Heat and Mass Transfer Simulation of Nano-Modified Oil-Immersed Transformer Based on Multi-Scale

Abstract

The fast and accurate calculation of the internal temperature rise in the oil-immersed transformer is the premise to realize the thermal health management and load energy evaluation of the in-service transformer. In view of the influence of nanofluids on the heat transfer process of transformer, a numerical simulation algorithm based on lattice Boltzmann method (LBM) and finite difference method (FDM) is proposed to study the heat and mass transfer process inside nano-modified oil-immersed transformer. Firstly, the D2Q9 lattice model is used to solve the fluid and thermal lattice Boltzmann equations inside the oil-immersed transformer at the mesoscopic scale, and the temperature field and velocity field are obtained by macroscopic transformation. Secondly, the electric field distribution inside the oil-immersed transformer is calculated by FDM. The viscous resistance in LBM analysis and the electric field force in FDM analysis, as well as the gravity and buoyancy of particles, are used to explore the motion characteristics of nanoparticles and metal particles. Finally, compared with the thermal ring method and the finite volume method (FVM), the relative error is less than 5%, which verifies the effectiveness of the numerical model and provides a method for studying the internal electrothermal convection of nano-modified oil-immersed transformers.

1. Introduction

When the oil-immersed transformer is in operation, the internal winding and trans-former oil will rise in temperature, which will affect the insulation life and operation safety of the equipment under high-temperature conditions. Therefore, improving the heat dissipation of oil-immersed transformer has become a research hotspot. In 1995, Choi et al. proposed the concept of nanofluids for the first time [1] and verified that the addition of nanoparticles to a fluid will significantly improve its heat dissipation capacity [2,3]. In recent years, with the increasing application of nanofluids, the hypothesis has emerged that adding nanoparticles to transformer oil to form a stable colloidal suspension, can improve its heat transfer characteristics [4]. Nanoparticles enhance thermal conductivity through Brownian motion and microconvection effects, while increasing fluid viscosity via particle-fluid interactions and agglomeration phenomena. These combined mechanisms ultimately improve convective heat transfer performance, resulting in an elevated Nusselt number. To validate this theoretical framework, systematic investigation of heat transfer characteristics in nanoparticle-enhanced oil-immersed transformers is required.

At present, the research on the temperature rise in oil-immersed transformers mainly includes the thermal circuit method, traditional finite element analysis, and computational fluid dynamics [5,6,7,8,9,10,11]. However, as an implicit algorithm, finite element analysis requires multiple iterations in each sub-step when solving transient temperature rise, which wastes a lot of computing resources and time, making it difficult to meet the rapid calculation requirements of transformer temperature rise. In recent years, some scholars have begun to use deep learning to predict the temperature rise in transformers [12,13]. Although speed and accuracy have greatly improved, because machine learning models must be trained on specific objects with a large amount of data, it is difficult to apply to on-site operation of transformers. Due to the complexity of the above methods, the efficiency is generally low when calculating the temperature rise in the transformer. Furthermore, the particle distribution and migration characteristics inside oil-immersed transformer will affect its electrothermal characteristics, especially the partial discharge and insulation deterioration caused by charged metal particles [14,15,16]. Therefore, constructing a model that can quickly and accurately calculate the temperature rise in the transformer while accounting for the influence of the particle distribution in transformer oil on electrothermal characteristics is a current research challenge.

In recent years, with the rapid development of LBM, as an explicit solution algorithm, it has shown natural advantages in simulating fluid flow and heat transfer processes [17]. In particular, the simplicity of programming and the ease of parallel computing have made LBM increasingly popular and used by many scholars for fluid flow and heat transfer simulations [18,19,20,21]. Although LBM has been used in many heat transfer studies, and some scholars have used LBM to study the heat transfer characteristics of nanofluids in different cavities, few scholars have used LBM to study the effect of adding nanoparticles on the heat transfer characteristics of oil-immersed transformers [22,23,24].

