Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids

applied sciences

Communication Electromagnetic and Thermal Phenomena Modeling of Electrical Discharges in Liquids

Marcin Wesołowski 1,*, Sylwester Tabor 2, Paweł Kiełbasa 2 and Sławomir Kurpaska 2

1 Faculty of Electrical Engineering, Warsaw University of Technology, 00-661 Warsaw, Poland 2 Faculty of Production and Energy Engineering, Cracow, University of Agriculture in Krakow, 30-149 Krakow, Poland; [email protected] (S.T.); [email protected] (P.K.); [email protected] (S.K.) * Correspondence: [email protected]

Received: 4 May 2020; Accepted: 2 June 2020; Published: 4 June 2020

Abstract: Electrical discharges in liquids have received lots of attention with respect to their potential applications in various techniques and technical processes. Exemplary, they are useful for water treatment, chemical and thermal processes acceleration, or nanoparticles production. In this paper the special utility of discharges for cold pasteurization of fruit juices is presented. Development of devices for its implementation is a significant engineering problem and should be performed using modeling and simulation techniques to determine the real parameters of discharges. Unfortunately, there is a lack of clear and uniform description of breakdown phenomena in liquids. To overcome this limitation, new methods and algorithms for streamers propagation and breakdown phase analysis are presented in the paper. All solutions were tested in “active area” in the form of liquid material model, placed between two flat electrodes. Electromagnetic and thermal-coupled field analysis were performed to determine all the factors that affect the discharge propagation. Additionally, some circuit models were used to include the power source cooperation with discharge region. In general, presented solutions can be defined as universal and one can use them for numerical simulation of other types of discharges.

Keywords: electrical discharges in liquids; modeling; streamer propagation; discharge energy; electromagnetic ﬁeld; thermal ﬁeld

1. Introduction Electrical discharges generated in liquids have many potential advantages and can be used in some low-temperature plasma techniques. Most popular utilities of such a devices are connected with water treatment [1,2], sterilization, and pasteurization of liquid foods [1,3,4] and surface treatment [1,5]. Basic beneﬁts of HVED (high voltage electrical discharges) technique are reduced process time and temperature, minimal degradation of thermosensitive compounds, and minimization of energy consumption in comparison to classic treatment methods [4,6]. A very promising utility of HVED in agri-food industry is microbial inactivation as a result of discharges in liquid media [7,8]. The exact mechanism of the inﬂuence of the electric discharges on the inactivation of microorganisms has not yet been fully described [4,9]. Nevertheless, it can be concluded, that there are some important factors leading to a reduction in the number of microbials. These factors include production of ozone, hydrogen peroxide, hydroxyl and superoxide free radicals [4,9]. Additionally, UV radiation generated in the discharge, mechanical stresses, and waves can help reducing some microbials [9,10]. Electric breakdown process results from liquid sample ionization under high voltage applied between electrodes. Generation of the discharge process can be divided into three phases: the pulse initiation, pre-breakdown phase (streamers), and breakdown stage (arc formation) [11,12].

Appl. Sci. 2020, 10, 3900; doi:10.3390/app10113900 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 3900 2 of 20

The characteristics of the shape of the electric discharge and the discharge propagation of processed liquid are not recognized in a satisfactory manner. Because of the varied physical parameters of liquids (juices in cases analyzed by authors), it is not possible to use universal devices and programs to implement this technology. Additionally, discharge duration, shape, and other parameters depend on electrodes system geometry and electrical parameters of power source [10]. The development of HVED technique is possible through the usage of modeling methods, facilitating the selection of discharge parameters to a specific demand. There are currently no models that allow precise description of the development of the discharge in liquid environments [13]. Main aim of the work is developing numerical models and procedures to compute pulse discharges in liquids. Mentioned task requires to account models of streamers propagation, electrical breakdown (arc) stage, and models for thermal energy propagation in the time between next discharges. Authors’ calculation procedures that use field and circuit models are described in the paper. Despite of some simplifications, presented models and procedures enable to effective modeling of electrical discharges distributions in liquids.

2. Discharge Models in Liquids When a high voltage of appropriate value is applied to an electrode system filled with dielectric material, filamentary discharges called “streamers” appear at the first stage of breakdown in gases, liquids, and solids. Environmental conditions of the discharges determine the streamers structures including their shape, speed of propagation, current values, electric field intensity at the streamer’s head, plasma channel radius, and degree of ionization [14]. According to the large number of factors that have influence on electric breakdown process, uniform theory for the development of discharges in liquids cannot be presented. The discharge process depends on the type and purity of the liquid material, electrostatic pressure, viscosity, conductivity, thermal conductivity, and dielectric permittivity [14–16]. Therefore, a large number of theories have been proposed [12,14,16]. Discharges in liquids are similar, under certain conditions, to discharge process in gas [16]. In gaseous environments most important factor for discharge process is impact ionization that depend on kinetic energy of electrons in the discharge area [17,18]. In most cases it has been assumed that the discharges can be initiated by the absorption of radiation quantum [19]. These assumptions are generally acceptable for gaseous environments [14,19]. In liquids, however, such correlations cannot be found so far, because of the difficult performance of studies of the mentioned effects [20]. Therefore, the proposed descriptions are burdened with greater uncertainty. The most popular descriptions of discharges development, important from their applicability point of view, are presented below.

2.1. Direct Impact Ionization Description of the pre-breakdown (streamer) phenomenon in liquids, resulting from direct impact ionization, is similar to the electrical breakdown process in gases [13,14]. Nevertheless, because of the higher liquid densities, electrons free paths are significantly shorter, which also affects the increased electron beam scattering [16]. Acceleration of free electrons to reach required ionization energy in a short time and free paths is difficult in such conditions and few requirements must be fulfilled. The most important is to provide appropriate voltage rise rate on the electrodes. It allows rapid acceleration of free electrons. As shown, among others, in [14,16], this process requires extremely high intensities of the electric field at the level of 220 MV/cm. Pulse rise time should be maintained at picoseconds level [20,21]. The second possibility of impact ionization is to create areas characterized by reduced mass density. In such areas (gas bubbles), electrons free paths between next collisions are longer. In addition, in the areas of lower density, the electrical breakdown voltages are reduced [6,14,16].

2.1.1. Theory of Gas Bubbles Numerous studies [22–25] have shown that the liquid phase may be unstable during the impulse discharges. Plasma channels are formed along the electric ﬁeld gradients [6,16]. Such channels, because of their lower density and diﬀerent dielectric permittivity, allow easier development of discharges Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 19

2.1.1. Theory of Gas Bubbles 2.1.1. Theory of Gas Bubbles Numerous studies [22–25] have shown that the liquid phase may be unstable during the Numerous studies [22–25] have shown that the liquid phase may be unstable during the impulse discharges. Plasma channels are formed along the electric field gradients [6,16]. Such Appl.impulse Sci. 2020 discharges., 10, 3900 Plasma channels are formed along the electric field gradients [6,16]. 3Such of 20 channels, because of their lower density and different dielectric permittivity, allow easier channels, because of their lower density and different dielectric permittivity, allow easier development of discharges (especially local discharges) leading to the electrical breakdown inside development of discharges (especially local discharges) leading to the electrical breakdown inside the liquid [16,26]. Models that use gas bubbles theory usually assume the presence of micro-bubbles (especiallythe liquid [16,26]. local discharges) Models that leading use gas to bubbles the electrical theory breakdown usually assume inside the liquidpresence [16 of,26 micro-bubbles]. Models that in non-gaseous media or the self-ignition of gas bubbles. The generation of gas bubbles in liquids usein non-gaseous gas bubbles media theory or usually the self-ignition assume the of presence gas bubbles. of micro-bubbles The generation in non-gaseous of gas bubbles media in liquids or the may result from local overheating of the medium because of Joule effect in areas characterized by self-ignitionmay result from of gas local bubbles. overheating The generation of the medium of gas bubbles because in liquidsof Joule may effect result in areas from localcharacterized overheating by high intensity of electric field (near electrodes), electrostatic expansion of micro-bubbles occurring in ofhigh the intensity medium of because electric offield Joule (near eff ectelectrodes), in areas characterizedelectrostatic expansion by high intensity of micro-bubbles of electric occurring field (near in liquid or chemical processes [20,27,28]. The process of gas bubble development has been electrodes),liquid or chemical electrostatic processes expansion [20,27,28]. of micro-bubbles The process occurring of ingas liquid bubble or chemical development processes has [20, 27been,28]. schematically shown in Figure 1. However, in fast discharges (aprox.10 ns), characterized by fast Theschematically process of shown gas bubble in Figure development 1. However, has been in fast schematically discharges shown(aprox.10 in Figurens), characterized1. However, by in fastfast pulses rise times (approx. 150 ps), gas bubbles are not observed. As it was mentioned in [26], gas dischargespulses rise (aprox.10times (approx. ns), characterized 150 ps), gas bybubbles fast pulses are not rise observed. times (approx. As it was 150 ps),mentioned gas bubbles in [26], are gas not bubbles can exist in such conditions, but their dimensions are lower than the visible size. It has been observed.bubbles can As exist it was in such mentioned conditions, in [26 ],but gas their bubbles dimensions can exist are in lower such conditions,than the visible but their size. dimensionsIt has been generally accepted that the theory of gas bubbles can be used to describe discharges with significant aregenerally lower accepted than the visiblethat the size. theory It hasof gas been bubbles generally can acceptedbe used to that describe the theory discharges of gas with bubbles significant can be durations, larger than microseconds [16]. useddurations, to describe larger dischargesthan microseconds with significant [16]. durations, larger than microseconds [16].

