Empirical and Numerical Analysis of Conduction and Induction Charging of Droplets in a Three-Electrode System

Article Empirical and Numerical Analysis of Conduction and Induction Charging of Droplets in a Three-Electrode System

Adam Pelesz * and Tomasz Czapka *

Department of Electrical Engineering Fundamentals, Faculty of Electrical Engineering, Wrocław University of Science and Technology, 50-370 Wrocław, Poland * Correspondence: [email protected] (A.P.); [email protected] (T.C.)

Received: 20 November 2019; Accepted: 16 January 2020; Published: 18 January 2020

Abstract: The charging process of water droplets in different electrification systems using empirical and numerical analysis is described. A simplified model based on capacitance distribution is proposed. The results of the experimental studies and numerical calculations enable comparison of the induction and conduction charging methods in terms of the droplet charging level, described by the Q/m parameter. Research findings indicate that the conduction method in a 3-electrode system is the most effective among all the considered methods. It has been also shown that the application of a 2-electrode system under certain conditions provides greater Q/m parameter values in comparison to the system with three electrodes using the induction method. The research results can be used to develop charging systems (e.g., nozzles) that produce streams with electrically charged liquid particles.

Keywords: electroaerosol; electriﬁcation of droplets; induction electriﬁcation

1. Introduction Charging of aerosol particles is widely used in many technological processes [1–4]. Among the most common applications, one can mention agriculture (e.g., pesticide application and artificial pollination) [5–9], medicine (e.g., inhalation medications) [10,11], industry (e.g., painting, xerography, and food industries) [12–15], and others (e.g., thin film deposition, cosmetology, insecticide spraying) [16–18]. The use of electroaerosols improves the efficiency of a given process, for example providing more uniform object coverage and increasing the coating efficiency [19–21]. In general, there are several methods of aerosol charging, such as induction, conduction, corona discharge, postdispersion, and triboelectrification methods [2,12,13,22–27]. Induction and conduction charging methods are commonly used for conducting liquids. Furthermore, both methods can be implemented in two- or three-electrode systems. All mentioned methods, which require an external high voltage source, are schematically shown in Figure1. It should be noted that both conduction and induction methods are based on the principle of electrostatic induction. The key difference between them is the electric potential of the sprayed liquid. In the case of the induction method, the sprayed liquid is charged while being on ground potential. On the contrary, in the case of the conduction method, the liquid is charged at high potential (negative or positive). For a 2-electrode system, the induction and conduction methods are equivalent in terms of electric field distribution (i.e., if the influence of other surrounding objects, which may act as additional electrodes, are omitted). Moreover, in the case of a 3-electrode system, despite the same geometry (i.e., electrode configuration), there are some important differences, including in the electric field distribution, energy conversion, and the space charge effect. A more detailed description of the mentioned differences can be found in [25].

Energies 2020, 13, 469; doi:10.3390/en13020469 www.mdpi.com/journal/energies Energies 2020, 13, x FOR PEER REVIEW 2 of 15

field distribution, energy conversion, and the space charge effect. A more detailed description of the Energies 2020, 13, 469 2 of 15 mentioned differences can be found in [25].

Figure 1. SchematicSchematic view view of of aerosol aerosol charging charging methods, methods, requiring requiring external external high high voltage voltage source: source: (a) (inductiona) induction method method in in a a 2-electrode 2-electrode system, system, (b) induction(b) induction method method in a 3-electrode in a 3-electrode system, (c )system, conduction (c) methodconduction in a method 2-electrode in a 2 system,-electrode (d) system, conduction (d) conduction method in amethod 3-electrode in a 3 system,-electrode (e) system, corona discharge(e) corona method,discharge and method, (f) postdispersion and (f) postdispersion method. The method. main components The main components of the system of theare: system N—conducting are: N— nozzleconducting acting nozzle as electrode; acting as S—streamelectrode; S of—stream liquid; of A—aerosol; liquid; A— T—target;aerosol; T— HV—hightarget; HV voltage—high voltage source; C—coronasource; C— electrode;corona electrode R—ring; R electrode;—ring electrode G—grid; G electrode.—grid electrode.

To comparecompare didifferentfferent chargingcharging systems,systems, aa simplesimple capacitancecapacitance modelmodel andand electrostaticelectrostatic numericalnumerical simulations have have been been proposed. proposed. Furthermore, Furthermore, the suggested the suggested concept concept seems seems to be su toffi cient be sufficient to estimate to theestimate value the of Q value/m at lowof Q liquid/m at flowlow rates.liquid Theflow parameter rates. TheQ /parameterm provides Q an/m unambiguous provides an comparisonunambiguous of thecomparison individual of chargingthe individual systems charging from the systems point offrom view the of point their of effi viewciency. of their efficiency. InIn our our research, research, both both arrangements arrangements are are analyzed analyzed only only in terms in terms of the ofelectrostatic the electrostatic field. In field. general, In q E thegeneral, induced the surfaceinduced charge surface density charge sidensityof conducting qsi of conducting objects is objects proportional is proportional to electric to field electricacting field on E thatacting object. on that For object. an object For an in aobject dielectric in a dielectric medium medium such as air,such for as which air, for relative which permittivityrelative permittivity may be q assumedmay be assumed as unitary as εunitaryr = 1, siεrcan = 1, beqsi givencan be by: given by:

qsi푞푠푖= =ε0 E휀0.퐸. (1)(1)

To obtain the total charge Qi induced on the object (e.g., a droplet), one should integrate the To obtain the total charge Qi induced on the object (e.g., a droplet), one should integrate the charge overcharge the over entire the surface entire surfaces: s: Z Qi = qsids. (2) 푄 = ∫ 푞 d푠. (2) 푖 s 푠푖 푠 For a conducting object, the induced charge Qi is a result of the object–environment capacitance C and theFor resulting a conducting potential object, di fftheerence inducedV: charge Qi is a result of the object–environment capacitance C and the resulting potential difference V: Q = V C. (3) i · 푄 = 푉 ∙ 퐶. (3) 2. Capacitance Distribution in the Charging Systems푖 When spraying with water and aqueous solutions (e.g., pesticides), a droplet can be effectively 2. Capacitance Distribution in the Charging Systems treated as a conducting object. This is due to the fact that the droplet is affected by an electric field longerWhen than spraying its material with charge–transfer water and aqueous time solutions constant τ(e.g.[1,2, ].pesticides), For the mentioned a droplet liquids,can be effectively this time constanttreated as can a conducting be lower than object. 1 µs This [2]. As is due can beto seenthe fact from that Equation the droplet (3), the is affected electrical by capacitance an electric of field the systemlonger than used its during material the charging charge–transfer process istime the constant main factor τ [1,2] (i.e.,. apartFor the from mentioned the electric liquids, field relatedthis time to theconsta potentialnt can dibeff lowererence thanV) that 1 μs directly [2]. As influencescan be seen the from value Equation of the charge(3), the induced electrical on capacitance the droplet. of The the capacitancesystem used distributions during the charging in two- and process three-electrode is the main systems factor (i.e. are, schematicallyapart from the shown electric in field Figure related2. to the potential difference V) that directly influences the value of the charge induced on the droplet.

