Journal of Electrostatics 103 (2020) 103412 Contents lists available at ScienceDirect Journal of Electrostatics journal homepage: http://www.elsevier.com/locate/elstat Study of partial discharges in liquids S.M. Korobeynikov a, b, *, A.G. Ovsyannikov b, A.V. Ridel a, b, D.I. Karpov a, M.N. Lyutikova b, c, Yu. A. Kuznetsova d, V.B. Yassinskiy d a Lavrentyev Institute of Hydrodynamics of SB RAS, 15 Lavrentyev Prosp., Novosibirsk, 630090, Russia Novosibirsk State Technical University, 20 Karl Marx Avenue, Novosibirsk, 630073, Russia c Federal Grid Company of Unified Energy System, PJSC, Noyabrsk Division, Russia d Karaganda State Technical University, 56 Av. Nursultan Nazarbaev, 100027, Karaganda, Kazakhstan b A R T I C L E I N F O The paper was presented at ISEHD-2019 Keywords: Partial discharges Transformer oil X-rays Bubble Streamers A B S T R A C T Experimental and theoretical results concerning partial discharges in liquids are presented. Experiments on the initiation of partial discharges in floating up bubbles under low background radiation were performed. An additional source of ionizing radiation (X-rays) should be used for the inception of partial discharges under these conditions. Different dynamics of floating up bubbles after partial discharges were recorded and described. In particular, the inception and development of the streamer in transformer oil from the bubble surface was observed that led to the breakdown of the electrode gap in some cases. The influence of partial discharge in one bubble (or in a glass microsphere) on the discharge inception in a neighbor bubble was studied. The experimental measurements and theoretical calculations of optical and electrical characteristics of partial discharges in the bubbles were made. The formation of partial discharge in a microbubble after the cavitation in the extremely high electric field was proved with the analysis and numerical simulation of Kerr fringes. The systematic cal­ culations of “true” and “apparent” charges were performed for round and deformed bubbles. The theoretical dependencies obtained are in good agreement with the experimental measurements performed. 1. Introduction Usually, gas voids in solid electrical insulation are considered as the main source of partial discharges at high voltage action, which cause aging of insulation and defines the life-time of apparatus [1‒3]. The physical mechanism and the main characteristics of PD in cavities of solid dielectrics were studied in Refs. [4–8]. These works concluded that the ignition voltage of the PD inside the cavities obeys the Paschen’s law, and the extinction occurs due to the action of the opposite field of charges that appeared on the dielectric walls of the cavities. The de­ pendences of PD repetition rate on the voltage applied to the insulation were determined, also the processes of the destruction of dielectrics under the PD effect, and other important questions were investigated. Authors [9–18] studied a phenomenology of PD in liquids in the context of the electrical breakdown of liquids. Therefore, they used gaps with a strongly inhomogeneous field. One of the important questions was the question of the initial phase of liquid breakdown. It has been established that in the “tip-plane” gaps with a field strength of more than 10 MV/cm, the streamer is the first to form, and it leads to the formation of vapor-gas bubbles [16]. In turn, the appearance of PD in the bubbles leads to the growth of the discharge channel and finally to an electric breakdown of the entire gap. From a practical point of view, we should note that bubbles in the oil in high-voltage electrical equipment could appear not only as a result of streamer propagation but also for another reason. Most often, bubbles arise due to deficiencies in the technological processes of manufacturing equipment; besides, they can appear during operation due to local overheating in certain places of structures, during cavitation in the fluid flows created by the cooling system, etc. Bubbles in oil are one of the most dangerous inclusions because they can lead to power failure due to breakdown. It is considered that the origin of insulation breakdown is frequently a partial discharge (PDs) in vapor-gas bubbles that occurred due to a local increase of field in bubbles and their low electrical strength. This work aims to analyze the physical picture of PD, in bubble especially. The questions considered in this paper are as follows partial discharges in free bubbles and microsphere, the interaction of PDs in bubble and microsphere and also in pair of bubbles, breakdown of liquid * Corresponding author. Lavrentyev Institute of Hydrodynamics of SB RAS, 15 Lavrentyev prosp., Novosibirsk, 630090, Russia. E-mail address: korobeynikov@corp.nstu.ru (S.M. Korobeynikov). https://doi.org/10.1016/j.elstat.2019.103412 Received 27 September 2019; Received in revised form 5 December 2019; Accepted 9 December 2019 0304-3886/© 2019 Elsevier B.V. All rights reserved. S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 due to streamer initiated by PD in the bubble. The new is simultaneous registration of bubble geometry (shadow picture), electrical and elec­ trooptical PD signals as well as systematic simulations of electrical properties of PD in bubbles. Other new data provide information about X-rays’ effect on the PD process and inception voltage. The next question is the experimental study of PD and gassing due to PD in the point-plane electrode system in mineral oil and rapeseed oil. Additional useful information concerning PD could be obtained in the case of an electrooptical study of prebreakdown processes in nitro­ benzene. The simulation and experimental data were compared. It allowed us to determine the electric field in the region with or without partial discharges in bubbles inside this region. Another result of this part of work is an estimation of electrohydrodynamic (EHD) flow ve­ locity and space charge density at the injection from the electrode into liquid. The correlation between the breakdown voltage and the concentra­ tion of water in transformer oil is of great importance and it will be discussed in this work. Besides the role of water in breakdown due to the appearance of water-oil emulsion, the water molecules could be the source of initiating electrons in the bubbles. Fig. 2. Partial discharge occurs in the lower bubble in the second frame, transformer oil atU ¼ 16 kV, d ¼ 6:8 mm (Eamp ¼ 23:5 kV/cm). amplitude value of the applied voltage corresponding to the PD incep­ tion voltage in a bubble according to Pashen’s law gives 25 kV approximately for the interelectrode gap d ¼ 6:8mm (the mean value of the electric field in the gap is Eamp ¼ 36:8kV/cm). Nevertheless, no one PD was detected up to the voltage magnitude of 42 kV. In the next series, the experiments were performed for floating up helium bubbles of the same sizes and the behavior of these bubbles was studied. Theoretical estimation of the applied voltage corresponding to the Pashen’s discharge in a helium bubble give 6.4 kV approximately in this case (Eamp ¼ 9:4kV/cm). In our experiments, this required at least 15 kV. Moreover, this PD in a bubble was a very rare event. One PD was observed after several hours of exposure to voltage. The discharge inside the bubble in all registered cases leads to the elongation of the bubble and its separation into two charged parts (Fig. 2). At the action of alternating electric fields, these parts move in opposite directions and oscillate in opposite directions. Sometimes when one of the secondary bubbles approaches an electrode a secondary powerful PD takes place. Any dependence on the voltage phase of PD inception wasn’t registered even at the voltage two times higher than required by Pashen’s law. Oscillogram records of the electric and optical signals of the PD in the bubble are shown in Fig. 3. One can see that the pulses have almost the same shape, the leading edges of the pulses are the same, and the falling edges of the pulses are slightly different. The delay of the optical signal was due to different lengths of measuring cables and the transit time of electrons inside the photomultiplier. The rise time of both signals was about 20 ns (at own rise time of the scope of 15 ns). Sometimes at higher voltages, one could also see the formation of streamers at large bubble deformation after PD (Fig. 4). Here the mean electric field at the amplitude value of the voltage is Eamp ¼ 37 kV/cm. Streamers inside transformer oil are observed in the form of semi­ transparent filaments at the elongated parts of the bubble. Several experiments were performed with PD appearance in the glass hollow microsphere. These tests aimed to reveal a possible influence of 2. Methodology This paper presents the experimental and theoretical results con­ cerning partial discharges in liquids. In the first experiments, we studied PD in specially prepared helium bubbles in mineral oil under AC voltage action. Fig. 1 shows the experimental setup. Almost the same setup as described early in Refs. [19,20] was used in the experiments. The interelectrode gap was 6.8 mm usually. In the case of free-floating bubbles, we used a high-speed video camera (6) and a photomultiplier (7) for optical registration. They are located coaxially on one optical axis. The red color LED illuminates the gap from the upper part of the cell. To avoid a disturbance for photomultiplier we installed the light filter in blue-violet pass-band. In experiments with X-ray, the photomultiplier was absent. In this case, the experimental setup included only a high-speed video camera (6) installed coaxially with illuminator (7) for the optical detection of the bubble’s shadow pictures. In the experiments with transformer oils, Nitro GX, GK, BG types of oils were used. No noticeable difference in results was found. The moisture content of the oil was not specifically measured before each experiment. It is known from experience that when the oil was evacuated with stir­ ring with a magnetic stirrer and if there was an adsorbent in the flask, the water concentration did not exceed 5 ppm. 3. Experimental results 3.1. Partial discharges in free bubbles The bubble generation system created several bubbles per second automatically. The diameter of the bubbles was about 1.5 mm usually. In the first series of experiments, we studied air bubbles. Estimation of the Fig. 1. Experimental setup. Fig. 3. Signals of PD in bubble: upper-electrical, lower – optical. 