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The analysis of the charge on a cylindrical dielectric particle in unipolar corona field
Journal of Electrostatics 69 (2011) 176e179
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
The analysis of the charge on a cylindrical dielectric particle in unipolar corona field M. Koseoglu, H.Z. Alisoy* Department of Electrical and Electronics Engineering, Inonu University, 44280, Malatya, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 November 2010 Received in revised form 1 February 2011 Accepted 16 March 2011 Available online 31 March 2011
In this study, a nonlinear integro-differential equation is derived to express the charging kinetics of stationary and rotational cylindrical dielectric particle in unipolar corona field. An algorithm is developed to solve the derived equation numerically by using Euler method. The variation of charge on the dielectric cylindrical particle is presented and evaluated by considering the variable parameters of the particle, medium and field. Assessments of the numerical results demonstrate that the limit charge value on stationary cylindrical dielectric particle during the charging process in unipolar corona field is less in comparison with the rotational cylindrical particle in identical conditions. Also this is an important factor for electroneion technology in the aspect of charging kinetic problems. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Rotational cylindrical particle Unipolar corona field Charging kinetics
1. Introduction The charging process of dielectric particles in unipolar corona field is a considerable and interesting matter in the aspect of the analysis of the particle behavior in an electric field for electroneion technology and aerosol applications [1e6]. In ref. [7], a numerical method was proposed for solving the differential equations that describe the corona-electrostatic field by considering certain approximations. The basic principles governing the charging process of the particle are well described by considering its motion in corona field in the literature [5,6]. In references [8e16], it has been assumed that the charges, which are precipitated from the discharge region on the spherical particle, lead to a rotational motion of the particle because of the polarization in the electrical field, so the particle will have a uniform surface charge distribution. On the base of this assumption, an expression describing the charging kinetics for a spherical dielectric particle was derived. However, the effect of rotational direction on the charging kinetics of the particle is not considered sufficiently. Because the spherical particle can rotate in all directions, so the analysis of the effect of rotational direction on the charging kinetics becomes more complicated. On the other hand, if a cylindrical particle is considered, the problem can be partially simplified since it has a definite rotational direction. In this context, it will be interesting to analyze
* Corresponding author. E-mail address: [email protected] (H.Z. Alisoy). 0304-3886/$ e see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2011.03.006
the charging kinetics of rotational cylindrical dielectric particle in unipolar corona field. In this study, some expressions, which enable to determine the charging kinetics by considering the stationary and rotational cases of the cylindrical dielectric particles in unipolar corona field (Fig. 1), are derived and solved numerically by Euler and Simpson methods, respectively. The results are presented and evaluated comparatively. The results obtained can be used for expressing and analyzing the behaviors of dielectric particles in electrical fields, and also those can be applied for the problems of electron-ion technology and aerosol science. 2. Theoretical model In this section, two different cases were examined to analyze the effect of field charging of cylindrical dielectric particles on their charging kinetics: i) stationary case, ii) rotational case. The physical model of the mentioned problem was established by using the following assumptions: i) the charge distribution is uniform on the surface of the particle which has infinite length, ii) the space charge density (r) and the external electrical field (E0) have a uniform distribution, and ion mobility (m) is constant, iii) the electrical field deformation due to space charges in the ion flux is negligible small, iv) ions move along the electrical field lines, the effect of forces other than Coulomb forces is negligible. The first assumption implies that the insulating particles are free to rotate around their axis, so that each point on their surface has an equal probability of being exposed to the field charging current [6,9,10].
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
177
then we obtain a nonlinear integro-differential equation as following
l vYðq; sÞ ¼ cosq lþ1 vs
Fig. 1. Illustration of the cross-section of cylindrical dielectric particle in unipolar corona field. The circular and ellipsoidal regions represent the profile of EQ and E0 þ EP respectively.
