- Email: [email protected]
Modeling of charge neutralization by ionizer
ARTICLE IN PRESS
Journal of Electrostatics 63 (2005) 767–773 www.elsevier.com/locate/elstat
Modeling of charge neutralization by ionizer A. Ohsawa Physical Engineering Safety Research Group, National Institute of Industrial Safety, 1-4-6 Umezono, Kiyose, Tokyo 204-0024, Japan Available online 23 March 2005
Abstract This paper discusses the effect of the relationship between the discharge frequency and background airflow velocity of ac corona ionizers on charge neutralization investigated by a computer model. The one-dimensional fluid model consists of a model for ion sources, the continuity equations for ions, Poisson’s equation for the electric field, and the circuit equation for solving the potential of the object to be neutralized. The model shows that the neutralization strongly depends on the relationship and demonstrates that at relatively high airflow velocities, the charges of objects cannot be neutralized. It is determined that the essentials for sufficient neutralization are that in the region of ion transport, the density distributions of positive and negative ions have no fluctuation and their total space charges become almost zero at a steady state. r 2005 Elsevier B.V. All rights reserved. Keywords: Fluid model; ac corona ionizer; Charge neutralization; Discharge frequency; Airflow rate
1. Introduction Charge neutralization is often required in industry for increasing production rates and eliminating electrostatic hazards. For neutralization, corona ionizers are widely used. Despite their wide use, the understanding of the phenomena of charge neutralization with ionizers strongly depends on empirical knowledge. Therefore, an understanding based on theoretical approaches will result in better neutralization. Tel.: +81 424 94 6232; fax: +81 424 91 7846.
E-mail address: [email protected]. 0304-3886/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2005.03.043
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
768
In this study, we have developed a one-dimensional fluid model for charge neutralization with ionizers and chosen ac corona ionizers as a target to be modeled. Since airflow is often used in ionizers, it is included in the present model to investigate in detail the effect of the relationship between the discharge frequency and airflow velocity on the neutralization. Furthermore, the effect of the object capacitance to ground, which may affect not only the charge decay time but also other neutralization phenomena, will be investigated.
2. Model The one-dimensional fluid model presented in this paper is substantially similar to the model based on the probe theory for plasma diagnostics [1]. Our model solves self-consistently the motions of positive and negative ions in the transport region toward a charged object and consists of the following four parts: (i) an ion source model based on the classical theory for corona discharges; (ii) solving Poisson’s equation to obtain the electric field; (iii) solving the continuity equations for positive and negative ions; (iv) solving a circuit equation to obtain the potential of the object to be neutralized.
2.1. Model of the ion source To model the ion source using a corona discharge, we employ the following arguments regarding charge carriers in corona discharges by Raizer [2]: (I) charge carriers are produced only in the vicinity of a sharp corona electrode by a strong field there, and the field in the remaining region between electrodes is very weak; (II) no ionization, therefore, occurs in the region far from the corona discharge in the vicinity of the sharp electrode; (III) the carriers are positive ions in positive coronas and negative ions in negative coronas (or electrons in gases without electronegative components). The model of the ion sources gives the density, n, of positive or negative ions emitted from an ionizer and is given as the function of the applied voltage between electrodes of a corona ionizer. The model is on the basis of the derivation of the corona current [2]. For convenience, a coaxial cylinder geometry having radii r0 and R ð4r0 Þ is employed to obtain the density because an analytical field equation is necessary for obtaining the inception voltages for positive and negative coronas, V c ; as well as the corona current, i, in our model. To obtain the density, we use Peek’s empirical formulae for the corona inception fields surrounding the inner electrode in the ambient air [3], E c ; ( Ec ¼
pffiffiffiffi 33:7 105 ð1 þ 0:0241= r0 Þ pffiffiffiffi 31:0 105 ð1 þ 0:0308= r0 Þ
for positive corona; for negative corona
(1)
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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and the corona current per unit length of the cylinder [2], i¼
8pme0 V ðV V c Þ, R lnðR=r0 Þ
(2)
2
where m is the mobility of positive or negative ions, e0 is the permittivity in free space, and V is the voltage between the cylinders, respectively. Since the analytical field at a distance r from the axis, EðrÞ; is EðrÞ ¼ V =½r lnðR=r0 Þ ,
(3)
letting r ¼ r0 ; the inception voltage is expressed by V c ¼ E c r0 lnðR=r0 Þ.