Energies 2020, 13, x FOR PEER REVIEW 3 of 15

EnergiesThe capacitance2020, 13, 469 distributions in two- and three-electrode systems are schematically shown in Figure3 of 15 2.

Figure 2. Capacitance distribution in a (a) 2-electrode system and (b) 3-electrode system. Note: T—target; Figure 2. Capacitance distribution in a (a) 2-electrode system and (b) 3-electrode system. Note: T— R—ring electrode; D—droplet; N—nozzle; C2nt—capacitance between nozzle and target in a 2-electrode target; R—ring electrode; D—droplet; N—nozzle; C2nt—capacitance between nozzle and target in a 2- system; C2dt—capacitance between droplet and target in 2-electrode system; C3nt—capacitance between electrode system; C2dt—capacitance between droplet and target in 2-electrode system; C3nt— nozzle and target in a 3-electrode system; C3nr—capacitance between nozzle and ring electrode in capacitance between nozzle and target in a 3-electrode system; C3nr—capacitance between nozzle and a 3-electrode system; C3dr—capacitance between droplet and ring electrode in a 3-electrode system; ring electrode in a 3-electrode system; C3dr—capacitance between droplet and ring electrode in a 3- C3dt—capacitance between droplet and target in a 3-electrode system; C3rt—capacitance between ring electrodeelectrode and system target; C in3dt a— 3-electrodecapacitance system. between droplet and target in a 3-electrode system; C3rt— capacitance between ring electrode and target in a 3-electrode system. In the presented model, the influence of any other objects potentially appearing in the surrounds of theIn system the presented was omitted. model, To the simplify influence the of analysis, any other it was objects assumed potentially that all appearing other objects in the are surround far froms theof the electrification system was setup.omitted. Thus, To simplify capacitances the analysis, associated it was with assumed such objects that all were other negligibly objects are small far from and excludedthe electrification from further setup. analysis. Thus, capacitances In the real setup, associated however, with the such impact objects of suchwere capacitancesnegligibly small may and be significantexcluded from (especially further inanalysis. a 2-electrode In the systemreal setup, when however, the target the isimpact far away of such from capacitances a droplet, because may be thesignificant capacitance (especiallyC2dt can in thena 2-electrode be comparable system towh theen droplet–environmentthe target is far away from capacitance). a droplet, It because should the be notedcapacitance that charge C2dt can induced then be on comparable the droplet to depends the droplet only–environment on capacitances capacitance). associated It withshould the be droplet noted (i.e.,that Ccharge2dt, C3 dtinduced, C3dr). Theon the capacitances droplet dependsC2nt, C 3onlynt, C 3onnr, andcapacitancesC3rt do not associated directly influencewith the droplet the droplet (i.e., chargingC2dt, C3dt, process.C3dr). The However, capacitances those C additional2nt, C3nt, C3nr capacitances, and C3rt do were not directly marked influence because they the constitutedroplet charging part of theprocess. analyzed However, systems. those It wasadditional also assumed capacitances that the were conductive marked dropletbecause was they always constitute in contact part of with the theanalyzed conductive systems. nozzle It was (i.e., also a droplet assumed and that a nozzle the conduc togethertive form droplet a single was electrode), always in andcontact thus with droplet the potentialconductive was nozzle well determined.(i.e., a droplet and a nozzle together form a single electrode), and thus droplet potentialIn a 2-electrodewas well determined. system, regardless of the charging method, droplet-induced charge Q2el can be givenIn by: a 2-electrode system, regardless of the charging method, droplet-induced charge Q2el can be given by: Q = V C = V C , (4) 2el 2dt· 2dt · 2el where V2dt is the potential difference푄 between2푒푙 = 푉2 a푑푡 droplet ∙ 퐶2푑푡 and= 푉 a∙ target퐶2푒푙, in a 2-electrode system (i.e., the(4) charging voltage V), Q2el is the induced charge in a case of a 2-electrode system, and C2el is the total where V2dt is the potential difference between a droplet and a target in a 2-electrode system (i.e., the capacitance associated with a droplet in the 2-electrode system (in this case C2el = C2dt from Figure2). charging voltage V), Q2el is the induced charge in a case of a 2-electrode system, and C2el is the total In general, for a 3-electrode system, the expression for induced charge Q3el is given by: capacitance associated with a droplet in the 2-electrode system (in this case C2el = C2dt from Figure 2). In general, for a 3-electrode system, the expression for induced charge Q3el is given by: Q = V C + V C , (5) 3el dr· 3dr 3dt· 3dt 푄3푒푙 = 푉푑푟 ∙ 퐶3푑푟 + 푉3푑푡 ∙ 퐶3푑푡, (5) where Vdr is the potential difference between a droplet and a ring electrode, and V3dt is the potential diwherefference Vdr betweenis the potential a droplet difference and a target between in a 3-electrodea droplet and system. a ring electrode, and V3dt is the potential differenceIn the casebetween of the a inductiondroplet and method, a target there in a is3- noelectrode potential system. difference between a droplet and a target V3dt =In0, the and case the of potential the induction difference method, between there a droplet is no potential and a ring difference electrode between is equal a to droplet the charging and a voltagetarget VV3dtdr == 0,V .and Therefore, the potential the induced difference charge between is: a droplet and a ring electrode is equal to the charging voltage Vdr = V. Therefore, the induced charge is: Q3el_ind = Vdr C3dr = V C3el_ind. (6) 푄3푒푙_푖푛푑 = 푉푑푟 ·∙ 퐶3푑푟 = · 푉 ∙ 퐶3푒푙_푖푛푑. (6)

Energies 2020, 13, x FOR PEER REVIEW 4 of 15

On the other hand, in the case of conduction method, the potential difference between a droplet and a target is equal to the potential difference between a droplet and a ring electrode, and equal to the charging voltage (i.e., V3dt = Vdr = V). Consequently, the induced charge is given by:

푄3푒푙_푐표푛푑 = 푉 ∙ 퐶3푑푟 + 푉 ∙ 퐶3푑푡 = 푉 ∙ (퐶3푑푟 + 퐶3푑푡) = 푉 ∙ 퐶3푒푙_푐표푛푑, (7) where Q3el_ind is the induced charge in a case of a 3-electrode system and induction method, Q3el_cond is the induced charge in a case of a 3-electrode system and conduction method, C3el_ind is the total capacitance associated with a droplet in the 3-electrode system for induction method (i.e., C3el_ind = C3dr from Figures 2 and 3), and C3el_cond is the total capacitance associated with a droplet in the 3-electrode Energies 2020, 13, 469 4 of 15 system for induction method (i.e., C3el_cond = (C3dr + C3dt) from Figures 2 and 3). Considering that a capacitance is a function of the geometry of the electrodes and the dielectric propertiesOn the of other the medium hand, in between the case them, of conduction it is important method, to thenote potential that the dipotentialﬀerence difference between abetween droplet andelectrodes a target does is equal not affect to the the potential capacitance diﬀerence distribution between in a the droplet system and or a the ring value electrode, of these and capacitances. equal to the chargingHowever, voltage if there (i.e., is noV potential3dt = Vdr =diVfference). Consequently, between the the electrodes, induced charge no charge is given can by: be induced, so the effect of this capacitance can be omitted. The concept of so-called effective capacitance distribution only applies to the capacitancesQ = VbetweenC + VelectrodesC = V with(C a+ potentialC ) = Vdifference.C , Alternatively, it can (7) 3el_cond · 3dr · 3dt · 3dr 3dt · 3el_cond be assumed that the electrodes on the same potential are in a galvanic contact with each other, whereformingQ 3oneel_ind singleis the electrode. induced charge The concept in a case of ofcapacitance a 3-electrode between system such and electrodes induction becomes method, virtuallyQ3el_cond ismeaningless. the induced charge in a case of a 3-electrode system and conduction method, C3el_ind is the total capacitanceEffective associated capacitance with adistributi droplet inons the for 3-electrode induction system and conductionfor induction methods method (i.e., are C presented3el_ind = C 3 indr fromFigure Figures 3. In 2 the and following3), and C considerations,3el_cond is the total it capacitance is assumed associated that 3-electrode with a droplet systems in have the 3-electrode the same systemgeometry. for induction method (i.e., C3el_cond = (C3dr + C3dt) from Figures2 and3).

