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Charging kinetics of spherical dielectric particles in a unipolar corona field
ARTICLE IN PRESS
Journal of Electrostatics 63 (2005) 1095–1103 www.elsevier.com/locate/elstat
Charging kinetics of spherical dielectric particles in a unipolar corona field H.Z. Alisoya,, G.T. Alisoyb, M. Koseoglua a
Department of Electric and Electronics Engineering, Inonu University, 44280 Malatya, Turkey b Department of Mathematical Sciences, Inonu University, 44280 Malatya, Turkey
Received 17 February 2004; received in revised form 26 September 2004; accepted 19 February 2005 Available online 22 March 2005
Abstract In this paper, we investigate the charging of spherical dielectric particles in an electric field and unipolar volume charge. A differential equation expressing the charging process is derived and solved numerically using a modified Euler method. The results obtained allow one to determine the charge on the particle for arbitrary values of the particle’s radius and permittivity as well as the electric field intensity and surrounding volume charge density. These parameters all influence the charging process considerably. We demonstrate that a stationary particle acquires less charge than a rotating particle. r 2005 Elsevier B.V. All rights reserved. Keywords: Charging kinetics; Unipolar corona field; Spherical dielectric particles; Legendre polynomials
1. Introduction The charging kinetics of conductive particles having radii larger than about 3 microns were examined previously by Pauthenier and Moreau-Hanot [1] for the case of a unipolar corona field. In a similar paper by Zevenhoven [4], the effects of Brownian motion are neglected, while the ions are assumed to be driven towards the particle by an external electric field. In such a situation, the net electric field intensity Corresponding author.
E-mail address: [email protected] (H.Z. Alisoy). 0304-3886/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2005.02.003
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consists of the superposition of the external electric field E0 and the field created by the charge on the particle. The distribution of charge that is precipitated on a conductive sphere is uniform. In the case of an insulating particle, the charge density need not be uniformly distributed over the surface. On the other hand, a dielectric particle may become uniformly charged for any one of the following reasons: 1. The particle may have finite volume conductivity, in which case the surface distribution of charge will be uniform after a time on the order of t1 0 r1 v [2,3]. In this expression, is relative dielectric constant of the particle, 0 is dielectric constant of vacuum, and rv is the volume conductivity of the particle. 2. The charges may be distributed over the particle surface due to surface conductivity. The time constant of this mechanism is t2 0 rr1 [3–7], where r s is the radius of the particle and rs is its surface conductivity. 3. During the time that an insulating particle is charged, rotational particle motion may occur, thereby allowing, the charge on the surface of the particle to attain a uniform distribution over the time scale of t3 ðr=E 0 Þðr=0 Þ1=2 [7] where E0 is electric field intensity, and r is the density of the particle. If the dielectric particle resides in charging region for less than the smallest of the time constants mentioned above, then the charging process of the particle will have a specific character often encountered in electrostatic precipitators. For this reason, the main purpose in this work is to calculate the charging dynamics of the particle and the limit value of charge on the particle in such situations. From the numerical results obtained, we demonstrate that the charge of a stationary particle is smaller than the limit charge value for a rotating particle, even under the conditions (for example occurrence of tangential fields resulting from the non-uniform distribution of the charges and moisture) influencing the charging process in a negative manner.
2. Theory Let us assume that the residence time of the dielectric particle in the charging region is smaller than the time constants t1 ; t2 and t3 : In this case, we suppose that the calculation of the charging kinetics and limit charge value on the particle will be interesting and contribute to analysis of charging process. The charging of the spherical particle in a unipolar corona field may be expressed by the following equation: ( rn mE n ðy; tÞ if E n 40; qsðy; tÞ ¼ (1) 0 if E n o0: qt Here sðy; tÞ is the density of charges precipitated from the charging region onto the surface of the spherical particle in the period of t; E n ðy; tÞ is normal inward component of the total electric field intensity on the exterior surface of the particle,
ARTICLE IN PRESS H.Z. Alisoy et al. / Journal of Electrostatics 63 (2005) 1095–1103
q
(1,1)
1097
(,)
l
R
R
E0
E0 (a)
(b)
Fig. 1. (a) Spherical dielectric particle in the field of a point charge q. (b) Particle geometry and coordinates when the charge was located on the surface of the particle.