(4)
Assuming that the distortion of the field distribution of the ion space charge is not too large, the corona current of Eq. (2) is equal to i ¼ 2pRenmEðRÞ; where e is the electron charge. Hence, the density is expressed by 8 < 4e0 jV V j; jV j4jV j; c c n ¼ eR2 (5) : 0; jV jpjV j: c
We assume that the density of positive or negative ions coming out of an ionizer having any electrode geometry (e.g., point-to-plane) is equivalent to the density obtained above. Since ac corona ionizers are investigated in this paper, we use V ¼ V a sinð2pftÞ: Then, at V 40; n corresponds to the density of positive ions and, at V o0; n corresponds to that of negative ions. 2.2. Fluid model As described in Section 2.1, electrons produced in the vicinity of a corona needle tip may be ignored in the transport region between an ionizer and an object because of a high attachment rate at atmospheric pressure in air involving electronegative gases, e.g., oxygen. In addition, for ions in the transport region, no production is assumed but the loss by ion–ion recombination is considered. The equations to be solved in the fluid model are the continuity equations for positive and negative ions coupled with Poisson’s equation for the electric field, qnp þ r ðnp vp Þ Dp r2 np ¼ bnp nn , qt
(6)
qnn þ r ðnn vn Þ Dn r2 nn ¼ bnp nn , qt
(7)
r2 f ¼ eðnp nn Þ=e0 ,
(8)
where np and nn are the positive and negative ion densities, vp ¼ wp þ va ; vn ¼ wn þ va ; wp ; wn are the corresponding drift velocities, and va is the airflow velocity, respectively. The symbols D and b denote the diffusion and ion–ion recombination coefficients, respectively, and f is the potential. We use the mobilities for the positive
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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and negative ions and the recombination coefficient in air given by Morrow and Lowke [4] and the diffusion coefficients in [5]. In this version of the one-dimensional model, the airflow velocity is assumed to be constant throughout the transport region to eliminate the calculation regarding the airflow. To solve Eqs. (6) and (7), the flux-corrected transport scheme [6,7] is used, and the Thomas tridiagonal algorithm is used to solve Eq. (8). For the computation, a uniform grid separation, dx; in the direction of x from an ac ionizer to a charged object is used, and is 1 1 determined by dx ¼ minð104 ; dx1 ; dx2 Þ; where dx1 p 10 va =ð2f Þ and dx2 p 10 jvd j=f ; respectively, and va =ð2f Þ is the interval between air-conveying positive and negative ions alternatively emitted from the ion source, and vd is the faster drift velocity among positive and negative ions in the Laplacian field with the initial potential of a charged object. The time step, dt; is determined by dtp minðdtcfl ; dtf Þ; where dtcfl is the minimum time step obtained from the Courant–Friedrichs–Lewy condition for 1 all grid points and dtf ¼ 100 f 1 : 2.3. Boundary conditions and circuit equation Positive or negative ions having the density obtained from Eq. (5) time by time are set at x ¼ 0; and a conductive object with a capacitance to ground, C, and the initial potential of fL0 is placed at x ¼ L: We assumed that f ¼ 0 at x ¼ 0 because the body of most ac ionizers is grounded. The charges of the ions reached at the object are assumed to be absorbed there, then, np ¼ 0 and nn ¼ 0 at x ¼ L: To obtain the potential of the object and its decay, we apply a simultaneous circuit equation based on current continuity at the boundary of the object surface shown below. The total current in the transport region toward the object, it ; is obtained from the following equation considering the displacement current as well [8]: Z Z eA L eA L it ¼ ðnp up nn un Þ El dx ¼ ðnp up nn un Þ el dx, (9) fL 0 L 0 where A is the area of the object, np up and nn un are the corresponding fluxes of positive and negative ions, El is the Laplacian field, and el is the unit vector of the Laplacian field. In the one-dimensional coordinates, the current, it ; is equal to the neutralization current to the object, in ; in ¼ C
dfL . dt
(10)
3. Results and discussion In this paper, the model uses L ¼ 0:1 m; A ¼ 0:01 m2 ; fL0 ¼ 1 kV; and C ¼ 20 pF usually used in charged plate monitors; in addition, we use C ¼ e0 A=L ð 0:885 pFÞ as a representative value for ESD-sensitive devices. For the ion source, r0 ¼ 0:1 mm; R ¼ 1 cm; and V a ¼ 7 kV are used. The computation is repeated until the neutralization current reaches a steady state.
