Design and Experimental Veriﬁcation of a PZT Pump with Streamlined Flow Tubes

: In this paper, a streamlined ﬂow tube valveless piezoelectric pump (SLFT PZT pump) is proposed to modify the single ﬂow trend and improve the ﬂuid ﬂow stability. Firstly, the structural and working principle of the streamlined ﬂow tube, which accounts for changing the ﬂow trend and improving the ﬂow stability, were analyzed. The ﬂow resistance and ﬂow rate equations were established. Secondly, the pressure and velocity ﬁelds of the tube were simulated. These simulated results were consistent with the theoretical results. Thirdly, the ﬂow resistance of the ﬂow tube was tested with pressure di ﬀ erences of 1000 Pa, 1200 Pa, 1400 Pa and 1600 Pa respectively. The trend of the result curves was consistent with the simulated results. The amplitude-frequency relationship and the ﬂow-rate-frequency relationship were also tested, both result curves highly corelate. The maximum amplitude was 0.228 mm (10 Hz, 120 V), and the maximum ﬂow rate was 17.01 mL / min (10 Hz, 100 V). Finally, the theoretical ﬂow rate of the SLFT PZT pump was calculated at 100 V and 120 V. These results roughly ﬁtted with the experimental results. The streamlined ﬂow tube could change the internal ﬂow trend that remarkably improved the ﬂow stability. Therefore, it promoted the application of the valveless PZT pump in living cells, biomedical and polymer delivery.


Introduction
As a functional material, piezoelectric ceramics play an irreplaceable role in modern intelligent industry [1][2][3]. Piezoelectric ceramics have been widely used in modern intelligent devices, such as pressure sensors [4], vibration energy collecting devices [5], intelligent wearable devices with human-computer interaction [6], and so on. At the same time, with the advent of MEMS (Micro-Electro-Mechanical System), the inverse piezoelectric driving technology that converts electrical energy into mechanical energy has become a research hotspot in research concerning modern micromachines [7,8]. The actuator has not only decreased in size but also enhances conversion efficiency in ultrasonic motors [9], ultrasonic transducers [10], piezoelectric pumps [11], and so on. Among them, the piezoelectric pump is further divided into valve piezoelectric pumps and valveless piezoelectric pumps.
As an emerging piezoelectric-driving-fluid technology, the valveless piezoelectric (PZT) pump has shown ever-increasing application potential in aerospace, biomedical, microfluidics and sensor fields [12][13][14]. It has many advantages when compared with traditional motor pumps, like its simple structure, compact size, lack of electromagnetic interference and easy integration,

Structural Principle
To simplify the principle, three sections were defined for the streamlined flow tube, as shown in Figure 1.

Structural Principle
To simplify the principle, three sections were defined for the streamlined flow tube, as shown in Figure 1. When the fluid flows in the streamlined flow tube, the main reason for the backflow vortex is the singular flow trend of boundary layer flows, which leads to the separation of boundary layer flows [33]. Therefore, the flow trend should be fully considered when designing the flow tube. The flow tube designed consists of two parts: divergence and contraction parts. Here, forward flow direction is defined in Sections 1.1-3.3, and backward flow direction is in Sections 3.3-1.1, as shown in Figure 1. Due to fluid flows forward through the tube (positive x axis), the pressure gradient changes in the boundary layer in the x direction are much larger than that in normal direction (y axis). Therefore, the pressure gradient in normal direction is neglected, and the Navier-Stokes equation can be simplified as [34,35] Since the flow area from Sections 1.1-2.2 gradually increases, and the pressure gradient constantly increases, then the boundary layer equation with different pressure gradients is When the cross-sectional area of the fluid gradually increases, it is bound to cause the pressure to increase while the flow rate decreases. According to Equations (1) and (2), it can be seen that the boundary layer fluid should satisfy the following conditions when flow area gradually increases According to Formula (3), for a single divergent flow, the flow velocity would gradually decrease, then the boundary layer thickness would increase. When the velocity is reduced to zero, there are backflow vortexes to disturb fluid delivery.
On the contrary, when flow area gradually decreases, the boundary layer fluid should satisfy the follow expression as When the fluid flows in the streamlined flow tube, the main reason for the backflow vortex is the singular flow trend of boundary layer flows, which leads to the separation of boundary layer flows [33]. Therefore, the flow trend should be fully considered when designing the flow tube. The flow tube designed consists of two parts: divergence and contraction parts. Here, forward flow direction is defined in Sections 1.1-3.3, and backward flow direction is in Sections 3.3-1.1, as shown in Figure 1. Due to fluid flows forward through the tube (positive x axis), the pressure gradient changes in the boundary layer in the x direction are much larger than that in normal direction (y axis). Therefore, the pressure gradient in normal direction is neglected, and the Navier-Stokes equation can be simplified as [34,35]  Since the flow area from Sections 1.1-2.2 gradually increases, and the pressure gradient constantly increases, then the boundary layer equation with different pressure gradients is When the cross-sectional area of the fluid gradually increases, it is bound to cause the pressure to increase while the flow rate decreases. According to Equations (1) and (2), it can be seen that the boundary layer fluid should satisfy the following conditions when flow area gradually increases According to Formula (3), for a single divergent flow, the flow velocity would gradually decrease, then the boundary layer thickness would increase. When the velocity is reduced to zero, there are backflow vortexes to disturb fluid delivery.
On the contrary, when flow area gradually decreases, the boundary layer fluid should satisfy the follow expression as ∂p ∂x < 0 ∂ 2 u y According to Formula (4), for a single convergent flow the internal pressure decreases, then the flow velocity increases and the boundary thickness decreases. Therefore, the flow tube structure designed should be able to change the fluid flow trend, whose principle should obey both Equations (3) and (4), for the improvement of flow stability.

