# Flow Analysis of a Novel, Three-Way Cartridge Flow Control Valve
## 4 Results of Numerical Simulations
### Results of CFD Simulations in Ansys  
In the first step, maps of pressure and velocity distribution were determined, assuming the typical value of the supply pressure \(p_{0}=11\,\mathrm{MPa}\) and the same load on both outputs \(A\) and \(B\) with the pressure \(p_{gA}=p_{gB}=10\,\mathrm{MPa}\). A default nozzle diameter \(d_{n1}=3.5\,\mathrm{mm}\) and a spool displacement \(x_{1}=0.5\,\mathrm{mm}\) were used. As a result, the average value of the fluid velocity in the supply port \(v_{p,avg}=19\,\mathrm{m}\)\(\,\mathrm{s}^{-1}\) was obtained. Figure 7a,b shows the pressure and velocity distributions in an isometric view on three mutually perpendicular planes drawn through the axes of the main valve channels. In turn, Figure 8a,b presents the same parameters in the cross-section relative to the longitudinal symmetry plane of the valve. Figure 9 shows an enlargement of the area around the flow slot of the \(B\) channel with an estimate of the jet angle value. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Turbulence** & **Turbulence** & **Turb. Viscosity** & **Kinetic Energy** & \multicolumn{3}{c}{**Kinetic Energy**} \\
**Intensity** & **Length Scale** & **Constant** & **Constant** & **Dissipation** & **Constants** \\ \hline \(I\) (\%) & \(\ell\) (mm) & C\({}_{\mu}\) (-) & \(s_{k}\) (-) & \(s_{\varepsilon}\) (-) & C\({}_{1\varepsilon}\) (-) & C\({}_{2\varepsilon}\) (-) \\
4.3–6.3 & 0.12–0.51 & 0.09 & 1.0 & 1.3 & 1.44 & 1.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the \(k\)-\(\varepsilon\) model. As presented in Figures 7a and 8a, the inlet pressure in the \(P\) port is slightly higher than the pressure resulting from the loading of the outlet ports. The valve divides the inlet fluid stream into two streams directed to \(A\) and \(B\) ports. Due to the differences in cross-sectional areas of the gaps and the occurrence of the \(n_{1}\) throttling nozzle, the division is uneven. The stream directed to channel \(A\) has an average velocity of 6.5 m/s, whereas to channel \(B\) it is 12.5 m/s. The fluid velocity increases locally through the \(B\) channel, and the stream bends. It significantly impacts the forces associated with the working medium flow. As seen in Figure 9, the cross-sectional area of the slot is perpendicular to the longitudinal axis of the valve. Hence, the fluid stream inflows to the gap at an angle \(\theta_{B,in}=84^{\circ}\), while the outflow bends at an angle of \(87^{\circ}\). Since the axial component of the flow forces is proportional to the cosine of the fluid stream jet angle, and the outflow angle is greater than the inflow one, the flow force in the \(B\) gap acts in the direction of the spring force. In the case of the \(A\) channel, the medium flows out through radially arranged holes (Figures 7b and 8b). Hence, the input and output jet angles are close to \(90^{\circ}\), and the impact of the flow forces on the spool in this gap is negligible. Flow through the flow control valve when ports A and B are loaded with different pressures are shown in Figures 10 and 11. Figure 8: CFD results for equal loads on both outputs in the longitudinal cross-section: (**a**) pressure distribution, (**b**) velocity distribution; P, A, B—connection ports. Figure 7: CFD results for equal loads on both outputs in the isometric view: (**a**) pressure distribution, (**b**) velocity distribution; P, A, B—connection ports. In this case, the pressure \(p_{0}=5.5\) MPa was assumed at the inlet to the valve, \(p_{2}=p_{A}=5.0\) MPa at port A, and \(p_{B}=p_{r}=1.0\) MPa at port \(B\). With these data, the average value of liquid flow velocity at the valve input was 31.6 m s\({}^{-1}\), in the \(A\) regulated channel \(v=5.2\) m s\({}^{-1}\) and in the \(B\) relief channel \(v=26.4\) m s\({}^{-1}\). As shown in Figure 10, despite a significant surplus of the supply flow rate over the controlled value in \(B\) channel, the inlet pressure is only 0.5 MPa higher.