Fractional Kinetic Models for Drying Using a Semi-Empirical Method in the Framework of Different Types of Kernels

Abstract

In this study, we analyze the Lewis model within the framework of the Caputo–Fabrizio fractional derivative in the sense of Caputo (CFC), the Caputo-type Atangana–Baleanu (ABC) fractional derivative and the generalized ABC with a three-parameter Mittag–Leffler kernel using a semi-empirical method. We derive some novel analytic solutions for fractional kinetic models with the help of Laplace transforms. We also provide comparative visual representations of the solutions through graphs, using kinetic data for soybean drying at temperatures of 50, 60, 70 and 80 °C. The comparative solutions derived from kinetic data reveal the fundamental symmetrical behavior of the drying process at different temperatures.

1. Introduction

The purpose of food drying processes is to help prevent microbial growth and deterioration and extend shelf life by reducing the moisture content of food materials. This also improves storage for easy transportation. The drying kinetic phenomenon, which deals with the level of moisture loss inside food, has been investigated to identify the long-term storage conditions of foods. Dehydration processes are crucial in the food processing and chemical industries. Drying food products means the removal of water from solids up to a certain level, greatly minimizing microbial spoilage and spoilage chemical reactions. This is performed primarily because moisture content in food is directly related to the reproduction of microbes, fungi, etc. Such an adverse effect reduces the shelf life of foods, thereby increasing cost burden.

Kinetic modeling is a mathematical approach used to understand how a system evolves over time or how a reaction progresses. This modeling is widely used in many fields such as chemical reactions, biological processes, diffusion and heat transfer. This modeling helps us to understand the dynamic behavior of a system by determining the rate of change in a process.

In general, drying kinetic modeling is based on differential equations to represent the temporal variability of moisture content. Mathematical modeling of food drying processes is of great importance in terms of studies on drying technologies. We obtained a model using fractional operators, which has shown to provide better fits and more reliable adjustments than classical models.

A first-order kinetic model describing the moisture transfer during drying is considered, as follows:

$${\displaystyle \frac{-dX}{dt}}=k\left(X\left(t\right)-X_e\right).$$

(1)

In this context, X represents the concentration or density of a substance (such as a chemical or material). It changes over time, and its value at any given time is represented by $X\left(t\right)$. This parameter represents the material’s moisture content on a dry basis during drying (kg water/kg dry solids), while $X_e$ is the equilibrium moisture content of the dehydrated material (kg water/kg dry solids). $X_e$ represents the equilibrium concentration (or a stable state) of the system. Processes like reactions or diffusion tend to approach this value over time. It signifies the steady-state or equilibrium position of the system. The parameter k denotes the drying rate constant (min¹), and t refers to the drying time (min) [1]. The k parameter determines the speed of the process and can affect the rate of reaction, diffusion, or similar processes. The value of k can change with factors such as temperature, environmental conditions, or other influences. Equation (1) describes how a system evolves over time toward equilibrium. There are two main components: the changing quantity $X\left(t\right)$ and the equilibrium value $X_e$. The system will change over time such that $X\left(t\right)$ moves toward $X_e$.

Semi-empirical models have been designed to ensure ease of use and better alignment with the drying data of the food material undergoing dehydration. The development of semi-theoretical models is of great importance in order to ensure the compatibility of theoretical knowledge in real-life applications. Semi-theoretical models are the simplified general series solutions of Fick’s second law, which explains rates of diffusion involving both time and space:

$${\displaystyle \frac{\partial X}{\partial t}}=D{\nabla }^{2}X,$$

where $X\left(t\right)$ is the moisture content of the food and D is the diffusion coefficient.

Classical diffusion theory is often used in tne modeling and simulation of mass transfer studies such as drying studies. The modeling of mass transfer can be obtained by Fick’s Law. Studies on mass transfer models obtained with the fractional version of Fick’s Law show that diffusion is anomalous. While the classical Lewis model explains the exponential behavior of diffusion, the fractional Lewis model captures the non-exponential behavior, commonly referred to as anomalous diffusion. The classical approach in the mathematical modeling of the diffusion process applies Fick’s Law, but when diffusion is anomalous, fractional calculus can better explain the anomalous diffusion process and data.