Figure 1. Formation of the streamer near the electrode. (a) Bubble cluster; (b) protrusion; (c) gas channels. FigureFigure 1. 1.FormationFormation of ofthe the streamer streamer near near the the electrode. ( a) BubbleBubble cluster;cluster; ( b(b)) protrusion; protrusion; (c ()c gas) gas channels. channels. 2.1.2.2.2.2. Electric Field Depe Dependentndent Molecular Ionization 2.2.2. Electric Field Dependent Molecular Ionization Electric field-dependent field-dependent molecular ionization (field (field ionization) does not require, in general, the Electric field-dependent molecular ionization (field ionization) does not require, in general, the assumption of free electrons and ions occurrence in thethe spacespace beforebefore dischargedischarge propagation.propagation. In the assumption of free electrons and ions occurrence in the space before discharge propagation. In the environment characterized by high values of the elec electrictric field field intensity, it is possible to separate environment characterized by high values of the electric field intensity, it is possible to separate electrons fromfrom inertinert particles particles [16 [16,29].,29]. As As a resulta result of this of this process, process, free electronsfree electrons and positive and positive ions appear ions electrons from inert particles [16,29]. As a result of this process, free electrons and positive ions inappear the inter-electrode in the inter-electrode space (right space panel (right of panel Figure of2). Figure Because 2). ofBecause the di ffoferent the mobilitydifferent ofmobility electrons of appear in the inter-electrode space (right panel of Figure 2). Because of the different mobility of andelectrons ions, localand ions, variations local invariations density ofin mentioned density of particles mentioned may particles occur, leading may occur, to the generationleading to andthe electrons and ions, local variations in density of mentioned particles may occur, leading to the propagationgeneration and of propagation streamers. Such of streamers. models areSuch commonly models are used commonly to describe used ionization to describe in ionization transformer in generation and propagation of streamers. Such models are commonly used to describe ionization in oilstransformer [30,31] and oils water [30,31] [32 and]. The water basic [32]. disadvantage The basic disadvantage in the practical in the utility practical of this utility theory of this is the theory lack ofis transformer oils [30,31] and water [32]. The basic disadvantage in the practical utility of this theory is analyticalthe lack of description analytical description of the ionization of the process ionization in liquids. process In in practice, liquids. Zener In practice, models Zener are in models use, but are they in the lack of analytical description of the ionization process in liquids. In practice, Zener models are in areuse, originally but they are used originally to describe used the to tunneling describe th processe tunneling in semiconductor process in semi materialsconductor [33]. materials [33]. use, but they are originally used to describe the tunneling process in semiconductor materials [33].

(a) (b) (a) (b) Figure 2. Ionization of atoms under the influence of the field. (a) No ionization at low intensity fields; Figure 2. IonizationIonization of of atoms atoms under under the the influence influence of the field. field. ( (aa)) No No ionization ionization at at low low intensity intensity fields; fields; (b) ionization in the area of high intensity. (b) ionization in the areaarea ofof highhigh intensity.intensity. The ionization process under the influence of the field depends on many factors, such as ionization potential, liquid density, or electric field intensity gradients [16,29]. This mechanism is characterized by many advantages, such as the lack of necessity to take into account the free energy carriers or pollutants (with lower ionization energy) during the initiation of the discharge. However, it should be mentioned that this type of ionization dominates in areas with very high electric field strengths, such Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 19

The ionization process under the influence of the field depends on many factors, such as ionization potential, liquid density, or electric field intensity gradients [16,29]. This mechanism is characterized by many advantages, such as the lack of necessity to take into account the free energy

Appl.carriers Sci. 2020or pollutants, 10, 3900 (with lower ionization energy) during the initiation of the discharge. However,4 of 20 it should be mentioned that this type of ionization dominates in areas with very high electric field strengths, such as in the vicinity of blades or leader’s head [16,34]. Because of the lack of theoretical asdescription in the vicinity of this of bladesprocess or for leader’s liquid head environments, [16,34]. Because utility of of the the lack proposed of theoretical model description is significantly of this processlimited. for liquid environments, utility of the proposed model is signiﬁcantly limited.

2.1.3.2.2.3. Electric Field Dependent Ionic Dissociation Formally, the ionic dissociation process is similar,similar, in terms of thethe description,description, to the above mentioned ionizationionization underunder thethe influenceinfluence of of the the electric electric field field [16 [16].]. To To describe describe this this process, process, one one has has to assumeto assume “active “active area” area” filled filled with with a mixture a mixture ofneutral of neutral particles, particles, positive positive and and negative negative ions. ions. Under Under the influencethe influence of the of external the external electric electric field, thefield, neutral the ne particlesutral particles are dissociated, are dissociated, which leadswhich to leads an increase to an inincrease the number in the of number ions (Figure of ions3)[ (Figure35,36]. This3) [35,36]. fact reduces This fact the breakdownreduces the voltagebreakdown and providesvoltage and the developmentprovides the development of the discharge. of the It shoulddischarge. be mentioned,It should be however, mentioned, that however, the mobility that ofthe positive mobility and of negativepositive and ions negative is negligible ions comparedis negligible to compared the rate of to electrical the rate dischargeof electrical development. discharge development. For this reason, For utilitythis reason, of this utility process of this to describe process the to describe development the development of the discharge of the is limiteddischarge [16 is,35 limited]. [16,35].

(a) (b)

Figure 3. IonIon dissociation dissociation under under the the influence influence of of the the field: field: (a) ( ano) noionization ionization at low at low intensity intensity fields; fields; (b) (ionizationb) ionization in the in thearea area of high of high intensity. intensity.

2.1.4.2.2.4. Electrostriction Mechanism The electrostriction mechanism is used to describedescribe discharges characterized by short duration. In the vicinityvicinity ofof electrodes,electrodes, an inhomogeneousinhomogeneous distribution of the high intensity electric fieldfield is created. Under conditions of significant significant gradients of the electric field field intensity, in poorly conducting liquids, therethere are are stresses stresses causing causing deformations deformations and and discontinuities discontinuities of the of liquidthe liquid [37,38 [37,38].]. In these In areas,these microporesareas, micropores filled with filled gas with are formed,gas are featured formed, by featured a characteristic by a characteristic extension toward extension the electric toward fields the forceelectric lines. fields In force these lines. structures, In thes thee structures, free electrons the can free obtain electrons energy can values obtain that energy enable values the initiationthat enable of the collisioninitiation ionization of the collision process ionization [16,38,39 ].process The forces [16,38 aff,39].ecting The the forces liquid affecting as a result the of liquid the electrostriction as a result of phenomenonthe electrostriction are described phenomenon by the are formula described (1) for by non-polar the formula and (2)(1) forfor polar non-polar liquids and [16 ,(2)38]. for polar liquids [16,38]. ε0 (ε 1)(ε + 2) 2 F ε ()()ε− − ε + E , (1) ≈ 2 0 31 2∇ 2 F ≈ ∇E , (1) 2 αεε0 3 F E2, (2) ≈ 2 ∇ αεε where: ε0—vacuum dielectric permittivity; ε—dielectric0 permittivity2 of the medium in which the F ≈ ∇E , (2) discharge occurs; α—a coefficient determined empirically2 for polar liquids The analysis of the given equations shows that the electrostriction mechanism is intensified in the conditionswhere: ε0—vacuum of fast-changing dielectric electromagnetic permittivity; fieldsε—dielectric with significant permittivity gradients. of the Therefore,medium in this which process the candischarge be used occurs; to describe α—a coefficient discharges determined of nanosecond empirically duration, for in polar the vicinity liquids of pierce electrodes [39,40]. For longerThe analysis discharges, of the the given hydrostatic equations pressure shows reduces that the the electrostriction formation of microporesmechanism [is40 intensified]. in the conditions of fast-changing electromagnetic fields with significant gradients. Therefore, this 2.2. Material and Method In the previous part of the paper, most popular models of electrical discharges formation and propagation in liquids are detailed. Presented models cannot be, in general, used interchangeably, Appl. Sci. 2020, 10, 3900 5 of 20 and utility of specific description depends on electrical parameters of the discharge and material parameters of the region, where breakdown process occurs. Main topic of the paper is developing universal models and calculating procedure for the complex analysis of high voltage electric discharges. Models consider the pre-breakdown and breakdown stage of the discharge. Additionally, calculations of thermal phenomena in the time between the discharges are realized. Before development of numerical models, some tests were performed using physical model of discharge system. Based on measurements of electrical parameters of discharges, it was possible to select appropriate model of ionization. These tasks are detailed in Section3 of the paper. In Section4, general concept of HVED modeling is presented. It is assumed, that the mentioned issue can be solved using coupled electromagnetic and thermal fields. Both field and circuit models are in use. Electrostatic calculations, described in Section5, are used to analyze the streamers propagation in models of liquids by using FEM (ANSYS) system. Electric field distribution and gas bubble models are used to determine streamers geometry in the pre-breakdown stage. Utility of additional analytical equations enables to determine speed of streamers propagations. In the Section6, procedure for analysis of breakdown (arc) stage of discharge is detailed. FEM (ANSYS) numerical model with plasma channel is used. Plasma conductivity and cross section of the channel are determined by analytical equations. Momentary values of voltage drop and current in plasma channel are computed using coupled circuit model (composed of resistor and capacitor) of power supply. Such solution enables to determine transient characteristics of discharge, including power supply capacitor discharge. Section7 presents the analysis of temperature distribution in the time between the discharges. The computational fluid dynamics procedures are used to guarantee the precise analysis of convection heat transfer in liquid environment. Based on the calculations results obtained in this stage, it is possible to determine the minimal break time between discharges.

