# Some Aspects of the Modelling of Dried Red Beets Rehydration Process
## 2 Materials and Methods

Table 2: Rehydration procedure. The moisture content (dry basis) is determined according to formula  
\[M=\frac{m-m_{d,m}}{m_{d,m}}. \tag{6}\]  
Variable \(Y\) is calculated for the mass gain and volume increase using equation  
\[Y=\frac{X-1}{X_{e}-1} \tag{7}\]  
and for dry matter loss, the formula is  
\[Y=X \tag{8}\]  
Equilibrium value \(X_{e}\) is evaluated from the Peleg model assuming that the rehydration lasted long enough (\(t\)\(\rightarrow\)\(\infty\))  
\[X_{e}=1\pm\frac{1}{A_{2}} \tag{9}\]  
The Peleg model was formulated for describing the process of rehydration. Although it is empirical model, its constants have physical meaning. Constant \(A_{1}\) gives the information about the water absorption rate especially during the early stage of the discussed process. Constant \(A_{2}\) allows us to predict the maximum capacity of water absorption. The mentioned model has been widely applied to predict the rehydration course of dried biological materials [39; 40; 41; 42; 43; 44]. Empirical models (2)-(5) given in Table 3 were formulated for describing the drying process. Their constant \(k\) also has a physical meaning because it gives information about the rate of the investigated process. The discussed models have been, however, applied also for describing the course of dried material rehydration. A first-order kinetics model, otherwise known as the [PERSON] (Newton) model, has been adopted to predict the rehydration process of such food materials as carrots [45], date plums [46], ginger [47], mangoes [48], onions [49], pumpkins [50], soybeans [51] and tomatoes [52]. The Page model has been used to describe the rewetting kinetics of such dried products as canola [53], lemon [54], pineapple [55], rough rice [56], Thompson seedless grape [54], and tiger nut [57]. The Henderson-Pabis and the modified Page models have been used to predict the course of the rehydration of dried apples [58]. Obtained from experiments, rehydration data (for dried red beet) were fitted to five investigated models (presented in Table 3). The Matlab R2018a software with the Levenberg-Marquardt algorithm was applied (Curve Fitting Toolbox2018a, MathWorks, Inc., Natick, MA, USA), whereas the quality of the considered models was statistically determined using the coefficient of determination R\({}^{2}\), adjusted R\({}^{2}\), and root mean square error RMSE (the highest values of R\({}^{2}\) and adjusted R\({}^{2}\) and the lowest RMSE values indicate the model with the best fit) [40; 59; 60]. In order to investigate the effect of drying and rehydration conditions on the constants of models (1)-(5), the following type of equation was used:  
\[\text{Constant}=a_{0}+a_{1}T_{d}+a_{2}v_{d}+a_{3}L+a_{4}T_{r}+a_{5}T_{d}{}^{2}+a _{6}v_{d}{}^{2}+a_{7}L^{2}+a_{8}T_{r}{}^{2}+a_{9}T_{d}v_{d}+a_{10}T_{d}L+a_{11} T_{d}T_{r}+a_{12}v_{d}L+a_{13}v_{d}T_{r}+a_{14}LT_{r} \tag{10}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model No.** & **Model Equation** & **Model Name** & **References** \\ \hline (1) & \(Y=1\pm\frac{t}{A_{1}+A_{2}^{2}}\)* & [PERSON] & [34] \\ (2) & \(Y=\exp(-kt)\) & [PERSON] ([PERSON]) & [35] \\ (3) & \(Y=\exp(-kt)\) & [PERSON]–Pabis & [36] \\ (4) & \(Y=\exp(-kt^{n})\) & Page & [37] \\ (5) & \(Y=\exp(-(kt)^{n})\) & Modified Page & [38] \\ \hline \hline \end{tabular}
* Sign minus refers to the dry matter of solid loss.  
\end{table} 
Table 3: Empirical models used for describing the course of the rehydration of dried red beets. where \(a_{0}\div a_{14}\) are constants, \(L\) is the characteristics dimension, \(T_{d}\) is the drying air temperature, \(T_{r}\) is the rehydration temperature, and \(v_{d}\) is the drying air velocity. The statistical analysis ANOVA was applied. The rehydration capacity of the dried red beet was determined using the following indices:  
* Rehydration index RI: \[\text{RI}=\frac{\text{mass of material after rehydration}}{\text{mass of dried material}}\] (11)  
The discussed index is very often applied for describing rehydration.