Fractional calculus is an alternative mathematical method to describe models in many disciplines such as chemistry, biology, physics, medicine, food engineering, etc., and to obtain more precise results [2,3]. In the literature, there are many definitions of fractional derivatives obtained by defining new ones and developing predefined derivatives, the earliest known fractional derivatives include the Riemann–Liouville (RL) and Liouville–Caputo (C) derivatives [4,5]. In recent years, Caputo–Fabrizio [6] and Atangana–Baleanu [7] have introduced valuable definitions of derivatives and integrals applicable to real-world problems, making these derivatives highly sought after by scientists. In the non-singular CF derivative, the singularities present in the RL and Caputo definitions are removed through the use of the exponential function. In contrast, The AB fractional derivative is formulated by replacing the exponential function in the CF definition with the Mittag–Leffler function. Abdeljawad and Baleanu [8] studied the right fractional derivative and its corresponding integral for the newly proposed non-local fractional derivative with a Mittag–Leffler kernel, along with the related integration-by-parts formula. Abdeljawad [9] extended fractional calculus to higher orders with non-singular exponential kernels. In [10], fractional integrals of arbitrary order corresponding to fractional operators of the type Riemann (ABR) and Caputo (ABC) were derived using the infinite binomial theorem with a three-parameter Mittag–Leffler kernel. Al-Refai et al. [11] considered linear and nonlinear fractional diffusion equations defined by the CF fractional derivative. In studies [12,13], the authors analyzed the groundwater pollution equation in terms of the CF derivative. The CF derivative has a structure similar to the Caputo derivative; however, this derivative offers a higher level of ‘memory’ and a more flexible structure. The difference from the Caputo derivative is that it is less dependent on the past and exhibits better numerical properties. The AB derivative has a more general structure compared to the Caputo and CF derivatives. This derivative allows for a more flexible and broader memory definition in different systems, enabling a more accurate modeling of physical processes. Bas et al. [14] investigated fractional Sturm–Liouville problems with non-singular operators, presenting different versions with exponential and Mittag–Leffler kernels in the RL and Caputo senses, and obtained analytical solutions using the Laplace transform. In [15], real-world modeling problems were analyzed using the newly defined Liouville–Caputo fractional conformable derivative and its modified form. Inspired by these studies, we investigated the kinetic equation in terms of CFC, ABC and generalized ABC to obtain more precise results. Fractional derivatives allow us to extend the analysis of mathematical models beyond integer orders to arbitrary orders, providing a more flexible framework for complex systems. The nonlinear systems of fractional-order differential equations, defined in the Caputo sense, were solved using the Laplace Adomian decomposition method in [16]. The studies given in [17,18] provide a relevant numerical approach for fractional partial differential equations.

The primary motivation for using the aforementioned derivatives is that the non-singular kernels of the ABC, CFC and generalized ABC fractional derivatives offer greater flexibility in selecting appropriate values for the fractional-order parameter. This flexibility enables us to achieve more accurate results compared to their classical counterparts.

The use of fractional derivatives in kinetic modeling enables the representation of ‘memory’ processes, where systems are influenced by their past states. This is particularly useful for situations such as drying kinetics and heating processes. The studies given in [19,20] introduce a new fractional-order kinetic model for drying, developed by generalizing Lewis’s equation using the Laplace transform. In study [21], solutions of fractional kinetic equations based on the generalized Hurwitz–Lerch Zeta function have been obtained using the Sumudu transform. The fractional kinetic model was studied in [22] to analyze various suspension profiles considering non-local effects in turbulent transport processes. In [23], a mathematical model based on the generalization of the first-order kinetic model was proposed for the modeling of soybean drying kinetics, considering that the moisture variation rate is expressed by a derivative of arbitrary order. Qadha et al. [24] presented the solution of a generalized fractional kinetic equation using extended hypergeometric logarithmic functions. The authors of [25] used the Sumudu transform technique to calculate the solutions of the fractional kinetic equations.