Working Principle
During fluid flowing in the forward direction, the initial pressure of the flow tube is P, the initial velocity is u, the radius of the flow tube at Section 2.2 is R, then the velocity and pressure are expressed as where x is the length of the flow tube, and y is the flow area radius. Therefore, according to the formulas (1)-(4), the relationship between the pressure change and the velocity are where L 1−2 is the length from Sections 1.1-2.2, L 2−3 is the length from Sections 2.2-3.3. During backwards fluid flow, similarly, the coordinate system is established with the Section 3.3 as the origin, the relationship between the pressure change and the velocity are According to Formulas (6) and (7), whether fluid flows forward or backward, the divergent or convergent trend is similar. For example, when fluid flows forward, the fluid begins with an adverse pressure gradient from Sections 1.1-2.2, where the internal pressure of the flow field increases while the velocity decreases. Then goes through a consequent pressure gradient from Sections 2.2-3.3, where the pressure decreases while velocity increases. Similarly, when it flows backwards, the adverse pressure gradient appears in Sections 3.3-2.2 and the consequent pressure gradient appears in Sections 2.2-1.1. The internal velocity and pressure change are opposite to the barotropic gradient. By changing the fluid flow trend, it is theoretically guaranteed that the velocity of the internal boundary layer will not appear to be zero, avoiding the occurrence of backflow and vortex, thereby improving the flow stability.
When the piezoelectric vibrator is oscillating periodically, the volume of the pump chamber changes correspondingly. Scholars have divided this periodic change into two stages: one is the suction phase (suction stroke) when the internal volume of the pump chamber increases from minimum to maximum. The other one is the discharge phase (compression stroke) when the internal volume reduces from maximum to minimum. In both the suction and compression strokes, the flow trend changes in the streamlined flow tube of the SLFT PZT pump are always opposite to each other. Thus, at any stage, there are always obvious characteristics of unequal flow resistance, which can realize macroscopic fluid transportation.
The three-dimensional model of the SLFT PZT pump, shown in Figure 2, was fabricated by Sterolithography Apparatus (iSLA660, ZRapid, Suzhou, China). The size parameters of the pump are shown in Table 1. The resonance impedance and resonance frequency of the piezoelectric vibrator (Dong Guan Cosson Electric Co, LTD, Dongguan, China) adopted in this paper are 1.9 ± 0.5 KHz and 400 Ω respectively. The size parameters of the piezoelectric vibrator are shown in Table 2. (Dong Guan Cosson Electric Co, LTD, Dongguan, China) adopted in this paper are 1.9 ± 0.5 KHz and 400 Ω respectively. The size parameters of the piezoelectric vibrator are shown in Table 2.

Theoretical Analysis
In order to determine the internal pressure loss, under the conditions of mass conservation, the change of internal mass flow is independent on the flow tube size. Integrating the streamwise equation over a unit span control volume extending from 0 to x yields where is the density, is the section averaged pressure, is the wall shear stress, and u is the streamwise x velocity, x is the length of the flow tube, Q is the flow rate [36][37][38].
For a sinusoidal oscillation with angular frequency .

