ARTICLE IN PRESS
0304-3886/$ - se
doi:10.1016/j.el
�CorrespondE-mail addr
(P. Zamankhan
Journal of Electrostatics 65 (2007) 709–720
www.elsevier.com/locate/elstat
Effects of corotron size and parameters on thedielectric substrate surface charge
Parsa Zamankhana,�, Goodarz Ahmadia,�, Fa-Gung Fanb
aDepartment of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USAbJ.C. Wilson Center for Research and Technology, Xerox Corporation, Webster, NY 14580, USA
Received 26 February 2006; received in revised form 6 December 2006; accepted 18 May 2007
Available online 26 June 2007
Abstract
Effects of variation of parameters of a corona device (corotron) used in electro-photographic machines on the amount of surface
charge build-up on the surface of dielectric substrate were studied. Particular attention was given to the effect of corotron dimension
including wire–shields and wire–plate distances, substrate thickness, shields insulation and the substrate speed on the amount of substrate
surface charge. The computational analyses were performed for a two-dimensional cross-section of the corotron under steady-state
condition. The Maxwell equations were solved and the electrical quantities in a rectangular positive single wire corotron were evaluated.
The simulation results showed that for a fixed wire voltage, the corotron size, the substrate thickness, insulation of shields and the
substrate speed will affect the distributions of electrical quantities in the corotron. The wire–substrate distance and the substrate speed,
however, were found to be the main parameters that control the amount of surface charge build-up on the substrate.
r 2007 Elsevier B.V. All rights reserved.
Keywords: Corotron; Dielectric substrate; Surface charge; Charge density; Electrical potential
1. Introduction
Xerography is based on controlled transfer of chargedtoner particles between surfaces. Corotrons are the keydevices of electro-photographic machines for charging thephotoreceptor and the paper surfaces to desired surfacecharge levels for the purpose of toner transfer. A genericcorotron is a long rectangular box with all its sides beinggrounded except the one facing the substrate (photorecep-tor/paper). A corotron includes one or several thin wiresserving as the charging elements. During the operation ofthe corotron, a high voltage is applied to the wires. Coronadischarge occurs when the intensity of the electric field nearthe wire exceeds the threshold of air breakdown. The gasmolecules near the wire in a region called ‘‘ionized sheath’’are ionized. The wire then repels the ions with the samepolarity and a unipolar charge current is generated fromthe wire to the shield and the substrate. The ions deposit on
e front matter r 2007 Elsevier B.V. All rights reserved.
stat.2007.05.007
ing authors.
esses: [email protected]
), [email protected] (G. Ahmadi).
the dielectric substrate and increase the level of substratesurface charge. Corotrons are designed to elevate the levelof the surface charge on the substrate to designed valuesfor different applications.The amount of the surface charge on the substrates
(which is proportional with the substrate voltage) exposedto a corotron is a function of several variables includingwire voltage, wire diameter, corotron size, substrate speed,insulation of corotron shields and the substrate thickness.Therefore, studying the effects of variations of theseparameters on the amount of the surface charge build-upon the substrate is of interest for understanding theperformance of the corotron and will provide valuableinformation for the designers of these devices.Corona discharge and its associated electro-hydro-
dynamic (EHD) flows and wire oscillation in wire–planegeometries such as electrostatic precipitators, high-voltageDC transmission lines, wire–shield, wire–cylinder andwire–plane devices have been studied experimentally,theoretically and numerically by a number of authors[1–28]. There have been, however, very few publishedworks on charging a moving dielectric substrate by
ARTICLE IN PRESSP. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720710
a corotron in the open literature. In [29], the electricalquantities were evaluated and compared favorably with theexperimental results. In [30], in addition to the electricalquantities, the flow field inside and outside of a corotronwas analyzed numerically.
Earlier simulation studies [29,30] showed that the wirevoltage and the substrate speed have significant effects onthe level of the voltage/surface charge of the substrate. Itwas also found that the size of the gap between the sideshields and the substrate has marginal effects on the level ofthe voltage/surface charge on the substrate [30]. In thispaper, the effects of other important parameters such ascorotron size, shields insulation and substrate thickness onthe amount of the surface charge build-up on the substratesurface are studied, and additional information concerningthe effects of the substrate speed is provided. To ourknowledge the effects of these parameters on the distribu-tion of surface charge along the substrate have not beenreported in the open literature.
A series of numerical simulation for a rectangularpositive single wire corotron under steady-state conditionswere performed. Electrostatics set of Maxwell’s equationswere solved using FIDAPTM code for a cross-section of thecorotron. The simulation results showed that the wire–shield distance, wire–substrate distance, substrate thick-ness, shield insulation and the substrate speed significantlyaffect the distribution of the electrical quantities inside ofthe corotron. The wire–substrate distance and the substratespeed, however, were found to be the most importantfactors that control the level of substrate surface charge.