The modeling presented in this article aims to examine the fractional kinetic model in terms of CFC, ABC and generalized ABC derivatives and to obtain more precise solutions by utilizing the experimental data from the study [19]. The ABC, CFC and generalized ABC fractional derivatives are highly useful for kinetic equations. These derivatives are particularly used for modeling memory processes and anomalous behaviors. The physical interpretation of this model is related to the ability of fractional derivatives to better capture memory effects and anomalous diffusion in physical processes such as drying kinetics. The examined fractional derivatives indicate that the system depends not only on its present state but also on its past states. This is particularly important for drying kinetics, heat transfer and other transport processes. Kinetic equations can be applied to a broader range of applications with these fractional derivatives, leading to more accurate results.

This paper introduces a novel approach that uses fractional-order calculus to enhance the accuracy and precision of soybean seed drying modeling. At first, the new fractional versions of the Lewis drying kinetic model are proposed. An effective analytical approach is employed to derive the solutions of the fractional Lewis kinetic model. A comparative analysis of the new solutions for the problems examined with different derivatives is presented using graphical representations. This study also illustrates, using figures, that the kinetic model solutions obtained through different fractional derivatives exhibit symmetric behaviors at certain temperature values.

The remainder of this paper is structured as follows: Section 2 presents fundamental definitions and theorems. Section 3 presents analytical solutions for fractional-order versions of the Lewis kinetic model within fractional derivatives ABC, CFC and generalized ABC, including the Mittag–Leffler function via the Laplace transform. Some significant discussions are presented with the help of graphs in Section 4. Finally, Section 5 provides a summary of the key findings of this study.

2. Preliminaries

In this part, we provide key theorems, definitions and properties of non-local operators that will be applied in the following sections.

Definition 1 

([6]). The left- and right-sided Caputo–Fabrizio fractional derivatives in the Caputo sense are defined by

$${}_a{}^{\mathit{CFC}}D^{\alpha }f\left(t\right)={\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}\underset{a}{\overset{t}{\int }}{f}^{\prime }\left(s\right)exp\left({\displaystyle \frac{-\alpha }{1-\alpha }}\left(t-s\right)\right)ds,$$

(2)

$${}^{\mathit{CFC}}D_b^{\alpha }f\left(t\right)={\displaystyle \frac{-M\left(\alpha \right)}{1-\alpha }}\underset{t}{\overset{b}{\int }}{f}^{\prime }\left(s\right)exp\left({\displaystyle \frac{-\alpha }{1-\alpha }}\left(s-t\right)\right)ds,$$

(3)

where $M\left(\alpha \right)>0$ is a normalization function with $M\left(0\right)=M\left(1\right)=1$, and $f\in$$H^1\left(a,b\right),$$a<b,$$\alpha \in \left[0,1\right]$.

Theorem 1 

([6]). The Laplace transform (LT) incorporating an exponential kernel is expressed as

$$\mathcal{L}\left\{\left({}_a{}^{\mathit{CFC}}D^{\alpha }f\right)\left(t\right)\right\}\left(s\right)={\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{s\mathcal{L}\left\{f\left(t\right)\right\}\left(s\right)}{s+\frac{\alpha }{1-\alpha }}}-{\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}f\left(a\right)e^{-as}{\displaystyle \frac{1}{s+\frac{\alpha }{1-\alpha }}}.$$

(4)

Definition 2 

([7]). The Atangana–Baleanu fractional derivatives in the Caputo sense are defined for the left and right sides, respectively, as follows:

$${}_a{}^{\mathit{ABC}}D^{\alpha }f\left(t\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}\underset{a}{\overset{t}{\int }}{f}^{\prime }\left(s\right)E_{\alpha }\left({\displaystyle \frac{-\alpha }{1-\alpha }}{\left(t-s\right)}^{\alpha }\right)ds,$$

(5)

$${}^{\mathit{ABC}}D_b^{\alpha }f\left(t\right)={\displaystyle \frac{-B\left(\alpha \right)}{1-\alpha }}\underset{t}{\overset{b}{\int }}{f}^{\prime }\left(s\right)E_{\alpha }\left({\displaystyle \frac{-\alpha }{1-\alpha }}{\left(s-t\right)}^{\alpha }\right)ds,$$

(6)

where $B\left(\alpha \right)>0$ is a normalization function with $B\left(0\right)=B\left(1\right)=1$ and $f\in$$H^1\left(a,b\right),$$a<b,$$\alpha \in \left[0,1\right].$

