Experimental and statistical investigation on the dielectric breakdown of magneto nanofluids for power applications

Abstract

The insulating oil serves the dual purpose of providing insulation and cooling within transformers. This investigation aims to explore the impact of various nanoparticles on the dielectric breakdown voltage (BDV) of dielectric oils. The study examines the effect of the concentration of magnetic nanoparticles on the dielectric breakdown voltage of insulating oils. Nanoparticles such as iron (II, III) oxide (Fe₃O₄), cobalt (II, III) oxide (CO₃O₄), and ferrous phosphide (Fe₃P) were utilized to create nanofluids with carrier mediums consisting of mineral oil and synthetic ester oil. BDV determination was conducted using a VDE and S–S electrode system according to IEC 60156 standards. Nanofluid were prepared using a two-step method, and their concentrations ranged from 0.01 g/L, 0.02 g/L, and 0.04 g/L in base oils. Twelve iterations were conducted for each prepared nanofluid, and breakdown voltage measurements were recorded. The results indicate a noteworthy enhancement in the breakdown voltage of nanofluids. The statistical analysis was performed on the dielectric property of nanofluid samples for better breakdown accuracy. The maximum enhancement at specific nanoparticle concentrations was shown by each nanofluid. The results show that under the S–S electrode configuration, the greatest overall enhancement was observed for Fe₃P in mineral oil, with an enhancement of 70.05%, and Fe₃O₄ in synthetic ester oil, with an enhancement of 46.29%.

Introduction

The electrical industries nowadays are focused on developing machines of the same rating but with reduced size and weight. In numerous high-voltage engineering applications, both electrical and thermal stresses intensify as product dimensions continue to decrease over time. It is desirable to have a new kind of liquid insulation to fulfil this need. The desirable insulation should have better dielectric properties and use a lesser volume of insulating oil when compared to conventional liquid insulation (Rajnak et al. 2019). To achieve this goal, nanofluids are being used and investigated more profoundly. Nanofluids are colloidal suspensions consisting of finely dispersed nano-sized solid particles within a liquid medium (Jamshed et al. 2021 , 2021; Ouni et al. 2022; Shahzad et al. 2021; Ashraf et al. 2024; Zhang et al. 2023; Shah et al. 2021; Wakif et al. 2024). The addition of nanoparticles boosts the dielectric constant and thermal conductivity of insulating liquid (Jamshed et al. 2021; Wakif et al. 2024; Rasool et al. 2023). The term “nanofluid” refers to fluid substances appropriately mixed with particles with diameters in the nanometre range, typically around 10–100 nm (Bacha et al. 2024). These particles exhibit enhanced optical, mechanical, thermal, chemical, magnetic, and electrical characteristics compared to ordinary solids due to their significant surface-area-to-volume ratio (Salih and Mahmood 2023). As industries grapple with increasing heat generation and concerns regarding cooling and product maintenance, scientists have explored the unique thermal properties of various fluids produced by adding solid particles, saving energy and processing time on micrometre and millimetre scales (Dhumal et al. 2023). It has been reported in the previous research the ZnO and CuO-based nanofluids show improved BDV compared to pure oils with maximum enhancement shown by ZnO nanoparticles (Gangadhar et al. 2024). The research investigated the influence of incorporating C₆₀ nanoparticles into the natural ester MIDEL eN 1204 and the synthetic ester MIDEL 7131 on their BDV. Findings indicated a significant enhancement, with the BDV increasing by 32.5% compared to the baseline value for both liquids. This improvement occurred at an optimal nanoparticle concentration of 0.01% wt./wt. The findings indicate that the BDV increased up to a maximum concentration of 0.008% wt./wt. when using commercial Fe₃O₄ powder and up to 0.012% wt./wt. for nanofluids containing oleate-coated colloidal Fe₃O₄ (Gangadhar et al. 2024, 2023; Alghamdi et al. 2024; Harfouf et al. 2024). The thermal efficiency of Cu-EO over Al₂O₃-EO is seen with a minimum of 1.5% and a maximum enhancement of 9.1% (Elboughdiri et al. 2023). Several experimental and numerical investigations were conducted on the nanofluids Cu(np) C₂H₆O₂(bf), Al₂O₃(np) C₂H₆O₂(bf), and TiO₂(np) C₂H₆O₂(bf). A notable finding was that the use of the nanofluid mixture Al₂O₃(np) C₂H₆O₂(bf) significantly reduces the strength of the braking torque (Gangadhar et al. 2024, 2023; Rasool et al. 2023; Areekara et al. 2023; Jamshed and Nisar 2021).