2 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 Fig. 4. Streamer formation (the right frame), U ¼ 25 kV (Eamp ¼ 37 kV/cm). Fig. 7. Partial discharges in helium bubbles exposed to X-rays without screen, U ¼ 8:8 (Eamp ¼ 13 kV/cm). PD in microsphere on PD appearance inside the bubble close to the microsphere (up to the contact of one with the other). It was established that PD in the microsphere has usual properties like conventional PD in gas inclusion inside solid dielectrics. The PD patterns and the applied voltage necessary for the initiation of Pashen’s discharge in the micro­ sphere were the same. But the influence of frequent PD in the micro­ sphere on the rare occurrence of PD in a free bubble was not registered. An interesting result was obtained in the case of the simultaneous floating of two bubbles. The distance between them was less than the bubble radius (Fig. 5) Several PD were registered. Every PD in one bubble led to PD in the second one. shown in Fig. 8. One could see streamer formation at the poles of the most elongated bubble. Interestingly, the structures emerging after PD are very slow (velocity ~1 m/s) developing streamers. Sometimes the next frames show the breakdown of the gap at very low voltage. Breakdown took place on the next frame of Fig. 9 (not shown here). The electrical strength of transformer oil at AC action is more than 200 kV/ cm usually. In our case, breakdown takes place at average field strength less than 20 kV/cm. It means that there is significant field amplification at the apex of the elongated bubble. Sometimes streamer formation doesn’t lead to breakdown (Fig. 9). 3.2. Experiments with X-rays 3.3. PD in point-plane electrode system The main hypothesis explaining the rarity of PD inception even under the action of increased voltage is the lack of initiating (starting) electrons. For testing, we decided to affect the gap (with floating up bubbles) by X-rays (Fig. 6). Under this condition, PDs appeared in all bubbles that were crossing the region of high electric field simultaneously. When the voltage de­ creases, PDs ceased at U ¼ 6:4kV that corresponds to Paschen’s limit. In case of X-rays other interesting effects were observed (Figs. 7–9). The first one is the instability of bubbles after PD. Their shape looks like the so-called “Tailor cone” [21]. Usually, Tailor cone appears if instability of charged interface between liquid and gas phases develops in the directions of the gaseous phase. From Figs. 7 and 8 we can conclude that in the case of a partial discharge inside the bubble, instability of the Taylor cone type, sometimes developing in the direc­ tion of the liquid phase, is realized. The examples of bubbles instabilities after PD in all of them are Another type of PD occurs in a liquid in the presence of local strong electric fields (Fig. 10). In our experiments [22] mineral transformer oil and rapeseed oil were used. The “Point-plane” electrode system was used to generate a strong electric field. The curvature radius of the point electrode was less than 2 mm. The estimated electric field strength in liquids close to the needle tip exceeded 15 MV/cm. Comparative characteristics of PD pulses in mineral (MO) and rapeseed (RO) oils are given in Table 1. It was found that the gas formation rates of hydrogen and methane at the PD in rapeseed and transformer oils are a close one to others. 3.4. Electrooptical study of prebreakdown processes in nitrobenzene: the simulation Our previous experimental data concerning electrooptical registra­ tion of prebreakdown processes (see e.g Refs. [23,24]) were reconsid­ ered from the PD point of view. The Kerr electrooptical effect is a Fig. 5. Generation and floating of bubble’s pairs. U ¼ 27 kV (Eamp ¼ 39:6 kV/cm). Fig. 6. Partial discharges in helium bubbles exposed to X-rays, U ¼ 16:3kV (Eamp ¼ 24 kV/cm). Fig. 8. Partial discharges in helium bubbles exposed to X-rays without lead screen, U ¼ 18 kV (Eamp ¼ 18 kV/cm). 3 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 Table 1 PD characteristics in mineral and rapeseed oil. Characteristic MO RO Pulse width, [ns] Duration of leading edge, [ns] The maximum current, [μA] Apparent charge, [pC] 500 200–400 39–69 36–62 500 200–500 27–57 19–53 Fig. 9. Partial discharges in helium bubbles exposed to X-rays without screen, U ¼ 20 kV (Eamp ¼ 20 kV/cm). powerful tool for studying prebreakdown processes in liquids with large induced birefringence, such as nitrobenzene [24]. The obtained exper­ imental material was not fully analyzed either because of the high laboriousness of the processing and because of the lack of an adequate mathematical model of the processes under study at that time. The analyzed picture with PD is presented in Fig. 11. Here it is the scan of the polarized light passed through optical slit and analyzer (kerrogram) Rl Kerr fringes are lines of equal value of the integral l12 E2? ðx; z; tÞ d z. Therefore, the redistribution of the electric field leads to a redistribution of the position of the Kerr fringes. To analyze the available experimental electrooptical measurements [23,24], a mathematical model of a real measuring cell (tip-plane) was built, and the finite-element method was used to calculate the distribution of intensity, changes in the intensity of light transmitted through the measuring cell, and other physical char­ acteristics [25]. The computations were carried out in the absence or presence of various inhomogeneities in the form of a space charge, unionized or ionized bubbles. By ionization of the bubble after PD that has arisen, the distribution of the electric field changes. This is reflected in the picture of the Kerr bands by the violation of their regularity near the electrode. Moreover, at a certain distance from the electrode, the regularity of the fringes should be restored. To verify this statement, within the framework of our mathematical model [25], we obtained the pictures of Kerr bands near the point electrode by calculation. For this purpose, the intensity dis­ tribution of the light transmitted through the Kerr cell was calculated (Fig. 12). � � Z Iðx; zÞ ¼ I0 ⋅sin2 π ⋅ B ⋅ E2? ðyÞ dy : (1) Fig. 11. Rotating-prism scan of prebreakdown events near the cathode. t1 ; t2 are the moments of PD inside bubble zone, t3 is the moment of break­ down initiation. We placed a bubble with a diameter of 20 μm directly near the electrode and visualized the Kerr band patterns for the air bubble before and after its