2. Governing equations and boundary conditions
In an earlier study [30] it was shown that inclusion of theion transport by the flow in the governing equations does
Fig. 1. Schematic o
not affect the distribution of electrical quantities incorotrons significantly. Therefore, in this study which isfocused on the sensitivity of the electrical quantities incorotrons due to changes in the various parameters, the iontransport term by convection was neglected for sake ofeconomy of the computational effort. As a result, theelectrostatic governing equations were decoupled from theairflow equations and the computational time decreasedsignificantly.Since the corona discharge along a positive wire is steady
and uniform and the dimension of a corotron along its wiredirection is much larger than its cross-section, a two-dimensional model was used in the analysis. For constantphysical properties, the steady-state electrostatics govern-ing equations for corotrons are given as
Gauss law : r2V ¼ �q
�, (1)
Conservation of charge : Dr2qþ br � ðqrV Þ ¼ 0, (2)
Electric field–electrical potential relation:
E ¼ �rV . (3)
In Eqs. (1)–(3), V is the electrical potential, E isthe electric field vector, q is the space charge density,D is ion diffusivity in the air, b is the ion mobility and� is the air permittivity. In the performed simulations apermittivity of e ¼ 8.85� 10�12 C/V �m, an ion mobilityof b ¼ 2� 104m2/V � s and an ion diffusivity of D ¼ 5.32�10�6m2/s were assumed.For the corotron geometry shown in Fig. 1 the boundary
conditions for the electrostatics equations are
At the wire surface : V ¼ Vwire. (4)
At the shields : V ¼ 0;qq
qn¼ 0. (5)
f the corotron.
ARTICLE IN PRESS
1In a computational element, the computational electric Peclet number
is defined by PeE ¼ UEh/D in which UE is the ion drift velocity and h is the
computational grid size. The ion drift velocity is defined by
UE ¼ b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðqV=qxÞ2 þ ðqV=qyÞ2
q.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720 711
Along A2B; B2C; D2E and E2F :qV
qn¼ 0;
qn¼ 0.
(6)
At the substrate :qq
qn¼ 0. (7)
In Eqs. (4)–(7), n represents the local unit normal pointingoutward from the corresponding surface boundary.
Since ion transport by diffusion is included in the chargeconservation equation given by (2), boundary conditions atall the boundaries are required. The details of the physicalsituation for the space charge at the solid boundaries suchas the shields and the substrate are not known. Feng [22,23]and Medlin et al. [18,19] used the zero gradient boundarycondition on the walls. Feng et al. [29] used the sameboundary condition for the shields and the substrate incorotron. The results of Feng and his coworkers andMedlin et al. showed no noticeable differences with theearlier results [2,3,7,8,12,13,16,24–27] obtained for thecases that the diffusion term was neglected. This wasbecause the diffusion term in the charge transport equationwas small comparing to the drift term. In this study,however, to follow the format used in FIDAPTM, thediffusion term was included in the analysis and theboundary conditions given by Eqs. (5)–(7) were used.
To estimate the charge density on the surface of the wire,Kaptsov’s [31] hypothesis was used. Kaptsov’s hypothesisrelates the gradient of electrical potential at the wiresurface to the onset electric field strength. That is,
n � rV ¼ Constant ¼ Eonset at the wire surface; (8)
where n is the local unit normal vector that points into thewire, rV is the gradient of electrical potential on the wiresurface and Eonset is the electric field threshold strength forthe corona onset. Here, Eonset can be calculated fromPeek’s formula [32] given as
Eonset ¼ Adþ B
ffiffiffiffiffiffiffid
Rw
s !. (9)
In Eq. (9) Rw is the radius of the wire, d is the gas densityconsidered as 1.225 kg/m3 in this study and A ¼ 32.3�105V/m, B ¼ 8.46� 104V/m1/2 are the constants of Peek’sformula.
Based on Peek’s formula given by Eq. (9), an iterativeprocedure through an FIDAPTM user-defined subroutinefor applying the charge density boundary condition on thewire was developed. The iterative procedure for imple-menting Kaptsov’s hypothesis has been discussed in detailin [30].
