Our experiments on a transformer oil-based nanofluid (NF) with ZnO nanoparticles reveal a higher relative permittivity than that of pure transformer oil. Meanwhile, the relative permittivity of ZnO NF presents a linear increase with nanoparticle volumetric concentration and a linear decrease with ambient temperature. A model based on nanoparticle polarization is proposed to investigate the mechanisms of NF relative permittivity. Analysis of the presented polarization model suggests that the value of the NF relative permittivity is dominantly determined by transformer oil, while the higher relative permittivity of NFs compared with that of pure oil is mainly caused by nanoparticle inner polarization.

## I. INTRODUCTION

Oil-paper insulation is widely used in electrical power equipment, such as transformers and inductors. However, faults and degradations caused by thermal ageing have emerged in large numbers due to the low cooling efficiency of insulating systems. In recent years, an innovative kind of dielectric, transformer oil-based nanofluids (NFs), has attracted much attention for its great improvement in both heat transfer efficiency and insulating properties.^{1–4} Transformer oil-based NFs are a colloidal suspension of nanoparticles in transformer oil. The average size of these particles is in the range of a few nanometers.^{5}

Researches have shown that both the AC breakdown voltage and the partial discharge inception voltage of transformer oil-based NFs were remarkably higher than those of pure oil.^{6} The withstand voltage of transformer oil increased and the streamer propagation velocity reduced for the presence of nanoparticles under positive lightning impulse.^{5,7} According to the nanoparticle charging theory of Hwang,^{8,9} nanoparticles in NFs could capture free electrons and convert these fast moving electrons to slower negatively charged particles, which reduces the streamer and increases the breakdown voltage of transformer oil. Meanwhile, Du^{10} believed that nanoparticles could increase the shallow trap density in transformer oil, and reduce the speed of streamer propagation through electron trapping and de-trapping process in shallow traps.

This paper focuses on the relative permittivity of the transformer oil-based NF with ZnO nanoparticles, which has not been described. Experiments are performed to study the effect of nanoparticle volumetric concentration and ambient temperature on the NF relative permittivity. A model based on nanoparticle polarization is proposed to investigate the mechanisms of NF relative permittivity, since the Maxwell-Garnett formula commonly used in describing and calculating the relative permittivity of mixed dielectric is found not appropriate in this case. Further analysis of the nanoparticle polarization model is then performed.

## II. EXPERIMENTAL DETAILS

The NF samples used in the experiments were prepared by dispersing ZnO nanoparticles (Sigma-Aldrich 677450) into transformer oil (Karamay 25#) at 30 °C by ultrasonic vibration treatment. The micromorphology of the ZnO nanoparticles under transmission electron microscope (TEM) is given in Fig. 1, and the average size of the nanoparticles is about 30 nm. The nanoparticle volumetric concentrations of the NF samples were 0.025%, 0.1%, and 0.2%, respectively. Fig. 2 is the image of these ZnO NF samples.

To decrease the negative effects of microbubbles generated during ultrasonic processing, all the NF samples were left to rest for more than 12 h. The relative permittivity was measured in a three-terminal experimental cell by a relative permittivity tester, in accordance with IEC 60247.^{11} The applied AC voltage was 2000 V (50 Hz), and the electrode distance of the cell was 2 mm. Each set of the measured data was repeated three times.

## III. RESULTS AND DISCUSSION

### A. Relative permittivity

The experiments reveal that the relative permittivity of the transformer oil-based ZnO NF is linearly relative to nanoparticle volumetric concentration and ambient temperature. As shown in Fig. 3(a), the NF relative permittivity increases with ZnO nanoparticle volumetric concentration. Under the temperature of 20 °C, it increases by nearly 2.11% to 2.180% when the concentration reaches 0.2%, while that of pure transformer oil is 2.135.

Fig. 3(b) shows the variation of relative permittivity as a function of ambient temperature for pure transformer oil and transformer oil-based ZnO NF with different volumetric concentrations. A linear decrease of the relative permittivity with temperature is revealed. For 0.1% ZnO NF, the relative permittivity drops from 2.159 to 2.079, as the temperature rises from 20 °C to 90 °C.