Theoretical Analysis
In order to determine the internal pressure loss, under the conditions of mass conservation, the change of internal mass flow is independent on the flow tube size. Integrating the streamwise equation over a unit span control volume extending from 0 to x yields where ρ is the density, p is the section averaged pressure, τ w is the wall shear stress, and u is the streamwise x velocity, x is the length of the flow tube, Q is the flow rate [36][37][38]. For a sinusoidal oscillation with angular frequency ω. Q(t) = Q max sin(ωt), Therefore, pressure loss can be expressed as is the skin friction coefficient, q j = 0.5ρU 2 j , ∆p u is the unsteady, linear inertia term, ∆p s is the viscous loss, ∆p m is the nonlinear minor loss.
According to the pressure loss equation where ρ is the fluid density, ξ is the flow resistance coefficient, and u is the average flow velocity. Therefore, the flow resistance coefficient in the forward flow was derived by Equations (8) and (9), as follow where ∆P is the forward flow pressure loss, u is the forward flow average velocity. Similarly, the flow resistance coefficient in the backwards direction is where ∆P is the backward flow pressure loss, u is the average velocity of backwards flow. When the vibrator is working, its cross section likes a paraboloid as shown in Figure 3. is the Strouhal number, is the inlet or outlet diameter, = is the skin friction coefficient, , ∆ is the unsteady, linear inertia term, ∆ is the viscous loss, ∆ is the nonlinear minor loss.
According to the pressure loss equation where is the fluid density, is the flow resistance coefficient, and is the average flow velocity. Therefore, the flow resistance coefficient in the forward flow was derived by Equations (8) and (9), as follow where Δ is the forward flow pressure loss, is the forward flow average velocity. Similarly, the flow resistance coefficient in the backwards direction is where Δ is the backward flow pressure loss, is the average velocity of backwards flow. When the vibrator is working, its cross section likes a paraboloid as shown in Figure 3. The equation of the working section is where R is the working radius of the piezoelectric vibrator, is the maximum amplitude of the vibrator, is the constant which is related to the size and properties of the piezoelectric vibrator, is the piezoelectric constant, ( ) is the driving voltage, ℎ is the thickness of piezoelectric vibrator [39,40].
Therefore, the volume change [41,42] caused by the PZT vibrator is The equation of the working section is where R is the working radius of the piezoelectric vibrator, ω 0 is the maximum amplitude of the vibrator, K is the constant which is related to the size and properties of the piezoelectric vibrator, d 31 is the piezoelectric constant, V (ac) is the driving voltage, h is the thickness of piezoelectric vibrator [39,40]. Therefore, the volume change [41,42] caused by the PZT vibrator is According to the relationship between volume change, flow rate and frequency [43,44], the flow rate equation of the SLFT PZT pump is where Q V is the pumping flow rate, and f is the driving frequency.

Flow Field Simulation
The 3D model of the fluid area was established by Solidworks (Solidworks2016, Dassault Systemes S.A, Waltham, MA, USA), and imported into Design Modeler (ANSYS, Canonsburg, PA, USA) to mesh. The meshing model was swept with a size of 0.1 mm, and the grids obtained were about 584,664 nodes and 569,800 elements. Finally, FLUENT (Fluent 18.1, ANSYS, Canonsburg, PA, USA) was used to conduct fluid simulation. The inlet and outlet pressure were set as 1500 Pa and 0 Pa respectively, during simulating fluid flow in forward direction. Vice versa in backward direction. The steady-state with k − ε calculation model was used; and the fluid medium was room-temperature liquid water.
The pressure distribution and velocity streamline diagram of the streamlined flow tube are shown in Figures 4 and 5, respectively. It can be seen that no matter whether the fluid flowed in the forward or backward direction, the pressure distribution inside the flow tube began with increase, and then decrease. The trend of the simulated pressure results was basically consistent with that of the theoretical analysis. The trend of velocity change in the flow field was opposite to the trend of pressure change. In Figure 5, the flow rate decreased first and then increased, no matter whether the simulation was in forward or backward flow. The relationship between pressure and velocity satisfies the Bernoulli equation, and was consistent with the theoretical analysis. The simulation results of the backward flow show that a small number of vortices appeared near the maximum flow zone (Figure 5b), but there was no obvious vortex phenomenon in the forward flow ( Figure 5a). It shows that the problem of vortex can be alleviated by changing the flow tendency of the streamlined flow tube.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 15 According to the relationship between volume change, flow rate and frequency [43,44], the flow rate equation of the SLFT PZT pump is where is the pumping flow rate, and is the driving frequency.