Nanofluids consist of nano-sized particles suspended within a fluid, enhancing its heat capacity and heat transfer capabilities through particle collisions (Jamshed et al. 2022). Among the various types of nanofluids, nano-suspensions are widely used and have garnered significant attention in the realm of heat transfer and energy applications. One of the most notable applications of nanotechnology lies in the production of nanofluids with exceptionally high thermal conductivity (Modassir et al. 2022; Jamshed et al. 2021; Akram et al. 2022; Shahzad et al. 2022; Zaydan et al. 2024; Alkathiri et al. 2022). When these nanofluids are introduced into base fluids, they exhibit a remarkable enhancement in thermal properties (Jiosseu et al. 2023; Siddique and Basak 2024; Yang et al. 2023; Apmann et al. 2022).

The role of nanofluids in the medical field, particularly in drug delivery systems, antibiotic activities, and imaging. Magnetic nanofluids are employed for targeted drug delivery and differential diagnosis (Rasool et al. 2023; Shah et al. 2022 , 2022).

These fluids typically comprise a base fluid mixed with nanocrystalline additives, such as metals, carbides, oxides, and carbon nanotubes. Nanofluids have garnered significant attention from researchers due to their enhanced thermal properties and diverse applications. The literature on nanofluids encompasses a wide range of studies, including experimental findings, theoretical investigations, and intriguing phenomena (Wakif 2023).

Hybrid nanoparticles are composed of two or more distinct nanometre-sized materials, forming what are known as hybrid nanofluids (Scott et al. 2022). These nanocomposites offer a novel approach to heat transfer by combining traditional heat transfer fluids with the addition of nanoparticles (Bai et al. 2024; Abdullaev et al. 2022). Hybrid nanofluids present significant potential for thermodynamic applications, providing enhanced control over the conductivity of the resulting fluids (Shahzd et al. 2022). These innovative liquids demonstrate superior heat transfer performance compared to conventional fluids commonly used for heat transfer applications, such as water, ethylene glycol, and oil, as well as single-component nanofluids (Zhang et al. 2024).

The consistent dispersion of nanoparticles is crucial for studying the dielectric characteristics of nanofluids. Nonetheless, maintaining the nanometre-scale size of nanoparticles can be difficult due to the attractive forces between them, leading to agglomeration and settling at the bottom (Jeong et al. 2013). Magnetic fluids have demonstrated advantages in both thermal and dielectric aspects. They hold promise in improving cooling by optimizing fluid circulation within transformer windings. Furthermore, they can strengthen a transformer's resilience against lightning impulses while mitigating the effects of moisture in traditional insulating fluids. Leveraging the benefits of magnetic fluids could lead to the creation of more compact and efficient transformers or enhance the longevity and load-carrying capabilities of existing units (Jamshed et al. 2021).

Given that magnetic fluid undergoes magnetically induced flow, unlike conventional oil, one would anticipate efficient heat dissipation with such fluid. Nevertheless, the findings indicating magnetic nano-fluid's heightened dielectric strength compared to pure insulating oil were unexpected (Jamshed et al. 2022). External particles significantly influence the dielectric breakdown resilience of liquid insulators. Magnetic particles with polarizable properties, possessing a higher permittivity than the surrounding liquid, undergo an electrical force directed toward the region of maximum stress. When using electrodes with a uniform field, particle movement is believed to initiate because of irregularities on the electrode surfaces, generating localized field gradients. This accumulation of particles continues, potentially forming a bridge across the gap, eventually resulting in breakdown (Mansour et al. 2016). Additionally, the magnetic dipole–dipole interaction between particles needs to be considered. The clustering of magnetic particles within an external magnetic field produced by the transformer windings influences the dielectric breakdown strength of a transformer oil-based magnetic fluid (Rajnak et al. 2017).

Mineral oil refers to a type of oil derived from petroleum, typically used in various industrial applications, including as a coolant and insulator in transformers and electrical equipment. Synthetic ester oil is a type of lubricating oil that is artificially produced through the esterification process. It is commonly used in various industrial applications, including as a coolant and insulating fluid in transformers and electrical equipment. Synthetic ester oils offer advantages such as high biodegradability and improved fire safety compared to traditional mineral oils (Rajnak et al. 2018).

In the previous studies, the experimental investigation on ester oils has been performed using the present nanoparticles, but the comparison of dielectric characteristics of synthetic ester oils with mineral oil has not been performed yet using the present magnetic nanoparticles, and also the statistical analysis on the breakdown voltages of nanofluids has been performed in the present study that outlines the importance of this study (Khan et al. 2022).

The main objective of this paper is to enhance the dielectric breakdown strength of magnetic nanofluids, intending to employ them as liquid insulation in transformers. The previous studies only investigate the breakdown behaviour of mineral oil with magnetic nanoparticles. However, synthetic esters have not been investigated. Moreover, statistical analysis has been performed on the BDV results to indicate their better accuracy for industrial application. Magnetic nanoparticles (Fe₃O₄, CO₃O₄, and Fe₃P) are employed in this study to enhance the breakdown resilience of both mineral oil and synthetic ester oils (Suhaimi et al. 2022). Regression analysis was conducted on oil breakdown voltages to enhance the accuracy of experimental outcomes. Additionally, the influence of electrode configurations was investigated to assess the effect of electrode shape on the breakdown strength of oils. Additionally, a theoretical model has been developed to explain the mechanism of the enhancement of BDV with the addition of nanoparticles under both electrode configurations (Hamid et al. 2016).

