## 3 Mathematical Model of Thermal Balance in HCS  
### Solution for Heating Power  
The heat-generation power of HCS mainly comes from closed hydraulic pumps, hydraulic drive motors, hydraulic pipelines, the oil refilling system, etc. The leakage rate of the closed hydraulic pump is  
\[q_{\text{lp}}=\frac{C_{\text{lp}}\Delta p_{p}}{\mu_{\text{t0}}\text{e}^{-\lambda (t-t_{0})}} \tag{2}\]  
where \(C_{\text{lp}}\) is the leakage coefficient of the variable pump, \(\Delta p_{p}\) is the pressure differential between the inlet and outlet of the hydraulic pump, \(\lambda\) is the coefficient of viscosity temperature of hydraulic oil, and \(\mu_{\text{t0}}\) is the dynamic viscosity of hydraulic oil at \(t=t_{0}\).  
The heat power generated by the volumetric loss in a closed hydraulic pump is  
\[P_{H1}=\frac{C_{\text{lp}}\Delta{p_{p}}^{2}}{\mu_{\text{t0}}\text{e}^{-\lambda (t-t_{0})}} \tag{3}\]  
The mechanical efficiency of a closed hydraulic pump is  
\[\eta_{\text{pm}}=\frac{1}{1+\left(0.034+\frac{0.386}{\Delta p_{p}}\right) \frac{1}{\beta}} \tag{4}\]  
The heat power generated by the mechanical loss in a closed hydraulic pump is  
\[P_{H2}=\Delta pq\left(0.034+\frac{0.386}{\Delta p_{p}}\right)\frac{1}{\beta} \tag{5}\]  
where \(\beta\) denotes the volumetric elasticity modulus of hydraulic oil.  
Similar to hydraulic pumps, the heat-generation mechanism of hydraulic drive motors can be categorized into two types, namely heat power generated by volumetric loss and heat power generated by mechanical loss. The corresponding calculation formulas are as follows.  
Figure 2: The principle of HCS (note: the red line represents the high-pressure oil hose, the blue line represents the low-pressure oil hose, and the green line represents the pressure transmission oil hose). 1—power source, 2—coupler, 3—hydraulic closed pump, 4—oil refilling pump, 5—displacement control cylinder of closed pump, 6—displacement control valve, 7—filter, 8—relief valve of oil refilling, 9—safety valve of main circuit1, 10—one-way valve1, 11—safety valve of main circuit2, 12—one-way valve2, 13—flushing shuttle valve, 14—relief valve of cooling circuit, 15—cooler, 16—hydraulic motor, and 17—rotary load.  
\[P_{H3}=\frac{C_{\rm Im}\Delta p_{m}^{2}}{\mu_{\rm u0}{\rm e}^{-\lambda(l-t_{0})}} \tag{6}\]  
\[P_{H4}=\Delta pq\left(0.034+\frac{0.386}{\Delta p_{m}}\right)\frac{1}{\beta} \tag{7}\]  
where \(C_{\rm Im}\) denotes the leakage coefficient of the hydraulic motor and \(\Delta p_{m}\) denotes the pressure differential between the inlet and outlet of the hydraulic motor.  
The pressure drop along the hydraulic pipeline can be determined using the Darcy-Weisbach formula. The resulting heating power in hydraulic rigid tubes and flexible tubes due to this pressure drop can be calculated by Formulas (8) and (9), respectively.  
\[P_{H5}=q_{1}{\sum_{1}}^{m}\frac{75\rho l_{i}vV_{i}}{2d_{\rm Hi}^{2}} \tag{8}\]  
\[P_{H6}=q_{1}{\sum_{1}}^{m}\frac{40\rho l_{i}vV_{i}}{d_{\rm Hi}^{2}} \tag{9}\]  
where is the main circuit flow rate, \(d_{\rm Hi}\) and \(l_{i}\), are the hydraulic diameters and lengths of the pipes, \(\rho\) is the density of the hydraulic oil, \(v\) is the viscosity of the hydraulic oil, and \(V_{i}\) is the average flow velocity of the hydraulic oil in the pipes.  
The heating power of pressure loss in hydraulic bending pipe can be calculated by Formula (10).  
\[P_{\rm H7}=q_{1}{\sum_{1}}^{m}\zeta_{i}\frac{\rho V_{i}^{2}}{2} \tag{10}\]  
where \(\zeta_{i}\) is the local resistance coefficient, which is specific to the type of local resistance device and related to the Reynolds number. When the Reynolds number is greater than \(10^{5}\), \(\zeta_{i}\) is not affected by the Reynolds number.  
The heating power generated by the oil refilling pressure can be calculated by using Formula (11).  
\[P_{\rm H8}=\frac{q_{1}p_{s}}{60} \tag{11}\]  
where \(p_{s}\) denotes the oil refilling pressure.  
The total heat-generation power of a closed hydraulic circuit can be calculated using Formula (12).  
\[P_{\rm H}={\sum_{i=1}^{8}P_{Hi}} \tag{12}\]