Let us consider the charging process of cylindrical dielectric particle in the electric field including positive space charges. Generally, the total electrical field (ET) on the surface of the particle is the sum of the following components: i) the external electrical field (E0), ii) the electrical field resulted from the polarization of the particle (EP), iii) the electrical field created by the space charges (EQ), iv) the electrical field created by the image charges (EI). The last component (EI) is negligible small in the field charging process (rp > 1 mm). In the field charging process, only radial component of ET must be considered. In this case, the charging equation is given as follows [14,16],
rmEn ðq; tÞ; if En > 0 if En < 0
0;
(1)
where s(q,t) is the charge density on the particle, En(q,t) is the normal component of the total electric field intensity at the surface of the particle, r is the unipolar corona charge density (space charge density) in the medium, m is the mobility of the ions and q is the polar angle. Polar axis is directed opposite to the external electric field. In the light of mentioned definitions and assumptions, En(q,t) can be written as the sum of two components:
En ðq; tÞ ¼ En1 þ En2
(2a)
where the first component En1 , is the electric field intensity resulted from external electric field E0 and polarization of the particle. This component can be written as [17e20]
En1
¼ E0 þ Ep ¼ 2ep E0 cosq= ep þ em
En2
Zp=2 p=2
sðq; tÞ sðq; tÞdq e0 ep þ em
(2c)
where eo is the permittivity of free space. In the case of En>0, if we arrange Eq. (1) by considering some simplifications and conversions given below
ep þ em sðq; tÞ ep rmt Yðq; tÞ ¼ ; s ¼ ; ¼ l 2pe0 em em 4pem e0 ep
p=2
2p Y ¼ ðq; sÞ lþ1
(4)
In the case that the right-hand side of Eq. (4) is negative, vY(q,s)/ vs will be equal to zero. Eq. (4) can be numerically solved for the values of l ¼ 2, 3, 5, 10 by using Euler method. For this purpose a step width of h, which is very small in comparison with s, is chosen. Then the solution of Eq. (4) for s ¼ 0 is obtained by considering the initial condition of Y(q,0) ¼ Y0 ¼ 0 as
vYðq; sÞ j ¼ Y 0 ðq; 0Þ ¼ Y0’ ¼ cosq vs s¼0
Yðq; hÞ ¼ Y1 ¼ Y0 þ hY00 ¼ hcosq
(5a)
(3)
(5b)
In order to solve Eq. (5b) the relative charge on the particle must be determined. If the relative charge on the particle is expressed as
ep þ em Q s q ¼ 4pe0 em ep E0 rp s
(6)
where the upper index (s) of q represents the stationary case, Qs is the charge precipitated on the particle from discharge region in stationary case, then the relative charge accumulating on the particle in the first step of h can be expressed as
Dqs1
Zp=2 ¼ h
Y00 dq
Zp=2 ¼ h
p=2
cosqdq ¼ 2h
(7)
p=2
And the total charge on the particle in the period of h is given as
qs1 ¼ qs0 þ Dqs1 ¼ Dqs1
(8)
qs0
where is the initial relative charge value on the particle, and qs0 ¼ 0. Thus, for the first step of h Eq. (5b) can be expressed as
vYðq; sÞ j ¼ Y10 ¼ vs s¼h
2ph 2lh 1 ,cosq1 lþ1 lþ1
(9)
And in the following step, the region, where Eq. (10) is satisfied, will be charged.
(2b)
where em and ep are dielectric permittivities of the medium and the particle, respectively. The second componentEn2 is the electric field intensity created by the charges precipitated onto the surface of the particle from the discharge region, and it is expressed as
e p , ¼ 2pe0 em ep þ em
Yðq; sÞdq
The expressions for the next steps are obtained by considering the step width of h as following
2.1. The field charging of stationary cylindrical dielectric particle
vsðq; tÞ ¼ vt
þ Zp=2
cosq cosq1 ¼
2lh l þ 1 2ph
(10)
In this equation q1 is the polar angle corresponding to the charged region in the first step. If Eq. (10) is satisfied, then the right-hand side of the charging equation given by Eq. (4) will be positive in the mentioned step. In the second step, appropriately
Dqs2 ¼ h
Zq
Y10 dq
(11)
q
and qs2 ¼ qs1 þ Dqs2 . As seen, in the ith step
Yi ¼ Ai cosq Bi
(12a)
Yi0 ¼ Ci cosq Di
(12b)
178
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
For the values of Ai, Bi, Ci, Di, qsi ,Dqsiþ1 ,qi, the following recurrent expressions can be derived:
Aiþ1 ¼ Ai þ hCi 8B iþ1 ¼ Bi þ hDi > > > 2p > > A C > iþ1 ¼ 1 > l þ 1 iþ1 > > < l s 2p qiþ1 B Diþ1 ¼ l l þ 1 þ 1 iþ1 > > s s s > > qiþ1 ¼ qi þ Dqiþ1 > > > s > > Dqiþ1 ¼ 2hðCi sinqi Di qi Þ : D qiþ1 ¼ cos1 iþ1 Ciþ1
an appropriate value which satisfies the condition of vY(q,s)/vs > 0. The results of numerical calculations obtained for l ¼ 2; 3; 5; 10 are presented in Fig. 3.