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
771
Fig. 1 shows the time evolution of the distributions for positive and negative ion densities at f ¼ 1 kHz; va ¼ 1 m=s; and C ¼ 0:885 pF: The decay that corresponds to this evolution is curve 2 in Fig. 3. Both the positive and negative ions push toward the object by the airflow, besides the fraction of the negative ions moves ahead because of the external field created by the potential of the object positively charged, as shown in Figs. 1a and b. Finally, as shown in Fig. 1c, the space charge created by both ions becomes quasi-neutral in most of the ion transport region at the steady state and also the fluctuation of the ion densities disappears. On the other hand, at f ¼ 50 Hz and va ¼ 1 m=s; a large fluctuation in the distributions appears, as shown in Fig. 2a. However, at a slower airflow velocity (va ¼ 0:1 m=s), the fluctuation of both ion densities disappears and the space charges by them are quasi-neutralized again, as shown in Fig. 2b. This is because the interval between positive and negative ions pushed out by the slower airflow becomes shorter. The quasi-neutralization and density fluctuation are properly caused by the drift and diffusion of ions, and the drift is predominant because the simulation with Dn ¼ Dp ¼ 0; at f ¼ 1 kHz and va ¼ 1 m=s has demonstrated it. The decrease in the ion densities with increasing distance x is caused by the ion–ion recombination because the simulation with b ¼ 0 has shown that the distributions become uniform in the transport region. The charge decays of the object investigated are shown in Fig. 3. The decay of curve 1 for f ¼ 50 Hz and C ¼ 0:885 pF is very different from the others in terms of the fluctuation of the offset voltage ( 20 V in amplitude). This is caused by the 4x1014
3 2 1 0
(a)
4x1014
t = 55.7 ms φL = 99.99 V Positive ion Negative ion
3
Ion density (m-3)
t = 25.5 ms φL= 521.5 V Positive ion Negative ion
Ion density (m-3)
Ion density (m-3)
4x1014
2 1 0
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
(b)
t = 0.113 s φL = -64 mV Positive ion Negative ion
3 2 1 0
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
(c)
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
t = 0.22 s Positive ion Negative ion
6x1013 5 4 3 2 1 0
t = 8.34 s Positive ion Negative ion
4x1013 3 2 1 0
0.0
(a)
Ion density (m-3)
Ion density (m-3)
Fig. 1. Time evolution of the density distributions of positive and negative ions at f ¼ 1 kHz; va ¼ 1 m=s; and C ¼ 0:885 pF: Note that the maximum of vertical axes is limited by 4 1014 m3 :
0.2
0.4 0.6 x (m)
0.8 1.0x10-1
0.0
(b)
0.2
0.4 0.6 x (m)
0.8 1.0x10-1
Fig. 2. Density distributions of positive and negative ions at the steady state for f ¼ 50 Hz; (a) va ¼ 1 m=s and (b) va ¼ 0:1 m=s:
ARTICLE IN PRESS
Potential of object (V)
1000 800 600 1
400 200
(a)
0.885 pF 1: 50 Hz 2: 1 kHz
2
0 0.00 0.05 0.10 0.15 0.20 Time (s)
Potential of object (V)
A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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1000 800 600 3
400
4
200 0 0.0
(b)
20 pF 3: 50 Hz 4: 1 kHz
0.5
1.0 1.5 2.0 Time (s)
2.5
Fig. 3. Charge decays of the object at va ¼ 1 m=s for different frequencies and capacitances.
