## Modeling
### Steady State Heat Transfer Model of Connection Section  
The connecting section of the back plate heat pipe system includes the rising gas pipe, the falling liquid pipe, the gas collecting pipe and the liquid collecting pipe in the evaporator and the condenser. These adiabatic sections are considered to have no heat exchange with the indoor environment, no heat loss, and the refrigerant flows in the adiabatic section with equal enthalpy, only pressure loss.  
(1) Riser and downcomer.  
The resistance of riser and downcomer mainly includes the local resistance and gravity resistance of the working fluid flow. The calculation formula of pressure drop is as follows:  
\[\Delta P=\Delta P_{c}+\Delta P_{g} \tag{39}\]  
\[\Delta P_{c}=(\lambda l+\sum\xi)\frac{\rho_{r}u_{r}^{2}}{2g} \tag{40}\]  
\[\Delta P_{g}=\rho_{r}gH. \tag{41}\]  
The refrigerant mass in the riser and downcomer:  
\[M_{g1}=\rho_{r}A_{g1}L_{g1}. \tag{42}\]  
(2) Gas collection pipe and liquid collection pipe.  
There are many flow channels in the evaporator and condenser. During refrigerant distribution and collection, there is a pressure drop loss of sudden expansion and contraction. The pressure drop loss can be calculated by the following formula:  
\[\Delta P_{e}=\Big{[}(1-\delta)^{2}-0.4\delta\Big{]}\frac{\rho V^{2}}{2} \tag{43}\]  
\[\Delta P_{C}=(0.8-0.4\delta^{2})\frac{\rho V^{2}}{2}. \tag{44}\]  
The mass of refrigerant in the gas collecting pipe or liquid collecting pipe:  
\[M_{g2}=\rho_{r}A_{g2}L_{g2}. \tag{45}\]  
Programming Calculation of Steady-State Heat Transfer of the Micro-Channel Heat Pipe Backplane System  
The operation of the whole system meets the three conservation laws of energy conservation, momentum conservation and mass conservation. Specifically, the temperature, pressure and enthalpy of the refrigerant in each component do not change after the completion of a cycle, which is equal to the value before the completion of the cycle. The mass flow of the refrigerant is consistent in the system. The specific calculation steps and calculation flowchart Figure 4 are as follows:1. Input the structural dimension parameters of each component of the system, input the given working condition conditions, including the charging capacity M0, the temperature and flow at the inlet of the backplane, and CDU.
2. Start the calculation from the evaporator, assuming that in the initial calculation, the inlet enthalpy value is H0, the inlet pressure P0 and the mass flow G0 of the evaporator.
3. Use the refrigerant flow sequence to calculate the heat exchange, pressure drop and refrigerant mass of evaporator, gas collector, rising gas pipe, condenser, liquid collector and falling liquid pipe, respectively, according to the component model. The refrigerant outlet pressure, temperature and enthalpy of each component are assigned to the next component as the inlet parameter, and the refrigerant enthalpy H1 and pressure at the outlet are calculated after calculating the last component falling liquid pipe Force P1 and total mass M1.
4. Compare refrigerant enthalpy H1, pressure P1 and initial assumed values H0 and P0. If accuracy requirements are met, continue to compare mass M1 and initial assumed value M0. If not, correct initial assumed value H0 and return to step (2) for calculation. If mass M1 and initial assumed value M0 do not meet accuracy requirements, correct initial assumed value P0 and return to step (2) for calculation.
5. Repeat steps (2), (3) and (4) until the error convergence conditions are met, and output the import and export parameters of each component.  
Figure 4: Flow chart of steady-state heat transfer calculation for the micro-channel backplane heat pipe system.