## 2 Experiment  
### Materials  
#### 2.1.1 Thermophysical Properties of the Adsorbents (MSC-30)  
The physical properties of MSC-30 in this work were measured and compared with some previously reported data such as those of [PERSON] et al. [30], presenting methane's adsorption characteristics on MSC-30 using the gravimetric method and showing good agreement with the published specifications. [PERSON] et al. [30] evaluated essential aspects for the adsorption process on MSC-30, such as pore surface area, micropore volume, and pore size distribution, using the standard procedure of nitrogen adsorption at a cryogenic temperature of liquid nitrogen under atmospheric pressure (\(-196\)\({}^{\circ}\)C/1 atm). The specific heat capacity necessary for the system performance analysis was adopted from the work of [PERSON] et al. [21]. The surface characteristics of the MSC-30 were examined through the standard nitrogen adsorption on a surface characterization instrument (3 Flex, Micromertics, Norcross, GA, USA) at \(-196\)\({}^{\circ}\)C. The samples' preparation consisted only of a degassing step at 200 \({}^{\circ}\)C for 8 h in a high vacuum prior to the measurement. The SEM images were taken by the field emission scanning electron microscope (JSM-7900F, JOEL, Peabody, MA, USA). The heat capacity was measured by a differential scanning calorimetry (DSC) instrument (DSC-60, Shimadzu, Kyoto, Japan) by comparing the measured values to a known standard sample of alpha-alumina according to the method in [PERSON] et al. [31]. Furthermore, to obtain the temperature-dependent relation, a polynomial function in a simple form of \(c_{p,Ads}(T)=a_{0}+a_{1}T+a_{2}T^{2}+a_{3}T^{3}+a_{4}T^{4}\) was used in this study. The nitrogen adsorption/desorption isotherm on MSC-30 is given in Figure 1. Based on these adsorption data, the surface area has been obtained through the BET isotherm analysis given Equation (1), which is the linear transformation of BET isotherm as a function of relative pressure:  
\[\frac{1}{c\left(\frac{P_{0}}{P}-1\right)}=\frac{1}{c_{m}}+\frac{C-1}{c_{m}C} \left(\frac{P}{P_{0}}\right) \tag{1}\]  
where \(c\) is the adsorbed amount, \(c_{m}\) is the monolayer adsorbed amount, and \(P\) and \(P_{0}\) are the absolute pressure and saturation pressure, respectively. From filling in the data of adsorbed amount \(c\) and respective absolute pressures \(P\), we can obtain the \(y\)-axis intercept \(Y_{Itc}\), the first term of the right side in Equation (1), and the slope \(Y_{Slp}\), the second term of the right side. The standard model then calculates the surface area \(S_{BET}\) using the terms \(Y_{Slp}\) and \(Y_{Itc}\) as follows:  
\[S_{BET}=\frac{a_{N2}N_{A}}{V_{M}\left(Y_{Slp}+Y_{Itc}\right)} \tag{2}\]  
where \(N_{A}\) is the Avogadro constant, \(V_{M}\) is the standard molar volume (\(V_{M}\)= 22.414 dm\({}^{3}\)/mol), and \(a_{N2}\) is the cross-sectional area of the adsorbed molecule planar to the adsorbed surface, here the nitrogen molecule. We choose to follow the general recommendation for the nitrogen molecular cross-sectional area of nitrogen molecules \(a_{N2}\) = 0.162 nm\({}^{2}\) on carbon-based adsorbents [32]. The resulting BET isotherm parameters are calculated based on the linear fit in the relative pressure region of \(P/P_{0}\) = 0.05 to 0.3. The resulting BET constant is \(C\) = 147.968, with the monolayer adsorption equal to \(c_{m}\) = 31.29 mmol/g. The correlation coefficient of the BET linear fitting showed a very good agreement with the experimental data and was estimated to be 0.99924. The correlation coefficient \(R\) was calculated using the standard formula:  
\[R=\frac{\sum(x_{i}-\overline{x})(y_{i}-\overline{y})/N}{\sigma_{x}\sigma_{y}} \sigma_{p}=\sqrt{\frac{\sum(p_{i}-\overline{p})^{2}}{N}} \tag{3}\]  
where, \(x_{i}\) and \(y_{i}\) are the individual values of the \(x\)-axis and \(y\)-axis, while \(\overline{x}\) and \(\overline{y}\) are the mean values of the sample sets. \(N\) states the size of the sample set.