## 3 Drying Model
### Diffusion Model
Therefore, in this study, to examine the impact of ultrasound pre-treatment on drying kinetics, the moisture-dependent effective diffusivity model is adapted to incorporate the additional influence of temperature as  
\[D_{eff}=D_{0}exp\left(-\frac{E_{a}}{RT}\right)\left(\frac{1+X_{0}}{1+X}\frac{ \rho_{1}(1+X)+\rho_{0}X}{\rho_{1}(1+X_{0})+\rho_{0}X_{0}}\right)^{2}, \tag{5}\]  
with \(D_{ref}=D_{0}\exp\left(-\frac{E_{a}}{RT}\right)\). The energy balance equation is written as the change in temperature caused by the heat conduction in the medium and the enthalpy flow of the water diffusion, namely [29; 30]  
\[\frac{\partial}{\partial t}\left[\left(\rho_{0}c_{p,s}+\rho_{0}c_{p,l}X\right) T\right]-\
abla\cdot\left[D_{eff}c_{p,l}T\
abla(\rho_{0}X)\right]-\
abla\cdot \left[\lambda_{eff}\
abla(T)\right]=0. \tag{6}\]  
In Equation (6), \(c_{eff}\) (J/kg.K) and \(\rho_{eff}\) (kg/m\({}^{3}\)), respectively, denote the effective heat capacity and density of the medium; \(c_{p,l}\) (J/kg.K) denotes the specific heat capacity of liquid water; and \(\lambda_{eff}\) (W/m.K) denotes the effective thermal conductivity of the porous medium. These thermal properties of carrots are determined from their constituents, according to [31]  
\[\rho=\frac{1}{\sum\frac{x_{i}}{\rho_{i}}};\ c=\sum x_{i}c_{i};\ \text{and}\ \lambda_{ eff}=\frac{\sum\frac{x_{i}\lambda_{i}}{\rho_{i}}}{\sum\frac{x_{i}}{\rho_{i}}}, \tag{7}\]  
where \(x_{i}\) is the mass fraction of food components including protein, fat, carbohydrate, fiber, ash, and gas. Particularly, the mass fraction of the components of fresh carrots is presented in Table 1. A detailed description of the physical properties (density, specific heat capacity, and thermal conductivity) of the constituents can be found in [32]. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Constituent** & **Moisture** & **Protein** & **Fat** & **Carbohydrate** & **Fiber** & **Ash** \\ \hline Mass fraction (\%) & 88.29 & 0.93 & 0.24 & 6.58 & 3.00 & 0.96 \\ \hline \end{tabular}
\end{table} 
Table 1: The mass fraction of the constituents of a fresh carrot sample [4]. To solve the differential equation system of the diffusion model (Equations (2) and (6)), boundary conditions are needed. Particularly, a thin layer of carrot with a radius of \(d\) and thickness of \(\delta\) is subjected to atmospheric pressure hot-air drying with a constant drying temperature of \(T_{g}\) and a relative humidity of RH (c.f. Figure 1). The carrot slice is initially at a uniform temperature of \(T_{0}\) and moisture content of \(X_{0}\) (kg water/kg dry solid). Since the diameter of the slice is more than ten times the thickness, the heat and mass transfer processes are assumed to be one-dimensional. A one-dimensional coordinate system with the origin located at the sample center is used to solve the drying model, the boundary equations of the developed heat and mass transfer model at the sample surface, i.e., \(z=\frac{\delta}{2}\), are written as  
\[j_{w}\cdot\mathbf{n}=-\beta\Big{(}\rho_{v,surf}-\rho_{v,g}\Big{)}, \tag{8}\]  
\[j_{e}\cdot\mathbf{n}=-\alpha\Big{(}T_{g}-T_{surf}\Big{)}+\Delta h_{evp}\beta\Big{(} \rho_{v,surf}-\rho_{v,g}\Big{)} \tag{9}\]  
In Equations (8) and (9), \(\mathbf{n}\) denotes the outward unit normal vector pointing out of the sample surface, and \(j_{w}\) and \(j_{e}\) are the vapor and heat fluxes transferred from the sample surface towards the bulk steam, respectively. For a single slice of carrot with the flow configuration presented in Figure 1, the heat and mass transfer coefficients (\(\alpha\) and \(\beta\), respectively) are determined from the correlations  
\[Nu=2+0.616 Re^{0.52}Pr^{\frac{1}{3}}, \tag{10}\]  
\[Sh=2+0.616 Re^{0.52}Sc^{\frac{1}{3}} \tag{11}\]  
In Equations (10) and (11), \(Nu=\frac{\alpha d}{\lambda_{g}}\), \(Re=\frac{\omega_{g}d}{\upsilon_{g}}\) and \(Pr=\frac{c_{p,g}\mu_{g}}{\lambda_{g}}\) are the dimensionless Nusselt, Reynolds, and Prandtl numbers, respectively.