## 4 Results and Discussion
### Influence of Ultrasound Pre-Treatment on Drying Kinetics of a Single Slice  
In addition to the quality properties, the influence of US pre-treatment on the mass transport property is explored. The effective moisture diffusivity is determined numerically by using the inverse method. An objective optimizing function \(g\Big{(}D_{ref}\Big{)}\) is defined as the sum of the square of the difference between numerical and experimental moisture content, i.e., \(X_{i,sim}\) and \(X_{i,\text{exp}}\), as  
\[g\Big{(}D_{ref}\Big{)}=\sum_{i=1\leq md}\big{(}X_{exp}(t_{i})-X_{sim}(t_{i}) \big{)}^{2}, \tag{30}\]  
where \(t_{i}\) denotes the time interval \(i\) of the data sampling. Thus, the value of reference moisture diffusivity \(D_{ref}\) is obtained while the discrepancy between numerical and experimental moisture-content evolutions is minimized. Both optimized numerical and experimental moisture-content evolution of carrot slices during drying processes are plotted together in Figure 5. As can be seen, a good agreement between numerical and experimental drying histories indicates the reliability of the obtained reference moisture diffusivity. Based on the determined values of reference moisture diffusivity, the activation energy \(E_{n}\) and Arrhenius factor \(D_{0}\) are fitted, these parameters for US pre-treatment and non-pre-treatment drying processes are presented in Table 2. As can be seen in Figure 6, the effective moisture diffusivity has been enhanced significantly by the US pre-treatment. The reference moisture diffusivity increases by more than 30% when the US pre-treatment is applied. The obtained benefits of US pre-treatment on drying kinetics and product color align with previous studies where the effective moisture diffusivity was shown to increase by up to 120%, resulting in a drying time reduction of approximately 20% and improved product quality for carrots and different fruits [35; 36; 37; 38; 39]. The improved heat transfer in cellular materials due to US pre-treatment can be attributed to cavitation, the sponge effect, and the formation of microchannels. The formation of microchannels, resulting from the ruptures of tissues and cells caused by acoustic cavitation, is a significant contributing factor [16]. This implies that the duration of pre-treatment, the nature of the propagation solution, and the temperature, can directly influence the drying acceleration. This hypothesis should be investigated through X-ray micro-computed tomography measurements in future research. Figure 5: Numerical and experimental moisture content evolution over time during the hot-air-drying process with (**a**) and without (**b**) US pre-treatment. Due to the penetration of liquid water into the sample during the soaking time, the initial moisture content also increases, i.e., the average initial moisture content of a fresh carrot of 8.60 kg water/kg dry solid, and the average moisture content of a soaked carrot of 8.97 kg water/kg dry solid, resulting in the prolongation of drying time. For a comfortable comparison, the numerical simulations are additionally performed with identical initial conditions of \(X_{0}\) = 8.8 kg water/kg dry solid, \(T_{0}\) = 20 \({}^{\circ}\)C, and \(T_{\mathrm{g}}\) = 50 \({}^{\circ}\)C, \(p_{\mathrm{vg}}\) = 2000 Pa for both US pre-treatment and non-pre-treatment drying processes. The results are plotted together in Figure 7. As can be seen, for the US pre-treated case, the sample surface remains wet longer due to the larger moisture diffusive flux. Thus, the surface temperature remains at the saturated adiabatic temperature of the drying agent for a longer time. This may be a reason helping to explain the lower color change obtained with carrot slices pre-treated by the US compared to the non-pre-treated product. \begin{table}
\begin{tabular}{c c c} \hline
**Drying Process** & \(D_{0}\)\(\times\)**10\({}^{3}\) (m\({}^{2}\)/s)** & \(E_{a}\) (KJ/mol.K)** \\ \hline With US pre-treatment & 87.76 & 1.01 \\ \hline Without US pre-treatment & 41.62 & 49.53 \\ \hline \end{tabular}
\end{table}