## 2 Materials and Methods
### Technological Methods  
The technological process is presented in Figure 1. The material was pre-treated using ultrasound (US) (ultrasonic bath, MKD Ultrasonic, Stary Konik, Poland) with a frequency of 21 kHz and a power of 180 W. The process was carried out for 10 min, 20 min, and 30 min in a glass beaker with distilled water in a ratio of 1:4. During the treatment, the temperature was kept at 25 \({}^{\circ}\)C \(\pm\) 1 \({}^{\circ}\)C. The parameters were chosen based on our earlier study.  
After treatment, apples were dried on filter paper (slices were placed on the filter paper, once on each side), and 100 g of material was placed on a sieve of a prototype hybrid dryer (Promise Tech Inc., Wroclaw, Poland) equipped with an ultrasound generator (contact ultrasound strainer). Drying was carried out at a drying air temperature of 70 \({}^{\circ}\)C. The air flowed perpendicular to the material at a speed of 1.5 m/s. The process was supported by sonication with an ultrasound power of 120 W, 160 W, or 200 W and a frequency of 36 kHz. Every 5 min, the process was automatically stopped, and the system weighed the screen with the material with an accuracy of \(\pm\)0.1 g. The temporary interruption of the drying process was necessary to minimize the impact of airflow on the mass measurement. To ensure precise measurements, the drying process was briefly paused, only for the time required to take accurate mass readings. This brief interruption did not significantly affect the overall drying process or the dynamics of mass and energy transfer, as the drying conditions were consistently maintained between measurements. The experiment was designed using the response surface methodology (RSM-CCF), with one central point (C) with 3 repetitions and two variables on three levels (factor 1--time of initial sonication; factor 2--ultrasound power during drying), which is shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Run**} & \multicolumn{2}{c}{**Factor 1**} & \multicolumn{2}{c}{**Factor 2**} & \\ \cline{2-6}  & **US Pre-Treatment** & & & **Ultrasound Power** & **Coded Value** & **Sample Code** \\  & **Time T [min]** & **Coded Value** & & **During Drying D [W]** & **Coded Value** & \\ \hline
1 & 10 & \(-1\) & 120 & \(-1\) & T\_10\_D\_120W * \\ \hline
2 & 10 & \(-1\) & 200 & 1 & T\_10\_D\_200W \\ \hline
3 & 30 & 1 & 120 & \(-1\) & T\_30\_D\_120W \\ \hline
4 & 30 & 1 & 200 & 1 & T\_30\_D\_200W \\ \hline
5 & 10 & \(-1\) & 160 & 0 & T\_10\_D\_160W \\ \hline
6 & 30 & 0 & 160 & 0 & T\_30\_D\_160W \\ \hline
7 & 20 & 0 & 120 & \(-1\) & T\_20\_D\_120W \\ \hline
8 & 20 & 0 & 200 & 1 & T\_20\_D\_200W \\ \hline
9 (C) & 20 & 0 & 160 & 0 & T\_20\_D\_160W \\ \hline \hline \end{tabular}  
* T\_10\_D\_120W stands for apples subjected to 10 min of US pre-treatment and hot-air-dried with 120 W of US assistance; T\_10\_D\_200W stands for apples subjected to 10 min of US pre-treatment and hot-air-dried with 200 W of US assistance; T\_30\_D\_120W stands for apples subjected to 30 min of US pre-treatment and hot-air-dried with 120 W of US assistance, etc.  
\end{table}
Table 1: The scheme of a two-factorial design.  
Figure 1: The technological process of the ultrasound-treated apples that were dried with a hybrid method based on ultrasound-assisted hot-air drying.  
Drying kinetics were plotted as dimensionless water content (MR) as a function of time. The relative moisture ratio was calculated based on the following equation [28]:  
\[MR=\frac{u_{\tau}}{u_{0}} \tag{1}\]  
where \(u_{\tau}\) is the moisture content during a given moment of the drying process (g water/g d.m.), and \(u_{0}\) is the initial moisture content (g water/g d.m.).