## 3 Numerical Method for the PDEs  
The NMOL is a well-established technique (or rather a semianalytical method) for solving partial differential equations by typically using a finite difference relationship for the spatial derivatives and ordinary differential equations for the time derivative [27; 40]. This is essentially a two-step process: the spatial (boundary value) derivatives are approximated algebraically, for example, by finite difference or a flux limiter, on a spatial grid. The resulting system of ODEs in the initial value variable (typically time) is then integrated by an initial value ODE integrator (e.g., Euler's method, RK45, Backward Differentiation Formulae method). All of the major classes of PDEs can be accommodated (elliptic, hyperbolic, and parabolic) [27]. The semianalytical character of the formulation leads to a simple and compact algorithm, which yields accurate results with less computational effort than other techniques. In NMOL, by separating discretization of space and time, it is easy to establish stability and convergence for a wide range of problems [27; 41]. \begin{table}
\begin{tabular}{c c c} \hline Parameters in fluid heat & \(A_{f}=\frac{2a_{ref}}{R_{2}\rho/c_{ref}}\) & \(B_{f}=-\frac{\dot{m}}{\pi R_{0}^{2}\rho}\) & \(C_{f}=\frac{\lambda_{f}}{\rho/c_{ref}}\) \\ \hline Parameters in metal tube heat & \(A_{m}=\frac{2R_{0}a_{m}}{\left(R_{2}^{2}-R_{1}^{2}\right)\rho=c_{ref}}\) & \(B_{m}=-\frac{2R_{0}a_{ref}}{\left(R_{2}^{2}-R_{1}^{2}\right)\rho=c_{ref}}\) & \(C_{m}=\frac{\lambda_{m}}{\rho a_{ref}}\) \\ \hline Parameters in adsorbent heat & \(A_{a}=\frac{\Delta H}{c_{ref}}\) & \(B_{a}=-\frac{2R_{0}a_{m}}{\left(R_{2}^{2}-R_{2}^{2}\right)\rho,c_{ref}}\) & \(C_{a}=\frac{\lambda_{a}}{\rho a_{ref}}\) \\ and mass balance Equation (3) & \(A_{a}=\frac{\Delta H}{c_{ref}}\) & \(B_{a}=-\frac{2R_{0}a_{m}}{\left(R_{2}^{2}-R_{2}^{2}\right)\rho,c_{ref}}\) & \(C_{a}=\frac{\lambda_{a}}{\rho a_{ref}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Constant parameters from the set of heat and mass balance partial differential equations of the adsorbent model. The mathematical model was established comprising the set of heat and mass balance PDEs (Equations (1)-(4)), together with the initial and boundary conditions (Equations (5)-(12)). The model was solved by the NMOL. The first step was discretization of the \(x\)-variable (Figure 3). The region was divided into strips by \(N\) sections with \(N+1\) nodes, where \(dx\) was the spacing between the discretized line:  
\[dx=\frac{L}{N}. \tag{13}\]  
To avoid meaningless results (characteristic in adsorption equation oscillation) or numerical diffusion, especially with respect to convection segments [27], in this work, first- and second-order derivatives (1-4) used the five-point difference scheme of four-order truncation error discretization. Let \(F\) be any of the functions \(T_{f},\ T_{m},\ T_{a},\ a\), and \(F(i)\) means here \(F(x_{i})\). Replacing the first- and second-order derivatives at nodes \(i=3\) to \(N-1\) with respect to \(x\) using a central five-point difference formula becomes [27]  
\[\frac{dF(i)}{dx}=\frac{2F(i-2)-16F(i-1)+16F(i+1)-2F(i+2)}{24 dx}, \tag{14}\]  
\[\frac{d^{2}F(i)}{dx^{2}}=\frac{-2F(i-2)+32F(i-1)-60F(i)+32F(i+1)-2F(i+2)}{24 dx^{2}}. \tag{15}\]  
Replacing the first- and second-order derivatives at nodes \(i=1\) and \(i=2\) using a forward five-point difference formula becomes, respectively,  
\[\frac{dF(1)}{dx}=\frac{-50F(1)+96F(2)-72F(3)+32F(4)-6F(5)}{24 dx}, \tag{16}\]  
\[\frac{d^{2}F(1)}{dx^{2}}=\frac{-\frac{415}{3}F(1)+192F(2)-72F(3)+\frac{64}{3} F(4)-3F(5)-100\frac{dF(1)}{dx}dx}{24 dx^{2}}, \tag{17}\]  
\[\frac{dF(2)}{dx}=\frac{-6F(1)-20F(2)+36F(3)-12F(4)+2F(5)}{24 dx}, \tag{18}\]  
\[\frac{d^{2}F(2)}{dx^{2}}=\frac{20F(1)-30F(2)-8F(3)+28F(4)-12F(5)+2F(6)}{24 dx^ {2}}. \tag{19}\]  
Figure 3: Illustration of adsorber discretization in the \(x\)-direction (numerical method of lines, NMOL).