Theorem 2 

([13]). The fractional LT with Mittag–Leffler kernel is given by $0<\alpha \le 1:$

$$\mathcal{L}\left\{{}_a{}^{\mathit{ABC}}D^{\alpha }f\left(t\right)\right\}\left(s\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{s\mathcal{L}\left\{f\left(t\right)\right\}\left(s\right)-s^{\alpha -1}f\left(a\right)}{s^{\alpha }+\frac{\alpha }{1-\alpha }}}.$$

(7)

Definition 3 

([6]). The left- and right-sided ABC operator involving the generalized Mittag–Leffler function $E_{\alpha ,\mu }^{\gamma }\left(\lambda ,z^{\alpha }\right)$ such that $\gamma \in \mathbb{R},$$Re\left(\mu \right)>0,$$\alpha \in \left(0,1\right)$ and $\lambda ={\displaystyle \frac{-\alpha }{1-\alpha }}$ are given by

$${}_a{}^{\mathit{ABC}}D^{\alpha ,\mu ,\gamma }f\left(z\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}\underset{a}{\overset{t}{\int }}{E}_{\alpha ,\mu }^{\gamma }\left(\lambda ,{\left(z-x\right)}^{\alpha }\right)f^{\prime }\left(x\right)dx$$

(8)

and

$${}^{\mathit{ABC}}D_b^{\mathit{\alpha ,\mu ,\gamma }}f\left(z\right)={\displaystyle \frac{-B\left(\alpha \right)}{1-\alpha }}\underset{a}{\overset{t}{\int }}{E}_{\alpha ,\mu }^{\gamma }\left(\lambda ,{\left(x-z\right)}^{\alpha }\right)f^{\prime }\left(x\right)dx.$$

Theorem 3 

([10]). The LT of the generalized Atangana–Baleanu (ABC) operator can be expressed as

$$\mathcal{L}\left\{{}^{\mathit{ABC}}D^{\mathit{\alpha ,\mu ,\gamma }}f\left(t\right)\right\}\left(s\right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{s}^{1-\mu }\mathcal{L}\left\{f\left(t\right)\right\}{\left(1-\lambda s^{-a}\right)}^{-\gamma }-{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}f\left(0\right)s^{-\mu }{\left(1-\lambda s^{-a}\right)}^{-\gamma }.$$

(9)

Definition 4 

([5]). The Mittag–Leffler function $E_{\delta }\left(z\right)$ is given by the series representation as follows:

$$E_{\delta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }\left(\delta k+1\right)}},\left(z, \delta \in \mathbb{C}, \mathrm{Re}\left(\delta \right)>0\right),$$

(10)

and the Mittag–Leffler function with two parameters is defined by

$$E_{\delta ,\theta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathrm{\Gamma }\left(\delta k+\theta \right)}},\left(z, \delta , \theta \in \mathbb{C}, \mathrm{Re}\left(\delta \right)>0\right).$$

On the other hand, a generalized Mittag–Leffler function is given by

$$E_{\delta ,\theta }^{\rho }\left(\lambda ,z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{z}^{\delta k+\theta -1}{\left(\rho \right)}_k}{\mathrm{\Gamma }\left(\delta k+\theta \right)k!}},\left(z,\delta ,\theta \in \mathbb{C},\mathrm{Re}\left(\delta \right)>0\right)$$

where $\rho$ is an additional parameter, which is a positive real number, and ${\left(\rho \right)}_k$ represents the Pochhammer symbol and is known as the rising factorial such that ${\left(\rho \right)}_k=\rho \left(\rho +1\right)\dots \left(\rho +k-1\right).$ $\lambda$ is a real or complex parameter and is defined as a scaling factor in some special cases.

