Experimental analysis of the process of purifying transformer oil from various impurities under the influence of a constant electric field and assessment of mathematically modeled results

4.1. Development of a mathematical model of the process of transformer oil purification in a constant electric field

The mathematical model of the process of cleaning transformer oil using an electric field is compiled based on the following equations: Charged particles move under the influence of an electric field according to the following differential equation: $m \frac{d^{2}x}{{dt}^2} =qE- \gamma \frac{dx}{dt}$, where: $m$ – particle mass, $q$ – particle charge, $E$ – electric field strength, $\gamma$ – coefficient of friction.

The oil's electrical conductivity ($\sigma$) and dielectric constant ($\epsilon$) are related by the following relationship: $\sigma =\omega \epsilon \mathrm{t}\mathrm{a}\mathrm{n}\left(\delta \right)$, where: $\omega$ – frequency, $\mathrm{t}\mathrm{g}\left(\delta \right)$ – tangent of the dielectric loss angle.

The optimal electric field strength satisfies the following condition: $E_{opt}= \sqrt{\frac{\sigma }{\epsilon }}$.

The total field strength of the mechanical mixture layer, taking into account the electric field strength $E$${\overrightarrow{E}}_0$ in the $\zeta$ – layer and the conditional correction field strength ${\overrightarrow{E}}^{\left(1\right)}$, is determined as follows:

It should also be taken into account that due to the imbalance of volumetric charges between two layers, the time of electrical relaxation (${\tau }_1=\frac{{\epsilon }_0{\epsilon }_1}{{\delta }_1}$ and ${\tau }_2=\frac{{\epsilon }_0{\epsilon }_2}{{\delta }_2}$, and ${\tau }_2$ – relaxation times of the first and second media, respectively; ${\epsilon }_1$ and${\epsilon }_2$ – electric field permeability of the first and second media, respectively; ${\epsilon }_0$ – as a constant of the electric field: ${\epsilon }_0=\frac{1}{4\pi \bullet {299792458}^2\times {10}^{-7}}=$ 8.85418781∙10^-12 Ф/m^-1 equal to) and surface charge is formed between the layers [12].

For the first medium, i.e., for solid mechanical impurities, the charge density in differential form is expressed as follows:

Here, the function of the dependence of the charge density $D_1^{\left(1\right)}\left(x\right)$ on the layer thickness is $x$-layer thickness [m].

For the second medium, i.e., for dense mechanical impurities, the charge density in differential form is expressed as follows:

The surface density function and electric field strength at the boundary between the layers of the solid and dense mechanical mixture are expressed as follows:

If we express the $x$ – variable in terms of $\zeta$, then we get the following expression:

In this expression $E_0$ is expressed taking into account the potential difference and relaxation as follows:

In the layers, the electric field strength equals the internal electric field strength, creating a dipole moment, resulting in polarization forces, namely:

If we assume that the resulting dipole moment is directly proportional to the external electric field, and the charge density is directly proportional to the potential difference, the above expression takes the following form:

If the concentration of mechanical impurities in transformer oil is very small and the generated external electric field is in the form of a flat capacitor, $\alpha$ can be neglected.

If, $\frac{{\gamma }_1^{*}l}{b}\bullet \frac{{d\zeta }^{*}}{[{\phi }_0+{\gamma }_1^{*}\ln (1-{\zeta }^{*})]f}=dt$, taking into account Eq. (5) and (8), we can take into account the following relationship:

We introduce the following notation, i.e., we introduce $A$, which reflects the surface density, length, and dielectric properties of the first layer: $A=-\frac{{\sigma }_{1}l}{{\epsilon }_0{\epsilon }_2}$.

