Ion Flow Effects on Negative Direct Current Corona in Air

Transcription

1 Plasma Chem Plasma Process (2010) 30:55 73 DOI /s ORIGINAL PAPER Ion Flow Effects on Negative Direct Current Corona in Air Yuesheng Zheng Jinliang He Bo Zhang Wei Li Rong Zeng Received: 24 October 2009 / Accepted: 11 December 2009 / Published online: 5 January 2010 Ó Springer Science+Business Media, LLC 2009 Abstract Numerical simulations are presented for the ion flow effects on the negative direct current corona in air. One dimensional equivalent steady corona model containing the continuity equations for electrons and ions on the basis of Townsend theory coupled with Poisson s equation is applied. The conductor radius and the electric field intensity on conductor surface keep constant under the standard atmosphere condition in this paper. The results suggest that the space charges within the plasma region have no influence on the electric field distribution throughout the interelectrode gap, which is only governed by the ion flow. The influences of the gap distance and the corona current on the corona discharge within the plasma region have been investigated. If the voltage-current characteristics of the negative corona discharge are measured, the ion flow region and the plasma region can be investigated individually. Based on Kaptzov s hypothesis, the plasma region can be investigated solely. Keywords Corona discharge Corona plasma Space charge Ion flow Introduction Atmosphere corona discharge has been extensively used in electrostatic precipitators, photocopiers, indoor air cleaners, jet printers for decades [1 3]. New applications have been continuously emerging, for example gas treatment [4, 5], ozone production [6 8], surface processing [9], as well as plasma enhanced chemical vapor deposition [10 12]. However, the electromagnetic interference and the audible noise caused by corona must be avoided in some occasion, such as high voltage power transmission line [13]. Accurate control of corona discharge plays an important role in a host of emerging technologies. Y. Zheng J. He (&) B. Zhang W. Li R. Zeng State Key Lab of Power Systems, Department of Electrical Engineering, Tsinghua University, Haidian District, Beijing, People s Republic of China hejl@mail.tsinghua.edu.cn Y. Zheng yuesheng.zheng@gmail.com

2 56 Plasma Chem Plasma Process (2010) 30:55 73 Different from the alternating current (AC) corona, the direct current (DC) corona generates an ion flow in the broad surrounding air, which distorts the electric field throughout the interelectrode gap. Although the corona plasma is the source of ion flow, it was always neglected compared with the broad ion flow region in calculating the ion flow distribution [14 18]. Sarma and Janischewskyj [19] first modeled the electric field distribution within the plasma region and the onset condition of smooth conductors was studied. Chen and Davidson [20, 21] extended this model to study the space charge distribution also within the plasma region based on Kaptzov s hypothesis. The corona plasma and the ion flow were investigated separately. Lander [22] modeled the current density and the space charge distribution throughout the interelectrode gap with different conductor radius using measured values of the supplied voltages and corona currents. However, the influence of the ion flow on the plasma region has not yet been investigated. Corona onset fields are almost the same between the positive and negative coronas, while the processes of the secondary electron emission are greatly different [23, 24]. Compared with the positive corona, the energetic electron distribution in the negative corona is much wider [21, 22]. And the number of electrons in the negative corona is an order of magnitude greater than that in the positive corona [21]. The negative corona produces more ozone [8] and less radio interference and audible noise [13]. The discharge electrode material has little effect on the positive corona, but it has an impact on the negative corona [25]. What is more, the negative corona is superior for industry application for higher voltage leading to spark discharge [26]. Due to the difference in the generation mechanisms of the positive and negative coronas, only the negative corona is investigated in this paper. The objective of this paper is to investigate the ion flow effects on the negative DC corona plasma in air by numerical simulations throughout the interelectrode gap. A parametric study is conducted to examine the effects of the ion flow thickness and the ion flow density on the corona discharge through controlling the gap distance and the corona current. The next section briefly describes different modes of the negative corona. Section Numberical Simulation explains the physical model, the governing equations, the boundary conditions and the numerical methods. The results are presented in the next section. Discussion of recombination and diffusion in the plasma region follows in the last before section. Negative Corona Modes The negative DC corona discharge in air exhibits different modes. Its appearance depends on the atmospheric conditions, the electric field intensity, the non-uniformity degree of electric field and the cathode surface conditions. Figure 1 is an example of the appearance of the negative corona discharge in air, which is deduced from the literature [27, 28] for the cathode with a spherical protrusion. The negative corona can develop over a wide range of the supplied voltage in different modes prior to the complete breakdown of the gap. With the supplied voltage increased, these are Trichel streamer, pulseless glow and negative streamer, respectively. With a short gap distance, not all negative corona modes exist. With the gap distance increased, the threshold of different negative corona discharge modes become distinct. The corona current also increases as the supplied voltage increased. The negative corona discharge is random and irregular in time at the threshold. But with the supplied voltage further increased, it becomes quite regular in time, customarily named Trichel streamer or Trichel pulse [29]. The Trichel streamer develops along a narrow