2.2. The field charging of rotational cylindrical dielectric particle
(13)
The required expressions to solve Eq. (5b) are obtained from Eq. (13) by considering the initial conditions of
A0 ¼ 0; B0 ¼ 0; C0 ¼ 1; D0 ¼ 0; qs0 ¼ 0; Dqs0 ¼ 0; q0 ¼ p=2 (14) The algorithm used for numerical solution of Eq. (4) is given Fig. 2. In the algorithm maximum number of iterations imax is set to
As a result of the mutual interaction between the electric field and the charges precipitated on the particle from the discharge region, the particle of the radius rp has a rotational motion. Assuming that the precipitated charges distribute all over the cylinder surface uniformly, the normal component of the electric field is expressed as
En ¼
ep
ep þ em
2E0 cosq
Qr
(16)
2pe0 em rp
where Qr is the charge precipitated on the particle from discharge region in rotational case. In this case, the charging equation can be written as,
dQ r ðtÞ=dt ¼ rm
Z
En ðq; tÞds
(17)
and the integration is done with respect to q over the whole region where the condition of En > 0 is satisfied. This region can be determined for any moment of t by using Eq. (16) as follows,
cos q < cosqlim ¼
ep þ em Q r 4pe0 em ep E0 rp
(18)
When En ¼ 0, the polar angle q, which corresponds to charging region, reaches to its limit value (qlim), and in this case the charge on the particle will be maximum. If qlim ¼ 0, the maximum value of the charge on the particle can be determined from Eq. (18) as
r Qmax ¼ 4pe0 rp E0 ep em = ep þ em
(19)
If Eq. (17) is rearranged by considering the Eqs. (3), (6) and (16), then it yields
ZQlim
dqr ¼ ds
ðcosq qr Þdq
(20)
Qlim
1 0.9 0.8 0.7
q
0.6 0.5 0.4
λ1=2
0.3
λ2=3 λ3=5
0.2
λ4=10
0.1 0
qr 0
0.5
1
1.5
2
2.5
3
τ Fig. 2. The algorithm used for numerical solution of Eq. (4).
Fig. 3. The change of normalized charge on the particle versus s for different l values. Solid and marked curves represent the rotational and stationary cases respectively.
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
where the upper index (r) of relative charge (q) represents the rotational case. The integration of Eq. (20) is found by considering Eq. (18) as
dqr ¼ 2 ds
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðqr Þ2 qr cos1 qr
(21)
The solution of this first order differential equation, Eq. (21), for the initial condition of qr(0) ¼ 0 is found as
1 sðq Þ ¼ 2
Zq
r
0
r
dqr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðqr Þ2 qr cos1 ðqr Þ
(22)
3. Results and discussion The solution of Eq. (4) corresponding to the initial condition of Y(q,0) ¼ Y0 ¼ 0 is numerically determined for the values of l ¼ 2, 3, 5, 10 by using Euler method. For this purpose, a step width of h, which is small enough in comparison with s, is chosen, and then the required equations are written and solved numerically by considering the defined initial conditions. Eq. (22), which expresses the charging kinetics of a rotational cylindrical dielectric particle in a unipolar corona field, is solved by using Simpson method, and the relative charge on the particle for any s is presented in Fig. 3. As seen in Fig. 3, the relative charge value on the particle increases with the increase in dielectric permittivity of the particle in identical conditions for stationary case. However, the limit relative charge value of the rotational particle is bigger than that of the stationary particle. The distribution of surface charge density on the particle is presented versus the value of cosq for different l (l ¼ ep e1 m ) values in Fig. 4. As seen in this figure, the distribution of surface charge on the particle changes nonlinearly according to the rotational position of the particle. This nonlinear change in surface charge distribution is consistent with the previous studies [14,16]. The curves of the surface charge density seen in Fig. 4 can be separated into two regions. In the first region, where cosq is bigger than w0.6, the effect of ep on the surface charge density is observed more evident in comparison with the second region, where cosq is
λ1=2 λ2=3
0.5
λ3=5 λ4=10
Y(θ,τ)
0.4
0.3
0.2
0.1
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cos(θ) Fig. 4. The distribution of surface charge density on the stationary particle for different l values.
