# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 2 Materials and Methods
The Pilosof-Boquet-Batholomai model has been used to predict the rehydration of water uptake to food powders [61], water absorption of wheat starch, whey protein concentrate, and whey protein isolate [49]. The [PERSON] and [PERSON] model, in turn, has been applied to predict the rehydration kinetics of soybean and pigeonpea grains [62], wheat starch, whey protein concentrate, and whey protein isolate [49]. Empirical models (4)-(7) (Table 1) were developed to describe the drying process; however, two of them, namely, the [PERSON] (Newton) model and the Page model, have been adopted to predict the rehydration process. The Lewis model (a first-order kinetics model) gave good results in the rehydration prediction of dried products such as mangoes [70], paraboiled rice [68], pumpkin [71], rice porridge [72], and soybean curd [73]. The Page model has been applied to predict the rewetting of dried pumpkin [71] and rough rice [74]. The Henderson and Pabis model and the modified Page model have not yet been used for description of dried product rehydration. Constant \(k\) in models (4)-(7) has physical meaning and informs us about the rate of the dried material rehydration. Rehydration curves were fitted to seven considered models (Table 1). The Matlab (Curve Fitting Toolbox 2018a, MathWorks, Inc., Natick, MA, USA) software (with the Levenberg-Marquardt algorithm) was used. The following statistical test methods were applied to statistically evaluate the suitability of the discussed models for predicting the rehydration kinetics of dried apples: sum squared error (SSE) coefficient of determination R\({}^{2}\), adjusted R\({}^{2}\), and root mean square error RMSE. These statistical criteria are widely used in the literature. In this work, the effect of characteristic dimension \(L\), drying air temperature \(T_{d}\), drying air velocity \(v_{d}\), and rehydration temperature \(T_{r}\) on the model constants were also studied. The constants of the models (1)-(7) (Table 1) involving the mentioned drying and rehydration conditions were determined by investigating the following type of equation:  
\[ \begin{array}{l}\text{Constant}=a_{0}+a_{1}L+a_{2}T_{d}+a_{3}v_{d}+a_{4}T_{ r}+a_{5}L^{2}+a_{6}{T_{d}}^{2}+a_{7}v_{d}{}^{2}\\ +a_{8}{T_{r}}^{2}+a_{9}T_{d}v_{d}+a_{10}T_{d}L+a_{11}T_{d}T_{r}+a_{12}v_{d}L+a_ {13}v_{d}T_{r}+a_{14}LT_{r},\end{array}  \tag{13}\]  
where \(a_{0}\div a_{14}\) are constants. The statistical analysis ANOVA was used. The following quality indices were applied to describe rehydration of dried apples. * Index RI\({}_{1}\) (in kg\({}_{\text{reh}}\)/kg\({}_{\text{dry}}\)): \[\text{RI}_{1}=\frac{\text{mass of material after rehydration}}{\text{ mass of dried material}},\] (14)
* Index RI\({}_{2}\) (in kg\({}_{\text{reh}}\)/kg\({}_{\text{raw}}\)): \[\text{RI}_{2}=\frac{\text{mass of material after rehydration}}{\text{ mass of raw material}},\] (15)
* water absorption capacity WAC: \[\text{WAC}=\frac{m_{r}(100-s_{r})-m_{d}(100-s_{d})}{m_{0}(100-s_{0})-m_{d}(100-s_{d})},\] (16)  
where \(m\) is the mass (kg); \(s\) is the dry matter content (%); and subscripts \(d\), \(0\), and \(r\) refer to dry, before drying, and rehydrated, respectively. The WAC index has been widely applied to express the rehydration characteristics [22; 75]. The rehydration indices RI\({}_{1}\), RI\({}_{2}\), and WAC were predicted with feedforward multilayer perceptron (MLP) artificial neural networks (ANNs) in Matlab (Neural Network Toolbox 2018a, MathWorks, Inc.). The inputs \(L\), \(T_{d}\), \(v_{d}\), and \(T_{r}\) and outputs RI\({}_{1}\), RI\({}_{2}\), and WAC were normalised to range 0-1 using the _minmax_ method. The cases were randomly divided into training set--70%, validation set--15%, and testing set--15%. The following statistical test methods were used to evaluate statistically the suitability of applied model: SSE, R\({}^{2}\), adjusted R\({}^{2}\), and RMSE.