Experimental method

Selecting a nanoparticle and oils

For the experimental phase, three nanoparticles were employed: Fe₃O₄, CO₃O₄, and Fe₃P. These nanoparticles were prepared utilizing a Ball milling machine situated at the Interdisciplinary Nanotechnology Centre, AMU Aligarh, employing a top-down nanotechnology approach. The specifications of these nanoparticles are outlined in Table 1. Earlier studies conducted by researchers predominantly concentrated on traditional mineral oil; nevertheless, our investigation delves into contemporary biodegradable insulating oils, which are commercially established. The insulating oils employed comprise mineral oil and synthetic ester oil, with certain properties outlined in Table 2.

Table 1 Specifications of nanoparticles

Table 2 Properties of insulating oils

Synthesis of nanofluids

The synthesis methods for transformer-oil-based nanofluids can be categorized into one-step or two-step processes. In the one-step method, nanoparticles are prepared and dispersed in the host oil simultaneously. This approach bypasses the need for separate drying, storage, and transportation of the nanoparticles, thereby reducing agglomeration and enhancing the stability of nanoparticle dispersion in the oil.

In this experimental setup, nanofluids were created through a two-step process outlined in Fig. 1. The materials employed in the nanofluid preparation included nanoparticles along with oleic acid acting as a surfactant and an ester-based dielectric liquid. Initially, a 10% oleic acid solution was prepared by adding 50 ml of oleic acid to 450 ml of insulating oil (the carrier liquid) in a beaker. To ensure thorough mixing, a magnetic stirrer was employed (Khelifa et al. 2022). A magnetic stirrer, also known as a magnetic mixer, is a piece of laboratory equipment that utilizes a rotating magnetic field to induce rapid spinning of a stir bar (sometimes referred to as a “flea”) immersed in a liquid, thereby stirring it effectively. This rotating field can be generated either by a rotating magnet or by a set of stationary electromagnets positioned beneath the vessel containing the liquid. A magnetic bar is placed inside the beaker, which spins to ensure proper solution mixing. Typically, the stirring process lasts for approximately 30 min. Once the solution is thoroughly mixed, nanoparticles are added (Khan et al. 2021). During the experimental phase, I created three distinct concentrations of nanofluids for each of the three nanoparticles. The quantities of nanoparticles utilized to achieve these different concentrations were 0.01 g/L, 0.02 g/L, and 0.04 g/L. However, an issue of stability arose as the concentrations of nanofluids increased.

Fig. 1

Block diagram for preparation of nanofluids

Once the nanoparticles are introduced into the solution, it is essential to ensure thorough mixing as they tend to settle at the bottom of the beaker. Simply stirring the sample is not adequate to achieve complete mixing. To disperse the nanoparticles evenly throughout the solution, ultrasonic bath treatment is employed. Sonication, a technique commonly utilized in nanotechnology, is effective in achieving uniform dispersion of nanoparticles in liquids. Furthermore, it serves to break apart aggregates of micron-sized colloidal particles. In my research, the sonication process lasts approximately three hours. Following sonication, the solution is securely covered and left undisturbed for a day. It is crucial to isolate the nanofluids completely to prevent moisture infiltration. Moisture presence in the insulating oil can severely compromise breakdown strength. Prolonged usage of nanoparticles may also lead to moisture accumulation and agglomeration. To mitigate the clustering risk, the nanoparticles are placed inside a digital incubator at an optimal temperature to eliminate moisture.

AC breakdown voltage

The breakdown voltage testing adheres to the guidelines outlined in the IEC 60156 standard. The measurement of breakdown strength is conducted using a Fully Automatic Oil BDV Tester (see Fig. 3). These oil testers yield precise and consistent results of the measured breakdown voltage. Their rapid high-voltage switch-off time enables efficient testing of dielectric liquids. The internal structure of the tester, coupled with the implementation of automatic high-voltage breakers, ensures operator safety by cutting off the supplied voltage if the tester’s lid is opened during the test.

