# Numerical Analysis of Fluid Stream Division to Supply Multiple Hydraulic Receivers
## 5 Modeling and Simulations in Matlab/Simulink
### Simulations with Constant Load Torque  
In the first stage, the system operation was analyzed under the condition that all hydraulic motors are loaded with the same torque of a constant value. Taking into account technical parameters of the modeled system components, two load levels were adopted, a low moment value of \(T_{Lm}=5.0\) N m and a high one \(T_{Lm}=75.0\) N m. The obtained results in both cases are shown in Figure 11 and Figure 12, respectively.  
In each case, the initial tension of springs in all flow control valves was adjusted to obtain the required rotation speed at the given load regarding the valve characteristics: an increase in pressure causes a slight decrease in the flow rate. Pump delivery was set to \(108\)\(\text{ }\text{ }\text{dm}^{3}\,\text{min}^{-1}\), which provides the necessary flow rate to all receivers with a slight surplus. The pressure in the supply line was limited to \(p_{0}=16\)\(\text{ }\text{MPa}\) through a relief valve. This value ensures the sufficiently high driving torque of all hydraulic motors. It should be noted that maintaining the supply line pressure too high leads to significant energy losses.  
The subsequent test involved loading each hydraulic motor with a different loading moment. The assumed values were \(T_{Lm1}=5\) N m, \(T_{Lm2}=25\) N m, \(T_{Lm3}=50\) N m and \(T_{Lm4}=75\) N m. The flow rates and pressures in the individual lines of the system are shown in Figure 13a, while the graphs of loading moments and the obtained rotational speeds are demonstrated in Figure 13b, respectively.  
Figure 11: Results with constant load moment of low value \(T_{Lm}=5.0\) N m: (**a**) flow rates and pressures; (**b**) rotational speed of hydraulic motors and load moments.  
Figure 12: Results with constant load moment of high value \(T_{Lm}=75.0\) N m: (**a**) flow rates and pressures; (**b**) rotational speed of hydraulic motors and load moments.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Start-Up** & \multicolumn{3}{c}{**Flow Control Valve**} & \multicolumn{3}{c}{**Hydraulic Motor**} \\
**Time** & **Spool Diam.** & **\(n\)1 Diam.** & **\(n\)2 Diam.** & **Spring Rate** & **Vol. Disp.** & **Mom. Inertia** \\ \hline \(t_{start}\) & \(\varnothing D_{vj}\) & \(\varnothing D_{n1j}\) & \(\varnothing D_{n2j}\) & \(k_{spj}\) & \(D_{n1i}\) & \(l_{m2j}\) \\ s & mm & mm & mm & N mm\({}^{-1}\) & cm\({}^{3}\) rot\({}^{-1}\) & kg m\({}^{2}\) \\ \hline
0.25 & 12.0 & 3.5 & 3.0 & 20.0 & 5.02 & 0.008 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Fixed values of simulation model parameters.  
To ensure the correct system operation in a wide range of load torques, it was necessary to modify the settings of the valve springs. In particular, the initial tension of all springs, in this case, had to ensure the required flow rate at maximum load. The obtained results show that this causes a slight increase in the rotational speed of the engines with the least load. However, from the view of the system's proper operation, it is acceptable, particularly since the maximum difference in the rotational speed values does not exceed 5%.  
As shown by the analysis carried out at this stage, flow control valves can provide the sufficient flow rate required to obtain the appropriate speed of rotation of the hydraulic motors, regardless of the pressure differences caused by various load torque values.