# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 2 Materials and Methods  
Apple var. Ligol were bought from Warsaw market, Poland. Ligol is a dessert variety. Its flesh is cream-coloured and very juicy and tasty [50]. In order to assure the homogenous fruits, maturity indicators such as apple appearance and size were taken into consideration [51]. Before the experiments, fruits were washed, hand peeled, cut into 3-milimetere and 10-milimetere thick slices and 10-milimetere thick cubes. The apple characteristic dimension \(L\) was determined according to [52] as for slices, \(L=s\);
* for cubes, \(L^{-2}=3\,s^{-2}\); where \(s\) is the half of the slice thickness and cube thickness (mm). The initial moisture content of the fresh apples was around 85% w.b. (5.7 d.b.). Drying was conducted the same day the fresh apple samples were prepared. The following drying methods were applied: natural convection in chamber dryer (thin layer; drying air velocity \(v_{d}\) = 0.01 m/s; the laboratory dryer KCW-100 (PREMED, Marki, Poland)), forced convection in tunnel dryer (thin layer; \(v_{d}\) = 0.5 and 2 m/s; the laboratory dryer constructed in the Department of Fundamental Engineering and Energetics, Warsaw University of Life Sciences, Warsaw, Poland), and fluid bed drying in a laboratory-fluidised bed dryer (\(v_{d}\) = 6 m/s; the laboratory dryer constructed in the Department of Fundamental Engineering and Energetics, WULS, Warsaw, Poland). The drying methods used in this study are widely applied in drying, especially of agriculture products at industrial scales. The experiments were conducted at drying air temperatures \(T_{d}\) = 50, 60, and 70 \({}^{\circ}\)C (RH was in the range of 20-60%). The used range of \(T_{d}\) is typical for agriculture product drying [17]. The final moisture content of dried apples was about 9% w.b. (0.1 d.b.). The drying equipment used and the way of carrying out the experiments was described in the papers by [PERSON] and [PERSON] [53], [PERSON] and [PERSON] [54], and [PERSON] et al. [55]. Dried material (the same drying conditions) obtained from three independent drying experiments was mixed and next stored in a sealed container (without exposure to sunlight) for seven days at the temperature 20 \({}^{\circ}\)C.  
The rehydration process of dried apples was carried out in distilled water at the temperature \(T_{r}\) = 20, 45, 70, and 95 \({}^{\circ}\)C (applied \(T_{r}\) are widely used in the literature, the value of the \(T_{r}\) is dependent on the further use of rehydrated product). The temperature conditions were warranted with a water bath. The process lasted 2 h at medium temperature 95 \({}^{\circ}\)C, 4 h at 70 \({}^{\circ}\)C, 5 h at 45 \({}^{\circ}\)C, and 6 h at 20 \({}^{\circ}\)C. The mass of dried apple sample at the beginning of each experiment was approximately 10 g. The distilled water temperature was maintained constant, and water was not stirred during the experiment. Mass of dried sample to rehydration medium mass ratio was 1:20 (at the beginning of process). The sample mass \(m\) was measured applying the WPE 300 scales (RADWAG, Radom, Poland) with \(\pm\)0.001 g accuracy. The sample dry matter \(m_{d.m. }\) was determined in accordance with AOC standards [56]. The sample volume \(V\) was determined by applying the buoyancy method in petroleum benzine [57], and the relative error was lower than 5%. The experiments of mass, dry matter, and volume determination were conducted in three repetitions. The scanner (CanoScan 5600F, Canon Inc., Tokyo, Japan, 4800 \(\times\) 9600 dpi) was applied for determining the colour attributes for fresh, dried, and rehydrated apple sample. Obtained colour images (20 images for each apple batch) were loaded into the sRGB colour space. The mean brightness of pixels in each RGB channel of the image was used to express colour parameters. The ImageJ (ver.47i, Madison, WI, USA) software was used. A way of carrying out the colour measurements was described in the paper by [PERSON] et al. [58].