The Automatic Oil BDV Tester depicted in Fig. 3 serves a dual purpose: it can be utilized for either breakdown testing or withstand testing. The apparatus is equipped with a touchscreen panel for instrument operation and features three distinct time intervals: stand time, stirring time, and intermediate time. Two different electrode configurations are employed to conduct the breakdown test: VDE electrodes and S–S electrodes (refer to Fig. 2). The electrode gap measures 2.5 mm for both configurations, and the oil capacity is set at 500 ml. Nanofluids are poured into the electrode configuration until reaching the oil level mark. Subsequently, the assembly is placed inside the fully automatic Oil BDV Tester by opening the instrument’s lid (Fig. 3). During the experiment, the stand time, stirring time, and intermediate time are adjusted to 4 min, 2 min, and 3 min, respectively. Each experiment is repeated three times. The duration and repetitions remain consistent across all nanofluids and electrode configurations tested. The empirical formula of enhancement of BDV is in Eq. (1) (Khan et al. 2022).

$$Enhancement\left(\%\right)\;=\left(\frac{Average\;Breakdown\;voltage\;of\;nf\;\left(A\;to\;E\right)\;}{Breakdown\;voltage\;of\;insultating\;oil\;\left(A\right)}-1\right)\;\times\;100$$

(1)

where,

Fig. 2

Different electrode configurations for the breakdown of nanofluid a VDE and b S–S electrodes with 2.5-mm electrode gap

Fig. 3

AC breakdown voltage tester determines dielectric BDV with different electrode

A—average BDV of insulating oil.

B—average BDV of 10% oleic acid solution of insulating oil.

C—average BDV of 0.01 g/L nanofluids.

D—average BDV of 0.02 g/L nanofluids.

E—average BDV of 0.04 g/L nanofluids.

Result and discussion

The electrical breakdown of a dielectric liquid represents the ultimate phase in the electrical breakdown sequence, which encompasses several preceding stages. Breakdown occurs when an arc forms, constituting an electrical short circuit through the liquid. This enables the passage of substantial destructive currents between two terminals that would otherwise be insulated by the liquid. Prior to the formation of an arc, highly conductive structures known as electrical streamers emerge in dielectric liquids. A streamer originates at an electrode if the electric field intensity at that electrode surpasses a liquid-specific threshold, typically falling within the range of 1 × 10⁸–1 × 10⁹ (V/m) for most mineral oils. Once initiated, a streamer tends to propagate from the initiating electrode towards a grounding point or an electrode of opposite polarity. If a streamer successfully spans the entirety of the liquid insulation, it will create a highly conductive path across the liquid gap, ultimately resulting in the generation of an arc. Nevertheless, the commencement of a streamer does not automatically guarantee the formation of an arc and subsequent electrical BDV. If the excitation level at the initiating electrode fails to maintain a certain threshold, the propagating streamer will cease and dissipate. As depicted in Fig. 4, nanoparticles function as ideal dielectrics within the oil. When subjected to the applied electric field, nanoparticles undergo polarization, acquiring an electric dipole character. This conversion of numerous nanoparticles into electric dipoles enhances the breakdown resilience of the oil. The limited mobility of particles impedes the accumulation of net space charge at the tip of the streamer, thereby restraining streamer expansion within the oil and consequently increasing its breakdown strength (Qin et al. 2023).

Fig. 4

Polarization of Fe₃O₄ nanoparticles and electric charge distribution in mineral oil

AC BDV

The dielectric breakdown test is integral to the approval process for insulating oil used in transformers. The assessment of AC BDV is conducted using a Fully Automatic Oil BDV Tester, specifically the PE AOBDV.M100 model. This tester features a chamber designed to house the oil sample and electrodes. The accepted standard for determining AC BDV is ASTMD877. The electrodes, crafted from brass, feature a gap length of 2.5 mm and a diameter of 10 mm. The voltage is incremented at a rate of 2 kV/s. The experiment is conducted under ambient room temperature conditions. Nanofluid is introduced into the chamber with electrodes until it reaches the designated oil level marking. Subsequently, the chamber is placed inside the Fully Automatic Oil BDV Tester by opening the instrument’s closure. During the experiment, each nanofluid undergoes a standing time of 4 min, a stirring time of 2 min, and an intermediate time of 3 min, as specified by the standard procedure. The experiment is conducted using two distinct electrode configurations: S–S and M-M, as depicted in Fig. 2. Electrodes conforming to ASTM D-1816 standards are employed to analyse breakdown characteristics under these two standard electrodes S–S and VDE configurations. The breakdown of nanofluid leads to the creation of an arc between the electrodes. Twelve breakdown readings are recorded for each sample, and the final value is determined by averaging all the readings. Two separate batches of each oil are tested for accuracy and consistency. The AC breakdown tester used to assess the dielectric breakdown of nanofluids is depicted in Figs. 5, 6, 7, 8, 9, and 10. The AOBDV tester provides BDV measurements with an accuracy of 1% (Siddique et al. 2021; Primo et al. 2019).