3. Preliminary Research of Discharge Parameters in Juices In the previous section of the paper, most popular mechanisms describing the nature of the high voltage electrical discharges in liquids are detailed. These mechanisms usually describe specific electrical discharges, classified primarily, because of the speed of their propagation (fast and slow discharges) [27,39]. In order to select the model for further analysis of electrical breakdown process in juices, basic parameters of physical model are measured. All presented measurements are performed to determine the energy values of pulsed discharges and specific discharge periods. The test stand, consisting of a high voltage pulse generator, working chamber, and a control system, is used. Sample of juice is placed between the two flat electrodes. Figure4 shows the design of working chamber

(FigureAppl. Sci.4 2020a) and, 10, blockx FOR PEER diagram REVIEW ofthe test stand (Figure4b). 6 of 19

FigureFigure 4.4. View ofof workingworking chamberchamber ((a)a) andand blockblock diagramdiagram forfor HVEDHVED testtest standstand ((b).b). 1—electrodes; 2—juice2—juice container.container.

The AS3-mini analyzer equipped with an oscilloscope to record voltage and current waveforms is used. During the measurements of high voltage pulses parameters, the initial voltage was kept at 20 kV. Recorded waveforms are presented in the Figure 5. Results show the oscillatory character of the voltage values of the generated pulses. Its initial amplitude depends on the applied voltage triggering the pulse.

Figure 5. Time course of high voltage impulse.

Tests results were used to determine the shape of the generated pulses, together with all the specific quantities (amplitudes, rise and fall times). They were used to model the electrical discharge process. According to the measurements results, it was assumed, that the theory of gas bubbles can be used to describe the discharge phenomena in the analyzed juice model [14,16,39]. In pursuance of the description of this process, the initiation of the discharge occurs as a result of the gas bubble generation because of the Joule heat effect or large field intensity gradient in vicinity of electrodes. Depending on the initial voltage and energy value, it is possible to develop “slow” (propagation speed within several km/s) or fast (propagation velocity from several to several tens km/s) discharges [27,39]. Based on the measurements of current and voltage values in the tested device, the approximate propagation velocity (Figure 5) of the discharge was determined based on the current value of the rise time. Estimated propagation speed was at 3470 m/s, which allowed us to classify the discharge as “slow.” Under these conditions, it has been assumed that the electrical discharge is a result of ionization process in the gas phase filling the plasma channel [19,41]. This fact is important in the discharge numerical modeling point of view. As it was mentioned above, the discharge parameters in liquids are determined by the value of the electric strength, the distance between the electrodes, their shape, temperature, and material parameters of the liquid (juice in analyzed case). When appropriate number of free electrons and positive ions occur in the discharge area, one or more plasma channels (streamers) can be created. Almost all electrons will flow through the channel characterized by minimal resistance value [40,41]. Streamers starts at the cathode toward the anode. When the anode area is reached by plasma channel, the impedance of the discharge area decreases dramatically, which leads to a significant increment of the current. This stage, called the breakdown, is characterized by the occurrence of a plasma channel with almost cylindrical cross-section and Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 19

Appl. Sci.Figure2020 ,4.10 View, 3900 of working chamber (a) and block diagram for HVED test stand (b). 1—electrodes;6 of 20 2—juice container.

The AS3-mini analyzer equipped with an oscilloscope to record voltage and current waveforms is The AS3-mini analyzer equipped with an oscilloscope to record voltage and current waveforms used. During the measurements of high voltage pulses parameters, the initial voltage was kept at 20 is used. During the measurements of high voltage pulses parameters, the initial voltage was kept at kV. Recorded waveforms are presented in the Figure5. Results show the oscillatory character of the 20 kV. Recorded waveforms are presented in the Figure 5. Results show the oscillatory character of voltage values of the generated pulses. Its initial amplitude depends on the applied voltage triggering the voltage values of the generated pulses. Its initial amplitude depends on the applied voltage the pulse. triggering the pulse.

Figure 5. TimeTime course of high voltage impulse.

Tests results were used to determinedetermine the shape of the generated pulses, together with all the specificspecific quantities (amplitudes, rise and fall times).times). They were used to model the electrical discharge process. According toto thethe measurementsmeasurements results, results, it it was was assumed, assumed, that that the the theory theory of of gas gas bubbles bubbles can can be beused used to describeto describe the the discharge discharge phenomena phenomena in the in th analyzede analyzed juice juice model model [14, 16[14,,3916,39].]. In pursuance In pursuance of the of thedescription description of this of process,this process, the initiation the initiation of the dischargeof the discharge occurs as occu a resultrs as of a theresult gas of bubble the gas generation bubble generationbecause of thebecause Joule of heat the eJouleffect orheat large effect field or intensity large field gradient intensity in vicinitygradient of in electrodes. vicinity of Dependingelectrodes. Dependingon the initial on voltage the initial and voltage energy value,and energy it is possible value, it to is develop possible “slow” to develop (propagation “slow” speed(propagation within speedseveral within km/s) orseveral fast (propagation km/s) or velocityfast (propagation from several velocity to several from tens several km/s) discharges to several [27 tens,39]. Basedkm/s) dischargeson the measurements [27,39]. Based of current on the andmeasurements voltage values of cu inrrent the testedand voltage device, values the approximate in the tested propagation device, the velocityapproximate (Figure propagation5) of the discharge velocity was(Figure determined 5) of the baseddischarge on the was current determined value of based the rise on time.the current valueEstimated of the rise propagationtime. speed was at 3470 m/s, which allowed us to classify the discharge as “slow.”Estimated Under these propagation conditions, speed it has was been at assumed 3470 m/s, that which the electrical allowed discharge us to classify is a result the discharge of ionization as process“slow.” inUnder the gas these phase conditions, filling the it plasmahas been channel assume [19d, 41that]. Thisthe electrical fact is important discharge in theis a dischargeresult of ionizationnumerical process modeling in pointthe gas of view.phase Asfilling it was the mentioned plasma channel above, [19,41]. the discharge This fact parameters is important in liquids in the dischargeare determined numerical by the modeling value of point the electric of view. strength, As it was the distancementioned between above, the the electrodes, discharge theirparameters shape, intemperature, liquids are and determined material parametersby the value of of the the liquid electric (juice strength, in analyzed the distance case). When between appropriate the electrodes, number theirof free shape, electrons temperature, and positive and ions material occur in parameters the discharge of area, the oneliquid or more(juice plasma in analyzed channels case). (streamers) When appropriatecan be created. number Almost of all free electrons electrons will and flow positive through ions the channeloccur in characterized the discharge by area, minimal one resistance or more valueplasma [40 channels,41]. Streamers (streamers) starts can at the be cathodecreated. toward Almost the all anode. electrons When will the flow anode through area is the reached channel by plasmacharacterized channel, by the minimal impedance resistance of the value discharge [40,41]. area Streamers decreases dramatically,starts at the cathode which leads toward to a the significant anode. Whenincrement the anode of the area current. is reached This stage, by plasma called channe the breakdown,l, the impedance is characterized of the discharge by the occurrence area decreases of a plasmadramatically, channel which with leads almost to a cylindrical significant cross-section increment of and the durationcurrent. This depending stage, called on the the capacity breakdown, in the issystem characterized [28,42]. The by dischargethe occurrence energy of can a plasma be simply channel defined with as: almost cylindrical cross-section and

C W = U2 U2 , (3) 2 i − f where: Ui—voltage before breakdown, Uf—voltage after breakdown, C—capacitance The formula was used during determination of energy balance of the discharge presented in the next section of the paper. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 19 duration depending on the capacity in the system [28,42]. The discharge energy can be simply defined as:

= C ()2 − 2 W U i U f , (3) 2 where: Ui—voltage before breakdown, Uf—voltage after breakdown, C—capacitance The formula was used during determination of energy balance of the discharge presented in the Appl. Sci. 2020, 10, 3900 7 of 20 next section of the paper.