### B. Maxwell-Garnett formula

Maxwell-Garnett formula,^{12} which is commonly used in describing and calculating the relative permittivity of mixed dielectric, is introduced to explain the variation of relative permittivity of the ZnO NF. The formula is given in the following equation:

where *ε*_{r1} and *ε*_{r2} are the relative permittivities of continuous media and spherical dielectric, respectively. *ε*_{r,n} is the relative permittivity of the mixed dielectric, and *φ* is the spherical dielectric volumetric concentration.

Fig. 4 shows the comparison between the relative permittivity measured in the experiments and that calculated with Maxwell-Garnett formula at 20 °C and 60 °C, respectively. It is obvious that the calculations are much smaller than the measured data, which indicates that the Maxwell-Garnett formula might not be appropriate to explain the variation of NF relative permittivity.

In Eq. (1), the contribution of spherical dielectric to the relative permittivity of mixed dielectric can almost be neglected when the volumetric concentration *φ* is very small. Virtually, for transformer oil-based NFs, the nanoparticle volumetric concentration is indeed very low, and the effect of nanoparticles on NF relative permittivity is neglected in Maxwell-Garnett formula. To take the effect of nanoparticle into consideration, a model based on nanoparticle polarization is proposed.

### C. Nanoparticle polarization model

#### 1. Polarization in NFs

For transformer oil-based NFs under external electric field, both transformer oil molecules and nanoparticles will be polarized. Positive and negative surface polarization charges will gather at two sides of a nanoparticle, making the particle charged. Considered at microscopic level, the charged nanoparticle under electric field has the similar motion tendency with that of a polar molecule, namely, random thermal motion and orientation along electric field, which forms orientational polarization.

Consequently, there could be three types of polarization in transformer oil-based NFs: the polarization of transformer oil molecules, the inner polarization of nanoparticles, and the orientational polarization of charged nanoparticles as polar molecules. Fig. 5 illustrates the first two types of polarization in NFs.

Clausius-Mossotti equation is used to describe the relationship between macroscopic relative permittivity and microscopic polarization parameters. The relative permittivity of transformer oil-based NFs, *ε*_{r,n}, can be expressed by Clausius-Mossotti equation as follows:

where ε_{0} is the permittivity value in vacuum, *α*_{1} is the polarizability of a transformer oil molecule, *α*_{2} is the polarizability of a nanoparticle caused by inner polarization, and *α*_{3} is the orientational polarizability of a charged nanoparticle treated as a polar molecule. *N*_{1} and *N*_{2} are the numbers of transformer oil molecules and nanoparticles in per unit volume.

#### 2. Surface polarization charge on nanoparticle

Fig. 6 shows a spherical nanoparticle (diameter 2*a*, relative permittivity *ε*_{r2}) in the continuous dielectric (transformer oil with relative permittivity *ε*_{r1}). An external electric field *E*_{0} is applied in the direction of x-axis. Polarization charges gather on the surface of the nanoparticle: positive charges in the angle range of −π/2 < *φ*_{p} < +π/2 and negative charges in the angle range of +π/2 < *φ*_{p} < +3π/2. The surface polarization charge density *σ*_{p} can be expressed as

The total positive charges on the nanoparticle, *Q*_{+}, can be calculated by Eq. (4) as

The negative charges *Q*_{−} has the same magnitude with *Q*_{+}. To model the charged nanoparticle as a polar molecule, the particle is substituted by an electric dipole with a distance of 2*a*, shown in Fig. 7.

Assuming that, the permanent dipole moment of the polar molecule, *μ*_{c} (C·m), is the electric dipole moment of the charged nanoparticle when it is orientational polarized. And the expression of *μ*_{c} is

#### 3. Polarizability of transformer oil molecule

For pure transformer oil, only the polarizability *α*_{1} and molecule number *N*_{1} are considered when Clausius-Mossotti equation is used to describe the relative permittivity of pure oil, given in the following equation:

To get the value of *N*_{1}*α*_{1}, Eq. (6) is thus transformed into

Substituting the measured relative permittivity of pure transformer oil into Eq. (7), we can get the value of *N*_{1}*α*_{1}.