Figure 3. 3.E Effectiveﬀective capacitance capacitance distribution: distribution: (a) induction(a) induction method method in a 2-electrode in a 2-electrode system, (b system) induction, (b) methodinduction in method a 3-electrode in a system,3-electrode and system, (c) conduction and (c) method conduction in a 3-electrode method in system. a 3-electrode Capacitances system. associatedCapacitances directly associated with dropletsdirectly with are marked droplet withs are amarked thicker with line. a thicker line.

ConsideringThe capacitance that distributiona capacitance in is a a 2function-electrode of the system geometry is the of same the electrodes regardless and of the dielectric charging propertiesmethod. In of the the case medium of a 3- betweenelectrode them, system, it is capacitance important tovalues note do that not the change, potential however difference the betweeneffective electrodescapacitance does distribution not affect varies the capacitance because of distribution the change inin thethe systempotential or values. the value In ofa 3 these-electrode capacitances. system However,where the ifinduction there is nomethod potential is due diff toerence the lack between of potential the electrodes, difference no between charge cana droplet be induced, and a target, so the effect of this capacitance can be omitted. The concept of so-called effective capacitance distribution the capacitance Cdt can be omitted, as shown in Figure 3b. onlyAs applies capacitances to the capacitances have always between positive electrodes values, it with is obvious a potential that: di fference. Alternatively, it can be assumed that the electrodes on the same potential are in a galvanic contact with each other, forming one single electrode. The concept of capacitance(퐶3푑푟 between + 퐶3푑푡 such) > 퐶 electrodes3푑푟. becomes virtually meaningless.(8) Effective capacitance distributions for induction and conduction methods are presented in Figure3. Therefore, in a 3-electrode system, based on Equations (6) and (7): In the following considerations, it is assumed that 3-electrode systems have the same geometry. The capacitance distribution in a 2-electrode퐶3푒푙_푐표푛푑 system > 퐶3 is푒푙_ the푖푛푑. same regardless of the charging method.(9) In the case of a 3-electrode system, capacitance values do not change, however the effective capacitance Thus, if the same value of voltage is applied, it can be shown from Equation (3) that: distribution varies because of the change in the potential values. In a 3-electrode system where the induction method is due to the lack of potential difference between a droplet and a target, the capacitance Cdt can be omitted, as shown in Figure3b. As capacitances have always positive values, it is obvious that:

(C3dr + C3dt) > C3dr. (8)

Therefore, in a 3-electrode system, based on Equations (6) and (7):

C3el_cond > C3el_ind. (9) Energies 2020, 13, 469 5 of 15

Thus, if the same value of voltage is applied, it can be shown from Equation (3) that:

Q3el_cond > Q3el_ind. (10)

If C3dt << C3dr (e.g., the target is far away from the nozzle, and the ring electrode is close to the nozzle), charges induced in the particular systems can be given by:

Q3el_cond Q3elind . (11)

It should be noted that the capacitances C2dt and C3dt are not equal. Geometries in a 2-electrode system and a 3-electrode one are diﬀerent due to the presence of the ring electrode. The capacitance C2dt is related to the total droplet surface. However, in the case of a 3-electrode system, the capacitance C3dt is associated only with part of the droplet surface because of the presence of the C3dr capacitance, as shown in Figure2b. Because the capacitance is directly proportional to the electrode surface area, the relation between these two capacitances is:

C2dt > C3dt. (12)

In the case of the conduction method, the electric ﬁeld associated with a droplet aﬀects the target or the ring electrode. If the ring is closer to the droplet than the target it can be hypothesized that:

(C3dr + C3dt) > C2dt. (13)

Considering the conduction method in Equation (13):

C C , (14) 3el_cond ≥ 2el Thus, assuming that the same voltage is applied:

Q Q . (15) 3el_cond ≥ 2el

The case Q3el_cond Q2el refers to the situation where the ring electrode is far away from the nozzle in the 3-electrode system with the conduction method, and thus the electric ﬁeld distribution in that system is similar to that in the 2-electrode system. In general, the value of C3dr can be greater, smaller, or equal to the value of C2dt, so there are possible electrode conﬁgurations where:

C2dt > C3dr. (16)

In the case of the induction method, Equation (16) can be given by:

C2el > C3el_ind, (17)

Thus, assuming that the same voltage is applied:

Q2el > Q3el_ind. (18)

Equations (16)–(18) are only valid for one of three possible situations (e.g., C2dt > C3dr), which occur when the surface of the ring electrode is much smaller than the target surface. As a consequence of the above-mentioned remark, there may be some electrode conﬁgurations in which charge induced in a 2-electrode system is higher than charge induced with the induction method in a 3-electrode system for the same applied voltage. Energies 2020, 13, 469 6 of 15

3. Computer Simulations of the Droplet Charging Process To show the electric field distribution for cases described in the previous section, numerical finite element method (FEM) simulations were performed in the COMSOL Multiphysics 4.3 software using 2D axisymmetric geometry. Thus, the presented model employs an analysis of the distribution of electrical capacitances in the electrification system with different methods of droplets charging. The area of the calculated electric field distribution was restricted to a cylinder with a height of 150 mm and a diameter of 100 mm. On the external boundaries of the model (excluding the grounded floor, i.e., the target) only the tangential component of the electric field could have existed. The permittivity of the mediumEnergies 20 filling20, 13, x theFOR space PEER REVIEW was assumed to be εr = 1 (i.e., air). Electric field distributions for induction6 of 15 andEnergies conduction 2020, 13, x FOR methods PEER REVIEW in the 3-electrode system are presented in Figures4 and5. 6 of 15

Figure 4. Sketch of electric field lines associated with the droplet (a) and direction of the electric field Figure 4. Sketch of electric ﬁeld lines associated with the droplet (a) and direction of the electric ﬁeld Figure(b) for 4.the Sketch induction of electric method. field A lines target associated and a nozzle with with the droplet a droplet (a) areand grounded direction andof the the electric polarity field of (b) for the induction method. A target and a nozzle with a droplet are grounded and the polarity of the (theb) forvoltage the induction applied to method. the ring A electrode target and is negative,a nozzle withso the a induceddroplet arecharge grounded is positive. and the polarity of voltage applied to the ring electrode is negative, so the induced charge is positive. the voltage applied to the ring electrode is negative, so the induced charge is positive.