fluctuation of the ion density distributions shown in Fig. 2a. However, when the object capacitance is large, the fluctuation is absorbed by it, as shown in curve 3 in Fig. 3b, while almost the same fluctuation exists in the current. This situation can be explained by Eq. (10) because when it has the same fluctuation, dfL =dt becomes large at smaller capacitances. This indicates that neutralization at low frequencies and high airflow rates may be insufficient for objects with low capacitances to ground, e.g., ESD-sensitive devices. This situation has practically been observed at very low discharge frequencies or high airflow rates [9]. As a result, the model shows that, when the steady-state density distributions of positive and negative ions have no fluctuation and their space charges are quasi-neutralized, as shown in Figs. 1c and 2b, the charges of the objects can be sufficiently neutralized, resulting in very small fluctuations in offset voltages. It is of interest that the displacement current by the ions approaching the object (Fig. 1a) reduces a large portion of the object potential (induced voltage) at the low capacitance of 0.885 pF, even though no true charges by the ions from the ionizer exist on the object. However, at a large capacitance, 20 pF, the potential decays caused by the induced voltage are small. For example, the decays until the ions reach it are 628.3 V at C ¼ 0:885 pF (curve 2) and 42.3 V at C ¼ 20 pF (curve 4). Further, a slow decay is initially observed in curve 1. Since the period of the slow decay corresponds to the first one-cycle of the discharge frequency, this is represented by the motion of ions in the period. However, in practical tests using a charged plate monitor, such a decay may not be observed because the tests are performed under the continuous operation of ionizers, but it is not given initially in this model. The model will be modified in this regard in future to discuss the practical decay time.
4. Conclusions The effect of the relationship between the discharge frequency and background airflow rate of ac corona ionizers on the charge neutralization is investigated by the one-dimensional fluid model developed here. The model shows that the neutralization strongly depends on this relationship and demonstrates that air-blowing ac ionizers with relatively high airflow rates or low discharge frequencies cannot
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
773
sufficiently neutralize the charges of objects, in particular, of objects with a low capacitance to ground. For sufficient neutralization with ac ionizers, it was determined to be important that the steady-state density distributions of positive and negative ions have no fluctuation and the space charges created by them be quasi-neutralized in the region of ion transport. References [1] J.-S. Chang, IEEE Trans. Ind. Appl. 37 (6) (2001) 1641–1645. [2] Y.P. Raizer, Gas Discharge Physics, Springer, Berlin, 1991. [3] K. Asano in: Institute of Electrostatics Japan (Ed.), Electrostatics Handbook, Ohmsha, Tokyo, 1998 (in Japanese). [4] R. Morrow, J.J. Lowke, J. Phys. D 30 (1997) 614–627. [5] K. Honda, Gas Discharge Phenomena, Tokyo Denki University Press, Tokyo, 1988 (in Japanese). [6] S.T. Zalesak, J. Comput. Phys. 31 (1979) 335–362. [7] E.E. Kundhardt, C. Wu, J. Comput. Phys. 68 (1987) 127–150. [8] N. Sato, J. Phys. D 13 (1980) L3–L6. [9] K. Izumi, T. Suzuki, T. Takahashi, private communications.