In this paper, a simulation model based on LBM is constructed at the mesoscopic scale for nano-modified oil-immersed transformers. The internal temperature rise in oil-immersed transformers is solved and compared with the thermal circuit method and the finite volume method to verify the validity of the model. Based on this model, the heat transfer process inside nano-modified transformers with different particles and different volume fractions was analyzed, and the migration characteristics of nanoparticles and metal particles were simulated. This provides a reference for studying the heat and mass transfer process of oil-immersed transformers under electrothermal convection conditions.

2. Methodology

The whole calculation process of this study is shown in Figure 1. Firstly, the simulation model of oil-immersed transformer is constructed, and the D2Q9 model is used to calculate the heat–flow coupling field inside the oil-immersed transformer. Secondly, based on this model, the heat transfer characteristics of transformer oil with different volume fractions and different particle types are studied. Finally, FDM is used to calculate the electric field distribution in the transformer, and then the electric field force of the particles is calculated. Combined with gravity, viscous force and buoyancy, the resultant force is calculated to simulate the motion characteristics of AlN nanoparticles and metal particles, and the motion trajectories of the particles are obtained.

2.1. Lattice Boltzmann Method

Unlike traditional numerical methods for solving the macroscopic variable equations, LBM is an explicit algorithm that uses the distribution function to describe the macroscopic physical quantities at the mesoscopic scale, based on the dynamic theory. In this paper, the simulation of the internal flow–thermal coupling field process of the oil-immersed transformer is presented by the D2Q9 lattice model with a fully rebound boundary scheme. The directions and weights of the specific migrations are shown in Figure 2.

The concrete expression of discrete velocities c_i are defined as follows:

$${\mathit{c}}_{\mathit{i}}=\left\{\begin{array}{l}(0,0) i=0\\ (\cos {\theta }_i,\sin {\theta }_i)c i=1,2,3,4\\ \sqrt{2}(\cos {\theta }_i,\sin {\theta }_i)c i=5,6,7,8\end{array}\right.$$

(1)

$${\theta }_i=\left\{\begin{array}{l}{\displaystyle \frac{(i-1)}{2}}\mathrm{\pi } i=1,2,3,4\\ {\displaystyle \frac{(i-5)}{2}}\mathrm{\pi }+{\displaystyle \frac{\mathrm{\pi }}{4}} i=5,6,7,8\end{array}\right.$$

(2)

where c = Δx/Δt, Δx is the lattice space, and Δt is the lattice time step size.

The natural convection heat transfer process inside the oil-immersed transformer includes two physical fields of flow and heat, which are coupled by buoyancy. The mesoscopic momentum and heat distribution functions f_i (x, t) and g_i (x, t) can be obtained by solving LBEs.

$$\left\{\begin{array}{l}{f}_i\left(\mathit{x}+{\mathit{c}}_{\mathit{i}}\mathrm{\Delta }t,t+\mathrm{\Delta }t\right)=f_i(\mathit{x},t)\left(1-{\omega }_f\right)+{\omega }_f{f}_i^{\mathrm{e}\mathrm{q}}(\mathit{x},t)+\mathit{F}\\ g_i\left(\mathit{x}+{\mathit{c}}_{\mathit{i}}\mathrm{\Delta }t,t+\mathrm{\Delta }t\right)={\mathrm{g}}_i(\mathit{x},t)\left(1-{\omega }_{\mathrm{g}}\right)+{\omega }_{\mathrm{g}}{\mathrm{g}}_i^{\mathrm{e}\mathrm{q}}(\mathit{x},t)\end{array}\right.$$

(3)

where ɷ_f = Δt/τ_f and ɷ_g = Δt/τ_g are the fluid relaxation factor and thermal relaxation factor, respectively; τ_f and τ_g are the relaxation times of velocity fields and temperature fields; F is the external force on the particles, which is calculated by the Boussinesq approximation (the internal temperature rise in the transformer is usually less than 40K) [25]:

$$\mathit{F}=3{\omega }_i\rho {\mathrm{g}}_{\mathrm{y}}\beta {\displaystyle \frac{T-T_{\mathrm{c}}}{T_{\mathrm{h}}-T_{\mathrm{c}}}}{c}_{\mathrm{y}}$$

(4)

where ρ is the density, g_y is the acceleration of gravity along the y direction, β is the coefficient of thermal expansion, T_h and T_c are the temperatures of the solid boundary (hot wall) and the shell (cold wall), and c_y is the velocity component along the −y direction.