Flow Field Simulation
The

Flow Resistance Experiment
The flow resistance of the streamlined flow tube was measured under different pressure differences in both the forward flow direction and backward flow direction. The schematic diagram is shown in Figure 6. The left orifice of the flow tube was connected to the water storage by a silicone tube, and the water clip cut off water without obvious silicone tube deformation. The beaker was placed on the balance to collect the water outflowing from the other orifice. The pressure difference was changed by changing the height (H) between the outlet of the streamlined tube and the liquid level of the water storage, and the total outflow mass was recorded within one minute when the water clip was removed.

Flow Resistance Experiment
The flow resistance of the streamlined flow tube was measured under different pressure differences in both the forward flow direction and backward flow direction. The schematic diagram is shown in Figure 6. The left orifice of the flow tube was connected to the water storage by a silicone tube, and the water clip cut off water without obvious silicone tube deformation. The beaker was placed on the balance to collect the water outflowing from the other orifice. The pressure difference was changed by changing the height (H) between the outlet of the streamlined tube and the liquid level of the water storage, and the total outflow mass was recorded within one minute when the water clip was removed.

Flow Resistance Experiment
The flow resistance of the streamlined flow tube was measured under different pressure differences in both the forward flow direction and backward flow direction. The schematic diagram is shown in Figure 6. The left orifice of the flow tube was connected to the water storage by a silicone tube, and the water clip cut off water without obvious silicone tube deformation. The beaker was placed on the balance to collect the water outflowing from the other orifice. The pressure difference was changed by changing the height (H) between the outlet of the streamlined tube and the liquid level of the water storage, and the total outflow mass was recorded within one minute when the water clip was removed.

Amplitude and Flow Rate Measurement
The fluid medium used in the experiment was room-temperature liquid water. The main experimental equipment included a function signal generator (AFG1062, Tektronix, Beaverton, WA,
The amplitude of the vibrator was measured by the laser displacement sensor by focusing the laser on its center, as shown in Figure 7. The amplitude results, which would be displayed on a monitor screen after procession by the host, would vary with different applied voltage and frequency, which were adjusted using the signal generator and power amplifier. At the same time, during the amplitude test of the piezoelectric vibrator, liquid was injected into the valveless PZT pump through a syringe, to eliminate the influence of internal bubbles on the experiment. laser on its center, as shown in Figure 7. The amplitude results, which would be displayed on a monitor screen after procession by the host, would vary with different applied voltage and frequency, which were adjusted using the signal generator and power amplifier. At the same time, during the amplitude test of the piezoelectric vibrator, liquid was injected into the valveless PZT pump through a syringe, to eliminate the influence of internal bubbles on the experiment.
The flow rate was measured using the weight increase of the beaker on the electronic balance within one minute, during pumping fluid from the left beaker to the right using the PZT pump, as shown in Figure 8. Before the flow rate measurement, we ensured that the liquid levels of the two beakers were the same using the siphon effect, and that there were no bubbles in the tube. The actual flow rate measuring system is shown in Figure 9. To ensure the accuracy of the measurement, we quickly injected the liquid into the valveless PZT pump to elimainate the internal bubbles.    The flow rate was measured using the weight increase of the beaker on the electronic balance within one minute, during pumping fluid from the left beaker to the right using the PZT pump, as shown in Figure 8. Before the flow rate measurement, we ensured that the liquid levels of the two beakers were the same using the siphon effect, and that there were no bubbles in the tube. The actual flow rate measuring system is shown in Figure 9. To ensure the accuracy of the measurement, we quickly injected the liquid into the valveless PZT pump to elimainate the internal bubbles.

