CHAPTER 1 INTRODUCTION
1.3 Objective of the present study
The objective of the present study is to develop an innovative multiple-vibrating fan system capable of providing a large air flow rate without extra power consumption or cost. Thus, such a system could be used to cool electronic devices at a low cost.
Moreover, the system could be used with a finned heat sink to construct a cooling system without increasing either its volume or the number of piezoelectric plates used.
Hence, the cooling system can be applied to electronic devices both compactly and economically.
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CHAPTER 2 DEVELOPMENT OF THE VIBRATING-FAN SYSTEMS
2.1 Characteristics of a piezoelectric fan
A piezoelectric fan is usually made of a piezoelectric plate and a mylar sheet.
The characteristics of a piezoelectric fan include low power consumption, a long lifetime and a simple structure. The foundational construction of the piezoelectric plate will be introduced in this section.
2.1.1 Multilayer piezoelectric plate for a piezoelectric fan
The large deformation of a piezoelectric plate depends on the application of a sufficiently high electric field across it. Unfortunately, the application of high voltage is not preferred in small electronic devices, due to cost, safety and heat dissipation issues. However, piezoelectric materials can be stacked to linearly increase the overall deflection while maintaining the low voltage requirement. As shown in Fig.2-1 [34, 35], the thickness of a piezoelectric plate can be decreased to maintain the electric field at a lower voltage. Then, the thin piezoelectric plates can be stacked together to increase the overall deflection. This structure can be further explained by the use of an electric circuit diagram. A single-layer piezoelectric plate can be represented as a series-connected circuit, in which the electric field depends on the thickness of the piezoelectric plate. However, a multi-layer piezoelectric plate can be represented as a parallel-connected circuit; that is, the electric field across each layer can be maintained at a low value while providing large deflection. According to this design, the deflection of a piezoelectric plate can be increased while maintaining a low input voltage. Table 3 [31] shows the performance of two piezoelectric fans with the same geometric configuration and indicates the effectiveness of decreasing the driving voltage.
2.1.2 Power consumption of a piezoelectric fan
The power consumption of a piezoelectric plate depends on its volume, the input frequency and the input voltage. Fig.2-2 shows the power consumption of a piezoelectric plate operated at 60 Hz. The power consumption increases exponentially as the input voltage is increased. The power consumption reaches 0.4 W when the input voltage is increased to 100 V. Fig.2-3 shows the power consumption of a piezoelectric plate operated at 60 V. The power consumption increases approximately linearly as the input frequency is increased. The power consumption reaches 0.1 W when the input frequency is increased to 100 Hz. Fig.2-4 shows the power consumption of piezoelectric plates with different volumes at a fixed input voltage.
The power consumption increases approximately linearly as the volume is increased.
According to Table 4 [6], a 6.4 mm-wide piezoelectric plate consumes approximately twice the power of a 3.2 mm-wide piezoelectric plate. According to the data shown in this section, the approximate power consumption of the piezoelectric plates can be assessed quickly by the input frequency and the volume, because both show a linear dependence. However, the effect of the input voltage on the power consumption should be measured by the use of a device.
2.1.3 Lifetime of a piezoelectric fan
According to a previous study [36], the lifetime of a piezoelectric actuator depends on its driving voltage, the room temperature and the environmental humidity.
S. Nakamura et al. [37, 38] have performed fatigue tests to acquire the fundamental data necessary for predicting the lifetime of a piggyback piezoelectric actuator operated at 3,000 Hz. The piezoelectric actuator studied was 5.4 mm wide, 6.3 mm long and 0.2 mm thick. The lifetime of a piggyback piezoelectric actuator is defined as the number of driving cycles necessary to produce 95% of the initial displacement
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2.1.4 Structure of a piezoelectric fan
A piezoelectric fan is manufactured by attaching a piezoelectric plate to a cantilever beam made of either metal or plastic (e.g., PVF2, Mylar). When an alternating voltage is applied, the piezoelectric plate expands and contracts in an alternating form at the same frequency as the applied voltage. This results in the vibration of the cantilever beam, which produces air flow. In this manner, the piezoelectric fan is capable of vibrating very quickly with a high operating frequency.