For a particular function, the Mittag–Leffler function is expressed as

$$E_{\delta }\left(\lambda ,z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{z}^{\delta k}}{\mathrm{\Gamma }\left(\delta k+1\right)}},$$

and

$$\begin{array}{c}\hfill E_{\delta ,\theta }\left(\lambda ,z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\lambda }^k{z}^{\delta k+\theta -1}}{\mathrm{\Gamma }\left(\delta k+\theta \right)}},\lambda \in \mathbb{R},z\in \mathbb{C},\mathrm{Re}\left(\delta \right)>0.\end{array}$$

(11)

Lemma 1 

([10]). Let $\alpha ,\mu ,\gamma ,\lambda ,s\in \mathbb{C},$$\mathrm{Re}\left(\mu \right)>0,$$\mathrm{Re}\left(s\right)>0,$$\left|\lambda s^{-\alpha }\right|<1.$ Then, the LT of $E_{\alpha ,\mu }^{\gamma }\left(\lambda ,t^{\alpha }\right)$ is given by

$$\mathcal{L}\left\{E_{\alpha ,\mu }^{\gamma }\left(\lambda ,t^{\alpha }\right)\right\}=s^{-\mu }{\left(1-{\lambda }^{-\alpha }\right)}^{-\gamma }.$$

3. Main Results

In this part, we examine the Lewis kinetic model given in Equation (1) by means of non-local fractional operators, including ABC, CFC and generalized ABC, with a three-parameter Mittag–Leffler function. Analytical solutions of the abovementioned fractional models have been obtained using Laplace transforms. Some studies about integer-order and fractional-order drying kinetic models can be found in studies [19,20]. The Caputo fractional derivative is widely utilized in real-world applications because it allows for integer-order initial conditions, and the derivative of a constant is zero. In this context, kinetic models have been considered by CF and AB in the Caputo sense (CFC and ABC, respectively), and ABC with a generalized Mittag–Leffler kernel.

3.1. Kinetic Model in the Framework of CFC Derivatives

The fractional kinetic model in the Caputo–Fabrizio sense is formulated as follows:

$$\begin{array}{cc}\hfill {}_0{}^{\mathit{CFC}}D^{\alpha }Y\left(t\right)& =-k\left(Y\left(t\right)-Y_e\right),0<\alpha <1,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill Y\left(0\right)& =c_1.\hfill \end{array}$$

(13)

If the is LT applied to Equation (12) and considering the initial condition Equation (13) $,$ we arrive at

$$\mathcal{L}\left\{{}_0{}^{\mathit{CFC}}D^{\alpha }Y\left(t\right)\right\}=\mathcal{L}\left\{-k\left(Y\left(t\right)-Y_e\right)\right\}.$$

(14)

Thus, we have

$${\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{s\mathcal{L}\left\{Y\left(t\right)\right\}-Y\left(0\right)}{s^{\alpha }+\frac{\alpha }{1-\alpha }}}=-k\mathcal{L}\left\{Y\left(t\right)-Y_e\right\},$$

(15)

and from here, by applying the inverse LT to Equation (15), the analytical solution Equations (12) and (13) can be expresses as follows:

$$Y\left(t\right)=M\left(\alpha \right)e^{{\displaystyle \frac{\alpha kt}{\left(\alpha -1\right)k-M\left(\alpha \right)}}}\left(Y_e-Y\left(0\right)\right)+Y_e.$$

(16)

Example 1. 

Let us examine the following example for the case of $\alpha =0.7$:

$$\begin{array}{cc}\hfill {}_0{}^{\mathit{CFC}}D^{0.7}Y\left(t\right)& =-k\left(Y\left(t\right)-Y_e\right),0<\alpha <1,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Y\left(0\right)& =1.\hfill \end{array}$$

(18)

If the is LT is applied to Equation (17), and considering the initial condition Equation (18)$,$ we arrive at

$$\mathcal{L}\left\{{}_0{}^{\mathit{CFC}}D^{0.7}Y\left(t\right)\right\}={\displaystyle \frac{M\left(0.7\right)}{1-0.7}}·{\displaystyle \frac{s\mathcal{L}\left\{Y\right(t\left)\right\}-Y\left(0\right)}{s^{0.7}+\frac{0.7}{1-0.7}}}.$$

By making the necessary adjustments here, the following equation is obtained:

$$Y\left(s\right)={\displaystyle \frac{\frac{k{Y}_e}{s}+\frac{M\left(0.7\right)}{0.3}·\frac{1}{s^{0.7}+\frac{7}{3}}}{k+\frac{M\left(0.7\right)}{0.3}·\frac{s}{s^{0.7}+\frac{7}{3}}}}.$$