3 Plasma Chem Plasma Process (2010) 30: Fig. 1 Areas of existence of the different negative corona discharge modes in air. The dimension is not to scale channel and continuously changes its position on the cathode. Increasing the supplied voltage can cause an increase of the corona current and the repeat rate of pulses before the discharge mode transforming into a pulseless glow. The pulseless glow is characterized by a pulseless discharge current. The wandering of the discharge ceases and becomes fixed at one point. The discharge is particularly stable and preserves the same feature as the glow discharges. With a further increase of the supplied voltage, the pulseless glow will change over into a negative streamer, whose current consists of a DC component and superposed pulses. Numerical Simulation Physical Model The negative corona only possibly occurs in the electronegative gases. Figure 2 describes the model in detail. One dimension cylindrical symmetry is assumed and the conductor radius keeps 0.01 m invariable in this paper. The outer electrode is grounded. When the discharge electrode is at a negative potential and the electric field on its surface is strong enough, the electron avalanches are initiated on the conductor surface, and develop in a continuously decreasing field towards the anode. The whole interelectrode is in the domain of computation. When transmission lines operate under the normal voltage, the electron-positive ions pairs are assumed only generated by the impact ionization. The negative ions are formed by electrons attaching themselves to the oxygen molecules. The thermal contact between electrons and heavy species is poor, so the mean temperatures of all space charges are assumed to be constant. It is reasonable that the ionization coefficient and the attachment coefficient of electrons are assumed only depending on the local electric field and the atmospheric condition [30]. Likewise, the mobility of space charges depends solely on the local electric field and the atmospheric condition.

4 58 Plasma Chem Plasma Process (2010) 30:55 73 Fig. 2 Sketch of the physical model The negative ions and electrons move to the anode and the positive ions move to the cathode in different velocities driven by the applied field. The plasma region maintains conservation of charges for positive ions, negative ions and electrons. The positive ions and electrons exist only in the plasma region, while the negative ions fill the whole interelectrode region. As a result of varying equilibrium between the accumulation and removal of the ionic space charges throughout the whole interelectrode region, different modes of the negative corona discharge described in Negative Corona Modes can occur as a function of the supplied voltage. As the supplied voltage is higher than the onset voltage of the whole conductor, the corona discharge distribution becomes uniform. So the assumption of one dimensional cylindrical symmetry is reasonable. Either steady or pulsating discharge on conductor, the conductor potential and the total corona current remain practically constant. Only the average value of the corona current contributes to the power loss. A possible procedure for the theoretical determination of the corona discharge process is to consider the corona plasma to be in an equivalent steady state. However, the radio interference and the audible noise generated by pulsating discharge (mainly by Trichel streamer in the negative corona) are not considered in this paper. The secondary electrons to sustain the negative corona discharge may be produced by photoemission from the discharge electrode, bombardment of the discharge surface by positive ions, or photoionization in the gas. The process of the secondary electron emission can not be considered in simulations directly in this paper. If the voltage-current characteristics of the corona on conductor have been measured, the number density of electrons indicating the secondary electron emission from the cathode over a wide range of operating current can be determined.