Figure 5. Sketch of electric field lines associated with the droplet (a) and direction of the electric field Figure 5. Sketch of electric field lines associated with the droplet (a) and direction of the electric field (b) for the conduction method. A ring electrode and a target are grounded and the polarity of the Figure(b) for 5.the Sketch conduction of electric method. field linesA ring associated electrode with and the a targetdroplet are (a )grounded and direction and ofthe the polarity electric of field the voltage applied to the droplet is positive, so the induced charge is also positive. (voltageb) for the applied conduction to the dropletmethod. is Apositive, ring electrode so the induced and a target charge are is alsogrounded positive and. the polarity of the voltage applied to the droplet is positive, so the induced charge is also positive. In Figure5, it can be seen that in the case of the conduction method, the electric field lines In Figure 5, it can be seen that in the case of the conduction method, the electric field lines (associated with the droplet) are linked to the ring electrode and the target. In contrast, in the case of (associatedIn Figure with 5, the it candroplet) be seen are thatlinked in to the the case ring of electrode the conduction and the target. method In, thecontrast, electric in the field case lines of the induction method, the electric field lines (associated with the droplet) are linked only to the ring (associatedthe induction with method, the droplet) the electric are linked field tolines the (associated ring electrode with and the the droplet) target. are In linkedcontrast, only in theto the case ring of electrode, as shown in Figure4. theelectrode, induction as shown method, in theFigure electric 4. field lines (associated with the droplet) are linked only to the ring To illustrate the influence of different electrode configurations on the charging process, another electrode,To illustrate as shown the in influence Figure 4. of different electrode configurations on the charging process, another FEM model was created using 2D axisymmetric geometry. The geometry of the assumed setup is FEMTo model illustrate was thecreated influence using of 2D different axisymmetric electrode geometry. configuration The geometrys on the charging of the assumed process, anothersetup is shown in Figure6. The simulation area was again restricted to a cylinder with a height of 500 mm and FEMshown model in Figure was created6. The simulation using 2D axisymmetricarea was again geometry. restricted The to a geometry cylinder withof the a heightassumed of setup500 mm is a diameter of 200 mm. Other parameters of the simulation (i.e., external boundary condition, medium shownand a diameterin Figure of 6. 200The mm.simulation Other pareaarameters was again of the restricted simulation to a (i.e.cylinder, external with boundary a height of condition, 500 mm permittivity) were unchanged in comparison to the simulation described previously (Figures4 and5). andmedium a diameter permittivity) of 200 mm. were Other unchanged parameters in comparison of the simulation to the (i.e. simulation, external describedboundary previouslycondition, medium(Figures 4 permittivity) and 5). were unchanged in comparison to the simulation described previously (Figures 4 and 5).

Energies 2020, 13, 469 7 of 15 Energies 2020, 13, x FOR PEER REVIEW 7 of 15

Figure 6. Sketch of the droplet charging geometry used in ﬁnite element method (FEM) numerical Figure 6. Sketch of the droplet charging geometry used in finite element method (FEM) numerical simulation. Note: N—a nozzle (e.g., conducting needle with water supply); D—droplet with a diameter simulation. Note: N—a nozzle (e.g., conducting needle with water supply); D—droplet with a of 2 mm. All dimensions are in mm. diameter of 2 mm. All dimensions are in mm. In the analyzed model, the ring electrode can be in a position below (h < 100 mm) or above (h > 100In the mm) analyzed the droplet, model because, the ring the electrode position can of the be in ring a position electrode below does ( noth < 100 aﬀect mm) the or capacitance above (h > distribution100 mm) the in droplet, the system because (presented the position in Figures of 1 the and ring2), rather electrode only does slightly not changes affect the the capacitance values of thesedistribution capacitances. in the system (presented in Figures 1 and 2), rather only slightly changes the values of theseIn capacitances. the described above model, various scenarios were represented by changing the diameter of In the described above model, various scenarios were represented by changing the diameter of the ring electrode eld or the target td. The following cases were analyzed: the ring electrode eld or the target td. The following cases were analyzed: A. eld = 40 mm, td = 20 mm (i.e., a narrow ring electrode and a small target); A. eld = 40 mm, td = 20 mm (i.e., a narrow ring electrode and a small target); B. eld = 40 mm, td = 200 mm (i.e., a narrow ring electrode and a big target); B. eld = 40 mm, td = 200 mm (i.e., a narrow ring electrode and a big target); C. eld = 80 mm, td = 200 mm (i.e., a wide ring electrode and a big target). C. eld = 80 mm, td = 200 mm (i.e., a wide ring electrode and a big target). TheThe termsterms “small”,“small”, “big”,“big”, “narrow”,“narrow”, andand “wide”“wide” shouldshould bebe treatedtreated inin aa relativerelative ratherrather thanthan anan absoluteabsolute manner.manner. InIn analyzed analyzed situations, situations, the the droplet–target droplet–target distance distance was was kept kept constant constant and and equal equal to 100 to mm.100 mm. In the In physicalthe physical system system realization, realization, this parameter this parameter also plays also an plays important an important role. However, role. However, when the when droplet the isdroplet far away is far from away the from target the and target the ringand electrodethe ring electrode is close to is theclose nozzle, to the the nozzle, capacitance the capacitanceC3dt should C3dt beshould negligible, be negligible, and then and both then methods both methods (i.e., conduction (i.e., conduction and induction) and induction) should giveshould similar give results.similar Theresults. presented The presented analysis focuses analysis only focuses on cases only where on cases the whe nozzlere the is relatively nozzle is close relatively to the close grounded to the target.grounded In all target. three cases,In all thethree ring cases, electrode the ring height electrodeh was changedheight h inwas the ch rangeanged of in 25–200 the range mm. of The 25 total–200 capacitancemm. The total between capacitance the droplet between and the the droplet environment and the was environment numerically was calculated, numerically and calculated, the obtained and valuesthe obtained were divided values were by the divided droplet-environment by the droplet- capacitanceenvironment for capacitance the 2-electrode for the system, 2-electrode which system, is the samewhich regardless is the same of theregardless charging of method.the charging method. TakingTaking into into account account the the capacitance capacitance distributions distributions shown shown inin FigureFigure3 ,3, particular particular capacitances capacitances can can bebe givengiven by:by: C3el_cond = (C3dr + C3dt), (19) 퐶3푒푙_푐표푛푑 = (퐶3푑푟 + 퐶3푑푡), (19) C3el_ind = C3dr, (20)

C =퐶(3C푒푙_푖푛푑 = C 퐶3푑푟, ).( (21)20) 3dt 3el_cond − 3el_ind

퐶3푑푡 = (퐶3푒푙_푐표푛푑 − 퐶3푒푙_푖푛푑). (21)

