## 3 Numerical Implementation
### Piezolectric Pump Optimization  
Analyses were done between 100 K to 500 K mesh elements for coupled analysis to prove mesh independency. After mesh convergence analyses, \(<\)1% for structure-based error and \(-\)8% flow-based error were calculated for 200 K. The analysis time measured a minimum 2.5 h to 262 h for 100 K to 500 K grid analysis with 1% relative tolerance and 1/40 step size (40 time steps per cycle); as a result, the 200 K grid had an acceptable error, so it was used. There were many analyses done, as well, for different  
Figure 3: (**a**) Displacement of the piezoelectric actuator at pumping mode, (**b**) suction mode, (**c**) inlet velocity illustration, (**d**) outlet velocity illustration at the same time, (**e**) velocity field streamlines, (**f**) cell Reynold’s number. time-steps from 20 steps per cycle to 1600 steps per cycle. These analyses took a minimum of 1.5 h to 48 h for 100 K grid size and 1% relative tolerance. Error was calculated with respect to 1600 steps per cycle. The error was, respectively, >41% for 20 step, 8.5% for 40 step, 7.5% for 100 step, and <1.5% for 400 step per one cycle. i7-7700-32 GB hardware was used for this analysis and time optimization. Nozzle/diffuser divergence angle, nozzle/diffuser wide radius, nozzle/diffuser length, and chamber height were optimized. According to [13] and [30], when the chamber diameter increases, the efficiency of nozzle/diffuser and the net flow increases. When diameter of the piezoelectric disk increases, the expected deformation of the piezoelectric disk increases as well. These parameters were optimized for best efficiency, maximum deflection, maximum velocity, and net flow, as well as manufacturability considerations. Analyses were performed to achieve the most efficient nozzle and diffuser design. Since chamber diameter was fixed by piezoelectric actuator selection, only chamber height (\(mu\)) was optimized in a range of 0.2 mm to 2.5 mm, and the one with the highest net flow rate using Equation (4) was 1 mm, shown in Figure 4a. Figure 4: Numerical simulation results for net flow rate versus (**a**) chamber height; (**b**) length of nozzle/diffuser, (**c**) divergence angle. As seen in Figure 4b, nozzle/diffuser length affects the net flow rate, so more than 10 different lengths were analyzed. The net flow rate significantly changed around 1 mm. There are many published pieces of research about nozzle and diffuser design [13; 30; 36; 37] that are typically above 4 degrees half angle. When the nozzle and diffuser angle increases, the net flow rate increases too. However, there are some velocity values and venturi effects that may constrict the flow, so analyses were done to have the most efficient planar nozzle/diffuser. As seen in Figure 4c above, the most efficient nozzle and diffuser selected half-angle value \(\theta\) was in between 12 and 14 degrees. In addition to that, voltage and frequency parameters sweep was done. The piezoelectric actuator terminal was connected to an electrical circuit, and there was a voltage source in the circuit. Different voltages and frequencies were applied to the voltage source (see Figure 5). When the voltage increased, displacement and net flow increased explicitly. It was observed that the flow rate peaked at 100 Hz for the frequency range between 20 Hz and 200 Hz. (see Figure 5). Another case considered was the calculation of the current drawn by the piezoelectric disk. There is an electrical circuit defined in COMSOL workspace, and the cable connected between the electrical source and the piezoelectric actuator was used as a resistor; the current through the cable was calculated. As expected, it increased when the voltage increased. Using the voltage and the current values, power consumption was calculated for different voltage and frequencies applied to the system. It is shown in Figure 6 below. Last but not least, outlet and inlet flow rates were recorded, as illustrated in Figure 7 for 3 \(\upmu\)m displacement amplitude of the piezoelectric disc.