179
smaller than w0.6. The surface charge density increases with the increase of ep in the first region. In the second region, the change of ep has a negligible effect on the distribution of surface charge density; however the surface charge density decreases with the increase of ep for those values.
4. Conclusion Some mathematical expressions are derived and solved numerically to determine the charging kinetics and the limit charge value of stationary and rotational cylindrical dielectric particles in unipolar corona field. For stationary and rotational cases, the change in the charging kinetics of the cylindrical dielectric particle is presented and evaluated. Assessments of the numerical results demonstrate that the limit charge value on the stationary particle during the charging process is less in comparison with the rotational particle in the identical conditions. The surface charge density on the particle increases nonlinearly with the increase of cosq. For any value of s, the increase of dielectric permittivity results in an increase of limit charge value on the particle. The results of this study are considerable in the aspect of aerosol science and electroneion technology, especially for the electrostatic precipitators used in dielectric separation process.
References [1] A. Mizuno, Electrostatic precipitation, IEEE Trans. Dielect. El. Insul. 7 (5) (2000) 615e624. [2] P. Intra, N. Tippayawong, Progress in unipolar corona discharger designs for airborne particle charging: a literature review, J. Electrostat. 67 (4) (2009) 605e615. [3] L.M. Dumitran, O. Blejan, P. Notingher, A. Samuila, L. Dascalescu, Particle Charging in Combined Corona-Electrostatic Fields. IEEE, 2005, IAS 21429e1434. [4] A. Samuila, L. Dascalescu, Unipolar charging of cylindrical insulating particles near electrode surfaces, IEEE Trans. Ind. Appl. 34 (1) (1998) 51e56. [5] M.N. Horenstein, Computation of corona space charge, electric field and VeI characteristics using equipotential charge shells, IEEE Trans. Ind. Appl. 20 (1984) 1607e1612. [6] L. Dascalescu, L.M. Dumitran, A. Samuila, Charging of one or several cylindrical particles by monopolar ions in electric fields, IEEE Trans. Ind. Appl. 39 (2) (2003) 362e367. [7] A. Caron, L. Dascalescu, Numerical modeling of combined corona electrostatic fields, J. Electrostat. 61 (2004) 43e55. [8] W.R. Smythe, Static and Dynamic Electricity. Mc Graw-Hill, New York, 1968. [9] P.A. Lawless, Particle charging bounds, symmetry relations, and an analytic charging rate model for the continuum regime, J. Aerosol Sci. 27 (22) (1996) 191e215. [10] 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. [11] T.B. Jones, Electromechanics of Particles. Cambridge University Press, New York, 1995. [12] G.T. Alisoy, H.Z. Alisoy, M. Koseoglu, Calculation of electrical field of spherical and cylindrical gas voids in dielectrics by taking surface conductivity into consideration, COMPEL 24 (4) (2005) 1152e1163. [13] H.Z. Alisoy, G.T. Alisoy, A. Sahin, C. Yeroglu, The evaluations of the influence of surface conductivity to the energy of particles in discharge channel and interaction force in contact charging process, Phy. Let. A 360 (2006) 14e21. [14] C.A.P. Zevenhoven, Unipolar field charging of particles: effects of particle conductivity and rotation, J. Electrostat. 46 (1) (1999) 1e12. [15] L. Unger, D. Boulaud, J.P. Borra, Unipolar field charging of particles by electrical discharge: effect of particle shape, J. Aerosol Sci. 35 (2004) 965e979. [16] H.Z. Alisoy, G.T. Alisoy, M. Koseoglu, Charging kinetics of spherical dielectric particles in a unipolar corona field, J. Electrostat. 63 (2005) 1095e1103. [17] P.M. Morse, H. Feshbach, Methods of Theoretical Physics. Mc Graw-Hill, New York, 1953. [18] A.A. Tikhonov, A.A. Samarsky, Equations of Mathematical Physics. Nauka, Moscow, 1977. [19] I.M. Bortnik, I.P. Vereshchagin, Yu N. Vershinin, et al., Electrophysical Fundamentals of High Voltage Technique. Energoatomizdat, Moscow, 1993, (in Russian). [20] H.Z. Alisoy, B.B. Alagoz, G.H. Alisoy, An analysis of corona field charging kinetics for polydisperse aerosol particles by considering concentration and mobility, J. Phys. D: Appl. Phys. 43 (2010) 365205.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
The analysis of the charge on a cylindrical dielectric particle in unipolar corona field M. Koseoglu, H.Z. Alisoy* Department of Electrical and Electronics Engineering, Inonu University, 44280, Malatya, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 November 2010 Received in revised form 1 February 2011 Accepted 16 March 2011 Available online 31 March 2011