5 Plasma Chem Plasma Process (2010) 30: Set of Hydrodynamic Equations The condition of charge carriers allows the application of one dimensional static hydrodynamic approach, based on the continuity equations for the electrons, the positive ions and the negative ions coupled with Poisson s equation [19, 21, 22]. The signs on the right hand side of (1) (5) reflect the direction in the negative corona. The different parameters mean: a Electron ionization coefficient; g Electron attachment coefficient; e Permittivity of air; u Potential; E Intensity of electric field; p Atmospheric pressure; l e Electron mobility; l p Positive ion mobility; l n Negative ion mobility; n e Electron density; n p Positive ion density; n n Negative ion density; u e Electron velocity; u p Positive ion velocity; u n Negative ion velocity; Equations (1) (3) are continuity equations for the electrons, the positive ions and the negative ions, respectively. The corona plasma and the ion flow can be described by (1) (3) together. The terms on the right-hand side is the source terms, considering the ionization and the attachment of electrons. The recombination is neglected for only playing a minor role [19 22]. The diffusion of electrons is also neglected [19 22, 31]. Roles of recombination and diffusion will be discussed in the last before section. Poisson s equation is given by (4). Densities of the electrons, the positive ions and the negative ions are present in the term on right-hand side. The electric field is computed using (5). The relative permittivity of air is set to be 1 in analysis. The electron ionization coefficient is given by (6) [19, 21, 22] and the electron attachment coefficient is given by (7) [19]. p is set to be 760 torr in calculation. The mobilities of the space charges are given by (8) (10). l e, l p and l n are set to be 500, 1.5, and 1.8 cm 2 V -1 s -1, respectively [32]. But the data are restricted to a limited range of the electric field values deduced from experiment results. Outside this range, the same relation is assumed. d rdr dðrn e u e Þ rdr rd/ dr dðrn p u p Þ rdr ¼ða gþn e u e ¼ an e u e dðrn n u n Þ ¼ gn e u e rdr ¼ eðn p n n n e Þ e ð1þ ð2þ ð3þ ð4þ

6 60 Plasma Chem Plasma Process (2010) 30:55 73 E ¼ d/ dr a=p ¼ 4:7786 expð 221p=EÞ cm 1 torr 1 25 E=p 60 V cm 1 torr 1 g=p ¼ 0:013 0: ðe=pþþ0: ðe=pþ 2 cm 1 torr 1 25 E=p 60 V cm 1 torr 1 u e ¼ l e E u p ¼ l p E u n ¼ l n E ð5þ ð6þ ð7þ ð8þ ð9þ ð10þ Cathode Boundary The cathode boundary is also the inner boundary of the plasma region. There is no need to specify the outer boundary of the plasma region, which is movable. It can be distinguished from the calculated results. The electron from the cathode is the plasma source, but also the ion flow source. It is generated by the secondary electron emission, but the generation mechanism is still debatable. Its value is difficult to be decided, which may be a function of the electrode material and the surface condition for the negative corona. So the electron density on the cathode is varied from 10 8 to m -3 in analysis relative to the corresponding operating current per meter on conductor. It is just an average value for the pulsating corona. The cathode is assumed to be a perfect sink for the positive ion flow. This means the impinging positive ion flow condenses completely at the anode. Its value does not need to be set at the cathode. The negative ion density on the cathode is assumed to be zero, which implies that the electrons created on the cathode have to travel a certain distance then form negative ions. The cathode potential is related to the supplied voltage. The electric field intensity on the cathode in the absent of the space charges should be higher than that estimated by Peek s empirical formula for coaxial cylinders using (11) [33]. E o is the corona onset field in kv/cm, m the roughness coefficient of conductor (m = 1 for smooth surface), r 0 the conductor radius in cm. The electric field intensity on the cathode keeps 31.2 kv/cm invariable in this paper. E o ¼ 31:0m 1 þ 0:308 pffiffiffiffi kv=cm ð11þ r 0 Anode Boundary The anode boundary is also the outer boundary of the uniform ion flow region. The anode is assumed to be a perfect sink for the negative ion flow. Outside the plasma region, there is no electron. The positive ion density on the anode is set to be zero, which also implies that the positive ion density on the outer boundary of the plasma region is zero. The outer cylinder is grounded, so the anode potential is set to be zero.