4. The Concept of Discharges Modeling in Liquids Discharge model, designed to analyze the phenomenaphenomena occurring in devices using HVED for the preservation of liquid food products, should enableenable analysis of coupled electromagnetic and thermal ﬁelds.fields. Primarily, the model was used during the designdesign of the laboratorylaboratory discharges generator and planning future tests to implement thisthis techniquetechnique inin industry.industry. InIn particular,particular, thethe modelmodel shouldshould allow:allow:

- Selection of the electrodes system in terms of their dimensions and distance; - Determination of the voltage values to initiate the discharge; - Determination of discharges energy; - Determination of discharges energy; - Determination of the duration of discharges; - Determination of the duration of discharges; - Determination of the temperature distribution after discharge in order to analyze the time - Determination of the temperature distribution after discharge in order to analyze the time required required to initiate next discharge. This task was aimed to eliminate the possibility of to initiate next discharge. This task was aimed to eliminate the possibility of subsequent electrical subsequent electrical discharges in the same channel; discharges in the same channel; - Determination of the power supply circuit parameters based on the required voltage and - Determination of the power supply circuit parameters based on the required voltage and energy energy of discharges. of discharges. Characterization of all above mentioned parameters requires advanced models and calculating proceduresCharacterization using both of field all above and circuit mentioned models. parameters Field computations, requires advanced carried models out in the and commercial calculating proceduresFEM system, using were both used ﬁeld to compute and circuit the models.electrodes Field system computations, variables, stream carrieders out propagation in the commercial scheme, FEMand temperature system, were distributions. used to compute Power the electrodessource pa systemrameters variables, have been streamers determined propagation usingscheme, circuit andmodels. temperature The calculation distributions. algorithm Power is sourceshown parametersin Figure 6. have Both been field determined and circuit using models circuit have models. been Theextended calculation with algorithmAuthor’s isprocedures shown in Figureand algorithms,6. Both ﬁeld characterized and circuit models in subsequent have been section extended of with this Author’spaper. procedures and algorithms, characterized in subsequent section of this paper.

Figure 6. Schematic diagram of the calculation algorithm.

5. Modeling of the Discharge Development Numerical calculations of discharges in liquids are divided into three basic stages, allowing for the implementation of the modeling objectivesobjectives describeddescribed inin thethe previousprevious section:section: - Determination of the streamers propagation (pre-breakdown); - Determination of of discharge breakdown (arc) st stageage parameters (voltage, current and energy values); - Modeling ofof thethe dischargedischarge system system including including the the power power source source to select to select appropriate appropriate capacitor capacitor used usedduring during discharge discharge process. process.

The analysis presented in this paper uses a model consisting of two ﬂat electrodes of 50 mm diameter. Active area (model of liquid juice), wherein the discharge occurs, was placed between the electrodes. Distance between the electrodes was 60 mm. The geometry of the model is shown in Figure7a. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 19

The analysis presented in this paper uses a model consisting of two flat electrodes of 50 mm diameter. Active area (model of liquid juice), wherein the discharge occurs, was placed between the electrodes. Distance between the electrodes was 60 mm. The geometry of the model is shown in Figure 7a. The gas bubble theory was used to determine streamers propagation in the active area. This assumption was introduced, because of the results of preliminary analysis presented in previous section of the paper. According to the obtained results, one can classify the discharge as “slow” [39] and theoretical description of gas bubble theory is applicable in such conditions [14]. The static models implemented in ANSYS program were used to determine electric field distributions. According to the proposed procedure, streamers development was computed using a special “virtual” surfaces arranged in parallel to electrodes (Figure 7b). Adopted “virtual” surfaces form a kind of differential division of the active area. All streamers heads can be located only on these surfaces and streamers spread are computed using a “frame by frame” method. So that, resolution of calculation results depends directly on the number of surfaces. The calculation procedure has been detailed further in the paper. Computations of streamers development could not be performed using only commercial FEM system (ANSYS). Some additional procedures were implemented in Authors’ Appl. Sci. 2020subprograms., 10, 3900 Mentioned procedures were used to compute electric field gradients, voltage values at 8 of 20 streamers heads, and to generate geometry of streamers paths and gas bubbles.

Figure 7. GeometryFigure 7. Geometry of the of system the system (a) ( anda) and division division of of the the discharge discharge space spacewith “virtual” with “virtual”surfaces (b). surfaces (b). 1—electrodes; 2—juice; 3—“virtual” surfaces dividing the discharge area. 1—electrodes; 2—juice; 3—“virtual” surfaces dividing the discharge area. According to the description of the HVED technique, a series of several tens to several hundred The gasdischarges bubble is used theory [4,9]. Therefore, was used the procedur to determinee for determining streamers the streamers propagation propagation in have the to active area. This assumptionbe performed was introduced,many times, regarding because the of number the results of analyzed of preliminary impulses. The analysis initial conditions presented for in previous each impulse, in the form of variable physical parameters of the liquid environment, have been section of the paper. According to the obtained results, one can classify the discharge as “slow” [39] introduced based on the temperature distribution that occurs after the previous pulse was generated and theoretical(Figure description 6). During the of gasmodeling bubble of theorythe first isdischa applicablerge, it was in suchassumed conditions that the active [14]. Thearea staticis models implementedcharacterized in ANSYS by programhomogeneous were temperature, used to determineconductivity, electricand concentration field distributions. of charges. The According to the proposedsolution procedure, of the electric streamers field intensity development equation and was the computedelectric potential using (4) afor special the electrostatic “virtual” surfaces problem in such a system always leads to results, where the electric field intensity is homogeneous arranged inthroughout parallel the to analyzed electrodes area (Figureand the electric7b). Adopted potential distribution “virtual” is surfaces characterized form by aiso-surfaces kind ofdi fferential division ofparallel the active to the electrodes area. All (Figure streamers 8a). heads can be located only on these surfaces and streamers spread are computed using a “frame by frame”div()εE method.= 4πρ So that, resolution of calculation(4) results depends directly on the number of surfaces. The calculation procedure has been detailed further in the where: E—electric field intensity; ε—dielectric permittivity; ρ—charge density paper. Computations of streamers development could not be performed using only commercial FEM Because of homogeneous electric field distribution, discharge initiation point cannot be clearly system (ANSYS).determined. Some In additionalorder to initiate procedures first discha wererge, implemented the random innumber Authors’ generator subprograms. was used, Mentioned proceduresimplemented were used into a MathCAD compute [43] electric program. field Node gradients, numbers (of voltage the finite valueselement atmesh) streamers lying on the heads, and to generate geometrycathode surface of streamers were exported paths to this and software. gas bubbles. Random node number was generated and re-entered into the ANSYS program to initiate the discharge. The gas bubble model with spherical geometry of According10 μm diameter to the description was created around of the the HVED selected technique, node (Figurea 8b). series According of several to the Equation tens to (5), several the hundred dischargesintensity is used of [ 4the,9]. electric Therefore, field in the the bubble procedure is higher for than determining in its direct vicinity the streamers [39]. The phenomenon propagation have to be performed many times, regarding the number of analyzed impulses. The initial conditions for each impulse, in the form of variable physical parameters of the liquid environment, have been introduced based on the temperature distribution that occurs after the previous pulse was generated (Figure6). During the modeling of the first discharge, it was assumed that the active area is characterized by homogeneousAppl. Sci. temperature, 2020, 10, x FOR PEER conductivity, REVIEW and concentration of charges. The solution9 of of 19 the electric field intensity equation and the electric potential (4) for the electrostatic problem in such a system was considered during calculations by introducing the value resulting from the formula (5) as an always leadsadditional to results, source where of electric the field electric in the numerical field intensity model (Figure is homogeneous 8c) [17]. throughout the analyzed area and the electric potential distribution is characterizedε by iso-surfaces parallel to the electrodes = 3 r (Figure8a). EB Eia , (5) 2ε +1 div(εE) =r 4πρ (4)

where: Eia—the intensity of the electric field in the vicinity of the gas bubble; εr—electrical dielectric where: E—electricpermittivity ﬁeld intensity; ε—dielectric permittivity; ρ—charge density

Figure 8. DistributionFigure 8. Distribution of the of electric the electric potential potential inin the model model not not including including (a) and( includinga) and including the gas the gas bubble (c) and the bubble model on the surface of the upper electrode (b). 1—electrode; 2—discharge c b bubble ( ) andarea; the3—gas bubble bubble. model on the surface of the upper electrode ( ). 1—electrode; 2—discharge area; 3—gas bubble. The plasma channel can be developed only if the local electric field intensity on the streamer’s head exceeds the critical value specified for breakdown of the dielectric material (juice model in the analyzed case) (E *) [17]. This condition was used to determine the streamer’s propagation path, by determination of electric field intensity gradients (Figure 9a). However, in many cases, the main discharge streamer can be branched. The condition for creation of new streamers may be presented basing on the field fluctuation criterion, leading to formula (6) [17]. > −δ Ei E* i , (6)

where: Ei—the gradient of the electric field intensity in the direction in which a new streamer can propagate; δi—random variable, taking into account the fluctuations associated with the local heterogeneity of the critical values of the breakdown voltage (randomness of the discharge). Computations of streamers propagation scheme were performed by the analysis of electric field intensity gradients distributions between individual nodes located on the successive “virtual” surfaces of the discharge area (Figure 7). In cases, where gradients were characterized by similar values (differences lower than 10%) (Figure 9b), the condition (6) was analyzed and a new streamer was introduced (Figure 9c,d).

