## 3 Drying Model
### Conveyor-Belt-Dryer Model  
As discussed, the diffusion model considers the heat and mass transfer process of a single slice, so it cannot be applied to modeling the drying process of an actual dryer where thousands of slices interact with the drying agent. In this section, a mathematical model of a co-current conveyor-belt dryer is developed. A systematic sketch of the belt dryer is presented in Figure 2. The steady-state operation of the dryer is assumed herein. The interaction between the wet solid and the drying agent can be described based on the principle that the mass flow rate of dry solid remains unchanged, while the change in the mass of water is attributed to liquid vaporization. The convective thermal energy absorbed by the wet solid from the drying agent serves the dual purpose of heating the wet solid and facilitating evaporation. Thus, the heat and mass balance equations for the wet solid are written as  
\[G_{s}dX=-g_{v}dF, \tag{18}\]  
\[(G_{s}c_{s}+G_{s}Xc_{f})dT=\alpha_{b}(T_{a}-T_{s})dF-\Delta h_{exp}g_{v}dF \tag{19}\]  
Figure 2: The sketch of the co-current conveyor-belt dryer. where \(G_{s}\)(kg dry solid/s) is the mass flow rate of the dry solid, \(\alpha_{b}\) (W/m\({}^{2}\)K) is the convective heat transfer between the wet solid and the airflow, \(dF=\) W.\(dl\) (m\({}^{2}\)) is a differential area with \(W\) (m) as the width of the belt, \(l\) is the length of the belt, and \(g_{v}\) is the evaporation flux from the wet solid (kg/m\({}^{2}\)s). Similarly, the moisture content of the airflow varies as a result of the vapor evaporated from the wet solid. Additionally, alterations in the total energy of the airflow occur due to the convective thermal energy transferred to the wet solid and the enthalpy flow of the vapor transported from the wet solid to the airflow. Consequently, the mass and energy conservation equations for the drying agent are expressed as  
\[G_{a}dY_{a}=g_{v}dF, \tag{20}\]  
\[G_{a}dh_{a}=\Delta h_{evp}g_{v}dF-\alpha_{b}\Big{(}T_{a}-T_{f}\Big{)}dF \tag{21}\]  
where \(G_{a}\) (kg dry air/s) is the mass flow rate of dry air, and \(h_{a}=c_{p,a}T_{a}+Y_{a}\left(\Delta h_{evp}+c_{p,v}T_{a}\right)\) (J/kg dry air) is the specific enthalpy of the humid air. After arrangement, the heat and mass transfer equations of the belt-dryer model can be rewritten in the form of an ordinary differential equation system as  
\[\frac{dX}{dl}=\frac{-g_{v}W}{G_{s}}, \tag{22}\]  
\[\frac{dT}{dl}=\frac{\alpha_{b}(T_{a}-T_{s})W}{G_{s}c_{s}+G_{s}Xc_{l}}-\frac{ \Delta h_{evp}g_{v}W}{G_{s}c_{s}+G_{s}Xc_{l}} \tag{23}\]  
\[\frac{dY_{a}}{dl}=\frac{g_{v}}{G_{a}}W \tag{24}\]  
\[\frac{dh_{a}}{dl}=\frac{\Delta h_{evp}g_{v}W-\alpha_{b}\Big{(}T_{a}-T_{f} \Big{)}W}{G_{a}} \tag{25}\]  
For solving this system of differential equations, the evaporation flux from the product's surface needs to be known. Theoretically, the diffusion model can be applied to accurately describe the heat and mass interaction of the product, however, the computational cost of solving the partial differential equation system of the diffusion model is expensive. This high computation cost becomes a serious problem when the calculation needs to be performed simultaneously for thousands of slides. Thus, this complicated diffusion model is reduced to a semi-empirical model, named the characteristic drying-curve model. The internal mass transfer resistance is omitted, and the drying process is controlled solely by the external mass transfer resistance between the airflow and the product's surface. This simplification is acceptable for thin slices. The drying process is divided into the first and the second drying periods. During the first drying period, when the moisture content X is high, the liquid is strongly pumped to the surface under capillary action. It helps to keep the sample surface wet enough that it can be considered as a purely liquid surface. Thus, the evaporation flux can be computed as  
\[G_{s}dX=-g_{v}dF, \tag{18}\]0  
The second drying period commences when the sample surface is partly dried.