rv is volume charge density of the medium surrounding the particle, m is the mobility of the corona ions, and y is the angle relative to the polar axis. The polar axis is defined as being opposite to the electric field direction, as illustrated in Fig 1(a). The electric field intensity E n ðy; tÞ in Eq. (1) consists of two components. E n ðy; tÞ ¼ E n1 þ E n2:
(2)
The component E n1 is the electric field intensity caused by the uniform external electric field E 0 : For a dielectric particle, E n1 can be found from the following expression [2,3]: E n1 ¼
31 E 0 cos y, 1 þ 22
(3)
where 1 and 2 are the dielectric constants of the dielectric particle and the surrounding medium, respectively. The second component En2 is the electric field intensity produced by the charges that have precipitated on the surface of the particle from the discharge region. In order to determine this field component, En2, we utilize the electric field intensity, E 0n2 ; which was created on the surface of dielectric spherical particle by the point charge q located at the distance of ‘ from the centre of the spherical particle. This electric field intensity, E 0n2 ; is expressed as [2,3,7] E 0n2 ¼
1 X q1 nð2n þ 1Þ hn Pn ðcos yÞ, 2 4p0 2 ‘R n¼0 nð1 þ lÞ þ 1
(4)
where R is the radius of the particle, l ¼ 1 1 2 ; Pn are Legendre polynomials and h ¼ R ‘1 : In order to remove the divergence of the series in Eq. (4) while h-1 (this expresses physically that the point charge q comes around the surface of the spherical particle), the following expression is used to evaluate the generating functions of Legendre polynomials: 1 X n¼0
hn Pn ðxÞ ¼ ð1 2xh þ h2 Þ1=2 .
(5)
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By differentiating and integrating Eq. (5), the following equations are, respectively, obtained [8–10]: 1 X
nhn1 Pn ðxÞ ¼ ðx hÞð1 2xh þ h2 Þ3=2
(6a)
n¼0
and 1 X n¼0
1 nþ1
ðn þ 1Þ h
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2xh þ h2 þ h x : Pn ðxÞ ¼ ln 1x
(6b)
We can write the series included in Eq. (4) by utilizing Eqs. (5) and (6) as follows: " # 1=2 1 ha3 ka4 X Pn ðxÞhn 3=2 S ¼ a1 a2 a3 þ . (7) khð1 kÞ ðn þ 1Þðn þ kÞ n¼0 In this equation, k ¼ 1 þ l1 ; x ¼ cos y and 2khðx hÞ a2 ¼ ; a pffiffiffiffiffi 1 a3 þ h x a4 ¼ ln . 1x
a1 ¼ k2 ð1 kÞð1 2kÞ; a3 ¼ 1 2xh þ h2 ;
In the following steps, we must know the normal component of electric field intensity on the surface of the spherical particle which is created by the charge located on the surface of the particle at ‘ ¼ R and y ¼ 0: So, we must accept h ¼ 1; namely the charge is on the surface of the spherical particle. In order to improve the convergence of the series given in Eq. (7), we utilize the following expansion [10]. 1 X Pn ðcos yÞ y ¼ 1 2 ln 1 þ sin : (8) nðn þ 1Þ 2 n¼1 Assuming that h ¼ 1 in Eq. (7), and rearranging Eq. (7) to remove the divergence of the series by using Eq. (8), we obtain the following equation: " # 1 X Pn ðcos yÞ 1 , (9) S ¼ b1 0:5b2 þ b3 b4 þ b5 b6 nðn þ 1Þðn þ kÞ n¼1 where b1 ¼ 2k2 ; b2 ¼ sinðy=2Þ; b3 ¼ ðk 0:5Þ ln ðb2 Þ; b4 ¼ 2ð0:5 kÞ2 ln ð1 þ b2 Þ; b5 ¼ k1 ð0:5 kÞð1 k2 Þ and b6 ¼ kð0:5 kÞð1 kÞ: Thus, the normal component of the electric field intensity occurred by the point charge q on the spherical particle may be expressed as q E 0n2 ð‘; R; yÞ ‘¼R ¼ cðyÞ, (10) 2 4p y¼0 0R 0 where cðyÞ ¼ 1 2 2 S: In this case, the expression obtained for E n2 ðR; R; yÞ becomes a Green function for the examined problem. In other words, the expression of E 0n2 ðR; R; yÞ allow us to determine the normal component of electric field intensity on
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any point of the surface by utilizing the charges precipitating on the other points of the surface of the spherical particle. If the arbitrary distribution of the charge over the surface of the spherical particle is sðy; jÞ; then the normal component of the electric field intensity E n2 can be expressed as (see Fig. 1(b)) Z 2p Z p=2 1 sðy; jÞ , (11) dj1 dy1 sin y1 sðy1 ; j1 ÞcðaÞ E n2 ðy; jÞ ¼ 4p0 0 ð1 þ 2 Þ 0 0 where a is the angle between the directions ðy; jÞ and ðy1 ; j1 Þ as expressed by cos a ¼ cos y cos y1 þ sin y sin y1 cosðj j1 Þ.
