# Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
## 2 Materials and Methods
Although several mechanisms have been proposed in the literature to describe rehydration process on the basis of different mass transports, it is not clear which is the predominant mechanism [59]. Many theoretical and empirical approaches have been employed, and in some cases, empirical models were preferred. Among the various proposed models, empirical ones are the most widely used because of their mathematical simplicity and utility [23]. The discussed models are not only very useful but their interpretation also provides valuable information about the course of the rehydration because model constants have physical meaning. Models employed to characterise the rehydration kinetics of dried apple are shown in Table 1.  
Nomenclature for the equations given in Table 1 is as follows: \(t\) is the time (h); \(A_{1}\div A_{6}\); and \(a\), \(k\), and \(n\) are the constants (and they have a physical meaning, informing them about the rate of the described process). Variable \(X\) denotes the ratio: mass of rehydrated sample to the mass of dried material used for rehydration (\(m/m_{0}\)), volume of the rehydrated sample to the volume of the dried material used for rehydration (\(V/V_{0}\)), and dry matter of the rehydrated sample to the dry matter of dried material used for rehydration (\(m_{d.m.}/m_{d.m.0}\)). The variable \(Y\) (the values of mass gain, dry matter loss, and the volume increase were normalised to the range 0-1) is determined according to the following equations:  
* for the mass gain and the volume increase \[Y=\frac{X-1}{X_{e}-1},\] (8)
* for the dry matter loss. \[Y=X, \tag{9}\]  
where \(X_{e}\) denotes the equilibrium value. It can be determined from the Peleg model, Pilosof-Boquet-Batholomai model, and the Singh and Kulshrestha model, making the assumption that the process of rehydration lasted long enough and therefore \(t\to\infty\). The discussed value can be determined according to the following equations:  
* for the Peleg model \[X_{e}=1\pm\frac{1}{A_{2}},\] (10)
* for the Pilosof-Boquet-Batholomai model \[X_{e}=1+A_{3},\] (11)
* for the Singh and Kulshrestha model \[X_{e}=1+A_{5}.\] (12)  
It turned out from the calculations that the error of the difference between the values of \(X_{e}\) determined from the abovementioned models was not greater than 0.2%; therefore, to count the variable \(Y\), the equilibrium value determined from the Peleg model was taken. Empirical models (1)-(3) (Table 1) were developed to describe the rehydration process. As was mentioned above, they were easy to apply, and, very importantly, their constants have physical meaning. Constants \(A_{1}\), \(A_{4}\), and \(A_{6}\) provide the information about the rate of water absorption during the process, especially in its early stage, whereas constants \(A_{2}\), \(A_{3}\), and \(A_{5}\) allow for the prediction of water absorption kinetics for a relatively long period of absorption and inform us about the maximum capacity of water absorption. The [PERSON], Pilosof-Boquet-Batholomai, and Singh and Kulshrestha models have been widely \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model No.** & **Model Equation** & **Model Name** & **References** \\ \hline (1) & \(X=1\pm\frac{l}{A_{1}+A_{2}}\)* & [PERSON] & [60] \\ (2) & \(X=1+\frac{A_{3}l}{A_{4}+l}\) & Pilosof–Boquet–Batholomai & [61] \\ (3) & \(X=1+\frac{A_{5}l}{A_{6}l+l}\) & [PERSON] and [PERSON] & [62] \\ (4) & \(Y=\exp(-kl)\) & [PERSON] ([PERSON]) & [63] \\ (5) & \(Y=a\exp(-kl)\) & [PERSON]–Pabis & [64] \\ (6) & \(Y=\exp(-kl^{n})\) & Page & [65] \\ (7) & \(Y=\exp[-(kl)^{n}]\) & Modified Page & [66] \\ \hline \hline \end{tabular}
* Sing minus for the dry matter of solid loss.  
\end{table} 
Table 1: Empirical models applied to rehydration kinetics of dried apples. used for describing the relational process of biological materials. The Peleg model has been applied to describe, among others, the rehydration process of the following dried materials: kiwi [67], paraboiled rice [68], and squid fillets [69].