Journal of Electrostatics 63 (2005) 767–773 www.elsevier.com/locate/elstat
Modeling of charge neutralization by ionizer A. Ohsawa Physical Engineering Safety Research Group, National Institute of Industrial Safety, 1-4-6 Umezono, Kiyose, Tokyo 204-0024, Japan Available online 23 March 2005
Abstract This paper discusses the effect of the relationship between the discharge frequency and background airflow velocity of ac corona ionizers on charge neutralization investigated by a computer model. The one-dimensional fluid model consists of a model for ion sources, the continuity equations for ions, Poisson’s equation for the electric field, and the circuit equation for solving the potential of the object to be neutralized. The model shows that the neutralization strongly depends on the relationship and demonstrates that at relatively high airflow velocities, the charges of objects cannot be neutralized. It is determined that the essentials for sufficient neutralization are that in the region of ion transport, the density distributions of positive and negative ions have no fluctuation and their total space charges become almost zero at a steady state. r 2005 Elsevier B.V. All rights reserved. Keywords: Fluid model; ac corona ionizer; Charge neutralization; Discharge frequency; Airflow rate
1. Introduction Charge neutralization is often required in industry for increasing production rates and eliminating electrostatic hazards. For neutralization, corona ionizers are widely used. Despite their wide use, the understanding of the phenomena of charge neutralization with ionizers strongly depends on empirical knowledge. Therefore, an understanding based on theoretical approaches will result in better neutralization. Tel.: +81 424 94 6232; fax: +81 424 91 7846.
E-mail address: [email protected]. 0304-3886/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2005.03.043
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
768
In this study, we have developed a one-dimensional fluid model for charge neutralization with ionizers and chosen ac corona ionizers as a target to be modeled. Since airflow is often used in ionizers, it is included in the present model to investigate in detail the effect of the relationship between the discharge frequency and airflow velocity on the neutralization. Furthermore, the effect of the object capacitance to ground, which may affect not only the charge decay time but also other neutralization phenomena, will be investigated.
2. Model The one-dimensional fluid model presented in this paper is substantially similar to the model based on the probe theory for plasma diagnostics [1]. Our model solves self-consistently the motions of positive and negative ions in the transport region toward a charged object and consists of the following four parts: (i) an ion source model based on the classical theory for corona discharges; (ii) solving Poisson’s equation to obtain the electric field; (iii) solving the continuity equations for positive and negative ions; (iv) solving a circuit equation to obtain the potential of the object to be neutralized.
2.1. Model of the ion source To model the ion source using a corona discharge, we employ the following arguments regarding charge carriers in corona discharges by Raizer [2]: (I) charge carriers are produced only in the vicinity of a sharp corona electrode by a strong field there, and the field in the remaining region between electrodes is very weak; (II) no ionization, therefore, occurs in the region far from the corona discharge in the vicinity of the sharp electrode; (III) the carriers are positive ions in positive coronas and negative ions in negative coronas (or electrons in gases without electronegative components). The model of the ion sources gives the density, n, of positive or negative ions emitted from an ionizer and is given as the function of the applied voltage between electrodes of a corona ionizer. The model is on the basis of the derivation of the corona current [2]. For convenience, a coaxial cylinder geometry having radii r0 and R ð4r0 Þ is employed to obtain the density because an analytical field equation is necessary for obtaining the inception voltages for positive and negative coronas, V c ; as well as the corona current, i, in our model. To obtain the density, we use Peek’s empirical formulae for the corona inception fields surrounding the inner electrode in the ambient air [3], E c ; ( Ec ¼
pffiffiffiffi 33:7 105 ð1 þ 0:0241= r0 Þ pffiffiffiffi 31:0 105 ð1 þ 0:0308= r0 Þ
for positive corona; for negative corona
(1)
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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and the corona current per unit length of the cylinder [2], i¼
8pme0 V ðV V c Þ, R lnðR=r0 Þ
(2)
2
where m is the mobility of positive or negative ions, e0 is the permittivity in free space, and V is the voltage between the cylinders, respectively. Since the analytical field at a distance r from the axis, EðrÞ; is EðrÞ ¼ V =½r lnðR=r0 Þ ,
(3)
letting r ¼ r0 ; the inception voltage is expressed by V c ¼ E c r0 lnðR=r0 Þ.