In this study, a nonlinear integro-differential equation is derived to express the charging kinetics of stationary and rotational cylindrical dielectric particle in unipolar corona field. An algorithm is developed to solve the derived equation numerically by using Euler method. The variation of charge on the dielectric cylindrical particle is presented and evaluated by considering the variable parameters of the particle, medium and field. Assessments of the numerical results demonstrate that the limit charge value on stationary cylindrical dielectric particle during the charging process in unipolar corona field is less in comparison with the rotational cylindrical particle in identical conditions. Also this is an important factor for electroneion technology in the aspect of charging kinetic problems. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Rotational cylindrical particle Unipolar corona field Charging kinetics
1. Introduction The charging process of dielectric particles in unipolar corona field is a considerable and interesting matter in the aspect of the analysis of the particle behavior in an electric field for electroneion technology and aerosol applications [1e6]. In ref. [7], a numerical method was proposed for solving the differential equations that describe the corona-electrostatic field by considering certain approximations. The basic principles governing the charging process of the particle are well described by considering its motion in corona field in the literature [5,6]. In references [8e16], it has been assumed that the charges, which are precipitated from the discharge region on the spherical particle, lead to a rotational motion of the particle because of the polarization in the electrical field, so the particle will have a uniform surface charge distribution. On the base of this assumption, an expression describing the charging kinetics for a spherical dielectric particle was derived. However, the effect of rotational direction on the charging kinetics of the particle is not considered sufficiently. Because the spherical particle can rotate in all directions, so the analysis of the effect of rotational direction on the charging kinetics becomes more complicated. On the other hand, if a cylindrical particle is considered, the problem can be partially simplified since it has a definite rotational direction. In this context, it will be interesting to analyze
* Corresponding author. E-mail address: [email protected] (H.Z. Alisoy). 0304-3886/$ e see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2011.03.006
the charging kinetics of rotational cylindrical dielectric particle in unipolar corona field. In this study, some expressions, which enable to determine the charging kinetics by considering the stationary and rotational cases of the cylindrical dielectric particles in unipolar corona field (Fig. 1), are derived and solved numerically by Euler and Simpson methods, respectively. The results are presented and evaluated comparatively. The results obtained can be used for expressing and analyzing the behaviors of dielectric particles in electrical fields, and also those can be applied for the problems of electron-ion technology and aerosol science. 2. Theoretical model In this section, two different cases were examined to analyze the effect of field charging of cylindrical dielectric particles on their charging kinetics: i) stationary case, ii) rotational case. The physical model of the mentioned problem was established by using the following assumptions: i) the charge distribution is uniform on the surface of the particle which has infinite length, ii) the space charge density (r) and the external electrical field (E0) have a uniform distribution, and ion mobility (m) is constant, iii) the electrical field deformation due to space charges in the ion flux is negligible small, iv) ions move along the electrical field lines, the effect of forces other than Coulomb forces is negligible. The first assumption implies that the insulating particles are free to rotate around their axis, so that each point on their surface has an equal probability of being exposed to the field charging current [6,9,10].
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
177
then we obtain a nonlinear integro-differential equation as following
l vYðq; sÞ ¼ cosq lþ1 vs
Fig. 1. Illustration of the cross-section of cylindrical dielectric particle in unipolar corona field. The circular and ellipsoidal regions represent the profile of EQ and E0 þ EP respectively.