#### 4. Polarizability of nanoparticle

For a spherical nanoparticle in transformer oil, as shown in Fig. 5, the electric field *E*_{i} within the nanoparticle can be expressed as Eq. (8) when an external uniform electric field *E*_{0} is applied

The polarization vector *P*_{i} (C/m^{2}) in the nanoparticle is then

*μ*_{i} (C·m) is the vector sum of electric dipole moment within the nanoparticle, shown in Fig. 5. Equation (10) is the expression of *μ*_{i}, where *V* is the volume of the nanoparticle

According to the definition of polarizability, we can get the nanoparticle polarizability *α*_{2}

In addition, the nanoparticle number *N*_{2} can be calculated with volumetric concentration *φ* and nanoparticle diameter 2*a* as follows:

#### 5. Orientational polarizability of charged nanoparticle

Based on the theory of dipolar orientational polarization,^{13} the orientational polarizability of a charged nanoparticle can be expressed by the electric dipole moment of the charged nanoparticle, *μ*_{c}, given in the following equation:

where k is the Boltzmann constant and *T* is the thermodynamic temperature.

Combining Eqs. (2), (4), (7), (11), (12), and (13), we can get the nanoparticle polarization model of NF relative permittivity

The relative permittivity of transformer oil, *ε*_{r1}, at different temperatures can be measured in the experiments. And the relative permittivity of ZnO nanoparticles, *ε*_{r2}, remains a constant, *ε*_{r2} = 11.0. *E*_{0} is the electric field strength in the experimental cell, which is decided by the applied voltage and the electrode distance, *E*_{0} = 1 × 10^{6} V/m. The nanoparticle radius *a* is 15 nm.

Fig. 8 shows the comparison between the relative permittivity values calculated from the model and the values measured in the experiments. Fig. 8(a) demonstrates the variation of relative permittivity as a function of nanoparticle volumetric concentration at 20 °C, 60 °C, and 80 °C, respectively. Fig. 8(b) depicts the change of relative permittivity versus temperature under different volumetric concentrations. It is obvious that the calculated values, when the nanoparticle polarization is taken into account in the model, are in a fairly good agreement with the experimental data.

A further analysis of the influences of transformer oil, nanoparticle inner polarization, and charged nanoparticle orientational polarization shows that the magnitude of *N*_{2}*α*_{2} and *N*_{2}*α*_{3} is much smaller than that of *N*_{1}*α*_{1} because the nanoparticle volumetric concentration value is very small. This indicates a dominant influence of transformer oil on NF relative permittivity. Meanwhile, *N*_{2}*α*_{2} is also much larger than *N*_{2}*α*_{3}, indicating that the higher relative permittivity of NFs compared with that of pure transformer oil is mainly caused by nanoparticle inner polarization. Taking ZnO NF with 0.1% volumetric concentration at 60 °C as an example, *N*_{1}*α*_{1}, *N*_{2}*α*_{2}, and *N*_{2}*α*_{3} are calculated to be 7.088 × 10^{−12}, 8.854 × 10^{−14}, and 6.225 × 10^{−17}, respectively.

## IV. CONCLUSIONS

The transformer oil-based ZnO NF has a slightly higher relative permittivity than that of pure transformer oil at a low nanoparticle volumetric concentration. The relative permittivity increases linearly with nanoparticle volumetric concentration and decreases linearly with temperature. A model based on nanoparticle polarization is put forward to explain the mechanisms of NF relative permittivity, and the calculations fit well with the values measured in the experiments. It is suggested that the transformer oil has a dominant influence on the relative permittivity of NFs, and the higher relative permittivity of NFs compared with that of pure transformer oil is largely caused by nanoparticle inner polarization.

## ACKNOWLEDGMENTS

We thank for the National Natural Science Foundation of China for supporting this research under Contract No. 50907051, the Fundamental Research Funds for the Central Universities, and State Key Laboratory of Electrical Insulation and Power Equipment (EIPE13311).