Results and Discussion
Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 15 USA), a power amplifier (HVP-300D, NJFN, Nanjing, China), an oscilloscope (DSO-X2004A, Keysight, Santa Rose, CA, USA), and a laser displacement sensor (LK-H020, Keyence, Osaka, Japan). The amplitude of the vibrator was measured by the laser displacement sensor by focusing the laser on its center, as shown in Figure 7. The amplitude results, which would be displayed on a monitor screen after procession by the host, would vary with different applied voltage and frequency, which were adjusted using the signal generator and power amplifier. At the same time, during the amplitude test of the piezoelectric vibrator, liquid was injected into the valveless PZT pump through a syringe, to eliminate the influence of internal bubbles on the experiment.
The flow rate was measured using the weight increase of the beaker on the electronic balance within one minute, during pumping fluid from the left beaker to the right using the PZT pump, as shown in Figure 8. Before the flow rate measurement, we ensured that the liquid levels of the two beakers were the same using the siphon effect, and that there were no bubbles in the tube. The actual flow rate measuring system is shown in Figure 9. To ensure the accuracy of the measurement, we quickly injected the liquid into the valveless PZT pump to elimainate the internal bubbles.    Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 15 USA), a power amplifier (HVP-300D, NJFN, Nanjing, China), an oscilloscope (DSO-X2004A, Keysight, Santa Rose, CA, USA), and a laser displacement sensor (LK-H020, Keyence, Osaka, Japan). The amplitude of the vibrator was measured by the laser displacement sensor by focusing the laser on its center, as shown in Figure 7. The amplitude results, which would be displayed on a monitor screen after procession by the host, would vary with different applied voltage and frequency, which were adjusted using the signal generator and power amplifier. At the same time, during the amplitude test of the piezoelectric vibrator, liquid was injected into the valveless PZT pump through a syringe, to eliminate the influence of internal bubbles on the experiment.

Results and Discussion
The flow rate was measured using the weight increase of the beaker on the electronic balance within one minute, during pumping fluid from the left beaker to the right using the PZT pump, as shown in Figure 8. Before the flow rate measurement, we ensured that the liquid levels of the two beakers were the same using the siphon effect, and that there were no bubbles in the tube. The actual flow rate measuring system is shown in Figure 9. To ensure the accuracy of the measurement, we quickly injected the liquid into the valveless PZT pump to elimainate the internal bubbles.