ionization using the obtained normalized values of the in­ tensity of light Iðx; zÞ=I0 transmitted through the cell. A phase shift distribution as Z ℓ2 Φðx; zÞ ¼ 2π ⋅ B ⋅ E2? ðyÞ dy ℓ1 was obtained for both cases (Fig. 13). An analysis of the obtained kerrograms and distribution phase shift showed that the perturbations introduced by the ionization of the bub­ ble completely disappear already at the distance of 65 μm from the tip. It confirms that here it is partial discharge. What is the reason for bubble appearance in experiments [23,24]? Bubbles are in the zone of space charge. One could see that the presence of such a zone leads to the initiation of cavitation processes inside. To this end, an assessment was made of the internal pressure arising in the near-electrode zone near the tip in the case of the appearance of a space charge there. From the conditions of mechanical equilibrium the space charge will act as a piston that moves fluid away from the electrode In this case, due to the strong inhomogeneity of the electric field near the tip, a sharp jump in negative pressure arose, leading to the appearance of a cavitation bubble. The magnitude of the pressure jump was estimated by the expression [26]. Here B is the Kerr constant; E? is the projection of the electric field in­ tensity vector on the z axis. Fig. 10. PD in transformer oil (1) - photomultiplier; (2) – electric signal. Fig. 12. Kerrogram. Applied voltage was 120 kV, tip radius was 300 μm. 4 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 molecules can attach electrons and be in oil in the form of negative ions. At the action of strong electric field these ions can lose electrons. It leads to forming free electrons capable of becoming so-called initiating elec­ trons. The mentioned role of negative ions as supplier of initiating electrons was proved in Refs. [7,8]. 4. Calculations of electrical characteristics of PD in bubbles 4.1. “Apparent” and “true” charges When we discuss the activity of PDs in insulating systems we need a quantitative criterion to describe the intensity of the electrical process. A good idea is to use the value of the electrical charge deposited on the walls of the bubbles after PD. The value of free positive charge on the bubble wall is called a “true” charge. Unfortunately, this value can not be measured directly in the case when there is no contact between the bubble and one of the electrodes. Nevertheless, PD in any bubble is accompanied by a current pulse. When PD takes place in a bubble that is on the electrode surface we register the current of discharge directly in the bubble and, hence, the charge in the bubble. If the bubble has no galvanic contact with the electrodes we measure a current signal induced in an external electric circuit by the processes of bubble po­ larization. The integral of this current over time gives us the so-called “apparent” charge that is, in fact, an image charge of the polarized bubble. Many factors influence the value of “apparent” charge such as some bubbles, the position of each bubble in an electrode gap, the shapes and the sizes of the bubbles. The influence of bubble size and bubble position for a spherical bubble on the relation between the “apparent” and “true” charges was first studied in Ref. [27] using two-dimensional approximation. The conductivity of a bubble was simulated by a very high value of dielectric permittivity inside the bubble. Theoretically, a bubble after PD is an electrical dipole that induced the image charges on the electrodes. The value of the “true” charge should be proportional to the area of the bubble-liquid interface approximately. From this point of view, the value of the “apparent” charge should be directly proportional to bubble volume. This relation is expected for small bubbles far enough from the electrodes. Nevertheless, it is not obvious for the bubbles of large sizes comparable to the gap distance since the interaction of bubble charges with the electrodes can be noticeable in this case. In this work, we performed the systematic calculations of “apparent” and “true” charge on large lattices in the three-dimensional case for spherical and deformed bubbles. The results of numerical simulations were compared with the analytical calculations of the charges on the bubble wall ob­ tained within the model of a single conducting bubble in an infinite dielectric. Fig. 13. Phase shift distribution (white color corresponds to the maximum value). Applied voltage was 120 kV, tip radius was 300 μm. Z z2 ρ E? dz; pint ¼ p0 z1 (2) where ρ is the space charge density, pint and p0 are the internal and external pressure, z1 and z2 are the coordinates (along the axis of sym­ metry of the electrode system z) marking the boundaries of the region with a homogeneous space charge, E? is the projection of the electric field intensity vector on the z axis. The range of space charge density values was selected based on es­ timates made in Ref. [23]. In Fig. 14 it can be seen that an increase in the value of a homoge­ neous charge leads to an increase in the tensile forces arising in nitro­ benzene due to a sharp decrease in internal pressure. This can lead to the appearance of cavitation bubbles. 