Fig. 5

AC BDV for Mineral oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under a S–S and b VDE electrodes configuration

Fig. 6

AC BDV for Synthetic ester oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under a S–S and b VDE electrodes configuration

Fig. 7

AC BDV for Mineral oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under a S–S and b VDE electrodes configuration

Fig. 8

AC BDV for Synthetic Ester oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under a S–S and b VDE electrodes configuration

Fig. 9

AC BDV for Mineral oil and corresponding nanofluids incorporating Fe₃P nanoparticles under a S–S and b VDE electrodes configuration

Fig. 10

AC BDV for Synthetic Ester oil and corresponding nanofluids incorporating Fe₃P nanoparticles under a S–S and b VDE electrodes configuration

The graph illustrates that the BDV reaches its peak at a specific concentration of nanoparticles. Beyond this concentration, there is a decline in the BDV due to nanoparticle agglomeration. At higher concentrations, nanoparticles tend to aggregate, forming a chain-like structure that acts as a conducting path in the oil, resulting in a reduction of BDV. Other factors contributing to the decrease in BDV may include the unintentional introduction of moisture during sample preparation and the heating of the sample during magnetic stirring. Therefore, it is crucial to consider the nanoparticle concentration that yields the maximum BDV for a specific type of oil.

The comparative results for all nanofluids with different electrode configurations are shown in Figs. 5, 6, 7, 8, and 9. This figure illustrates the variation in average breakdown voltage with different nanofluid concentrations. The breakdown voltage and standard deviation are notable for repetitive breakdown testing, as depicted in Figs. 5, 6, 7, 8, and 9. The enhancement in breakdown voltage for each nanofluid at specific nanoparticle concentrations is indicated in the graph, along with the range of breakdown strength at each point. The VDE electrode system is depicted as the Mushroom-Mushroom (M-M) electrode system, while the sphere-sphere (S–S) electrode system is also shown in all figures. The gap between the M-M electrodes is larger than that of the S–S electrode system, resulting in a higher BDV for the M-M electrode system for each insulating fluid.

Each nanofluid (NF) can sustain a specific nanoparticle (NP) concentration, which is considered its saturation or critical point. Beyond this point, adding more NPs negatively influences its characteristics. The enhancement in the BDV of the prepared magnetic NF is attributed to the high electron scavenging properties of the dispersed nanoparticles during the application of high voltage. These slow-moving nanoparticles reduce the rapid motion of electrons, suppressing streamer formation and consequently increasing the breakdown strength of the prepared NFs. The variation in BDV for iron oxide (Fe₃O₄) nanoparticle (NP)-based nanofluids (NFs) is illustrated in Figs. 5 and 6a and b. Co₃O₄ NP-based NFs initially exhibit a positive enhancement in BDV with the addition of up to 0.02 g/L NP concentration when the mineral oil and synthetic ester oil are used as the base fluid. Beyond this concentration, the BDV shows a negative enhancement, indicating a drop in BDV for NP concentrations greater than 0.02 g/L with mineral oil and synthetic ester oil as the base fluid. This suggests that nanoparticles tend to agglomerate near the electrode, creating a high electric field and resulting in a lowered breakdown voltage of the NF. Saturation of NPs is achieved above 0.02 g/L, as depicted in Figs. 5, 6, 7, and 8. Breakdown strength is not measured above 0.04 g/L, as BDV drops below the BDV of the base oil.

When transitioning the base fluid in NF to mineral oil and natural ester oil, the saturation point exceeds 0.04 g/L NP, as illustrated in Figs. 5a, b and 6a, b for both setups. Furthermore, the breakdown strength escalates with NP concentration when utilizing synthetic ester oil. mineral oil displays a greater tendency for NP accumulation than synthetic ester oil, consequently leading to a stable dispersion at higher concentrations. Similar tendencies are shown by iron phosphate NPs with synthetic ester oil as a base fluid in NF, as shown in Figs. 9a, b and 10a, b, except that iron phosphide-based NF shows a larger drop as compared with iron oxide-based NF. This is due to the similar size and magnetic nature of both compounds. Maximum enhancement in breakdown voltage is up to 0.02 g/L concentration and then drops with synthetic ester oil as base fluid. The maximum BDV is achieved at 0.02 g/L concentration under the VDE electrode system, while maximum enhancement is found in the spherical electrode system. The maximum enhancement of 70.05% is achieved with iron phosphide-based NFs as compared with iron oxide-based NFs.

Mechanism of dielectric fluid alteration

The process of dielectric modification due to conductive, semiconducting, and insulating nanoparticles is explained as follows. The primary reason for the improved breakdown strength is electron trapping by the nanoparticles. Essentially, nanoparticles act as electron searchers within the base oil. When an electric field is applied, nanoparticles become rapidly polarized as shown in Fig. 11. Consequently, fast-moving electrons generated in high fields are captured by the nanoparticles and then quickly released, transforming into slower-moving negatively charged particles. This increases the charge decay rate and effectively reduces the mobility of charged electrons. As a result, the propagation of streamers is hindered, thereby enhancing the breakdown strength.