If the concentration of mechanical impurities does not exceed 10 % (as is usually the case) $\ln \left(1-{\zeta }^{*}\right)\cong -{\zeta }^{*}$ then the time function of the mechanical impurity layer, according to Eq. (9), has the form:

We accept the following terms:

and we obtain Eq. (10):

To simplify the above expression, we introduce the following notations:

As a result, we get the following expression:

$\frac{{\gamma }_1^{*}l}{b}\bullet \frac{{d\zeta }^{*}}{[{\phi }_0+{\gamma }_1^{*}\ln (1-{\zeta }^{*})]f}=dt$ and Eq. (12) and introducing the notation ${\phi }^{*}=\phi /{\phi }_0$ – we obtain Eq. (13):

If a mandatory amount of transformer oil $Q$ is purified from the electrostatic precipitator per unit of time, i.e., expressed through the mandatory flow of transformer oil, then the following expression is obtained:

where: $S$ – effective cross-sectional area of the electrostatic precipitator; $Q$ – transformer oil consumption; $v_0$ – transformer oil flow rate to the electrostatic precipitator [m/s].

If the forced circulation of transformer oil is not carried out, then we reduce Eq. (13) for this condition to the following form:

where: $h^{*}$ is the coefficient of linear dimensions of the electrostatic precipitator, calculated depending on whether the coefficient of forced flow is greater or less than 1:

The relative consumption of transformer oil is calculated as follows:

where: $L$ – effective length of the electrostatic precipitator [m]; $H$ – effective height of the electrostatic precipitator [m]; $u$ – transverse component of the flow rate of mechanical impurities [m/s]; $v_{eff}$ – effective velocity of transformer oil in the electrostatic precipitator [m/s].

The transverse component of the flow velocity of mechanical impurities in the general case is expressed as follows:

where: $u_E-E_0$ is the speed of transformer oil deviating from the main direction under the influence of the electric field; $v_j$ – velocity of transformer oil caused by the generated electric field and convection; $v_{⫠}$ – convection and velocity perpendicular to the forced transformer oil flow:

where: $m$ – coefficient depending on the type of fluid flow, taking a value from 0.5 to 1. In particular, $m=$ 0.5 for turbulent flow, $m=$ 1 for laminar flow; $j=\frac{I}{\pi d_{0}L}$ – characteristic value of current density under the influence of turbulent flow (here: $I$ – current flowing through the conductor, $d_0$– conductor diameter); $c$ – coefficient of proportionality $(c\approx \mathrm{0,85})$; $k$ – mobility of charged particles generated by electroconvection; $\varrho$ – characteristic size of transformer oil; $E_0\approx \frac{U}{{\varrho }_e}$ – average field strength between the electrodes; Re – Reynolds number, $Re=\frac{v_0\varrho }{\nu }$; ($Re=$2320) boundary of turbulent and laminar flow; $c_0\approx 1$ empirical proportionality coefficient.

If $t\to \infty$, then if we express Eq. (14) through Eq. (15), (16), (17) and (19), it will take the following form:

where: ${\psi }_{00}^{*}$ – coefficient indicating the degree of purification of transformer oil from mechanical deposits, which, as experiments show, can increase to the value ${\psi }_{00}^{*}=\mathrm{0,1}$ (equals ${\psi }_{00}^{*}+{\phi }_{00}^{*}=1$) and is expressed as follows:

Taking into account the interpretation of Eq. (20) and Eq. (21), we obtain the following expression:

If the voltage supplied to the electrostatic precipitator remains constant, then the values inside the parentheses of Eq. (22) remain constant, and the parameters $B$, $L$, $H$ outside the parentheses are also considered constant parameters. Accordingly, the following notations are introduced:

from Eqs. (22) and (23), we obtain the following expression:

If there is a sufficient cleaning time, i.e., $t\to \infty$, it becomes possible to compare the degree of cleaning of mechanical sediments with experimental results. For this, we will use the least squares method.

$U$/$Q$ is not a linearly variable function, therefore we introduce the following notation for this ratio, let's say $U/Q=\chi$. Thus, we express ${\chi }^m\to \chi$ through the relationship:

If we logarithm this expression by exponent, we get the following expression:

where we introduce the following notations: $\ln \chi \equiv k$; $\ln M\equiv n$; $\ln {{(\phi }_{00}^{*}-\phi }_{\infty }^{*})\equiv N$.

