# Optimal Design of Droplet Ejection for PZT Printhead Based on Surrogate Model
## 3 PZT Printed CFD Modeling
### Droplet Ejection Process Model of PZT Printed  
The mathematical model of the droplet forms the basis and research foundation for the accurate construction of the CFD numerical model of the printhead. In this type of printhead, the PZT membrane is attached to the top of the ink chamber, and under the action of the driving pulse electrical signal, the membrane will be deformed by a small amount of bending. This deformation transmits pressure into the ink chamber to form a pressure wave, causing the liquid to be ejected from the nozzle as droplets. According to the above-mentioned working state of the printhead, this process can be simplified into a physical model for analysis. Firstly, according to the piezoelectric material characteristic formula:  
\[\left. \begin{array}{l}S_{1}=S_{11}^{E}T_{1}+d_{31}E_{3}\\ D_{3}=d_{31}T_{1}+\varepsilon_{33}^{T}E_{3}\end{array} \right\} \tag{1}\]  
where \(S_{1}\) and \(T\) are the strain and stress in the transverse direction, respectively; \(D_{3}\) and \(E_{3}\) are the electric field and stress in the longitudinal direction, respectively; \(\varepsilon_{33}^{T}\) is the dielectric constant when the external stress is zero or constant, and \(d_{31}\) is the PZT constant and the elastic compliance coefficient when the electric field is either constant or equal to zero. [PERSON] [21] proposed that if the electric field applied to the PZT membrane is uniform, then \(E_{3}\) is a linear function of the voltage, \(U\), that is loaded on the PZT membrane. If the maximum displacement of the PZT membrane in the deformation direction is denoted as h, then the relationship between the force and the displacement of the PZT driver can be expressed as:  
\[h=\alpha U-\beta F \tag{2}\]  
where \(\alpha\) and \(\beta\) are constants, \(U\) is the driving voltage value, and \(F\) is the external force. As shown in Figure 4, the electric field and displacement movement changes of the PZT membrane are simplified into an elastic model. When the pulse signal is applied to the PZT membrane, the membrane bends and squeezes the ink chamber with a certain acceleration under the action of the electric field's force, and the liquid in the ink chamber also exerts a certain pressure, \(P(t)\), on the PZT membrane. According to [PERSON]'s laws:  
\[m\frac{d^{2}h}{dt^{2}}=F(t)-P(t)A-kh \tag{3}\]  
Among them, \(A\), \(k\) and \(m\) are divided into the area, elastic coefficient and mass of the PZT membrane, and \(F(t)\) is the force generated by the accelerated motion and deformation of the membrane. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Observation Module** & \multicolumn{2}{c}{**Droplet Generation**} \\ \hline Pixel & 1032 \(\times\) 778 & Printed type & QS256/80 AAA \\ Voltage amplitude & 5 V (TTL) & Voltage amplitude & 100 V \\ Frequency & 80 kHz & Frequency & 8 kHz \\ Focal length & 6:1 & Ink type & Furan resin \\ Pulse width & 0.25 \(\mu\)s & & \\ Supply voltage & 220 V & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental parameters. Substituting the above formula into Equation (2), the force equation of the PZT membrane can be obtained by:  
\[m\frac{d^{2}h}{dt^{2}}+(k+\frac{1}{\beta})h=\frac{\alpha}{\beta}\phi-PA \tag{4}\]  
In the process of droplet injection and observing the principle of energy conservation, the Naiver-Stokes equations are established as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho v)=0 \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho v)+\
abla\cdot(\rho vv)=-\
abla p+\
abla \cdot[\eta(\
abla v+\
abla v^{T})]+\rho g+F \tag{6}\]  
where \(\rho\) is the density, \(v\) is the speed, \(p\) is the pressure, \(\eta\) is the viscosity and \(F\) is the volume force. Through the combined expression of Equations (4) and (6), the complete process from droplet generation to injection in the working process of the PZT printhead can be described.