(4)
Assuming that the distortion of the field distribution of the ion space charge is not too large, the corona current of Eq. (2) is equal to i ¼ 2pRenmEðRÞ; where e is the electron charge. Hence, the density is expressed by 8 < 4e0 jV V j; jV j4jV j; c c n ¼ eR2 (5) : 0; jV jpjV j: c
We assume that the density of positive or negative ions coming out of an ionizer having any electrode geometry (e.g., point-to-plane) is equivalent to the density obtained above. Since ac corona ionizers are investigated in this paper, we use V ¼ V a sinð2pftÞ: Then, at V 40; n corresponds to the density of positive ions and, at V o0; n corresponds to that of negative ions. 2.2. Fluid model As described in Section 2.1, electrons produced in the vicinity of a corona needle tip may be ignored in the transport region between an ionizer and an object because of a high attachment rate at atmospheric pressure in air involving electronegative gases, e.g., oxygen. In addition, for ions in the transport region, no production is assumed but the loss by ion–ion recombination is considered. The equations to be solved in the fluid model are the continuity equations for positive and negative ions coupled with Poisson’s equation for the electric field, qnp þ r ðnp vp Þ Dp r2 np ¼ bnp nn , qt
(6)
qnn þ r ðnn vn Þ Dn r2 nn ¼ bnp nn , qt
(7)
r2 f ¼ eðnp nn Þ=e0 ,
(8)
where np and nn are the positive and negative ion densities, vp ¼ wp þ va ; vn ¼ wn þ va ; wp ; wn are the corresponding drift velocities, and va is the airflow velocity, respectively. The symbols D and b denote the diffusion and ion–ion recombination coefficients, respectively, and f is the potential. We use the mobilities for the positive
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
770
and negative ions and the recombination coefficient in air given by Morrow and Lowke [4] and the diffusion coefficients in [5]. In this version of the one-dimensional model, the airflow velocity is assumed to be constant throughout the transport region to eliminate the calculation regarding the airflow. To solve Eqs. (6) and (7), the flux-corrected transport scheme [6,7] is used, and the Thomas tridiagonal algorithm is used to solve Eq. (8). For the computation, a uniform grid separation, dx; in the direction of x from an ac ionizer to a charged object is used, and is 1 1 determined by dx ¼ minð104 ; dx1 ; dx2 Þ; where dx1 p 10 va =ð2f Þ and dx2 p 10 jvd j=f ; respectively, and va =ð2f Þ is the interval between air-conveying positive and negative ions alternatively emitted from the ion source, and vd is the faster drift velocity among positive and negative ions in the Laplacian field with the initial potential of a charged object. The time step, dt; is determined by dtp minðdtcfl ; dtf Þ; where dtcfl is the minimum time step obtained from the Courant–Friedrichs–Lewy condition for 1 all grid points and dtf ¼ 100 f 1 : 2.3. Boundary conditions and circuit equation Positive or negative ions having the density obtained from Eq. (5) time by time are set at x ¼ 0; and a conductive object with a capacitance to ground, C, and the initial potential of fL0 is placed at x ¼ L: We assumed that f ¼ 0 at x ¼ 0 because the body of most ac ionizers is grounded. The charges of the ions reached at the object are assumed to be absorbed there, then, np ¼ 0 and nn ¼ 0 at x ¼ L: To obtain the potential of the object and its decay, we apply a simultaneous circuit equation based on current continuity at the boundary of the object surface shown below. The total current in the transport region toward the object, it ; is obtained from the following equation considering the displacement current as well [8]: Z Z eA L eA L it ¼ ðnp up nn un Þ El dx ¼ ðnp up nn un Þ el dx, (9) fL 0 L 0 where A is the area of the object, np up and nn un are the corresponding fluxes of positive and negative ions, El is the Laplacian field, and el is the unit vector of the Laplacian field. In the one-dimensional coordinates, the current, it ; is equal to the neutralization current to the object, in ; in ¼ C
dfL . dt
(10)
3. Results and discussion In this paper, the model uses L ¼ 0:1 m; A ¼ 0:01 m2 ; fL0 ¼ 1 kV; and C ¼ 20 pF usually used in charged plate monitors; in addition, we use C ¼ e0 A=L ð 0:885 pFÞ as a representative value for ESD-sensitive devices. For the ion source, r0 ¼ 0:1 mm; R ¼ 1 cm; and V a ¼ 7 kV are used. The computation is repeated until the neutralization current reaches a steady state.