The voltage boundary condition on the moving substratefollows from the conservation of surface charge on themoving substrate surface. Accordingly [29,30], in thesteady-state condition the surface charge conservation
may be expressed as
uphqsqx¼ bq
qV
qy. (10)
In Eq. (10) uph is the photoreceptor speed and s is thesurface charge density on the photoreceptor. Based onGauss’s law, the surface charge at the dielectric substratecan be expressed as
s ¼ �subEin � �Esurf , (11)
where Ein is the magnitude of electric field intensity acrossthe substrate, esub is the substrate permittivity and Esurf isthe magnitude of normal component of the electric field atthe surface of substrate exposed to the corotron. Since thephotoreceptor thickness is very small, Ein is much greaterthan Esurf in most part of the substrate exposed to thecorotron. Thus, the second term in Eq. (11) can beneglected and the surface charge at the substrate can beexpressed as
sffi �subEin ¼ �V
td=k. (12)
In Eq. (12), td is the substrate thickness and k (k ¼ esub/e)is the substrate dielectric constant. Here, it is assumed thatthe other side of the substrate is grounded and k ¼ 3.3 wasused. Eq. (12) shows that for a fixed substrate thickness,the substrate surface charge and voltage are proportional.Using (12) in Eq. (10), the voltage boundary condition
for the photoreceptor is given as
qV
qx¼ðbqtd=kÞ
uph�
qV
qy. (13)
The numerical procedure for implementing of theboundary condition given by Eq. (13) through anFIDAPTM user-defined subroutine has been discussed indetail in [30]. A detailed proof for Eq. (13) was provided in[30] as well.
3. Numerical schemes and implementation in FIDAPTM
To solve the electrostatic governing equations and theircorresponding boundary conditions, FIDAPTM which is afinite element code was used. To overcome the possiblenumerical instabilities for high values of computationalelectrical Peclet number,1 the governing equations werediscretized by the upwind Petrov–Galerkin formulation[33]. The discretized form of the governing equations wassolved by a segregated solver that solves the equations foreach quantity separately, and reduces the computationaltime. Additional information regarding the procedure toset up the problem in FIDAPTM can be found in [34].
ARTICLE IN PRESS
Fig. 3. Charge density distributions on the wire for different wire–top
shield distances. Wire voltage ¼ 6000V.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720712
To impose the boundary conditions for the chargedensity on the wire and voltage on the photoreceptor, twoFORTRAN routines based on the procedure described in[30] were developed and implemented into a user-definedsubroutine of the package. It was found that the procedurewas somewhat sensitive to the initial guess for the chargedensity on the wire, and a large over-estimation may causethe solution to diverge. In those cases, the computationswere repeated with a smaller value of the initial guess. Thecomputational procedure was verified through comparisonwith the experimental data that was presented in [30].
4. Results and discussions
In this section, the numerical results are presented andthe effects of various parameters are discussed.
4.1. Effect of wire–top shield distance on substrate surface
charge
To analyze the effect of the distance between the wireand the top shield on the amount of substrate voltage/surface charge, a series of simulations were performed. Thegeometry of the corotron as shown in Fig. 1 was kept fixedwhile the distance from the wire to the top shield wasvaried between 4 and 13.75mm. The study was performedfor wire voltages of 5500 and 6000V. All the shields wereassumed to be grounded and their voltages were assumedto be zero. Unless stated otherwise, the substrate speed waskept fixed at 0.5m/s moving from the left to the right.
Figs. 2a and b show the voltage distribution along thesubstrate for different distances for wire voltages of 5500and 6000V, respectively. For both wire voltages, thevoltage distributions along the substrate are roughly thesame for the different distances. That is, while the distancedecreases to nearly one-third from 13.75 to 4mm, thesubstrate saturation voltage increases about 7% for boththe wire voltages. The increase for the wire voltage of5500V is slightly higher in percentage.
Fig. 2. Voltage distributions along the substrate for different wire–top sh
Fig. 3 shows the distributions of the charge density onthe wire for different wire–top shield distances for the wirevoltage of 6000V. These figures show that the distancebetween the wire and the top shield has significant effect onthe distribution of the charge density on the wire. As thedistance decreases, the level of the charge density even inthe bottom part of the wire (1801ofo3601), whichfaces the substrate, increases but the increase in the wirecharge density has little effects on the substrate voltagedistribution.To understand the reason for this seemingly contra-
dictory trend, the electric field lines for wire voltage of6000V and wire–top shield distances of 13.75, 6 and 4mmwere plotted in Figs. 4a–c, respectively. The magnifiedregions near to the wire are also shown in these figures.
ield distances. (a) Wire voltage ¼ 5500V. (b) Wire voltage ¼ 6000V.
ARTICLE IN PRESS
Fig. 5. Electrical potential and charge density contours for wire voltage of 6000V and different wire–top shield distances. Solid lines are for 13.75mm and
dash lines are for 4mm. (a) Electrical potential contours. Contours are between 0 and 6000V with an increment of 545V (b) Charge density contours.
Contours are between 0.1 and 1.3mC/m3 with an increment of 0.27mC/m3.
