# Design and Experimental Verification of a PZT Pump with Streamlined Flow Tubes
## 3 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  
\[\Big{(}\overline{p}_{0}-\overline{p}_{x}\Big{)}d=\rho x\frac{\partial Q}{ \partial t}+\int_{0}^{x}\tau_{w}(x)dx+\rho\int_{-\frac{\pi}{2}}^{\frac{4}{2}} \!\!\Big{(}u_{x}^{2}-u_{0}^{2}\Big{)}dy \tag{8}\]  
where \(\rho\) is the density, \(\overline{p}\) is the section averaged pressure, \(\tau_{w}\) 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 \(\omega\). \(Q(t)=Q_{max}\sin(\omega t)\), Therefore, pressure loss can be expressed as  
\[\Delta P=\frac{\Big{(}\overline{p}_{0}-\overline{p}_{x}\Big{)}}{q_{j}}=\underbrace {\pi\!\Big{(}St\frac{x}{d}\Big{)}\!\cos(\omega t)}_{\Delta p_{u}}+ \underbrace{\int_{0}^{\frac{\pi}{2}}C_{f}\!d\!\Big{(}\frac{x}{d}\Big{)}}_{ \Delta p_{s}}+\underbrace{2\int_{-\frac{1}{2}}^{\frac{1}{2}}\!\!\Bigg{(}\frac {u_{x}^{2}-u_{0}^{2}}{\overline{U}_{j}^{2}}\Bigg{)}d\!\!\Big{(}\frac{y}{d} \Big{)}}_{\Delta p_{m}} \tag{9}\] \begin{table}
\begin{tabular}{c c} \hline
**Item** & **Size (mm)** \\ \hline d & 2.5 \\ L\({}_{1-2}\) & 40 \\ L\({}_{2-3}\) & 13.8 \\ R\({}_{1}\) & 270 \\ R\({}_{2}\) & 4 \\ Pump chamber diameter & 35 \\ Pump chamber height & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: The size of the SLFT PZT pump. \begin{table}
\begin{tabular}{c c} \hline
**Item** & **Size (mm)** \\ \hline Brass substrate diameter & 41 \\ Brass substrate thickness & 0.2 \\ Piezoelectric ceramics diameter & 35 \\ Piezoelectric ceramics thickness & 0.15 \\ \hline \end{tabular}
\end{table} 
Table 2: The size parameters of the PZT vibrator. Figure 2: The streamlined flow tube valveless piezoelectric (SLFT PZT) pump. where \(\overline{U_{j}}=\frac{2}{d}d\int_{0}^{\overline{T}}Q(t)dt\) is the time- and spatial-averaged expulsion velocity. Furthermore, \(St=\frac{\omega t}{U_{j}}\) is the Strouhal number, \(d\) is the inlet or outlet diameter, \(C_{f}=\frac{\tau_{u}}{q_{j}}\) is the skin friction coefficient, \(q_{j}=0.5\rho\overline{U_{j}^{2}}\), \(\Delta p_{u}\) is the unsteady, linear inertia term, \(\Delta p_{s}\) is the viscous loss, \(\Delta p_{m}\) is the nonlinear minor loss. According to the pressure loss equation  
\[\Delta P=\Delta p_{u}+\Delta p_{s}+\Delta p_{m}=\xi\frac{\rho u^{2}}{2} \tag{10}\]  
where \(\rho\) is the fluid density, \(\xi\) 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  
\[\xi_{+}=\frac{\Delta P}{\frac{\rho u^{2}}{2}}=\frac{\Delta p_{u}+\Delta p_{s}+ \Delta p_{m}}{\frac{\rho u^{2}}{2}}=\frac{2(\Delta p_{u}+\Delta p_{s}+\Delta p _{m})}{\rho u^{2}} \tag{11}\]  
where \(\Delta P\) is the forward flow pressure loss, \(u\) is the forward flow average velocity. Similarly, the flow resistance coefficient in the backwards direction is  
\[\xi_{-}=\frac{\Delta P}{\frac{\rho u^{2}}{2}}=\frac{\Delta p_{u}+\Delta p_{s}+ \Delta p_{m}}{\frac{\rho u^{2}}{2}}=\frac{2(\Delta p_{u}+\Delta p_{s}+\Delta p _{m})}{\rho u^{2}} \tag{12}\]  
where \(\Delta P\) 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.  
The equation of the working section is  
\[\left\{ \begin{array}{c}\omega(r)=\omega_{0}\Big{(}1-\frac{r^{2}}{R^{2}} \Big{)}\left(|r|\leq R\right)\\ \omega_{0}=\frac{Kd_{31}V_{(\alpha)}}{h^{2}}R^{2}\end{array} \right.