# Research on a Piezoelectric Pump with Flexible Valves
## 3 Theoretical Analysis  
When the PZT pump works, the piezoelectric vibrator is bent up and down. The volume of the pump chamber changes periodically so that the fluid in the pump chamber moves and acts on the valves. Assuming that the flexible valves are subjected to a uniformly distributed load \(q(t)\) at a certain moment, the free end of the leaflet can be simplified into three continuous cantilevers, as shown in Figure 3. Combining the chain algorithm and the load increasing technique [20], the deformed clearance \(Y\)' can be calculated, and its solution formula includes \(q(t)\). The specific derivation process is in Appendix A.  
When the vibrator moves to the limit position, its radial section is arched. The deformation of the middle point of the vibrator is the largest, and the amplitude of the radial points decreases with the increase of the radius \(r\). The deformation curve \(\omega(r)\) can be fitted by an amplitude test experiment of the vibrator. The volume changes when the pump chamber moves from the equilibrium position to the limit position can be obtained by integrating \(\omega(r)\), as follows:  
\[\Delta V=2\pi\int_{0}^{R}\omega(r)rdr \tag{1}\]  
Figure 3: The simplified model and deformation of the valve.  
where \(\Delta V\) is the volume change quantity in a quarter period.  
Whether in the valved state or the valveless state, the inlet/outlet of the PPFV corresponds to a nominal valve body. The valve body can be opened and closed by deformation, and then the fluid can be delivered in one direction. Ignoring the fluid loss caused by the gap between the flexible valve and the flow channel, for flexible valves, the deformation of the valved state is the strongest, according to Bernoulli equation, as follows:  
\[p_{1}+\frac{1}{2}{\rho{v_{1}}^{2}}+\rho gh_{1}=p_{2}+\frac{1}{2}{\rho{v_{2}}^{ 2}}+\rho gh_{2} \tag{2}\]  
where \(h_{1}=h_{2}\).  
\[q(t_{0})=8\rho f^{2}\Delta V^{2}(\frac{1}{A_{1}}+\frac{1}{A_{2}})(\frac{1}{A_{ 1}}-\frac{1}{A_{2}}) \tag{3}\]  
where \(t_{0}\) represents the time when the PZT vibrator moves to its limit position, \(p_{1}\) and \(p_{2}\) are the pump chamber's pressure and the outlet's pressure, respectively, \(\rho\) is the density of the working medium, \(v_{1}\) and \(v_{2}\) are, respectively, the fluid velocity of the pump chamber and the outlet fluid velocity, \(h_{1}\) and \(h_{2}\) are the heights of the fluid in the pump chamber and in the outlet, respectively, \(f\) represents the driving frequency, and \(A_{1}\) and \(A_{2}\) respectively represent the cross-sectional area of the channel in the chamber and the cross-sectional area of the outlet channel.  
According to the above calculation formulas, the deformation of the flexible valve is not only related to its structure and material parameters, but also related to the uniformly distributed load \(q(t)\). The magnitude and direction of the uniformly distributed load are related to the input signal of the PZT vibrator, which mainly depends on the frequency and the voltage. Therefore, the deformation of flexible valves can be changed by controlling the input signal, and the conversion between the valved state and the valveless state can be realized.