## 3 Numerical Modeling of HDS
### Desiccant Wheel  
Some general assumptions are as follows. (1) The length of the airflow direction far exceeds the thickness of the runner wall so that the model can be simplified to one dimension. (2) The axial heat conduction and mass diffusion inside the solid desiccant is much lower than the wall thickness direction and can be ignored. (3) The size of the airflow channel is evenly distributed. (4) The thermophysical properties of the desiccant can be treated as constant.  
According to the conservation laws of energy and mass, and the heat and mass transfer between the air and the desiccant wheel, the following control equation can be obtained [27].  
\[A\rho_{a}\left(\frac{\partial Y}{\partial\tau}+V\frac{\partial Y}{\partial z }\right)+\frac{1}{2}f_{d}\frac{\partial w}{\partial\tau}=0 \tag{6}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Evaporator** & **Condenser** \\ \hline Outer diameter of tube & 9.53 mm & 7 mm \\ Tube wall thickness & 0.35 mm & 0.32 mm \\ Fin distance & 1.58 mm & 1.4 mm \\ Fin thickness & 0.12 mm & 0.115 mm \\ Tube distance & 25.4 mm & 20 mm \\ Arrangement of the tubes & Regular triangle & Regular triangle \\ Number of tube rows & 3 & 5 \\ Number of tubes per row & 10 & 14 \\ Heat transfer area & 6.92 m\({}^{2}\) & 9.92 m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structural parameters of evaporator and condenser.  
\[\frac{1}{2}f_{d}\frac{\partial w}{\partial\tau}=K_{y}s(Y-Y_{w}) \tag{7}\]  
\[A\rho_{a}\big{(}C_{pa}+YC_{pv}\big{)}\bigg{(}\frac{\partial T_{a}}{\partial\tau}+V \frac{\partial T_{a}}{\partial z}\bigg{)}+\frac{1}{2}\Big{[}f_{d}\Big{(}C_{pd} +wC_{pl}\Big{)}+f_{m}C_{pm}\Big{]}\frac{\partial T_{m}}{\partial\tau}=K_{y}s(Y -Y_{w})q_{st} \tag{8}\]  
\[\frac{1}{2}\Big{[}f_{d}\Big{(}C_{pd}+wC_{pl}\Big{)}+f_{d}C_{pm}\Big{]}\frac{ \partial T_{m}}{\partial\tau}=\alpha s(T_{a}-T_{m})+K_{y}s(Y-Y_{w})q_{st}+K_{y }sC_{pv}(Y-Y_{w})(T_{a}-T_{m}) \tag{9}\]  
where \(\tau\) indicates time. \(A\) is the channel cross-sectional area, and \(s\) is the boundary length of a channel. \(\rho\) is the density. The \(f_{d}\) and \(f_{m}\) are the specific desiccant mass and specific matrix material mass, respectively. \(V\) is the velocity of airflow. \(T\) is the temperature. \(\alpha\) is the heat transfer coefficient. \(K_{y}\) is the mass transfer coefficient. \(Y_{w}\) is the air moisture content on the surface of the desiccant. \(Y\) is the air moisture content. \(w\) is the desiccant water content. \(C_{pa}\), \(C_{pv}\), \(C_{pd}\), \(C_{pl}\) and \(C_{pm}\) are the specific heat at a constant pressure of air, water vapor, silica gel, saturated water and matrix materials, respectively. The subscript \(m\) indicates the property of the matrix material. \(q_{st}\) is the water/silica adsorption heat.  
\(\alpha\) and \(K_{y}\) are calculated by the following equations [28]:  
\[\alpha=\frac{Nu\lambda s}{4A} \tag{10}\]  
\[K_{y}=\rho_{a}\frac{ShDs}{4A} \tag{11}\]  
\[D=2.303\times 10^{-5}\frac{0.98\times 10^{5}}{P_{a}}(\frac{T_{a}}{256})^{1.81} \tag{12}\]  
where \(\lambda\) is the thermal conductivity of air, and \(D\) is the mass diffusion coefficient of water vapor in the air. \(Nu\) and \(Sh\) are Nusselt numbers and Sherwood numbers, respectively.  
The water/silica adsorption heat is calculated by Equation (13) [29]:  
\[q_{st}=h_{v}\Big{(}1.0+0.2843e^{-10.28w}\Big{)} \tag{13}\]  
where the \(h_{v}\) is the evaporation's latent heat.  
RH of the silica gel surface can be calculated by Equation (14) [30]:  
\[\phi_{w}=0.0078-0.05759w+24.16554w^{2}-124.78w^{3}+204.226w^{4} \tag{14}\]  
The Antonine equation gives the pressure of saturated water vapor [31]:  
\[P_{ws}=\exp(23.196-\frac{3816.44}{T_{m}-46.13}) \tag{15}\]  
The relationship between moisture content and RH can be described by Equation (16) [28]:  
\[Y_{w}=\frac{0.62188p_{w}}{p_{atm}-p_{w}}=\frac{0.62188\phi_{w}}{\frac{p_{atm}} {p_{ws}}-\phi_{w}} \tag{16}\]  
The values of the parameters used in the numerical calculation of the desiccant wheel are shown in Table 2.