Let us consider the charging process of cylindrical dielectric particle in the electric field including positive space charges. Generally, the total electrical field (ET) on the surface of the particle is the sum of the following components: i) the external electrical field (E0), ii) the electrical field resulted from the polarization of the particle (EP), iii) the electrical field created by the space charges (EQ), iv) the electrical field created by the image charges (EI). The last component (EI) is negligible small in the field charging process (rp > 1 mm). In the field charging process, only radial component of ET must be considered. In this case, the charging equation is given as follows [14,16],
rmEn ðq; tÞ; if En > 0 if En < 0
0;
(1)
where s(q,t) is the charge density on the particle, En(q,t) is the normal component of the total electric field intensity at the surface of the particle, r is the unipolar corona charge density (space charge density) in the medium, m is the mobility of the ions and q is the polar angle. Polar axis is directed opposite to the external electric field. In the light of mentioned definitions and assumptions, En(q,t) can be written as the sum of two components:
En ðq; tÞ ¼ En1 þ En2
(2a)
where the first component En1 , is the electric field intensity resulted from external electric field E0 and polarization of the particle. This component can be written as [17e20]
En1
¼ E0 þ Ep ¼ 2ep E0 cosq= ep þ em
En2
Zp=2 p=2
sðq; tÞ sðq; tÞdq e0 ep þ em
(2c)
where eo is the permittivity of free space. In the case of En>0, if we arrange Eq. (1) by considering some simplifications and conversions given below
ep þ em sðq; tÞ ep rmt Yðq; tÞ ¼ ; s ¼ ; ¼ l 2pe0 em em 4pem e0 ep
p=2
2p Y ¼ ðq; sÞ lþ1
(4)
In the case that the right-hand side of Eq. (4) is negative, vY(q,s)/ vs will be equal to zero. Eq. (4) can be numerically solved for the values of l ¼ 2, 3, 5, 10 by using Euler method. For this purpose a step width of h, which is very small in comparison with s, is chosen. Then the solution of Eq. (4) for s ¼ 0 is obtained by considering the initial condition of Y(q,0) ¼ Y0 ¼ 0 as
vYðq; sÞ j ¼ Y 0 ðq; 0Þ ¼ Y0’ ¼ cosq vs s¼0
Yðq; hÞ ¼ Y1 ¼ Y0 þ hY00 ¼ hcosq
(5a)
(3)
(5b)
In order to solve Eq. (5b) the relative charge on the particle must be determined. If the relative charge on the particle is expressed as
ep þ em Q s q ¼ 4pe0 em ep E0 rp s
(6)
where the upper index (s) of q represents the stationary case, Qs is the charge precipitated on the particle from discharge region in stationary case, then the relative charge accumulating on the particle in the first step of h can be expressed as
Dqs1
Zp=2 ¼ h
Y00 dq
Zp=2 ¼ h
p=2
cosqdq ¼ 2h
(7)
p=2
And the total charge on the particle in the period of h is given as
qs1 ¼ qs0 þ Dqs1 ¼ Dqs1
(8)
qs0
where is the initial relative charge value on the particle, and qs0 ¼ 0. Thus, for the first step of h Eq. (5b) can be expressed as
vYðq; sÞ j ¼ Y10 ¼ vs s¼h
2ph 2lh 1 ,cosq1 lþ1 lþ1
(9)
And in the following step, the region, where Eq. (10) is satisfied, will be charged.
(2b)
where em and ep are dielectric permittivities of the medium and the particle, respectively. The second componentEn2 is the electric field intensity created by the charges precipitated onto the surface of the particle from the discharge region, and it is expressed as
e p , ¼ 2pe0 em ep þ em
Yðq; sÞdq
The expressions for the next steps are obtained by considering the step width of h as following
2.1. The field charging of stationary cylindrical dielectric particle
vsðq; tÞ ¼ vt
þ Zp=2
cosq cosq1 ¼
2lh l þ 1 2ph
(10)
In this equation q1 is the polar angle corresponding to the charged region in the first step. If Eq. (10) is satisfied, then the right-hand side of the charging equation given by Eq. (4) will be positive in the mentioned step. In the second step, appropriately
Dqs2 ¼ h
Zq
Y10 dq
(11)
q
and qs2 ¼ qs1 þ Dqs2 . As seen, in the ith step
Yi ¼ Ai cosq Bi
(12a)
Yi0 ¼ Ci cosq Di
(12b)
178
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
For the values of Ai, Bi, Ci, Di, qsi ,Dqsiþ1 ,qi, the following recurrent expressions can be derived:
Aiþ1 ¼ Ai þ hCi 8B iþ1 ¼ Bi þ hDi > > > 2p > > A C > iþ1 ¼ 1 > l þ 1 iþ1 > > < l s 2p qiþ1 B Diþ1 ¼ l l þ 1 þ 1 iþ1 > > s s s > > qiþ1 ¼ qi þ Dqiþ1 > > > s > > Dqiþ1 ¼ 2hðCi sinqi Di qi Þ : D qiþ1 ¼ cos1 iþ1 Ciþ1
an appropriate value which satisfies the condition of vY(q,s)/vs > 0. The results of numerical calculations obtained for l ¼ 2; 3; 5; 10 are presented in Fig. 3.
