## 5 Results
The variables studied were: temperature of desorption, which was equal to the hot driving water temperature \(T_{des}=363\) K; temperature of the adsorption process, equal to the cooling water temperature \(T_{ads}=313\) K; temperature of evaporation \(T_{evap}=288\) K; and temperature of condensation \(T_{cond}=313\) K. Cycle duration in the examined case was \(\tau_{cycle}=\tau_{ads}+\tau_{des}=360\) s, as well as heating/cooling medium (water) flow rate \(\dot{m}_{ads}=\dot{m}_{des}=0.01\) kg/s. Parameters of adsorption equilibrium described by the Dubinin-Astakhov (D-A) equation used for the calculations were: limiting adsorption amount \(a_{0}=0.35\) kg kg\({}^{-1}\), characteristic energy of the adsorbent \(\beta\)\(E_{0}=3780.8\) mol\({}^{-1}\), constant in the D-A Equation \(n=1.016\), process constant \(D_{0}=0.000254\) m\({}^{2}\) s\({}^{-1}\), and activation energy \(E_{a}=42,000\) J mol\({}^{-1}\)[15; 34]. Other parameter values used for the calculations are given in Table 2. Due to the initial starter cycles and the characteristics of the adsorption devices, the calculations were carried out for 10 adsorber/desorber cycles. To compare the results of the numerical methods in the last two cycles, stable operation of the modeled adsorption column (adsorber/desorber) was used. \begin{table}
\begin{tabular}{c c c c} \hline
**Description** & **Symbol** & **Unit** & **Value** \\ \hline Particle radius & \(r_{a}\) & mm & 0.1 \\ Bulk density of the bed particles & \(\rho_{a}\) & kg m\({}^{-3}\) & 600 \\ Bulk density of the tube material & \(\dot{p}_{m}\) & kg m\({}^{-3}\) & 8936 \\ Average specific heat of the bed & \(c_{ps}\) & J kg\({}^{-1}\) k\({}^{-1}\) & 924 \\ Average specific heat of the tube & \(c_{pm}\) & J kg\({}^{-1}\) K\({}^{-1}\) & 383 \\ Average specific heat of the water & \(c_{pf}\) & J kg\({}^{-1}\) K\({}^{-1}\) & 4182 \\ Thermal conductivity of the bed & \(\lambda_{a}\) & W m\({}^{-1}\) K\({}^{-1}\) & 0.175 \\ Thermal conductivity of the tube & \(\lambda_{m}\) & W m\({}^{-1}\) K\({}^{-1}\) & 401 \\ Heat transfer coefficient (fluid–metal tube) & \(a_{mf}\) & W m\({}^{-2}\) K\({}^{-1}\) & 1000 \\ Heat transfer coefficient (metal tube–adsorbent) & \(a_{am}\) & W m\({}^{-2}\) K\({}^{-1}\) & 1000 \\ Length of the tube & \(L_{t}\) & m & 1.5 \\ Duration of the cycle (\(\tau_{ads}+\tau_{des}\)) & \(\tau_{cycle}\) & s & 180 + 180 \\ Tube internal radius R1 & R1 & mm & 10 \\ Tube outer radius R2 & R2 & mm & 11 \\ Adsorber outer radius R3 & R3 & mm & 26 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the calculation (sourced from [15; 34; 44; 45]). Figure 4: Comparison between the time variations of the presented numerical methods and experimental averaged bed temperatures. ([PERSON] et al. [21]). Figure 5 shows the results comparison of the calculated temperatures of the modeled fluid (water) \(T_{f}\), metal tube \(T_{m}\), and adsorbent \(T_{a}\) in function of time based on the ODE numerical methods RKfixed, RK45, and Gear BDF for the described one-dimensional model of the adsorber. Temperatures at the beginning of the adsorption column are presented in Figure 5, while temperatures in the middle and at the end of the adsorption column are presented in Figure 5,c, respectively. The graphs in Figure 5 show the temperatures curves in the following order (as viewed in desorption--the upper part of the graph): the highest curves are fluid temperatures, the middle curves shows the metal tube temperatures, and the bottom curves show the adsorbent temperatures. It can be observed that when the change of \(T_{f}\) for \(x=0\), \(i=1\) takes place, the oscillations of \(T_{f}\), \(T_{m}\), and \(T_{a}\) appear. Such oscillations are characteristic of numerical solutions of so-called stiff problems. Stiffness is a property of a system of differential equations rather than its solution. For such problems, an implicit multistep method (e.g., Gear BDF) rather than explicit multistep methods (e.g., RKfixed or RK45) should be used.