(12)
We can use Eqs. (3) and (11) to simplify Eq. (2). For the case E n 40; the charging expressed by Eq. (1) can be rearranged, and after some simple conversions, we can arrive at the following equation: qyðx; tÞ 3l 1 ¼ x y qt lþ2 1þl ( Z 1 l x dx1 yðx1 ; tÞ pffiffiffiffiffi KðxÞ þ c3 c4 2 c2 pð1 þ lÞ 0 " Z p l1 l1 a ln 1 þ sin dj1 2ðl þ 1Þ 0 lþ1 2 #) 1 l X ðn þ 1Þ1 þ Pn ðcos aÞ , ð13Þ 2 l þ 1 n¼1 n ðl þ 1Þ þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where t ¼ rn mt=ð0 2 Þ; y ¼ s=ð0 2 E 0 Þ and x ¼ 2c1 ðc1 þ c2 Þ1 ; KðxÞ is first type of elliptic integral; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðl 1Þ lðl þ 2Þ þ ln 2 ; c1 ¼ ð1 x2 Þð1 x21 Þ; c2 ¼ 1 xx1 ; c3 ¼ 2ðl þ 1Þ l þ 1 pðl 1Þ lnðc2 þ jx x1 jÞ. c4 ¼ 4ðl þ 1Þ 3. Results and discussion In order to facilitate the calculation of the double integral in Eq. (13), following approach is used as an alternative to computing the term given in square brackets in Eq. (13): F ðcos aÞ ¼ d 1 cos a þ d 2
d3 . d 4 cos a
(14)
This approach allows us to calculate the expression given in curly brackets in Eq. (13) with an error of 0.2%. In this expression, the di coefficients are determined for different values of l:
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In this case, the integral calculation over j1 in Eq. (13) can be obtained as Z p Z p dj1 F ðcos aÞ ¼ dj1 F ðxx1 þ c2 cos jÞ 0 20 3 d3 6 7 ¼ p4xx1 d 1 þ d 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 2 ðd 1 xx1 Þ c2
ð15Þ
After these conversions, the solution of charging dynamic equation (13) becomes greatly simplified. As seen in Eq. (13), for the case x1 ¼ x; the integral function will be discontinuous. For this reason, while the integral is being calculated for the righthand side of Eq. (13), the integral interval from 0 to 1 must be chosen as follows: First, the integral is numerically calculated for the intervals of ð0; x oÞ and ðx þ o; 1Þ (here, o is an arbitrary chosen small number to begin numerical calculation). These calculated integrals are added together. Second, the sum of the integrals calculated for the intervals of (x o; x o=2) and (x þ o=2; x þ o) is obtained; this sum is added to the previous sum. This process is continued until 2% numeric error is reached. Eq. (13), which implements the initial condition (no precipitated charge) yðx; 0Þ ¼ 0; has been solved by using a modified Euler method [11]. It is difficult to use the Euler method to solve Eq. (13), because y depends on t and also on the parameter x. For this reason, the right-hand side of the charging equation is calculated, and yðx; tÞ is determined for definite values of x (the calculations are performed for 20 points). Values of yðx; tÞ for x values between the chosen 20 points are found by interpolation [11]. Thus, the charge of spherical particle at any instantaneous time t may be expressed as Z Z 1 2 Q ¼ s ds ¼ 2p0 2 E 0 R yðx; tÞ dx: (16) 0
From Eq. (16), the numerical results calculated for l ¼ 2; 3; 4; 5; and 7 are shown in Fig. 2. The characteristic limit charge density versus the parameter x for the different values of l is given in Fig. 3. The limit charge on the particle mentioned here is the instantaneous limit charge value according to the position of the particle. So, it depends on x. For different l values, the limit value of the uniform charge distribution on spherical dielectric particle is expressed as [4,12] Qlim ¼ 4p0 2
3l E 0 R2 . lþ2
(17)
The normalized charge ðQ=Qlim Þ numerically calculated by utilizing the Eqs. (16) and (17) is plotted versus l ¼ 1 1 2 in Fig. 4. As seen in Fig. 4, for l ¼ 2; Q=Qlim ¼ 0:35; and the normalized charge increases slowly by the increase of l: For l ¼ 7; the normalized charge is 0.51, and for l ! 1; Q=Qlim ! 1: Let us compare the numerical results obtained from Eqs. (16) and (17) with existing experimental results. The charging process of stationary and rotational
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15 =7
=5 =4
10 Q/ε0ε2E0R2
=3
=2 5
0
2
4
6
8
10
12
14
16
18
20
τ Fig. 2. Charging kinetics of particle versus time as calculated from Eq. (16).