Energies 2020, 13, x FOR PEER REVIEW 8 of 15 Energies 2020, 13, 469 8 of 15 Energies 2020, 13, x FOR PEER REVIEW 8 of 15 It should be also pointed out that induced charge is directly proportional to corresponding capacitancesIt shouldIt should , be namely be also also pointed in pointed Equations out out that (3), that induced(4), induced (6), and charge charge(7). Capacitances is isdirectly directly proportional C proportional2el, C3el_cond, and to to corresponding C corresponding3el_ind are called 3el_cond capacitancesdropletcapacitances, capacitances, namely namely in becausein Equations Equations of their (3), (3), (4), corresponding (4), (6) (6),, and and (7). (7). chargingCapacitances Capacitances methods. CC22elel, ,C CValues3el_cond3el_cond, and of, and capacitances C3Cel_ind3el_ind areare called C called 3el_ind 2el dropletanddroplet C capacitances capacitances relative becauseto because C for of differentoftheir their corresponding corresponding electrification charging charging conditions methods. methods. are shown Values Values in of ofFigure capacitances capacitances 7 (i.e., caseCC3el_cond3el_cond A: a andnarrowand C3el_indC3el_ind ringrelativerelative electrode to C to2el forCand2el different fora small diﬀerent target),electrification electriﬁcation Figure conditions 8 (i.e. conditions, case are B: showna are narrow shown in Figurering in Figureelectrode 7 (i.e.7(, casei.e., and caseA: a abig A: narrowtarget),a narrow ring and ringelectrode Figure electrode 9 and(i.e. ,and acase small a C: small atarget), wide target), ring Figure Figureelectrode 8 (i.e.8 (i.e.,, andcase case a B:big B:a target).narrow a narrow ring ring electrode electrode and and a big a big target),target), and and Figure Figure 9 (i.e.9 (i.e.,, case case C: C: a wide a wide ring ring electrode electrode and and a abig big target). target).

Figure 7. Droplet capacitance C3el values for induction and conductive charging methods , relative to Figure 7. Droplet capacitance C3el values for induction and conductive charging methods, relative the capacitance of the 2-elect3elrode system C2el, as a function of ring electrode height h. Simulation Figureto the 7. capacitanceDroplet capacitance of the 2-electrode C values system for inductionC2el, as aand function conductive of ring charging electrode methods height,h relative. Simulation to parameters: rd = 40 mm, td = 20 mm (i.e., narrow ring electrode and a small target). theparameters: capacitancer d of= the40 mm,2-electtd rode= 20 mm system (i.e., C narrow2el, as a ring function electrode of ring and electrode a small target). height h. Simulation parameters: rd = 40 mm, td = 20 mm (i.e., narrow ring electrode and a small target).

Figure 8. Droplet capacitance C3el values for induction and conductive charging methods, relative Figure 8. Droplet capacitance C3el values for induction and conductive charging methods, relative to to the capacitance of the 2-electrode system C2el, as a function of ring electrode height h. Simulation Figurethe capacitance8. Droplet capacitance of the 2-electrode C3el values system for induction C2el, as a and function conductive of ring charging electrode methods, height h relative. Simulation to parameters: rd = 40 mm, td = 200 mm (i.e., narrow ring electrode and a big target). theparameters: capacitance r d of = 40the mm, 2-electrode td = 200 mm system (i.e. ,C narrow2el, as a ring function electrode of ring and electrode a big target). height h. Simulation parameters:Results for rd = a 40 setup mm, withtd = 200 a narrowmm (i.e. ring, narrow electrode ring electrode (rd = 40 and mm) a big and target). a small target (td = 20 mm) are

shown in Figure7. It can be seen from Figure7 that both droplet capacitances C3el_cond and C3el_ind have similar values. The diﬀerences are most signiﬁcant when the ring electrode is high above the droplet (h > 120 mm). In this case, C3dt is small and C3dr is dominant when conduction charging is considered, such as in Equations (20) and (21). The maximum values of capacitances C3el_cond and C3el_ind were obtained for the height of the ring electrode h equal to 100 mm (i.e., when the ring electrode is near the droplet). These capacitance values for 3-electrode systems were about 3 times higher than the capacitance C2el calculated for the setup with two electrodes. Based on the presented results, similar values of the induced charge for both methods using a 3-electrode system can be expected. The presented case corresponds to a system with a well-designed ring electrode.

Energies 2020, 13, 469 9 of 15 Energies 2020, 13, x FOR PEER REVIEW 9 of 15

Figure 9. Droplet capacitance C3el values for induction and conductive charging methods, relative Figure 9. Droplet capacitance C3el values for induction and conductive charging methods, relative to to the capacitance of the 2-electrode system C2el, as a function of ring electrode height h. Simulation the capacitance of the 2-electrode system C2el, as a function of ring electrode height h. Simulation parameters: rd = 80 mm, td = 200 mm (i.e., a wide ring electrode and a big target). parameters: rd = 80 mm, td = 200 mm (i.e., a wide ring electrode and a big target).