7 Plasma Chem Plasma Process (2010) 30: Methods of Numerical Solution The mesh size is 0.01 mm within the region of radius 5r 0 and 1 mm outside this region. If the mesh size is smaller than that in this paper, the results are not sensitive to the mesh size, with the relative error smaller than 1%. Equations (1) (3) were calculated by the Runge Kutta method and (4) and (5) were calculated by the central finite difference method. Calculations of the electric field and the space charge density were in two separating loops. When every variable was convergent, the calculation stopped. The relative error between the last two steps is smaller than 0.1% for all parameters. Results The Role of Space Charge The distributions of the space charge density, the corona current and the electric field throughout the interelectrode gap are shown in Fig. 3. The electron density on the cathode is set as 10 9 m -3. The radius of the outer electrode is 1 m. The corona currents contributed by space charges were calculated using (12) (15). I e, I p, I n and I are magnitude of the electron current, the positive ion current, the negative ion current and the total current, respectively. I e ¼ 2pren e u e ð12þ I p ¼ 2pren p u p ð13þ I n ¼ 2pren n u n ð14þ I ¼ I e þ I n þ I p ð15þ Substituting (12) (14) into (1) (3), (16) (18) can be obtained. Electrons are initiated near the cathode, increasing exponentially within the region a [ g and decreasing quickly in the region a \ g. From (12), it can be found that I e reaches its peak when a = g, and meanwhile the slope of I n is equal to that of I p. As shown in Fig. 3b, the electron current increases at first, and then decreases after reaching its peak. On the cathode, n n (or I n )is zero and n p (or I p ) reaches its maximum. On the outer boundary of the plasma region, n e (or I e ), n p (or I p ) are zero and n n (or I n ) reaches its maximum. As shown in Fig. 3a and b, the densities and the currents of space charges on the boundary coincide well with the boundary conditions in Numerical Simulation. Outside the plasma region, only negative ions exist. So it can be found that the plasma region thickness is about two times of the conductor radius. Compared with the ion flow region, the plasma region thickness can be neglected. di e dr ¼ða gþi e di p dr ¼ ai e ð16þ ð17þ di n dr ¼ gi e ð18þ The respective electric field distributions in the absence and presence of space charges are compared in Fig. 3c. The electric field is weakened on conductor surface and

8 62 Plasma Chem Plasma Process (2010) 30:55 73 Fig. 3 Distributions of space charge density, current and electric field throughout the interelectrode gap. Electron density on cathode is 10 9 m -3. Radius of outer electrode is 1 m. a Space charge density distributions. b Current distributions. c Electric field distributions

