## 5 Results
Figure 6 shows calculated uptake for the nodes \(i=1\) (lowest curve), \(i=11\) (in the middle), and \(i=21\) (highest curve) of the adsorber column. For the above-introduced calculation case of the adsorption heat pump, there was no significant difference in results based on the discussed fixed RK method as well as adaptive step size numerical fourth-order methods. The absolute temperature difference was smaller than 1% and was irrelevant from the perspective of engineering applications. In our experience, the Runge-Kutta and Gear routines give adequate results for relatively simple one-dimensional adsorption models. Figure 5: Results comparison of the Runge–Kutta fixed step size fourth-order (RKfixed), the Runge–Kutta–Fehlberg 4.5 th-order method with a variable step size (RK45), and the Gear Backward Differentiation Formulae (Gear BDF) methods, calculated temperatures of the modeled fluid (water) \(T_{f}\), metal tube \(T_{m}\), and adsorbent \(T_{a}\), \(N=20\): (**a**) node \(i=1\), (**b**) node \(i=11\), and (**c**) node \(i=21\). Table 3 presents additional comparisons of discretization, accuracy, CPU (Central Processing Unit) process time, and number of steps for the RKfixed, RK45, Gear BDF numerical methods as well as some of limitations. The Gear BDF routine \"ode15s\" (BDF \"on\") integrated a set of equations in the smallest number of steps and had the shortest processing time. For example, for the 20 sections in MOL (\(N=20\)) and accuracy (1e-7), it takes 9525 steps in 10 s of CPU time. By contrast, the RK45 routine \"ode45\" requires 16,776 steps, while RKfixed time step size (\(dt=0.1\) s) takes 36,010 steps in 122 s of CPU time. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{2}{c}{**Description**} & \multicolumn{1}{c}{**RKfixed (Classical)**} & \multicolumn{1}{c}{**RK45**} & \multicolumn{1}{c}{**Gear BDF**} \\ \hline \multirow{3}{*}{\(N=20\)} & accuracy & time step 0.5 s & 1e-5 & 1e-4 \\ \cline{2-5}  & CPU time & 12.6 s & 5.6 s & 4.7 s \\ \cline{2-5}  & steps & 7201 & 9537 & 3633 \\ \hline \multirow{3}{*}{\(N=30\)} & accuracy & time step 0.5 s & 1e-5 & 1e-4 \\ \cline{2-5}  & CPU time & 14.1 s & 6.5 s & 6.6 s \\ \cline{2-5}  & steps & 7201 & 10,953 & 5207 \\ \hline \multirow{3}{*}{\(N=20\)} & accuracy & time step 0.1 s & 1e-7 & 1e-7 \\ \cline{2-5}  & CPU time & 122.2 s & 10.1 s & 10.7 s \\ \cline{2-5}  & steps & 36,001 & 16,776 & 9525 \\ \hline \multirow{3}{*}{\(N=20\)} & accuracy & time step 1 s & 1e-4 no results & 1e-2 \\ \cline{2-5}  & CPU time & 6.1 s & 6.6 s & 2.3 s \\ \cline{2-5}  & steps & 3601 & 9345 & 782 \\ \hline \multirow{3}{*}{Examples of limitations} & \multicolumn{2}{c}{\(N<5\) no results,} & \multicolumn{2}{c}{\(N<18\) no results,} & \multicolumn{2}{c}{\(N<5\) no results,} \\  & accuracy \(\leq(16\)-4) no results, & \(\underline{ \begin{array}{c}N<18\end{array} \) no results,} & \multicolumn{2}{c}{\(N=20\) (1e-1) results without oscillation,} \\  & \(5\leq N\leq 10\) significant oscillations, & \(\underline{\begin{array}{c}\underline{\begin{array}{c}\underline{\begin{array}{c} \underlineAdditionally, the research identified restrictions on the use of all these numerical methods. Examples of limitations are presented in Table 3. In the Runge-Kutta fourth-order fixed time step method, for the above-introduced calculation case of adsorption, it was not possible to obtain results for discretization \(N<5\), fixed time step \(>1\) s or accuracy \(\leq\) (1e-4). In the range of \(5\leq N\leq 10\), there was no realistic result. In the Runge-Kutta-Fehlberg 4.5 th-order adaptive step method, there were no results for \(N<18\) or \(\dot{m_{f}}>0.3\ \frac{\text{kg}}{\text{s}}\). Since this method is mainly used for nonstiff systems of equations--and apparently for listed parameters, the stiffness of solving systems of equations is growing--there is a decreasing possibility of obtaining real results with this method. For use in a wider operating range, there is the Gear BDF method.