By using the partial fraction decomposition method and then applying the inverse Laplace transform, the analytical solution Equations (17) and (18) can be expresses as follows:

$$Y\left(t\right)=Y_e+M\left(0.7\right)e^{{\displaystyle \frac{0.7kt}{-0.3k-M\left(0.7\right)}}}(Y_e-1).$$

3.2. Kinetic Model in the Framework of ABC Derivative

The fractional kinetic model formulated within the ABC fractional framework is presented as follows:

$$\begin{array}{cc}\hfill {}_0{}^{\mathit{ABC}}D^{\alpha }Y\left(t\right)& =-k\left(Y\left(t\right)-Y_e\right),0<\alpha <1,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill Y\left(0\right)& =c_2.\hfill \end{array}$$

(20)

By performing the LT to Equation (19) under the initial condition Equation (20), we obtain

$$\mathcal{L}\left\{{}_0{}^{\mathit{ABC}}D^{\alpha }Y\left(t\right)\right\}=\mathcal{L}\left\{-k\left(Y\left(t\right)-Y_e\right)\right\}.$$

Thus, one can obtain

$${\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\displaystyle \frac{s\mathcal{L}\left\{Y\left(t\right)\right\}-s^{\alpha -1}Y\left(0\right)}{s^{\alpha }+\frac{\alpha }{1-\alpha }}}=\mathcal{L}\left\{-k\left(Y\left(t\right)-Y_e\right)\right\}.$$

(21)

By applying the inverse LT to Equation (21), the genaral solution Equations (19) and (20) can be expressed as

$$Y\left(t\right)={\displaystyle \frac{Y_{e}k\alpha }{B\left(\alpha \right)+k\left(1-\alpha \right)}}+{\displaystyle \frac{B\left(\alpha \right)Y\left(0\right)+Y_{e}k\left(1-2\alpha \right)}{B\left(\alpha \right)+k\left(1-\alpha \right)}}{E}_{\alpha }\left(-{\displaystyle \frac{\alpha k{t}^{\alpha }}{B\left(\alpha \right)+k\left(1-\alpha \right)}}\right).$$

(22)

3.3. Kinetic Model in the Framework of Generalized ABC Derivative

Consider the fractional kinetic model in the sense of a right-sided generalized ABC operator as follows:

$$\begin{array}{cc}\hfill {}_0{}^{\mathit{ABC}}D^{\mathit{\alpha ,\mu ,\gamma }}Y\left(t\right)& =-k\left(Y\left(t\right)-Y_e\right),0<\alpha <1,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill Y\left(0\right)& =c.\hfill \end{array}$$

(24)

If the LT is applied to Equation (23) and evaluated with the initial condition Equation (24), we obtain

$$\mathcal{L}\left\{{}_0{}^{\mathit{ABC}}D^{\mathit{\alpha ,\mu ,\gamma }}Y\left(t\right)\right\}=\mathcal{L}\left\{-k\left(Y\left(t\right)-Y_e\right)\right\},$$

and then, it can be easily determined that

$${\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{s}^{1-\mu }\mathcal{L}\left\{Y\left(t\right)\right\}{\left(1-\lambda s^{-\alpha }\right)}^{-\gamma }-{\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}Y\left(0\right)s^{-\mu }{\left(1-\lambda s^{-\alpha }\right)}^{-\gamma }+k\mathcal{L}\left\{Y\left(t\right)\right\}=k{\displaystyle \frac{Y_e}{s}}.$$

(25)

If Equation (25) is simplified,

$$\mathcal{L}\left\{Y\left(t\right)\right\}={\displaystyle \frac{Y\left(0\right)}{s+\frac{k\left(1-\alpha \right)}{B\left(\alpha \right)s^{-\mu }{\left(1-\lambda s^{-\alpha }\right)}^{-\gamma }}}}+{\displaystyle \frac{k\frac{Y_e}{s}}{\frac{B\left(\alpha \right)}{1-\alpha }{s}^{1-\mu }{\left(1-\lambda s^{-\alpha }\right)}^{-\gamma }+k}}.$$

(26)

