# Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation

^{*}

## Abstract

**:**

## 1. Introduction

- -
- Thus, the following requirements are imposed on the CB:
- -
- Low resistance in normal conditions (in the normally closed contact);
- -
- High-voltage proof of external and internal insulation, which makes it possible to withstand lightning and switching overvoltage, as well as transient recovering voltage (TRV) after the arc is extinguished;
- -
- The ability of both making and breaking the short-circuit currents —the CB must reliably extinguish the arc without its re-ignition;
- -
- Ensuring fast transition from the closed to open position and vice versa, especially in automatic reclosing cycles.

## 2. Switching of SF6 Circuit Breakers

#### 2.1. Interrupter Types in SF6 Circuit Breakers

**Single pressure SF6 circuit breaker operating.**

**Double-pressure SF6 circuit breakers.**

#### 2.2. Models of Switching Arc Interaction with SF6 Gas Flow

- -
- Continuity equation (law of conservation of mass);
- -
- Equation of the second law (law of conservation of momentum);
- -
- Energy equation (law of conservation of energy).

**Analytical models.**

**Modified arc models.**

**Experimental KEMA model.**

**Magnetohydrodynamic model.**

**Hydrokinetic model.**

**Kinetic model.**

- 1.

#### 2.3. Methods for Calculating the Processes of Interaction between Arc and the SF6 Flow

**Analytical methods.**

**Numerical methods.**

**Non-Numerical methods.**

## 3. Analytical Calculation of SF6 Circuit Breaker Breaking

- (1)
- There was no supply and removal of heat during the outflow of gas (adiabatic process);
- (2)
- The process of gas outflow had a steady character;
- (3)
- There were no friction losses;
- (4)
- The gas was considered ideal;

#### 3.1. SF6 Circuit Breaker under Study

#### 3.2. Computational Model of the Circuit Breaker under Study

- -
- Full contact separation ${L}_{max}=120\mathrm{m}\mathrm{m}$;
- -
- The contact separation before blast start is ${L}_{ext}=18\mathrm{m}\mathrm{m}$;
- -
- Piston cross section ${S}_{p}=8.953{\mathrm{m}\mathrm{m}}^{2}$;
- -
- Ambient medium temperature $\vartheta =40\xb0\mathrm{C}=313\mathrm{K};$
- -
- Pressure inside CB ${p}_{0}=0.42\mathrm{M}\mathrm{p}\mathrm{a}$;
- -
- The flow coefficient μ at all stages of the outflow was assumed to be 0.9 (the outflow coefficient, which took into account the decrease in the actual cross section of the hole due to the compression of the jet in it);
- -
- Adiabatic exponent for SF6 gas ${k}_{a}=1.086$;
- -
- We set the discretization step of the calculation; for this, we divided the entire piston stroke into 20 identical sections: $n=20$. The contact separation in each section would be:$$\u2206l=\frac{{L}_{max}}{n}=\frac{120}{20}=6\mathrm{m}\mathrm{m}$$

#### 3.3. Calculation Results

## 4. Numerical Calculation of SF6 Gas Circuit Breaker Switching

- -
- The Boussinessq model;
- -
- The Spallart–Allmaras model;
- -
- The $k-\epsilon $ model;
- -
- The $k-\omega $ model;
- -
- The Reynolds stress model;
- -
- The direct numerical simulation (DNS);
- -
- The large eddy simulation.

#### 4.1. The $k-\epsilon $ Model of Turbulent Flow

^{2}/s

^{3}; ${C}_{\mu}$ is coefficient of the $k-\epsilon $ model; and $k$ is turbulent kinetic energy, m

^{2}/s

^{2}.

#### 4.2. The $k-\omega $ Model of Turbulent Flow

#### 4.3. The Computational Model of the Object under Study

**n**—boundary normal, with direction outside the region; ${\mathbf{u}}_{\mathbf{r}\mathbf{e}\mathbf{l}}$ is relative velocity; and ${\mathbf{u}}_{\mathbf{t}\mathbf{r}}$ is translation velocity.

#### 4.4. The Proposed Model of Interaction between SF6 Gas Flow and Arc

#### 4.5. Calculation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Puffer-type interrupter: 1—moving main contact; 2—moving arcing contact; 3—fixed main contact; 4—arc; 5—fixed arcing contact; 6—PTFE main nozzle; 7—PTFE auxiliary nozzle; 8—piston; A—above-piston volume; B—under-piston volume.