3.5. The effect of water presence in transformer oil on its breakdown voltage In our opinion the water could play two roles in prebreakdown processes. First, it is considered traditionally that little water drops leads to field increase in the oil volumes close to the poles of the bubble as in the case of ionized bubble (Fig. 8). In the case of a single drop in the bulk of liquid, the increase in the local field strength is about 3, for a drop on the electrode it is up to 4. Under the action of an electric field, the drop is deformed and the electric field near its tips increases even more signif­ icantly. Another option, but also leading to a local field enhancement, is the emergence of instability of the charged drop surface, with the for­ mation of, so-called “Taylor cone”. These processes could explain the decrease of electric field strength in case of drop formation, detailed mechanism of PD generation and subsequent breakdown remains un­ clear. In our opinion water could play other role in PD inside bubbles. Water molecules have great electron affinity. Therefore, a number of 4.2. Numerical model The Poisson theorem of electrodynamics was used to calculate the electric field potential in the gap between plane electrodes. The dielectric with the dielectric permittivity εe ¼ 2:3 (corresponds to the transformer oil used in our experiments) filled the gap. The dielectric permittivity of gas in the bubbles wasεi ¼ 1. The continuity equation was used for the calculation of the electric charge transfer inside the bubble during PD. We did not simulate the process of the avalanche development in the bubble during PD since our interest was focused on the integral characteristics such as the total “true” charge, the “apparent” charge and the electric fields around the bubble. Thus, the simple model of constant conductivity of the substance inside the bubble during PD was applied. The value of the conductivity σ ¼ 0:018 S=cmwas derived from the Maxwellian relaxation time that was estimated from our experiments (decay time of PD, for example in Fig. 3). Since the hydrodynamic processes accompanying discharges have the characteristic time of the order of milliseconds we considered that Fig. 14. The dependence of the pressure pint on the value of a homogeneous charge in the region of 0.0–0.1 mm from the tip. Applied voltage was 120 kV. External (atmospheric) pressure was p0 ¼ 101.3 kPa. 5 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 cavity volumeQapp � bn , wheren ¼ 2:89 � 0:01 as in the case of a spherical cavity. Nevertheless, the apparent charge reduced to the bubble volume Qapp =Vcan depend on the deformation of the bubble. We plotted the relationship between Qapp =V and the coefficient of the deformation b=a. This dependence is linear when the deformation coefficient is higher than 1.2 as it is represented in Fig. 17. It should be noted that we observed a good agreement of our nu­ merical simulations of “apparent” charge to our experimental mea­ surements [28]. the bubble shape is constant during PD event that lasts for tens of nanoseconds. The three-dimensional calculations were performed on the lattice of size 256 � 256 � 256 nodes. The exact calculations of “apparent” charges took the relative accuracy for the field computation not worse than 10 11 that was achieved with the use of the graphic processing units (GPU) as the computing devices. In this paper, we consider that total charge relaxation in a bubble during PD takes place. 4.3. PD simulations for spherical bubbles 4.5. Simulation of mutual influence of the bubbles during discharge Numerical simulations of electric fields and charges were performed for a single spherical bubble placed to the center of the electrode gap. The values of “true” and “apparent” charges were calculated for different sizes of the bubble. Note, that the voltage should reach the value of the Pashen limit for the discharge in gas. The larger the bubble is the larger the Pashen voltage. In our simulations, we used the Pashen curve for helium to get the PD voltage in the bubble and then set the corresponding voltage applied to the gap. Fig. 15 shows that both the dependencies (for “true” and “apparent” charge) are linear in log-log plot. The slope coefficient is about 2 (1:91 � 0:01) for the “true” charge graph and about 3 (2:93� 0:01) for the “apparent” charge graph. The gap length was 6.8 mm in these simulations. Thus, the “true” charge is approximately proportional to the bubble surface and the “apparent” charge increases proportionally to the volume of the bubble even for the large bubble of the size of about 3 mm. The experiments discussed in section 3.1, Fig. 5 showed that the PD in one bubble can stimulate the PD in neighbor bubbles. We simulated simultaneous PDs in two bubbles of the same radiuses placed along the electric field line symmetrically to the electrodes. The “true” charges on the walls of the bubbles were calculated. The “true” charge per a bubble in dependence on the distance between bubbles is shown in Fig. 18. Here the radiuses of the bubbles were equal to 0:08 d. The graph shows that the discharge in neighbor bubbles can influence one another only if the spacing between bubbles is smaller than the bubble diameter. 4.6. Analytical solution for true charge of PD for deformed bubble The problem of the electric field around the prolate conducting ellipsoid of revolution placed to the uniform electric has analytical so­ lution (see, for example, [5,29]). We applied this problem to find the “true” charge on the wall of an elongated bubble after PD and compare it with our numerical simulations for large bubbles and our experiments. The electric field-potential distribution is given by the expression pffiffiffiffiffiffiffiffiffiffiffiffi φ ¼ E0 ⋅b⋅ 1 λ2 ⋅τ⋅σ ⋅Fðσ Þ; (3) 4.4. PD simulations for deformed bubble The electric forces at the gas-liquid interface deform the bubble in a high enough field [20] such as the bubble becomes longer in the di­ rection of the electric field. The deformation affects the electric char­ acteristics of PD. We simulated the “true” and “apparent” charge