# Analysis of Rewetting Characteristics and Process Parameters in Tobacco Strip Redrying Stage
## 2 Model Construction
### Mathematical Models
#### 2.2.1 Mass Equation  
According to the assumption that the geometry of the solid phase does not change with time, the mass conservations of the liquid phase and gas phase need to be analyzed separately. Here, the liquid phase refers to the liquid water and the capillary pressure depends on the concentration (\(c_{l}\)) and temperature (\(T\)) of the tobacco strips [19]; however, the effect of temperature is negligible in this work, and the mass equation is:  
\[\frac{\partial}{\partial t}(\varphi S_{l}\rho_{l})+\
abla\cdot(\overrightarrow {n}_{w})=-R_{cm} \tag{1}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Name** & **Value** \\ \hline porosity \(\varphi\) & 0.55 [17] \\ density of tobacco \(\rho\) & kg/m\({}^{3}\)[8] \\ initial vapor mass fraction \(\omega_{v}\) & 0.85 \\ air heat capacity \(c_{p,a}\) & 1006 J/kg/K [18] \\ vapor heat capacity \(c_{p,p}\) & 2029 J/kg/K [18] \\ condensation constant \(K_{cm}\) & 1000 s\({}^{-1}\) \\ universal gas constant \(R\) & 8.314 J/mol/K \\ initial moisture content \(c_{ll(l=0)}\) & 8.5\% \\ initial pressure \(P\) & 1 atm \\ \hline \hline \end{tabular} The unsourced data are obtained from the measurements.  
\end{table} 
Table 1: Physical properties of the heaped tobacco strips. Figure 2: Heat and mass transfer within the heaped tobacco strips in the rewetting session. where \(t\) is time (s), \(\varphi\) is porosity, \(S_{l}\) is the saturation of the liquid-phase water, \(\rho_{l}\) is the density of the liquid phase (kg/m\({}^{3}\)), \(R_{con}\) is a condensation rate (kg/m\({}^{3}\)/s), and \(\vec{n_{w}}\) indicates the liquid-phase flux (kg/m\({}^{2}\)/s\({}^{1}\)) as given by [PERSON]'s law:  
\[\vec{n}_{w}=-\rho_{l}\frac{k_{l}k_{l,r}}{\mu_{l}}\
abla p_{l}=-\rho_{l}\frac{k _{l}k_{l,r}}{\mu_{l}}\
abla p+\rho_{l}\frac{k_{l}k_{l,r}}{\mu_{l}}\
abla p_{c} \tag{2}\]  
where \(\mu_{l}\) is the viscosity of the liquid phase (Pa.s); \(k_{l}\) indicates the fixed permeability of the liquid phase (m\({}^{2}\)); \(k_{l,r}\) indicates the relative permeability of the liquid phase, which is the ratio of the effective permeability of the liquid phase to absolute permeability and is a dimensionless number; \(p_{l}\) is the pressure in the liquid phase (Pa); \(P\) is the total pressure (Pa); and \(p_{c}\) is the capillary pressure (Pa). The first term represents the flow due to the gradients in the gas pressure. The second term represents the flow due to the capillary pressure. The capillary pressure depends on the concentration (\(c_{l}\)) and temperature (\(T\)) of the tobacco strips [19]. Therefore, it can be rewritten as:  
\[\vec{n}_{w}=-\rho_{l}\frac{k_{l}k_{l,r}}{\mu_{l}}\
abla P+\rho_{l}\frac{k_{l}k _{l,r}}{\mu_{l}}\frac{\partial p_{c}}{\partial c_{l}}\
abla c_{l}+\rho_{l}\frac {k_{l}k_{l,r}}{\mu_{l}}\frac{\partial p_{c}}{\partial T}\
abla T \tag{3}\]  
where \(c_{l}\) indicates the concentration of the liquid-phase water(kg/m\({}^{3}\)) and \(T\) indicates the temperature of the tobacco strips (\({}^{\circ}\)C). The capillary diffusivity due to the temperature gradient is known as the Soret effect and is often less significant than the diffusivity due to the concentration gradients, and thus, will be neglected in this work. Therefore, Equation (1) can be rewritten as:  
\[\frac{\partial}{\partial t}(\varphi S_{l}\rho_{l})+\
abla\cdot(-\rho_{l}\frac {k_{l}k_{l,r}}{\mu_{l}}\
abla P+\rho_{l}\frac{k_{l}k_{l,r}}{\mu_{l}}\frac{ \partial p_{c}}{\partial c_{l}}\
abla c_{l})=-R_{con} \tag{4}\]  
Based on the above equations, an appropriate closure equation linking \(S_{l}\) to \(p_{c}\) is provided, as the capillary pressure and a retention model need to be introduced [20].