Results and Discussion
The experimental and simulating flow rate results of the flow tube, under four different orifice pressure differentials of 1000 Pa, 1200 Pa, 1400 Pa, and 1600 Pa, were determined for fluid flow in forward and backward directions. According to the flow resistance test schematic diagram, shown in Figure 6, the pressure differentials can be adjusted by changing the liquid level height H between liquid level and flow pipes. The relationship between liquid level height and pressure difference is where P is the pressure difference between liquid level and flow tube, ρ is the liquid density, g is the gravitational acceleration. The corresponding detail values are shown in Table 3. The experimental and simulated flow rate results, under different pressure differences, are shown in Figure 10. As the pressure difference increased, the flow rate increased. Under any pressure difference, the forward flow rate was always greater than the backward flow rate, which proves that the forward flow resistance of the flow tube is less than the backward. According to Figure 10, the trend of the forward and backward flow rate of the streamlined flow tube obtained by simulation was basically consistent with that by experiment. However, the simulated flow rate was larger than the experimental flow rate, as there was significant pressure loss in the peripheral pipe connection system in the actual experiment. The experimental and simulating flow rate results of the flow tube, under four different orifice pressure differentials of 1000 Pa, 1200 Pa, 1400 Pa, and 1600 Pa, were determined for fluid flow in forward and backward directions. According to the flow resistance test schematic diagram, shown in Figure 6, the pressure differentials can be adjusted by changing the liquid level height H between liquid level and flow pipes. The relationship between liquid level height and pressure difference is where is the pressure difference between liquid level and flow tube, is the liquid density, is the gravitational acceleration.
The corresponding detail values are shown in Table 3. The experimental and simulated flow rate results, under different pressure differences, are shown in Figure 10. As the pressure difference increased, the flow rate increased. Under any pressure difference, the forward flow rate was always greater than the backward flow rate, which proves that the forward flow resistance of the flow tube is less than the backward. According to Figure 10, the trend of the forward and backward flow rate of the streamlined flow tube obtained by simulation was basically consistent with that by experiment. However, the simulated flow rate was larger than the experimental flow rate, as there was significant pressure loss in the peripheral pipe connection system in the actual experiment. The amplitude-frequency relationship of the SLFT PZT pump, form 40-120 V with a step of 20 V, is shown in Figure 11. It can be seen that the amplitude of the center point of the vibrator was positively correlated with the voltage, and these maximum amplitudes were 0.062 mm (40 V, 10 Hz), 0.095 mm (60 V, 10 Hz), 0.136 mm (80 V, 10 Hz), 0.217 mm (100 V, 11 Hz), and 0.228 mm (120 V, 10 Hz), as shown in Figure 11. In the low frequency region (1-25Hz), firstly, the amplitude of the piezoelectric vibrator increased with the increasing frequency (1-10 Hz), then, when the amplitude increased to a certain value, the amplitude decreased with the increasing frequency (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25), and was approximately zero after 25 Hz. In the high frequency region (>25 Hz), the amplitude decayed with increasing frequency and was accompanied by a few amplitude peak points. These peak points were related to the characteristic properties of the piezoelectric vibrator [16]. The amplitude-frequency relationship of the SLFT PZT pump, form 40-120 V with a step of 20 V, is shown in Figure 11. It can be seen that the amplitude of the center point of the vibrator was positively correlated with the voltage, and these maximum amplitudes were 0.062 mm (40 V, 10 Hz), 0.095 mm (60 V, 10 Hz), 0.136 mm (80 V, 10 Hz), 0.217 mm (100 V, 11 Hz), and 0.228 mm (120 V, 10 Hz), as shown in Figure 11. In the low frequency region (1-25Hz), firstly, the amplitude of the piezoelectric vibrator increased with the increasing frequency (1-10 Hz), then, when the amplitude increased to a certain value, the amplitude decreased with the increasing frequency (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25), and was approximately zero after 25 Hz. In the high frequency region (>25 Hz), the amplitude decayed with increasing frequency and was accompanied by a few amplitude peak points. These peak points were related to the characteristic properties of the piezoelectric vibrator [16]. The measured flow-rate-frequency relationships of the SLFT PZT pump at different voltages are shown in Figure 12. It is obvious that the flow rate was positively related to the voltage, and that the overall trends of flow rate were basically consistent with the amplitude-frequency trend. The output flow rate increased with the driving frequency increase, before 11 Hz, and the output flow decreased with the frequency increase after 11 Hz. After the frequency reached 35 Hz, the output flow remained zero. These maximum flow rates of different driving voltages were 4.79 mL/min (40 V, 10 Hz), 7.61 mL/min (60 V, 10 Hz), 10.5 mL/min (80 V, 10 Hz), 16.45 mL/min (100 V, 11 Hz) and 17.01 mL/min (120 V, 10 Hz) respectively. The low frequency region (1-23Hz), where the maximum flow rate of different voltages occurred, was exactly the same as the frequency region where the maximum amplitude appeared, indicating that the pumping flow rate of the SLFT PZT pump is closely related to the vibration characteristics of the piezoelectric vibrator. In the high frequency region (>25 Hz), the pumping flow rate did not show more peak points as the amplitude changed. This is because the amplitude was so small; the pressure difference caused by the volume change cannot noticeably manipulate the liquid in the pump chamber. The local enlarged displacement-frequency relationship and flow-rate-frequency relationship are shown in the dash line rectangle  in Figures 11 and 12 respectively. It can be clearly seen that when the working frequency is higher than 25 Hz, the vibration amplitude and the flow rate are rather small (less than 4 μm and 0.8 mL/min, respectively) and approach zero fast. When the working frequency is about 10 Hz, the vibration amplitude and flow rate reach the maximum at different voltages. Thus, we assume that this phenomenon is caused by pump system resonance. The measured flow-rate-frequency relationships of the SLFT PZT pump at different voltages are shown in Figure 12. It is obvious that the flow rate was positively related to the voltage, and that the overall trends of flow rate were basically consistent with the amplitude-frequency trend. The output flow rate increased with the driving frequency increase, before 11 Hz, and the output flow decreased with the frequency increase after 11 Hz. After the frequency reached 35 Hz, the output flow remained zero. These maximum flow rates of different driving voltages were 4.79 mL/min (40 V, 10 Hz), 7.61 mL/min (60 V, 10 Hz), 10.5 mL/min (80 V, 10 Hz), 16.45 mL/min (100 V, 11 Hz) and 17.01 mL/min (120 V, 10 Hz) respectively. The low frequency region (1-23Hz), where the maximum flow rate of different voltages occurred, was exactly the same as the frequency region where the maximum amplitude appeared, indicating that the pumping flow rate of the SLFT PZT pump is closely related to the vibration characteristics of the piezoelectric vibrator. In the high frequency region (>25 Hz), the pumping flow rate did not show more peak points as the amplitude changed. This is because the amplitude was so small; the pressure difference caused by the volume change cannot noticeably manipulate the liquid in the pump chamber. The measured flow-rate-frequency relationships of the SLFT PZT pump at