quartz spheres ðl ¼ 3:75Þ of radii 3 to 10 mm in the electric field has been experimentally investigated in Ref. [13], and it has been observed that the limit charge values measured on stationary spheres are 0.63 times smaller than the limit charge values for rotating spheres. However, in accordance with the numeric results obtained by using Eqs. (16) and (17) (Fig. 4), we see that Q/Qlim is equal to 0.43 for the value of l ¼ 3:75: The difference between the experimental results and numerical results probably results from the variation of the surface resistivity of the quartz over a wide interval in accordance with the humidity rate. For example, if the surface resistivity of quartz is 109 O, the time constant associated with the migration of the charges over the surface of the particles will be comparable with their charging time. Additionally, as a result of the non-uniform distribution of charge over the surface of the particle, strong tangential fields will occur, and these fields will increase the surface conductivity considerably.
4. Conclusion A nonlinear differential equation, which allows us to calculate the limit charge value and charging process of spherical dielectric particle in unipolar corona field, is
ARTICLE IN PRESS H.Z. Alisoy et al. / Journal of Electrostatics 63 (2005) 1095–1103
1102
10 9 8 7
y
7 6
5
5
4
4
3
3 =2
2 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x = Cos Fig. 3. The variation of limit charge density over the surface of the particle versus x for different l values. (y is the limit charge density, x ¼ cos y; l ¼ 1 1 2 ).
0.52 0.5 0.48
Q / Qlim
0.46 0.44 0.42 0.4 0.38 0.36 0.34 2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
= 12-1 Fig. 4. The dependence of normalized charge (Q=Qlim ) on l:
7
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derived and solved by using a modified Euler method. Numerically calculated results allow one to determine the charge of the particle for arbitrary values of the particle radius, particle dielectric permittivity, electric field intensity, and volume charge density. All these parameters influence the charging process considerably. We have determined that the limit charge value on the particle increases with increasing dielectric constant of the particle as shown in Fig. 2. In accordance with the numerical results obtained, we have determined that the limit charge of a stationary particle during the charging process is smaller less than the limit charge of a rotating particle. References [1] M.M. Pauthenier, M. Moreau-Hanot, Charging of spherical particles in an ionizing field, J. Phys. Radium 7 (1932) 590–613. [2] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [3] W.R. Smythe, Static and Dynamic Electricity, second ed., McGraw-Hill, New York, 1950. [4] C.A.P. Zevenhoven, Uni-polar field charging of particles: effects of particle conductivity and rotation, J. Electrostat. 46 (1999) 1–12. [5] P.A. Lawless, Particle charging bounds symmetry relations and an analytic charging rate model for the continuum regime, J. Aerosol Sci. 27 (2) (1996) 191–215. [6] A.G. Bailey, The charging of insulator surfaces, J. Electrostat. 51–52 (2001) 82–90. [7] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Course of Theoretical Physics, vol. 8, Pergamon Press, New York, 1982. [8] H.B. Dwight, Tables of Integrals and Other Math Data, Macmillan, New York, 1961. [9] A.P. Prudnikov, Yu.A. Britchkov, O.I. Marichev, Integrals and Series, Nauka, Moscow, 1981 (in Russian). [10] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed., Academic Press, New York, 2000. [11] D.M. Etter, Engineering Problem Solving with MATLAB, Prentice-Hall, Englewood Cliffs, NJ, 1993. [12] H.Z. Alisoy, G.T. Alisoy, S.E. Hamamci, M. Koseoglu, Combined kinetic charging of particles on the precipitating electrode in a corona field, J. Phys. D: Appl. Phys. 37 (2004) 1459–1466. [13] V.Y. Steinschreiber, Calculation of forces influencing the aerosol particles in electric field, Ph.D. Thesis, Physics Institute of Science Academy, Baku, 1972 (in Russian).