Results for the setup with a narrow ring electrode (rd = 40 mm) and a big target (td = 200 mm) are Results for a setup with a narrow ring electrode (rd = 40 mm) and a small target (td = 20 mm) are shown in Figure8. Figure8 shows that the capacitance C3el_cond always significantly exceeds C3el_ind, shown in Figure 7. It can be seen from Figure 7 that both droplet capacitances C3el_cond and C3el_ind have especially when the ring electrode is far away from the droplet position. This effect indicates that for similar values. The differences are most significant when the ring electrode is high above the droplet conduction charging, the influence of both capacitances (i.e., C3dr and C3dt) is comparable. Based on (h > 120 mm). In this case, C3dt is small and C3dr is dominant when conduction charging is considered, these results and Equation (21), it can be concluded that an increase in the distance between the ring such as in Equations (20) and (21). The maximum values of capacitances C3el_cond and C3el_ind were electrode and the droplet leads to an increase in the capacitance C3dt influence. It was also found that obtained for the height of the ring electrode h equal to 100 mm (i.e., when the ring electrode is near there are some positions of the ring electrode h for which the capacitance C3el_ind is lower than the the droplet). These capacitance values for 3-electrode systems were about 3 times higher than the droplet capacitance of a 2-electrode system C2el (i.e., values lower than 1 in Figure8). Moreover, it was capacitanceobserved that C2 forel calculated a certain narrowfor the setup range with of height two electrodes.h (i.e., 85 < Basedh < 105 on mm), the presented a 3-electrode results, system similar with valuesinduction of thecharging induced provides charge higher for bothQ/m methodsvalues (i.e., using values a 3 greater-electrode than system 1 in Figure can be8). expected. The presented case corresponds to a system with a well-designed ring electrode. Results for the setup with a wide ring electrode (rd = 80 mm) and a big target (td = 200 mm) are Results for the setup with a narrow ring electrode (rd = 40 mm) and a big target (td = 200 mm) are shown in Figure9. As can be seen in Figure9, the capacitance C3el_cond is always significantly greater shown in Figure 8. Figure 8 shows that the capacitance C3el_cond always significantly exceeds C3el_ind, than C3el_ind, which is a similar situation to the previous case. In addition, it can be noticed that the especiallyinduction when charging the methodring electrode in a 3-electrode is far away configuration from the droplet is always position. less This efficient effect than indicates electrification that for conductionin the 2-electrode charging system, the (i.e.,influence values of lower both capacitances than 1 in Figure (i.e.9,). C The3dr and discussed C3dt) is comparable. case corresponds Based to on a thesesystem results with aand poorly Equation designed (21), electrodeit can be concluded arrangement. that an increase in the distance between the ring electrodeThe followingand the droplet findings leads can to be an made increase for all in analyzedthe capacitance cases: C3dt influence. It was also found that there are some positions of the ring electrode h for which the capacitance C3el_ind is lower than the dropletdroplet capacitance capacitance of a 2 in-electrode a 3-electrode system system C2el (i.e. is the, values highest lower when than the 1 ringin Figure electrode 8). Moreover, is slightly it under was • observedthe droplet; that for a certain narrow range of height h (i.e., 85 < h < 105 mm), a 3-electrode system with inductiondroplet charging capacitance provides in the higher 3-electrode Q/m values conduction (i.e., values method greaterC thanis 1 alwaysin Figure greater 8). than in the • 3el_cond Results3-electrode for the induction setup with method a wideC3el_ind ringor electrode 2-electrode (rd = system 80 mm)C2 eland; a big target (td = 200 mm) are shownfor in certain Figure electrode 9. As can configurations be seen in Figure (i.e., the9, the position capacitance of the ringC3el_cond electrode), is always droplet signific capacitanceantly greater in • than theC3el_ind 3-electrode, which is induction a similar methodsituationC 3toel_ind theis previous lower than case. in In the addition, 2-electrode it can system be noticedC2el, even that if the inductionring electrode charging ismethod at the samein a 3 height-electrode as the configuration droplet (Figure is always9); less efficient than electrification in thedroplet 2-electrode capacitance system in(i.e. the, values 3-electrode lower conduction than 1 in Figure method 9).C The discussedis less dependentcase corresponds on the ringto a • 3el_cond systemelectrode with a poorly position. designed electrode arrangement. The following findings can be made for all analyzed cases: Considering that the induced charge is directly proportional to droplet capacitance, as stated •in Equationdroplet (3),capacitance and assuming in a 3-electrode the same system value ofis the applied highest voltage when for the 2- rin andg electrode 3-electrode is slightly systems, under the above-mentionedthe droplet; observations lead to the conclusion that a 3-electrode conduction charging method •always droplet gives capacitance higher values in the of 3 droplet-electrode charge conduction than a 3-electrodemethod C3el_cond induction is always method greater or than a 2-electrode in the 3- electrode induction method C3el_ind or 2-electrode system C2el;

Energies 2020, 13, 469 10 of 15 system. Moreover, the 3-electrode induction method may sometimes be less efficient in terms of the induced charge than the 2-electrode system. It should also be noted that the above examples were given to illustrate the influence of different electrode arrangements on the charging levels of droplets, and are not related to any real geometry. When designing the electrode system, other factors should also be taken into account (e.g., electrical strength of the system). In the real dynamic case of the charging process, the droplet (i.e., water or other liquid) is deformed due to the electric field. Since in each setup the electric field is different (even at the same voltage), the droplet deformation may be different. This effect causes further changes in the electric field distribution, and thus leads to a different induced charge. Other factors affecting the value of the Q/m ratio are space charge [28] and flow rate, as well as sprayed liquid properties [29]. The charged droplets may also shield the sprayed liquid from the electric field, reducing the charging level [30]. It should also be mentioned that the charge of a droplet is limited due to the electric field strength of the medium or due to the Rayleigh limit [1,2]. For that reason, one droplet can separate into smaller droplets [1,2], and very complex numerical simulation techniques (i.e., including particle tracking, moving mesh, and computational fluid dynamics CFD) [31–34] are often used to model the droplet charging process.

4. Measurements and Calculations of the Q/m Parameter To conﬁrm the validity of the previous theoretical considerations, a simple experimental setup for the measurements of the Q/m parameter was proposed. The main parts of the setup were the pump (P) equipped with a rubber pipe (RP) for pumping water, a metallic needle as a nozzle (N) with a needle holder (NH) and a conducting ring electrode (R) (for which height above the ground h was adjusted during the experiment), and a Faraday cup (F) in a grounded shield (GS). The Faraday cup was almost completely shielded and only had a hole allowing the charged droplets to enter it. In the charging process, a high-voltage DC power supply was used. The switch S was used to change the electriﬁcation method (i.e., induction or conduction). The measurement setup is shown in Figure 10. The capacitance of the cable and Faraday cup Cn was equal to 1.18 nF. The voltage on Cn was measured usingEnergies electrometer. 2020, 13, x FOR PEER REVIEW 11 of 15

Figure 10. SchemeScheme of of the the measurement measurement setup. setup. Note: Note: P—pump; P—pump; RP—rubber RP—rubber pipe; N pipe;—nozzle; N—nozzle; NH— NH—needleneedle holder; holder; S—switch; S—switch; R—ring R—ringelectrode; electrode; F—Faraday F—Faraday cup; GS— cup;grounded GS—grounded shield; HV shield;DC—high HV- DC—high-voltagevoltage DC power DCsupply. power supply.

The diameter and the mass of the droplets formed during the charging process were equal to The diameter and the mass of the droplets formed during the charging process were equal to 2.7 2.7 mm and 10.3 mg, respectively. The ratio of droplet generation was equal to 100 droplets per minute mm and 10.3 mg, respectively. The ratio of droplet generation was equal to 100 droplets per minute toto avoidavoid a shieldingshielding eeffectﬀect fromfrom otherother chargedcharged droplets.droplets. The applied voltage was equal to about 5 kV (positive(positive forfor thethe conductionconduction method and negative for the inductioninduction method),method), soso thethe dropletdroplet chargecharge waswas alwaysalways positive.positive. AA speciﬁcspecific voltage voltage level level was was chosen chosen to to avoid avoid deformation deformation of of the the liquid liquid droplet droplet or or droplet breakdown into smaller droplets. For this reason, the obtained values of Q/m were relatively low. Distilled water with a conductivity of 1.4 µS/cm (measured at 23 °C) was used in the studies. Based on the measurement setup, another FEM simulation model was proposed, as outlined in Figure 11. The simulation area was a cylinder measuring 100 mm in height and 1000 mm in diameter. Other simulation parameters (i.e., external boundary condition, medium permittivity) were unchanged in relation to the previously described simulations. The total induced charge on droplet Qd was numerically calculated. To obtain the Q/m parameter, the value of Qd was divided by the experimentally measured droplet mass.

Figure 11. View of the model used in finite element method (FEM) computer simulation, based on the measurement setup shown in Figure 10. Note: N—nozzle (needle); NH—needle holder; D—water droplet; R—ring electrode; T—target. The droplet diameter was equal to 2.7 mm. All dimensions are in mm.

Energies 2020, 13, x FOR PEER REVIEW 11 of 15

Figure 10. Scheme of the measurement setup. Note: P—pump; RP—rubber pipe; N—nozzle; NH— needle holder; S—switch; R—ring electrode; F—Faraday cup; GS—grounded shield; HV DC—high- voltage DC power supply.

