# Simulation and Experimental Analysis of Pressure Pulsation Characteristics of Pump Source Fluid
## 2 Theoretical Analysis of Fluid Pressure Pulsation of the Plunger Pump
### Analysis of Fluid Pressure Pulsation of the Plunger Pump  
The fluid pressure pulsation at the outlet of the plunger pump is closely related to the movement of the single plunger structure. The rotating speed of the cylinder, the inclination angle of the swash plate, the diameter of the single plunger, and the radius of the distribution circle of the plunger hole will affect the fluid pressure pulsation characteristics of the plunger pump. The movement diagram of the single plunger structure is shown in Figure 2.  
The theoretical instantaneous flow of the single-plunger structure from position A to position B is:  
\[q_{t}\ =\ \frac{\pi}{4}d^{2}\omega R\eta_{v}\tan\alpha\sin\phi \tag{8}\]  
where:  
\(\omega\) is the angular velocity of cylinder rotation (rad/s);  
\(\eta_{v}\) is the volumetric efficiency of the plunger pump;  
\(d\) is the plunger diameter (mm);  
\(R\) is the diameter of the plunger axis distribution circle (mm);  
\(\alpha\) is the inclination of the swash plate (\({}^{\circ}\)); and  
\(\phi\) is the cylinder angular displacement (rad).  
The angle between each plunger is \(\beta\ =\ \frac{2\pi}{Z}\), where \(Z\) is the number of plungers. The total output flow of the plunger pump is the sum of the flow of each single plunger, and the total output flow of the plunger pump is:  
\[\left\{\begin{array}{c}\mathrm{q}\ =\ \sum\limits_{i=\ 1}^{Z}\mathrm{q}\ =\ \frac{\pi}{8}d^{2}\omega R\eta_{v}\tan\alpha\frac{\cos(\varphi-\frac{ \delta}{2})}{\sin\frac{\delta}{2}}\cdot\cdot\cdot\cdot\cdot\cdot\mathrm{0}\leq \varphi\leq\frac{\delta}{2}\\ \mathrm{q}\ =\ \sum\limits_{i=\ 1}^{Z}\mathrm{q}\ =\ \frac{\pi}{8}d^{2}\omega R \eta_{v}\tan\alpha\frac{\cos(\varphi-\frac{\delta}{2})}{\sin\frac{\delta}{2} }\cdot\cdot\cdot\cdot\cdot\frac{\delta}{2}\leq\varphi\leq\beta\end{array}\right. \tag{9}\]  
When the single-plunger structure moves in the plunger hole of the cylinder, the volume of the enclosed plunger chamber changes with time, which causes the pressure of  
Figure 2: Movement diagram of the single plunger structure.  
the fluid to fluctuate. According to the basic principles of fluid mechanics, as the flow rate of the closed cavity between the plunger and the cylinder changes, it will inevitably cause a change in pressure [26], and the instantaneous pressure can be obtained as:  
\[p\ =\ \frac{E}{\gamma V}\int_{0}^{t}q\mathrm{dt} \tag{10}\]  
where:  
\(E\) is the fluid modulus of elasticity (MPa); and  
\(V\) is the volume of the plunger cavity (m\({}^{3}\)).  
From Equations (9) and (10), it can be taken:  
\[\left\{\begin{array}{l}\mathrm{p}\ =\ K\sin\left[\left(\omega\mathrm{t}-\frac{ \pi\pi}{22}\right)+\sin\left(\frac{\pi\pi}{22}\right)\right]\cdot\dots\cdot 0\leq\mathrm{t}\leq\frac{\pi}{2\omega}\\ \mathrm{p}\ =\ K\sin\left[\left(\omega\mathrm{t}-\frac{3\pi}{22}\right)+\sin \left(\frac{\pi\pi}{22}\right)\right]\cdot\dots\cdot\frac{\pi}{2\omega}\leq \mathrm{t}\leq\frac{2\pi}{2\omega}\end{array}\right. \tag{11}\]  
where \(K\ =\ \frac{E}{\gamma V}\times\frac{\pi\mathrm{d}^{3}\eta_{c}R\tan a}{8\sin \frac{\pi}{22}}\).