CHAPTER 2 DEVELOPMENT OF VIBRATING FAN SYSTEMS
2.4 Study of the resonant frequency on the multiple-vibrating fan system
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 derived by using the weighting method, as shown in Equation (3.4),
T8 ＝! ∑¡v¢‚ s]Ÿt,v∗Os]Ÿt,vQW∗∑¡v¢‚ Ÿv‘t,v∗OŸv‘t,v
O—˜™ (3.4)
where T !", is the temperature measured by the thermocouples of each section at the bottom fin surface, and T! #", is the temperature measured by the thermocouples of each section at the side fin surface. Additionally, A !", is the area of each section at the bottom fin surface, and A! #", is the area of each section at the side fin surface.
To demonstrate the improvement in the convection ability by using the piezoelectric fan, the dimensionless PZT-convection number ?M is defined as shown in Equation (3.7) to assess the total convection ability.
M J¥8£¤š¥8M J 2 0"# 02¦§"06 2¦ 02"++ 0 "¦6 ¨ 6¥ 6¥" + ¦ number and the Nusselt number can be expressed as Equation (3.8) and Equation (3.9) [43], respectively, as follows:
Re J©O\ªl£¤š (3.8)
Nu J¥8£¤š…l£¤š (3.9)
where ν is dynamic viscosity of air, A is fan amplitude, ω is fan vibrating frequency, k is thermal conductivity of air, and L is the characteristic length of the fan. The characteristic length is chosen by employing the hydraulic diameter of the vibrating fan envelope, as shown in Equation (3.10) [44],
L JW?OZO\n
\Qn (3.10)
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where W is the width of the piezoelectric fan. Additionally, the cooling effect of this cooling system is a combination of the natural convection and the forced convection.
The importance of the natural convection relative to the forced convection (Ri is defined as Equation (3.11) by the Grashof number and the Reynolds number [45].
Ri J®"
-£¤šN (3.11)
By replacing the notations of Gr and Re , Ri can be expressed as
Ri J'¯y8•,£¤š±Nl£¤š( Nžzl°m (3.12)
In Equation (3.12), the characteristic length (L0) is defined as the height of the fin (H ) for buoyancy effect, and β is the expansion coefficient. For an ideal gas, the thermal expansion coefficient β is expressed as 1/T, where T is assumed to be 273 K.
Therefore, Ri can be rewritten as Equation (3.13).
Ri J'¯5©—˜™NOmy8•,£¤š( žz
\Nl£¤šN (3.13)
3.2 Simulation model for a single piezoelectric fan cooling system
Using CFD-GEOM and CFD-ACE+, a three-dimensional, transitional model was built to account for the flow field of a piezoelectric fan cooling system, as shown in Fig.3-2. To simplify the model, the vibrating motion of the piezoelectric fan was defined by an equation possessing a sine wave, expressed as Equation (3.14), in which ω is the vibrating frequency and L is the length of the piezoelectric fan.
y?z JOW\∗ •lŒƒWsin?2πωp (3.14)
The vibration period is divided into ten equal time steps, which can be set as Equation
(3.15).
∆t J U%©U (3.15)
Other major assumptions of the model are as follows:
(1) The stable temperature of the cooling system without a piezoelectric fan is set as the initial condition of simulation.
(2) An isothermal temperature of 340 K is set as the boundary condition of the heat source.
(3) The density and the viscosity of the fluid are constants.
(4) There is no slip condition on the wall.
(5) The effect of gravity is considered.
(6) The inlet and outlet pressures of the control surfaces are fixed pressures.
In the simulation model, the dimensions of each part and the mesh density of the part both play important roles in the accuracy of the simulation results. The use of a suitable mesh density prevents truncation errors and rounding errors. To build a robust model, a mesh independence study was performed.
