Discussions

To generalize above cases, figure 3.17 shows the effect of viscosities of base fluids on the thermal conductivity of Fe3O4 nanofluids with 1% and 2% volume fractions of Fe3O4 nanoparticles. As the curve shows, the thermal conductivity of Fe3O4 nanofluids decreases with increasing viscosity and finally approaches a constant value. And it is observed that the viscosity around 100cP is a critical value of demarcation. Below viscosity of 100cP, the Brownian motion and the convectionlike behavior are more active so that the thermal conductivity of dynamic part is more apparent, and the measured thermal conductivity has a higher value than that of Maxwell prediction model. Over the viscosity of 100cP, the Brownian motion and the convectionlike behavior become inactive so that the thermal conductivity of dynamic part vanishes. In other words, the measured thermal conductivity of nanofluids only presents that of static part of which the value is the same with that predicted by the Maxwell prediction model.

These results improve the confidence in believing that the viscosity of base fluid does influence the thermal conductivity of nanofluids, and the thermal conductivity of static

part should be based on the Maxwell prediction model.

On the other hand, because the base fluid used in this experiment is composed of two different fluids, diesel oil and PDMS, the difference in the interfacial thermal resistance [84] should be considered. However, the surface of the Fe3O4 nanoparticle is modified by the surfactant, which makes strong bonding strength between nanoparticles and base fluids. And the contact angles between components of viscous base fluids and bulk Fe3O4 are less than 5° as shown in Fig. 3.18, which indicates a wetting system [85].

For the reason above, the interfacial thermal resistance is speculated to be small and play a minor role in this experiment.

Figure 3.1 FESEM SEI of Al2O3 nanoparticles

Figure 3.2 The Measured viscosities of the water-EG base fluids as a function of volume fraction of EG at 25℃

Figure 3.3 The Measured viscosities of the EG-glycerol base fluids as a function of volume fraction of glycerol at 25℃

Figure 3.4 The thermal conductivity of the water-EG base fluid and Al2O3 nanofluid versus the volume fraction of EG

Figure 3.5 The thermal conductivity of the EG-glycerol base fluid and Al2O3 nanofluid versus the volume fraction of glycerol

Figure 3.6 The thermal conductivity ratio of the water-EG based Al2O3 nanofluid versus different viscosities

Figure 3.7 The thermal conductivity ratio of the EG-glycerol based Al2O3 nanofluid versus different viscosities

Figure 3.8 The magnetic effects on the Fe3O4 nanofluid

Figure 3.9 The TEM photo of Fe3O4 nanoparticles

Figure 3.10 The crystalline phases of the Fe3O4 nanoparticles

Figure 3.11 The magnetized curve of the Fe3O4 nanofluid measured by a VSM

-15000 -10000 -5000 0 5000 10000 15000

M (emu/g)

Figure 3.12 The measured viscosities of the diesel oil-PDMS base fluids as a function of volume fraction of PDMS at 25℃

Figure 3.13 The thermal conductivity ratios, knano/kbf and kMaxwell/kbf of viscous nanofluids versus volume fraction of Fe3O4 nanoparticles at the viscosity of 4.18 cP

Figure 3.14 The thermal conductivity ratios, knano/kbf and kMaxwell/kbf of viscous nanofluids versus volume fraction of Fe3O4 nanoparticles at the viscosity of 31.8 cP

Figure 3.15 The thermal conductivity ratios, knano/kbf and kMaxwell/kbf of viscous nanofluids versus volume fraction of Fe3O4 nanoparticles at the viscosity of 140.4 cP

Figure 3.16 The thermal conductivity ratios, knano/kbf and kMaxwell/kbf of viscous nanofluids versus volume fraction of Fe3O4 nanoparticles at the viscosity of 648 cP

(a)

(b)

Figure 3.17 Effect of the viscosity of the base fluid on the thermal conductivity ratio of nanofluids: (a) knanofluid/kbf and (b) knanofluid/kMaxwell