2.2. The field charging of rotational cylindrical dielectric particle
(13)
The required expressions to solve Eq. (5b) are obtained from Eq. (13) by considering the initial conditions of
A0 ¼ 0; B0 ¼ 0; C0 ¼ 1; D0 ¼ 0; qs0 ¼ 0; Dqs0 ¼ 0; q0 ¼ p=2 (14) The algorithm used for numerical solution of Eq. (4) is given Fig. 2. In the algorithm maximum number of iterations imax is set to
As a result of the mutual interaction between the electric field and the charges precipitated on the particle from the discharge region, the particle of the radius rp has a rotational motion. Assuming that the precipitated charges distribute all over the cylinder surface uniformly, the normal component of the electric field is expressed as
En ¼
ep
ep þ em
2E0 cosq
Qr
(16)
2pe0 em rp
where Qr is the charge precipitated on the particle from discharge region in rotational case. In this case, the charging equation can be written as,
dQ r ðtÞ=dt ¼ rm
Z
En ðq; tÞds
(17)
and the integration is done with respect to q over the whole region where the condition of En > 0 is satisfied. This region can be determined for any moment of t by using Eq. (16) as follows,
cos q < cosqlim ¼
ep þ em Q r 4pe0 em ep E0 rp
(18)
When En ¼ 0, the polar angle q, which corresponds to charging region, reaches to its limit value (qlim), and in this case the charge on the particle will be maximum. If qlim ¼ 0, the maximum value of the charge on the particle can be determined from Eq. (18) as
r Qmax ¼ 4pe0 rp E0 ep em = ep þ em
(19)
If Eq. (17) is rearranged by considering the Eqs. (3), (6) and (16), then it yields
ZQlim
dqr ¼ ds
ðcosq qr Þdq
(20)
Qlim
1 0.9 0.8 0.7
q
0.6 0.5 0.4
λ1=2
0.3
λ2=3 λ3=5
0.2
λ4=10
0.1 0
qr 0
0.5
1
1.5
2
2.5
3
τ Fig. 2. The algorithm used for numerical solution of Eq. (4).
Fig. 3. The change of normalized charge on the particle versus s for different l values. Solid and marked curves represent the rotational and stationary cases respectively.
M. Koseoglu, H.Z. Alisoy / Journal of Electrostatics 69 (2011) 176e179
where the upper index (r) of relative charge (q) represents the rotational case. The integration of Eq. (20) is found by considering Eq. (18) as
dqr ¼ 2 ds
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðqr Þ2 qr cos1 qr
(21)
The solution of this first order differential equation, Eq. (21), for the initial condition of qr(0) ¼ 0 is found as
1 sðq Þ ¼ 2
Zq
r
0
r
dqr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðqr Þ2 qr cos1 ðqr Þ
(22)
3. Results and discussion The solution of Eq. (4) corresponding to the initial condition of Y(q,0) ¼ Y0 ¼ 0 is numerically determined for the values of l ¼ 2, 3, 5, 10 by using Euler method. For this purpose, a step width of h, which is small enough in comparison with s, is chosen, and then the required equations are written and solved numerically by considering the defined initial conditions. Eq. (22), which expresses the charging kinetics of a rotational cylindrical dielectric particle in a unipolar corona field, is solved by using Simpson method, and the relative charge on the particle for any s is presented in Fig. 3. As seen in Fig. 3, the relative charge value on the particle increases with the increase in dielectric permittivity of the particle in identical conditions for stationary case. However, the limit relative charge value of the rotational particle is bigger than that of the stationary particle. The distribution of surface charge density on the particle is presented versus the value of cosq for different l (l ¼ ep e1 m ) values in Fig. 4. As seen in this figure, the distribution of surface charge on the particle changes nonlinearly according to the rotational position of the particle. This nonlinear change in surface charge distribution is consistent with the previous studies [14,16]. The curves of the surface charge density seen in Fig. 4 can be separated into two regions. In the first region, where cosq is bigger than w0.6, the effect of ep on the surface charge density is observed more evident in comparison with the second region, where cosq is
λ1=2 λ2=3
0.5
λ3=5 λ4=10
Y(θ,τ)
0.4
0.3
0.2
0.1
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cos(θ) Fig. 4. The distribution of surface charge density on the stationary particle for different l values.
