## 2 Methods
### Governing Equations
#### 2.2.8 Numerical Procedure and Model Validation  
The system of equations described in this study was solved simultaneously by adopting six physics in COMSOL Multiphysics software. The coupling among the equations for the solved parameters was ensured via the Multiphysics options in which non-isothermal flow was solved for the internal flow in the tubes and the flow in porous and free spaces domains. Furthermore, the time variations of the pressures of the condenser and evaporator were reflected at the valve opening boundaries using adjustable physics namely Ordinary Differential Equations. The source terms in modified physics were programmed and imposed in the conventional equations using user-defined variable options. The number of mesh used was about 506,664 mesh elements with average mesh quality of 0.86 and considering an adequate number of meshes at the boundaries where the variables are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Ref** \\ \hline \multicolumn{4}{c}{**Thermophysical Properties and Isotherms’ Parameters**} \\ \hline Activated carbon density \((\rho_{S})\) & 2200 & \((\text{kg}\cdot\text{m}^{-3})\) & [22] \\ Polyvinyl alcohol density \((\rho_{pv})\) & 1290 & \((\text{kg}\cdot\text{m}^{-3})\) & [31] \\ \hline Nanoplatelets density \((\rho_{\text{nano}})\) & 2300 & \((\text{kg}\cdot\text{m}^{-3})\) & [32,33] \\ \hline Bulk density \((\rho_{bulk})\) & 614 & \((\text{kg}\cdot\text{m}^{-3})\) & [26] \\ \hline Composite specific heat \(\left(C_{p,com}\right)\) & 790 & \(\left(\text{J}\cdot\text{kg}^{-1}\cdot\text{k}^{-1}\right)\) & [25] \\ \hline Bed permeability \((K_{p})\) & \(4\times 10^{-11}\) & \(\text{m}^{2}\) & [16] \\ \hline Total porosity \((\varepsilon_{t})\) & 0.3483 & & \\ \hline Thermal conductivity of composite \((k_{com})\) & 1.550 & \(\left(\text{W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}\right)\) & \\ \hline Pre-exponent constant \(\left(\frac{D_{\text{fa}}}{r_{s}^{2}}\right)\) & \(2.84\times 10^{-2}\) & \(\left(\text{s}^{-1}\right)\) & \\ \hline Activation energy \((E_{a})\) & 272.4050 & \(\left(\text{kJ}\cdot\text{kg}^{-1}\right)\) & \\ \hline Isosteric heat of adsorption at \((H_{ads})\) & 1056.445 & \(\left(\text{kJ}\cdot\text{kg}^{-1}\right)\) & [26] \\ \hline Maximum uptake \((X_{max})\) & 0.62 & \(\left(\text{kg}\cdot\text{kg}^{-1}\right)\) & \\ \hline Adsorption characteristics parameter \((E)\) & 134.07 & \(\left(\text{kJ}\cdot\text{kg}^{-1}\right)\) & \\ \hline Heterogeneity parameter \((n)\) & 1.88 & & \\ \hline [PERSON]’s equation constant \((A)\) & 8.1122 & & \\ \hline [PERSON]’s equation constant \((B)\) & 1592.8641 & & \\ \hline [PERSON]’s equation constant \((C)\) & 226.184 & & \\ \hline \multicolumn{4}{c}{**The adsorbent beds**} \\ \hline Tube length \((L)\) & 0.4 & m & \\ \hline Tube inner radius \((R_{i})\) & 3.15 & mm & \\ \hline Tube outer radius \((R_{o})\) & 3.96 & mm & \\ \hline Bed height \((H_{bd})\) & 2.5 & mm & \\ \hline Vacuum chamber space \((L_{cb})\) & 2 & mm & \\ \hline Vapour in/out opening length & 8 & mm & \\ \hline Average hot water velocity \((u_{ave,hot,in})\) & 0.56 & m\(\cdot\text{s}^{-1}\) & \\ \hline Average cold-water velocity \((u_{ave,cool,in})\) & 0.792 & m\(\cdot\text{s}^{-1}\) & \\ \hline Regeneration temperature \((T_{hw,i})\) & 90 & \(\dot{}^{\text{`C}}\) & \\ \hline adsorption temperature \((T_{cw,i})\) & 25 & \(\dot{}^{\text{`C}}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of the parameters set in the simulation model. undergoing high gradients. The segregated technique with MUMPS and PARDISO solvers was employed to establish the final solutions with acceptable accuracy. The time step in the switching modes was as low as 0.1 s to catch the rapid change in the main variables during the short switching mode time while it was set to 1 s during the adsorption and regeneration modes.