Figure 3.18 The contact angles between bulk Fe3O4 and components of viscous base fluids: (a) diesel oil; (b) PDMS

Table 3.1 Physical properties of Al2O3 nanoparticles

Properties Unit Typical value

Appearance － White powder

Al2O3 content % ≧99.8

Average diameter nm 13

Specific surface area m²/g ≧100

Apparent density g/cm³ 0.05

Thermal conductivity W/m^.k 37~39

Refraction index － 1.48~1.51

Emissivity and range of wavelength － 0.915~0.920 at 5~20μm

Table 3.2 Physical properties of Fe3O4 nanofluids

Volume fraction

(%)

Density (g/cm³)

Msaturation

(emu/g)

Initial relative permeability

Diesel oil 0 0.7381 0 1

0.25M 1.12 0.7709 3.85 1.055 0.5M 2.24 0.8203 8.85 1.120 0.75M 3.36 0.8612 12.7 1.188

1M 4.48 0.9313 16.7 1.251

Chapter 4 Application of Fe

O

Nanofluids on Transformers

This chapter focuses on the performance of transformers which applies the ferrofluids as the magnetic core. One transformer is constructed on a capillary, and the other one is constructed on a wafer by the MEMS process. Firstly, the definitions of coupling coefficient and quality factor are described. Then, the inductance, resistance, coupling coefficient and quality factor of transformers are measured and simulated.

4.1 Definitions of Coupling Coefficient and Quality Factor

For a simple transformer, there are three major parts: the primary winding, the secondary winding and the magnetic core. For the ideal transformer, all the magnetic flux produced from the primary winding will pass through the secondary winding without any leakage. But actually, part of magnetic flux will not pass through the secondary winding. This is due to the imperfect coupling of windings, which results from the design of the windings and magnetic core. The leakage flux stores and discharges magnetic energy alternately, which acts effectively as inductors in series with the primary and secondary windings and causes the voltage drop. So the coupling coefficient is an important factor of transformer performance. In this study, according to the Japanese industrial standard (JIS) C6435, the coupling coefficient K is defined as

follow:

open sc

L

K = 1− L (4.1)

As Fig. 4.1 shown, the leakage inductance Lsc is measured from the one side while the other side is closed, and the inductance Lopen is measured form the one side while the other side is open. The leakage inductances of the primary and secondary sides, Lsc1 and Lsc2, are respectively shown as follow:

1

The coupling coefficient K measured from primary and secondary side should be the same. In order to avoid the error of measurement, the precise coupling coefficient should be measured form the side with higher inductance. Besides, a simple transformer is composed of two inductors. For an ideal inductor, there are no energy loss whiles the current goes through the coil. However, real inductors include the resistance in series and the capacitance in parallel with the inductance, as shown in Fig. 4.2. The series resistance of inductor will consumes energy and causes energy loss. The criterion of efficiency of inductors is the quality factor (Q) which is the ratio of inductive reactance to resistance at a given frequency. The quality factor of an inductor is shown as follow:

R

Q₌ωL (4.4)

inductor.

4.2 Transformer on a Capillary

In this section, different magnetic cores, the air, bulk Fe3O4 and Fe3O4 nanofluids of 0.25M, 0.5M, 0.75M and 1M, are applied as the magnetic core of transformers. The self-inductance Lopen, leakage inductance Lsc, coupling coefficient K, resistance and quality factor Q are measured and calculated at the frequency ranging from 100 kHz to 100 MHz.