**Figure 4.**Block diagram of the analytical method: ${S}_{p}$—piston cross section; $l$—full contact separation; ${p}_{0}$—pressure in the under-piston volume during the calculation; ${k}_{a}$—adiabatic index for SF6 gas; ${V}_{0}$—initial volume of gas under the piston; M0—initial mass of SF6 gas under the piston; ${\gamma}_{0}$—initial density of SF6 gas in the under-piston volume; ${\gamma}_{i}$—SF6 density in the under-piston volume during the calculation; $Y={p}_{0}/{p}_{i}$—relative backpressure.

**Figure 5.**A 110 kV dead-tank SF6 CB: 1—moving main contact; 2—moving arcing contact; 3—fixed main contact; 4—arc; 5—fixed arcing contact; 6—PTFE main nozzle; 7—PTFE auxiliary nozzle; 8—piston; 9—tube of fixed main contact; 10—tube of moving main contact; 11—valve in the piston; 12—valve closed when contacts are opened; A—above-piston volume; B—under-piston volume.

**Figure 6.**(

**a**) Dependence of the SF6 outlet cross section on the contact separation; (

**b**) Dependence of the piston velocity on the contact separation.

**Figure 8.**Computational model for numerical calculation (axial symmetry): 1—fixed arcing contact; 2—moving arcing contact; 3—auxiliary PTFE nozzle; 4—main PTFE nozzle; A—above-piston volume; B—under-piston volume.

**Figure 9.**(

**a**) Temperature change according to [76]; (

**b**) Arc model with additional line 5 between arcing contacts for adaptive heat release (along the moving contact system): 1—fixed arcing contact; 2—moving arcing contact; 3—auxiliary PTFE nozzle; 4—main PTFE nozzle; A—above-piston volume; B—under-piston volume.

**Figure 10.**Velocity field without taking arc into account for turbulent model $k-\epsilon $ (on the left—the contact separation and time).

**Figure 11.**Pressure field without taking arc into account for turbulent model $k-\epsilon $ (on the left—the contact separation and time).

**Figure 14.**Velocity changes in the smallest cross-section of the nozzle depending on the time and contact separation.

**Figure 15.**Velocity field taking arc into account for turbulent model $k-\epsilon $ (on the left—the contact separation and time).

**Figure 16.**Pressure field taking arc into account for turbulent model $k-\epsilon $ (on the left—the contact separation and time).

**Figure 18.**Pressure change in under-piston volume (comparison with [82]).

Parameter Description | Parameter | |
---|---|---|

Symbol | Formula | |

Arc parameters (varying from test to test) | ||

Time constant | ${\tau}_{1}$ | $\frac{{k}_{t}}{{l}_{a}-{l}_{T}}$ |

Cooling power constants | ${B}_{1}$ | – |

${B}_{2}$ | – | |

Parameters related to CB design | ||

Distance between arcing contacts | ${l}_{a}$ | – |

Empirical constant (depends on tested CB and for conditions of the short-line fault) | ${l}_{t},{k}_{t}$ | According to [3] |

Time constants | ${\tau}_{2}$ | $\frac{{\tau}_{1}}{{k}_{1}}$ |

${\tau}_{3}$ | $\frac{{\tau}_{2}}{{k}_{2}}$ | |

Constants representing the breaker design | ${k}_{1}$ | According to [37] |

${k}_{2}$ | According to [37] | |

${k}_{3}$ | According to [37] | |

Cooling power | ${P}_{1}$ | ${B}_{1}\xb7{g}_{1}^{0.6}$ |

${P}_{2}$ | ${B}_{2}\xb7{g}_{2}^{0.1}$ | |

${P}_{3}$ | $\frac{{B}_{2}}{{k}_{3}}$ |

№ | Ref. | Problem under Study | The Model of Arc Interaction with SF6 Flow | Computational Numerical Model |
---|---|---|---|---|

1 | [32] | Predicting arc extinction by simulating outgassing with nozzle ablation | Conservation equations, Joule heating, and radiation transfer | A two-dimensional axisymmetric |

2 | [33] | Exploration of the arc extinguishing process, when the capacitive current is turned off by a self-generating switch | Conservation equations, Joule heating, radiation transfer | A two-dimensional axisymmetric |

3 | [47] | Exploration of the nozzle ablation process for breaking capacity | Conservation equations, radiation transfer | A two-dimensional planar |

4 | [54] | Elimination of an impulse wave in front of a stationary arcing contact inside the nozzle, causing a decrease in the flow rate of SF6 gas in the nozzle | Conservation equations | A two-dimensional planar |

5 | [55] | Arc re-ignition prediction | Conservation equations, Joule heating and radiation transfer | A two-dimensional axisymmetric |

6 | [56] | Influence of impurities, arising in the process of nozzle ablation on the process of arc quenching | Conservation equations | A two-dimensional planar |

7 | [57] | The reconstruction of a digital model of an arc in cylindrical nozzles | Conservation equations | A two-dimensional planar |