The calculation formula for the relaxation time of velocity fields and temperature fields are:

$${\tau }_f=2c_{\mathrm{s}}^2\mathrm{\Delta }t\left(2v+c_{\mathrm{s}}^2\mathrm{\Delta }t\right)={\displaystyle \frac{v}{c_{\mathrm{s}}^2\mathrm{\Delta }t}}+{\displaystyle \frac{1}{2}}$$

(5)

$${\tau }_{\mathrm{g}}=2c_{\mathrm{s}}^2\mathrm{\Delta }t\left({\displaystyle \frac{2\lambda }{\rho c_{\mathrm{p}}}}+c_{\mathrm{s}}^2\mathrm{\Delta }t\right)={\displaystyle \frac{\alpha }{c_{\mathrm{s}}^2\mathrm{\Delta }t}}+{\displaystyle \frac{1}{2}}$$

(6)

where v is kinematic viscosity, α is the thermal diffusion rate, and c_s is the lattice sound velocity. For the D2Q9 model in this paper, it is $c_s^2$ = c²/3.

The equilibrium momentum and heat distribution functions are:

$$\left\{\begin{array}{l}{f}_i^{\mathrm{e}\mathrm{q}}={\omega }_i\rho \left[1+{\displaystyle \frac{3\left({\mathit{c}}_{\mathit{i}}\mathit{u}\right)}{c^2}}+{\displaystyle \frac{9{\left({\mathit{c}}_{\mathit{i}}\mathit{u}\right)}^2}{2c^4}}-{\displaystyle \frac{3{\mathit{u}}^2}{2c^2}}\right]\\ g_i^{\mathrm{e}\mathrm{q}}={\omega }_{i}T\left[1+{\displaystyle \frac{3\left({\mathit{c}}_{\mathit{i}}\mathit{u}\right)}{c^2}}\right]\end{array}\right.$$

(7)

where u is the macroscopic flow rate of the insulating oil fluid.

When LBM is used to solve heat and mass transfer problems, the treatment of boundary conditions is necessary. Remarkably, a uniform heat flux boundary is used here as the heat source. According to Fourier’s Law and the finite difference approximation, the heat flux can be expressed as:

$$q=-k{\displaystyle \frac{\partial T}{\partial y}}=-k{\displaystyle \frac{T_1-T_0}{\mathrm{\Delta }y}}$$

(8)

where k is the thermal conductivity of the material.

The density, fluid flow velocity, and temperature of the macroscopic insulating oil can be calculated by the sum of the momentum and heat balance functions along different lattice directions:

$$\rho ={\displaystyle \sum _{i=0}^8{f}_i=}{\displaystyle \sum _{i=0}^8{f}_i^{\mathrm{e}\mathrm{q}}}$$

(9)

$$\rho u={\displaystyle \sum _{i=0}^8{\mathit{c}}_{\mathit{i}}{f}_i=}{\displaystyle \sum _{i=0}^8{\mathit{c}}_{\mathit{i}}{f}_i^{\mathrm{e}\mathrm{q}}}$$

(10)

$$T={\displaystyle \sum _{i=0}^8{g}_i=}{\displaystyle \sum _{i=0}^8{g}_i^{\mathrm{e}\mathrm{q}}}$$

(11)

2.2. Finite Difference Method for Electric Field

Using the finite difference method to calculate the electromagnetic field can make up for the disadvantages of the analytical method. In the 2D field, according to Poisson’s equation, the potential function φ satisfies the following conditions:

$${\nabla }^2\phi ={\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\phi }{\partial y^2}}=-{\displaystyle \frac{\sigma (x,y)}{\epsilon }}=\mathit{F}$$

(12)

$${\phi |}_C=f\left(p\right)$$

(13)

where σ is the bulk charge density and ε is the dielectric constant of a certain point.

