# Determination of Hydrogen's Thermophysical Properties Using a Statistical Thermodynamic Method
## 3 Results and Discussion
### Convenience of the Statistical Thermodynamic Method  
Convenience is another advantage of the statistical thermodynamic method. Three tests are conducted on the same computer with MATLAB 2022a to assess the convenience of the statistical thermodynamic method. The running speeds of programs under different methods for determining hydrogen's thermophysical properties are compared. To guarantee that all methods for determining hydrogen's thermophysical properties are compared under the same conditions, no parallel toolbox calculations or other acceleration modules are used in any of calculations, so that the interference of all acceleration measures in computational speed can be eliminated. During these tests, the computer maintains the same operating conditions. The hardware characteristics of the computer described in the present study are as follows: the volume and frequency of RAM are 16.0 GB and 2.60 GHz, respectively; the type of CPU is 11 th Gen Intel(R) Core(TM) i5-11400F; the type of GPU is NVIDIA GeForce GT 730. The most direct way to judge the running speed of a MATLAB program is to measure its running time by adding \"tic\" and \"toc\" (two timing functions of MATLAB) to the start and the end of the program, respectively. \"toc\" reads the elapsed time expressed in seconds since the stopwatch timer started by the call to the \"tic\" function. If the running time is much less than 0.1 s, the program will run too fast for \"tic\" and \"toc\" to provide useful data. In this case, the timed section in the program must be run in a loop, and the total running time of the loop is measured. To ensure the reliability of measuring the running time, every program under a specific method must be run 20 times under the same condition. The average running time indicates the running speed of the program. The first test is to calculate para-H\({}_{2}\) enthalpies at 0.1 MPa from 20 K to 300 K in 0.02 K or 0.01 K increments, and the corresponding circulation numbers are 14,001 and 28,001, respectively. The comparison of the running speeds of programs with three methods is listed in Table 4. It is indicated that reading REFPROP data in a program for para-H\({}_{2}\) enthalpy calculation is the most time-consuming, requiring about 23 times as much time as statistical thermodynamic calculation. Statistical thermodynamic calculation can save much more time if the circulation numbers are multiplied. The second test is to calculate the heat absorption of the hydrogen vapor in a VCS tube, or the hydrogen enthalpy difference at the outlet and inlet of the tube. The inlet temperature is 20 K. The outlet temperature varies from 100 K to 300 K in 0.02 K or 0.01 K increments, and the corresponding circulation numbers are 10,001 or 20,001. Two cases are considered: (1) the hydrogen vapor always maintains an equilibrium state if para-ortho H\({}_{2}\) conversion is employed; (2) the hydrogen vapor contains para-H\({}_{2}\) if para-ortho H\({}_{2}\) conversion is not employed. Table 5 shows that much more time is consumed with the methods of calling enthalpy data from REFPROP and integrating \(c_{\rm p}\). It is worth noting that acquiring equilibrium H\({}_{2}\) enthalpies is much more time-consuming than acquiring para-H\({}_{2}\) enthalpies. Therefore, reading REFPROP data is not recommended when para-ortho H\({}_{2}\) conversion is considered. Statistical thermodynamic calculation, however, is more efficient and faster. Whether para-ortho H\({}_{2}\) conversion is considered or not, the statistical thermodynamic method saves the most time. For instance, the average running time of \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Method**} & **Average Running Time/s** & **Average Running Time/s** \\  & **(14,001 Circulations)** & **(28,001 Circulations)** \\ \hline Spline Interpolation & 0.1735 & 0.3356 \\ REFPROP Data & 1.6971 & 3.3626 \\ Statistical Thermodynamic Calculation & 0.0716 & 0.1463 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average running time of para-H\({}_{2}\) enthalpy calculation programs.