Journal of Electrostatics 63 (2005) 1095–1103 www.elsevier.com/locate/elstat
Charging kinetics of spherical dielectric particles in a unipolar corona field H.Z. Alisoya,, G.T. Alisoyb, M. Koseoglua a
Department of Electric and Electronics Engineering, Inonu University, 44280 Malatya, Turkey b Department of Mathematical Sciences, Inonu University, 44280 Malatya, Turkey
Received 17 February 2004; received in revised form 26 September 2004; accepted 19 February 2005 Available online 22 March 2005
Abstract In this paper, we investigate the charging of spherical dielectric particles in an electric field and unipolar volume charge. A differential equation expressing the charging process is derived and solved numerically using a modified Euler method. The results obtained allow one to determine the charge on the particle for arbitrary values of the particle’s radius and permittivity as well as the electric field intensity and surrounding volume charge density. These parameters all influence the charging process considerably. We demonstrate that a stationary particle acquires less charge than a rotating particle. r 2005 Elsevier B.V. All rights reserved. Keywords: Charging kinetics; Unipolar corona field; Spherical dielectric particles; Legendre polynomials
1. Introduction The charging kinetics of conductive particles having radii larger than about 3 microns were examined previously by Pauthenier and Moreau-Hanot [1] for the case of a unipolar corona field. In a similar paper by Zevenhoven [4], the effects of Brownian motion are neglected, while the ions are assumed to be driven towards the particle by an external electric field. In such a situation, the net electric field intensity Corresponding author.
E-mail address: [email protected] (H.Z. Alisoy). 0304-3886/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2005.02.003
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consists of the superposition of the external electric field E0 and the field created by the charge on the particle. The distribution of charge that is precipitated on a conductive sphere is uniform. In the case of an insulating particle, the charge density need not be uniformly distributed over the surface. On the other hand, a dielectric particle may become uniformly charged for any one of the following reasons: 1. The particle may have finite volume conductivity, in which case the surface distribution of charge will be uniform after a time on the order of t1 0 r1 v [2,3]. In this expression, is relative dielectric constant of the particle, 0 is dielectric constant of vacuum, and rv is the volume conductivity of the particle. 2. The charges may be distributed over the particle surface due to surface conductivity. The time constant of this mechanism is t2 0 rr1 [3–7], where r s is the radius of the particle and rs is its surface conductivity. 3. During the time that an insulating particle is charged, rotational particle motion may occur, thereby allowing, the charge on the surface of the particle to attain a uniform distribution over the time scale of t3 ðr=E 0 Þðr=0 Þ1=2 [7] where E0 is electric field intensity, and r is the density of the particle. If the dielectric particle resides in charging region for less than the smallest of the time constants mentioned above, then the charging process of the particle will have a specific character often encountered in electrostatic precipitators. For this reason, the main purpose in this work is to calculate the charging dynamics of the particle and the limit value of charge on the particle in such situations. From the numerical results obtained, we demonstrate that the charge of a stationary particle is smaller than the limit charge value for a rotating particle, even under the conditions (for example occurrence of tangential fields resulting from the non-uniform distribution of the charges and moisture) influencing the charging process in a negative manner.
2. Theory Let us assume that the residence time of the dielectric particle in the charging region is smaller than the time constants t1 ; t2 and t3 : In this case, we suppose that the calculation of the charging kinetics and limit charge value on the particle will be interesting and contribute to analysis of charging process. The charging of the spherical particle in a unipolar corona field may be expressed by the following equation: ( rn mE n ðy; tÞ if E n 40; qsðy; tÞ ¼ (1) 0 if E n o0: qt Here sðy; tÞ is the density of charges precipitated from the charging region onto the surface of the spherical particle in the period of t; E n ðy; tÞ is normal inward component of the total electric field intensity on the exterior surface of the particle,
ARTICLE IN PRESS H.Z. Alisoy et al. / Journal of Electrostatics 63 (2005) 1095–1103
q
(1,1)
1097
(,)
l
R
R
E0
E0 (a)
(b)
Fig. 1. (a) Spherical dielectric particle in the field of a point charge q. (b) Particle geometry and coordinates when the charge was located on the surface of the particle.
