.


Introduction
Two-dimensional digital model for analysis of direct current corona field and induced electrohydrodynamic air flow field in wire-to-plane electrode system is presented.The model based on the finite-difference method in polar coordinate system for corona field computation is suitable to use in usual personal computers.Results of computation of current-voltage discharge characteristic are compared with experimental one.The digital model of electrohydrodynamic air flow field consists of finitedifference approximation of the Navier-Stokes equation and the continuity equation in Cartesian coordinates.Digital velocity values are checked experimentally.Movement of charges in outer zone of the discharge is complicated due to drag of neutral air molecules.This is the reason of electrohydrodynamic flow rise named as electric wind.Electric wind phenomenon can be used for heat and mass enhancement, in ionic air cleaners etc.Many results of experimental and theoretical research are published [1,2].Because of high voltage conditions in the discharge gap authors use for experimental research laser-Doppler anemometers, particle image velocimetry systems or hot-wire anemometers [3][4][5].Theoretical research is related to simultaneous analysis of electric field in the electrode system with direct-current corona discharge and air flow field in the system.Analytical study of this complex problem can be performed only for onedimensional field of coaxial electrode system.Analysis of the complex field in all practical electrode systems, such as wire-to-plate, pin-to-plate etc, is available only by use of numerical methods.

Modeling direct current corona field
We use the system of corona field equations in wireto-plane electrode system reduced to the Poisson's equation and the charge conservation equation.The finite difference approximation of the Poisson's equation in polar system of coordinates is the following [6] The partial derivatives of the charge conservation equation are of the form: where potential is represented by V and space charge density correspondingly by .Distances from the central node of polar grid O to the neighbor nodes P, Q, R and S are denoted by a P , a Q , a R and a S .Boundary conditions for potential are of Dirichlet and Neumann type.Boundary condition for space charge density on the surface of the wire is determined iteratively until Kaptzov's condition is established [7].The number of nodes in polar grid is 518 for the values of geometrical dimensions of the electrode system (Fig. 1) r 0  0,05 mm and h  12,0 mm.
Correctness of the model is checked by comparison of the numerical solution of Laplace's equation with analytical solution of an electrostatic field in wire-to-plane electrode system.Average difference between numerical and analytical values of electrostatic field potential on the axis of symmetry (x = 0) is 0,2 % for the number of nodes of polar grid n = 518.Additional test of the numerical corona field model is comparison of numerical data of the current-voltage characteristic with the experimental one.
Maximum difference between numerical and experimental values of linear current density corresponding to the voltage value U  7,0 kV is 150 A/m.Spatial components of Coulomb force calculated in rectangular coordinate system is shown in Fig. 2.

Numerical model of electro hydrodynamic air flow field
Electrohydrodynamic air flow is determined by the system of equations comprising of the Navier-Stokes and flow continuity equations.Emitting electrode with the ionization zone cross section area is negligible in comparison with an overall area of the field therefore the wire electrode can be represented as a point.Cartesian system of coordinates can be used to reduce the number of nodes.Finite-difference approximation of the Navier-Stokes and the flow continuity equations for this system of coordinates contains the Coulomb force components F x , F y and the flow velocity components w x , w y [7]:   w is the x axis component of average velocity in the square (Fig. 3),   -variation of time between iterations, a -distance between nodes, regular in all the grid (a  2 mm for Fig. 4).Hot-wire anemometer DO 9847K is used for experimental test of computed data.The diameter of the anemometer probe is 8 mm whereas the spacing between electrodes is 12 mm.Therefore it is no possibility to measure an air flow velocity in the volume between electrodes for the risk of spark discharge and essential distortion not only the corona field, but also the air flow field.Measured value of the airflow velocity in the point x = 60 mm at the surface of plane electrode is 1,02 m/s, and the numerical value is 1,57 m/s, the difference between the results is 35 %.Coincidence of computed and measured results is only qualitative.This is in agreement with the results of other researchers [3].

Conclusions
Polar grid is used for computation of corona field to reduce the number of nodes in computational area at the sufficient amount of information on the distribution of the field quantities near the emitting electrode.
Two-dimensional numerical model of direct current corona field analysis in wire-to-plane electrode system is based upon the finite-difference method in polar system of coordinates.
It is investigated that Coulomb force strength has an influence to air velocity, but almost no influence to vectors direction, because the ratio of spatial force components vary insignificantly, only changes their module.
Maximum value of air flow velocity above the wire corresponding to the U = 10 kV is approximately 4 m/s.Maximum value of the total volume flow-rate is near the 60•10 -3 m 3 /s at the mentioned conditions.
Measured and computed values of air flow velocities coincide qualitatively because of the difference between measured and computed values totals about 30%.Two-dimensional digital model for analysis of direct current corona field and induced electrohydrodynamic air flow field in wireto-plane electrode system is presented.The model based on the finite-difference method in polar coordinate system for corona field computation is suitable to use in usual personal computers.Results of computation of current-voltage discharge characteristic are compared with experimental one.The digital model of electrohydrodynamic air flow field consists of finite-difference approximation of the Navier-Stokes equation and the continuity equation in Cartesian coordinates.Digital velocity values are checked experimentally.Ill. 10, bibl.7 (in English; abstracts in English and Lithuanian).Analizuojamas elektrodų sistemos "laidas šalia plokštumos" dvimačio vienpolio vainikinio išlydžio elektrinio lauko ir jo sukelto oro tekėjimo lauko analizės skaitinis modelis.Vainikinio išlydžio elektrinis laukas skaičiuojamas baigtinių skirtumų metodu polinėje koordinačių sistemoje.Skaitinio modeliavimo rezultatai yra palyginti su eksperimento rezultatais.Oro tekėjimo lauko lygčių sistemos skaitinį modelį sudaro Naviero ir Stokeso, taip pat tolydumo lygties skirtuminė aproksimacija Dekarto koordinačių sistemoje.Il. 10, bibl.7 (anglų kalba; santraukos anglų ir lietuvių k.).

FFig. 1 .
Fig. 1.Coulomb force components Components of the Coulomb force per unit volume are determined as the product of space charge density and the field strength:

Fig. 2 .
Fig. 2. Spatial force vectors, recounted by rectangular grid force projections in elements of rectangular grid

Fig. 3 .
Fig. 3. Computational grid with square elements It is clear from Fig. 4 and Fig. 5 that Coulomb force strength has an influence to air velocity, but almost no influence to vectors direction, because the ratio of spatial