# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 2 Theoretical Model and Implementation
### Analytical Model for VCS-MLI Combined Structure
#### 2.3.2 VCS-MLI Heat Transfer Model and Its Verification  
The heat fluxes in a VCS-MLI must meet the following conservation condition [21; 22; 25]:  
\[Q_{1}=Q_{2}+Q_{3} \tag{11}\]  
where \(Q_{1}\), \(Q_{2}\) and \(Q_{3}\) represent the heat fluxes flowing through the outer MLI, absorbed by hydrogen vapor in the VCS tube, and transferred through the inner MLI to the liquid hydrogen tank. They can be written as [22; 26]  
\[Q_{1}=\frac{T_{\rm h}-T_{\rm VCS}}{R_{\rm MLI-out}} \tag{12}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Parameters of VDMLI** & **Value** \\ \hline Material of Radiation Shields & Double-Aluminized Mylar \\ Residual Gas Pressure, \(p\) & 0.001 Pa \\ Accomodation Coefficient of Residual Gas, \(\alpha\) & 0.9 \\ Material of Spaces & Dacron Net \\ Empirical Coefficient of Spaces, \(C_{2}\) & 0.008 \\ Relative Density of Spaces, \(f\) & 0.02 \\ Total Number of Radiation Shields, \(m\) & 43 \\ Total Number of Spaces, \(n\) & 126 \\ Total Thickness & 33.2 mm \\ Warm Boundary Temperature, \(T_{\rm h}\) & 300 K \\ Cold Boundary Temperature, \(T_{\rm c}\) & 20 K \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main physical parameters of VDMLI [24].  
Figure 3: Diagram of VDMLI configuration.  
\[Q_{2}=\dot{m}(h_{\text{out}}-h_{\text{in}}) \tag{13}\]  
\[Q_{3}=\frac{T_{\text{VCS}}-T_{\text{c}}}{R_{\text{MLI-in}}} \tag{14}\]  
where \(T_{\text{h}}\), \(T_{\text{c}}\) and \(T_{\text{VCS}}\) are the temperatures of the warm boundary, the cold boundary and the VCS. \(R_{\text{MLI-out}}\) and \(R_{\text{MLI-in}}\) represent the total thermal resistance of the outer MLI and the inner MLI, which can be acquired using the layer-by-layer (LBL) model [22; 24; 27; 28]. \(\dot{m}\) is the mass flow rate of hydrogen vapor in the VCS tube. \(H_{\text{out}}\) and \(h_{\text{in}}\) are the enthalpies of the hydrogen vapor at the outlet and inlet of the tube. If para-ortho H\({}_{2}\) conversion happens, \(h\) is equilibrium H\({}_{2}\) enthalpy. If there is no para-ortho hydrogen conversion, \(h\) is para-H\({}_{2}\) enthalpy. In the latter case, the heat flux absorbed in the tube can be calculated as [22]  
\[Q_{2}=\dot{m}\int_{T_{\text{in}}}^{T_{\text{out}}}c_{\text{p}}\text{d}T \tag{15}\]  
where \(c_{\text{p}}\) is para-H\({}_{2}\) isobaric heat capacity; \(T_{\text{in}}\) and \(T_{\text{out}}\) are the temperatures of the tube inlet and outlet, respectively.  
The LBL model is verified to calculate the thermal resistances of the inner and outer MLI. The temperature profile is consistent with [PERSON]'s [24] experiment and calculation, as shown in Figure 4. Moreover, the calculated temperature is closer to the experimental data in the low-layer-density zone. The analytical model for the VCS-MLI structure is then validated. A 40 mm thick foam is added between the tank and the cold boundary in accordance with the insulation structure that [PERSON] [22] studied. Figure 5 shows that the temperature distributions agree well with [PERSON]'s results [22]. The maximum deviations are not more than 6% whether or not the VCS is applied. The analytical model for the VCS-MLI is applicable to subsequent analysis.  
Figure 4: Temperature profiles in VDMLI [24].