Fig. 4.3 shows the self-inductances of coils of transformers with different magnetic cores. It illustrated that the overall inductances increase with the increase of Fe3O4

concentration. At the frequency ranging from 100 kHz to 15 MHz, the inductances decrease rapidly due to the skin effect of coils. The inductance formula of a solenoid is shown as follow:

where μ* is the permeability, N is the number of coil turns, A is the cross-section area, and l is the length of solenoid. When the frequency increases, the current gathers at the skin of coils, which decreases the effective cross-section area A of solenoid and results in the decrease of inductance. At the frequency ranging from 15 MHz to 100 MHz, the

inductances increase gradually and approach the maximum inductance at the resonance frequency. Over the resonance frequency, the inductor will drop sharply and transfer the inductive to the capacitive. Although the inductances with the core of air, Fe3O4

nanofluids of 0.25M, 0.5M, 0.75M and 1M increase linearly with the increase of Fe3O4

concentration, the inductance with the core of bulk Fe3O4 does not increase by several times as we expect. One of the reasons to explain this phenomenon is that a lot of cavities are formed in the bulk Fe3O4 during the fabricating process, and the density of bulk Fe3O4 is not as high as it is expected. The measured density of bulk Fe3O4 is 0.34472 g/cm³ which is much smaller than that of magnetite, and the bulk Fe3O4 has high porosity up to 0.9337. Fig. 4.4 shows the leakage inductances of coils of transformers with different magnetic cores. Similar with the self-inductance of coils, the leakage inductance decreases at lower frequency due to the skin effect and increases at higher frequency due to the resonance. Fig 4.5 shows the coupling coefficient of transformers with different magnetic cores. The coupling coefficient increases with the increase of Fe3O4 concentration. The coupling coefficient increases rapidly below the frequency of 5 MHz and increases gently over the frequency of 5 MHz.

The experimental results mentioned above shows improvements on the inductance and coupling coefficient of transformers. However, as shown in Fig. 4.6, the resistance of each pattern increases as a function of frequency, and also increases with the increase

of Fe3O4 concentration. At the frequency of 100 MHz, the resistance with the magnetic core of 0.25 M and 1 M Fe3O4 nanofluids is respectively two and five times to the resistance with the air core. It is because that when the magnetic core is magnetized alternately, the magnetization of material always lags behind the external magnetic field.

At the low frequency, the external magnetic field and the magnetization of material can be regarded as proportional to each other through the scalar permeability. But at the high frequency, they will react to each other with some lag so that the permeability can be expressed as a complex number:

''

* μ' μ

μ = −^j (4.6)

The real part of complex permeability, μ’, which has the same phase with the magnetic field, presents the stored energy coefficient when the material is magnetized. And the imaginary part of complex permeability, μ’’, which has the phase of 90 degree delay comparing to the magnetic field, presents the consumed energy coefficient when the material is magnetized. The impedance of a real inductor is expressed as follow:

L j R

Z = + ω (4.7)

Substituting (4.5) and (4.6) into (4.7) and rearranging, the real part of impedance is related to the frequency:

l

The imaginary part of complex permeability will reflect dramatically on the resistance

when the frequency increases. Moreover, the resistance with bulk Fe3O4 shows an exceptional value. The resistance with bulk Fe3O4 is lower than that with 1M Fe3O4

nanofluid below the frequency of 100 MHz. But it can be observed that the slope of curve of bulk Fe3O4 increases faster than that of 1M Fe3O4 nanofluid, and it is speculated that the imaginary part of frequency-dependent complex permeability of bulk Fe3O4 will be larger than that of 1M Fe3O4 nanofluid at higher frequency.

Fig. 4.7 shows the quality factor of coils of transformers with different magnetic cores. Because the increase of resistance is larger and faster than the increase of inductance with the present of Fe3O4 nanofluids, the quality factor decreases with the increase of Fe3O4 concentration. With the present of bulk Fe3O4, the quality factor still lower than that with the air core at high frequency. However, below the frequency of 4 MHz, the quality factor with the magnetic core of bulk Fe3O4 is higher than that with the air core as shown in Fig. 4.8.