The dimensions of the air domain may affect the simulation results, because the heat sink is surrounded by air. Therefore, it was necessary to check the influence of the air domain. To do so, the air domain was separated into three parts, including the upstream domain, the downstream domain, and the side domain. Fig.3-2 shows the
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schematic view of the simulation model. Fig. 3-3 shows the pressure field of the downstream domain at different thicknesses. When the thickness exceeds 30 mm, the pressure fields are nearly equivalent. However, when the thickness is 10 mm or 20 mm, the pressure fields are clearly affected by the boundary conditions. Thus, a thickness of 40 mm was chosen for the downstream domain. The pressure fields of the upstream domain at different thicknesses and the side domain at different thicknesses were also determined in the same way. To prevent the occurrence of unusual pressure fields, 20 mm was chosen as the thickness of the upstream domain and 30 mm was chosen as the thickness of the side domain.
After confirming the dimensions of the simulation model, the mesh independence test was performed. The mesh independence test of the model was established based on the heat flux of the heated surface. The results were considered to be mesh-independent when the variance in heat flux was less than 0.01%.
According to the result of the gap air domain test, the variance was only 0.005% when the mesh density was 15 cell/per 4.5 mm. Eight mesh independence tests were carried out to determine the suitable mesh density for the simulation model, including the upstream domain, the downstream domain, the side domain, the gap air domain, the fin height, the fin length, the fin thickness, and the piezoelectric fan thickness.
3.3 Theoretical model for a multiple-vibrating fan cooling system
In this study, the overall heat convection coefficient of the multiple-vibrating fan cooling system, h , , can be assessed by two methods, including a direct method and an approximation method.
3.3.1 Direct method for calculating ¶̅¸¹º,»¼
The cooling ability of a vibrating fan depends on its operating frequency, fan
amplitude and orientation. According to Fig.3-4, the orientations of the magnetic fans are the same as that of the piezoelectric fan. Thus, the cooling ability of a magnetic fan can be considered as that of a piezoelectric fan and can be analyzed by the method presented in section 3.1. There are five vibrating fans in the multiple-vibrating fan system, thus 75 thermocouples are needed to analyze a system with five vibrating fans.
The overall heat convection coefficient of the multiple-vibrating fan cooling system can still be calculated by Equation (3.16), where h , is the overall heat
The overall heat convection coefficient of the multiple-vibrating fan cooling system can be calculated by this equation. However, in application, the set-up is difficult to achieve, because it possesses 75 thermocouples. Thus, an easier method must be investigated.
3.3.2 Approximation method for calculating ¶̅¸¹º,¿¼
As mentioned in section 3.1, the Reynolds number and the Nusselt number can
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be expressed as Equation (3.8) and Equation (3.9), and the correlation between Re and Nu can be expressed as Equation (3.18) [29], which is an equation of an heat convection coefficient of the system can be expressed as Equation (3.21).
h , J… ∑ }CÉJ5ÉJ1 1?Ã\ÈÆÄNÃ\ÈÄ[ VC2•ÂÃ\ÈÅ ƒ~ (3.21)
By means of the approximation method, h , can be estimated depending on the operating frequency and the fan amplitude.
CHAPTER 4 EXPERIMENTAL DESIGN AND APPARATUS
In this chapter, the devices used to drive a piezoelectric fan will be introduced.
The method to observe the characteristics of a piezoelectric fan cooling system, including motion and temperature, also will be introduced. Additionally, the manufacturing process of the multiple-vibrating fan cooling system will be presented.
4.1 Apparatus for a piezoelectric fan cooling system
The apparatus for driving and observing a single piezoelectric fan cooling system and a multiple-vibrating fan cooling system are nearly the same. Further, the manufacturing processes are almost identical. The primary devices and the manufacturing processes will be introduced in the sections below.
4.1.1 Temperature measurement and record
The temperature of the heat sink surface is measured with a type-T thermocouple.