Fig. 11

Polarization and surface charge distribution of a Fe₃O₄ nanoparticle in mineral oil

A model (Lv et al. 2014) is proposed that plays a pivotal role in dielectric modification. In this model, the relaxation time constant $(\Gamma\mathrm{r})$ of the (nanoparticle/oil) system is determined using the formula provided in Eq. (2).

(2)

where ε is the oil permittivity and σ is the DC conductivity of the mineral oil and nanoparticle, respectively.

If the system’s time constant value is shorter than a microsecond, which is the duration involved in the propagation of a streamer, then the nanoparticles will capture the fast-moving electrons present in the oil and convert these electrons into low-mobility negatively charged particles (Lv et al. 2014).

As previously explained, the applied electric field produces fast electrons, which rapidly polarize the nanoparticles. Magnetic nanoparticles like Fe₃O₄, having a low time constant, capture these fast electrons in transformer oil and reduce their mobility (Lv et al. 2014). This results in an increased potential drop along the streamer length, thereby enhancing the breakdown strength. Consequently, adding magnetic nanoparticles improves the dielectric performance.

Statistical analysis

• Statistical analysis in engineering involves the application of statistical methods and techniques to analyse data and solve engineering problems. This type of analysis is crucial in engineering for making data-driven decisions, optimizing processes, improving product quality, and ensuring reliability and safety. Here are some common applications of statistical analysis in engineering:

• Quality control and process optimization: Statistical methods such as control charts, process capability analysis, and design of experiments (DOE) are used to monitor and improve manufacturing processes, ensuring that products meet quality standards and specifications.

• Reliability analysis: Engineers use statistical techniques such as reliability modelling, survival analysis, and Weibull analysis to assess the reliability and durability of components, systems, and infrastructure over time.

• Experimental design and analysis: DOE techniques are employed to systematically plan, conduct, and analyse experiments in order to identify significant factors affecting a process or product performance and optimize design parameters.

• Failure analysis: Statistical methods help engineers analyse failure data, identify root causes of failures, and develop preventive measures to enhance product reliability and safety.

• Risk assessment and decision-making: Probabilistic risk assessment techniques, such as fault tree analysis and reliability-centred maintenance (RCM), are used to evaluate and mitigate risks associated with engineering systems, equipment, and operations.

• Statistical modelling and simulation: Engineers develop mathematical models and use simulation techniques to predict the behaviour of complex systems, assess design alternatives, and optimize performance parameters.

• Environmental monitoring and impact assessment: Statistical analysis is applied to environmental data to assess the impact of engineering activities on the environment, comply with regulatory requirements, and develop sustainable solutions.

• In essence, statistical analysis holds significant importance across different phases of the engineering lifecycle, spanning from initial design and development through manufacturing, operation, and maintenance. Its utilization contributes to enhancing the efficiency, reliability, and sustainability of engineering systems and processes.

Standard deviation

The standard deviation is a commonly utilized metric in statistics and probability theory to gauge variability or diversity within a dataset. It quantifies the extent of variation or “dispersion” from the “average” (mean or expected value). A low standard deviation suggests that the data points are closely clustered around the mean, while a high standard deviation signifies that the data is widely spread across a broader range of values (Figs. 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23; Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14).

$$\mathrm S=\frac{\sqrt{\sum_{\mathrm i=1}^{\mathrm N}\left({\mathrm x}_{\mathrm i}-\mathrm x\right)^2}}{\mathrm N-1}$$

(3)

where,

Fig. 12

Average BDV vs Fe₃O₄ nanoparticles concentration

Fig. 13

Average BDV vs Fe₃O₄ nanoparticles concentration

Fig. 14

Average BDV vs Fe₃O₄ nanoparticles concentration

Fig. 15

Average BDV vs Fe₃O₄ nanoparticles concentration

Fig. 16

Average BDV vs CO₃O₄ nanoparticles concentration

Fig. 17

Average BDV vs CO₃O₄ nanoparticles concentration

Fig. 18

Average BDV vs CO₃O₄ nanoparticles concentration

Fig. 19

Average BDV vs CO₃O₄ nanoparticles concentration

Fig. 20

Average BDV vs Fe₃P nanoparticles concentration

Fig. 21

Average BDV vs Fe₃P nanoparticles concentration

Fig. 22

Average BDV vs Fe₃P nanoparticles concentration

Fig. 23

Average BDV vs Fe₃P nanoparticles concentration

Table 3 BVD for mineral oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under S–S electrode configuration

Table 4 BDV for mineral oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under VDE electrodes configuration

Table 5 BDV for Synthetic ester oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under S–S electrodes configuration

Table 6 BDV for synthetic ester oil and corresponding nanofluids incorporating Fe₃O₄ nanoparticles under VDE electrodes configuration

