1. Introduction
Piezoelectric pumps can be used in many fields (e.g., medicine, MEMS and fuel cells) for their advantages of simple structure, high actuation strength, easy miniaturization, no magnetic influence and low noise [
1,
2,
3,
4,
5,
6,
7,
8,
9]. In recent years, with the continuous expansion of the application field of piezoelectric pumps, many new application fields that require piezoelectric pumps to have higher output performance have appeared. Unfortunately, the existing output performance of piezoelectric pumps cannot fully satisfy the application requirements in many emerging fields. Therefore, the output performance of piezoelectric pumps should be urgently improved. Improving the output performance of piezoelectric pumps is a very complex and difficult challenge, especially without increasing the volume and input power of the pump. It is worth noting that increasing the operating frequency is one of the most simple and convenient potential ways to improve the performance of piezoelectric pumps without increasing the volume and input power of the pumps.
However, the high driving frequency will cause considerable problems, seriously decreasing the output performance and efficiency of piezoelectric pump. These considerable problems mainly focus on the following three aspects. The first aspect is that the hysteresis of the check valve at high frequency seriously reduces the output flow of piezoelectric pumps. The hysteresis means that the vibration of check valves lags behind that of the piezoelectric actuator [
10]. Zhang et al. studied the hysteresis of check valves and found that the hysteresis caused the flow rate of piezoelectric pumps to decrease rapidly at high driving frequency [
11]. Since there is a time delay when the fluid in the pump chamber transmits the vibrational mechanical energy, the opening and closing of passive check valves lags behind the vibration of the piezoelectric actuator. The hysteresis of the valves was not obvious at low driving frequency, but it was significant at high driving frequency. Obviously, when the piezoelectric pump sucks liquid, the hysteresis of opening the inlet valve reduces the amount of liquid sucked into the pump chamber, and the hysteresis of closing the outlet valve causes the liquid in the outlet pipe to flow back into the pump chamber. When the pump discharges liquid, the hysteresis of opening the outlet valve reduces the amount of liquid discharged from the pump chamber, and the hysteresis of closing the inlet valve makes the liquid in the pump chamber return to the inlet pipe again. Researchers have been committed to designing valves with high-frequency response to solve this problem [
12,
13,
14,
15,
16]. The existing research in this field has become mature, and the response frequency of some high-frequency valves reachs over 1 kHz. The second aspect is that piezoelectric pumps are prone to severe cavitation at high driving frequency. Zhang et al. [
17] studied the cavitation phenomenon of piezoelectric pumps and found that the cavitation phenomenon became more serious as the driving frequency increased. When the piezoelectric pump sucks a liquid, the volume of the pump chamber becomes larger and the pressure in the chamber decreases, so the gas dissolved in the liquid escapes and produces cavitation. He et al. [
18] studied the harm of cavitation to piezoelectric pumps. Cavitation leads to a decrease in the bulk modulus of the working fluid in the piezoelectric pump, thereby severely reducing the efficiency of the pump. To inhibit cavitation, Kim Gie downregulated the amount of gas in the fluid by presetting the pressure on the fluid [
19]. As suggested from the experimental results, this method is feasible. The third aspect is that some local structures of the piezoelectric pump weaken the high-frequency output performance. For example, the structure and volume of the pump chamber, the structure of the valve hole, and the diameter of the pipeline all have an important influence on the high-frequency output performance of the piezoelectric pump [
20,
21,
22]. The problem of structure weakening high-frequency performance is particularly prominent in valveless piezoelectric pumps. The valveless piezoelectric pump eliminates the effect of the check valve on performance, so it is suitable for working at high driving frequency. However, some local structures of the pump reduce its high-frequency output performance. For example, valveless piezoelectric pumps have poor cut-off properties of inlet and outlet of pump chamber, resulting in low output flow rate and pressure at high frequency. To improve the high-frequency output performance of valveless piezoelectric pumps, some scholars optimized the structure of pipelines and pump chamber. For example, scholars proposed a valveless piezoelectric pump with Cone-shaped tubes [
23], a valveless piezoelectric pump with Y-shape tubes [
24] and a valveless piezoelectric pump with unsymmetrical slope chamber bottom [
25]. The optimization of the local structure of the valveless piezoelectric pump greatly improves the high-frequency output performance.