However, a piezoelectric fan should be operated at its resonant frequency. Thus, the piezoelectric fan can be operated at a high ratio of fan tip deflection to power consumption [15] and efficiently produces an oscillating flow.
Because piezoelectric fans are used for electronic cooling and energy saving applications, the important issues are the quantity of the induced air flow and the power consumption. The operating frequency and the amplitude of a piezoelectric fan have a significant influence on the flow rate and power consumption. Thus, the geometry and the materials should be considered carefully, because they directly affect the resonant frequency. Table 6 shows the different geometries and the materials of the vibrating fans used in this research.
2.2 Design of the multiple-vibrating fan system
The multiple-vibrating fan system takes advantage of the magnetic force and the
piezoelectric force to operate in high performance applications. However, a previously investigated piezoelectric fan does not include the component for taking advantage of the magnetic force. Thus, the vibrating fans used in this research should be designed differently from the previous piezoelectric fan for the application of magnetic force to the system.
2.2.1 Piezoelectric fan of the multiple-vibrating fan system
Fig.2-5 shows a schematic view of the piezoelectric fan used in this research. A cantilever beam made of phosphor copper or plastic is attached to a piezoelectric plate, as in the previous piezoelectric fan. The only difference between this piezoelectric fan and the previous one is the circular magnet incorporated on the fan tip. The magnet can be attached on the tip of the fan either by physical or chemical methods means that the magnetic force can be applied with the direction of motion while the piezoelectric fan is vibrating.
2.2.2 Magnetic fan of the multiple-vibrating fan system
Fig.2-6 shows another vibrating fan used in this work, named the magnetic fan.
The magnetic fan is compeletly made of phosphor copper or plastic. A cylindrical magnet is attached on the tip of the fan such that the magnetic force can be applied to the fan by the magnet while the other magnets also move toward the magnet. This fan has a simple structure and and can be fabricated at a low cost. The geometries of the fans used in each case are shown in Table 6.
2.2.3 Structure of the multiple-vibrating fan system
Fig.2-7 shows a schematic view of the multiple-vibrating fan system. Four
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magnetic fans (No.2、No.3、No.4、No.5) and one piezoelectric fan (No.1) are used in this system. The pole of a given magnet is opposite that of the adjacent magnets. Thus, all magnetic forces applied to these magnets are repulsive. The resonant frequency of each fan should be adjusted to the same value by varying the geometries of the fans.
Thus, the fans can be operated at the highest ratio of fan tip deflection to power consumption [16]. By this design, the magnetic fans can receive the force applied by the piezoelectric fan through the magnets on their fan tips. Thus, these fans are able to vibrate simultaneously with the vibrating piezoelectric fan in the middle.
2.3 Force analysis of the multiple-vibrating fan system
The interactive magnetic forces in the multiple-vibrating fan system are very complex, hence it is necessary to simplify the forces to more easily analyze the system.
Toward this end, an experimental plate, as shown in Fig.2-8, was used. In the figure, a magnetic fan is fixed on a clamp end that can be moved along the moving direction.
Thus, the vibrating fans can be easily adjusted to the desired position, and the influence of the distance between the magnets on the interactive magnetic forces can be readily investigated.
2.3.1 Magnetic forces between the vibrating fans
Fig.2-9 shows the interactive magnetic forces between the fans. This is a complex system in which the magnetic forces affect one another. The problem is how to simplify the magnetic forces such that the system can be more easily analyzed.
Equation (2.1) expresses the magnetic force between the two cylindrical magnets [39], as shown in Fig.2-10, where B0 is the magnetic flux density, A is the area of the magnets, t is the thickness of the magnet, r is the radius of the magnet, x is the distance between two magnets, and µ0 is the permeability of the intervening medium.