In FDM calculations, the region needs to be discretized into grids and nodes, and the partial differential equation is transformed into a different equation by using the difference quotient instead of the derivatives. Thus, the 2D Poisson Equation (12) can be approximately expressed as:

$${\phi }_1+{\phi }_2+{\phi }_3+{\phi }_4-4{\phi }_0=h^2\mathit{F}$$

(14)

where h is the grid step.

2.3. Calculation of Heat Transfer Parameters

The heat flux density of heat source q and the heat transfer coefficient of the trans-former tank hf are needed in LBM. The heat flux of the heat source is closely related to the loss of windings and the iron core. Moreover, the winding loss is affected by temperature, which is expressed as:

$$P_T=P_0[1+k(T-T_0)]$$

(15)

where P₀ is the winding loss at temperature T₀, k is the temperature coefficient of the conductor, and T is the real-time temperature. Then the heat flux density can be written as:

$$q_i={\displaystyle \frac{P}{A_i}}$$

(16)

where q_i is the heat flux density of one heating surface, P is the total loss, and A_i is the heat transfer area of one heating surface.

The heat transfer coefficient on different tanks is different, calculated by:

$$h_f={\displaystyle \frac{Nu·\lambda }{H}}$$

(17)

where Nu is the Nusselt number, λ is the thermal conductivity of the medium, and H is the characteristic length. For Nusselt number, according to the empirical formula, its relationship with Prandtl number (Pr) and Rayleigh number (Ra) is as follows:

$$Nu={\left\{0.825+{\displaystyle \frac{0.837R{a}^{\frac{1}{6}}}{{\left[1+{\left(\frac{0.429}{Pr}\right)}^{\frac{9}{16}}\right]}^{\frac{8}{27}}}}\right\}}^2$$

(18)

2.4. Force Calculation of Particles

The Lagrangian motion of particles in oil-immersed transformers is analyzed through a force balance incorporating gravitational, buoyant, viscous drag, and electrostatic interactions. The Stokesian viscous drag force acting on a spherical particle is expressed as:

$${\mathit{F}}_{\mathbf{v}}={\displaystyle \frac{1}{{\tau }_{\mathrm{p}}}}m{v}_{\mathbf{p}\mathbf{f}}$$

(19)

$${\tau }_p={\displaystyle \frac{{\rho }_{\mathrm{p}}{d}^2}{18\mu }}$$

(20)

where m is the mass of the particle, v_pf is the relative velocity between particle and fluid, τ_p is the time constant, ρ_p is the density of the particle, d is the diameter of the particle, and μ is the viscosity.

Since nanoparticles and metal particles are subjected to different forces in the electric field, they need to be calculated separately. For nanoparticles, the electric field force can be expressed as:

$${\mathit{F}}_{\mathbf{e}}=\mathit{E}q$$

(21)

where E is the electric field intensity, and q is the charge on the nanoparticle, which can be calculated by the following formula:

$$q=4\mathrm{\pi }{\epsilon }_0{\epsilon }_{r}r{U}_0$$

(22)

where ε₀ and ε_r are the dielectric constant and relative dielectric constant of vacuum, r is the radius of the particle, and U₀ is the Zeta potential of the particle. According to Ma et al. [26], the electric field force on metal particles can be expressed as:

$${\mathit{F}}_{\mathbf{e}}={\displaystyle \frac{2}{3}}k{\mathrm{\pi }}^3{r}^2{\epsilon }_0{\epsilon }_r{E}_m^2\sin \left(\omega t+\theta \right)\sin \theta$$

(23)

where k is the turbulent kinetic energy, E_m is the electric field intensity at the position of the metal particle, ω is the angular frequency, and θ is the phase angle of the sinewave voltage.

$${\mathit{F}}_{\mathbf{g}}-{\mathit{F}}_{\mathbf{b}}={\rho }_{p}gV-{\rho }_{l}gV={\displaystyle \frac{4}{3}}\mathrm{\pi }{r}^{3}g({\rho }_p-{\rho }_l)$$

(24)

where F_g and F_b is gravity and buoyancy, V is the volume of the particle, and ρ_l is the density of the liquid.

