^{1,a)}, Markus Zahn

^{1,b)}, Francis M. O’Sullivan

^{2}, Leif A. A. Pettersson

^{3}, Olof Hjortstam

^{4}and Rongsheng Liu

^{3}

### Abstract

Transformer oil-based nanofluids with conductive nanoparticle suspensions defy conventional wisdom as past experimental work showed that such nanofluids have substantially higher positive voltage breakdown levels with slower positive streamer velocities than that of pure transformer oil. This paradoxical superior electrical breakdown performance compared to that of pure oil is due to the electron charging of the nanoparticles to convert fast electrons from field ionization to slow negatively chargednanoparticlecharge carriers with effective mobility reduction by a factor of about . The charging dynamics of a nanoparticle in transformer oil with both infinite and finite conductivities shows that this electron trapping is the cause of the decrease in positive streamer velocity, resulting in higher electrical breakdown strength. Analysis derives the electric field in the vicinity of the nanoparticles, electron trajectories on electric field lines that chargenanoparticles, and expressions for the charging characteristics of the nanoparticles as a function of time and dielectric permittivity and conductivity of nanoparticles and the surrounding transformer oil. This chargednanoparticle model is used with a comprehensive electrodynamic analysis for the charge generation, recombination, and transport of positive and negative ions, electrons, and chargednanoparticles between a positive high voltage sharp needle electrode and a large spherical ground electrode. Case studies show that transformer oil molecular ionization without nanoparticles cause an electric field and space charge wave to propagate between electrodes, generating heat that can cause transformer oil to vaporize, creating the positive streamer. With nanoparticles as electron scavengers, the speed of the streamer is reduced, offering improved high voltage equipment performance and reliability.

I. INTRODUCTION

II. CHARGERELAXATION TIME

III. MODELING THE CHARGING DYNAMICS OF A NANOPARTICLE

A. Perfectly conducting nanoparticle

B. Finitely conducting nanoparticle

C. Electric field lines

1. Perfectly conducting nanoparticle

2. Finitely conducting nanoparticle

D. Evaluating nanoparticlecharging

IV. MODEL OF STREAMER PROPAGATION IN TRANSFORMER OIL-BASED NANOFLUIDS

A. Governing equations

B. Electron mobility

C. Recombination and electron attachment

D. Charge generation via field ionization

E. Modeling nanoparticlecharging

V. RESULTS AND DISCUSSION

A. Electric field dynamics

B. Charge density dynamics

C. Electric potential

VI. CONCLUSION

### Key Topics

- Nanoparticles
- 192.0
- Electric fields
- 80.0
- Electrodes
- 23.0
- Permittivity
- 23.0
- Electron mobility
- 22.0

## Figures

Nanoparticle of an arbitrary material with a radius , permittivity , and conductivity , surrounded by transformer oil with a permittivity of and conductivity stressed by a uniform -directed electric field turned on at .

Nanoparticle of an arbitrary material with a radius , permittivity , and conductivity , surrounded by transformer oil with a permittivity of and conductivity stressed by a uniform -directed electric field turned on at .

Electric field lines for various times after a uniform -directed electric field is turned on at around a perfectly conducting spherical nanoparticle of radius surrounded by transformer oil with permittivity , conductivity , and free electrons with uniform charge density and mobility . The thick electric field lines terminate on the particle at and , where and separate field lines that terminate on the nanoparticle from field lines that go around the particle. The cylindrical radius of Eq. (27) of the separation field line at defines the charging current in Eq. (29). The cylindrical radius of Eq. (28) defines the separation field line at . The dominant charge carrier in charging the nanoparticles are electrons because of their much higher mobilities than positive and negative ions. The conductivity of transformer oil, , is much less than the effective conductivity of the electrons, . The electrons charge each nanoparticle to saturation, as given in Eq. (8) with time constant given in Eq. (12). The electric field lines in this figure were plotted using MATHEMATICA StreamPlot (Ref. 18). (a) (b) (c) (d) .

Electric field lines for various times after a uniform -directed electric field is turned on at around a perfectly conducting spherical nanoparticle of radius surrounded by transformer oil with permittivity , conductivity , and free electrons with uniform charge density and mobility . The thick electric field lines terminate on the particle at and , where and separate field lines that terminate on the nanoparticle from field lines that go around the particle. The cylindrical radius of Eq. (27) of the separation field line at defines the charging current in Eq. (29). The cylindrical radius of Eq. (28) defines the separation field line at . The dominant charge carrier in charging the nanoparticles are electrons because of their much higher mobilities than positive and negative ions. The conductivity of transformer oil, , is much less than the effective conductivity of the electrons, . The electrons charge each nanoparticle to saturation, as given in Eq. (8) with time constant given in Eq. (12). The electric field lines in this figure were plotted using MATHEMATICA StreamPlot (Ref. 18). (a) (b) (c) (d) .