The mentioned studies have noticeably contributed to enhancing effective frequency of the pump, and the output performance of the pump has been effectively enhanced. Therefore, studying various problems caused by high driving frequency is of great significance to improve the working frequency and the pump performance.
The reciprocating motion of piezoelectric vibrator results in high frequency vibration of liquid in the pump system [
26]. The vibration of the liquid causes its velocity pulsation phenomenon. Moreover, the vibration of the liquid will generate inertial force. The inertial force is regarded as a dynamic load that increases the power loss of the pump system. In this study, the dynamic load of the liquid in the inlet and outlet pipelines is proposed as one of the vital reasons for reducing the output performance of piezoelectric pumps. Since the liquid in the pump system is primarily distributed in the inlet and outlet pipelines, the dynamic load of the liquid in the pipelines largely causes the performance of piezoelectric pumps to decrease at high frequency. To decline the dynamic load of liquid, a piezoelectric pump with two elastic chambers was proposed in the present study. The elastic chamber in this pump is capable of cutting off the rigid connection between the liquid in the pump chamber and the liquid in the pipelines. Thus, the elastic chamber is similar to accumulator in fluid transmission system, so the vibration of the liquid can be effectively reduced. Thus, the velocity pulsation is smoothed, and the dynamic load of the liquid in the pipelines decreases. Obviously, this study uses a method different from those mentioned above to improve the performance of piezoelectric pumps.
We have conducted continuous research on the dynamic load of piezoelectric pumps. In our previous study [
27], we had preliminarily verified that the elastic chamber can reduce the dynamic load. In that paper we proposed a piezoelectric pump with two elastic chambers and studied the effectiveness of the elastic chamber in reducing dynamic load with the needle valve fully open and 50% needle valve opening. In addition, the effect of the elastic chamber height on its function was also studied. In the present paper we focus on the effect of different structural parameters on the performance of a piezoelectric pump with two elastic chambers. The performance of the piezoelectric pump with two elastic chambers was studied by theoretical analyses and experimental tests. Firstly, a mathematical model of a piezoelectric pump with two elastic chambers was established. Based on this model, the performance of the pump and the effect of different structural parameters on the performance were analyzed theoretically. Then we developed prototypes with a range of structural parameters and conducted tests on these prototypes.
2. Theoretical Analysis
To introduce the design concept of the piezoelectric pump with two elastic chambers, the concept of dynamic load should be explained first. According to the structure and working principle of piezoelectric pumps, the piezoelectric pump system can be considered a vibration system and can be simplified as a vibration model in which a piezoelectric actuator drives the liquid to conduct simple harmonic vibration. The inlet and outlet check valves make the liquid in the inlet and outlet pipelines move in one direction, as shown in
Figure 1a. In the present study, the inertial force of the liquid during vibration is termed the dynamic load. The dynamic load refers to an additional load on the piezoelectric actuator, causing an energy loss and reducing the output performance of the piezoelectric pump. Since the liquid mass in the pipelines is larger than the liquid mass in the pump chamber, the dynamic load of the liquid in the pipelines is larger than that of the liquid in the pump chamber. Accordingly, the dynamic load of the liquid in the pipelines primarily restricts the further enhancement of the output performance of the piezoelectric pump.
The vibration theory of mechanical system is followed to analyze the generation mechanism of the dynamic load. When the piezoelectric vibrator is driven by a sinusoidal signal, and the amplitude of the vibrator is assumed as
, the vibration displacement
of the liquid in the pump chamber is defined as [
27]:
The liquid in the pump system is assumed to be incompressible. When the pump is working, the liquid in the pump chamber and the liquid in the outlet pipeline are regarded as rigid connections, so the vibration displacement
of the liquid in the inlet pipeline is:
In Equation (2),
L2 denotes the amplitude of the liquid in the inlet pipeline. If it is assumed that there is no leakage in the pump chamber, all the liquid sucked from the inlet pipeline flows into the pump chamber within each suction cycle. Assume that the cross-sectional areas of the pump chamber and inlet pipeline are
and
, respectively. Thus,
and
satisfy the following equation:
In Equation (3),
is liquid density. Equation (4) can be obtained by substituting Equation (1) and Equation (2) into Equation (3):
Assuming that the mass of the liquid in the inlet pipeline is
,
satisfies the following equation:
In Equation (5),
is the length of the inlet pipeline. According to Newton’s second law, the dynamic load
of the liquid in the inlet pipeline is:
Next, Equation (7) can be obtained by substituting Equations (2), (4) and (5) into Equation (6):
Using a similar derivation process, we can obtain the dynamic load
of the liquid in the outlet pipeline:
In Equation (8),
is the length of the outlet pipeline, so the resultant force of
and
acting on the piezoelectric vibrator is:
Obviously, the amplitude of is proportional to the square of the driving frequency , and proportional to the length of the pipelines and vibrator amplitude . Accordingly, the dynamic load increases quadratically with the driving frequency .