The diameter and the mass of the droplets formed during the charging process were equal to 2.7 mm and 10.3 mg, respectively. The ratio of droplet generation was equal to 100 droplets per minute to avoid a shielding effect from other charged droplets. The applied voltage was equal to about 5 kV (pEnergiesositive2020 for, 13 the, 469 conduction method and negative for the induction method), so the droplet charge11 of 15 was always positive. A specific voltage level was chosen to avoid deformation of the liquid droplet or droplet breakdown into smaller droplets. For this reason, the obtained values of Q/m were relativelydroplet breakdown low. Distilled into water smaller with droplets. a conductivity For this reason,of 1.4 µS/cm the obtained (measured values at 23 of °C)Q/m waswere used relatively in the studies.low. Distilled water with a conductivity of 1.4 µS/cm (measured at 23 ◦C) was used in the studies. BasedBased on on the the measurement measurement setup, setup, another another FEM FEM simulation simulation model model was was proposed proposed,, as outlined as outlined in Figurein Figure 11. The11. simulation The simulation area was area a wascylinder a cylinder measuring measuring 100 mm 100in height mm in and height 1000 andmm in 1000 diameter. mm in Otherdiameter. simulation Other simulation parameters parameters (i.e., external (i.e., external boundary boundary condition, condition, medium medium permittivity) permittivity) were were unchangedunchanged in in relation relation to to the the previously previously described described simulations. simulations. The The total total induced induced charge charge on on droplet droplet QQdd waswas numerically numerically calculated. calculated. To To obtain obtain the the QQ/m/m parameter,parameter, the the value value of of QQdd waswas divided divided by by the the experiexperimentallymentally measured measured droplet droplet mass. mass.

FigureFigure 11. 11. ViewView of of the the model model used used in in finite ﬁnite element element method method ( (FEM)FEM) computer computer simulation, simulation, based based on on the the measurementmeasurement setup setup shown shown in in Figure 10. Note: Note: N N—nozzle—nozzle (needle) (needle);; NH NH—needle—needle holder; holder; D D—water—water dropletdroplet;; R R—ring—ring electrode electrode;; T T—target.—target. The The droplet droplet diameter diameter was was equal equal to to 2.7 2.7 mm. mm. All All dimensions dimensions are are inin mm. mm.

The first experiment was carried out for a 2-electrode setup (i.e., without the ring electrode). The measured and calculated values of the Q/m parameter were equal to 15.6 µC/kg and 20.4 µC/kg, respectively. This means that the calculated (Q/m)2el_sim was 1.31 times greater than the measured (Q/m)2el_exp. In the next part of the experimental study, the ring electrode was introduced into the system, which enabled the charging of droplets in a 3-electrode configuration. The height of the ring electrode h was in the range of 180 to 330 mm. The results of Q/m measurements and simulation-based calculations are shown in Figure 12. It can be seen from Figure 12 that the differences between numerically calculated and measured values do not exceed 20%. EnergiesEnergies 2020 20, 2013,, 1x3 FOR, x FOR PEER PEER REVIEW REVIEW 12 12of of15 15

TheThe first first experiment experiment was was carried carried out out for for a 2 a- electrode2-electrode setup setup (i.e. (i.e., without, without the the ring ring electrode). electrode). The The measuredmeasured and and calculated calculated values values of ofthe the Q /Qm/ mparameter parameter were were equal equal to to15. 15.6 µC/kg6 µC/kg an and 20.d 20.4 µC/kg,4 µC/kg, respectively.respectively. This This means means that that the the calculated calculated (Q (/Qm/)m2el_sim)2el_sim was was 1.31 1.31 times times greater greater than than the the measured measured (Q(/Qm/)m2el_exp)2el_exp. . In Inthe the next next part part of ofthe the experimental experimental study, study, the the ring ring electrode electrode was was introduced introduced into into the the system, system, whichwhich enabled enabled the the charging charging of ofdrop dropletslets in ina 3 a- electrode3-electrode configuration. configuration. The The height height of ofthe the ring ring electrode electrode h wash was in in the the range range of of 180 180 to to 330 330 mm. mm. The The results results of ofQ /Qm/ mmeasurements measurements and and simulation simulation-based-based calculationscalculationsEnergies 2020 ,a13re a, re 469 shown shown in in Figure Figure 12. 12. It Itcan can be be seen seen from from Figure Figure 12 12 that that the the differences differences between between12 of 15 numericallynumerically calculated calculated and and measured measured values values do do not not exceed exceed 20%. 20%.

Figure 12. Measured (“exp”) and calculated (“sim”) (Q/m)3el parameter values for both electriﬁcation FigureFigure 12. 12. Measured Measured (“exp”) (“exp”) and and calculated calculated (“sim”) (“sim”) (Q /(mQ)/3mel )parameter3el parameter values values for for both both electrification electrification methods in the 3-electrode system, as a function of the ring electrode height h. methodsmethods in thein the 3-electrode 3-electrode system, system, as asa function a function of ofthe the ring ring electrode electrode height height h. h.

Figure 13 shows the measured and calculated (Q/m)3el parameter values for conduction and FigureFigure 13 13shows shows the the measured measured and and calculated calculated (Q (/Qm/)m3el) 3parameterel parameter values values for for conduction conduction and and induction methods in the 3-electrode system divided by the (Q/m)2el parameter values obtained in induction methods in the 3-electrode system divided by the (Q/m)2el parameter values obtained in 2- induction2-electrode methods systems in the (i.e., 3- measuredelectrode system and calculated divided for by correspondingthe (Q/m)2el parameter cases). values obtained in 2- electrodeelectrode systems systems (i.e. (i.e., measured, measured and and calculated calculated for for corresponding corresponding cases). cases).

FigureFigure 13. 13. Measured Measured (“exp”) (“exp”) and and calculated calculated (“sim”) (“sim”) (Q (/mQ/)m3el ))parameter3el parameterparameter values values values for for forboth both electrification electrification electriﬁcation methodsmethods (“ind” (“ind” and and “cond”) “cond”) in ina in 3a - electrode3 3-electrode-electrode system, system, relative relative relative to toQ to /mQ //mof ofa 2a - electrode2 2-electrode-electrode system, system, as as a a