different voltages are shown in Figure 12. It is obvious that the flow rate was positively related to the voltage, and that the overall trends of flow rate were basically consistent with the amplitude-frequency trend. The output flow rate increased with the driving frequency increase, before 11 Hz, and the output flow decreased with the frequency increase after 11 Hz. After the frequency reached 35 Hz, the output flow remained zero. These maximum flow rates of different driving voltages were 4.79 mL/min (40 V, 10 Hz), 7.61 mL/min (60 V, 10 Hz), 10.5 mL/min (80 V, 10 Hz), 16.45 mL/min (100 V, 11 Hz) and 17.01 mL/min (120 V, 10 Hz) respectively. The low frequency region (1-23Hz), where the maximum flow rate of different voltages occurred, was exactly the same as the frequency region where the maximum amplitude appeared, indicating that the pumping flow rate of the SLFT PZT pump is closely related to the vibration characteristics of the piezoelectric vibrator. In the high frequency region (>25 Hz), the pumping flow rate did not show more peak points as the amplitude changed. This is because the amplitude was so small; the pressure difference caused by the volume change cannot noticeably manipulate the liquid in the pump chamber. The local enlarged displacement-frequency relationship and flow-rate-frequency relationship are shown in the dash line rectangle  in Figures 11 and 12 respectively. It can be clearly seen that when the working frequency is higher than 25 Hz, the vibration amplitude and the flow rate are rather small (less than 4 μm and 0.8 mL/min, respectively) and approach zero fast. When the working frequency is about 10 Hz, the vibration amplitude and flow rate reach the maximum at different voltages. Thus, we assume that this phenomenon is caused by pump system resonance. The local enlarged displacement-frequency relationship and flow-rate-frequency relationship are shown in the dash line rectangle  in Figures 11 and 12 respectively. It can be clearly seen that when the working frequency is higher than 25 Hz, the vibration amplitude and the flow rate are rather small (less than 4 µm and 0.8 mL/min, respectively) and approach zero fast. When the working frequency is about 10 Hz, the vibration amplitude and flow rate reach the maximum at different voltages. Thus, we assume that this phenomenon is caused by pump system resonance.
According to the literature [45,46] and the characteristics of the piezoelectric vibrator, another resonance may exist in the high-frequency region (around 1.8 kHz-2.1 kHz). By studying the amplitude of the piezoelectric vibrator, it was found that the amplitude decreases with the frequency increase, as shown in Figure 11, and the amplitude transformation period also decreases. Although there may be another resonance in the high frequency region, the resonance amplitude in the high frequency region will be smaller than the first peak. The working medium of the piezoelectric pump is liquid, and when compared with gas, liquid has the characteristics of high density, high viscosity and incompressibility. The amplitude of the piezoelectric vibrator in the high frequency region is so small that it would cause a small change in the pump chamber volume, and that the pressure difference generated is not enough to make up for the pressure loss caused by the liquid discharged from the pump. Therefore, there is little flow rate output in the high frequency region.
According to the theoretical analysis, the resistance coefficients of forward and backward flow were obtained by simulation and substituted into Equations (10)- (12). The amplitudes were measured by laser displacement sensor and input into Equation (13), thereby calculating the theoretical flow rate. The theoretical flow rate was calculated by the amplitude of 100 V and 120 V, whose results were compared with the experiment, as shown in Figure 13.
It can be seen that the maximum theoretical flow was 16.89 mL/min (100 V, 10 Hz) and 18.75 mL/min (120 V, 9 Hz), respectively, and the maximum experimental flow rate was 16.45 mL/min (100 V, 11 Hz) and 17.01 mL/min (120 V, 10 Hz), respectively. In the low frequency region (1-25 Hz), the theoretical flow rate was consistent with the experimental flow. In the high frequency region (>25 Hz), the experimental flow was close to zero, and the theoretical flow decreased in fluctuation corresponding to the fluctuation of amplitude, further proving the direct relationship between amplitude and pumping flow rate. By comparison, it was found that the frequency of maximum flow rate in the experiment lagged behind the frequency of maximum flow rate by theory. Several reasons account for the frequency delay, when compared with the theoretical results, and are illustrated as follows: 1. The vibrator was attached to pump chamber by silica gel bonding, which indirectly increases the volume and reduces the volume change quantity of the pump chamber to some extent.
2. The connection between the flow tube and the pump body is prone to pressure loss, resulting in increased internal pressure loss and decreased pumping efficiency.
3. The viscosity of the experimental water is inevitable, which also causes fluid flowing hysteresis. According to the literature [45,46] and the characteristics of the piezoelectric vibrator, another resonance may exist in the high-frequency region (around 1.8 kHz-2.1 kHz). By studying the amplitude of the piezoelectric vibrator, it was found that the amplitude decreases with the frequency increase, as shown in Figure 11, and the amplitude transformation period also decreases. Although there may be another resonance in the high frequency region, the resonance amplitude in the high frequency region will be smaller than the first peak. The working medium of the piezoelectric pump is liquid, and when compared with gas, liquid has the characteristics of high density, high viscosity and incompressibility. The amplitude of the piezoelectric vibrator in the high frequency region is so small that it would cause a small change in the pump chamber volume, and that the pressure difference generated is not enough to make up for the pressure loss caused by the liquid discharged from the pump. Therefore, there is little flow rate output in the high frequency region.
According to the theoretical analysis, the resistance coefficients of forward and backward flow were obtained by simulation and substituted into Equations (10)- (12). The amplitudes were measured by laser displacement sensor and input into Equation (13), thereby calculating the theoretical flow rate. The theoretical flow rate was calculated by the amplitude of 100 V and 120 V, whose results were compared with the experiment, as shown in Figure 13.
It can be seen that the maximum theoretical flow was 16.89 mL/min (100 V, 10 Hz) and 18.75 mL/min (120 V, 9 Hz), respectively, and the maximum experimental flow rate was 16.45 mL/min (100 V, 11 Hz) and 17.01 mL/min (120 V, 10 Hz), respectively. In the low frequency region (1-25 Hz), the theoretical flow rate was consistent with the experimental flow. In the high frequency region (>25 Hz), the experimental flow was close to zero, and the theoretical flow decreased in fluctuation corresponding to the fluctuation of amplitude, further proving the direct relationship between amplitude and pumping flow rate. By comparison, it was found that the frequency of maximum flow rate in the experiment lagged behind the frequency of maximum flow rate by theory. Several reasons account for the frequency delay, when compared with the theoretical results, and are illustrated as follows: 1. The vibrator was attached to pump chamber by silica gel bonding, which indirectly increases the volume and reduces the volume change quantity of the pump chamber to some extent.
2. The connection between the flow tube and the pump body is prone to pressure loss, resulting in increased internal pressure loss and decreased pumping efficiency.
3. The viscosity of the experimental water is inevitable, which also causes fluid flowing hysteresis. The flow rate at 120 V.