of PD depending on the bubble elongation and volume of the deformed bub­ ble. The shape of the bubble was approximated with the ellipsoid of revolution with the large half-axis and the small half-axisa. The elliptic cavity was placed at the center of the interelectrode gap. The “true” charge increases with the area of the bubble surface for the deformed bubble Qtrue � bm (, where m ¼ 1:88 � 0:01) as it was observed for the bubble of a spherical shape. Fig. 16 shows also that the “apparent” charge value increases approximately linearly with the where the function Fðσ Þ is � � 1 σþ1 2 FðσÞ ¼ 1 ln : BðλÞ σ 1 σ pffiffiffiffiffiffiffiffi 2 Here BðλÞ ¼ ln 1þpffi1ffiffiffiffiλffiffi2ffi 1 1 λ (4) pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 λ2 , σ and τ are the coordinate of the coordinate system of the prolate ellipsoid of revolution related to Car­ tesian coordinates as Fig. 15. Dependence of the true charge of PD in a gas-filled spherical cavity on the diameter of the cavity. (–■–) The apparent charge, (–●–) true charge, numerical calculations, (– � –) true charge, calculations with the expres­ sion (11). Fig. 16. Dependence of the “apparent” and “true” charges of PD in a gas-filled elliptic cavity on the cavity size db ¼ 2b at ðb =aÞ � 1:5. (–■–) The apparent charge, (–●–) true charge, numerical calculations, (– � –) true charge, calcu­ lations with the expression (10). 6 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 discharge in the bubbles. As a result, we need to calculate the “external” electric field E0 in expressions (6), (7) for the real values of electric field in the bubble corresponding to the Pashen’s voltage. For spherical bubble, the well-known relation gives the electric field very far from bubble E0 ¼ 2εe þ εi UPash 3 εe db (8) For the approximation of the bubble with an ellipsoid of revolution, we have E0 ¼ UPash � 2b λ2 εi 2 ðεe � pffiffiffiffiffiffiffiffiffiffiffi2ffi 1 λ � 2 εe 1 Fig. 17. Dependence of the value of the “apparent” charge per bubble volume on the coefficient of bubble deformation ðb=aÞ. εi Þλ2 ln ðεe λ2 �3=2 1þ 1 pffiffiffiffiffiffiffiffiffiffiffiffi ! 1 λ2 pffiffiffiffiffiffiffiffiffiffiffiffi 1 λ2 (9) All the calculations of the “true” and “apparent” charges listed below are made with this field correction. The integration of the density of free electric charge over the part of the bubble surface S carrying the positive charge gave us the true charge of PD Z Z Dn εe ! ! Qtrue ¼ dS ¼ En dS; with D ¼ εe E 4π 4π Sþ Here En ¼ E0 Sþ 2 τ ð1 λ2 Þ λ BðλÞ 3=2 pffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi. Then, the value of true charge on the 2 1 ð1 λ Þ τ bubble surface after PD is Qtrue ¼ εe E0 b2 ð1 3=2 λ2 Þ 2 BðλÞ : (10) It gives for a spherical bubble Qtrue ¼ ð3 = 4Þ εe E0 b2 : Fig. 18. “True” charge depending on the position of the center of a bubble for the case of the two spherical symmetrically placed bubbles. pffiffiffiffiffiffiffiffiffiffiffiffi x ¼ b⋅τ⋅σ ⋅ 1 � λ2 ; y2 ¼ b2 1 λ2 σ2 � z2 ¼ b2 1 λ2 σ 2 � 1 1 � 1 1 � τ2� cos2 α; τ2 sin2 α; It is clear that the value of true charge of PD in a single bubble is pro­ portional to the surface area of the bubble � �pffiffiffiffiffiffiffiffiffiffiffiffi � . pffiffiffiffiffiffiffiffiffiffiffiffi � S ¼ 2π b2 λ λ þ arcsin 1 λ2 (12) 1 λ2 (5) provided that the deformation of the bubble is the same. Fig. 19 shows that the value of “true” charge on the bubble – liquid interface reduced to the bubble surface (expression (10) divided by (12)) changes only slightly with the deformation of the bubble in contrast to “apparent” charge that changes significantly with the deformation of the bubble of the same volume (Fig. 17). The comparison of the value of “true” charge obtained with the ex­ pressions (10), (11) and numerical simulation was made (Figs. 15 and 16) that shows very good agreement both for small and large bubbles. where α is the azimuthal angle. The physical scales of the problem are the electric field at large distance (at infinity) from the bubble E0 , the large hemi-axis of the ellipsoid b , and the bubble deformation. To characterize the deforma­ tion, we used the parameter λ ¼ a=b, where a is the small hemi-axis of the ellipsoid. Thus, the ellipsoidal coordinate σs ¼ 1= pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λ2 corresponds to the surface of the bubble after PD. This solution gives the electric field on the symmetry axis of the system as � � 2 Ex ¼ E0 FðσÞ þ : (6) 2 σ ⋅ðσ 1Þ⋅BðλÞ 5. The streamer initiation from the top of the bubble In some experiments, we observed the generation of filamentary streamers in the liquid phase in transformer oil just after PD in a bubble (Figs. 4, 7 and 8). The streamers started from the tops of the bubble. The free electric charge (“true” charge) at the apexes of the bubble amplifies the electric field at the tops. To determine this field, we made the cal­ culations of the electric field distribution around an elongated bubble of the shape of an ellipsoid of revolution with the different degrees of deformation. Three-dimensional calculations were performed on the lattice having the shape of a parallelepiped with the sizes of 448 � 256 � 256 lattice steps. Plane electrodes were on the small faces of the parallelepiped. The cavity was simulated with the elliptic region with the dielectric permittivity placed at the center of the electrode gap. We This allowed us to get the maximal electric field on the bubble apex in dependence on the bubble deformation as 3=2 Emax ¼ 2E0 ð1 λ2 Þ λ2 ⋅BðλÞ ; (11) (7) with Emax ¼ 3⋅E0 for spherical surface (λ ¼ 1). The electric field in a spherical or deformed bubble is somewhat higher than the average electric field in the gap. Thus, we have to take this fact into account when we find the Pashen’s voltage UPash for the 7 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 Fig. 20. Electric field stress along the symmetry axis of the electrode gap before PD (green line), after PD calculated numerically (red) and analytically (blue). Elliptic bubble with the deformation coefficient 5.57. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) Fig. 19. “True” charge reduced to the bubble surface at different deformation of an ellipsoidal bubble. scaled the calculation to the size of the gap distance d ¼ 6:8mm. This means that the spatial resolution for the field calculation was about 15 μ m. Periodic boundary conditions were used at the sides of the lattice. The electrohydrodynamic simulation showed that the deformation of a dielectric gas bubble in an external electric field is about b= a ¼ 1:5at equilibrium [4]. This deformation increases after PD. Since the defor­ mation of the bubble at the moment of streamer initiation in liquid can be different (here we refer to the conditions of significant overvoltage as in Fig. 4) the values of the electric field at the apex of the bubble were found for different bubble elongation. The results of these calculations are shown in Table 2. Fig. 20 shows the component of the electric field stress along the symmetry axis of the gap before and after PD for an elliptic bubble with the deformation ðb =aÞ ¼ 5:57. We considered the case of the complete charge relaxation in the bubble. The distribution of the component of the electric field stress along the symmetry axis of the gap obtained with the expression (6) is also shown in Fig. 20. The relation of the calculated values of the maximal electric field stress obtained in our calculation Ecalc to the average electric stress in the gap Eav ¼ Vapp =d is shown in the forth column of Table 2. It is seen that the maximal electric field increases significantly with the deformation (Table 2, fifth column). It should be noted that the value of Ecalc is smaller than the real maximal value Emax . Indeed, the value of the calculated field is Ecalc ¼ φs h φl , where φs and φl are the values of the electric potential in a lattice node on the bubble surface and in the nearest node in the dielectric liquid. Real field is somewhat higher than that obtained in numerical calculations. This difference is clear from Fig. 20 (analytical and numerical curves). Nevertheless, this field is still significantly smaller then that is necessary for the filamentary streamer initiation in a liquid phase Ef . Numerous well-known estimations of Ef give the values not less than ~ 1–5 MV/cm. The question is how this field Ef can be reached at the bubble surface? In order to obtain the values of the maximal field Emax � Ef we may assume that the region of charge concentration is at least several times smaller than the bubble size in our calculations. That is we may suppose that the charge on the bubble surface is localized in a small spot at the apex of the bubble. Particularly, it is possible if the discharge inside the bubble has the shape of a thin enough streamer channel in a gas phase. Let us use the approximation of the streamer channel in gas inside the bubble with the very elongated ellipsoid of revolution with the complete charge relaxation inside the channel. Then, the value of the electric field ~2–5 MV/cm is reached if the size of channel at the apex is about 6 μ m in accordance with (7) for the bubble of size of 10–20 μm near the apex. This partly confirms the assumption that the discharge in a large enough bubble can develop in the form of thin filaments. 6. Discussion There are a lot of types of partial discharges in liquids. What they have in common is that the discharge decays shortly after initiation. The two following questions are of the greatest interest. What were the reasons and conditions for discharge initiation? And what were the reasons and conditions for discharge interruption? For initiating here should be initiating electron, the electric field strength sufficient for impact ionization and the conditions for secondary electrons appear­ ance. When studying PD in floating up gas bubbles in transformer oil, we noted large discrepancies between the expected and actual electric fields of the occurrence of PD [1]. The discharges ignited in the bubbles when the electric field strength was two to three times higher than calculated according to the Paschen’s law, and the delay in the appearance of the PD reached several hours. The delay of the PD was a consequence of the absence of initial electrons. Recall that the Paschen law formulates the condition for self-sustaining discharges in gases, and it coincides with the breakdown voltage of gases with the obligatory presence of an initial electron near the cathode. In other words, the Paschen’s law formulates the necessary conditions and the presence of the initial electron is a sufficient condition for gas breakdown, which, by default, is usually omitted. The problem of the initial electron is not new. In Ref. [30], the discharge delay time was proposed to be determined as Table 2 Electric field stress at the apex of the ellipsoidal bubble for different bubble deformations. a, mm b, mm b=a Ecalc =Eav Emax , kV/cm 0.21 0.21 0.21 0.21 0.21 0.21 0.27 0.32 0.43 0.53 0.64 1.18 1.29 1.5 2.0 2.5 3.0 5.57 3.18 3.52 4.35 5.18 9.50 14.95 118 130 161 192 351 553 td ¼ 1 ; w⋅N0 (13) where N0 is the number of initial electrons appearing at the cathode in 1 s, is the probability that the initial electron will create an avalanche initiating the breakdown of the gap. The generation of