8 | [58] | Exploration of the influence of the aperiodic component of the tripping current on the process of arcing | Magnetohydrodynamic: conservation equations, Maxwell’s equations | A two-dimensional axisymmetric |

9 | [59] | Creation of a software package for modeling arc extinguishing processes | Conservation equations, Joule heating | A two-dimensional planar |

10 | [60] | Exploration of the process of arc extinguishing by a self-blast CB, taking into account the ablation of the nozzle | Conservation equations | A two-dimensional axisymmetric |

11 | [61] | Investigation of the process of arc extinguishing by a self-generated switch, taking into account the ablation of the nozzle | Conservation equations, radiation transfer | A two-dimensional axisymmetric |

12 | [62] | Exploration of the arc extinguishing process in a supersonic nozzle | Conservation equations | A two-dimensional axisymmetric |

13 | [63] | Improved accuracy at low breaking currents (wire arc). | Magnetohydrodynamic: conservation equations, Maxwell’s equations | 3D |

l, mm | V_{i}, mm^{3} | p_{i}, MPa | Y_{i} | Ψ_{i} | v_{avg.i}, m/s | G_{i}, kg/s | $\mathsf{\Delta}{\mathbf{M}}_{\mathbf{i}},\text{}\mathbf{kg}\mathbf{\xb7}{\mathbf{10}}^{\mathbf{3}}$ | ${\mathbf{M}}_{\mathbf{i}}\mathbf{,}\text{}\mathbf{kg}\mathbf{\xb7}{\mathbf{10}}^{\mathbf{3}}$ |
---|---|---|---|---|---|---|---|---|