Table 7 BDV for mineral oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under S–S electrodes configuration

Table 8 BDV for mineral oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under VDE electrodes configuration

Table 9 BDV for synthetic ester oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under S–S electrodes configuration

Table 10 BDV for synthetic ester oil and corresponding nanofluids incorporating CO₃O₄ nanoparticles under VDE electrodes configuration

Table 11 BDV for mineral oil and corresponding nanofluids incorporating Fe₃P nanoparticles under S–S electrodes configuration

Table 12 BDV for mineral oil and corresponding nanofluids incorporating Fe₃P nanoparticles under VDE electrodes configuration

Table 13 BDV for synthetic ester oil and corresponding nanofluids incorporating Fe₃P nanoparticles under S–S electrodes configuration

Table 14 BDV for synthetic ester oil and corresponding nanofluids incorporating Fe₃P nanoparticles under VDE electrodes configuration

S—sample standard deviation.

N—the number of observations.

x _i—the observed values of a sample item.

x—the mean value of the observations.

For Fe₃O₄ nanofluids

Mineral oil

The minimum standard error for the experimental data for the S–S electrode configuration is 0.44, and the VDE electrode configuration is 0.64.

Synthetic ester oil

The minimum standard error for the experimental data for the S–S electrode configuration is 0.54, and the VDE electrode configuration is 0.89.

For CO₃O₄ nanofluids

Mineral oil

The minimum standard error for the experimental data for the S–S electrode configuration is 0.44, and the VDE electrode configuration is 0.68.

Synthetic ester oil

The minimum standard error for the experimental data for the S–S electrode configuration is 0.68, and the VDE electrode configuration is 0.89.

For Fe₃P nanofluids

Mineral oil

The minimum standard error for the experimental data for the S–S electrode configuration is 0.44, and the VDE electrode configuration is 0.64.

Synthetic ester oil

The minimum standard error for the experimental data for S–S electrode configuration is 0.70, and VDE electrode configuration is 0.89.

It has been shown the BDV performed for 12 iterating across the VDE electrode is high compared to S–S electrodes, and the standard error is high for the VDE electrode is high compared to the S–S electrode, which shows better accuracy of BDV for the VDE electrode. However, the BDV of nanofluids is lower for the S–S electrode compared to the VDE electrode.

Enhancement in AC BDV

Enhancement in AC BDV refers to the increase in the voltage at which electrical insulation breaks down under alternating current (AC) conditions. This enhancement is often achieved through various means, such as the use of improved insulating materials, innovative design techniques, or the addition of additives like nanofluids. In engineering, particularly in applications involving high-voltage equipment such as transformers, cables, and switchgear, achieving a higher BDV is critical for ensuring the reliability and safety of the electrical system. By enhancing the AC BDV, engineers can reduce the risk of electrical breakdown or insulation failure, which could lead to equipment damage, power outages, or even safety hazards. The enhancement in AC BDV can be studied through experimental testing, where different factors such as insulation material composition, geometry, temperature, and the presence of additives are systematically varied and analysed to identify the most effective methods for improving BDV performance. Overall, enhancing AC BDV is an important aspect of electrical engineering aimed at improving the efficiency, reliability, and safety of electrical systems operating under AC conditions.

The graphs illustrate that the voltage attains its peak at a particular (optimal) concentration of nanoparticles. The rise in AC BDV with nanoparticle concentration can be attributed to the concept of the interfacial layer. Additionally, the heightened permittivity of nanoparticles may contribute to the enhanced relative permittivity of nanofluids shown in Figs. 24, 25, 26, 27, 28, and 29. The interfacial layer existing between nanoparticles and the oil significantly boosts the dielectric strength of nanofluids (Rao et al. 2022). At lower nanoparticle concentrations, a larger interfacial volume introduces more traps into the oil, leading to a uniform distribution of nanoparticles and consequently yielding higher BDV (Hussain et al. 2022). However, at higher concentrations, nanoparticles tend to aggregate, losing their individual properties and increasing in size to become micrometre-sized nanoparticles (Rafiq et al. 2019). Therefore, the reduction in interfacial volume leads to a decrease in the number of traps, subsequently resulting in a decrease in breakdown voltage, as elucidated in Section III. Additionally, at higher concentrations of nanoparticles, there is a reduction in the inter-particle distance among the nanoparticles introduced, leading to the formation of conductive paths or chain-like structures above a specific threshold value (Kurimský et al. 2018). A portion of leakage currents may traverse through the conducting paths within the oil, leading to a decrease in AC BDV at elevated nanoparticle concentrations (Thabet et al. 2016). Additionally, other factors contributing to the decline in breakdown voltage may include the presence of moisture within the oil samples (Ibrahim et al. 2016).