functionfunction of of a the a thethe ring ringring electrode electrodeelectrode height heightheight h. h.h.Simulation SimulationSimulation results resultsresults relative relativerelative to to to(Q ( Q/(mQ/m)/2mel_sim)2)el_sim2el_sim, empirical,, empirical results results Q m relativerelative to (to Q /( mQ)/2mel_exp.)2el_exp.el_exp . It can be seen from Figure 13 that the differences between simulation and experimental results are It canIt can be be seen seen from from Figure Figure 13 13that that the the differences differences between between simulation simulation and and experimental experimental results results clearly visible. This is the result of different values of (Q/m)2el being obtained from measurements and areare clearly clearly visible. visible. This This is theis the result result of ofdifferent different values values of of(Q (/Qm/)m2el) being2el being obtained obtained from from measur measurementsements calculations (15.6 µC/kg and 20.4 µC/kg, respectively). It is also clear that for most of the tested ring andand calculations calculations (15.6 (15.6 µC/kg µC/kg and and 20. 20.4 µC/kg,4 µC/kg, respectively). respectively). It isIt alsois also clear clear that that for for most most of ofthe the tested tested electrode positions, the induction method in a 3-electrode system is less efficient than a 2-electrode ringring electrode electrode positions, positions, the the induction induction method method in in a 3 a- electrode 3-electrode system system is is less less efficient efficient than than a 2 a- 2- charging system (i.e., relative values of Q/m are lower than 1, as shown in Figure 13). Only for the case electrodeelectrode charging charging system system (i.e. (i.e., relative, relative values values of ofQ /Qm/ mare are lower lower than than 1, as1, asshown shown in inFigure Figure 13). 13). Only Only in which a ring electrode is close to the droplet is the induction method better than a 2-electrode system forfor the the case case in inwhich which a ringa ring electrode electrode is closeis close to tothe the droplet droplet is theis the induction induction method method better better than than a 2a- 2- in terms of charging efficiency. On the other hand, the conduction method in a 3-electrode system electrodeelectrode system system in interms terms of ofcharging charging ef ficiency.efficiency. On On the the other other hand, hand, the the conduction conduction method method in ina 3a- 3- is always more effective than a 2-electrode charging system (i.e., relative values of Q/m parameter electrodeelectrode system system is alwaysis always more more effective effective than than a 2a- electrode2-electrode charging charging system system (i.e. (i.e., relative, relative values values of of are greater than 1, as presented in Figure 13). The obtained results are consistent with those shown

in Figure8. It should be noted that according to the simulation data, in a 3-electrode system the Q/m parameter should be less dependent on the ring electrode position in the case of conduction method than in the system with the induction method. However, the results of the measurements show that Q/m greatly increases in the case of the conduction method when the ring electrode is close to the droplet. Ratios of Q/m parameters for both charging methods are presented in Figure 14. It can be noticed in Figure 14 that the conduction method is always more eﬀective than the induction method Energies 2020, 13, x FOR PEER REVIEW 13 of 15

Q/m parameter are greater than 1, as presented in Figure 13). The obtained results are consistent with those shown in Figure 8. It should be noted that according to the simulation data, in a 3-electrode system the Q/m parameter should be less dependent on the ring electrode position in the case of conduction method than in the system with the induction method. However, the results of the measurements show that Q/m greatly increases in the case of the conduction method when the ring Energies 2020, 13, 469 13 of 15 electrode is close to the droplet. Ratios of Q/m parameters for both charging methods are presented in Figure 14. It can be noticed in (Figurei.e., 1.5–2.5 14 that times the greaterconductionQ/m methodparameter is always values aremore obtained). effective Thethan results the induction of the numerical method ( simulationi.e., 1.5– 2.5and times measurements greater Q/m areparameter consistent values with are each obtained other.). A The noticeable results of di ﬀtheerence numerical for cases simulation when the and ring measurementselectrode is far are from consistent the droplet with may each be dueother. to theA noticeable external factors difference (i.e., the for appearance cases when of the additional ring electrodecoupling is capacitances).far from the droplet may be due to the external factors (i.e., the appearance of additional coupling capacitances).

Figure 14. Ratio of measured (“exp”) and calculated (“sim”) Q/m parameter values for both conduction Figure 14. Ratio of measured (“exp”) and calculated (“sim”) Q/m parameter values for both (Q/m)3el_cond and induction (Q/m)3el_ind methods in the 3-electrode system. conduction (Q/m)3el_cond and induction (Q/m)3el_ind methods in the 3-electrode system. 5. Discussion and Conclusions 5. Discussion and Conclusions The proposed simplified model based on the capacitance distribution in different electrification systemsThe proposed can be used simplified to study model the process based ofon electrification the capacitance of liquiddistribution droplets. in different For a 2-electrode electrification system, systemsonly the can capacitance be used to study between the process the nozzle of electrification and the target of isliquid responsible droplets. for For the a 2 charging-electrode level system, of the onlydroplets. the capacitance In the 3-electrode between systems,the nozzle the and capacitance the target distribution is responsible is more for the complex. charging In thelevel case of ofthe the droplets.induction In the method, 3-electrode the value systems, of the the charge capacitance induced distribution on the droplet is more depends complex. only In on the the case capacitance of the inductionbetween method, the droplet the and value the of ring the electrode. charge induced When the on conduction the droplet method depends is considered,only on the the capacitance capacitance betweenbetween the the droplet droplet and and the the ring target electrode. has an additionalWhen the influence conduction on the method charging is considered, level. The the total capacitancecapacitance between responsible the droplet for inducing and the charge target on has the an droplet additional is the influence highest inon the the latter charging case, level. making The the total3-electrode capacitance system responsible with the conductionfor inducing method charge the on most the droplet efficient is from the the highest point in of the view la oftter charging case, makinglevel, assumingthe 3-electrode that thesystem same with voltage the conduction is applied in method all systems. the most efficient from the point of view of chargingBased level, on the assuming model, itthat is possible the same to voltage determine is applied the Q/m inparameter all systems. describing the efficiency of the electrificationBased on the systems. model, Itit canis possible be noticed to determine that in a 3-electrode the Q/m parameter system, the describing conduction the charging efficiency method of thecan electrification give significantly systems. higher It canQ/m bevalues noticed than that in thein a induction 3-electrode method, system, especially the conduction when the charging grounded methodtarget can is close give to significantly the nozzle, higher which greatlyQ/m values affects than the in electric the induction field near method, the droplet. especially Similar when findings the groundedwere established target is close during to experiments.the nozzle, which A significant greatly affects coherence the electric of simulation field near and the measurement droplet. Similar results findingsconfirms were the validityestablished of the proposed during experiments. model, which A can significant be helpful when coherence designing of simulationa real-life charging and measurementsystem (i.e., settingresults the confirms distance the between validity the of ring the electrode proposed and model the nozzle)., which can be helpful when designingIt has a real also-life been charging proven thatsystem for a(i.e. specific, setting electrode the distance configuration between using the thering induction electrode method, and the the nozzle2-electrode). system is more effective than the 3-electrode setup, and in this case, the measured values of theItQ has/m parameteralso been proven were even that smaller for a specific than for electrode the 3-electrode configuration system. using the induction method, the 2-electrodeFuture studies system will is more focus effect on developingive than the a model 3-electrode that considers setup, and the dynamicin this case, processes the measured occurring valuesduring of the droplet Q/m electrification,parameter were including even smaller the screening than for etheffect 3- andelectrode deformation system. of droplets. Future studies will focus on developing a model that considers the dynamic processes occurring duringAuthor droplet Contributions: electrification,Conceptualization, including the A.P.; screening methodology, effect A.P.and anddeformation T.C.; validation, of droplets. A.P. and T.C.; formal analysis, A.P. and T.C.; investigation, A.P. and T.C.; resources, T.C.; writing—original draft preparation, A.P.; writing—review and editing, A.P. and T.C.; visualization, A.P.; funding acquisition, T.C. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding. This work was financed by a statutory activity subsidy from the Polish Ministry of Science and Higher Education (PMSHE) for the Department of Electrical Engineering Fundamentals of Wroclaw University of Science and Technology. Conflicts of Interest: The authors declare no competing financial interest. Energies 2020, 13, 469 14 of 15

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