The dynamic load of liquid in the pipelines results from the liquid vibration, so reducing the liquid vibration in the pipelines helps decrease the dynamic load. To decrease the dynamic load in the inlet and outlet pipelines, a piezoelectric pump with two elastic chambers was designed. In brief, the design concept of the piezoelectric pump with two elastic chambers is that an elastic chamber is respectively built outside the inlet and outlet valves to decrease the dynamic load in the pipelines. The elastic chamber can reduce the liquid vibration in the inlet and outlet pipelines and smooth the output flow rate, as shown in
Figure 1b.
The structure of piezoelectric pump with two elastic chambers is illustrated in
Figure 2. The two elastic chambers consist of an inlet elastic chamber and an outlet elastic chamber, respectively located below the inlet check valve and the outlet check valve. The elastic chamber is composed of a chamber below the check valve and an elastic diaphragm.
When the pump with two elastic chambers works, the inlet and outlet elastic chambers block the rigid connection between the liquid in the pump chamber and the liquid in the pipelines. The flow resistance caused by the inlet and outlet check valves is ignored. Subsequently, the simplified vibration model of the pump system can be built, as illustrated in
Figure 3. In the simplified vibration model,
denotes the liquid mass in the pump chamber,
represents the liquid mass in the inlet pipeline and
represents the liquid mass in the outlet pipeline. Elastic element
and damping element
cf2 represent the function of the inlet elastic chamber. Likewise, elastic element
and damping element
represent the function of the outlet elastic chamber.
represents the vibration displacement of the liquid in the pump chamber.
and
respectively represent the vibration displacement of the liquid in the inlet and outlet pipelines.
According to D’Alembert’s principle, a force analysis was conducted on the liquid mass
, and the motion differential equation of the liquid in the inlet pipeline was built, as expressed in Equation (10). For a detailed derivation of Equation (10), please refer to our previous article [
27]:
The vibration displacement
can be obtained from Equation (10):
where:
H2 denotes the amplitude amplification factor,
denotes the frequency ratio,
denotes phase angle, and
denotes the damping ratio.
is the natural frequency of the vibration model (
Figure 3a).
is the natural angular frequency corresponding to
.
According to Newton’s second law and Equation (11), we can obtain the new dynamic load
of the liquid in the inlet pipeline, as shown in Equation (17):
Likewise, the new dynamic load
of the liquid in the outlet pipeline is:
The resultant force of
and
acting on the piezoelectric vibrator is a new dynamic load
. So the new dynamic load
of the piezoelectric pump with the inlet and outlet elastic chambers is:
After comparing and F1, it can be found that is less than F1. So the dynamic loads in the inlet and outlet pipelines and their resultant force on the piezoelectric vibrator are both reduced.
In addition, the dynamic loads of the piezoelectric pump with the inlet elastic chamber and the piezoelectric pump with the outlet elastic chamber are also analyzed. According to Equations (9) and (19), we can easily obtain the dynamic load
of the piezoelectric pump with the inlet elastic chamber and the dynamic load
of the piezoelectric pump with the outlet elastic chamber:
Assuming that the structural parameters of the inlet and outlet pipelines are the same, A2 is equal to A3, and l2 is equal to l3. Assuming that the structural parameters of the inlet and outlet elastic chambers are the same, H2 is equal to H3, and is equal to . After analyzing Equations (20) and (21), it can be found that the power loss caused by is equal to the power loss caused by .
