# Flow Analysis of a Novel, Three-Way Cartridge Flow Control Valve
## 3 Methodology
### Mathematical Model  
The formulated mathematical model includes a variable speed pump, a studied flow control valve and two relief valves (Figure 4). A vane pump with eleven vanes was used as the supply unit. The theoretical flow rate against the rotational angle \(\alpha\) depends on the eccentricity \(e_{p}\). It can be calculated from the following equation:  
\[Q_{0}(\alpha)=[r-\frac{e_{p}}{2}\cdot(1+cos(\alpha))]\cdot\omega_{0}\cdot b \cdot e_{p}\cdot(1-cos(\alpha)), \tag{1}\]  
where the pump design parameters are \(r=43\,\mathrm{mm}\) and \(b=20\,\mathrm{mm}\). The required flow rate of \(Q_{0}=30\,\mathrm{dm}^{3}\cdot\mathrm{min}^{-1}\) is obtained for the eccentricity value \(e_{p}=2.5\,\mathrm{mm}\). The resulting non-uniformity of the flow is \(2.13\,\%\). Nominal value of rotational speed  
Figure 3: Model of the working fluid; P, A, B—connection ports. Figure 2: Cartridge flow regulator: (**a**) view, (**b**) section; 1—body, 2—adjuster, 3—sleeve, 4—spool, 5—nozzle, 6—spring, 7—sealing; P—supply port, A—output port, B—relief port. \(\omega_{nom}=1440\) rot \(\cdot\) min\({}^{-1}\) is reached after the starting time \(t_{start}\), when it increases linearly from zero:  
\[\omega_{0}(t)= \begin{cases}\omega_{nom}\cdot t/t_{start}&\text{for}\,t<t_{start}\\ \omega_{nom}&\text{otherwise}.\end{cases}  \tag{2}\]  
Having determined pump flow rate \(Q_{0}\) as the input, a mass conservation equation can be formulated for the supply line of volume \(V_{0}\):  
\[\frac{dp_{0}(t)}{dt}=\frac{B_{f}}{V_{0}}\cdot(Q_{0}(t)-Q_{r}(t)-Q_{1}(t)-Q_{B}( t)). \tag{3}\]  
The safety relief valve (2), for which a detailed model one can find, e.g., in [23] is used to protect the system against excessive pressure build-up. The load-generating relief valve (4) has an analogous structure. The model uses the safety valve characteristics described by the equation for the range of its nominal operation:  
\[Q_{r}(t)=\frac{p_{0}-p_{r}}{\Delta p_{v2}}\cdot Q_{v2,nom}, \tag{4}\]  
\[Q_{2}(t)=\frac{p_{2}-p_{r}}{\Delta p_{v4}}\cdot Q_{v4,nom}. \tag{5}\]  
The flow rates through the inlet throttle nozzle \(n_{1}\), as well as the \(g_{A}\) and \(g_{B}\) gaps of the flow control valve (3), can be determined from Equations (6)-(8), respectively:  
\[Q_{1}(t)=A_{n1}\cdot\mu_{n1}\cdot\sqrt{2/\rho\cdot|p_{0}(t)-p_{1}(t)|}, \tag{6}\]  
\[Q_{A}(t)=A_{gA}(x_{1})\cdot\mu_{gA}(x_{1})\cdot\sqrt{2/\rho\cdot|p_{1}(t)-p_{2 }(t)|}. \tag{7}\]  
\[Q_{B}(t)=A_{gB}(x_{1})\cdot\mu_{gB}(x_{1})\cdot\sqrt{2/\rho\cdot|p_{1}(t)-p_{r }(t)|}. \tag{8}\]  
The input nozzle is changeable. The following diameters were established for studies: \(d_{1}=2.5,3.5,4.5,5.5\) mm. However, for the given diameter, the nozzle area is constant. Therefore, the discharge coefficient was also assumed to be constant, \(\mu_{n1}=0.7\). The areas of \(gA\) and \(gB\) gaps depend on the \(x_{1}\) spool position, according to Figure 5. Discharge coefficients of these gaps were assumed as \(\mu_{gA}=\mu_{gB}=0.6\). The main outlet gap of the valve \(A_{gA}\) is created by exposing four radial holes in the spool and the sleeve. Its cross-section is a non-linear function of the spool displacement. Similarly, the \(A_{gB}\) discharge slot area of the annular shape also depends on the spool position, shown in Figure 5.  
Figure 4: Hydraulic system diagram with denotations for mathematical model formulation: 1—pump, 2—safety relief valve, 3—flow control valve, 4—load line relief valve; P, A, B—connection ports. The flow control valve spool of the \(m_{s}\) mass is subjected to the hydrostatic force \(F_{hs}\) resulting from the difference between \(p_{0}\) and \(p_{1}\) pressures acting on the spool area \(A_{s}\), the spring \(F_{spr}\), viscous friction \(F_{\phi}\) and hydrodynamic reactions caused by the change of flow direction in the \(g_{A}\) and \(g_{B}\) gaps, \(F_{hdA}\) and \(F_{hdB}\), respectively. The hydrostatic force \(F_{hs}(t)=(p_{0}(t)-p_{1}(t))\cdot A_{s}\).