Simulation Study on Acetylene Gas Diffusion Process in Large Transformers

To discover the influence of the diffusion process of acetylene gas in large oil filled power transformers on the dissolved gases analysis results, the diffusion process of acetylene gas over long distances and the influencing factors have been investigated by simulation. In this paper, based on the convection-diffusion equation of acetylene gas in oil, a one-dimensional model for the diffusion of acetylene gas in a 10 m long oil duct is established and numerically solved by the backward difference method. The simulation found that in right-angle coordinates, the concentration of acetylene gas in the oil is uniformly diffused outward, and the arrival time of the gas is related to the flow rate; in polar coordinates, the acetylene gas is mainly concentrated in the area near the defects and the tank wall, and the distribution of the gas content is extremely inhomogeneous, and the homogeneous state can not be reached for a long time. The results show that the simulation of gas production from local defects in large transformers is closer to reality by using polar coordinate form and boundary flux conditions; because the diffusion coefficient is very small in magnitude, the slow flow of transformer oil leads to the predominant convective transfer of acetylene gas.


Introduction
Power transformer is one of the most important equipment in power grid.In recent years, a number of transformer burning and explosion accidents occurs, and made Serious damage to personal safety and economic interests.Therefore, it is necessary to make deeper the research and application on advanced technology for condition monitoring to improve the quality of transformer operation and maintenance, while dissolved gas in oil monitoring and analyse is one of the most reliable methods in application [1].When an internal failure occurs, the transformer oil will be subjected to electricity, heat, oxidation and other effects and decomposition of hydrogen, acetylene, methane and other small molecule gases, the generated gas will be dissolved in the transformer oil.Among all gas components, acetylene (C2H2) is is a characteristic gas for identifying overheating and discharge faults in oil-immersed transformers.Its concentration in oil is important in determining the type and severity of internal insulation faults in power transformers [2].The volume fraction of acetylene gas dissolved in the transformer oil and the rate of gas production have a strong correlation with the type of fault in oil-immersed transformers, so it is possible to monitor the insulation condition of the transformer in real time or to diagnose the type of latent faults based on the changes in the state of the acetylene gas separated in the oil (including the gas volume fraction, the rate of gas production, etc.) [3].
Gases from the decomposition of the insulating oil due to the transformer fault will again dissolve in the oil and spread further, resulting in different concentrations and components of gases in the oil with time and spatial location, which affects the results of dissolved gas analyse (DGA).A. Shahsiah et al. experimentally studied the diffusion distribution of dissolved gases in oil in oil filled transformers and found that diffusion coefficients in oil are larger than those in insulating paper, and with the rise in temperature they increases [4].Ruijin Liao et al. carried out molecular dynamics simulation on the diffusion behaviour of seven characteristic gases, and the results showed that the diffusion coefficient of small gas molecules in cellulose was one order of magnitude smaller than that in mineral oil [5]; while Chenmeng Xiang et al. analysed the diffusion behaviour of seven characteristic gas molecules in camellia seed insulating oil by simulation, and established the diffusion mathematical model of gases by combining with Fick's second law [6].The mathematical model of gas diffusion in transformer oil was established by combining Fick's second law [6].Although a large number of studies have been conducted on the diffusion coefficients of gases in transformer oils, there is a lack of studies on the diffusion process of gases in large transformers, and the effect of transformer oil flow is less considered.
In this paper, based on the convection-diffusion equation of gas in transformer oil, a onedimensional simulation model in right-angle coordinates and polar coordinates is established and compared with the analytical solution for verification, and then the influence laws of boundary conditions, flow rate and diffusion coefficient on the distribution of the C2H2 concentration are analysed.

Convection-diffusion equation of gas
When the gas diffuses in transformer oil, ignoring the adsorption effect and the process of gas spillage in the form of bubbles, the convection-diffusion process of gases in oil follows the law of mass conservation as follows: the amount of mass change per unit volume is equal to the amount of mass change due to convection and the amount of mass change due to diffusion of [7].Combined with the first Fick's law of gas diffusion, the convection-diffusion equation for gases can be expressed as [6,7].
Where c is the gas content, t is the time, D is the diffusion coefficient, and u is the flow rate of the transformer oil.