3. Model Parameters and Validity Verification

During the operation of the oil-immersed transformer, the main heat sources come from core and winding loss, and the transformer oil inside the transformer forms a circular flow dominated by natural convection. This study mainly focuses on the qualitative analysis of the influence of nano-modified transformer oil on the heat transfer inside the transformer. Therefore, the computational model only considers core losses and winding losses, while ignoring power losses caused by structural components and eddy currents. The heat transfer process is that the core and winding transferred heat to the surface through conduction, and then the heat is transferred to the tank through the natural convection of the transformer oil. The transformer tank and the air exchange heat through the large-space natural convection and radiation. The model constructed is shown in Figure 3. In this paper, LBM-FDM is applied to solve the problem of heat and mass transfer inside the nano-modified transformer. A 2D axisymmetric thermal-fluid-particle tracking multi-physical field coupling model composed of iron core, primary winding, and secondary winding groups is established. Through simulation, the validity of the results is verified by different methods.

3.1. Model Parameters

In addition to the methods introduced above, the physical parameters and boundary conditions of the simulation need to be introduced. At the same time, because this study is to explore the internal heat transfer characteristics of transformer oil after adding nanoparticles, rather than analyzing the physical properties of nanofluid, we make certain assumptions: nanofluid and transformer oil form a whole, only single-phase flow simulation is considered, and the variability of nanofluid is not considered. When studying its heat transfer process, its heat transfer coefficient is treated as equivalent.

The density of the nanofluid is given as follows [27]:

$${\rho }_{\mathrm{n}\mathrm{f}}=(1-\varphi ){\rho }_{\mathrm{f}}+\varphi {\rho }_{\mathrm{p}}$$

(25)

where subscripts f, nf, and p stand for base fluid, nanofluid, and solid, respectively. ϕ is the solid volume fraction of the nanofluid. The specific heat capacity at constant pressure is described by [28]:

$${(\rho c_{\mathrm{p}})}_{\mathrm{n}\mathrm{f}}=(1-\varphi ){(\rho c_{\mathrm{p}})}_{\mathrm{f}}+\varphi {(\rho c_{\mathrm{p}})}_{\mathrm{p}}$$

(26)

Considering Brownian motion, the dynamic viscosity of the nanofluid can be expressed as [29]:

$${\displaystyle \frac{{\mu }_{\mathrm{n}\mathrm{f}}}{{\mu }_{\mathrm{f}}}}=1+2.5\varphi +6.5{\varphi }^2$$

(27)

According to Maxwell [30], the thermal conductivity of the nanofluid can be approximated by:

$$k_{\mathrm{n}\mathrm{f}}=k_{\mathrm{f}}{\displaystyle \frac{k_{\mathrm{p}}+2k_{\mathrm{f}}-2\varphi (k_{\mathrm{f}}-k_{\mathrm{p}})}{k_{\mathrm{p}}+2k_{\mathrm{f}}+\varphi (k_{\mathrm{f}}-k_{\mathrm{p}})}}$$

(28)

With the above thermophysical properties, the thermal diffusivity, and Prandtl number of nanofluids can be calculated:

$$\left\{\begin{array}{l}{\alpha }_{\mathrm{n}\mathrm{f}}={\displaystyle \frac{k_{\mathrm{n}\mathrm{f}}}{{(\rho c_{\mathrm{p}})}_{\mathrm{n}\mathrm{f}}}}\\ P{r}_{\mathrm{n}\mathrm{f}}={\displaystyle \frac{{(\mu c_{\mathrm{p}})}_{\mathrm{n}\mathrm{f}}}{k_{\mathrm{n}\mathrm{f}}}}\end{array}\right.$$

(29)

The physical parameters of transformer oil are shown in

Table 1. Thermophysical properties of transformer oil.

