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## Cooling performance

The idea of transformer cooling by using ferrofluids is not recent. Raj and Moskowitz [77] patented the principle of electromagnetic system cooling by using ferrofluid in 1995. The patent is based on the fact that, in an electromagnetic system such as a transformer, the source of magnetic field is the same as the source of heat. The magnetic field intensity gradient and the temperature gradient are parallel. Thus, the configuration is basically that of Figure 2.10 and thermomagnetic convection can arise. The use of temperaturesensitive ferrofluids is already mentioned in the patent. The authors specify that the magnetic material of the nanoparticles should have a Curie temperature slightly above the operating temperature. If the Curie temperature is too much above the operating temperature, thermomagnetic convection will not develop. If it is lower than the operating temperature, the ferrofluid will lose its magnetization before it arrives at the hot spot.

Several magnetic materials are cited, including the Mn-Zn ferrites. Since this patent, several research teams have studied by experiments and / or numerical approaches the benefit of transformer oil-based ferrofluid for transformer cooling. Segal and Raj [1] experimented the use of ferrofluid for power transformer cooling on two small distribution transformers (10 and 50 kVA). Various transformer oil-based ferrofluids with different saturation magnetization were tested. Passive or active cooling systems were used. The temperatures in the system for transformer oil-based ferrofluid and pure transformer oil were compared, showing a contrasted influence of the magnetic nanoparticles.

At most, a temperature decrease of approximately 10 was observed in the windings using ferrofluid, see Figure 2.15. Other measurements in this work showed a negative influence of the magnetic nanoparticles though.

### Suitability as insulating liquid

Transformer oil is an insulating medium as much as a cooling fluid. The dielectric properties of transformer oil-based ferrofluids have been measured to assess the suitability of such a mixture for transformer cooling [1, 79, 80, 81, 82, 83, 84, 85, 86, 87]. Main studies use ferrofluids with a low volume fraction of magnetic nanoparticles ( 1%). The magnetic nanoparticles are often made of magnetite. The measured properties are the breakdown voltage, the voltage above which an insulating medium becomes electrically conductive, and the electrical resistivity. Surprisingly, ferrofluids show a breakdown voltage of the same order of magnitude, and even higher sometimes, than pure transformer oil. Figure 2.19 shows the results of Lee and Kim [85] for instance. As pointed out by Segal et al. [79], transformer oil is usually purified from any particle to obtain optimal dielectric properties, while a ferrofluid naturally contains nanoparticles (even though the nanoparticles are two or three orders of magnitude smaller than the particles usually found in transformer oil). Moreover, the nanoparticles in a ferrofluid are made of metallic materials, which have much higher electrical conductivity than transformer oil. To explain the increase of the breakdown voltage, several authors mention the role of electron scavenger of the magnetic nanoparticles. As a matter of fact, Hang et al. [88] have proposed a theoretical explanation based on the trapping of the free electrons by the magnetic nanoparticles; the nanoparticles being much slower than the free electrons, it results in the reduction of the streamer velocity and a higher breakdown voltage. Kudelcik et al. [83] measured the breakdown voltage of transformer oil-based ferrofluids with a volume fraction of magnetic nanoparticles up to 2%. For = 2%, they measured a lower breakdown voltage than that of pure transformer oil. Apparently, 1% is the maximum volume fraction leading to an enhancement of the breakdown voltage. Regarding the electrical resistivity, the measurements show that the magnetic nanoparticles strongly reduce the electrical resistivity of the transformer oil (division by 10 to 100). The question is whether the electrical resistivity of the ferrofluid still respects the transformer norms. Note that Segal [89] patented the use of ferrofluid in transformers, for its enhanced insulating as well as cooling properties, for the transformer manufacturer ABB in 1999.

#### Robin boundary conditions

The problem is solved in a hollow cylinder of rectangular meridian section = n (r, , z) 2 R3; 0.5 r < 1, 0 < 2, 0 < z < 1 o . We solve the temperature equations (3.52) and the equations of fluid dynamics (3.55) in . We have f = T = . The fluid is not a ferrofluid. The magnetic body force and the pyromagnetic coefficient term are absent from the momentum equation and the temperature equation, respectively (Cp = Cm = 0). We enforce Dirichlet conditions on the temperature on the bottom and interior sides: @ T,d = @ bot [ @ int. We enforce Robin boundary conditions on the temperature on the top and exterior sides: @ T,n = @ top [ @ ext. The convection coefficient on the exterior side is hc,1 = 5. The convection coefficient on the top side is hc,2 = 2. The exterior temperatures on both sides are Tr,1 = Tr,2 = 3. The Boussinesq force is not considered. The viscosity is constant. The values of the dimensionless coefficients are given in Table 3.10.