9 Plasma Chem Plasma Process (2010) 30: strengthened on grounded cylinder by space charges. The Coulomb effect of space charges on the electric field is weakened with the distance from axis (or conductor surface) increased. Compared with the ion densities, the electron density can be neglected as shown in Fig. 3a, for the electron velocity is much higher than the ion velocity. When only space charges within the plasma region are considered in (4) and (5), the calculated electric field distribution coincides completely with that in the absence of space charges in Fig. 3c, so the space charges within the plasma region do not change the electric field distribution throughout the interelectrode gap. When only the negative ions within the ion flow region are considered in (4) and (5), the calculated electric field distribution coincides completely with that in the presence of space charges in Fig. 3c, so the distortion of the electric field in the presence of space charges owns a great deal to the ion flow. Influence of Gap Distance Because the plasma boundary keeps moving, the thickness of the ion flow is an uncertain parameter. As the results shown in The Role of Space Charge, the ion flow thickness is almost equal to the gap distance. The influence of the ion flow thickness on the corona discharge can be conducted through controlling the gap distance. The distributions of the electric field and the negative ion density throughout the interelectrode gap under different gap distances are shown in Fig. 4. The radii of the outer electrodes are 1, 5, and 10 m, corresponding to the gap distance 0.99, 4.99, and 9.99 m, respectively. The electron densities on the cathode are all 10 9 m -3. Under different gap distances, the negative ion density distributions within the overlapped region are the same as shown in Fig. 4. The electric field distributions within the overlapped region are also the same under different gap distances. As found in The Role of Space Charge, the electric field distribution throughout the interelectrode gap is governed by the negative ion density distribution. So the electric field distributions within the overlapped region can be controlled the same under different gap distances, if the Fig. 4 The distributions of electric field and negative ion density throughout the interelectrode gap under different gap distances. Electron density on cathode is 10 9 m -3. Radii of outer electrodes are 1, 5, and 10 m, corresponding to gap distance 0.99, 4.99, and 9.99 m, respectively

10 64 Plasma Chem Plasma Process (2010) 30:55 73 Fig. 5 Ion flow effects on electric field distributions in plasma region under different gap distances. Electron density on cathode is 10 9 m -3 in the presence of space charges. There is not any space charge between electrodes in the absence of space charges. Radii of outer electrodes are 1, 5, and 10 m, corresponding to gap distance 0.99, 4.99, and 9.99 m, respectively negative ion density distributions within the overlapped region keep the same under different gap distances. The electric field distributions within the plasma region are the same under different gap distances as shown in Fig. 4. From these governing equations, it can be found that the space charges densities only depend on the electric field distribution, so the distributions of the space charge density and the current under different gap distances also keep unchanged, which are the same as shown in Fig. 3a and b. The same distributions of the negative ion density under different gap distances shown in Fig. 4 also validate the fact. So the plasma characteristics can be controlled all the same under different gap distances by satisfying given conditions. When the space charges between electrodes are vacuumed, on the assumption that there is not any space charge between electrodes, the electric field in the plasma region is strengthened again. The ion flow effects on the electric field distributions in the plasma region under different gap distances are shown in Fig. 5. The electric field in the plasma region is weakened by the ion flow. Comparing the results of the electric field intensities in the presence and absence of the space charges, it can be found that the suppression of the electric field by the ion flow is intensified with the increase of the gap distance. When the electric field intensity on the cathode keeps as 31.2 kv/cm unchanged, on the assumption that there is not any space charge between electrodes in vacuum, the ion flow effects on the supplied voltage under different gap distances are shown in Fig. 6. Comparing the results in air with those in vacuum, it can be found that much higher supplied voltage is needed with the gap distance increased. Influence of Corona Current As the distance from the axis (or conductor surface) increased, the corona current is constant as shown in Fig. 3b. Because the ion flow distribution monotonously decreases with the increase of the distance from axis, the corona current is an appropriate monitoring