initial electrons is usually associated with the ionization of neutral gas particles under the influence of the radiation 8 S.M. Korobeynikov et al. Journal of Electrostatics 103 (2020) 103412 background of the earth. It is known that in 1 cm3 of atmospheric air 4–10 pairs of ions are formed every second [30]. Given the small size of the gas bubble in the liquid, the delay times for the occurrence of PD should be expected to be very large [31]. To generate the initial elec­ trons, backlighting of the electrodes or the entire gap with ultraviolet [32] or X-ray radiation is used. So in Ref. [33], an X-ray unit (150 kV, 1200 mA) was used to control 138 kV GIS epoxy support insulators with controlled defects. Under the influence of X-ray irradiation in the tested insulators, the pulse repetition rate increased and the occurrence voltage sharply decreased: when irradiated with X-rays, its value was 25 kV, and without the X-ray, no PD was recorded up to a voltage of 400 kV. Based on experience, the ideal dose to maintain discharge activity was esti­ mated at 0.5 mR/s in the region of the cavity in the body of the insulator using continuous radiation. More intense radiation led to over-ionization and high conductivity of the internal volume of the cavity even in the absence of a high voltage at the insulator. Some manufacturers of electrical equipment use X-ray illumination of the internal volume of the switchgear during their high voltage tests [34]. At the same time, they reveal microcavities in bushing insulators in which PD does not occur with the standard test procedure. It was found that even a short duration (5 ns) pulsed X-ray illumination reduces the voltage of the occurrence of PD by 2–5 times (depending on the size of the cavities) at a standard rate of rising of the test voltage. In the case of PD in free bubbles, experiments with X-rays and without X-rays showed that the appearance of initiating electrons is the main problem of PD initiation. PD stops due to the decrease of the electric field inside a bubble because of the screening of the region inside the bubble by the charge deposited on the surface of the bubble after discharge. The Coulomb forces act on the charged surface of the bubble that leads to elongation and breaks up of the bubble into two charged parts. The action of the applied alternating voltage can initiate the next PD between a bubble and the closest electrode and, in some cases, the breakdown of the electrode gap at the subsequent half-periods. If we take into consideration the avalanche evolution then the streamer model of PD in a bubble is preferable. In all our experiments criterion of avalanche-streamer transition is fulfilled. PD in the point-plane electrode system occurs at the condition of the extreme electric field when impact ionization in the oil takes place [9, 11,16]. Then a tree of filament or bush-like streamers develops from the tip of the needle. During streamers propagation, the electric field near its tips decreases (the charge in tips decreases due to several streamers and voltage drop on its length), impact ionization stops and PD stops. As for PD in nitrobenzene at pulse voltage action one could see that the dark zone (see Fig. 11) close to the electrode appears approximately 0.5 μs before PD. This zone consists of bubbles. It isn’t single bubble, but is a cluster of smaller bubbles. Kerr fringes don’t change its position during development of bubble zone. Shifts of fringes at the times t1 and t2 show that PDs occur in the bubble zone. The development of this zone leads to breakdown initiation at the moment t3 . For the mixtures of water in transformer oil we can conclude that the colloidal form of water can initiate PD, and also water negative ions could give initiating electron. The systematic calculations of electrical “intensity” of PD charac­ terized with the value of “apparent” charge showed that “apparent” charge of PD in a bubble changes approximately proportional to bubble volume and the “true” charge increases approximately proportionally to the area of bubble liquid interface even for the bubbles of large sizes. The calculations of the electric field at the apex of the elliptic bubble led us to the hypothesis that the discharge has non-homogeneous spatial shape in the bubble and develops in the form of thin enough channels in some conditions. impact ionization in liquid in case of extreme fields. Optical and elec­ trical characteristics of PD of floating up bubbles were registered. Many effects such as different shapes of bubbles, bubble break up, streamer formation from the charged bubble surface to the transformer oil etc. were observed after PD. The systematic calculations of electrical fields, “true” and “apparent” charges were performed for spherical and deformed (elliptic) bubbles of large size (compared to the distance be­ tween electrodes). The calculations are in a good agreement with the performed analytical calculations and experimental measurements. The possibility of streamer form of PD in a large helium bubble was discussed. Declaration of competing interest Authors have no conflict of interest. Acknowledgment The work was supported by the Russian Science Foundation, grant No 16-19-10229 (experiments with floating up bubbles including X-ray initiation, measurements and calculations of electrical characteristics of PD). References [1] J.H. Mason, The deterioration and breakdown of dielectrics from internal discharges, Proc. IEEE 98 (1) (1951) 44–59. [2] F.H. Kreuger, Partial Discharge Detection in High-Voltage Equipment, Butterworths, London, UK, 1989. [3] P.H.F. 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