F '?x J KLMNORSPNM?6Q6N NT K;UNV?;QW6U NX?;Q6W NT (2.1)
According to Equation (2.1), the magnetic force depends on the characteristics of the magnets as well as the distance between two magnets. Fig.2-11 shows that FU(W is the magnetic force between magnets No.1 and No.2. Similarly, FU(Y is the interactive force between magnets No.1 and No.3. Thus, the total magnetic force applied to magnet No.1 can be expressed as Equation (2.2),
∆F ',U?x J FW(U?x V FY(U?x V FZ(U?x V F[(U?x (2.2)
Each applied force in Equation (2.2) can be calculated from Equation (2.1). The only unknown parameter is the distance between the two magnets. In this study, the distance between two adjacent magnets is set at a constant value. When the distance is set at 10 mm, XY(U is twice the value of XW(U. The ratio of FW(U to FY(U is calculated as 11.43, so the magnetic force, FY(U, can be eliminated. The condition is the same for F[(U. Thus, Equation (2.2) can be simplified to Equation (2.3).
∆F ',U?x J FW(U?x V FZ(U?x (2.3)
By means of this elimination, the magnetic force applied to No.3 can be simplified to Equation (2.4), as follows:
∆F ',Y?x J FY(W?x (2.4)
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After this procedure, there are only four effective magnetic forces in existence in this system. Thus, the complex system can be simplified as shown in Fig.2-11.
2.3.2 Bending force of a magnetic fan
Because the magnetic fan is bent by the repulsive magnetic force, it can be seen as a fixed-end beam in which a concentrated load can be applied at any point, as shown in Fig.2-12. The relationship between the maximum deflection (A ) and the repulsive force (F ') can be expressed as Equation (2.5) [40].
A J \]^_` N?3L X a (2.5)
In this equation, F ' is the force applied to the fan, L is the length of the magnetic fan, I is the area moment inertia of the fan, E is the Young’s modulus of the fan, A is the amplitude of the fan and a is the arm of the repulsive force. By substituting I J 1 12⁄ WHY and a J 0.93L into Equation (2.5), it can be simplified to Equation (2.6).
A J 3.58 `n5\]^lmm (2.6)
From this equation, it can be understood that the thickness and the length of the magnetic fan are the main factors that influence its deflection.
2.4 Study of the resonant frequency on the multiple-vibrating fan system
Generally, the best operating frequency of a vibrating fan is its resonant frequency. This condition minimizes the power consumption of a vibrating fan while
providing the maximum amplitude [16]. The multiple-vibrating fan system takes advantage of the resonant frequency to improve the performance of the system. The key aspect of this approach is that the fans must be precisely manufactured to ensure that their resonant frequencies are the same. In this manner, each fan is able to operate in its best performance.
2.4.1 Resonant frequency of a magnetic fan
The resonant frequency of the magnetic fan depends on the length, the thickness, the width and the end mass of the fan, as well as the repulsive force produced by the magnet on its tip. The motion of the magnetic fan can be represented as that of an end-loaded cantilever beam and its stiffness can be expressed by Equation (2.7) [41].
K " J `n5Y.[olmm (2.7)
In this equation, W is the width, H is the thickness, and L is the length of the fan. The
effective mass of the cantilever beam can be approximated as m"++. For the magnetic fans, the effective mass is equal to m 'V 0.24m [41], where m is the mass of However, the magnetic force introduces an additional stiffness within the system.
Thus, this further effect on the resonant frequency of the magnetic fan should be
15 can be given as a function of the distance between the magnets, as shown in Equation (2.10).
K '?x J |##;\]^| (2.10)
By inserting Equation (2.1) into Equation (2.10), K '?x can be expressed as Equation (2.11).
K '?x J KLMNORSPNM?6Q6N NT }X;WmX?;QW6W mV?;Q6Z m~ (2.11)
According to Fig.2-11, the magnetic force applied to the No.2 fan can be expressed as Equation (2.12).
∆F ',W?x J FW(U?x V FW(Y?x) (2.12)
Thus, the additional magnetic stiffness applied to fan No.2 can be expressed as
Equation (2.13).