Conclusions
Based on the existing research, a streamlined flow tube valveless piezoelectric pump (SLFT PZT pump) was fabricated and analyzed. Firstly, the structural and working principle of the flow tube were analyzed. At the same time, the theoretical flow rate and flow resistance coefficient equations were established. Secondly, the pressure and velocity field of the flow tube were further simulated, and the results fit the theoretical well, proving that the flow tube has the ability to modify the single

Conclusions
Based on the existing research, a streamlined flow tube valveless piezoelectric pump (SLFT PZT pump) was fabricated and analyzed. Firstly, the structural and working principle of the flow tube were analyzed. At the same time, the theoretical flow rate and flow resistance coefficient equations were established. Secondly, the pressure and velocity field of the flow tube were further simulated, and the results fit the theoretical well, proving that the flow tube has the ability to modify the single flow trend and improve the flow stability. Thirdly, the amplitude-frequency relationships of the piezoelectric vibrator, and the flow-rate-frequency relationships of the SLFT PZT pump were tested at 40, 60, 80, 100 and 120 V. The results show that both relations trends are basically the same, indicating that the flow rate was highly related to the vibrator amplitude. Finally, in order to verify the correctness of the theoretical analysis, the flow resistance coefficient of the flow tube, under 100 V and 120 V, was substituted into the flow rate equation to obtain the theoretical flow rate, and the results were basically the same as the experimental results.