## 2 Materials and Methods
### Mathematical Model
#### 2.2.1 Energy Balance Equations  
The model equations, describing the transient heat balances for each _i-th_ node, are defined under the following conditions:  
* The zeolite bed is modelled as a homogeneous porous medium, where the gas phase is constantly in thermal equilibrium with the solid phase.
* The water vapour is modelled as an ideal gas with temperature and pressure such as to be always below the saturation curve during the evolution of the process.
* Heat and mass transfer between solid and gas phases in the reactor are regulated by adsorption kinetics, whose equations are described in the following paragraph.
* The surrounding water vapour exchanges energy and mass with the zeolite bed through a steam flow entering or leaving the bed during the adsorption or desorption phase. This steam flow is continuously reintegrated by an evaporating or condensing flow coming from the pool below the reactor.  
HTF energy balance:  
\[C_{f}\ \frac{dT_{f}}{dt}=\Gamma_{HTF}\ c_{p}\Big{(}T_{in}-T_{f}\Big{)}-h_{f}A_{f} \Big{(}T_{f}-T_{he}\Big{)} \tag{3}\]  
where \(h_{f}\) is the convective heat transfer coefficient given by \(h_{f}=k\ N_{u}/D\) according to Dittus-Boelter empirical correlation.  
Heat-exchanger energy balance:  
\[C_{he}\ \frac{dT_{he}}{dt}=h_{f}A_{f}\Big{(}T_{f}-T_{he}\Big{)}-(ULA)_{he}(T_{he}- T_{r}) \tag{4}\]  
where \((ULA)_{he}\) is the heat capacity of heat-exchanger and zeolite, evaluated taking into account the conductive heat transfer and the effects of the fin.  
Zeolite energy balance for adsorption (5) and desorption (6):  
\[\frac{d}{dt}(C_{r}\ T_{r})=(ULA)_{he}(T_{he}-T_{r})-(ULA)_{r}(T_{r}-T_{v})+\ \Delta H\ M_{z}\frac{dx}{dt}+\Gamma_{v}\ c_{p,v}\ T_{v} \tag{5}\]  
\[\frac{d}{dt}(C_{r}\ T_{r})=(ULA)_{he}(T_{he}-T_{r})-(ULA)_{r}(T_{r}-T_{v})+\ \Delta H\ M_{z}\frac{dx}{dt}-\Gamma_{v}\ c_{p,v}\ T_{v} \tag{6}\]  
where the last two terms on the right side of both equations represent, respectively, the adsorption/desorption heat and the water vapour enthalpy entering (+ sign) or leaving (- sign) the zeolite bed. Similar to the previous equation, \((ULA)_{r}\) represents the heat capacity of zeolite bed and water vapour surrounding the zeolite reactor. Such parameter is estimated considering the natural convection heat transfer between the water vapour and the internal walls of the zeolite bed.  
Surrounding water vapour energy balance during adsorption (7) and desorption (8) processes:  
\[\frac{d}{dt}(C_{v}T_{v})=(ULA)_{r}(T_{r}-T_{v})-(ULA)_{B}(T_{v}-T_{B})-\Gamma_{ v}\ c_{p,v}\ T_{v}+\Gamma_{evap}\ c_{p,v}\ T_{B} \tag{7}\]  
\[\frac{d}{dt}(C_{v}T_{v})=(ULA)_{r}(T_{r}-T_{v})-(ULA)_{B}(T_{v}-T_{B})+\Gamma_{ v}\ c_{p,v}\ T_{v}-\Gamma_{cond}\ c_{p,v}\ T_{B} \tag{8}\]  
where \(\Gamma_{evap}\) and \(\Gamma_{cond}\) are respectively, the evaporating/condensing mass flow rate coming from (+ sign) or going to (- sign) the pool below the reactor. \((ULA)_{B}\) is the heat capacity of water vapour and the bath, also evaluated through an analysis of the heat exchanged in natural convection. Due to the high degree of insulation implemented, as stated in the previous paragraphs, the heat losses between the reactor and the environment has been neglected.  
Finally, the following mass balance equations are stated:  
\[\frac{dM_{vi}}{dt}=-\ M_{z}\frac{dx}{dt}+\Gamma_{v} \tag{9}\]  
\[\frac{dM_{ye}}{dt}=\Gamma_{evap}-\Gamma_{v} \tag{10}\]  
where the first equation refers to the mass balance of the water vapour inside the zeolite bed, while the second one refers to the mass balance of the water vapour in the external area.