# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 3 Results and Discussion
Parameter \(k\) for the [PERSON] (Newton) model, \(n\) for the Page model, and \(n\) for the modified Page model depended in a statistically significant manner only on characteristic dimension, whereas parameter \(k\) for the Henderson and Pabis model and \(k\) for the modified Page model depended in a statistically significant way on rehydration temperature. The influence of \(L\) and \(v_{d}\) on parameter \(a\) for the Henderson and Pabis model and \(T_{d}\) and \(v_{d}\) on parameter \(k\) for the Page model was statistically insignificant. Taking into consideration the results presented in Tables 2 and 5, it can be stated that the Peleg model and the Pilosof-Bouquet-Batholomai model enable the prediction of the dry matter loss kinetics during rehydration of dried apples, considering all drying and rehydration conditions, namely, characteristic dimension, drying air temperature, drying air velocity, and rehydration temperature. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model No.** & **Model Name** & **Parameter Equations** \\ \hline \multirow{2}{*}{(1)} & Peleg & \(A_{1}=1.199471+0.03494-L^{2}-0.01674-T_{r}-v_{d}+0.005957-T_{r}+0.345962-v_{d} -0.0122-L_{r}\) \\  & \(A_{2}=-0.09904+0.105514-L^{2}+0.018807-T_{d}-v_{d}-8.7\times 10^{-5}T_{r}-T_{r}-0.38681-v_{d}L\) \\ \hline \multirow{2}{*}{(2)} & Pilosof–Bouquet–Batholomai & \(A_{3}=0.65515-0.00546-L^{2}-2.9\cdot 10^{-5}T_{r}+0.003547-T_{r}-0.07356-v_{d}+L 0.00107-L_{r}\) \\  & \(A_{4}=-0.02683+0.063203-L^{2}+0.018608-T_{d}-v_{d}-5.4\times 10^{-5}T_{r}T_{r}-0.38314-v_{d}L\) \\ \hline \multirow{2}{*}{(3)} & [PERSON] and [PERSON] & \(A_{5}=0.658516+0.003843-T_{r}v_{d}-0.07913-v_{d}L\) \\  & \(A_{6}=3.234311-0.00966-T_{d}+0.000608-T_{d}T_{r}\) \\ \hline \multirow{2}{*}{(4)} & [PERSON] ([PERSON]) & \(k=2.388268-0.06332-L^{2}\) \\  & & \(k=1.048803+0.000156-T_{r}\) \\ \hline \multirow{2}{*}{(5)} & Henderson–Pabis & \(a=0.825358-4.6\cdot 10^{-5}T_{r}^{2}8.45\times 10^{-5}T_{r}T_{r}\) \\  & & \(k=1.353281-0.02825-L^{2}+6.64\times 10^{-5}T_{r}^{2}\) \\ \hline \multirow{2}{*}{(6)} & Page & \(n=0.510797+0.008584-L^{2}\) \\  & & \(k=1.098296+0.00019-T_{r}\) \\ \hline \multirow{2}{*}{(7)} & Modified Page & \(n=0.510797+0.008584-L^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter equations for the models applied to dry matter loss during rehydration of dried apples. The obtained results of modelling using [PERSON], [PERSON], [PERSON] and [PERSON], [PERSON] ([PERSON]), and [PERSON] models provides similar results to those reported in the literature [49; 61; 62; 67; 68; 69; 71; 72; 73], whereas the Henderson and Pabis model and modified Page model were used for the first time for the description of dried product rehydration. Although models 1-7 are not theoretical (they are empirical ones), the constants of these models have physical meaning and inform us about the rate of the dried material rehydration. The equations of the model constants obtained in the work allow, on the one hand, for the calculation of their values, and, on the other hand, their mathematical forms (performed statistical analysis) indicate the factors (parameters of drying and rehydration processes) influencing the apple rehydration process kinetics. The obtained results indicate that both drying and rehydration conditions influence (although to different degree) the constants of the rehydration models, and therefore mass, dry matter, and volume kinetics during rehydration. The kinetics of rehydration indicate the structural changes (and the changes in the composition) of rehydrated material as a result of pretreatment, drying, and rehydration. The literature data, e.g., [72; 73], confirm the obtained results. Different ANN structures were tested. The rehydration indices RI\({}_{1}\), RI\({}_{2}\), and WAC were predicted with ANNs. MLP 4-4-1 was used for RI\({}_{1}\) (Figure 1a), MLP 4-4-1 for RI\({}_{2}\) (Figure 2a), and MLP 4-3-1 for WAC (Figure 3a), with _logsig_ transfer function in the hidden layer and _purelin_ transfer function in the output layer.