Fig. 4. Electric field lines for wire voltage of 6000V. (a) Wire–top shield distance ¼ 13.75mm. (b) Wire–top shield distance ¼ 6mm. (c) Wire–top shield
distance ¼ 4mm.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720 713
Neglecting the effect of airflow and ion diffusivity (whichare reasonable assumptions for range of operation ofcorotrons), the ions cannot cross any electric field line.Therefore, the current passing between each two electricfield lines remains constant. Comparing Figs. 4a–c,it is seen that the portion of the wire charging the subs-trate shrinks as the wire–top shield distance decreases.(The region shrinks from 208.51ofo323.91 to 224.61ofo311.11 when the distance between the wire and the top
shield decreases from 13.75 to 4mm). Thus, while decrea-sing of the distance between the wire and the top shieldincreases the level of the charge density of the wire (even atits side facing the substrate), the electric field lines in thecorotron are changed in a way that a smaller portion of thewire charges the substrate. Consequently, the currentdensity received by the substrate and the substratevoltage/surface charge increase slightly. In other words,when the wire–top shield distance decreases, the ions
ARTICLE IN PRESSP. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720714
emitted form the lower part of the wire acquire a highertendency to move up toward the top shield rather than thesubstrate.
For wire voltage of 6000V, Figs. 5a and b show theelectrical potential and charge density contours fordifferent wire–top shield distances. The solid and dashlines in these figures are for 13.75 and 4mm wire–top shielddistances, respectively. Fig. 5a shows that the wire–topshield distance significantly affects the distribution ofelectrical potential in the region above the wire. However,the electrical potential in the region below the wire, inparticular near the substrate, is not affected substantiallyby the wire–top shield distance. Despite the significantdifference in charge density profiles on the wire, Fig. 5bshows that the charge density contours are just slightlydifferent near the substrate for both wire–top shielddistances. The charge density distributions in the otherregions of the corotron are, however, significantly different.
As noted before, the distance between the wire and thesubstrate for the cases studied in Figs. 2–5 was 8.3mm. Theanalyses were repeated when the distance was decreased to6.3mm, and the effect of variation of wire–top shielddistance on the voltage distribution on the substrate wasalso studied. For those cases again the simulation resultsshowed that the distance between the wire and the topshield does not have a significant effect on the voltage levelat the substrate, and therefore the results were not shownhere. It may then be concluded that for a rectangular singlewire corotron, the distance between the wire and the topshield affects the charge density distribution on the wirebut it does not affect the level of voltage/surface charge onthe substrate. This conclusion is of importance in that itprovides flexibility for designers in changing the distancebetween the wire and the top shield.
It should be noted that based on the corona voltage,there is a minimum value for the distance between the wireand the top shield to prevent the corona from sparking.
Fig. 6. Voltage and charge density distributions for different wire–substrate dis
(b) Charge density distribution on the wire.
This distance, however, cannot be determined directly fromthe computational model used in this study. Thus,experimental measurements or other theoretical model isneeded to determine the distance for initiation of sparkingin a corotron with a specified wire voltage. The presentedresults, however, show that the potential for sparkingincreases when the wire–top shield distance becomes short.More specifically, for the wire voltage of 6000V, Fig. 3shows that when the wire is at a distance of 4mm from thetop shield the level of charge density at the top of the wireincreases significantly. Because of that the distances lessthan 4mm were not analyzed in this study.
4.2. Effect of wire–substrate distance on substrate surface
charge
In this section the effect of the distance between the wireand the substrate on the voltage/surface charge profile onthe moving substrate is studied. Simulations were per-formed when the distance between the wire and thesubstrate was decreased from 8.3 to 7.3mm and 6.3mm,while the other dimensions of the corotron as shown inFig. 1 were kept the same. The wire voltage was assumed tobe 5500V for all the cases studied.Fig. 6a shows the voltage distribution on the substrate
for the different wire–substrate distances. It is seen that thedistance between the wire and the substrate significantlyaffects the substrate voltage. This figure shows that whenthe wire–substrate distance decreases by about 25%, thesubstrate saturation voltage increases more than 80%.Fig. 6b shows the distribution of the charge density on
the wire for the different cases. It is seen that decreasing thedistance between the wire and the substrate leads tosignificant increase in the level of the charge density at thebottom of the wire (1801ofo3601).For the wire–substrate distance of 8.3 and 6.3mm, the
examination of the electric field lines near to the wire
tances and wire voltage of 5500V. (a) Voltage distribution on the substrate.
ARTICLE IN PRESSP. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720 715
showed that for 8.3mm distance, the segment of thewire charging the substrate was 2091ofo3331, whilefor 6.3mm distance, the region was expanded to 1911ofo 3341.