Parameters	Fitting Formula
μf/Pa·s	0.08467 − 0.0004T + 5e − 7T2
(Cp)f/J·kg−1·K−1	807.163 + 3.58T
ρf/kg·m−3	1098.72 − 0.712T
kf/W·m−1·K−1	0.159 − 7.10le − 5T

[21].

The parameters of heat sources, including thermal conductivity kh and winding loss P, are shown in

Table 2. Physical parameters of heat sources.

Item	Iron Core	Primary Winding	Secondary Winding
kh/W·m−1·K−1	21	368	368
P/W	203.6	1269	1400

[21].

3.2. Grid Independence Test

In LBM calculation, the number of grids affects the calculation results. In order to obtain accurate results, a grid independence test is needed. The results are shown in Figure 4. As the number of grids increases, the temperature and local Nusselt number will gradually increase and converge, while the computational time and memory occupied will also increase rapidly. Under the above conditions, the 1250 × 420 grid number is selected for the final simulation.

3.3. Contrast with Thermal Circuit Method

Here, the thermal circuit model applied to calculate the temperature rise in the hot-point in the nano-modified transformer [31] and compared with the results obtained by LBM. In the same initial condition, the results of the two methods, including initial temperature T0, steady-state temperature T1, and temperature rise ΔT, are shown in

Table 3. Comparison of hot-point temperatures between thermal circuit method and lattice Boltzmann method.

Item	Thermal Circuit Method (°C)	Lattice Boltzmann Method (°C)	Relative Error (%)
T0	25	25	/
T1	61.75	63.13	2.23
ΔT	36.75	38.13	3.76

.

As the table shows, the temperature rise calculated by LBM is just a little higher than that calculated by the thermal circuit method, and the relative error is less than 5%. It is concluded that the result obtained using LBM is reliable. As an explicit calculation method, LBM is more predictive and real-time than the hot path method, which relies on empirical formulas.

3.4. Contrast with Finite Volume Method

In order to further verify the validity of the model, the calculation results of LBM and FVM are compared under the same initial conditions. The results are shown in Figure 5 and Figure 6. It can be seen from the figure that the temperature and velocity distribution calculated by FVM are basically consistent with the temperature and velocity distribution calculated by LBM. The temperature near the winding is the highest, the flow velocity is the largest, the top-oil temperature is higher than the bottom-oil temperature, and the hot-spot temperature appears near the high- and low-voltage windings.

Comparing the hot-spot temperature calculated by FVM and LBM, the results are shown in

Table 4. Comparison of hot-point temperatures between finite volume method and lattice Boltzmann method.

Item	Finite Volume Method (°C)	Lattice Boltzmann Method (°C)	Relative Error (%)
T0	25	25	/
T1	64.24	63.13	−1.73
ΔT	39.24	38.13	−2.83

. It can be seen that the calculated temperature rise in LBM is slightly lower than that of FVM, and the relative error is less than 3%. At the same time, we have carried out the local maximum error analysis and found that the maximum error appears below the low-voltage winding, which was 9.45%, further verifying the validity of the model.

4. Results and Discussion

4.1. Effect of Nanoparticles on Heat Transfer

The effect of adding nanoparticles on the internal heat transfer characteristics of the oil-immersed transformers can be studied by comparing the Nusselt number. Firstly, an area is selected above the winding, which forms a radiation line from the core to the secondary winding group, with a length of 18 cm. The heat sources are increased to 1.2 and 1.5 times their previous values. The local Nusselt number from the inside to the outside is calculated by increasing the heat source of the transformer to 1.2 times and 1.5 times the original, as shown in Figure 7. The results show that with the increase in heat source, the heat transfer becomes more intense and an evident rise in Nu can be seen.