**Time evolution of the velocity field in the magnetic oil case**

As shown in Figure 6a of the article, the time evolution of the kinetic energy of the magnetic oil is not regular: around t = 8000 s, the kinetic energy suddenly changes. In order to understand this feature, we look into the time evolution of the axial velocity between 8000 and 8300 s, see Figure 4.10. The convection cell above the coil at t = 10000 s, see Figure 4.7c, is not present at t = 8000 s, see Figure 4.10a. At t = 8000 s, the hot fluid coming from the coil does not flow toward the symmetry axis; it flows directly toward the top wall of the tank. Once cooled down, it flows downward along the symmetry axis and the lateral wall of the tank. The flow is actually modified during the next 300 s, see Figures 4.10b, 4.10c and 4.10d. The circulation of the hot fluid is brought toward the symmetry axis and the convection cell of the permanent regime is formed. Thus, the sudden change of kinetic energy is not due to a numerical issue, but to the transition of the convection pattern to the final convection pattern.

**3D computations with perturbation in the permanent regime**

Previous 3D computions show that the computations on mode 0 are relevant because modes greater or equal to 1 are negligible in the permanent regime. Here, we want to confirm the stability of the axisymmetric solution in the permanent regime. Simulations are performed starting from the axisymmetric solution at t = 10000 s with a perturbation on temperature modes greater or equal to 1. The perturbation is again of order 10−3.

We focus on the regular oil case assuming that results can be extended to the magnetic oil case. Results lead us to test several coil geometries. As a matter of fact, the coil geometry impacts the symmetry of the solution. The tested coil geometries are: the straight coil of Figure 4.2, the curved coil of Figure 4.12 and the slightly curved coil of Figure 4.17. The latter geometry is between the first two. It adopts the square shape of the coil section but the corners of the coil section are curved.

