# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 3 Results and Discussion
Figure 4 presents the example images of the following samples: fresh, dried in force convection, and rehydrated in distilled water at 45 \({}^{\circ}\)C for 5 h. The summary results of statistical analyses on the modelling of lightness \(L^{\star}\), yellownes \(b^{\star}\), hue angle \(h^{\star}\), and chroma \(C^{\star}\) changed during the rehydration of dried apples are shown in Table 9. As can be seen, the first-order model approximated the colour changes not quite  
The following results of investigation of the effects of characteristic dimension \(L\), drying air temperature \(T_{d}\), drying air velocity \(v_{d}\), and rehydration temperature \(T_{r}\) on the first-order model constant \(k_{c}\) were obtained:  
* for lightness \(L^{\star}\) \[k_{c}=1.452388+0.028556v_{d}-0.0004{T_{d}}^{2}-0.0047{T_{d}}^{2}+0.016592 LT_{d},\] (25)
* for yellowness \(b^{\star}\) \[k_{c}=0.00951,\] (26)
* for hue angle \(h^{\star}\) \[k_{c}=0.077,\] (27)
* for chroma \(C^{\star}\) \[k_{c}=-0.16013+0.001343 LT_{d}.\] (28)  
It can be stated that the rate colour degradation for lightness and chroma depended in a statistically significant manner on characteristic dimension and on drying conditions (on \(T_{d}\) and \(v_{d}\) for \(L^{\star}\) and only on \(T_{d}\) for \(C^{\star}\)), and depended in a statistically insignificant way on rehydration conditions. Dependence of the rate colour degradation for yellowness and hue angle on both drying and rehydration conditions and on characteristic dimension was, on the other hand, statistically insignificant. The degradation of \(b^{\star}\) indicated the occurrence of Maillard reaction, responsible for nonenzymatic browning [77]. The colour variation \(\Delta E\) was predicted with ANN. MLP 5-8-1 was in this study (Figure 5). The _logsig_ transfer function in hidden layer and _purelin_ transfer function in output layer and _trainlm_ training function (L-M) and _learmgdm_ adaptation function were used. The R value was equal to 0.9166 (for validation set). Weights and biases between input and hidden layer and between hidden and output layers for this ANN are given in Table 10. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{**No.**} & \multicolumn{3}{c}{**Weights and Biases between Input and Hidden Layer**} & \multicolumn{3}{c}{**Weights and Biases between Hidden and Output Layer**} \\ \hline \multicolumn{1}{c}{} & \multicolumn{3}{c}{**Weights**} & \multicolumn{3}{c}{**Bias**} & \multicolumn{3}{c}{**Weights**} & \multicolumn{3}{c}{**Bias**} \\ \hline \(i\) & \(D_{ii}\) & \(D_{2i}\) & \(D_{3i}\) & \(D_{4i}\) & \(D_{3i}\) & \(D_{i}\) & \(W_{i}\) & \(B_{i}\) \\ \hline
1 & \(-0.3936\) & \(0.068912\) & \(7.7881\) & \(-0.37068\) & \(-0.29036\) & \(-4.083\) & 9.4829 & \\
2 & \(5.5244\) & \(3.3242\) & \(0.99513\) & \(-1.4537\) & \(-2.1599\) & 1.3043 & \(-1.7557\) & \\
3 & \(5.143\) & \(-0.15515\) & \(5.6857\) & \(-2.9232\) & \(-7.4673\) & 0.73944 & \(-3.9588\) & \\
4 & \(-0.67089\) & \(-5.7064\) & \(2.1326\) & \(0.36501\) & \(2.9711\) & 0.87002 & \(-3.3684\) & \\
5 & \(-4.0698\) & \(3.8803\) & \(2.1008\) & \(1.5752\) & \(3.0415\) & 0.7498 & 1.5977 & \\
6 & \(2.7802\) & \(5.5689\) & \(3.5537\) & \(-2.3719\) & \(-3.6688\) & \(-1.6085\) & \(-2.9859\) & \\
7 & \(0.13428\) & \(-0.70078\) & \(0.3338\) & \(-7.8718\) & \(1.8079\) & \(-0.83412\) & 4.0532 & \\
8 & \(7.6823\) & \(0.47277\) & \(-0.4144\) & \(8.8348\) & \(6.2643\) & \(-0.78616\) & 5.2191 & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Weights and biases between input and hidden layer and between hidden and output layers for the colour variation \(\Delta E\).