# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 3 Results and Discussion
### Convenience of the Statistical Thermodynamic Method
the program with Method 1.3 is about 33% that of the program with Method 1.1 and about 0.18% that of the program with Method 1.2. The third test is to calculate the thermal insulation performance of the VCS-VDMLI structure. The running speeds of the entire calculation procedure with different methods are compared. The VCS temperature is set as a loop variable. The initial VCS temperature is 60 K, and the VCS temperature increases 0.01 K after each loop. The iteration stops when three heat fluxes in the VCS-VDMLI meet the conservation condition in Equation (11) and the temperature profile of the VDMLI no longer changes with iterations. The test results are shown in Table 6. It can be found that the statistical thermodynamic method still saves the most time when the numbers of iterations are about the same. The average running time of the program with Method 1.3 is about 0.28% that of the program with Method 1.2. Although spline interpolation costs much less time than reading REFPROP data, the running time is more than twice that of statistical thermodynamic calculation. It can be found that the statistical thermodynamic method is feasible and efficient in engineering practice. The developed computational programs based on this method can be adopted when hydrogen's thermophysical properties are needed. In addition to accuracy and convenience, the statistical thermodynamic method based on explicit Helmholtz free energy EOSs has thermodynamic unity. Thermophysical properties, such as enthalpy, entropy, specific heat capacity, speed of sound, etc., can be obtained by taking first-order or at most second-order derivatives of the Helmholtz free energy with respect to temperature or density, which is much simpler than calculating enthalpies from integrals using EOSs explicit in pressure. The programs for determining thermophysical properties of different fluids have the same frame. All the coefficients in the reduced Helmholtz free energy equations are known. They can be written as elements in arrays and conveniently replaced when the fluid changes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Method** & **Average Running Time/s** & **Number of Iterations** \\ \hline \multirow{4}{*}{With Para-ortho Conversion} & Spline Interpolation (1.1) & 0.2668 & 5198 \\  & REFPROP Data (1.2) & 37.0464 & 5224 \\  & Statistical Thermodynamic & 0.1030 & 5182 \\  & Calculation (1.3) & 0.1947 & 6108 \\  & Spline Interpolation (2.1) & 1.0806 & 6137 \\  & Integral of \(c_{\text{p}}\) with & 15.3030 & 6102 \\  & Temperature (2.3) & & \\  & Statistical Thermodynamic & 0.0859 & 6090 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average running time of programs for calculating VCS-VDMLI performances. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Method** & **Average Running Time/s** & **Average Running Time/s** \\  & & **(10,001 Circulations)** & **(20,001 Circulations)** \\ \hline \multirow{4}{*}{With Para-ortho Conversion} & Spline Interpolation (1.1) & 0.3253 & 0.6562 \\  & REFPROP Data (1.2) & 56.8849 & 113.5774 \\  & Statistical Thermodynamic & 0.1077 & 0.2124 \\  & Calculation (1.3) & 0.1731 & 0.3384 \\  & Spline Interpolation (2.1) & 1.212 & 2.3988 \\  & Integral of \(c_{\text{p}}\) with & & \\  & Temperature (2.3) & & \\  & Statistical Thermodynamic & 0.0518 & 0.1039 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average running time of programs for calculating hydrogen enthalpy differences.