# Development of a Nonlinear Piezoelectric Energy Harvester for Alternating Air Load
## 3 The Model of the Volume Compression  
The model of volume compression in Figure 2 presents the relationship between volume compression parameters and air pressure. Figure 2a shows the model of structure and compressed air. A certain quality atmosphere is in the sealed piezoelectric energy harvester. The pressure is produced by the impact of gas molecules, and the pressure in the sealed volume can be changed by moving the piston. Figure 2b shows the micro unit of volume compression. The theory of compressed air can be represented by the micro unit.  
The radius of the power chamber and compressed chamber are defined as \(R\). The moving distance of the piston is \(l\). \(P_{x}\) and \(P_{1}\) are the changing pressure in the power chamber and compressed air load, respectively. \(V_{x}\) is the instantaneous volume. In order to analyze the changing pressure, the internal air environment is defined based on the theory of compressible gas and the type of the ideal gas. It is assumed that there is no heat exchange in the system.  
In Figure 2, the quality of a single gas molecule is \(m_{0}\). In the sealed chamber, the pressure is produced by the impact of gas molecule. \(N\) is the number of the gas molecules in the sealed chamber, \(V_{0}\) is the initial volume of sealed chamber, and \(v\) is the velocity of gas molecule. The vector equation of velocity can be defined as follow:  
\[\overset{\rightarrow}{v}=\overset{\rightarrow}{v}_{x}+\overset{ \rightarrow}{v}_{y}+\overset{\rightarrow}{v}_{z} \tag{1}\]  
where \(v_{x}\) is the velocity component in \(x\) direction and the wall of the chamber is struck by the gas molecule. From the kinetic energy only in the \(x\) direction acting on the PZT patch, deformation is produced. Thus, the velocity in \(x\) direction has been analyzed. The impulse of a single gas molecule in \(x\) direction is \(2m_{0}v_{x}\). The model of a micro unit is shown in Figure 2b.  
According to the theory of gas dynamics, the speed of the piston is much less than the speed of gas molecule. So, it can be assumed that gas distribution could not be influenced by the speed of piston and the gas is equally distributed in two chambers due to the rapid motion speed of gas. The changing function of unit volume molecules\(f_{n}(t)\) can be expressed as:  
Figure 1: Structure and working principle of the presented harvester.  
Figure 2: The model of the volume compression: (**a**) the model of structure and compressed air; (**b**) the micro unit of the volume compression.  
\[f_{n}(t)=\frac{N}{V_{0}-S_{0}l} \tag{2}\]  
Where \(S_{0}\) is the area of piston.  
According to the theory of Maxwell velocity distribution law, the speed distribution function in the \(x\) direction \(f(v_{x})\) can be expressed as:  
\[f(v_{x})=4\pi(\frac{m_{0}}{2\pi kT})^{3/2}e^{-m_{0}v_{x}^{2}/2 kT}v^{2} \tag{3}\]  
According to the Equations (2) and (3), the differential of gas molecule number d\(N\) can be obtained as follow:  
\[\mathrm{d}N=f(v_{x})\mathrm{d}v_{x}f_{n}(t)v_{x}\mathrm{d}A\mathrm{d}t \tag{4}\]  
According to the Equation (4), the differential of impulse d\(I\) can be obtained  
\[\mathrm{d}I=2m_{0}v_{x}\mathrm{d}N \tag{5}\]  
According to the theory of gas-molecular dynamics, the velocity of the molecule is between zero and infinity. The impulse equation is  
\[I=2m_{0}f_{n}(t)\mathrm{d}A\mathrm{d}t\int_{0}^{\infty}v_{x}^{2}f(v_{x}) \mathrm{d}v_{x} \tag{6}\]  
According to the Equation (6), the changing pressure \(P_{x}\) can be obtained as follows:  
\[P_{x}=\frac{I}{\mathrm{d}A\mathrm{d}t}=\frac{2}{3}f_{n}(t)\overline{\omega} \tag{7}\]  
\[\overline{\omega}=\frac{1}{2}m_{0}\overline{v}^{2} \tag{8}\]  
where \(\overline{\omega}\) is the average kinetic energy of a gas molecule.