Initial margin condition
In order to solve the convection-diffusion equation shown in Equation (1), it is also necessary to give two fixed solution conditions, the initial value and the boundary conditions.For simplicity, the initial C2H2 content in the oil can be neglected, i.e., it is assumed that the C2H2 gas content in the oil at the initial moment is zero.
Regarding the boundary conditions, there are two commonly used ones: 1) constant concentration continuous injection, i.e.
Where S denotes the boundary and c0 denotes the constant injection concentration; 2) Constant injection at constant flux, i.e.
Where R0 represents the constant injection flux at the boundary.The use of different boundary conditions leads to differences in the distribution of gas content in the oil.The constant-concentration continuous injection condition indicates a constant concentration at the boundary and applies to scenarios where the gas injection at the boundary can approximately maintain a constant concentration; the constant-flux injection condition indicates a constant injected flux at the boundary and applies to scenarios where the gas injection rate at the boundary is constant.

Analytical solution of the model
The actual transformer structure is complex, the gas not only in the oil diffusion, but also in the insulating paper, cardboard and other media diffusion, diffusion process and path is more complex.For simplicity, this paper only considers the diffusion process of gas in oil, and only one-dimensional model.
It is assumed that there is gas injection at x = 0 (i.e., the defect location), and the diffusion at the other endpoint is not hindered, i.e. ∂c/∂x=0, x is the diffusion distance in the right-angle coordinate system.When the boundary condition is constant concentration injection, the gas content distribution in the oil is [8]: Where erfc is the complementary error function.
When the boundary condition is constant flux injection and c0 = R0/u, the gas content distribution in the oil is [8]: From Equation ( 4) and ( 5), it can be seen that the gas content distribution in oil is different when different boundary conditions are used.

Diffusion process of gas
The flow rate was taken as 0.02 m/s, the diffusion coefficient was taken as 0.37 × 10 -8 m 2 /s [9] , and the C2H2 concentration was taken as the noted value, i.e., 1 μL/L.The distribution of the gas content under the boundary conditions of constant concentration and constant flux is shown in Figure 1.From the figure, it can be seen that the gas diffusion process is the same under the two boundary conditions with the above parameters; the gas is injected from x=0 (at the defect) and then gradually migrates to the x>0 region, and the gas diffusion arrives at the region where the concentration immediately becomes 1 μL/L, and the transition between the region containing C2H2 gas and the region without C2H2 gas is very steep [6] .

Factors affecting gas diffusion
Figure 2 shows the C2H2 gas content change curve at a distance of 5 m from the defect when keeping the diffusion coefficient of 0.37×10 -8 m 2 /s and the flow rate of 0.02 m/s~2 m/s.From the figure, it can be seen that the C2H2 gas content at this place will change suddenly from 0 to 1 μ L/L at a certain moment, and the moment of sudden change is related to the transformer oil flow rate: the larger the flow rate is, the earlier the moment of sudden change of C2H2 gas content is.Due to the fact that the diffusion coefficient of C2H2 gas in oil is very small, and the size of large-scale oil-filled equipment, such as the transformer is usually at the level of metre, at this time, the second term of the right-hand side of the formula (4), and the first and the third terms of the right-hand side of the formula (5) are approximated to be 0. Therefore, when the boundary conditions of both satisfy c0=R0/u, the distribution of C2H2 gas content has the same form, and the time for the gas content to reach the steady state at different locations is related to the distance to the defects as well as the transformer oil flow rate, which is the reason why the gas diffusion curves are the same under the two boundary conditions in Figures 1 and 2.
Figure 3 further analyses the C2H2 gas content change curve at 5 m from the defect under the constant flux boundary condition, keeping the flow rate of 0.02 m/s and the diffusion coefficient of (0.01~1)×10-8 m2/s.It can be seen that even though the diffusion coefficient changes by a factor of 100, the time for the gas content at this point to reach equilibrium does not change, in agreement with the previous analysis.Therefore, the results for the constant concentration boundary condition are not shown here.From the above results, it can be seen that the diffusion process of C2H2 gas in oil is related to the flow rate of transformer oil, and the equilibrium concentration of gas is related to the boundary conditions: under the condition of fixed concentration, the equilibrium concentration is consistent with the concentration at the boundary; under the condition of fixed flux, the equilibrium concentration depends on the injection rate at the boundary and the flow rate of transformer oil.In the simulation, for comparison, the injection rate R0=c0-u, but usually the defective gas production rate is not proportional to the transformer oil flow rate.