rv is volume charge density of the medium surrounding the particle, m is the mobility of the corona ions, and y is the angle relative to the polar axis. The polar axis is defined as being opposite to the electric field direction, as illustrated in Fig 1(a). The electric field intensity E n ðy; tÞ in Eq. (1) consists of two components. E n ðy; tÞ ¼ E n1 þ E n2:
(2)
The component E n1 is the electric field intensity caused by the uniform external electric field E 0 : For a dielectric particle, E n1 can be found from the following expression [2,3]: E n1 ¼
31 E 0 cos y, 1 þ 22
(3)
where 1 and 2 are the dielectric constants of the dielectric particle and the surrounding medium, respectively. The second component En2 is the electric field intensity produced by the charges that have precipitated on the surface of the particle from the discharge region. In order to determine this field component, En2, we utilize the electric field intensity, E 0n2 ; which was created on the surface of dielectric spherical particle by the point charge q located at the distance of ‘ from the centre of the spherical particle. This electric field intensity, E 0n2 ; is expressed as [2,3,7] E 0n2 ¼
1 X q1 nð2n þ 1Þ hn Pn ðcos yÞ, 2 4p0 2 ‘R n¼0 nð1 þ lÞ þ 1
(4)
where R is the radius of the particle, l ¼ 1 1 2 ; Pn are Legendre polynomials and h ¼ R ‘1 : In order to remove the divergence of the series in Eq. (4) while h-1 (this expresses physically that the point charge q comes around the surface of the spherical particle), the following expression is used to evaluate the generating functions of Legendre polynomials: 1 X n¼0
hn Pn ðxÞ ¼ ð1 2xh þ h2 Þ1=2 .
(5)
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By differentiating and integrating Eq. (5), the following equations are, respectively, obtained [8–10]: 1 X
nhn1 Pn ðxÞ ¼ ðx hÞð1 2xh þ h2 Þ3=2
(6a)
n¼0
and 1 X n¼0
1 nþ1
ðn þ 1Þ h
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2xh þ h2 þ h x : Pn ðxÞ ¼ ln 1x
(6b)
We can write the series included in Eq. (4) by utilizing Eqs. (5) and (6) as follows: " # 1=2 1 ha3 ka4 X Pn ðxÞhn 3=2 S ¼ a1 a2 a3 þ . (7) khð1 kÞ ðn þ 1Þðn þ kÞ n¼0 In this equation, k ¼ 1 þ l1 ; x ¼ cos y and 2khðx hÞ a2 ¼ ; a pffiffiffiffiffi 1 a3 þ h x a4 ¼ ln . 1x
a1 ¼ k2 ð1 kÞð1 2kÞ; a3 ¼ 1 2xh þ h2 ;
In the following steps, we must know the normal component of electric field intensity on the surface of the spherical particle which is created by the charge located on the surface of the particle at ‘ ¼ R and y ¼ 0: So, we must accept h ¼ 1; namely the charge is on the surface of the spherical particle. In order to improve the convergence of the series given in Eq. (7), we utilize the following expansion [10]. 1 X Pn ðcos yÞ y ¼ 1 2 ln 1 þ sin : (8) nðn þ 1Þ 2 n¼1 Assuming that h ¼ 1 in Eq. (7), and rearranging Eq. (7) to remove the divergence of the series by using Eq. (8), we obtain the following equation: " # 1 X Pn ðcos yÞ 1 , (9) S ¼ b1 0:5b2 þ b3 b4 þ b5 b6 nðn þ 1Þðn þ kÞ n¼1 where b1 ¼ 2k2 ; b2 ¼ sinðy=2Þ; b3 ¼ ðk 0:5Þ ln ðb2 Þ; b4 ¼ 2ð0:5 kÞ2 ln ð1 þ b2 Þ; b5 ¼ k1 ð0:5 kÞð1 k2 Þ and b6 ¼ kð0:5 kÞð1 kÞ: Thus, the normal component of the electric field intensity occurred by the point charge q on the spherical particle may be expressed as q E 0n2 ð‘; R; yÞ ‘¼R ¼ cðyÞ, (10) 2 4p y¼0 0R 0 where cðyÞ ¼ 1 2 2 S: In this case, the expression obtained for E n2 ðR; R; yÞ becomes a Green function for the examined problem. In other words, the expression of E 0n2 ðR; R; yÞ allow us to determine the normal component of electric field intensity on
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any point of the surface by utilizing the charges precipitating on the other points of the surface of the spherical particle. If the arbitrary distribution of the charge over the surface of the spherical particle is sðy; jÞ; then the normal component of the electric field intensity E n2 can be expressed as (see Fig. 1(b)) Z 2p Z p=2 1 sðy; jÞ , (11) dj1 dy1 sin y1 sðy1 ; j1 ÞcðaÞ E n2 ðy; jÞ ¼ 4p0 0 ð1 þ 2 Þ 0 0 where a is the angle between the directions ðy; jÞ and ðy1 ; j1 Þ as expressed by cos a ¼ cos y cos y1 þ sin y sin y1 cosðj j1 Þ.