Therefore, decreasing the wire–substrate distance notonly increases the level of the charge density at the bottompart of the wire, but also the part of the wire surface thatcharges the substrate expands. As a result, the level ofvoltage/surface charge on the substrate increases signifi-cantly. Since the substrate surface charge is sensitive to thewire–substrate distance, the wire vibration control findsaspects in stabilizing the substrate surface charge as well.Information regarding the wire vibration by electric forcein positive corona devices can be found in [14].
4.3. Effect of wire–side shields distance on substrate surface
charge
In this section the effects of the distance between the wireand the side shields on the voltage/surface charge distribu-tion on the substrate is investigated. To analyze the effectof the wire–side shields distance on the level of the voltage/surface charge of the substrate, several simulations withdifferent distances in a range of 7–11mm for the corotronmodel shown in Fig. 1 were performed. The wire voltagewas assumed to be 5500V for all the simulated cases.
Fig. 7a shows the voltage profile along the substrate fordifferent wire–side shield distances. It is seen that bydecreasing the wire–side shields distance, the level of thevoltage on the substrate decreases slightly. That is, whenthe distance is decreased by about 36%, the substrate exitvoltage decreases by about 4%.
Fig. 7b shows that decreasing the distance between thewire and the side shields leads to substantial increase of thecharge density on the entire surface of the wire. To providean understanding of this discrepancy, the electric fieldlines for wire–side shield distances of 11 and 7mm were
Fig. 7. Voltage and charge density distributions for different wire–side shiel
substrate. (b) Charge density distribution on the wire.
evaluated. Examination of the electric lines shows that thepart of the wire that charges the substrate shrinks from202.21ofo330.51 for 11mm to 2361ofo301.81 for7mm. Thus, despite substantial increase in the level ofthe charge density on the wire, the voltage on the substratedecreases slightly.It may then be concluded that the wire–side shields
distance does not have a significant effect on the level of thevoltage/surface charge on the substrate.
4.4. Effect of insulated shields on substrate voltage
For certain applications, some of the shields of corotronsare electrically insulated. In this section effect of insulatingthe top shield on the substrate voltage is studied. For thecorotron model shown in Fig. 1, the top shield wasassumed to be electrically insulated (which mathematicallyis equivalent with qV=qn ¼ 0 at the shield surface), andsimulations were performed for wire–top shield distancesof 13.75, 6, 4 and 3.25mm. The wire voltage was assumedto be 6000V in these simulations.Figs. 8a and b show the voltage profile along the
substrate and the charge density distribution on the wirefor these four cases, respectively. It is seen that for theinsulated shield, decreasing the wire–top shield distancedoes not change the distribution of the voltage on thesubstrate to a noticeable extent. Fig. 8b, however, showsthat by decreasing the wire–top shield distance from 13.75to 6mm, the amount of the charge density on the top of thewire decreases significantly. Further decrease of thedistance to 4mm leads to the appearance of a non-discharging region on the top of the wire. The non-discharging region becomes larger when the wire–top shielddistance is reduced to 3.25mm. The amount of the chargedensity on the bottom of the wire, however, increases onlyslightly when the wire–top shield distance decreases. Closeexamination of the electric lines (not shown here due to
d distances for a wire voltage of 5500V. (a) Voltage distribution on the
ARTICLE IN PRESS
Fig. 8. Voltage and charge density distributions for insulated top shield with different wire–top shield distances for a wire voltage of 6000V. (a) Voltage
distribution on the substrate. (b) Charge density distribution on the wire.
Fig. 9. Electrical potential contours for the wire voltage of 6000V and wire–top shield distance of 4mm. Contours are between 0 and 6000V with an
increment of 600V. The solid lines are for the insulated top shield and the dash lines are for the grounded one.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720716
space limitation) shows that the moderate increase of thecharge density at the bottom of the wire is accompaniedwith the shrinking of the part of the wire charging thesubstrate, to the extent that the voltage distribution on thesubstrate remains roughly unchanged. It may then beconcluded that the wire–top shield distance for insulatedtop shields does not affect the voltage/surface chargeprofile along the substrate despite the substantial changeon the charge density profile on the top of the wire.
Comparing Fig. 8a with Fig. 2b shows that there is not asignificant difference in the substrate voltage for corotronswith grounded or insulated top shields when the otherparameters are the same.
