6.2 Dual-Servo Stage
7.1.2 Pump Design and Analysis
The schematic confi guration of the piezo pump is shown in Figure 7.1. The dia- phragm is directly connected to the piezoactuator. Thus, the control volume of the pump chamber can be controlled by the dynamic motion of the diaphragm. If the diaphragm is moved in an upward direction, the input fl ow is induced. In this case, the check valve installed in the outlet needs to be closed in order to prevent the output fl ow motion. The output fl ow is induced by activating the diaphragm in a downward direction and simultaneously closing the check valve installed in the inlet.
Consequently, the fl ow rate of the pump can be controlled by controlling the dynamic motion of the piezoactuator. It should be noted that THUNDER piezoactuator [15], which is commercially available, was used in this test.
By applying the continuity and energy equations to the control volume of the piezo pump, the following equation is derived [13,16]:
p p0
in out
d d
d d
V V P
Q Q
t t
− = +
∑ ∑
β (7.1)where
Vp is the volume of the pumping chamber Vp0 is the initial volume of the chamber β is the bulk modulus of the fl uid P is the pressure in the pumping chamber
The fi rst term on the right-hand side of Equation 7.1 is a boundary deformation term, which is prescribed by the motion of the diaphragm, and therefore of the piezoac- tuator. By assuming a harmonic excitation of the piezoactuator with a circular fre- quency, ω, this term can be expressed by
Input flow
Piezoelectric actuator
Diaphragm
Output flow Check valve
Host structure
FIGURE 7.1 Schematic confi guration of the piezo pump.
= Δ ⋅ ω
d p d
( sin )
d d
V V t
t t (7.2)
where ΔV is the volume change due to the diaphragm motion and is given by
π + +
Δ = ( 12 22 1 2)Δ 12
D D D D
V x (7.3)
where
Δx is the maximum displacement of the diaphragm D1 is the external diameter of the diaphragm
D2 is the diameter of the plastic plate attached to the center of the diaphragm Substituting Equations 7.2 and 7.3 into Equation 7.1 yields the following equation:
π + + ⋅ω ⋅ Δ
Δ = − = ω +
β
0
2 2
p
1 2 1 2
in out
( ) d
12 cos d
D D D D x V P
Q Q Q t
t (7.4)
The fl ow motion of the piezo pump is controlled by the check valve. The fl ow passing through the check valve can be expressed by three conditions:
in crack in q in crack
in crack out crack
out crack out q out crack
1. : ( )
2. : 0
3. : ( )
P P P Q Q C P P P
P P P P P Q
P P P Q Q C P P P
< − Δ = = − −
− ≤ < + Δ =
+ ≤ Δ = − = − − +
(7.5)
where
Pin is the pressure in the inlet Pout is the pressure in the outlet
Pcrack is the minimum pressure to open the check valve Cq is the fl ow rate constant of the check valve
In case (1), the pressure of the pumping chamber is lower than that of the inlet, and therefore the fl ow-in occurs by closing the check valve in the outlet. In case (2), both the check valves are closed and hence no fl ow motion occurs. In case (3), the outlet fl ow motion is induced by opening the check valve in the outlet and simultaneously closing the check valve in the inlet. Now, from Equations 7.4 and 7.5, the following equation of motion of the piezo pump can be obtained:
β π + + ωβ ω
= + Δ
− 0
0
2 2
q 1 2 1 2
p atom crack p
d ( ) cos
d 12
C
P D D D D t
P x
t V P P V (7.6)
It is noted that the linearization with respect to the pressure of the chamber has been undertaken to obtain Equation 7.6. By neglecting the nonlinear hysteresis behavior of the piezoactuator, the displacement of the diaphragm, Δx, is linearly related to the control voltage, V(t), to be applied to the piezoactuator as follows:
x cV t( )
Δ = (7.7)
where c is a constant to be experimentally determined. Consequently, the governing equation of the piezo pump is obtained as
1 2
d cos ( )
d
P P t V t
t = φ + φ ω ⋅ (7.8)
where
β π + + ωβ
φ = φ =
− 0
0
2 2
q 1 2 1 2
1 2
p atom crack p
( )
, 12
C c D D D D
V P P V
It is noted that the pressure of the pumping chamber heavily depends on the diameter of the diaphragm and the initial volume of the chamber.
Figure 7.2 presents the simulated maximum pressure of the piezo pump with two different important design parameters: diameter of the diaphragm and initial volume of the chamber. The result shown in Figure 7.2a is obtained by choosing the fol- lowing values: Vp0 =25.132 cm3, Pcrack = 0.02 bar. The maximum pressure sharply decreases as the diaphragm diameter increases. This is due to the decrement of the volume change in the chamber. The result presented in Figure 7.2b is obtained by choosing the following values: D1 = 8 cm, Pcrack = 0.02 bar. The maximum pressure of the chamber is decreased as the initial volume is increased. This is, of course, due to the decrement of the volume change in the chamber.
In this test, the piezo pump is designed by adjusting the maximum pressure to be 2 bar. Figure 7.3 presents the photograph of the manufactured piezo pump. The specifi c material and geometry properties are listed in Table 7.1. Figure 7.4 presents the chamber pressure and the output fl ow rate during one cycle. The inlet pressure is fi xed by 1 bar, while 1.2 bar for the outlet pressure. The output fl ow rate is increased as the chamber pressure increases. The variation of the output fl ow rate is measured by changing the voltage applied to the piezoactuator, and is compared with the simu- lated one in Figure 7.5. The output fl ow rate increases as the voltage increases, as expected. It is also seen that the agreement between the measured and the simulated results is excellent, validating the proposed governing model (7.8) of the piezo pump.
In order to investigate the durability of the piezo pump, it was operated for 40 min by applying the voltage of 220 V. The measured output fl ow rate is shown in Figure 7.6. It is clearly observed that the output fl ow rate almost remains to be constant for the fi rst 10 min, and decreases a little thereafter. This may be caused by certain uncertainties such as the hysteresis behavior of the piezoactuator. However, the
FIGURE 7.3 Photograph of the piezo pump.
1 2 3 4 5 6
Maximum pressure (bar)
D1 (cm) (a)
3 9 12 15
10 20 30 40
0 1 2 3 4 5
Maximum pressure (bar)
V0 (cm3) (b)
6
FIGURE 7.2 The variation of the maximum pressure of the piezo pump. (a) Diaphragm diameter and (b) initial volume.
desired output fl ow rate can be achieved by applying an appropriate control voltage to the piezoactuator in a real-time feedback manner.