(12)
We can use Eqs. (3) and (11) to simplify Eq. (2). For the case E n 40; the charging expressed by Eq. (1) can be rearranged, and after some simple conversions, we can arrive at the following equation: qyðx; tÞ 3l 1 ¼ x y qt lþ2 1þl ( Z 1 l x dx1 yðx1 ; tÞ pffiffiffiffiffi KðxÞ þ c3 c4 2 c2 pð1 þ lÞ 0 " Z p l1 l1 a ln 1 þ sin dj1 2ðl þ 1Þ 0 lþ1 2 #) 1 l X ðn þ 1Þ1 þ Pn ðcos aÞ , ð13Þ 2 l þ 1 n¼1 n ðl þ 1Þ þ n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where t ¼ rn mt=ð0 2 Þ; y ¼ s=ð0 2 E 0 Þ and x ¼ 2c1 ðc1 þ c2 Þ1 ; KðxÞ is first type of elliptic integral; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðl 1Þ lðl þ 2Þ þ ln 2 ; c1 ¼ ð1 x2 Þð1 x21 Þ; c2 ¼ 1 xx1 ; c3 ¼ 2ðl þ 1Þ l þ 1 pðl 1Þ lnðc2 þ jx x1 jÞ. c4 ¼ 4ðl þ 1Þ 3. Results and discussion In order to facilitate the calculation of the double integral in Eq. (13), following approach is used as an alternative to computing the term given in square brackets in Eq. (13): F ðcos aÞ ¼ d 1 cos a þ d 2
d3 . d 4 cos a
(14)
This approach allows us to calculate the expression given in curly brackets in Eq. (13) with an error of 0.2%. In this expression, the di coefficients are determined for different values of l:
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In this case, the integral calculation over j1 in Eq. (13) can be obtained as Z p Z p dj1 F ðcos aÞ ¼ dj1 F ðxx1 þ c2 cos jÞ 0 20 3 d3 6 7 ¼ p4xx1 d 1 þ d 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 2 ðd 1 xx1 Þ c2
ð15Þ
After these conversions, the solution of charging dynamic equation (13) becomes greatly simplified. As seen in Eq. (13), for the case x1 ¼ x; the integral function will be discontinuous. For this reason, while the integral is being calculated for the righthand side of Eq. (13), the integral interval from 0 to 1 must be chosen as follows: First, the integral is numerically calculated for the intervals of ð0; x oÞ and ðx þ o; 1Þ (here, o is an arbitrary chosen small number to begin numerical calculation). These calculated integrals are added together. Second, the sum of the integrals calculated for the intervals of (x o; x o=2) and (x þ o=2; x þ o) is obtained; this sum is added to the previous sum. This process is continued until 2% numeric error is reached. Eq. (13), which implements the initial condition (no precipitated charge) yðx; 0Þ ¼ 0; has been solved by using a modified Euler method [11]. It is difficult to use the Euler method to solve Eq. (13), because y depends on t and also on the parameter x. For this reason, the right-hand side of the charging equation is calculated, and yðx; tÞ is determined for definite values of x (the calculations are performed for 20 points). Values of yðx; tÞ for x values between the chosen 20 points are found by interpolation [11]. Thus, the charge of spherical particle at any instantaneous time t may be expressed as Z Z 1 2 Q ¼ s ds ¼ 2p0 2 E 0 R yðx; tÞ dx: (16) 0
From Eq. (16), the numerical results calculated for l ¼ 2; 3; 4; 5; and 7 are shown in Fig. 2. The characteristic limit charge density versus the parameter x for the different values of l is given in Fig. 3. The limit charge on the particle mentioned here is the instantaneous limit charge value according to the position of the particle. So, it depends on x. For different l values, the limit value of the uniform charge distribution on spherical dielectric particle is expressed as [4,12] Qlim ¼ 4p0 2
3l E 0 R2 . lþ2
(17)
The normalized charge ðQ=Qlim Þ numerically calculated by utilizing the Eqs. (16) and (17) is plotted versus l ¼ 1 1 2 in Fig. 4. As seen in Fig. 4, for l ¼ 2; Q=Qlim ¼ 0:35; and the normalized charge increases slowly by the increase of l: For l ¼ 7; the normalized charge is 0.51, and for l ! 1; Q=Qlim ! 1: Let us compare the numerical results obtained from Eqs. (16) and (17) with existing experimental results. The charging process of stationary and rotational
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15 =7
=5 =4
10 Q/ε0ε2E0R2
=3
=2 5
0
2
4
6
8
10
12
14
16
18
20
τ Fig. 2. Charging kinetics of particle versus time as calculated from Eq. (16).
