# Dynamic Simulation Model and Performance Optimization of a Pressurized Pulsed Water Jet Device
## 2 Formation Principle of PPWJ
### Analysis of the Pulse Parameters of a PPWJ  
The pulse parameters of PWJ encompass pulse pressure, pulse frequency, and pulse duration. These factors play a pivotal role in determining the rock breaking capability of PWJ. The determination of the pulse parameters of PPWJ is based on the water jet theory and the operational principle of the device. The formation of a high-pressure PWJ is achieved during the piston stroke stage, and its modeling method is based on the principles of Lumped Element Modeling [43]. When the water in chamber 1 is squeezed by the piston, there is an estimated linear correlation between the pressure and density of the water [44]:  
\[p_{\mathrm{w}}=K_{\mathrm{w}}\left(\frac{\rho_{\mathrm{w}}}{\rho_{\mathrm{w0} }}-1\right) \tag{3}\]  
where \(p_{\mathrm{w}}\) is the pressure of chamber 1 during pressurization process; \(K_{\mathrm{w}}\) is the bulk modulus of water; \(\rho_{\mathrm{w}}\) is the density of water; \(\rho_{\mathrm{w0}}\) is the original density of water; and \(\rho_{\mathrm{w}}=\rho_{\mathrm{w0}}\) when \(p_{\mathrm{w}}=0\). As a derivation of Equation (3) with respect to time, we have the following:  
\[\frac{\mathrm{d}\rho_{\mathrm{w}}}{\mathrm{d}t}=\frac{\rho_{\mathrm{w0}}}{K_{ \mathrm{w}}}\frac{\mathrm{d}p_{\mathrm{w}}}{\mathrm{d}t} \tag{4}\]Based on the Reynolds transport theorem [45], the equation of continuity or conservation of fluid mass can be derived:  
\[\frac{\mathrm{d}m_{\mathrm{w}}}{\mathrm{d}t}=\left(\frac{\partial}{\partial t} \int\int\int\rho_{\mathrm{w}}\cdot\mathrm{d}V\right)_{CV}+\left(\int\int\rho_{ \mathrm{w}}v_{\mathrm{w}}\cdot\mathbf{n}\mathrm{d}A\right)_{CS}=0 \tag{5}\]  
where \(m_{\mathrm{w}}\) and \(v_{\mathrm{w}}\) are the mass and velocity of water, respectively; \(\mathbf{n}\) is the outward normal to the control surface; \(V\) represents the volume inside the control volume domain; and \(A\) represents the boundary surface. The first term of Equation (5) indicates the rate at which the accumulated mass inside chamber 1 changes for compressible water:  
\[\left(\frac{\partial}{\partial t}\!\int\int\int\rho_{\mathrm{w}}\cdot\mathrm{d }V\right)_{CV}=\frac{\partial}{\partial t}\!\left(\rho_{\mathrm{w}}A\left(l-x _{\mathrm{sp}}\right)\right)=\frac{\pi}{4}d_{\mathrm{pl}}^{2}\!\left(\frac{ \mathrm{d}\rho_{\mathrm{w}}}{\mathrm{d}t}\left(l-x_{\mathrm{sp}}\right)-v_{ \mathrm{sp}}\rho_{\mathrm{w}}\right) \tag{6}\]  
where \(A\) is the cross sectional area of chamber 1; \(l\) is the length of chamber 1; and \(x_{\mathrm{sp}}\) and \(v_{\mathrm{sp}}\) are, respectively, the stroke displacement and stroke velocity of the piston. The second term of Equation (5) implies the rate of flow mass outflow through the nozzle:  
\[\left(\int\int\rho_{\mathrm{w}}v_{\mathrm{w}}\cdot n\mathrm{d}A\right)_{CS}= \rho_{\mathrm{w}}v_{\mathrm{w}}a=\frac{\pi}{4}d_{\mathrm{n}}^{2}\rho_{\mathrm{ w}}v_{\mathrm{w}} \tag{7}\]  
where \(a\) is the cross sectional area of the nozzle and \(d_{\mathrm{n}}\) is the nozzle diameter. In combination with Equations (5)-(7), we have the following:  
\[d_{\mathrm{P1}}^{2}\!\left(\frac{\mathrm{d}\rho_{\mathrm{w}}}{\mathrm{d}t} \left(l-x_{\mathrm{sp}}\right)-v_{\mathrm{sp}}\rho_{\mathrm{w}}\right)+d_{ \mathrm{n}}^{2}\rho_{\mathrm{w}}v_{\mathrm{w}}=0 \tag{8}\]  
The Bernoulli hydrodynamic equation is utilized to estimate the mean velocity of the jet as it exits the nozzle based on value of the pressure inside chamber 1:  
\[u_{\mathrm{w}}=c_{\mathrm{d}}\sqrt{\frac{2p_{\mathrm{w}}}{\rho_{\mathrm{w}}}} \tag{9}\]  
where \(c_{\mathrm{d}}\) is the discharge coefficient of the nozzle.