From the expansions, we have

$$\begin{array}{cc}\hfill \mathcal{L}\left\{Y\left(t\right)\right\}& ={\displaystyle \frac{Y\left(0\right)}{s}}\sum _{j=0}^{\infty }{\left(-k\right)}^j{\left({\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}\right)}^j{s}^{\left(\mu -1\right)j}{\left(1-\lambda s^{-\alpha }\right)}^{\gamma j}+\hfill \\ \hfill & +k{\displaystyle \frac{Y_e}{s}}\sum _{j=0}^{\infty }{\left(-k\right)}^j{\left({\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}\right)}^{j+1}{s}^{\left(\mu -1\right)\left(j+1\right)}{\left(1-\lambda s^{-\alpha }\right)}^{\gamma \left(j+1\right)}.\hfill \end{array}$$

(27)

By applying the inverse LT to Equation (27), the genaral solution Equations (23) and (24) can be written as

$$Y\left(t\right)=Y\left(0\right)\sum _{j=0}^{\infty }{\left(-k\right)}^j{\left({\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}\right)}^j{E}_{\alpha ,\left(1-\mu \right)j+1}^{-\gamma j}\left(\lambda ,t\right)+k{Y}_e\sum _{j=0}^{\infty }{\left(-k\right)}^j{\left({\displaystyle \frac{1-\alpha }{B\left(\alpha \right)}}\right)}^{j+1}{E}_{\alpha ,\left(1-\mu \right)\left(j+1\right)+1}^{-\gamma \left(j+1\right)}\left(\lambda ,t\right).$$

(28)

4. Discussion

In this section, Lewis fractional-order models, formulated using CFC, ABC and ABC with a generalized Mittag–Leffler kernel, were applied to the drying process at temperatures of 50, 60, 70 and 80 °C, respectively. The estimated parameters for each model are presented in Table 1, Table 2, Table 3 and Table 4. The kinetic data for soybean drying were obtained from the study conducted by [19]. Additionally, comparative graphs of the fractional Lewis models with the classical Lewis and Page models are provided.

This study aims to extend and refine the fractional-order results presented in [19] using non-local fractional operators. Through this approach, we provide a more comprehensive and precise analysis of the drying kinetics. As observed in the graphs, increasing the value of $\alpha$ and approaching 1 in fractional modeling yield more accurate and precise results compared to the Lewis and Page models. Our findings indicate that the fractional results derived from Lewis’s CFC and Lewis’s ABC models exhibit significantly better agreement with the experimental data than those obtained using Lewis’s ABC model with the generalized Mittag–Leffler kernel. This trend is consistently observed across all graphical representations, further supporting the effectiveness of these fractional models in accurately describing the drying behavior. The kinetic model solutions examined in terms of CFC, ABC and generalized ABC derivatives exhibit symmetric behavior at specific temperature values, as can be seen in all the graphs in this section.

Table 1. Parameter values of drying models at 50 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC	Lewis Gen. ABC
Figure 1a	k	$0.31$	$0.178$	$0.23$	$0.23$	$0.23$
	Ye	$0.10$	$0.12$	$0.11$	$0.11$	$0.11$
	n	$0.52$	–	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$	$0.55$

Table 1. Parameter values of drying models at 50 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC	Lewis Gen. ABC
Figure 1a	k	$0.31$	$0.178$	$0.23$	$0.23$	$0.23$
	Ye	$0.10$	$0.12$	$0.11$	$0.11$	$0.11$
	n	$0.52$	–	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$	$0.55$

Figure 1. Drying kinetic behavior at 50 °C for solution functions under different models in °C: (a) $\alpha =0.55$; (b) $\alpha =0.7$; and (c) $\alpha =0.9$.

Figure 1. Drying kinetic behavior at 50 °C for solution functions under different models in °C: (a) $\alpha =0.55$; (b) $\alpha =0.7$; and (c) $\alpha =0.9$.

Table 2. Parameter values of drying models at 60 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 2a	k	$0.39$	$0.23$	$0.31$	$0.31$
	Ye	$0.08$	$0.10$	$0.07$	$0.07$
	n	$0.46$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Table 2. Parameter values of drying models at 60 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 2a	k	$0.39$	$0.23$	$0.31$	$0.31$
	Ye	$0.08$	$0.10$	$0.07$	$0.07$
	n	$0.46$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Figure 2. Drying kinetic behavior at 60 °C for solution functions under different models in °C: (a) $\alpha =0.55, {\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9, {\beta }_2=0.5$, $c_1=c_2=1$.