∆K ',W?x J |KLMNORSPNM?•Q€6N NT K•X;Nw‚mW X?;Nw‚WQW6mV?;Nw‚ZQ6mƒ X
•X;NwmmW X?;NwmWQW6mV?;NwmZQ6 mƒT| (2.13)
However, the additional stiffness applied to fans No.3 and No.5 is different from that of fans No.2 and No.4, because they are each affected by only one magnet. Thus, the additional stiffness of No.3 can be expressed as Equation (2.14).
∆K ',Y?x J |KLMNORSPNM?6Q6N NT •X; W
NwmmX?; W
NwmQW6mV?; Z
NwmQ6 mƒ|(2.14)
The additional stiffness induced by the magnetic force is positive for the repulsive force and negative for the attractive force. However, all interactive magnetic forces in this system are repulsive forces such that the magnetic forces always contribute a positive stiffness to the total stiffness. With the added effect of the magnetic stiffness, the total stiffness of the fan can be expressed as Equation (2.15).
K"++ J K " V ΔK ' (2.15)
Thus, the resonant frequency is a function of the beam stiffness and the additional stiffness. The resonant frequency can be expressed as Equation (2.16).
*"++ JW„U q \]^…Q%.WZt†† s (2.16)
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2.4.2 Resonant frequency of a piezoelectric fan
A piezoelectric fan is manufactured by bonding a piezoelectric plate to a cantilever beam, which is made of metal or plastic; hence, the characteristics of the cantilever beam and the characteristics of the piezoelectric plate both influence its resonant frequency. Thus, the method for calculating the resonant frequency becomes very complex. It was found that the resonant frequency of a vibrating plate is primarily affected by its length, as shown in Equation (2.17) [42].
*€ ∝l5NqUWˆ?U(‰` N (2.17)
Where ρ, σ, E, L and H are the density, the Poisson’s ratio, the Young’s modulus, the length and the thickness of the cantilever beam, respectively. However, this formula is for an ideal cantilever beam with a perfectly fixed end. It cannot be used to calculate the resonant frequency of the piezoelectric fan directly because its end is not perfectly fixed, as shown in Fig.2-13. Instead, Yoo et al. [24] proposed Equation (2.18) to calculate the theoretical resonant frequency of a piezoelectric fan.
*€ J Gl5NqUWˆ?U(‰` N (2.18)
This equation is only concerned with the characteristics of the cantilever beam. The effects of the bonding glue and the characteristics of the piezoelectric plate are attributed to the factor G. In this equation, G and *€ are unknown numbers and the other numbers are constants. Although the factor G is a variable for many reasons, it can be calculated experimentally. Table 7 [24] shows the experimentally measured resonant frequencies of the fans, as shown in Fig.2-14, and the corresponding values
calculated by using Equation (2.18) with the factor G = 0.558. The results show the precision of this equation.
Table 8 shows the measured values from this experimental work. The factor G was calculated for a piezoelectric fan measuring 50 mm in length (basement). It was found that the error in the factor G was less than 6% when L is between 60 mm and 40 mm. However, the error was increased to 27.11% for the fan measuring 25 mm in length. The situation is the same when other fans are used, such as a plastic fan. This result implies that the equation can only be used to assess the resonant frequency of a piezoelectric fan with small variations in length.
2.4.3 Resonant frequency of the multiple-vibrating fan system
Because the system operates in a high performance mode by taking advantage of the resonance effect, each fan in this system should be actuated at its resonant frequency. However, there is only one piezoelectric plate used in this system, which means that the operating frequency of the system is a single number. Thus, the fans in this system should be precisely manufactured to ensure that their resonant frequencies are the same. Fig.2-15 shows both systems with uniform resonant frequency and non- uniform resonant frequency. The system with uniform resonant frequency is an ideal model. Each fan operates at its resonant frequency, 36.7 Hz. Thus, the system can operate at its optimal performance. However, the system with non-uniform resonant frequency is the normal condition, because the resonant frequencies of each fan are affected by the magnetic force, which adds additional stiffness. This non-uniform resonant frequency condition results in a reduced performance, because the fans cannot vibrate at their individual resonant frequencies. Although the issue reduces the performance of the system, the overall performance is still much better than that of a single piezoelectric fan.