quartz spheres ðl ¼ 3:75Þ of radii 3 to 10 mm in the electric field has been experimentally investigated in Ref. [13], and it has been observed that the limit charge values measured on stationary spheres are 0.63 times smaller than the limit charge values for rotating spheres. However, in accordance with the numeric results obtained by using Eqs. (16) and (17) (Fig. 4), we see that Q/Qlim is equal to 0.43 for the value of l ¼ 3:75: The difference between the experimental results and numerical results probably results from the variation of the surface resistivity of the quartz over a wide interval in accordance with the humidity rate. For example, if the surface resistivity of quartz is 109 O, the time constant associated with the migration of the charges over the surface of the particles will be comparable with their charging time. Additionally, as a result of the non-uniform distribution of charge over the surface of the particle, strong tangential fields will occur, and these fields will increase the surface conductivity considerably.
4. Conclusion A nonlinear differential equation, which allows us to calculate the limit charge value and charging process of spherical dielectric particle in unipolar corona field, is
ARTICLE IN PRESS H.Z. Alisoy et al. / Journal of Electrostatics 63 (2005) 1095–1103
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10 9 8 7
y
7 6
5
5
4
4
3
3 =2
2 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x = Cos Fig. 3. The variation of limit charge density over the surface of the particle versus x for different l values. (y is the limit charge density, x ¼ cos y; l ¼ 1 1 2 ).
0.52 0.5 0.48
Q / Qlim
0.46 0.44 0.42 0.4 0.38 0.36 0.34 2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
= 12-1 Fig. 4. The dependence of normalized charge (Q=Qlim ) on l:
7
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derived and solved by using a modified Euler method. Numerically calculated results allow one to determine the charge of the particle for arbitrary values of the particle radius, particle dielectric permittivity, electric field intensity, and volume charge density. All these parameters influence the charging process considerably. We have determined that the limit charge value on the particle increases with increasing dielectric constant of the particle as shown in Fig. 2. In accordance with the numerical results obtained, we have determined that the limit charge of a stationary particle during the charging process is smaller less than the limit charge of a rotating particle. References [1] M.M. Pauthenier, M. Moreau-Hanot, Charging of spherical particles in an ionizing field, J. Phys. Radium 7 (1932) 590–613. [2] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [3] W.R. Smythe, Static and Dynamic Electricity, second ed., McGraw-Hill, New York, 1950. [4] C.A.P. Zevenhoven, Uni-polar field charging of particles: effects of particle conductivity and rotation, J. Electrostat. 46 (1999) 1–12. [5] P.A. Lawless, Particle charging bounds symmetry relations and an analytic charging rate model for the continuum regime, J. Aerosol Sci. 27 (2) (1996) 191–215. [6] A.G. Bailey, The charging of insulator surfaces, J. Electrostat. 51–52 (2001) 82–90. [7] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Course of Theoretical Physics, vol. 8, Pergamon Press, New York, 1982. [8] H.B. Dwight, Tables of Integrals and Other Math Data, Macmillan, New York, 1961. [9] A.P. Prudnikov, Yu.A. Britchkov, O.I. Marichev, Integrals and Series, Nauka, Moscow, 1981 (in Russian). [10] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, sixth ed., Academic Press, New York, 2000. [11] D.M. Etter, Engineering Problem Solving with MATLAB, Prentice-Hall, Englewood Cliffs, NJ, 1993. [12] H.Z. Alisoy, G.T. Alisoy, S.E. Hamamci, M. Koseoglu, Combined kinetic charging of particles on the precipitating electrode in a corona field, J. Phys. D: Appl. Phys. 37 (2004) 1459–1466. [13] V.Y. Steinschreiber, Calculation of forces influencing the aerosol particles in electric field, Ph.D. Thesis, Physics Institute of Science Academy, Baku, 1972 (in Russian).