Figure 9. Vectors of the electric field intensity (a), (b) and streamer models (c), (d) in subsequent layers of the discharge space. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 19 was considered during calculations by introducing the value resulting from the formula (5) as an additional source of electric field in the numerical model (Figure 8c) [17]. 3ε E = r E B ε + ia , (5) 2 r 1 where: Eia—the intensity of the electric field in the vicinity of the gas bubble; εr—electrical dielectric permittivity

Appl. Sci. 2020, 10, 3900 9 of 20

Because of homogeneous electric field distribution, discharge initiation point cannot be clearly determined. In order to initiate first discharge, the random number generator was used, implemented in a MathCAD [43] program. Node numbers (of the finite element mesh) lying on the cathode surface were exported to this software. Random node number was generated and re-entered into the ANSYS program to initiate the discharge. The gas bubble model with spherical geometry of 10 µm diameter was created around the selected node (Figure8b). According to the Equation (5), the intensity of the electricFigure field 8. inDistribution the bubble of is the higher electric than potential in its directin the vicinitymodel not [39 including]. The phenomenon (a) and including was the considered gas duringbubble calculations (c) and the by bubble introducing model on the the value surface resulting of the upper from electrode the formula (b). 1—electrode; (5) as an additional 2—discharge source of electricarea; field 3—gas in the bubble. numerical model (Figure8c) [17].

The plasma channel can be developed only if3 εther local electric field intensity on the streamer’s EB = Eia, (5) head exceeds the critical value specified for breakdow2εr + 1n of the dielectric material (juice model in the analyzed case) (E *) [17]. This condition was used to determine the streamer’s propagation path, by where: E —the intensity of the electric field in the vicinity of the gas bubble; εr—electrical determinationia of electric field intensity gradients (Figure 9a). However, in many cases, the main dielectric permittivity discharge streamer can be branched. The condition for creation of new streamers may be presented The plasma channel can be developed only if the local electric field intensity on the streamer’s basing on the field fluctuation criterion, leading to formula (6) [17]. head exceeds the critical value specified for breakdown of the dielectric material (juice model in the > −δ analyzed case) (E *) [17]. This condition wasEi usedE* to determinei , the streamer’s propagation path,(6) by determination of electric field intensity gradients (Figure9a). However, in many cases, the main where:discharge Ei—the streamer gradient can beof branched.the electric The field condition intensity for in creation the direction of new in streamers which a new may streamer be presented can propagate;basing on the δi—random field fluctuation variable, criterion, taking leading into account to formula the (6) fluctuations [17]. associated with the local heterogeneity of the critical values of the breakdown voltage (randomness of the discharge). Computations of streamers propagationE schemei > E wereδi, performed by the analysis of electric field (6) ∗ − intensity gradients distributions between individual nodes located on the successive “virtual” where:surfaces Eofi—the the discharge gradient area of the (Figure electric 7). field In case intensitys, where in gradients the direction were in characterized which a new by streamer similar valuescan propagate; (differencesδi—random lower than variable, 10%) (Figure taking 9b), into the account condition the fluctuations(6) was analyzed associated and a withnew streamer the local washeterogeneity introduced of (Figure the critical 9c,d). values of the breakdown voltage (randomness of the discharge).

Figure 9.9. VectorsVectors ofof the the electric electric ﬁeld field intensity intensity (a), ( (ab),) and(b) and streamer streamer models models (c), ( d()c), in ( subsequentd) in subsequent layers layersof the dischargeof the discharge space. space.

Computations of streamers propagation scheme were performed by the analysis of electric field intensity gradients distributions between individual nodes located on the successive “virtual” surfaces of the discharge area (Figure7). In cases, where gradients were characterized by similar values (differences lower than 10%) (Figure9b), the condition (6) was analyzed and a new streamer was introduced (Figure9c,d). The proposed procedure uses static calculations. Considerable disadvantage of this solution is the inability to use it to transient analysis of the pre-breakdown stage. In order to enable such calculations, the Equation (7) [17,44] was used to determine the average speed of streamers propagation, based on the analysis of local values of the electric field intensity.

h ( E ) ( E ) v(E) = e− g∗ e g (7) τ Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 19

The proposed procedure uses static calculations. Considerable disadvantage of this solution is the inability to use it to transient analysis of the pre-breakdown stage. In order to enable such calculations, the Equation (7) [17,44] was used to determine the average speed of streamers propagation, based on the analysis of local values of the electric field intensity.

− E*   E  h     v( E ) = e  g e g  (7) τ where:Appl. Sci. τ2020—time, 10, 3900 step; h—distance (between subsequent surfaces of the differential division of10 ofthe 20 discharge space); g—width of the streamer; E—electric strength, E*—breakdown field strength Equation (7) was used to determine the time, after which, the streamer head reaches subsequent where: τ—time step; h—distance (between subsequent surfaces of the differential division of the “virtual” surfaces dividing the area of the discharge propagation (Figure 7b). Such computations can discharge space); g—width of the streamer; E—electric strength, E —breakdown field strength be performed for all calculation steps to determine the temporal* characteristics of the discharge Equation (7) was used to determine the time, after which, the streamer head reaches subsequent propagation path. “virtual” surfaces dividing the area of the discharge propagation (Figure7b). Such computations Procedure shown in Figure 9 was used to determine the locations of all nodes, where streamers can be performed for all calculation steps to determine the temporal characteristics of the discharge heads occur. Around these nodes, models of the gas bubbles were created, similarly to the above propagation path. described modeling of the first node located on the surface of the electrode (Figure 8b). The Procedure shown in Figure9 was used to determine the locations of all nodes, where streamers occurrence of the plasma channel between the gas bubble models is taken into account. The channel washeads simulated occur. Around in the form these of nodes, a cylinder models with of a the diameter gas bubbles of 10 μ werem (Figure created, 9c,d). similarly Inside tothe the channel, above thedescribed model modeling of the plasma of the firstof conductivity node located determ on theined surface byof Equation the electrode (8) was (Figure assumed8b). The [45–47]. occurrence This formulaof the plasma is useful channel for poorly between ionized the gas plasma. bubble During models calculations is taken into movements account. The of channel plasma was particles simulated was µ neglectedin the form [46]. of a cylinder with a diameter of 10 m (Figure9c,d). Inside the channel, the model of the plasma of conductivity determined by Equation (8) was assumed [45–47]. This formula is useful for poorly ionized plasma. During calculations movementsn ⋅ e2 () of8kT plasma1/ 2 particles was neglected [46]. σ = 0,85 e M 1/ 2 , (8) m ()πm2 1/⋅2n ⋅ s nee e (e8kT) o a σ = 0.85 · , (8) M 1/2 where: T—temperature, k-Boltzmann constant,me (mπem—electrone) no smass,a ne—concentration of electrons, · · no—concentration of electrically neutral particles, e—atomic unit of charge, sa—active cross-section in where: T—temperature, k-Boltzmann constant, m —electron mass, n —concentration of electrons, which collisions occur e e n —concentration of electrically neutral particles, e—atomic unit of charge, s —active cross-section in o Relating to Equation (8), it is necessary to know the plasma temperaturea to determine its which collisions occur conductivity. In general, the presented algorithm for streamers development stage, does not require Relating to Equation (8), it is necessary to know the plasma temperature to determine its to perform thermal calculations. This fact results directly from the analysis of bibliography conductivity. In general, the presented algorithm for streamers development stage, does not require to [39,44,47], where it has been proved that in pre-breakdown stage (before electrical breakdown), the perform thermal calculations. This fact results directly from the analysis of bibliography [39,44,47], value of dissipated energy is relatively small in comparison to the energy released during the where it has been proved that in pre-breakdown stage (before electrical breakdown), the value of breakdown (arc) stages of discharge. Using the equations for isothermally heated body (9), it was dissipated energy is relatively small in comparison to the energy released during the breakdown possible to develop an approximate dynamic temperature characteristic (Figure 10a) for the (arc) stages of discharge. Using the equations for isothermally heated body (9), it was possible to discharge propagation. The obtained temperatures were used to compute the non-linear value of develop an approximate dynamic temperature characteristic (Figure 10a) for the discharge propagation. plasma conductivity (Figure 10b) [48]. The proposed solution is simplified because of the assumed The obtained temperatures were used to compute the non-linear value of plasma conductivity constant power value during the streamer conductivity computations. To consider mentioned effects (Figure 10b) [48]. The proposed solution is simplified because of the assumed constant power value it is necessary to perform iterative calculations. during the streamer conductivity computations. To consider mentioned effects it is necessary to τ perform iterative calculations. ()τ ≈ ⋅ + t Pp τ t p , (9) t(τ) Pp c ⋅ m+ tp, (9) ≈ · c pl m pl pl · pl where: PpPp—power—power dissipated dissipated in plasma;in plasma;cpl—specific cpl—specific heat ofheat the plasma;of the mplasma;pl—plasma mpl—plasma mass; tp—initial mass; ttemperaturep—initial temperature

Figure 10. Plasma heating process (a) and characteristic of plasma conductivity vs. temperature (b).