Next, AIN nanoparticles were added to the transformer, and the local Nusselt numbers at volume fractions of 0, 1%, 5%, and 10% were calculated. The results are shown in Figure 8. With the increase in the volume fraction of AlN nanoparticles, the heat transfer increases slightly and the local Nu number increases. This phenomenon is primarily attributed to the formation of nanofluids, where nanoparticles interact with and bind to liquid molecules, creating a colloidal suspension with enhanced heat transfer properties. Compared to the base fluid, nanofluids exhibit significantly improved thermal conductivity and convective heat transfer efficiency. This is consistent with the results of the literature [32,33]. Lastly, the types of nanoparticles were changed to nano-silica (SiO₂) and alumina (Al₂O₃), and the results were compared with those obtained using AlN nanoparticles. The result is shown in Figure 9, and it can be seen that the Nu of AlN and Al₂O₃ is similar and slightly higher than that of SiO₂. This means that AlN and Al₂O₃ nanoparticles have a better ability to suppress the temperature rise in transformer oil compared to SiO₂ nanoparticles.

4.2. Transfer of Nanoparticles

In order to simplify the calculation of the electric field force, we used FDM to calculate the electric field distribution when the primary winding voltage reaches the peak, as shown in Figure 10. It can be seen that the electric field is mainly distributed around the primary winding. The electric field force on the particles at other times is calculated by the Formula (25).

Two nanoparticles at different positions were set as research objects. The trajectories of the nanoparticles under the force at 300 s, 600 s, 900 s, 1200 s, and 1500 s are shown in Figure 11. The blue line represents the trajectory, and the red circle represents the initial position of a particle. As shown in the figure, the nanoparticles at two different positions follow the flow of transformer oil and move periodically in the form of natural convection in the clockwise direction. This is because the radius of nanoparticles is small, and the viscous resistance is much larger than the electric field force and gravity, indicating that nanoparticles are mainly affected by the force of viscous resistance.

4.3. The Transfer of Metal Particles

Similarly, the research objects were replaced with three metal particles located at different positions, and their trajectories at 1500 s are shown in Figure 12.

The trajectories of metal particles and nanoparticles exhibit distinct differences, particularly in regions of elevated electric field strength and low oil flow velocity (Figure 12). The interaction between particles and flow field is mainly manifested as follows: the flow of transformer oil acts on the particles through oil flow resistance, and the particles affect the flow field through reaction force. Under high-speed flow conditions, both particle types flow with the transformer oil due to the dominance of viscous resistance. However, in low-velocity regions, gravitational sedimentation becomes significant for metal particles, attributable to their substantially higher mass density compared to nanoparticles. If too many particles are deposited at the bottom of the transformer, partial discharge and thermal accumulation may occur, thus affecting the stable operation of the transformer. In addition, when the particles are close to the winding, the effect of the electric field force is more obvious. Under the action of the power frequency alternating sinusoidal electric field, the trajectories of the particles have obvious periodic oscillation (dense tracks in Figure 12), which is consistent with the results of the literature [15,16].

5. Conclusions

In this study, a simulation model of the nano-modified oil-immersed transformer was constructed based on LBM, and the heat and mass transfer characteristics of nano-modified transformer oil were calculated. Compared with the calculation results of the same model by the thermal circuit method and the finite volume method, the results showed that the error is within 5%, which verifies the validity of the model.

(1)

As the particle volume fraction in the nano-modified oil-immersed transformer increases, the local Nusselt number rises, indicating enhanced convective heat transfer performance. The migration behavior of particles varies by type: nanoparticles follow the natural convection direction of the transformer oil. Metal particles are advected by oil flow in high-speed region but show gravitational settling and oscillation in the low-speed region, and because metal particles can carry electric charge, this may produce partial discharge phenomena, thus endangering the safe operation of the transformer.

(2)

The LBM is applied to model heat and mass transfer in nanoparticle-enhanced oil-immersed transformers, offering an explicit computational framework for thermal–fluid simulations. Relative to the FVM, LBM demonstrates superior computational efficiency and reduced memory requirements in transient thermal analysis, owing to its inherent parallel computing capabilities.

(3)

Limited by the constraint of the two-dimensional model constructed in this paper, the influence of the three-dimensional effect of transformer oil flow is not considered in the simulation. In the future, a three-dimensional simulation model of transformer will be constructed based on the proposed method to further study the heat transfer characteristics, including the axial heat transfer process and the motion characteristics, including multiple particles.