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
771
Fig. 1 shows the time evolution of the distributions for positive and negative ion densities at f ¼ 1 kHz; va ¼ 1 m=s; and C ¼ 0:885 pF: The decay that corresponds to this evolution is curve 2 in Fig. 3. Both the positive and negative ions push toward the object by the airflow, besides the fraction of the negative ions moves ahead because of the external field created by the potential of the object positively charged, as shown in Figs. 1a and b. Finally, as shown in Fig. 1c, the space charge created by both ions becomes quasi-neutral in most of the ion transport region at the steady state and also the fluctuation of the ion densities disappears. On the other hand, at f ¼ 50 Hz and va ¼ 1 m=s; a large fluctuation in the distributions appears, as shown in Fig. 2a. However, at a slower airflow velocity (va ¼ 0:1 m=s), the fluctuation of both ion densities disappears and the space charges by them are quasi-neutralized again, as shown in Fig. 2b. This is because the interval between positive and negative ions pushed out by the slower airflow becomes shorter. The quasi-neutralization and density fluctuation are properly caused by the drift and diffusion of ions, and the drift is predominant because the simulation with Dn ¼ Dp ¼ 0; at f ¼ 1 kHz and va ¼ 1 m=s has demonstrated it. The decrease in the ion densities with increasing distance x is caused by the ion–ion recombination because the simulation with b ¼ 0 has shown that the distributions become uniform in the transport region. The charge decays of the object investigated are shown in Fig. 3. The decay of curve 1 for f ¼ 50 Hz and C ¼ 0:885 pF is very different from the others in terms of the fluctuation of the offset voltage ( 20 V in amplitude). This is caused by the 4x1014
3 2 1 0
(a)
4x1014
t = 55.7 ms φL = 99.99 V Positive ion Negative ion
3
Ion density (m-3)
t = 25.5 ms φL= 521.5 V Positive ion Negative ion
Ion density (m-3)
Ion density (m-3)
4x1014
2 1 0
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
(b)
t = 0.113 s φL = -64 mV Positive ion Negative ion
3 2 1 0
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
(c)
0.0 0.2 0.4 0.6 0.81.0x10-1 x (m)
t = 0.22 s Positive ion Negative ion
6x1013 5 4 3 2 1 0
t = 8.34 s Positive ion Negative ion
4x1013 3 2 1 0
0.0
(a)
Ion density (m-3)
Ion density (m-3)
Fig. 1. Time evolution of the density distributions of positive and negative ions at f ¼ 1 kHz; va ¼ 1 m=s; and C ¼ 0:885 pF: Note that the maximum of vertical axes is limited by 4 1014 m3 :
0.2
0.4 0.6 x (m)
0.8 1.0x10-1
0.0
(b)
0.2
0.4 0.6 x (m)
0.8 1.0x10-1
Fig. 2. Density distributions of positive and negative ions at the steady state for f ¼ 50 Hz; (a) va ¼ 1 m=s and (b) va ¼ 0:1 m=s:
ARTICLE IN PRESS
Potential of object (V)
1000 800 600 1
400 200
(a)
0.885 pF 1: 50 Hz 2: 1 kHz
2
0 0.00 0.05 0.10 0.15 0.20 Time (s)
Potential of object (V)
A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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1000 800 600 3
400
4
200 0 0.0
(b)
20 pF 3: 50 Hz 4: 1 kHz
0.5
1.0 1.5 2.0 Time (s)
2.5
Fig. 3. Charge decays of the object at va ¼ 1 m=s for different frequencies and capacitances.