One-dimensional model in polar coordinates
The gas injection at the boundary in the one-dimensional model in right-angle coordinates is equivalent to a uniform gas injection in an infinite plane, but the defect gas-producing area in the actual transformer is usually in a small local area, while the one-dimensional model in polar coordinates is equivalent to the internal defect gas production and then diffuse along the radial direction, which is closer to the actual situation.Therefore, the next analysis is carried out in the onedimensional polar coordinate model.

Boundary conditions
From the previous analysis, it can be seen that the use of constant flux injection boundary conditions is closer to the actual situation.Figure 4 shows the C2H2 gas change curve of a ±800 kV converter transformer during the trial operation stage.The C2H2 in the oil of the three converter transformers A, B and C at the oil chromatography detection on the 10th day of the trial operation had a significant increase compared with the 4th day, and the absolute gas production rates of C2H2 gas content in the three oils from 13 to 19 April were 1.28 mL/d, 1.09 mL/d and 1.28 mL/d, respectively, which were all more than the DL/T 722-2014 Guidelines for the Analysis and Determination of Dissolved Gases in Transformer Oil standard's noted value of 0.2 mL/d.As of 7 June 2022, the C2H2 content of the three converters had approached and exceeded the attention value (1.00 μL/L), with 1.02 μL/L, 0.97 μL/L, and 1.00 μL/L, respectively, and it was found later by the internal inspection that there was a lowenergy discharge at the low-voltage.Literature [10] used the pin-plate defect model to simulate the gas production characteristics of the discharge in the transformer, with the discharge voltage of 45 kV and the volume of the test cavity of 140 L. The absolute gas production rate of the discounted C2H2 gas content in the discharge process is 1.43 mL/d, which is consistent with the absolute gas production rate of the C2H2 gas in the abovementioned actual converter transformer.Therefore, the insulated gas production rate of C2H2 gas at the boundary condition is taken as 1.3 mL/d in the simulation, and assuming that the discharge area is a sphere with a radius of 5 mm, the constant flux boundary condition is 191.7 μL/(m 2 •s).

Numerical calculation method
A sidewind diffusion model is added to the simulation to improve stability, and the time is backwarddifferenced.Since it is not easy to obtain the analytical solution in polar coordinates, in order to verify the validity of the algorithm, this paper first compares the numerical calculation results with the analytical solution in the right-angle coordinate system.Figure 5 shows the comparison between the two under the constant flux boundary condition, when the flux is taken as 19.94 μL/(m 2 •s), the flow velocity is taken as 0.02 m/s, and the diffusion coefficient is taken as 1×10 -8 m 2 /s.

Diffusion process of the gas
Figure 6 shows the concentration distribution of C2H2 gas when the flow rate is taken as 0.02 m/s and the diffusion coefficient is taken as 0.37×10 -8 m 2 /s.It can be seen that the concentration distribution of C2H2 gas is obviously different from that in Figure 1 in right-angle coordinates, and the gas content is mainly concentrated near the defects and slowly diffuses in all directions.Under the given boundary flux condition, the C2H2 gas content near the defect reached 1.6 μL/L in about 1 s, but after 500 s, the gas content at 1 m from the defect was less than 0.05 μL/L.The C2H2 gas content in the vicinity of the defect reached 1.6 μL/L in about 1 s, but the gas content at 1 m from the defect was less than 0.05 μL/L after 500 s.According to the set flow rate, the transformer oil flowed up to a distance of 10 m in 500 s and the C2H2 gas also diffused to this place, but the concentration of C2H2 gas in the transformer oil farther away from the defect was lower because the gas dispersed in all directions due to the one-dimensional polar co-ordinate model considered.

