## 2 Materials and Methods
### CO\({}_{2}\) Adsorption Test
#### 2.2.4 Mathematical Model  
The mathematical model for determining the amount of the adsorbed CO\({}_{2}\) was extensively described in [PERSON] et al. [45]. The blank test highlighted the domination in the response curve of a flow-mixing in the ancillary equipment and provided a means of confronting the gas outlet responses to the CO\({}_{2}\) capture characteristics of the bed itself. A simple descriptive FOPDT model (first order plus dead time model) of the gas phase allows extracting the response of the particle phase from the measured gas phase response of the entire system under CO\({}_{2}\) capture conditions. The response curve gives the molar total gas holdup, viz, the product of the total molar throughput of gas and the area enclosed by the response curve and normalized ordinate 1 (Figures 2 and 3).  
The response curve may be used to evaluate the total amount of CO\({}_{2}\) present in the whole system as a function of time (i.e., its holdup: the moles of CO\({}_{2}\) that entered the system at time \(t\) minus those that left). The CO\({}_{2}\) holdup in the solid phase of the bed (i.e., the CO\({}_{2}\) captured by the hydrochar) is merely the total holdup of the entire system minus the holdup of the gas phase of the entire system [45]. Adsorption performance in CO\({}_{2}\)/CH\({}_{4}\) tests was evaluated estimating recovery, purity, and selectivity [22]. The gas recovery rate is the ratio of the quantity of gas recovered after the column to the fed quantity:  
\[\text{Recovery}(i)=\left(\int_{0}^{t}(y_{i}\,q)_{\text{OUT}}\,dt\right)/\left( \int_{0}^{t}(y_{i}\,q)_{\text{BLANK}}\,dt\right) \tag{1}\]  
The purity is defined by the quantity of the gas during the saturation phase divided by total gas leaving the column at the same time.  
\[\text{Purity}(i)=\left(\int_{0}^{t}(y_{i}\,q)_{\text{OUT}}\,dt\right)/\left( \int_{0}^{t}q_{\text{OUT}}\,dt\right) \tag{2}\]  
where \(y_{i}\) is the molar fraction of the specific gas and \(q\) is the total gas flow.  
The selectivity is expressed as follows:  
\[\text{Selectivity}=\frac{\text{mol}_{\text{CO${}_{2}$}}/\text{kg}_{\text{adsorbent }}}{\text{mol}_{\text{CH${}_{4}$}}/\text{kg}_{\text{adsorbent}}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \multicolumn{2}{c}{**Hydrothermal Conditions**} & \multicolumn{2}{c}{**Adsorption Conditions**} & \multirow{2}{*}{**Activation**} \\ \cline{2-2} \cline{4-5}  & **Time [min]** & **P [bar]** & & \(\text{W}_{\text{BED}}\) [g] \\ \hline \multicolumn{5}{c}{**CO\({}_{2}\)/N\({}_{2}\)**} \\ \hline HC\_200\_120\_2 & 120 & 2 & 1.135 & no \\ HC\_200\_120\_3 & 120 & 3 & 1.135 & no \\ HC\_200\_120\_4 & 120 & 4 & 1.135 & no \\ HC\_200\_120\_5 & 120 & 5 & 1.135 & no \\ HCA\_200\_0\_2 & 0 & 2 & 0.695 & yes \\ HCA\_200\_0\_3 & 0 & 3 & 0.695 & yes \\ HCA\_200\_0\_4 & 0 & 4 & 0.695 & yes \\ HCA\_200\_0\_5 & 0 & 5 & 0.695 & yes \\ HCA\_200\_120\_2 & 120 & 2 & 1.135 & yes \\ HCA\_200\_120\_3 & 120 & 3 & 1.135 & yes \\ HCA\_200\_120\_4 & 120 & 4 & 1.135 & yes \\ HCA\_200\_120\_5 & 120 & 5 & 1.135 & yes \\ \hline \multicolumn{5}{c}{**CO\({}_{2}\)/CH\({}_{4}\)**} \\ \hline HCA\_200\_0\_2 & 0 & 2 & 0.695 & yes \\ HCA\_200\_0\_5 & 0 & 5 & 0.695 & yes \\ \hline \hline \end{tabular}
\end{table}
Table 1: Synopsis of tests conditions.