# Numerical Analysis of Fluid Stream Division to Supply Multiple Hydraulic Receivers
## 3 Mathematical Model of the System  
The mathematical model includes equations describing pressures in lumped volumes, flow rates at specific points and equations of motion of movable elements. The model contains a pump, relief valves, flow control valves, solenoid control valves and hydraulic motors. The input signal is the pump output \(Q_{0}\). A constant value of the flow rate was assumed, taking into account the linear increase in the start-up time and the pulsation characteristic for multi-piston pumps:  
\[Q_{0}(t)=q_{rev}\cdot\omega_{0}(t)\cdot\sum_{j=0}^{z/2-1}sin(\omega_{0}\cdot t +\frac{2\,\pi\,j}{z}), \tag{1}\]  
where the piston stroke rate per revolution \(q_{rev}\) depends on the design parameters of the pump \(k_{1}\) and \(k_{2}\). For the applied version, it can be written as a function of the control signal \(u_{pump}(t)\):  
\[q_{rev}(t)=q_{max}\cdot k_{1}\cdot sin(k_{2}\cdot u_{pump}(t)), \tag{2}\]  
Figure 2: Hydraulic system diagram: 1—supply unit with a single pump; 2—flow control valve; 3—electromagnetic control valve; 4, 6—relief valve; 5—hydraulic motor. and the angular speed \(\omega_{0}(t)\) is maintained constant except the start-up time \(t_{start}\) when it increases linearly:  
\[\omega_{0}(t)= \begin{cases}\omega_{nom}\cdot t/t_{start}&\text{for}\,t<t_{start}\\ \omega_{nom}&\text{otherwise}.\end{cases}  \tag{3}\]  
The mass conservation equation formulated for the supply line has the following form:  
\[\frac{dp_{0}(t)}{dt}=B_{f}/V_{0}\cdot(Q_{0}(t)-Q_{01}(t)-Q_{02}(t)-Q_{03}(t)-Q _{04}(t)-Q_{ret}(t)). \tag{4}\]  
The relief valve opens when supply line pressure exceeds the specified value:  
\[Q_{ret}(t)=f(p_{0}(t)). \tag{5}\]  
The flow control valve mathematical model was created following the denotations shown in the cross-sectional view (Figure 3). The inflow and outflow rates, respectively, \(Q_{0i}\) and \(Q_{2i}\), where \(i=1,...,4\) is the valve index, may be calculated using the following equations:  
\[Q_{0i}(t)=A_{n1,i}\cdot\mu_{n1,i}\cdot\sqrt{2/\rho\cdot|p_{0}(t)-p_{1i}(t)|}, \tag{6}\]  
\[Q_{1i}(t)=A_{n2,i}(x_{i})\cdot\mu_{n2,i}(x_{i})\cdot\sqrt{2/\rho\cdot|p_{1i}( t)-p_{2i}(t)|}. \tag{7}\]  
The input nozzle has a fixed area depending on the insert used. For the nominal diameter \(d_{1i}=3.5\) mm, the nozzle area is \(A_{n1,i}=9.62\) mm\({}^{2}\). Therefore, the discharge coefficient has also been assumed constant, \(\mu_{n1,i}=0.7\). On the contrary, the output nozzle area and discharge coefficient depend on the spool position. The nozzle area \(A_{n2,i}=f(x_{i})\) is shown in Figure 4, while the discharge coefficient \(\mu_{n2,i}=f(x_{i})\) is determined in the next section by means of CFD method. Figure 4: Output nozzle area of the flow control valve against the spool position. Figure 3: Cross-sectional view of a flow control valve: 1—body; 2—spring tension adjuster; 3—spool; 4—control insert with a nozzle; 5—spring; A—input; B—output. The equation of the valve spool motion of mass \(m_{i}\) takes into account hydrostatic forces, viscous friction and spring tension force:  
\[m_{i}\frac{d^{2}x_{i}(t)}{dt^{2}}=F_{hs1,i}-F_{hs2,i}-F_{\phi,i}-F_{spr,i}. \tag{8}\]  
The hydrostatic force comes from the pressure difference acting on the front area of the spool: \(F_{hs1,i}-F_{hs2,i}=A_{s}\cdot(p_{1i}(t)-p_{0}(t))\). The viscous friction force is proportional to the fluid viscosity \(\eta\), damping factor \(\phi_{i}\) and spool velocity: \(F_{\phi,i}(t)=\eta\cdot\phi_{i}\cdot\frac{dx_{i}}{dt}\). Spring force includes initial spring tension and deflection due to valve spool displacement: \(F_{spr,i}(t)=(x_{spr,i}+x_{i}(t))\cdot k_{spr,i}\). The required pressure inside the flow control valve \(p_{1i}(t)\) can be calculated from the mass conservation equation:  
\[\frac{dp_{1i}(t)}{dt}=B_{f}/V_{1i}\cdot(Q_{0i}(t)-Q_{1i}(t)). \tag{9}\]  
The 2-position directional valves control the inflow of the working medium to individual hydraulic motors based on the \(sw\Omega_{i}\) control signal:  
\[Q_{2i}(t)=