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2.5 Method to modulate the resonant frequency of vibrating fans
According to Equation (2.19), six parameters influence the resonant frequency of a beam. frequency of a piezoelectric fan or a magnetic fan. A frequency modulator was added to the fan as shown in Fig.2-16. The frequency modulator is a movable mass, and the effective mass can be calculated by the Rayleigh Method.
The maximum kinetic energy of the frequency modulator shown in Fig.2-16 is given by Equation (2.20), where y?zA is the transverse velocity and depends on the deflection of the beam, and where m 2# is the mass of the frequency modulator.
T ;J UWm 2#Šy?zA ‹W (2.20)
The force deflection formula for this beam can be expressed as Equation (2.21).
y?z J \]^_`ŒN?3a X z (2.21)
By substituting y ; into Equation (2.21), y(z) can be demonstrated as Equation (2.22).
y?z J y ; NN?Y (?Yl( (2.22)
Where y ;J \]^_` N?3L X a is the deflection of the fan tip. A similar relationship can be written for velocity and is given as Equation (2.23).
y?xA =y ;A ŒNN?Y (Œ?Yl( (2.23)
By substituting Equation (2.23) into Equation (2.20), the maximum kinetic energy of the frequency modulator can be demonstrated as Equation (2.24).
T ;J UWm 2#?ŒNN?Yl( ?Y (Œ Wy A ;W (2.24)
Thus, the equivalent mass of the frequency modulator at the fan tip is then given by Equation (2.25).
m 2#,"3 J m 2#?ŒNN?Yl( ?Y (Œ W (2.25)
This can be attributed to the effective mass of the fan as shown in Equation (2.26).
*" ∝ WRU •lm??ŽN `n5m
]N?m]wŽ
?m•w] N \•‘Q \]^Q%.WZ s (2.26)
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By substituting a = 0.93L into Equation (2.26), then Equation (2.26) can be simplified to Equation (2.27).
*" ∝ WRU •lm??ŽN?N.’•wŽ‚.’•m N \•‘`n5mQ \]^Q%.WZ s
(2.27)
By moving the frequency modulator, the resonant frequency of the fan can be adjusted to the value desired.
CHAPTER 3 THEORETICAL MODEL OF A PIEZOELECTRIC FAN COOLING SYSTEM
In this chapter, the thermal analysis method for a piezoelectric fan and the multiple-vibrating fan cooling system will be introduced. Additionally, a simulation model used for calculating the flow field will be presented.
3.1 Theoretical model for a single piezoelectric fan cooling system
First, a method for analyzing the convective heat transfer ability of a single piezoelectric fan will be presented in this section, because the cooling ability of the multiple-vibrating fan cooling system can be inferred from that of a single piezoelectric fan. To calculate the overall convection heat transfer coefficient, the surface of a finned heat sink was divided into fifteen equal sections. The thermocouples were arranged in each section as shown in Fig.3-1. Equation (3.1) shows the energy balance of the heat sink, in which Q5 is the input heat from the heat source, Q , 26 7 is the total heat dissipated from the fin surfaces by convection, and Q , is the heat dissipated from each section.
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overall convection heat transfer coefficient. Thus, the overall convection heat transfer coefficient can be computed by using Equation (3.3).
h J O—˜™–—˜™,š•›]œ? •( ž (3.3)
By dividing the bottom and the side fin surface into five equal sections, as shown in Fig.3-1, the central temperature of a section can be assumed to be the average temperature of the section. Thus, the average temperature of the inner surface can be
By dividing the bottom and the side fin surface into five equal sections, as shown in Fig.3-1, the central temperature of a section can be assumed to be the average temperature of the section. Thus, the average temperature of the inner surface can be