## 3 Drying Model  
### Diffusion Model  
The diffusion-drying model is used in this work to describe the drying process. The underlying idea of the diffusion model was that the mixture of liquid water and water vapor was treated as a single phase named moisture. Based on [PERSON]'s law of diffusion, the driving force of the moisture transport was assumed to be the gradient of the moisture content. The mass conservation of moisture was written as the evolution of moisture concentration due to the moisture diffusion flow  
\[\frac{\partial\rho_{0}X}{\partial t}=\
abla\cdot\Big{[}D_{eff}\
abla(\rho_{0}X )\Big{]}, \tag{2}\]  
where \(\rho_{0}=\frac{M_{dry}}{\gamma}\) (kg dry solid/m\({}^{3}\)) is the apparent density of the dried medium. The effective diffusivity of moisture in the porous medium \(D_{eff}\) (m\({}^{2}\)/s) is defined as the ratio between the moisture flux and the moisture-content gradient. Indeed, the convection-diffusion transport mechanism of the liquid-vapor mixture in the heterogeneous void space of the food during the process is lumped into this artificial effective diffusion coefficient concept. The Arrhenius equation is classically used to represent the relationship between the diffusivity and temperature to estimate the effective moisture diffusivity  
\[D_{eff}=D_{0}exp\bigg{(}-\frac{E_{a}}{RT}\bigg{)}, \tag{3}\]  
where \(D_{0}\) is the Arrhenius factor (m\({}^{2}\)/s), \(E_{a}\) denotes the activation energy [kJ/mol]. The relationship shown in Equation (3) is named the temperature-dependent effective diffusivity model. Recently, [PERSON] et al. proposed that the moisture diffusivity of food products can be represented as a function of their moisture content as [PERSON] et al. (2018)  
\[D_{eff}=D_{ref}\bigg{(}\frac{1+X_{0}}{1+X}\frac{\rho_{l}(1+X)+\rho_{0}X}{\rho _{l}(1+X_{0})+\rho_{0}X_{0}}\bigg{)}^{2}, \tag{4}\]  
Figure 1: The schematic of the experimental apparatus used for drying experiments. where \(D_{ref}\) and \(X_{0}\) are the reference moisture diffusivity and the initial moisture content, respectively. Equation (4), which is referred to as the moisture-dependent effective diffusivity model, shows that the effective moisture diffusivity reduces with the reduction in moisture content. This hypothesis can be explained as the liquid water accumulated in the large pores of the medium is removed sooner and easier than from the small pores due to the capillary action and Kelvin effect. Firstly, the liquid in the large pores is pumped to the small pores under the capillary force, and it thus keeps the small pores wet longer. Secondly, since the concave liquid curvatures are formed in the partly wetted pores of hygroscopic materials, the equilibrium vapor pressure at the liquid-gas interface becomes smaller with the reduction in pore size, which is known as the Kelvin effect. Therefore, the vapor pressure difference between the gas phase and the liquid surface reduces, and the internal vaporization is restrained in tiny pores. Conclusively, the effective moisture diffusivity becomes smaller with the order of the dehydration process from large to tiny pores. It is worth noting that the moisture-dependent effective diffusivity model does not directly consider the influence of drying temperatures on volume shrinkage. The experimental findings suggest that higher drying temperatures promote volume shrinkage, resulting in a reduction in pore size as the drying temperature increases. Consequently, it is reasonable to anticipate that the effective moisture diffusivity is influenced by both moisture content and drying temperature. In the context of the hot-air drying of carrots, the moisture diffusivity has been well represented as an exponential function of both moisture content and temperature.