# Optimal Design of Droplet Ejection for PZT Printhead Based on Surrogate Model
## 3 PZT Printed CFD Modeling
### Modeling of the PZT Printhead CFD Numerical Model  
In order to obtain the basic data of the surrogate model and optimization design process, a CFD numerical model of the PZT printhead was established to simulate the ejection process of droplets [22]. Different from the previous 3D or 2D printhead simulation models, this study adopted a modeling method that simplified the calculation of the 3D model of the PZT printhead by dividing it into two: a 3D chamber model and a 2D nozzle-air domain model. The bending deformation of the PZT membrane was simulated by using a dynamic mesh technique, and the change in the cross-sectional area of the chamber was recorded in real-time. The recorded cross-sectional area was fitted into a fifth-order function as the input boundary condition of the nozzle-air domain model to simulate the droplet ejection process. For the 2D nozzle-air domain model, in addition to establishing its input boundary conditions, it is also necessary to verify the mesh independence of the model to prove its accuracy and validity. The modeling process and boundary condition settings of the PZT printhead numerical model are shown in Figure 5. In this study, the ANSYS Fluent 19.2 commercial software was used to perform computational simulation, with only the fluid dynamics behavior considered during the simulation. Figure 4: Simplified model of the droplet injection process. The 3D chamber model's main function is to simulate the process of deformation movement of the PZT membrane and obtain the ink mass-flow curve in the chamber, which is used as the boundary input condition of the 2D nozzle-air domain model. When the numerical simulation model was established, the PZT membrane was set as the moving mesh. The DEFINE_GRID_MOTION macro function of the UDF method was used to drive the motion deformation of the membrane. The DEFINE_ADJUST macro was used to record the change in the cross-sectional area of the chamber, and the mass-flow curve of the flowing ink was calculated by the DEFINE_EXECUTE_AT_END macro. The other boundary conditions were set as WALL. During this simulation, the ink mass-change value in the chamber was recorded, and a mass-flow curve using Equation (7) was obtained. \[mass\_flow\_rate(t)= \begin{cases}\begin{matrix}at^{5}+bt^{4}-ct^{3}-dt^{2}+et,&0 \leq t<t_{1}\\ 0,&t_{1}\leq t<t_{2}\end{matrix}\\ a[t-(t_{2}-t_{1})]^{5}+b[t-(t_{2}-t_{1})]^{4}-c[t-(t_{2}-t_{1})]^{3}\\ -d[t-(t_{2}-t_{1})]^{2}+e[t-(t_{2}-t_{1})],&t_{2}\leq t<t_{3}\\ 0,&t>t_{3}\end{cases} \end{cases} \tag{7}\]  
For the 2D nozzle-air domain CFD model, the validity and correctness needed to be verified, including the verification of the independence of the mesh. In the microdroplet ejection technology, the criterion of stable ejection is mainly based on the number of satellite droplets. Therefore, in this study, the ejection process in which there are no more than two satellite droplets, and the satellite droplets can join the main droplet before the collision with the bottom plate, is considered a stable ejection. Before the simulation calculation, a convergence model that can stably eject droplets and include mesh independence verification was selected first to verify the validity of the model. The relevant parameters set are shown in Table 2.  
The comparison between the experimental and the CFD numerical simulation results are shown in Figure 6, and the ejection process of the two has a good consistency. It can be obtained from the evolution process of the droplet flow contours that the driving voltage signal increased from 0 to 100 V during the period of 0-3.85 \(\upmu\)s, and the PZT membrane moved to the maximum displacement to squeeze the ink in the chamber. Under the influence of pressure waves in the chamber, the ink formed a half-moon liquid column at the nozzle. During the period of 3.85-14.4 \(\upmu\)s, the voltage signal remained unchanged at 100 V, and the ink continued to be ejected from the nozzle and formed a long liquid column under the liquid tension.