Figure 2. Drying kinetic behavior at 60 °C for solution functions under different models in °C: (a) $\alpha =0.55, {\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9, {\beta }_2=0.5$, $c_1=c_2=1$.

Table 3. Parameter values of drying models at 70 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 3a	k	$0.35$	$0.21$	$0.28$	$0.28$
	Ye	$0.07$	$0.10$	$0.09$	$0.09$
	n	$0.49$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Table 3. Parameter values of drying models at 70 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 3a	k	$0.35$	$0.21$	$0.28$	$0.28$
	Ye	$0.07$	$0.10$	$0.09$	$0.09$
	n	$0.49$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Table 4. Parameter values of drying models at 80 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 4a	k	$0.35$	$0.24$	$0.32$	$0.32$
	Ye	$0.05$	$0.08$	$0.06$	$0.06$
	n	$0.48$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Table 4. Parameter values of drying models at 80 °C.

Figures	Parameters	Pages	Lewis Classic	Lewis CFC	Lewis ABC
Figure 4a	k	$0.35$	$0.24$	$0.32$	$0.32$
	Ye	$0.05$	$0.08$	$0.06$	$0.06$
	n	$0.48$	–	–	–
	$\alpha$	–	–	$0.55$	$0.55$

Figure 3. Drying kinetic behavior at 70 °C for solution functions under different models in °C: (a) $\alpha =0.55, {\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9, {\beta }_2=0.5$, $c_1=c_2=1$.

Figure 3. Drying kinetic behavior at 70 °C for solution functions under different models in °C: (a) $\alpha =0.55, {\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9, {\beta }_2=0.5$, $c_1=c_2=1$.

Figure 4. Drying kinetic behavior at 80 °C for solution functions under different models in °C: (a) $\alpha =0.55,{\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9,{\beta }_2=0.5$, $c_1=c_2=1$.

Figure 4. Drying kinetic behavior at 80 °C for solution functions under different models in °C: (a) $\alpha =0.55,{\beta }_1=0.5$; (b) $\alpha =0.7$; and (c) $\alpha =0.9,{\beta }_2=0.5$, $c_1=c_2=1$.

5. Conclusions

As a result, this study presents an in-depth analysis of the Lewis model by incorporating various fractional derivatives: the Caputo–Fabrizio fractional derivative, the Caputo-type Atangana–Baleanu fractional derivative and the generalized ABC fractional derivative with a Mittag–Leffler kernel using a three-parameter approach. This study employs a semi-empirical method to derive novel analytical solutions for these fractional kinetic models using Laplace transforms. We evaluated the findings based on scientific data from the study [19] and compared the results in terms of fractional derivatives of CFC, ABC and generalized ABC.

Upon examining Figure 1c, Figure 2c, Figure 3c and Figure 4c, we find that the ABC fractional derivative provides more precise results than the CFC fractional derivative as well as the Lewis and Page models. Furthermore, when we reduce the value of $\alpha$, the fractional derivative of CFC produces more accurate results than the others, as observed in Figure 1a,b, Figure 2a,b, Figure 3a,b and Figure 4a,b. Finally, as seen in Figure 5a,b, we found that the fractional derivatives of CFC and ABC produced more consistent results compared to the generalized fractional derivative of ABC under all conditions.

Ultimately, the fractional derivatives examined in this study produced consistently superior results across all conditions when applied to the grain drying process within the framework of the Lewis model. By incorporating the scientific data provided by Matias et al. [19], and supporting these findings with detailed graphical representations, we have demonstrated that the fractional models considered here outperform the traditional integer-order model in terms of accuracy and reliability. The figures clearly illustrate that the fractional-order models, which account for memory effects and non-local interactions in the drying process, provide a more precise description of the kinetics involved. This comparison highlights the advantages of using fractional derivatives over integer-order models, as they better capture the complexity and dynamics of the drying process, leading to more accurate predictions and a deeper understanding of the phenomena at play. Thus, the results presented strongly support the idea that fractional modeling offers a more effective approach for analyzing and optimizing grain drying processes. Additionally, by improving the accuracy of our fractional kinetic model approach, it can assist in the optimization of drying equipment and energy consumption in the food, pharmaceutical and material processing industries.