Fig. 24

Maximum percentage of enhancement vs Fe₃O₄ nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of mineral oil

Fig. 25

Maximum percentage of enhancement vs Fe₃O₄ nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of synthetic ester oil

Fig. 26

Maximum percentage of enhancement vs CO₃O₄ nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of mineral oil

Fig. 27

Maximum percentage of enhancement vs CO₃O₄ nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of synthetic ester oil

Fig. 28

Maximum percentage of enhancement vs Fe₃P nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of mineral oil

Fig. 29

Maximum percentage of enhancement vs Fe₃P nanoparticles concentration under a S–S and b VDE electrodes in AC BDV of synthetic ester oil

The moisture tolerance of synthetic ester oil is significantly higher compared to mineral oil, indicating its capacity to absorb a considerably larger quantity of water without compromising its breakdown strength. Consequently, experimental findings demonstrate that synthetic ester oil exhibits a higher maximum BDV than mineral oil and synthetic ester oil (Hamid et al. 2016). Therefore, it is crucial to carefully consider the nanoparticle concentration that optimally enhances the breakdown value for commercialization purposes (Izzularab et al. 2016).

In preparation for statistical analysis, the standard deviation of the dataset is computed, and subsequently, error bars are depicted for each combination of oils, nanoparticles, and electrode configurations, as shown in Fig. 2. Graphical representation with error bars is essential for the analysis and implementation of our results.

The maximum increase in BDV observed for transformer oil-based nanofluids and synthetic ester oil-based nanofluids is as follows:

• For mineral oil

  • Maximum enhancement for Fe3P at 0.02 g/L is 70.05% (S–S electrodes configuration).

  • Maximum enhancement for CO₃O₄ at 0.02 g/L is 13.24% (M-M electrode configuration).

• For synthetic ester oil

  • Maximum enhancement for Fe₃O₄ at 0.02 g/L is 46.29% (S–S electrodes configuration).

  • Maximum enhancement for CO3O4 is 19.86% (M-M electrode configuration).

After observing the variations in breakdown strength and conducting comparisons among different oils and nanoparticles, it is concluded that the AC breakdown strength exhibits the highest increase of 70.05% under the S–S configuration for Fe₃P nanoparticles and 13.24% under the VDE electrodes configuration for CO₃O₄ nanoparticles at a concentration of 0.02 g/L, compared to pure mineral oil. The AC BDV at VDE is highest due to the more uniform electric field, and the concentration of nanoparticles is uniformly mixed in base oil. Similarly, the AC breakdown strength demonstrates the maximum increase of 46.29% for Fe₃O₄ at 0.02 g/L concentration under the S–S configuration and 19.86% for CO₃O₄ at 0.02 g/L concentration under the VDE electrodes configuration in comparison with the base synthetic ester oil.

Conclusion

The dielectric breakdown of oils plays a vital role on their application as insulation in power transformers. Magnetic nanofluids exhibit higher breakdown voltage compared to pure mineral oil due to the high permittivity of magnetic nanoparticles. The enhancement of electrical properties may also result suppression of leakage current resulting in a reduction in hotspot formation in the oil, thereby enhancing the overall breakdown strength. The presence of nanoparticles helps in the uniform distribution of the electric field within the fluid, reducing the chances of localized breakdowns and increasing the breakdown voltage of oils. Magnetic nanoparticles can act as barriers to the movement of charge carriers within the oil, thereby obstructing the formation of conductive paths and enhancing the dielectric strength of the fluid. It has been observed that the BDV is higher for the S–S electrode configuration in comparison to the VDE electrode configuration across various nanoparticle concentrations.

Various nanoparticles demonstrate the highest enhancement in BDV across different oils and under varying electrode configurations.

For mineral oil, the greatest enhancement is exhibited by Fe₃P nanoparticles under the S–S electrode configuration and CO₃O₄ nanoparticles under the VDE electrode configuration.

In synthetic ester oil, the most significant enhancement is provided by Fe₃O₄ nanoparticles for the S–S electrode configuration and CO₃O₄ nanoparticles for the VDE electrode configuration.

So, Fe₃P gives maximum enhancement in the breakdown strength of mineral oil for S–S electrode configuration and synthetic ester oil for S–S electrode configuration. Therefore, Fe₃P nanoparticles yield the greatest enhancement in the breakdown strength of mineral oil under the S–S electrode configuration, while for synthetic ester oil, the highest enhancement occurs under the S–S electrode configuration as well.

Fe₃O₄ nanoparticles provide the highest enhancement in the breakdown strength of synthetic ester oil under the S–S electrode configuration, while for mineral Oil, the maximum enhancement is observed under the S–S electrode configuration as well.

CO₃O₄ nanoparticles offer the greatest enhancement in the breakdown strength of both mineral oil and synthetic ester oil under the S–S electrode configuration.

Among all the oils, mineral oil is found to possess the higher BDV at all concentrations of different nanoparticles under S–S electrode configuration.