179
smaller than w0.6. The surface charge density increases with the increase of ep in the first region. In the second region, the change of ep has a negligible effect on the distribution of surface charge density; however the surface charge density decreases with the increase of ep for those values.
4. Conclusion Some mathematical expressions are derived and solved numerically to determine the charging kinetics and the limit charge value of stationary and rotational cylindrical dielectric particles in unipolar corona field. For stationary and rotational cases, the change in the charging kinetics of the cylindrical dielectric particle is presented and evaluated. Assessments of the numerical results demonstrate that the limit charge value on the stationary particle during the charging process is less in comparison with the rotational particle in the identical conditions. The surface charge density on the particle increases nonlinearly with the increase of cosq. For any value of s, the increase of dielectric permittivity results in an increase of limit charge value on the particle. The results of this study are considerable in the aspect of aerosol science and electroneion technology, especially for the electrostatic precipitators used in dielectric separation process.
References [1] A. Mizuno, Electrostatic precipitation, IEEE Trans. Dielect. El. Insul. 7 (5) (2000) 615e624. [2] P. Intra, N. Tippayawong, Progress in unipolar corona discharger designs for airborne particle charging: a literature review, J. Electrostat. 67 (4) (2009) 605e615. [3] L.M. Dumitran, O. Blejan, P. Notingher, A. Samuila, L. Dascalescu, Particle Charging in Combined Corona-Electrostatic Fields. IEEE, 2005, IAS 21429e1434. [4] A. Samuila, L. Dascalescu, Unipolar charging of cylindrical insulating particles near electrode surfaces, IEEE Trans. Ind. Appl. 34 (1) (1998) 51e56. [5] M.N. Horenstein, Computation of corona space charge, electric field and VeI characteristics using equipotential charge shells, IEEE Trans. Ind. Appl. 20 (1984) 1607e1612. [6] L. Dascalescu, L.M. Dumitran, A. Samuila, Charging of one or several cylindrical particles by monopolar ions in electric fields, IEEE Trans. Ind. Appl. 39 (2) (2003) 362e367. [7] A. Caron, L. Dascalescu, Numerical modeling of combined corona electrostatic fields, J. Electrostat. 61 (2004) 43e55. [8] W.R. Smythe, Static and Dynamic Electricity. Mc Graw-Hill, New York, 1968. [9] P.A. Lawless, Particle charging bounds, symmetry relations, and an analytic charging rate model for the continuum regime, J. Aerosol Sci. 27 (22) (1996) 191e215. [10] 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. [11] T.B. Jones, Electromechanics of Particles. Cambridge University Press, New York, 1995. [12] G.T. Alisoy, H.Z. Alisoy, M. Koseoglu, Calculation of electrical field of spherical and cylindrical gas voids in dielectrics by taking surface conductivity into consideration, COMPEL 24 (4) (2005) 1152e1163. [13] H.Z. Alisoy, G.T. Alisoy, A. Sahin, C. Yeroglu, The evaluations of the influence of surface conductivity to the energy of particles in discharge channel and interaction force in contact charging process, Phy. Let. A 360 (2006) 14e21. [14] C.A.P. Zevenhoven, Unipolar field charging of particles: effects of particle conductivity and rotation, J. Electrostat. 46 (1) (1999) 1e12. [15] L. Unger, D. Boulaud, J.P. Borra, Unipolar field charging of particles by electrical discharge: effect of particle shape, J. Aerosol Sci. 35 (2004) 965e979. [16] H.Z. Alisoy, G.T. Alisoy, M. Koseoglu, Charging kinetics of spherical dielectric particles in a unipolar corona field, J. Electrostat. 63 (2005) 1095e1103. [17] P.M. Morse, H. Feshbach, Methods of Theoretical Physics. Mc Graw-Hill, New York, 1953. [18] A.A. Tikhonov, A.A. Samarsky, Equations of Mathematical Physics. Nauka, Moscow, 1977. [19] I.M. Bortnik, I.P. Vereshchagin, Yu N. Vershinin, et al., Electrophysical Fundamentals of High Voltage Technique. Energoatomizdat, Moscow, 1993, (in Russian). [20] H.Z. Alisoy, B.B. Alagoz, G.H. Alisoy, An analysis of corona field charging kinetics for polydisperse aerosol particles by considering concentration and mobility, J. Phys. D: Appl. Phys. 43 (2010) 365205.