Charging dynamics, , of a perfectly conducting nanoparticle vs time in transformer oil as given by Eq. (13) with (approximately equals 11 electrons) and .

Charging dynamics, , of a perfectly conducting nanoparticle vs time in transformer oil as given by Eq. (13) with (approximately equals 11 electrons) and .

Screenshot of the closed form solution for in Eq. (20) generated by MATHEMATICA when numerical values are given to each variable (e.g., , , , , , , , , , and ).

Screenshot of the closed form solution for in Eq. (20) generated by MATHEMATICA when numerical values are given to each variable (e.g., , , , , , , , , , and ).

Charging dynamics, , of a nanoparticle with constant conductivity and varying permittivity (○), (▽), and (×) in transformer oil (, ). The other charging parameter values used, such as , , , , , and , are the same as given in Sec. III A, just after Eq. (13). (a) . (b) . (c) .

Charging dynamics, , of a nanoparticle with constant conductivity and varying permittivity (○), (▽), and (×) in transformer oil (, ). The other charging parameter values used, such as , , , , , and , are the same as given in Sec. III A, just after Eq. (13). (a) . (b) . (c) .

Initial of the charging dynamics, , for particles with conductivity of and varying permittivity .

Initial of the charging dynamics, , for particles with conductivity of and varying permittivity .

Computer-aided design representation of the needle-sphere electrode geometry used for streamer simulation purposes and detailed in the IEC 60897 standard (Ref. 24).

Computer-aided design representation of the needle-sphere electrode geometry used for streamer simulation purposes and detailed in the IEC 60897 standard (Ref. 24).

(a) Laplacian electric field magnitude [V/m] spatial distribution (i.e., ) at for applied step voltage near the radius needle electrode apex at the origin. The sphere electrode (not shown) is at , . (b) Laplacian electric field magnitude distribution along the needle-sphere -axis. The field enhancement is largest near the sharp needle tip quickly decreasing as increases.

(a) Laplacian electric field magnitude [V/m] spatial distribution (i.e., ) at for applied step voltage near the radius needle electrode apex at the origin. The sphere electrode (not shown) is at , . (b) Laplacian electric field magnitude distribution along the needle-sphere -axis. The field enhancement is largest near the sharp needle tip quickly decreasing as increases.

Temporal dynamics along the needle-sphere electrode axis at intervals from to given by the solution to the streamer model of Eqs. (37)–(42) for and . The solution is identical to the pure oil case. (a) Electric field distribution. (b) Net space charge density distribution. (c) Temperature distribution. The oil temperature at time is .

Temporal dynamics along the needle-sphere electrode axis at intervals from to given by the solution to the streamer model of Eqs. (37)–(42) for and . The solution is identical to the pure oil case. (a) Electric field distribution. (b) Net space charge density distribution. (c) Temperature distribution. The oil temperature at time is .

Electric field distribution along the needle-sphere electrode axis at given by the solutions to the streamer model of Eqs. (37)–(42) for the three NF case studies with different nanoparticle attachment time constants and the pure transformer oil.

Electric field distribution along the needle-sphere electrode axis at given by the solutions to the streamer model of Eqs. (37)–(42) for the three NF case studies with different nanoparticle attachment time constants and the pure transformer oil.

(a) Charge density distributions along the needle-sphere electrode axis at time given by the solution to the streamer model for transformer oil and transformer oil-based NF with and varying . (a) Pure transformer oil. (b) transformer oil-based nanofluid. (c) transformer oil-based nanofluid (d) transformer oil-based nanofluid.

(a) Charge density distributions along the needle-sphere electrode axis at time given by the solution to the streamer model for transformer oil and transformer oil-based NF with and varying . (a) Pure transformer oil. (b) transformer oil-based nanofluid. (c) transformer oil-based nanofluid (d) transformer oil-based nanofluid.

Electric potential distribution along the needle-sphere electrode axis at given by the solution of the NF field ionization case studies and the equivalent solution in pure oil. The needle tip is at and the streamer tail is to the left of the knee, where the slope changes dramatically, in each electric potential plot. The location of the streamer tip is at the knee in each electric potential plot.

Electric potential distribution along the needle-sphere electrode axis at given by the solution of the NF field ionization case studies and the equivalent solution in pure oil. The needle tip is at and the streamer tail is to the left of the knee, where the slope changes dramatically, in each electric potential plot. The location of the streamer tip is at the knee in each electric potential plot.

## Tables

Results of impulse voltage withstand testing in electrode gap system (Ref. 2).

Results of impulse voltage withstand testing in electrode gap system (Ref. 2).

Electrical and thermal properties of representative insulating and conducting nanoparticle materials.

Electrical and thermal properties of representative insulating and conducting nanoparticle materials.

Field ionization parameter values.

Field ionization parameter values.

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