4.3 MEMS Transformer on a Chip

In this section, two magnetic cores, the air core and the magnetic core of 1 M Fe3O4 nanofluid are applied on a MEMS transformer. The self-inductance Lopen, leakage inductance Lsc, coupling coefficient K, resistance and quality factor Q are measured and

calculated under the frequency ranging from 100 kHz to 100 MHz.

Fig. 4.9 shows the self-inductances and leakage inductances of coils of MEMS transformers with the air core and magnetic core of 1 M Fe3O4 nanofluid. It illustrated that the self-inductances and leakage inductances increase with the present of 1M Fe3O4

ferrofluid. At the frequency ranging from 100 kHz to 20 MHz, the inductances slightly decrease due to the skin effect. Over the frequency of 20 MHz, the inductances increase gradually and approach the maximum inductance at the resonance frequency. Fig. 4.10 shows the coupling coefficient of MEMS transformers with the air core and magnetic core of 1 M Fe3O4 nanofluid. The coupling coefficient increases slightly with the present of 1M Fe3O4 nanofluid.

Fig. 4.11 and Fig. 4.12 show the resistance and quality factor of coils of MEMS transformer with the air core and magnetic core of 1 M Fe3O4 nanofluid, respectively.

The resistance with 1M Fe3O4 nanofluid is higher than that with the air core due to the imaginary part of complex permeability, which results in the lower quality factor with 1M Fe3O4 nanofluid.

4.4 HFSS Simulation

In this section, the performance of transformer on a capillary and MEMS transformer on a chip is simulated by Ansoft HFSS. To generate an electromagnetic field solution,

HFSS employs the finite element method. In general, the finite element method divides the full problem space into thousands of smaller regions and represents the field in each element with a local function. In HFSS, the geometric model is automatically divided into a large number of tetrahedra, where a single tetrahedron is a four-sided pyramid.

This collection of tetrahedra is referred to as the finite element mesh. The governing equations, Maxwell’s equations, are shown as follow:

t

where E is the electric field, B is the magnetic field , H is the magnetizing field, J is the free current density, D is the electric displacement field, and ρ is the free charge density.

To calculate the S-matrix associated with a structure with ports, HFSS does the following: [86]

1. The structure is divided into a finite element mesh

2. The modes is computed on each port of the structure that are supported by a transmission line having the same cross-section as the port

3. The full electromagnetic field pattern is computed inside the structure, assuming that one mode is excited at a time

4. The generalized S-matrix is computed from the amount of reflection and transmission that occurs

The resulting S-matrix allows the magnitude of transmitted and reflected signals to be computed directly from a given set of input signals, reducing the full 3D electromagnetic behavior of a structure to a set of high frequency circuit parameters.

4.4.1 Simulation of Transformer on a Capillary

For the case of transformer on a capillary, some assumptions are adopted to simplify the complex model and reduce the calculating time. Firstly, on the geometry, the circular cross-sections of the wires and capillary are simplified as a hexagon and a dodecagon, respectively, to reduce the number of elements. On the material property, the relative permittivity, relative permeability and magnetic loss tangent are assumed as frequency-independent constants.

Fig. 4.13 shows the model of transformer on a capillary which consists of two coils, the glass tube and the core. Before discussing the result of simulation, the convergence of simulation is verified. The default criterion used by HFSS to determine the convergence of solution is the maximum magnitude delta S which means the maximum difference of S-Matrix magnitude between two consecutive passes. Fig. 4.14 shows the convergence curve of simulation with the model of transformer on a capillary. The

maximum magnitude delta S decreases when the number of tetrahedra increases, which indicates that the simulation has a converged solution. When the number of tetrahedra reaches 205395, the maximum magnitude delta S is 0.010308, which is a half of 0.02 of the default value. And when the number of tetrahedral reaches 255112, the maximum magnitude delta S is 0.00185, which is a tenth of the default value. Considering the solving efficiency, the model is meshed into about 220000 tetrahedra.