**Table of contents :**

**1 Introduction **

1.1 Context and motivation

1.2 Objectives

1.3 Outline of the manuscript

**2 Bibliography study on ferrofluids **

2.1 Generalities

2.1.1 Definition

2.1.2 Composition

2.1.3 Stability

2.1.4 Applications

2.2 Modeling

2.2.1 Magnetic properties

2.2.2 Governing equations

2.2.3 Thermophysical properties

2.3 Thermomagnetic convection

2.3.1 Thermomagnetic instability

2.3.2 Literature review

2.3.3 Conclusive remarks

2.4 Transformer cooling

2.4.1 Cooling performance

2.4.2 Suitability as insulating liquid

2.5 Conclusion of the chapter

**3 New developments in SFEMaNS: ferrohydrodynamics applications **

3.1 Physical setting

3.1.1 Geometry

3.1.2 Equations

3.2 Numerical method

3.2.1 Fourier representation

3.2.2 Finite element approximation spaces

3.2.3 Time-marching algorithm

3.3 Validation

3.3.1 Nondimensionalized equations

3.3.2 Magnetostatics

3.3.3 Temperature computation in solid regions

3.3.4 Kelvin magnetic body force

3.3.5 Robin boundary conditions

3.3.6 Temperature-dependent viscosity

3.3.7 Helmholtz magnetic body force

3.3.8 Pyromagnetic coefficient term in the temperature equation

3.4 Conclusion of the chapter

**4 Thermomagnetic convection in an oil bath heated by a solenoid **

4.1 Article published in IEEE Transaction on Magnetics

4.2 Additional comments

4.2.1 Experimental setup

4.2.2 Governing equations

4.2.3 Physical properties

4.2.4 Mesh choice

4.2.5 Comparison with another experiment

4.2.6 Interface conditions on the magnetic field

4.2.7 Focus on the convective flow

4.2.8 Presence of oscillations in the regular oil case

4.2.9 Time evolution of the velocity field in the magnetic oil case

4.3 Complementary discussion

4.3.1 Visualization of the magnetic body force

4.3.2 Three-dimensional study

4.3.3 Use of a temperature-dependent viscosity

4.3.4 Comparison of the Kelvin and Helmholtz force models

4.4 Conclusion of the chapter

**5 Realistic ferrofluid thermophysical properties **

5.1 Article published in Journal of Magnetism and Magnetic Materials

5.2 Complementary discussion

5.2.1 Experimental setup

5.2.2 Physical properties

5.2.3 Experiment vs. numerics using pure transformer oil

5.2.4 Numerical simulations with ferrofluid

5.2.5 Experiment vs. numerics using transformer oil-based ferrofluid

5.3 Improvement of the ferrofluid modeling

5.3.1 Temperature-dependent saturation magnetization of the magnetic nanoparticles

5.3.2 Complete temperature equation of ferrofluids

5.3.3 Limit of the linear magnetic material approximation

5.4 Conclusion of the chapter

**6 Thermomagnetic convection in a transformer **

6.1 Transformer principle

6.1.1 Simplified transformer

6.1.2 Absence of load

6.1.3 Presence of a load

6.1.4 Superimposed coils

6.2 Physical problem and modeling

6.2.1 The considered electromagnetic system

6.2.2 Modeling

6.3 Regular oil vs. magnetic oil cooling

6.3.1 Time evolutions

6.3.2 Temperature and velocity fields

6.3.3 Magnetic field

6.3.4 Three-dimensional study

6.4 Variations of the model

6.4.1 Tank magnetic permeability

6.4.2 Curie temperature

6.4.3 Size of the tank

6.4.4 Distance between the coils

6.5 Conclusion of the chapter

**7 Conclusion **

7.1 Outcome

7.2 Perspectives

**8 Résumé en français **

8.1 Introduction

8.2 Etude bibliographique sur les ferrofluides

8.2.1 Généralités

8.2.2 Modélisation

8.2.3 Convection thermomagnétique

8.2.4 Refroidissement des transformateurs

8.3 Nouveaux développements dans SFEMaNS

8.3.1 Problème physique

8.3.2 Méthode numérique

8.4 Convection thermomagnétique dans un bain d’huile

8.4.1 Modélisation du problème

8.4.2 Avantage du refroidissement par ferrofluide

8.4.3 Comparaison des modèles de force de Kelvin et de Helmholtz

8.5 Vraies propriétés thermophysiques du ferrofluide

8.5.1 Adaptation du modèle

8.5.2 Résultats avec les propriétés modifiées

8.5.3 Amélioration de la modélisation du ferrofluide

8.6 Convection thermomagnétique dans un transformateur

8.6.1 Modélisation du problème

8.6.2 Refroidissement par huile classique versus par ferrofluide

8.7 Conclusion

8.7.1 Bilan

8.7.2 Perspectives

**A Governing equations in fluid mechanics **

A.1 General equations

A.1.1 Continuity equation

A.1.2 Momentum equation

A.1.3 Energy equation

A.1.4 Theorem of the kinetic energy

A.1.5 Internal energy and temperature equations

A.2 Incompressible Navier-Stokes equations

A.2.1 Continuity equation

A.2.2 Momentum equation

A.2.3 Temperature equation

A.3 Newtonian fluid under Boussinesq approximation

A.3.1 Applicability of the approximation

A.3.2 Governing equations

**B Approximations using Taylor expansions in SFEMaNS **

B.1 Backward Difference Formula of second order (BDF2)

B.2 Time extrapolation of second order

B.3 Time extrapolation of first order

**C Additional convergence tests **

C.1 Magnetostatics

C.2 Kelvin magnetic body force

C.2.1 Linear law of magnetic susceptibility

C.2.2 Periodic solution

C.2.3 Periodic solution with non-zero pressure

C.2.4 Inverse law of magnetic susceptibility

C.3 Temperature-dependent viscosity

C.3.1 Polynomial temperature

C.3.2 Exponential temperature

**D Preparatory study for the design of the experiment **

D.1 Modeling

D.2 Results

**E Transformer case: estimation of the convection coefficient in the absence of heat transfer fins **

E.1 Theory

E.2 Computations

E.3 Conclusion

**Bibliography **