Modelling the Drying Kinetics of Apple (Golab Variety): Fractional Calculus vs Semi-empirical Models

In this work, two novel models have been proposed based on semi-empirical and factional calculus incorporating non-integer time derivatives in the Fick’s first law of anomalous diffusion. The experimental data has been collected from literature of 15 kinetics investigated in a convective dryer under the effect of temperatures ranging from 40 to 80 °C at 10 °C interval, and thickness of the slices of 2 to 6 mm at 2 mm interval. The collected experimental dataset was of apple slices (Golab variety). Results from this study were compared with a set of 64 thin-layer drying models previously published in the literature. The fitting capability of the models was compared using the mean of root mean square error MRMSE (%) of all kinetics and the global determination coefficient R2. All models’ constants and coefficients were optimised by dragonfly algorithm programmed in MATLAB software. Results showed that the fractional model is highly capable of describing the drying curve of the apple slices with a determination coefficient (R2) of 0.99981, and average root mean square error (MRMSE) of 0.43 % in comparison to the best empirical models with R2 of 0.99968 and MRMSE of 0.61 %.


Introduction
Drying is an important unit operation the role of which is to reduce the water content of the treated products. 1 Aimed at neutralizing bacterial activities and fungi, drying allows food products to be preserved for a long time. 2 It is a technique used worldwide; it allows storing and enhancing agricultural production surpluses in order to market them in times of scarcity. Given the diversity of the products to be dried and industrial processes, each with its specific constraints, the multiplicity of industrial drying techniques has been developed. [3][4][5][6] Several models of thin-layer drying kinetics of agricultural products have been proposed in literature 7 based on their behaviour, as well as the laws that govern energy and mass conservation, adsorption and desorption rates, heat transfer, and humidity. 8,9 The goal of our work was to develop two models, fractional and semi-empirical models. The performance of these models has been investigated based on experimental dataset collected from previously published scientific papers of 15 experimental drying kinetics of apples of different layer thickness (2,4, and 6 mm), and diverse temperatures (40, 50, 60, 70, and 80 °C). The accuracy of these models has been compared with 64 models from literature for thin-layer drying kinetics.

Modelling of the phenomenon of solar drying
Several mathematical models are proposed in the literature to model the drying kinetics of different products, in particular the semi-empirical models, which come from the drying phenomenology. 10 Table 1 presents 64 models published in the literature and tested on the drying of agrifood products. The moisture content evolution in agricultural products is directly proportional to its own moisture content 11 (see Eq. 1). ( where X(0) = X 0 , k is the kinetic constant of the model, X 0 the initial moisture content, X e is the equilibrium moisture content, and α is the fractional time index where 0 < α < 1 with α ϵ R. 12 The moisture ratio can be expressed and simplified into the following formula and after considering that X e is too small compared to X(t) and X 0 .
X(t) is the moisture content at time t, which is given by , m h is the successive weightings change of the wet product until it becomes stationary, m s is the dry mass of the product and was determined by drying the samples at 105 °C until the weight became constant. 13 with α as the whole order between 1 < α < 2, K α as the fractional rate constant, and n as the order of the reaction. These models were tested on 15 solar drying kinetics taken from the literature, of apple in a thin layer under the effect of 3 different thicknesses {2, 4, and 6 mm}, and 5 different temperatures {40, 50, 60, 70, and 80 °C}. 15 The second developed model in this work is a semi-empirical model.
Demir et al.
Haghi and Angiz -I Haghi and Angiz -II Haghi and Angiz -III Haghi and Angiz -IV Modifie Henderson and Perry Parameter adjustments of selected models from literature (64 models), as well as the proposed model were performed using dragonfly and swarm algorithms (DA MATLAB function). To avoid convergence to local minima, optimisation using dragonfly and swarm algorithms was performed 20 times. The optimised parameters of each model were used to estimate the moisture ratio and compare it with the experimental moisture ratio. The performance of each model was measured using different statistical parameters, like root mean square error (RMSE), reduced chi-squared (X 2 ), modelling efficiency (EF), and determination coefficient (R 2 ) for each kinetic and for each model. The average of these parameters for about 276 experimental points covering the 15 kinetics are presented in Table 2. The best performing model was selected based on the smallest average error (≈0) and the largest coefficient of determination (≈1). The mathematical formulas of these errors are presented in the articles cited in these references. [17][18][19][20][21][22][23] In order to simplify the comparison, the performances of the best 16 models are shown in Table 3 and Fig. 1, after having classified them from the smallest value to the largest value of MRMSE. The comparison between the performances of the different models indicates that the model proposed in this work MR64 provides the best correlation performance, where the value of MRMSE is (≈ 0.43 %).
The second best correlation performance is given by the coded model MR57, where the MRMSE has a value of (≈ 0.61 %).    show a slight superiority of the MR64 model compared to the MR57 model when modelling the experimental data of apple drying. Another comparison was made in terms of linear regression between the predicted and experimental rate of humidity (MR) to assess the ability of both models MR64 and MR57 to model solar drying kinetics.  show the juxtaposition of the first bisector and the line of the best linear fit of the output with the target, as well as the distribution of the experimental points confirming the excellent agreement between the moisture rate calculated by the two models and that of the experiment under different thicknesses and different temperatures for the entire database. Table 4 summarizes the errors and the linear regression vectors that are very close to ideal for all kinetics.

Fractional modelling results
This methodology was applied to fifteen drying kinetics of apple pieces in a thin layer under the effect of different temperatures and thicknesses. The results of the fractional modelling were then compared to those of the two best performing models found. The comparison was made using the mean statistical parameters and in the form of graphical illustrations and tables. The parameters of the fractional model were optimised using the "dragonfly" optimisation method programmed in the MATLAB software.
Optimisation results are presented in Table 5. In order to avoid local minima caused by the optimisation method, each optimisation operation was repeated 20 times.  A comparison in terms of histograms between the three best models was made by the overall coefficient of determination and the average of the mean absolute relative errors of the 15 drying kinetics of the apple in a thin layer (see Table 6).   Fig. 8, the proposed empirical model MR64 shows to be the best in terms of MRMSE and R 2 ; followed by the fractional model and the MR57 model.

Conclusion
The aim of this work was to model the phenomenon of apple drying in a thin layer using empirical and fractional methods. For this purpose, a dataset of 15 kinetics tak-en from previously published papers was extracted from drying kinetics of apple using Digitizer software. Sixty-four semi-empirical models were tested firstly, and based on the structure and performance of the best model, a semi-empirical model was proposed and tested using the same dataset. In addition, a novel model has been proposed using fraction calculus based on Fick's first law of n order.
The proposed semi-empirical model proves to be the most efficient by modelling the fifteen kinetics with MRMSE of 0.43 %, R 2 of 0.9998, followed by the fractional mod-