Plasma conductivity values were used to determine the charge distribution (10). All steps described above were used to create the computational algorithm for the propagation of electrical discharges in liquids, as shown in Figure 11.

∂ρ + div(σM E) = 0, (10) ∂τ · Appl.Appl. Sci. Sci. 2020 2020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 1111 ofof 1919

FigureFigure 10. 10. Plasma Plasma heating heating process process ( (aa)) and and characteristic characteristic of of plasma plasma conductivity conductivity vs. vs. temperature temperature ( (bb).).

PlasmaPlasma conductivityconductivity valuesvalues werewere usedused toto determinedetermine thethe chargecharge distributiondistribution (10).(10). AllAll stepssteps describeddescribed aboveabove werewere usedused toto createcreate thethe computationalcomputational algorithmalgorithm forfor thethe propagationpropagation ofof electricalelectrical dischargesdischarges in in liquids, liquids, as as shown shown in in Figure Figure 11. 11. ∂ρ ∂ρ + ()σ ⋅ = Appl. Sci. 2020, 10, 3900 +divdiv()σM ⋅EE =00, 11(10) of 20 ∂∂ττ M , (10)

FigureFigure 11. 11. Algorithm Algorithm for for streamer streamer propagation propagation calculation. calculation.

TheThe electric electric potential potential distributions distributions inin the the mode modelmodell for for the the first ﬁrstfirst discharge discharge have have been been shown shown in in FigureFigure 12.1212.. In In terms terms of of quality, quality, the thethe results resultsresults confirm conﬁrmconfirm the thethe proper properproper operation operation of of thethe algorithm.algorithm.

Figure 12. Distributions of potential values during prebreakdown stage (streamers) for different Figure 12.12. Distributions of of potential values during pr prebreakdownebreakdown stage (streamers) for didifferentﬀerent timesteps. (a): τ = 3.6 μs; (b): τ = 9.6 μs; (c): τ = 12 μs timesteps. (a): (a): ττ == 3.63.6 μµs;s; (b): (b): ττ == 9.69.6 μµs;s; (c): (c): τ τ= =1212 μsµ s

The algorithm allows to calculate the density of currents in streamers and the power loses during the initiation stage of the discharge. Current densities in diﬀerent time steps are presented in Figure 13. One can use the algorithm, to perform full calculations of the pre-breakdown, both in geometrical and energy point of view. Appl. Appl.Sci. 2020 Sci. 2020, 10, ,x 10 FOR, x FOR PEER PEER REVIEW REVIEW 12 of 1219 of 19

The Thealgorithm algorithm allows allows to to calculate calculate thethe density ofof currentscurrents in in streamers streamers and and the thepower power loses loses Appl.during Sci. the2020 initiation, 10, 3900 stage of the discharge. Current densities in different time steps are presented12 of in 20 during the initiation stage of the discharge. Current densities in different time steps are presented in Figure 13. Figure 13.

Figure 13. Densities of currents in streamers at various times (a–d) of the discharge development. Figure 13. Densities of currents in streamers at various times (a–d) of the discharge development. 6.Figure ModelingOne 13.can Densities of use the Breakdownthe of algorithm, currents (Arc) in tostreamers Stage perform at variousfull calculations times (a– dof) of the the dischargepre-breakdown, development. both in geometrical and energy point of view. Further analyses were carried out for the breakdown stage, where the plasma channel occurs One can use the algorithm, to perform full calculations of the pre-breakdown, both in between the electrodes (channel “connects” the electrodes). It was assumed, during this stage, that 6. Modeling of the Breakdown (Arc) Stage geometricalall “secondary” and energy streamers point (streamers of view. that not reach the electrode surface) were neglected (Figure 14). To computeFurther theanalyses current were and carried voltage out values for the in thebreakdown time domain, stage, itwhere was necessarythe plasma to channel determine occurs the 6. Modelingelectricalbetween the conductivityof theelectrodes Breakdown of(channel the plasma, (Arc) “connects” Stage where chargethe electr carriersodes). areIt was in the assumed, form of during electrons this and stage, ionized that all “secondary” streamers (streamers that not reach the electrode surface) were neglected (Figure particles.Further Itanalyses was required were to carried establish out the for inﬂuence the breakdown of thermodynamic stage, where parameters the (mainlyplasma temperaturechannel occurs 14). To compute the current and voltage values in the time domain, it was necessary to determine the betweenand pressure)the electrodes on the conductivity(channel “connects” value. For the highly electr ionizedodes). plasma,It was assumed, the conductivity during value this canstage, be that computedelectrical conductivity using formulas of the (11), plasma, and (8) where for weakly charge ionized carriers plasma are in [the39,46 form]. of electrons and ionized all “secondary”particles. It wasstreamers required (streamers to establish that notthe reachinfluence the ofelectrode thermodynamic surface) parameterswere neglected (mainly (Figure 14). Totemperature compute andthe currentpressure) and on voltagethe conductivity values( in4 πεvalue. the)2( kTtime For)3/ 2domain,highly1 ionized it was plasma, necessary the toconductivity determine the σD = 0.591 , (11) electricalvalue conductivitycan be computed of theusing plasma, formulas where (11), andcharge (8)e2 mforcarriers1/ weakly2 lnareθ ionized in the plasma form of[39,46]. electrons and ionized particles. It was required to establish the influence of thermodynamic parameters (mainly where: T—temperature, k—Boltzmann constant, m—mass of the particle, θ—electron concentration, temperature and pressure) on the conductivity value. For highly ionized plasma, the conductivity e—elementary energy value can be computed using formulas (11), and (8) for weakly ionized plasma [39,46].

Figure 14. The geometry of the discharge obtained on the basis of streamers propagation analysis (a) and adopted for the analysis of the breakdown stage (b). 1—electrodes; 2—discharge.

()()πε 2 3 / 2 σ = 4 kT 1 Figure 14. The geometry of the dischargeD 0 obtained,591 on the basis of streamers, propagation analysis(11) (a) Figure 14. The geometry of the discharge obtained one 2m the1/ basis2 ofln streamersθ propagation analysis (a) and andadopted adopted for forthe the analysis analysis of of the the breakdown breakdown stagestage ( b(b).). 1—electrodes; 1—electrodes; 2—discharge. 2—discharge. where: T—temperature, k—Boltzmann constant, m—mass of the particle, θ—electron concentration, It was assumed that in the presented system there is a plasma characterized by medium ionization e—elementary energy ()()πε 2 3 / 2 degree.It was Electrical assumed conductivity that in oftheσ the presented plasma= was system4 estimated therekT as is the a averageplasma1 valuecharacterized from formulas by medium (8) and (11) [39,47]. Figure 15 presents the conductivityD 0,591 values as a function of the, plasma temperature. Because (11) ionization degree. Electrical conductivity of the plasmae 2m1 was/ 2 estimatedlnθ as the average value from offormulas the mentioned (8) and relation,(11) [39,47]. arc stageFigure analysis 15 presents requires the coupled conductivity electromagnetic values as a and function thermal of calculations.the plasma where:temperature. T—temperature, Because ofk—Boltzmann the mentioned constant, relation, armc—mass stage analysis of the requiresparticle, coupled θ—electron electromagnetic concentration, e—elementaryand thermal energy calculations. It was assumed that in the presented system there is a plasma characterized by medium ionization degree. Electrical conductivity of the plasma was estimated as the average value from formulas (8) and (11) [39,47]. Figure 15 presents the conductivity values as a function of the plasma temperature. Because of the mentioned relation, arc stage analysis requires coupled electromagnetic and thermal calculations. Appl. Sci. 2020, 10, 3900 13 of 20

Dimensions of the plasma channel were determined automatically. In the case of a circular arc, theAppl. equation Sci. 2020, 10 for, x FOR radius PEER of REVIEW the plasma channel can be speciﬁed as [47]: 13 of 19

1/4 !1/4 !1/8 2 1 ρa 1/8 1/2 rl = 2g z I , (12) π2 · σρh · ρ · ·

where: I—current, z—arc length, g—gravity, h—speciﬁc enthalpy of the arc plasma, ρ—arc Appl. plasmaSci. 2020, density 10, x FOR PEER REVIEW 13 of 19

Figure 15. Plasma conductivity as a function of temperature.