11 Plasma Chem Plasma Process (2010) 30: Fig. 6 Ion flow effects on supplied voltage under different gap distances. Electron density on cathode is 10 9 m -3 in the presence of space charges. There is not any space charge between electrodes in vacuum parameter. The influence of the ion flow intensity on the corona discharge can be conducted through controlling the corona current. Distributions of the electric field and the negative ion density throughout the interelectrode gap with different corona currents are shown in Fig. 7. The radius of the outer electrode is 1 m. The electron densities on cathode are 10 8,10 9, and m -3, corresponding to corona current 0.014, 0.141, and la/cm, respectively. The negative ion density throughout the interelectrode gap increases with the corona current increased as shown in Fig. 7a. The electric field intensity also increases throughout the interelectrode gap influenced by the ion flow as shown in Fig. 7b. The electric field near the anode is strengthened by the ion flow. The strengthening of the electric field is intensified with the corona current increased. Because the drift velocity of the negative ions only depends on the electric field intensity, the negative ion density with high corona current drops quickly under the strong electric field as shown in Fig. 7a. The electric field near the cathode is almost unchanged with the corona current increased, however, it tends to rise at the outer region with the corona current increasing continually. With the same distance from axis (or conductor surface), the normalized electric fields in the plasma region are obtained from the electric field intensities in air divided by that in vacuum (in the absence of space charges) with the same electric field intensities of 31.2 kv/cm on cathodes. Comparison of the normalized electric field distributions in the plasma region with different corona currents are shown in Fig. 8. Compared with the electric field distribution in vacuum within the plasma region, the electric field intensity in air decreases fractionally at first and then increased quickly to exceed that in vacuum with the increase of the corona current. For the practical range of the corona current on transmission line less than 1 la/cm [19], the electric field distribution in plasma region has nothing to do with the corona current. If the corona current is higher than about 10 la/cm, the ion flow effects can not be neglected. Because the electric field in the plasma region is slightly changed by the ion flow, the space charge distributions are also slightly changed. The normalized negative ion density distributions in the plasma region with different corona currents are obtained by the

12 66 Plasma Chem Plasma Process (2010) 30:55 73 Fig. 7 Distributions of electric field and negative ion density throughout the interelectrode gap with different corona currents. Radius of outer electrode is 1 m. Electron densities on cathode are 10 8,10 9, and m -3, corresponding to corona current 0.014, 0.141, and la/cm, respectively. a Negative ion density distributions. b Electric field distributions negative ion densities divided by their maximums. The comparison of the normalized negative ion density distributions in the plasma region with different corona currents is shown in Fig. 9. The plasma region thickness decreases with the corona current increased, but the effect is insignificant. So the plasma region thickness only depends on the electric field distribution, which is not influenced by the corona current. When the space between electrodes is vacuumed, on the assumption that there is not any space charge between electrodes, the electric field in the plasma region is strengthened again. The ion flow effects on the electric field distributions in the plasma region with different corona currents are shown in Fig. 10. It can be found that the suppression of the electric field is intensified with the corona current increased.

13 Plasma Chem Plasma Process (2010) 30: Fig. 8 Comparison of normalized electric field distributions in plasma region with different corona currents. Radius of outer electrode is 1 m. Electron densities on cathode are 10 9,10 10, and m -3, corresponding to corona current 0.141, 1.404, and la/cm, respectively Fig. 9 Comparison of normalized negative ion density distributions in plasma region with different corona currents. Radius of outer electrode is 1 m. Electron densities on cathode are 10 9 and m -3, corresponding to corona current and la/cm When the electric field intensity on the cathode keeps 31.2 kv/cm unchanged, on the assumption that there is not any space charge between electrodes in vacuum, the ion flow effects on supplied voltage with different corona currents are shown in Fig. 11. It can be observed that the higher supplied voltage is needed with the corona current increased. Discussion When the recombination among electrons and ions is taken into account, (1) (3) can be replaced by (19) (21). Based on the data calculated in The Role of Space Charge, the influence of the recombination on space charge density distributions within the plasma

14 68 Plasma Chem Plasma Process (2010) 30:55 73 Fig. 10 Ion flow effects on electric field distributions in plasma region with different corona currents. Radius of outer electrode is 1 m. There is not any space charge between electrodes in the absence of space charges. Electron densities on cathode are 10 8,10 9, and m -3, corresponding to corona current 0.014, 0.141, and la/cm, respectively Fig. 11 Ion flow effects on supplied voltage with different corona currents. Radius of outer electrode is 1 m. There is not any space charge between electrodes in vacuum region is shown in Fig. 12. It can be found that the influence of the recombination among electrons and ions on space charge density distributions is negligible comparing with the electron ionization and attachment processes in the negative corona discharge. dðrn e u e Þ rdr ¼ða gþn e u e n e n p b ð19þ