6 | 1.209 | 0.420 | 1.000 | 0 | 0 | 0 | 0 | 29.674 |

12 | 1.155 | 0.441 | 0.975 | 0.218 | 0.40 | 0 | 0 | 29.674 |

18 | 1.101 | 0.465 | 0.938 | 0.336 | 1.15 | 0 | 0 | 29.674 |

24 | 1.048 | 0.491 | 0.895 | 0.424 | 1.80 | 0.114 | 0.380 | 29.294 |

30 | 0.994 | 0.512 | 0.856 | 0.482 | 2.40 | 0.271 | 0.676 | 28.617 |

36 | 0.940 | 0.530 | 0.823 | 0.520 | 3.10 | 0.455 | 0.881 | 27.736 |

42 | 0.886 | 0.547 | 0.795 | 0.546 | 4.15 | 0.494 | 0.715 | 27.022 |

48 | 0.833 | 0.569 | 0.766 | 0.569 | 5.04 | 0.533 | 0.635 | 26.387 |

54 | 0.779 | 0.596 | 0.734 | 0.588 | 5.28 | 0.574 | 0.652 | 25.735 |

60 | 0.725 | 0.627 | 0.701 | 0.604 | 5.28 | 0.863 | 0.981 | 24.755 |

66 | 0.671 | 0.653 | 0.671 | 0.614 | 5.28 | 1.177 | 1.337 | 23.417 |

72 | 0.618 | 0.673 | 0.646 | 0.620 | 5.28 | 1.231 | 1.399 | 22.019 |

78 | 0.564 | 0.695 | 0.625 | 0.623 | 5.28 | 1.279 | 1.453 | 20.565 |

84 | 0.510 | 0.719 | 0.604 | 0.625 | 5.28 | 1.326 | 1.507 | 19.058 |

90 | 0.457 | 0.747 | 0.582 | 0.625 | 5.28 | 1.374 | 1.561 | 17.497 |

96 | 0.403 | 0.780 | 0.559 | 0.625 | 5.28 | 1.427 | 1.622 | 15.875 |

102 | 0.349 | 0.820 | 0.535 | 0.625 | 5.14 | 1.491 | 1.740 | 14.135 |

108 | 0.295 | 0.867 | 0.508 | 0.625 | 4.40 | 1.564 | 2.133 | 12.002 |

114 | 0.242 | 0.902 | 0.486 | 0.625 | 3.30 | 2.360 | 4.291 | 7.711 |

120 | 0.188 | 0.733 | 0.526 | 0.625 | 1.95 | 2.188 | 6.732 | 0.979 |

Description | Parameter | |
---|---|---|

Designation | Value | |

Pressure inside the interrupter | p | 0.42 MPa |

Initial gas flow velocity | u | 0 m/s |

Ambient temperature | T | 313 K |

Von Karman constant | ${k}_{v}$ | 0.41 |

Parameters of k-e turbulence model | ||

– | ${C}_{\epsilon 1}$ | 1.44 |

– | ${C}_{\epsilon 2}$ | 1.92 |

– | ${C}_{\mu}$ | 0.09 |

Turbulent kinetic energy | ${\sigma}_{k}$ | 1 |

Turbulent dissipation rate | ${\sigma}_{\epsilon}$ | 1.3 |

Constant parameters of k-w turbulence model | ||

– | $\alpha $ | 0.12 |

– | ${\beta}_{0}$ | 0.072 |

– | ${\beta}_{0}^{*}$ | 0.09 |

Turbulent kinetic energy | ${\sigma}_{k}^{*}$ | 0.5 |

Specific turbulent dissipation rate | ${\sigma}_{\omega}$ | 0.5 |

Number of Elements | Vertex Elements | Edge Elements | Average Element Quality | Automatic Remeshing | Relative Tolerance | Tolerance Factor | Termination Technique | Max Iterations |
---|---|---|---|---|---|---|---|---|

4737 | 92 | 1090 | 0.4474 | 0.08 | 0.1 | 1 | Tolerance | 20 |

l, mm | t, ms | ${\mathit{p}}_{\mathit{k}-\mathit{\epsilon}}$, MPa | ${\mathit{p}}_{\mathit{k}-\mathit{\omega}}$, MPa | ${\mathit{u}}_{\mathit{k}-\mathit{\epsilon}}$, m/s | ${\mathit{u}}_{\mathit{k}-\mathit{\omega}}$, m/s | ${\mathit{G}}_{\mathit{k}-\mathit{\epsilon}}$, kg/s | ${\mathit{G}}_{\mathit{k}-\mathit{\omega}}$, kg/s |
---|---|---|---|---|---|---|---|

0 | 0 | 0.420 | 0.420 | 0 | 0 | 0 | 0 |

6 | 5.70 | 0.441 | 0.441 | 0 | 0 | 0 | 0 |

12 | 8.50 | 0.462 | 0.462 | 0 | 0 | 0 | 0 |

18 | 10.80 | 0.484 | 0.484 | 6.0 | 6.0 | 0 | 0 |

24 | 12.95 | 0.507 | 0.507 | 31.0 | 30.0 | 0.367 | 0.360 |

30 | 15.00 | 0.523 | 0.523 | 59.0 | 55.5 | 0.699 | 0.656 |

36 | 16.50 | 0.533 | 0.533 | 83.0 | 74.5 | 0.855 | 0.787 |

42 | 17.95 | 0.552 | 0.554 | 105.0 | 89.5 | 0.973 | 0.902 |

48 | 19.13 | 0.572 | 0.575 | 121.0 | 103.0 | 1.005 | 0.936 |

54 | 20.20 | 0.590 | 0.596 | 121.0 | 105.0 | 1.143 | 1.092 |

60 | 21.30 | 0.607 | 0.613 | 123.0 | 106.0 | 1.380 | 1.301 |

66 | 22.35 | 0.626 | 0.631 | 105.0 | 103.0 | 1.550 | 1.442 |

72 | 23.42 | 0.644 | 0.665 | 119.0 | 124.0 | 1.595 | 1.520 |

78 | 24.53 | 0.659 | 0.666 | 126.0 | 111.0 | 1.693 | 1.650 |

84 | 25.70 | 0.673 | 0.681 | 119.0 | 118.0 | 1.888 | 1.838 |

90 | 27.00 | 0.682 | 0.690 | 145.0 | 129.0 | 2.083 | 2.007 |

96 | 28.55 | 0.676 | 0.687 | 98.0 | 105.0 | 2.196 | 2.105 |

102 | 30.93 | 0.645 | 0.658 | 105.0 | 105.0 | 2.303 | 2.327 |

108 | 33.90 | 0.590 | 0.600 | 83.0 | 95.0 | 2.391 | 2.486 |

114 | 37.46 | 0.511 | 0.514 | 65.0 | 75.0 | 2.036 | 2.151 |

120 | 45.00 | 0.383 | 0.378 | 20.0 | 21.0 | 0.727 | 0.710 |

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## Share and Cite

**MDPI and ACS Style**

Popovtsev, V.V.; Khalyasmaa, A.I.; Patrakov, Y.V.
Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation. *Axioms* **2023**, *12*, 623.
https://doi.org/10.3390/axioms12070623

**AMA Style**

Popovtsev VV, Khalyasmaa AI, Patrakov YV.
Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation. *Axioms*. 2023; 12(7):623.
https://doi.org/10.3390/axioms12070623

**Chicago/Turabian Style**

Popovtsev, Vladislav V., Alexandra I. Khalyasmaa, and Yurii V. Patrakov.
2023. "Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation" *Axioms* 12, no. 7: 623.
https://doi.org/10.3390/axioms12070623