The temperature signal can be recorded in computer by a thermal meter at a rate of 10 times per second. The thermal meter used in this research is an Agilent 34970A, as shown in Fig.4-1. By using this thermal meter, 40 thermocouples can be measured simultaneously.
4.1.2 Signal generator and amplifier
A signal generator (GW Instek SFG-2004 4MHz) as shown in Fig.4-2 is used to drive the piezoelectric fan. The range of operating frequency is between 0.1 Hz and 4 M Hz. However, the range of input voltage is between 2 mV and 10 V so the input voltage is increased to the necessary value by an amplifier, as shown in Fig.4-3 (A. A.
Lab-Systems A-303 High Voltage Amplifier). The input voltage is between 50 V and 100 V, and the operating frequency is between 25 Hz and 50 Hz.
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4.1.3 Finned heat sink
The geometries of the finned heat sink are shown as Fig.4-4. Fifteen thermocouples were adhered to the inner surfaces of the heat sink. Each thermocouple was adhered to the center of a rectangular section as shown in Fig.4-5. A heat source device, which can dissipate between 1 W and 3 W of power, was attached to the outer surface of the heat sink. The outer surface of the heat sink is covered by an insulation block to prevent the dissipation of heat from the outer surface. Thus, the influence of the piezoelectric fan on the inner surface can be verified without perturbation.
4.1.4 High speed camera
In this work, the motion of the piezoelectric fan can be readily observed by a low-cost (less than NT. 40000), high-speed camera (JVC-HM550), because the operating frequencies of the piezoelectric fan and the multiple-vibrating fan system are between 20 Hz and 40 Hz. The camera is able to record the motion of the piezoelectric fan and that of the multiple-vibrating fan system at 300 frames per second. Fig.4-6 shows a screenshot from a movie recorded in the high-speed mode.
No afterimage exists in this picture. However, Fig.4-7, the screenshot from the movie recorded in normal mode shows an afterimage in the picture.
4.1.5 Manufacturing the piezoelectric fan
The piezoelectric plate used in this research was made at APC International, Ltd.
The detailed specifications are shown as Table 9. The geometry of the polyvinyl chloride (PVC) sheet is 60 mm × 20 mm × 0.6 mm. The PVC sheet is manufactured by a CNC machine. The piezoelectric fan is manufactured by attaching the rectangular piezoelectric plate to the PVC sheet. Various methods can be used to bond the sheets, including twin adhesive, bonding glue and adhesive tape. In this experiment, adhesive
tape was chosen, because it is both convenient and reliable. The fixed-end of the piezoelectric fan was fixed by two screws, as shown in Fig.4-8. Fig.4-9 demonstrates that the piezoelectric fan was connected with an adjustable base, which allows the fan to move in both the horizontal direction and the vertical direction.
4.2 Experimental set-up for a single piezoelectric fan cooling system
A diagram of the experimental set-up is shown in Fig.4-10. A signal generator is used to drive the piezoelectric fan and to modulate the frequency of the input current.
A heat source (dummy heater), which is able to dissipate between 1 W and 3 W of power, is attached to the outer surface of the heat sink and is supplied by a DC power supply. The temperature of the thermocouples attached to the heat sink surface is detected by a thermal meter and is recorded by a computer. The single piezoelectric fan cooling system is placed in an acrylic box to prevent perturbation from any external convective currents.
4.3 Experimental set-up for a multiple-vibrating fan cooling system
The only difference between the fans used in the multiple-vibrating fan cooling system and the piezoelectric fans used in all previous studies is the magnet on the tip of the fans. However, the magnet is also the key component, allowing the five fans to vibrate simultaneously. Fig.4-11 shows the multiple-vibrating fan cooling system. A
The only difference between the fans used in the multiple-vibrating fan cooling system and the piezoelectric fans used in all previous studies is the magnet on the tip of the fans. However, the magnet is also the key component, allowing the five fans to vibrate simultaneously. Fig.4-11 shows the multiple-vibrating fan cooling system. A