If we add such markings as, then, according to the condition of the least squares method, we obtain the following expression:

As a result, the equation of the dependence of the concentration of residual mechanical impurities on the transformer oil flowing through the electrostatic precipitator and the constant voltage applied to the electrostatic precipitator for purification was obtained. A mathematical model of efficient purification of transformer oil from various mechanical impurities using a constant electric field was developed based on the magnetic induction, length and surface density of the electrode, height and efficiency of the electrostatic precipitator, activity of charged impurities, oil consumption, mass and characteristic size, proportionality coefficient for turbulent and laminar flows, and the value of the applied constant voltage [13-16].

4.2. Analysis of theoretical results based on the obtained mathematical expressions

There is no feasibility of complete purification when it comes to purifying transformer oil of mechanical impurities with a constant electric field as indicated above. Consequently, through the theoretical research, we get the values of the function of the degree of oil purification of mechanical impurities with the consideration of the constant voltage and the transformer oil consumption in the MS Excel program [15-18]. To do this, we present the following preliminary parameters. Table 1 presents the initial values for determining the function of transformer oil purification from mechanical impurities under constant oil flow conditions.

To construct the graph of the dependence of mechanical impurities in transformer oil on the voltage supplied to the electrofilter, the transformer oil flow rate $Q$, the coefficient $M$, and the parameter m (the power of $\chi$) were conditionally assumed to be constant. Based on the values presented in Table 1, the functional relationship graph shown in Fig. 1 was obtained. In this graph, a total of 14 voltage values arranged in ascending order are presented. As the applied voltage increases, the degree of purification of transformer oil from mechanical impurities at a given oil flow rate decreases; that is, supplying voltage above the optimal level has a negative effect on the purification efficiency [18].

According to the findings of the calculations, the theoretical studies demonstrated that the level of oil purification in transformers was (0.17 – 0.57) = 40 percent on average. This represents a gain of 16 percent in electrical power of the oil.

Fig. 1Graph of the dependence of the degree of purification on the voltage and oil consumption

4.3. Analysis of the practical results of the conducted experimental studies

The experimental research was done in lab conditions and had the purpose of a practical evaluation of the transformer oil purification process with the help of constant electric field. Transformer oil was experimented on in an electrostatic precipitation equipment, which was designed specially, at different electric field voltages [17]. The extent of the reduction in mechanical impurity and moisture content of the oil was determined during the course, as well as the parameters of electrical conductivity and dielectric strength of the oil were re-evaluated. Fig. 2 presents the processes of conducting experimental research and the analysis of the obtained results.

Fig. 2Transformer oil inspection process

The results obtained are also compared to the theoretical model and conclude on the effect of constant electric field on the cleaning efficiency. With the help of this method, the effectiveness of the cleaning process and the state of transformer oil are traced and evaluated scientifically. Table 2 presents the oil filtration results of a mobile oil purification device.

Fig. 3 presents a general analysis of the results from practical and theoretical research on the electrical strength of transformer oil.

Based on the obtained results, a practical study revealed that the breakdown voltage in transformers increased by an average of 21 %. A comparison of practical and theoretical studies, based on these results, showed a 5 % difference in the electrical strength indicator of the oil. Table 3 presents a comprehensive analysis of the oil’s electrical strength indicator, comparing the results of practical and theoretical studies based on the findings mentioned above.

Fig. 4 presents a general analysis of the oil’s electrical strength indicator based on the results obtained from comparing practical and theoretical studies.

Fig. 3General analysis of the results of practical and theoretical research of transformer oil

Fig. 4Analysis of the results of practical and theoretical correction of transformer oil

The above data reveal that the finding of theoretical studies indicated that the level of purification of transformer oil was 40, the electrical strength of the oil was raised by 16 % and in practicals the findings on an experimental set up indicated that the electrical strength of the oil was raised by 21 %. The rationalization of this difference is due to the terms taken in the theoretical model (oil temperature, mechanical impurities distribution, electrodes shape and size, laminar or turbulent flow rate) [20-22]. The gap between the theoretical model and the practical outcomes is by 5 % percent that proves the adequate reliability of the model. Moreover, the effectiveness of the application of a constant electric field to enhance the dielectric properties of oil is practically proven by experiments carried out under the conditions of the laboratory.