Fig. 4.15 shows the simulated self-inductance of coils of transformers with different magnetic cores, which shows the same tendency in Fig. 4.3. The simulated self-inductance increases linearly with the increase of permeability, and also shows the decrease due to the skin effect and the increase due to the resonance. Fig. 4.16 and Fig.

4.17 show the simulated leakage inductance and coupling coefficient with different magnetic cores. These simulation results are quite agree with the measured results in Fig 4.3, Fig. 4.4 and Fig. 4.5.

Fig. 4.18 shows the simulated resistance of coils of transformers with different magnetic cores. Because of the lag between the magnetization of material and the external magnetic field, the magnetic tangent loss, tanδ, should be considered:

' tan ''

μ

δ = μ (4.13)

The researches of Hrianca et al. [87] and Sutariya et al. [88] are referred to estimate the values of tanδ used in this study. Fig. 4.19 shows the simulated quality factor of coils of

transformers with different magnetic cores. And the quality factor decreases due to the dramatic increase of resistance as the same as the results in Fig. 4.7.

4.4.2 Simulation of MEMS Transformer on a Chip

For the case of MEMS transformer, some assumptions are also adopted to simplify the complex model and reduce the calculating time. On the geometry, the circular cross-section of connecting wires is simplified as a hexagon to reduce the number of elements. On the material property, the relative permittivity, relative permeability and magnetic loss tangent are also assumed as frequency-independent constants.

Fig. 4.20 and Fig. 4.21 show the models of MEMS transformer and the dummy PCB. The convergence curve of simulation with the model of MEMS transformer is shown in Fig. 4.22. The maximum magnitude delta S decreases when the number of tetrahedra increases, which indicates that the simulation has a converged solution. When the number of tetrahedra reaches 39606, the maximum magnitude delta S is 0.0086597, which is less than a half of default value. And when the number of tetrahedral reaches 74443, the maximum magnitude delta S is 0.0021021, which is a tenth of the default value. Considering the solving efficiency, the model is meshed into about 40000 tetrahedra.

Fig. 4.23 shows the simulated self-inductance and leakage inductance of coils of

MEMS transformer with the air core and magnetic core of 1 M Fe3O4 nanofluid.

Comparing to the measured results, the simulated results shows that the parasitical capacitance is underestimated so that a higher resonance frequency is obtained, and the simulated inductances at the frequency of 100MHz are lower than the measured. Fig.

4.24 shows the simulated coupling coefficient of MEMS transformer with the air core and magnetic core of 1 M Fe3O4 nanofluid. With the present of magnetic core of 1 M Fe3O4 nanofluid, the coupling coefficient increases slightly but still poor.

Fig. 4.25 and Fig. 4.26 show the simulated resistance and quality factor of coils of MEMS transformer with the air core and magnetic core of 1 M Fe3O4 nanofluid. The measured resistance is higher than the simulated, and it is speculated that the high resistance results from the bad contact between bonding wires and copper pads. Due to the lag between the magnetization of material and the external magnetic field, with the present of magnetic core of 1 M Fe3O4 nanofluid, the resistance increases and the quality factor decreases.

4.5 Discussions

As shown in Fig. 2.10, it illustrates the characteristic of super-paramagnetism of Fe3O4 nanofluid. Thus, the Fe3O4 nanofluid is applied as the magnetic core to reduce the

material and the external magnetic field, the resistance increases as a function of frequency, which results in a low quality factor. The reasons of lag are speculated as follow:

1. On the macroscopic view, a nanoparticle can be regard as a small magnetic dipole as shown in Fig. 4.27(a). While the external magnetic field is applied, the directions of the magnetic dipoles will turn to the direction of external magnetic field and

1. On the macroscopic view, a nanoparticle can be regard as a small magnetic dipole as shown in Fig. 4.27(a). While the external magnetic field is applied, the directions of the magnetic dipoles will turn to the direction of external magnetic field and