Although insulating the top shield does not affect thedistribution of the voltage along the substrate, it affects thedistribution of the charge density and the electricalpotential inside of the device significantly. Figs. 9–11 showelectrical potential and charge density distributions andelectric field lines for corotrons with grounded andinsulated top shield. For those figures, the wire–top shielddistance was 4mm and the wire voltage was kept fixed at6000V. It is seen that insulating the top shield substantiallyalters the electrical potential, the charge density and thepattern of the electric field lines in the region on the top ofthe wire, while it does not affect the distributions in the
region near the substrate significantly. This may cause asubstantial difference in the airflow inside of the corotron.Analysis of the airflow (which is important for corotroncontamination analysis, wire vibration and ozone transportanalysis), however, is not the subject of this study.Fig. 12 shows the distribution of the charge density for
the case with the wire–shield distance of 3.25mm. Due tothe non-discharging portion of the wire, a free of chargeregion at the top of the wire is seen in the figure.In this study Kaptsov’s hypothesis was implemented
through an iterative method described in [30]. The resultspresented in this section show that the method is capable ofcapturing the non-discharging regions on the electrodes incorona devices. (The non-discharging region may occurwhen an electrode and an insulator or two electrodes areclose to each other.) It should be noted here that theiterative method with different parameters was earlier usedby Medlin et al. [18,19] for electrostatic precipitators andhigh-voltage DC lines.
4.5. Effect of substrate dielectric thickness on substrate
surface charge
As mentioned in the Introduction, corotrons are used toelevate the substrate surface charge to a certain amount for
ARTICLE IN PRESS
Fig. 10. Charge density contours for wire voltage of 6000V and wire–top shield distance of 4mm. (a) Grounded top shield, contours are between 1 and
8mC/m3 with an increment of 1mC/m3. (b) Insulated top shield, contours are between 0.1 and 1.3 mC/m3 with an increment of 0.15mC/m3.
Fig. 11. Electric field lines for wire voltage of 6000V and wire–top shield distance of 4mm. The black lines are for the insulated shield and the gray lines
are for the grounded one.
Fig. 12. Charge density contours in mC/m3 for a case with wire voltage of 6000V and wire–top shield distance of 3.25mm.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720 717
the toner transfer process. The substrate thickness for thecases presented in the earlier sections was kept fixed. Thus,the substrate voltage and the surface charge were propor-tional to one another and the trend of variation of one wassimilar to the other one. The substrate thickness affectsEqs. (12) and (13) and thus the substrate voltage andsurface charge.
In all the cases described earlier, the substrate thicknesswas 25 mm. To study the effects of the thickness, twosimulations with different substrate thicknesses as 20 and30 mm were performed. The wire voltage was assumed to be6000V for all the thicknesses.Fig. 13a shows that an increase in the substrate thickness
leads to a proportional increase of the substrate voltage.
ARTICLE IN PRESS
Fig. 13. Effects of substrate thickness on the distribution of electrical quantities on the wire and the substrate. (a) Voltage on the substrate. (b) Surface
charge on the substrate. (c) Charge density on the wire.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720718
Fig. 13b, however, shows that by increasing the thickness,the substrate surface charge decreases. That is, thevariation of the substrate thickness has opposingeffects on the substrate surface charge and voltage. Itshould be noted here that the substrate surface charge(not voltage) is normally considered as the design para-meter for the corotrons. Fig. 13b shows that the increase inthe substrate surface charge is about 10% when thethickness decreases by about 50%. Thus, the substratesurface charge is moderately sensitive to variation ofsubstrate thickness.
Fig. 13c shows that increasing the substrate thicknessleads to a decrease in the wire charge density andconsequently the current issued by the wire. That isexpected, since the substrate voltage increases by thethickness and the higher voltage of the substrate creates astronger resistance for the ion injection from the wire. Sincethe substrate surface charge decreases by the thickness, itcan be concluded that the current issued by the wire andthe current received by the substrate both decrease by thethickness.
4.6. Effect of substrate speed on substrate surface charge
In this section the effect of the substrate speed on thesubstrate surface charge is discussed and additional detailscompared with the earlier works [29,30] are provided. Forthe corotron geometry shown in Fig. 1, with a wire–topshield distance of 6mm, several simulations were per-formed where the substrate speeds were 0.3, 0.4 and0.5m/s. Here the substrate thickness was kept fixed at25 mm. The simulation results for the substrate voltageprofile and the wire charge density are, respectively, shownin Figs. 14a and b.It is seen from Fig. 14a that decreasing the substrate
speed leads to proportional increase in the substratevoltage/surface charge. (That is, the voltage increases byabout 40% when the speed is reduced by 40%.). Fig. 14bshows at higher substrate speeds, the level of charge densityon the bottom of the corotron wire increases. This isexpected since by decreasing the substrate voltage, theresistance against ion injection decreases and consequentlythe current from the wire increases. In fact, both the
ARTICLE IN PRESS
Fig. 14. Effect of substrate speed on substrate voltage and wire charge density. (a) Substrate voltage profiles. (b) Wire charge density profiles.