15 Plasma Chem Plasma Process (2010) 30: Fig. 12 Influence of recombination on space charge density distributions within plasma region. a Continuity equation for electrons. b Continuity equation for positive ions. c Continuity equation for negative ions dðrn p u p Þ ¼ an e u e þ n e n p b þ n p n n b rdr dðrn n u n Þ rdr ¼ gn e u e n p n n b ð20þ ð21þ

16 70 Plasma Chem Plasma Process (2010) 30:55 73 When the electron diffusion is taken into account, (1) can be replaced by (22). Based on the date calculated in The Role of Space Charge, the influence of the electron diffusion on the space charge density distribution within the plasma region is shown in Fig. 13. It can be found that the influence of the electron diffusion is negligible comparing with the electron ionization and attachment processes in the negative corona discharge. dðrn e l e EÞ drddne dr ¼ða gþn e u e ð22þ rdr rdr The transport properties in air, such as the ionization and attachment coefficients as well as the electron drift speed, have been obtained through experiments or solution of the stationary Boltzmann equation, but the dates used by different researchers make a very big difference, which may affect the results somewhat [30]. An accurate modeling requires an extensive knowledge of the transport properties in air. The corona cage is an ideal tool to study the corona performance of AC transmission line, while the space charges only within the corona cage are considered for DC transmission line. So a deep understanding of the influence of the gap distance is important for the corona discharge control. When the conductor surface condition and the electric field distribution in the plasma region are the same, the number of negative ions generated from corona discharge will also be the same. When the electric field on the conductor surface keeps constant, the corona current can also be controlled the same under different gap distance. The results in Influence of Gap Distance suggest that the corona cage is available to predict corona loss generated from HVDC (high voltage direct current) transmission line. The results are promising. Appropriate experimental data are urgently needed to validate and refine the model in detail. The availability of the corona cage for predicting the radio interference and the audible noise generated from the HVDC transmission line was verified by Nakano and Sunaga [34, 35] through experiments. It is based on the assumption that generation quantities of the radio interference, the audible noise and the corona current under different gap distances are the same by keeping the electric field on conductor surface the same in the presence of space charge, and the corona plasma region is neglected in calculating the electric field. As suggested by the results in Influence of Gap Distance, when the electric field on the Fig. 13 Influence of electron diffusion on space charge density distributions within plasma region

17 Plasma Chem Plasma Process (2010) 30: conductor surface is constant, the electric field distribution in the plasma region is invariable. The average characteristics of the plasma region can also be controlled all the same under different gap distance. Although the results can not directly verify the availability of the corona cage for predicting the radio interference and the audible noise generated from HVDC transmission line, it provide more information for corona control. The model containing the continuity equations for the electrons, the positive ions and the negative ions coupled with Poisson s equations is used in this paper. All the equations were calculated throughout the whole gap. However, it is time consuming for the multiscale phenomena. Because of the different physical characteristics of the corona plasma and the ion flow, it is feasible to study the corona discharge separately under some conditions. It is very interesting that an asymptotic model is successfully used to study the steady wire-to-wire corona discharge [36]. Because the space charge within the plasma region has no influence on the electric field distribution suggested by The Role of Space Charge, the electric field can be calculated only by the ion flow governing equations. The distributions of the electric field, the negative ion density and the corona current can be calculated by (5), (10), (23), (24). The ion flow governing equations are the reduced plasma governing equations. d rdr dðrn n u n Þ ¼ 0 rdr rd/ dr ¼ en n e If the voltage-current characteristics of the negative corona discharge have been measured, the electric field distribution within the plasma region and the corona current can be obtained. Then the space charge distributions in the plasma region can be calculated by (1) (3) straightforwardly. The electric field distribution in the plasma region has nothing to do with the corona current. If the electric field on the cathode is obtained, then there is no need to calculate Poisson s equation again within the plasma region. The electric field intensity on the conductor surface always keeps constant based on Kaptzov s hypothesis [14 16, 18, 20, 21]. So the electric field distribution in the plasma region will be obtained by (25) directly. Equation (25) is Laplace s equation, which is not associated with the space charge. Then the space charge continuity equations can be calculated straightforwardly. However, the conductor surface condition influences the onset electric field on the conductor surface. More accurate modeling based on nature condition is needed to replace Kaptzov s hypothesis. d ðreþ ¼0 rdr ð25þ The cathode secondary emission to sustain the negative corona discharge has close relationship with the plasma characteristics. When the charged particle density in the plasma region is obtained, Townsend secondary ionization coefficient can be calculated. The secondary electron may be produced by the bombardment of positive ions or the photoemission, but both Townsend secondary ionization coefficient due to positive ion impact or photoemission is difficult to measure accurately [24]. The effective secondary ionization coefficient obtained by different researchers always varies over several orders. For all cases in this paper, n e /n p and I e /I p at the cathode are and 0.126, respectively. If cathode secondary emission is caused by positive ion impact, the Townsend secondary ionization coefficient due to positive ion impact is not sensitive to the gap ð23þ ð24þ