Dimensions of the plasma channel were determined automatically. In the case of a circular arc, the equation for radius of the plasma channel can be specified as [47]:

1/ 4 1/ 4 1/ 8  2   1   ρ  r =   ⋅   ⋅ 2g a  ⋅ z1/ 8 ⋅ I 1/ 2 (12) ł 2 σρ   ρ  ,  π   h    where: I—current, z—arcFigureFigure length, 15. 15. PlasmaPlasma g—gravity, conductivity conductivity h—specific as aa functionfunction enthalpy of of temperature. oftemperature. the arc plasma, ρ—arc plasma density DimensionsAccording of to the formula plasma (12), channel to determine were determined the plasmaplasma automatically. channel radius, In it the is necessary case of a tocircular input arc, the equationdischarge for currentcurrent radius value value of andthe and plasma iterative iterative channel calculations calculations can must be specifiedmust be performed be performed as [47]: to use it.to Theuse valuesit. The of values discharge of currentsdischarge were currents introduced were introduced on the basis on of thethe resultsbasis of obtained the results in previous obtained iteration in previous steps iteration of the conducted steps of analyzes. In the ﬁrst step, the discharge1/ 4 diameter1/ was4 assumedρ 1/ as8 10 µm (the value was the same as the conducted analyzes. In =the first2  step,⋅  the1 discharge ⋅  diametera  was⋅ 1 /assumed8 ⋅ 1/ 2 as 10 μm (the value inwas the the pre-breakdown same as in the modeling). pre-breakdownrł  The arc modeling). plasma of The the2g arc given plasma conductivityz of theI given was, introducedconductivity to was the (12)  π 2  σρh   ρ  numericalintroduced model to the innumerical the form model of a solid in the body form (Figure of a solid 14b). body The (Figure physical 14b). phenomena The physical occurring phenomena in the plasma have been omitted. Nevertheless, this simpliﬁcation is acceptable in relation to determination where:occurring I—current, in the zplasma—arc length,have been g—gravity, omitted. Neverthe h—specificless, enthalpythis simplification of the arc is acceptable plasma, ρ in—arc relation plasma of active power of plasma channel. Static calculations of the electromagnetic problem were carried densityto determination of active power of plasma channel. Static calculations of the electromagnetic out to compute the discharge conduction current (Figure 16a,b) and the heat sources distribution problemAccording were to carried formula out (12),to compute to determine the discharg the eplas conductionma channel current radius, (Figure it 16a,b)is necessary and the heatto input (Figuresources 16distributionc). These computations (Figure 16c). wereThese performed computations iteratively, were performed with the iteration iteratively, frequency with the depending iteration discharge current value and iterative calculations must be performed to use it. The values of onfrequency the thermal depending analysis on results the thermal discussed analysis further results in the discussed paper. further in the paper. discharge currents were introduced on the basis of the results obtained in previous iteration steps of the conducted analyzes. In the first step, the discharge diameter was assumed as 10 μm (the value was the same as in the pre-breakdown modeling). The arc plasma of the given conductivity was introduced to the numerical model in the form of a solid body (Figure 14b). The physical phenomena occurring in the plasma have been omitted. Nevertheless, this simplification is acceptable in relation to determination of active power of plasma channel. Static calculations of the electromagnetic problem were carried out to compute the discharge conduction current (Figure 16a,b) and the heat sources distribution (Figure 16c). These computations were performed iteratively, with the iteration frequency depending on the thermal analysis results discussed further in the paper.

Figure 16. Current densitydensity vectorsvectors (a(a),), current current density density distribution distribution (b (),b), and and heat heat sources sources distribution distribution (c) in(c) thein the breakdown breakdown (arc) (arc) stage stage of dischargeof discharge analysis. analysis.

Based on the conduction currents density (J) andand thethe distributiondistribution ofof heatheat sourcessources ((ppVV), it was possible toto computecompute thethe resistance resistance of of the the arc arc (13). (13). The The proposed proposed method method uses uses the the total total active active power power (P) and(P) and current current (I) values,(I) values, obtained obtained by addingby adding its elementaryits elementary values values in individual in individual ﬁnite finite elements elements in the in the discharge model (i = 1 ... N). The computed resistance values were necessary to design circuit models that allow to determine operational characteristics of discharge model equipped with power source.

Figure 16. Current density vectors (a), current density distribution (b), and heat sources distribution (c) in the breakdown (arc) stage of discharge analysis.

Based on the conduction currents density (J) and the distribution of heat sources (pV), it was possible to compute the resistance of the arc (13). The proposed method uses the total active power (P) and current (I) values, obtained by adding its elementary values in individual finite elements in the discharge model (i = 1 ... N). The computed resistance values were necessary to design circuit models that allow to determine operational characteristics of discharge model equipped with power source. Appl. Sci. 2020, 10, 3900 14 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 19

Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 19 discharge model (i = 1 ... N). The computed resistanceN values were necessary to design circuit models that allow to determine operational characteristics of discharge⋅ model equipped with power source.  pVi Vi P N R = = i=1 2 N p ⋅V 2 (13) PN Vi i , IP  i=1pVi Vi  R = P = i=1 ()·⋅ R = 2 =  Ji Fi ,2 (13) (13) I 2 " N= #2 , I PNi 1()⋅  (JJi i FiF) i  i=1 · where: P—active power value, pVi—Joule heat iof=1 i-th finite element, Vi—volume of i-th finite where:element,P —activeJi—current power density value, in pi-th—Joule finite element, heat of i -thFi—external finite element, area ifV i—volume-th finite element of i-th finite element, where: P—active power value,Vi pVi—Joule heat of i-th finite element,i Vi—volume of i-th finite J —currentThermal density calculations in i-th finitewere element,performedF —externalas transient, area using if i-th previously finite element computed heat sources and element,i Ji—current density in i-th finite element,i Fi—external area if i-th finite element boundary conditions characteristic for conduction and convection heat transfer. In the example, the ThermalThermal calculations calculations were were performed performed as as transient, transient, using using previously previously computed computed heat heat sources sources and and temperature distributions in time 0–0.001 s are shown in Figure 17. boundaryboundary conditions conditions characteristic characteristic for for conduction conduction and and convection convection heat heat transfer. transfer. In In the the example, example, the the temperaturetemperature distributions distributions in in time 0–0.001 s are shown in Figure 1717..

Figure 17. Temperature distribution in the sample in subsequent moments. (a) 0.0001 s; (b) 0.0005 s; (c) 0.001s. FigureFigure 17. TemperatureTemperature distributiondistribution inin thethe sample sample in in subsequent subsequent moments. moments. (a ()a 0.0001) 0.0001 s; s; (b )(b 0.0005) 0.0005 s; (s;c) (0.001s.Afterc) 0.001s. thermal calculations in any time step, it is necessary to change the material parameters (primarily the plasma conductivity) and the supply voltage in the breakdown stage. In current After thermal calculations in any time step, it is necessary to change the material parameters algorithm,After thermalvoltage source calculations was modeled in any timeas the step, RC ci itrcuit. is necessary Resistance to changevalues determined the material based parameters on the (primarily the plasma conductivity) and the supply voltage in the breakdown stage. In current (primarilyfield modeling the plasma (13) and conductivity) the capacitance and the (the supply value voltage assumed in theas breakdownequal to the stage. power In currentsource algorithm, voltage source was modeled as the RC circuit. Resistance values determined based on the algorithm,capacitor—Figure voltage 4) source were used was to modeled compute as the the time RC constant circuit. of Resistance RC circuit valuesand source determined voltage values based field modeling (13) and the capacitance (the value assumed as equal to the power source onin any the time ﬁeld step modeling of the (13)analysis and (14). the capacitanceFigure 18 shows (the valuethe voltage assumed over as time equal characteristic, to the power obtained source capacitor—Figure 4) were used to compute the time constant of RC circuit and source voltage values capacitor—Figureby the conducted 4analysis.) were used to compute the time constant of RC circuit and source voltage values in inany any time time step step of theof the analysis analysis (14). (14). Figure Figure 18 shows 18 shows the voltage the voltage over timeover characteristic,time characteristic, obtained obtained by the τ byconducted the conducted analysis. analysis. − ()τ = τRC (14) VV(τ) = VV00ee− RCτ (14) − ()τ = RC (14) where: V00—initial—initial voltage, voltage, ττ—time—time V V0e where: V0—initial voltage, τ—time

Figure 18. The voltage course during the simulated discharge. Figure 18. The voltage course during the simulated discharge.

The voltage valueFigure on the 18. Theelectrodes, voltage courseafter thermaduring thel calculations, simulated discharge. was introduced as the initial condition in the next cycle of static electromagnetic calculations. Presented algorithm can be realized The voltage value on the electrodes, after thermal calculations, was introduced as the initial iteratively until whole energy stored in the capacitor before discharge initiation is dissipated. From condition in the next cycle of static electromagnetic calculations. Presented algorithm can be realized iteratively until whole energy stored in the capacitor before discharge initiation is dissipated. From Appl. Sci. 2020, 10, 3900 15 of 20

The voltage value on the electrodes, after thermal calculations, was introduced as the initial condition in the next cycle of static electromagnetic calculations. Presented algorithm can be realized iteratively until whole energy stored in the capacitor before discharge initiation is dissipated. From the moment of capacitor discharge, no current will ﬂow in the arc channel. During the break time between next discharges, temperature distribution problem was analyzed.

7. Modeling of Temperature Distribution in the Time between Discharges The temperature distribution after the discharge extinction, can be used to re-analyze the generation of the next discharge, carried out by using the principles given in Section5 of this paper. For example, time required for the proposed solutions, despite the significant time consumption, enables a full analysis of discharges in liquid environments. It is worth to point out that for simplifications of numerical analysis, two different heat transfer models were used during discharge phase and break between the next discharges analysis: - The “static” approach without fluid movements modeling (15) was used to compute heat transfer process during breakdown (in the pre-breakdown (streamer) stage, thermal analysis was not performed). The special criterion value of thermal conductivity (16) was used to include thermal convection heat transfer in the fluid [49]; - The CFD model was used for heat and mass transfer analysis in the time between next discharges. Coupled equations for continuity of flow (18), momentum (19) (here presented in one dimension of Cartesians system), and energy (20) balances were computed in Ansys program. A typical turbulent k-ε model was in use. Temperature field distribution in the last step of discharge analysis was used as initial condition for CFD modeling.