Fig. 15. Effect of substrate speed on electrical potential and charge density contours in the corotron. Solid and dash lines are for substrate speeds of 0.3
and 0.5m/s, respectively. (a) Electrical potential contours from 500 to 6000V with an increment of 500V. (b) Charge density contours from 0.15 to
3.22mC/m3 with an increment of 0.28mC/m3.
P. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720 719
currents issued from the wire and received by the substrateincrease as the substrate speed increases. (Assuming thesubstrate surface charge is zero at point A in Fig. 1, byintegrating Eq. (10) along the substrate, the current perunit of wire length received by the substrate is uphsexit,where sexit is the substrate saturated surface charge. Thus,the currents per unit of wire length received by thesubstrate for substrate speeds of 0.5, 0.4 and 0.3m/s are,respectively, 0.402, 0.377 and 0.340mA/m.) By increasingthe speed, however, each point on the substrate isexposed to the discharging corotron wire for a shorterperiod of time and thus the amount of accumulated surface
charge/voltage on each point decreases. It should be notedthat substrate speed is typically an input design parameter,thus the other parameters in corotrons are adjusted toachieve the required level of surface charge on the substrate.Fig. 15 shows the distributions of electrical potential and
charge density in the corotron for different substratespeeds. From Fig. 15a, it is seen that the differencesbetween the electric potential contours for differentsubstrate speeds are significant in the lower right side ofthe corotron. This is due to the fact that this region is closerto the highest level of substrate voltage. Fig. 15b showsthat the charge density in the region below the wire
ARTICLE IN PRESSP. Zamankhan et al. / Journal of Electrostatics 65 (2007) 709–720720
increases moderately with the higher substrate speed. Asnoted before, this is due to the higher level of chargedensity at the lower portion of the wire.
5. Conclusions
In this paper the effects of different corotron parameterson the substrate surface charge were studied through aseries of numerical simulations. On the basis of the resultspresented for a positive rectangular single wire corotron,the following conclusions are drawn:
�
The distances between the wire and the shields andinsulation of the top shield alter the distribution of theelectrical quantities inside of the corotron substantially, butonly slightly affect the substrate voltage/surface charge level. � The distance between the wire and the substrate, in additionto the wire voltage and the substrate speed, affects the levelof the substrate voltage/surface charge significantly.
� The substrate surface charge varies moderately by thesubstrate thickness.
Acknowledgments
This work was supported in part by Xerox Foundationand by the New York State Office of Science, Technologyand Academic Research (NYSTAR) (through the Centerfor Advanced Materials Processing, CAMP, of ClarksonUniversity). The authors would also like to thank theFluent Corporation for making the FIDAPTM codeavailable. Thanks are also given to the anonymousreviewers for their constructive suggestions.
References
[1] G.W. Penney, R.E. Matick, Potentials in D-C corona fields, Trans.
AIEE. 79 (1960) 91–99.
[2] J.R. Macdonald, B.S. Wallace, H.W. Spencer III, L.E. Sparks,
A mathematical model for calculating electrical conditions in
wire–duct electrostatic precipitation devices, J. Appl. Phys. 48 (1977)
2231–2243.
[3] T. Yamamoto, H.R. Velkoff, Electrohydrodynamics in an electro-
static precipitator, J. Fluid Mech. 108 (1981) 1–18.
[4] T. Takuma, T. Tsutomu, T. Kawamoto, Calculation of ion flow fields
of HVDC transmission lines by the finite element method, IEEE
Trans. Power Appar. Syst. PAS-100 (1981) 4802–4810.
[5] M. Hara, N. Hayashi, K. Shiotsuki, M. Akazaki, Influence of wind
and conductor potential on distribution of electric field and ion
current density at ground level in DC high voltage line to plane
geometry, IEEE Trans. Power Appar. Syst. PAS-101 (1982) 803–811.
[6] G.L. Leonard, M. Mitchner, S.A. Self, An experimental study of the
electrohydrodynamic flow in electrostatic precipitators, J. Fluid
Mech. 127 (1983) 123–140.
[7] M. Abdel-Salam, M. Farghally, S. Abdel-Sattar, Finite element
solution of monopolar corona equation, IEEE Trans. Electr. Insul.
18 (2) (1983) 110–119.
[8] G.A. Kallio, D.E. Stock, Computation of electrical conditions inside
wire–duct electrostatic precipitator using a combined finite-element,
finite difference technique, J. Appl. Phys. 59 (1986) 1799–1806.
[9] J.L. Davis, J.F. Hoburg, VDC transmission line computations using
finite element and characteristics methods, J. Electrostat. 18 (1986) 1–22.
[10] P. Atten, F.M.J. McCluskey, A.C. Lahjomri, Electrohydrodynamics
origin of turbulence in electrostatic precipitators, IEEE Trans. Ind.