18 72 Plasma Chem Plasma Process (2010) 30:55 73 distance or the corona current. If cathode secondary emission is caused by photoemission, the plasma characteristics with different gap distances are all the same, Townsend secondary ionization coefficient due to photoemission is also the same. Townsend secondary ionization coefficient due to photoemission is also not sensitive to the corona current which is tested by Chen and Davidson [21]. Weather the secondary electron emission is caused by positive ion impact or photoemission, secondary ionization coefficient is constant in this paper, although it can not be considered directly in the model. When the voltage-current characteristics of the negative corona discharge are obtained, the cathode secondary emission can be investigated base on the plasma characteristics. Conclusion When the conductor radius and the electric field on the cathode keep constant in negative direct current corona in air under the standard atmosphere condition, the ion flow effects on the corona plasma have been investigated by one dimensional static state hydrodynamic approach. The results show the following conclusions: 1. The space charges within the plasma region have no influence on the electric field distribution throughout the interelectrode gap, which is only governed by the ion flow. The gap distance has no influence on the corona current when the conductor surface condition keeps the same. The electric field distribution in the plasma region has nothing to do with the corona current within the practical range. The thickness of the plasma region only depends on the electric field distribution, which has no relation to the corona current. The suppression of the electric field is intensified and higher supplied voltage is needed with the increase of the gap distance or the corona current. 2. If the voltage-current characteristics of the negative corona discharge are measured, the ion flow region and the plasma region can be investigated individually. The electric field distribution within the plasma region and the corona current can be calculated by the ion flow governing equations firstly. Then the space charge density distribution within the plasma region can be calculated by the space charge continuity equations straightforwardly. 3. Based on Kaptzov s hypothesis, the plasma region can be investigated solely. The electric field distribution within the plasma region can be calculated by Laplace s equation firstly, then the space charge density distribution within the plasma region can be calculated by the space charge continuity equations directly. Acknowledgments The authors would like to thank the financial support from State Key Laboratory of Control and Simulation of Power System and Generation Equipment in China (Grants No. SKLD09M03), National Nature Science Foundation of China (Grants No and No ), and National Basic Research Program of China (973 Program) (Grant No. 2009CB724504). References 1. Oglesby S Jr, Nichols GB (1978) Electrostatic precipitation. Marcel Dekker, New York 2. Moore AD (1973) Electrostatics and its applications. Wiley, New York 3. Chang J, Lawless PA, Yamamoto T (1991) IEEE Trans Plasma Sci 19: Mok YS, Lee HW, Hyun YJ (2001) J Electrost 53: van Veldhuizen EM, Rutgers WR (2002) J Phys D Appl Phys 35:

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