∂t p 1 = V + [ (λ t)] (15) ∂τ cρ cρ ∇ ∇

n ! C1Ras λe = 1 + λs (16) Ras + C2 where: t—temperature, τ—time, c—specific heat, ρ—mass density, λ—thermal conductivity, λe—equivalent thermal conductivity (including fluid velocity), Ras—the Rayleigh number, C1,C2, n—factors depended on fluid flow type (laminar, turbulent).

∂ρ + ρ→v = 0, (17) ∂τ ∇ ·

2 2 2 ! ! dwx ∂p ∂ wx ∂ wx ∂ wx 1 ∂ ∂wx ∂wy ∂wz ρ = ρgx + µ + + + µ + + (18) dτ − ∂x ∂x2 ∂y2 ∂z2 3 ∂x ∂x ∂y ∂z ∂ (ρh) + ρh→v = [(λ + λt) T] + pV, (19) ∂τ ∇ · ∇ ∇ where: ρ—fluid mass density; p—hydrostatic pressure; ρg—mass forces; µ—dynamic viscosity; h—enthalpy; λ—thermal conductivity; λt—thermal conductivity for turbulent flow; T—temperature; pV—volumetric density of heat sources and loses. Because of the fact that the electrical permittivity and resistivity values of analyzed liquid depend on the temperature value (both parameters decrease in temperature function), calculations of thermal energy propagation in the time between the discharges seems to be a very important factor [50]. If the break time between initiation of the next discharge is too short, a large heterogeneity of temperature distribution is observed in the liquid sample. In such conditions (Equations (4) and (5)), streamers will probably propagate in the channel of the previous discharge or its vicinity. Transient thermal analysis between discharges enables to determine minimal time interval, after which, next discharge will propagate randomly, independently from the pattern of previous breakdown process. Exemplary Appl. Sci. 2020, 10, 3900 16 of 20 temperature and fluid velocity distributions computed after first discharge (presented in Figures 16 and 17) are presented in Figure 19. Appl.Appl. Sci. Sci. 2020 2020, 10, 10, x, xFOR FOR PEER PEER REVIEW REVIEW 1616 of of 19 19

FigureFigure 19. 19. Temperature Temperature distribution distribution ( t()t ) and and ﬂuidfluid fluid velocity velocity vectors vectors ( v()v )in in analyzed analyzed sample sample in in different di differentﬀerent timetime steps. steps.

PerformedPerformed CFD CFD analysis analysis were were used used to to determine determine theth the e electric electric fieldfield field intensity intensity distributions distributions in in the the analyzedanalyzed workload workload (sample (sample of of liquid liquid juice).juice). juice). Temperature TemperatureTemperature distributions distributions were were introduced introduced as as initial initial conditionsconditions of ofof the thethe electrostatic electrostaticelectrostatic problem. problem.problem. Such approachSuchSuch approachapproach enables enables toenables consider toto the considerconsider temperature thethe temperature dependencetemperature ofdependencedependence material parameters of of material material that parameters parameters are particularly that that are are important part particularlyicularly for electromagneticimportant important for for electromagnetic simulationselectromagnetic (ε, ρsimulations ).simulations Based on the(ε(,ε , ρ presented).ρ). BasedBased on approach,on thethe presentedpresented it was possible approach,approach, to compute itit waswas break possiblepossible time toto between computecompute the break discharges,break timetime between tobetween guarantee thethe randomdischarges,discharges, propagations to to guarantee guarantee of random streamers random propagations propagations and new discharge of of streamers streamers channel. and and new In new Figure discharge discharge 20 electric channel. channel. field In In intensities Figure Figure 20 20 (a,electricelectric b) and field field the intensities intensities gradients (a, of(a, b) electric b) and and the the field gradients gradients intensity of of (c,electric electric d) in field vicinity field intensity intensity of electrodes (c, (c, d) d) in in (1)vicinity vicinity are shown of of electrodes electrodes for the break(1)(1) are are time shown shown value for for of the 0.5the break sbreak (a, c) time andtime 1value svalue (b, d)of offrom 0.5 0.5 s previous s(a, (a, c) c) and and discharge. 1 1 s s(b, (b, d) d) In from from the figureprevious previous geometry discharge. discharge. of previous In In the the dischargefigurefigure geometry geometry channel of ofis previous previous shown (2).discharge discharge channel channel is is shown shown (2). (2).

Figure 20. Electromagnetic field distribution (a,b) and electromagnetic field gradients (c,d) Figure 20.20.Electromagnetic Electromagnetic ﬁeld fiel distributiond distribution (a,b) and(a,b electromagnetic) and electromagnetic ﬁeld gradients field (cgradients,d) distribution (c,d) distribution for nonlinear parameters (ε, ρ) after 0.5 s (a,c) and 1 s (b,d) from previous discharge. fordistribution nonlinear for parameters nonlinear (parametersε, ρ) after 0.5 (ε, s ρ (a) ,cafter) and 0.5 1 ss ((ba,,dc) fromand 1 previous s (b,d) from discharge. previous 1—electrode; discharge. 1—electrode; 2—geometry of previous discharge channel. 2—geometry1—electrode; of2—geometry previous discharge of previous channel. discharge channel.

SimulationsSimulations results, results, presented presented above, above, show show that that br breakeak time time of of 0.5 0.5 s sis is too too short short for for initiation initiation of of nextnext discharge. discharge. In In this this case, case, heterogeneous heterogeneous distribu distributiontion of of temperature temperature in in the the analyzed analyzed fluid fluid leads leads to to thethe conclusion conclusion that that the the next next discharge discharge will will propag propagateate in in the the channel channel of of previous previous one one (Figure (Figure 20c). 20c). AfterAfter 1 1 s, s, temperature temperature distribution distribution and and electric electric fi fieldeld intensity intensity distribution distribution are are more more homogeneous, homogeneous, andand random random distribution distribution of of the the discharge discharge channel channel is is probable probable in in this this case. case. Appl. Sci. 2020, 10, 3900 17 of 20

Simulations results, presented above, show that break time of 0.5 s is too short for initiation of next discharge. In this case, heterogeneous distribution of temperature in the analyzed ﬂuid leads to the conclusion that the next discharge will propagate in the channel of previous one (Figure 20c). After 1 s, temperature distribution and electric ﬁeld intensity distribution are more homogeneous, and random distribution of the discharge channel is probable in this case.

8. Conclusions The article deals with modelling and simulation issues of pulse discharges occurring in liquid environments, used in many techniques. As it was mentioned in the introduction, there is a lack of universal and accurate models to simulate all stages of discharges (initiation, pre-breakdown, and arc). Most popular models of discharges in liquids (especially dielectric) were shortly described. Because of the lack of information about the parameters of discharges in juices, preliminary tests were performed. Based on the authors’ laboratory tests, most important electrical parameters of discharges were performed. Additionally, specific transient characteristic of discharge propagations were determined. Considering the obtained results, discharge was classified as “slow.” In such type of discharge, gas bubble theory model was used to describe the phenomena of streamers propagation paths. In the paper, the numerical method and algorithm for pre-breakdown were characterized. Subsequently, analysis of electric potential gradients in vicinity of the streamer’s heads were used to determine the streamer position. The proposed method, based on gradients determination and probability distribution, was used to establish total discharge path and time characteristic of discharge growth. The mentioned phase of discharge is described in Section5 of the paper. It is worth to highlight that the proposed method enables to determine energy values based on momentary values of current and voltage. In the Section6, authors’ solutions in the scope of modelling of breakdown (arc) stage of discharge were detailed. For the given geometry of discharge path (determined in previous steps of the analysis), the total energy generated in the arc plasma was established using iterative calculations. Plasma conductivity and the cross section of plasma channel were computed using analytical formulas described in Section6. Coupled analysis of electromagnetic and thermal fields (in FEM system) were performed to determine the total power of the arc and transient temperature characteristics. Circuit models were used to simulate the power source in the form of RC system. Different capacity values of series capacitor enable to form different arc current characteristics. The CFD models were used in the last stage, where temperature distribution in the time between next discharges was computed. Based on the results, it was possible to consider nonlinear material parameters for the next discharge distribution modelling. Results presented in the paper concern discharges in juices and other liquids. All algorithms can be classiﬁed as original Authors’ solutions. It has to be highlighted, that similar solutions are rarely presented in scientiﬁc papers. Obviously, discussed models and algorithms, can be extended to obtain results characterized by higher accuracy. However, obtained results are promising and are characterized by high versatility.

Author Contributions: Conceptualization, methodology, software, writing—original draft preparation, visualization: M.W.; methodology, validation, P.K.; formal analysis, resources: S.K., writing—review and editing, supervision: S.T. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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