fluctuation of the ion density distributions shown in Fig. 2a. However, when the object capacitance is large, the fluctuation is absorbed by it, as shown in curve 3 in Fig. 3b, while almost the same fluctuation exists in the current. This situation can be explained by Eq. (10) because when it has the same fluctuation, dfL =dt becomes large at smaller capacitances. This indicates that neutralization at low frequencies and high airflow rates may be insufficient for objects with low capacitances to ground, e.g., ESD-sensitive devices. This situation has practically been observed at very low discharge frequencies or high airflow rates [9]. As a result, the model shows that, when the steady-state density distributions of positive and negative ions have no fluctuation and their space charges are quasi-neutralized, as shown in Figs. 1c and 2b, the charges of the objects can be sufficiently neutralized, resulting in very small fluctuations in offset voltages. It is of interest that the displacement current by the ions approaching the object (Fig. 1a) reduces a large portion of the object potential (induced voltage) at the low capacitance of 0.885 pF, even though no true charges by the ions from the ionizer exist on the object. However, at a large capacitance, 20 pF, the potential decays caused by the induced voltage are small. For example, the decays until the ions reach it are 628.3 V at C ¼ 0:885 pF (curve 2) and 42.3 V at C ¼ 20 pF (curve 4). Further, a slow decay is initially observed in curve 1. Since the period of the slow decay corresponds to the first one-cycle of the discharge frequency, this is represented by the motion of ions in the period. However, in practical tests using a charged plate monitor, such a decay may not be observed because the tests are performed under the continuous operation of ionizers, but it is not given initially in this model. The model will be modified in this regard in future to discuss the practical decay time.
4. Conclusions The effect of the relationship between the discharge frequency and background airflow rate of ac corona ionizers on the charge neutralization is investigated by the one-dimensional fluid model developed here. The model shows that the neutralization strongly depends on this relationship and demonstrates that air-blowing ac ionizers with relatively high airflow rates or low discharge frequencies cannot
ARTICLE IN PRESS A. Ohsawa / Journal of Electrostatics 63 (2005) 767–773
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sufficiently neutralize the charges of objects, in particular, of objects with a low capacitance to ground. For sufficient neutralization with ac ionizers, it was determined to be important that the steady-state density distributions of positive and negative ions have no fluctuation and the space charges created by them be quasi-neutralized in the region of ion transport. References [1] J.-S. Chang, IEEE Trans. Ind. Appl. 37 (6) (2001) 1641–1645. [2] Y.P. Raizer, Gas Discharge Physics, Springer, Berlin, 1991. [3] K. Asano in: Institute of Electrostatics Japan (Ed.), Electrostatics Handbook, Ohmsha, Tokyo, 1998 (in Japanese). [4] R. Morrow, J.J. Lowke, J. Phys. D 30 (1997) 614–627. [5] K. Honda, Gas Discharge Phenomena, Tokyo Denki University Press, Tokyo, 1988 (in Japanese). [6] S.T. Zalesak, J. Comput. Phys. 31 (1979) 335–362. [7] E.E. Kundhardt, C. Wu, J. Comput. Phys. 68 (1987) 127–150. [8] N. Sato, J. Phys. D 13 (1980) L3–L6. [9] K. Izumi, T. Suzuki, T. Takahashi, private communications.