High driving frequency
f makes the frequency ratio far greater than 1, thus making
H2 and
H3 respectively satisfy
and
. So Equation (19) can be modified to:
Equation (22) shows that the elastic chamber almost eliminates the influence of driving frequency f and the liquid mass on the dynamic load. When the piezoelectric pump has high driving frequency f and small elastic stiffness (kf2 and kf3), we can obtain . So the mechanism analysis indicates that the dynamic load is effectively decreased by the elastic chamber.
Finally, the effect of the pump chamber height on the output performance of the piezoelectric pump is analyzed. Assuming that the initial volume of the pump chamber is
V0, and the volume change of the pump chamber in one cycle is
. Based on the definition of liquid bulk modulus in fluid mechanics, when the volume of the pump chamber is compressed, the pressure
P in the pump chamber is:
In Equation (23), βe is the equivalent bulk modulus of the liquid in the pump chamber. Under the condition of keeping other structural parameters of the pump chamber unchanged, reducing the height of the pump chamber will reduce the initial volume V0 of the pump chamber. According to Equation (23), reducing the initial volume V0 will increase the output pressure P of pump chamber, so reducing the height of the pump chamber helps to increase the output pressure P of pump chamber.
When the height of the pump chamber is small, the liquid flow in the pump chamber can be analyzed by using the model of gap flow in flat plate, so the pressure loss
in the pump chamber is:
In Equation (24), μ is the fluid kinematic viscosity coefficient, l is the length of the gap, h is the height of the gap, and v is the average flow rate of the fluid. Equation (24) shows that the pressure loss is inversely proportional to the square of height h. The height h here represents the height of the pump chamber. Therefore, if the pump chamber height is too small, the output performance of the pump will be seriously reduced. In total, the output performance of piezoelectric pump can be improved by properly reducing the height of pump chamber, but the output performance will be reduced if the chamber height is too low.
5. Conclusions
In this study, a piezoelectric pump with two elastic chambers was investigated, in which the inlet and outlet elastic chambers were used to decrease the dynamic load of the liquid in the inlet and outlet pipelines. We have established a mathematical model of the piezoelectric pump with two elastic chambers and theoretically analyzed the performance of the pump and the effect of different structural parameters on its performance. Subsequently, prototypes with different structural parameters were developed and tested. Experimental results are presented to analyze the effect of several key parameters on the output performance of the prototype with two elastic chambers, which helps optimize the pump system.
The elastic chamber decreased the dynamic load of the liquid in the pipelines and improved the flow rate of the prototype. The flow rate of the prototype with two elastic chambers was higher than that of the prototype with one elastic chamber and the prototype without an elastic chamber. The maximal flow rate of the prototype with two elastic chambers was nearly 36% higher than that of the prototype without elastic chamber, however, the elastic chamber did not increase the output backpressure of the pump.
Adopting an elastic diaphragm with smaller stiffness or larger diameter was beneficial to decrease the dynamic load of the liquid and increase the flow rate of the pump. As the length of the inlet and outlet pipelines increased, the dynamic load of the liquid in the pipelines gradually increased, so the output flow rate of the prototype gradually decreased. Long pipelines adversely affected the flow output of piezoelectric pumps under large dynamic load in the pipelines. Compared with conventional piezoelectric pump, the piezoelectric pump with two elastic chambers had better flow output and was more suitable for long pipeline transmission system.
The pump chamber height had an important effect on the output performance of the prototype with two elastic chambers. As the chamber height increased, the flow rate increased first and then declined, but the output backpressure always decreased. At the height of 0.2 mm, the flow rate was peaked at 7.7 mL/min; at the height of 0.05 mm, the output backpressure reached the maximum of 28.2 kPa.
The dynamic load of the liquid in the pipelines produced resistance to the piezoelectric vibrator, reducing the amplitude of the piezoelectric vibrator to a certain extent. However, the prototype with two elastic chambers could effectively reduce the adverse impact of dynamic load on the piezoelectric vibrator. The flow rate decreased almost linearly with the backpressure. The flow rate of the prototype with two elastic chambers was higher than that of the prototype without elastic chamber at the same backpressure, and the flow rate difference between the two prototypes gradually decreased with the backpressure.