Influence factors of gas diffusion
Figure 7 shows the distribution of C2H2 gas content at the moment of 100 s by keeping the diffusion coefficient as 0.37×10 -8 m 2 /s and the flow rate taken as 0.02 m/s~2 m/s.It can be seen that as the flow velocity increases, the concentration of gas accumulated near the defect decreases, mainly because the flow of transformer oil drives the gas to leave the area near the defect quickly.Figure 8 shows the variation curve of C2H2 gas content when the flow rate is kept at 0.02 m/s and the diffusion coefficient is taken as (0.01~1) × 10 -8 m 2 /s.It can be seen that there is no obvious change in the C2H2 gas content distribution even when the diffusion coefficient is changed by a factor of 100.In summary, in the case of local small defects gas production, C2H2 gas diffuses in all directions, and the C2H2 gas content in the position closer to the defects quickly reaches the standard attention value, while the concentration in the position far away from the defects remains small after a long time.As the diffusion coefficient of C2H2 gas in transformer oil is small, when the gas reaches the boundary before convection transfer is dominant, at this time the influence of diffusion coefficient is small.Considering that the oil is circulating in the actual transformer, the simulation area is limited to an area with a radius of 5 m, and an additional concentration boundary condition is added at the outer boundary, i.e., the concentration of C2H2 gas is equal to that at the defect.Figure 9 shows the changes in the distribution of C2H2 gas content over 20 h under the above conditions when the flow rate is taken as 0.02 m/s and the diffusion coefficient is taken as 1 × 10 -8 m 2 /s.It can be seen that due to the gas production at the defect and the blocking effect at the boundary, the C2H2 gas content in the region near the defect and the outer boundary is larger and gradually increases, while the C2H2 gas content in most of the intermediate region is smaller.(2) Due to the small diffusion coefficient, the C2H2 gas mainly diffuses through convective transfer in the transformer, so the flow rate has a large influence on the diffusion process and concentration distribution of C2H2 gas; (3) When the defect area is small, the diffusion process of C2H2 gas in the transformer is slow, and the concentration distribution of C2H2 gas is difficult to reach uniformity for a long time.The structure of actual large power transformer is complicated, the flow rate of transformer oil is not uniform, and there is also the barrier of insulating paper and cardboard, so the diffusion of C2H2 gas is more complicated, so it is necessary to study the diffusion coefficient and the diffusion process of C2H2 gas in a more in-depth manner.
(a) Given concentration boundary conditions (b) Given flux boundary condition Figure 1.Gas diffusion process.

2 .
(a) Given concentration boundary conditions (b) Given flux boundary condition Figure Variation of C2H2 volume fraction at 5 m from defects.Comparing Figure 2(a) and (b), it can be found that the change rule of C2H2 gas content at x=5 m under the two boundary conditions is the same, and the moment of reaching the steady state is t=x/u.

Figure 3 .
Figure 3. Variation of C2H2 volume fraction at 5 m from defects.From the above results, it can be seen that the diffusion process of C2H2 gas in oil is related to the flow rate of transformer oil, and the equilibrium concentration of gas is related to the boundary conditions: under the condition of fixed concentration, the equilibrium concentration is consistent with the concentration at the boundary; under the condition of fixed flux, the equilibrium concentration depends on the injection rate at the boundary and the flow rate of transformer oil.In the simulation, for comparison, the injection rate R0=c0-u, but usually the defective gas production rate is not proportional to the transformer oil flow rate.

Figure 4 .
Figure 4. History record of C2H2 gas content in a ±800 kV converter substation.Literature[10] used the pin-plate defect model to simulate the gas production characteristics of the discharge in the transformer, with the discharge voltage of 45 kV and the volume of the test cavity of 140 L. The absolute gas production rate of the discounted C2H2 gas content in the discharge process is 1.43 mL/d, which is consistent with the absolute gas production rate of the C2H2 gas in the abovementioned actual converter transformer.Therefore, the insulated gas production rate of C2H2 gas at the boundary condition is taken as 1.3 mL/d in the simulation, and assuming that the discharge area is a sphere with a radius of 5 mm, the constant flux boundary condition is 191.7 μL/(m 2 •s).

Figure 5 .
Figure 5.Comparison of numerical results with analytical solutions.

Figure 6 .
Figure 6.Diffusion processes of gases in polar coordinates.

Figure 7 .
Figure 7. Effect of flow velocity on gas diffusion.

( 1 )
In reality, it is the defective region that produces gas continuously, and it is appropriate to use the boundary flux model in the simulation, and the flux during the partial discharge of the spiked metal is about 191.7 μL/(m 2 •s); ICMSOA-2023 Journal of Physics: Conference Series 2755 (2024) 012011 IOP Publishing doi:10.1088/1742-6596/2755/1/0120119