Appl. IA-23 (1987) 705–711.
[11] G. Giovanni, R.D. Graglia, Two dimensional finite boxes analysis of
monopolar corona fields including ion diffusion, IEEE Trans. Magn.
26 (2) (1990) 567–570.
[12] G.A. Kallio, D.E. Stock, Interaction of electrostatic and fluid
dynamic fields in wire–plate electrostatic precipitators, J. Fluid
Mech. 240 (1992) 133–166.
[13] M. Yu, E. Kuffel, J. Poltz, A new algorithm for calculating HVDC
corona with presence of wind, IEEE Trans. Magn. 28 (5) (1992)
2802–2804.
[14] R.K. Klechner, G.A. Domoto, Dynamics of a wire in positive corona
discharge, Proc. SPIE 1912 (1993) 210–237.
[15] M. Abdel-Salam, D. Wiitanen, Calculation of corona onset voltage for
duct-type precipitators, IEEE Trans. Ind. Appl. 29 (2) (1993) 274–280.
[16] K. Adamiak, Adaptive approach to finite element modeling of corona
fields, IEEE Trans. Ind. Appl. 30 (2) (1994) 387–393.
[17] M. Abdel-Salam, Z. Al-Hamouz, Analysis of monopolar ionized field as
influenced by ion diffusion, IEEE Trans. Ind. Appl. 31 (3) (1995) 484–493.
[18] A.J. Medlin, C.A.J. Fletcher, R. Morrow, A pseudotransient
approach to steady state solution of electric field–space charge
coupled problems, J. Electrostat. 43 (1998) 39–60.
[19] A.J. Medlin, R. Morrow, C.A.J. Fletcher, Calculation of monopolar
corona at a high voltage DC transmission line with crosswinds,
J. Electrostat. 43 (1998) 61–77.
[20] A. Soldati, S. Banerjee, Turbulence modification by large scale organized
electrohydrodynamic flows, Phys. Fluids 10 (7) (1998) 1742–1756.
[21] X. Li, I.R. Ciric, M.R. Raghuveer, Investigation of ionized fields due
to bundled unipolar DC transmission lines in presence of wind, IEEE
Trans. Power Deliv. 14 (1) (1999) 211–217.
[22] J.Q. Feng, Application of Galerkin finite-element method with
Newton iterations in computing steady-state solutions of unipolar
charge currents in corona devices, J. Comput. Phys. 151 (1999)
969–989.
[23] J.Q. Feng, Electrohydrodynamic flow associated with unipolar
charge current due to corona discharge from a wire enclosed in a
rectangular shield, J. Appl. Phys. 86 (5) (1999) 2412–2418.
[24] Z.M. Al-Hamouz, A combined algorithm based on finite elements
and a modified method of characteristics for the analysis of the
corona in wire–duct electrostatic precipitators, IEEE Trans. Ind.
Appl. 38 (1) (2002) 43–49.
[25] C.U. Bottner, The role of the space charge density in particulate
processes in the example of the electrostatic precipitator, Powder
Technol. 135–136 (2003) 285–294.
[26] H. Lei, L.Z. Wang, Z.N. Wu, Applications of upwind and downwind
schemes for calculating electrical conditions in a wire–plate electro-
static precipitator, J. Comput. Phys. 193 (2004) 697–707.
[27] A. Caron, L. Dascalescu, Numerical modeling of combined
corona–electrostatic fields, J. Electrostat. 61 (2004) 43–55.
[28] P. Zamankhan, G. Ahmadi, F.-G. Fan, Coupling effects of the flow
and electric fields in electrostatic precipitators, J. Appl. Phys. 96
(2004) 7002–7010.
[29] J.Q. Feng, P.W. Morehouse Jr., J.S. Facci, Steady state corona
charging behavior of a corotron over a moving dielectric substrate,
IEEE Trans. Ind. Appl. 37 (6) (2001) 1651–1657.
[30] P. Zamankhan, G. Ahmadi, F.-G. Fan, Variation of airflow and
electric fields in a corona device charging against a moving dielectric
substrate, J. Imaging Sci. Technol. 50 (4) (2006) 375–385.
[31] N.A. Kaptsov, Elktricheskie Yavleniya v Gazakh Vakuume, OGIZ,
Moscow, 1947.
[32] F.W. Peek Jr., Dielectric Phenomena in High Voltage Engineering,
McGraw-Hill, New York, 1929.
[33] O.C. Zienkiewicz, R.L. Taylor, Finite Element Method, vol. 3, Fluid
Mechanics, Butterworth-Heinemann, London, 2000.
[34] FIDAP User’s Manual, Version 8.7, Fluent Inc., Evanston, IL, 2003.
