## 2 Numerical Simulation of Flow Forces in Slide Valves
### Calculation of Steady Flow Force  
The calculation of steady flow force can be performed either by integrating the pressure on the valve spool or by calculating the momentum of the fluid within the control volume. Figure 2 shows a schematic of the forces acting on the valve chamber in the inlet throttling mode. Focusing on the spool and considering the fluid viscosity, the steady flow force is  
Figure 1: Pressure relief valve model use in a type of armored vehicle. (**a**) flow path model; (**b**) simplified model; (**c**) structural schematic.  
the sum of the pressure differential across the shoulders of the spool and the viscous force acting on the valve stem:  
\[F_{spool}=\int\limits_{A_{l}}P_{l}dA-\int\limits_{A_{r}}P_{r}dA+\int\limits_{A_{ stem}}\tau_{stem}dA \tag{1}\]  
where \(F_{spool}\) represents the steady flow force acting on the spool, in N; \(P_{l}\) is the pressure on the left shoulder, in Pa; \(P_{r}\) is the pressure on the right shoulder, in Pa; \(\tau_{stem}\) is the viscous shear force acting on the valve stem, in N; \(A_{l}\) is the annular area of the left shoulder, in m\({}^{2}\); \(A_{r}\) is the annular area of the right shoulder, in m\({}^{2}\); and \(A_{stem}\) is the cylindrical surface area of the valve stem, in m\({}^{2}\).  
Considering the control volume (CV) formed by the inlet and outlet flow cross-sections of the valve chamber and the inner wall of the chamber, the momentum theorem is applied as follows:  
\[-F_{spool}-\int\limits_{A_{slave}}\tau_{slave}dA=\frac{d}{dt}\int\limits_{\text {CV}}\rho v_{x}dV+\int\limits_{\text{CV}}\rho v_{x}\textbf{v}\cdot\textbf{n}dA \tag{2}\]  
where **v** is the fluid velocity, in m/s; \(v_{x}\) is the axial component of the fluid velocity, in m/s; **n** is the normal vector of the area element \(dA\); and \(A_{slave}\) is the cylindrical surface area of the valve body in m\({}^{2}\). The left-hand side of the equation represents the force exerted by the spool and valve body on the fluid within the control volume, while the right-hand side represents the rate of change in fluid momentum over time and the net flux within the control volume. For steady flow, neglecting the instantaneous change in fluid momentum, we have  
\[F_{spool}=\int\limits_{inlet}\rho v_{x}\textbf{v}\cdot\textbf{n}dA+\int \limits_{outlet}\rho v_{x}\textbf{v}\cdot\textbf{n}dA-\int\limits_{A_{slave}} \tau_{slave}dA \tag{3}\]  
The steady flow force of a slide valve is usually estimated in engineering using an empirical formula based on the momentum theorem:  
\[F_{spool}=\rho qv_{1}\cos\theta_{1}-\rho qv_{2}\cos\theta_{2} \tag{4}\]  
where \(v_{1}\) and \(v_{2}\) are the inflow and outflow velocities of the fluid and \(\theta_{1}\) and \(\theta_{2}\) are the inflow and outflow jet angles, respectively. Additionally, it is commonly assumed that one of the fluid inflow or outflow angles approaches \(90^{\circ}\) (depending on the throttling type). Combined with the valve flow equation, we have  
\[q=Av=C_{q}A(x_{p})\sqrt{2\Delta p/\rho} \tag{5}\]  
Figure 2: Force schematic of inlet throttling slide valve chamber.  
\[F_{spool}\approx\rho qv_{l}\cos\theta_{i}=2C_{q}A(x_{v})\cdot\Delta p\cdot\cos \theta_{i} \tag{6}\]  
where \(i=1,2\); \(C_{q}\) is the flow coefficient, typically taken as 0.62; \(A\) is the flow area; \(\Delta p\) is the pressure differential across the valve port; and the jet angle is commonly taken as \(69^{\circ}\).