Drying Kinetics of Industrial Pineapple Waste: Effective Diffusivity and Thermodynamic Properties Resulting from New Mathematical Models Derived from the Fick Equation

Abstract

This research focuses on the drying kinetics of industrial pineapple processing waste on a flat plate, revealing a two-phase drying process: an initial phase with a constant drying rate followed by a phase with a decreasing drying rate. During the constant rate phase, the convective mass transfer coefficient, influenced by temperature variations from 40 to 70 °C, ranged from 5.69 × 10⁻⁷ to 2.79 × 10⁻⁷ m s⁻¹. The study introduced a novel approach to modeling the decreasing drying rate phase, applying equations derived from the Fick equation. This process involved determining the activation energy and thermodynamic properties of drying using an experimental forced convection dryer at temperatures of 40, 50, 60, and 70 °C, and an air velocity of 1.5 m/s. Data were fitted to several mathematical models, including Fick’s with four series terms, and versions of the Henderson–Pabis and Page models modified by Cavalcanti-Mata, among others. The Cavalcanti-Mata and modified Page models provided the most accurate fit to the experimental data. Results showed that diffusion coefficients vary per model yet align with literature values. Additionally, enthalpy (ΔH) and entropy (ΔS) values decreased with temperature, while Gibbs free energy (ΔG) increased, indicating that drying is an energy-dependent, non-spontaneous process.

1. Introduction

Fruit cultivation is one of the important sectors of the Brazilian economy that has been increasingly prominent in recent years and continues to evolve, encompassing both the production of fresh fruits and the processing of juices and nectars [1]. Brazil is the world’s third-largest fruit producer, yielding approximately 45 million tons per year and utilizing an average of 2.6 million hectares, which accounts for only 0.3% of the national territory. In contrast, other crops occupy 7.8% of Brazil’s land. Fruit cultivation is a significant activity for the country’s economy, generating revenues of USD 6.45 billion in 2016 and USD 7.82 billion in 2021, thereby creating numerous job opportunities [2].

Pineapple farming in Brazil spans an area of 63,589 hectares, yielding 1,545,036 million fruits annually. These fruits are commonly eaten fresh or processed. Fresh pineapples have a shorter shelf life and tend to be consumed more quickly. They are often discarded when not consumed, contributing to a waste problem that reaches approximately 40% in Brazil. This high figure highlights the need for improved logistics within the sector [3].

A portion of pineapple production involves processing into juice, pulps, concentrates, jams, and other products, which extends the shelf life of these items, thus mitigating significant waste within the industry. However, this activity generates organic waste in the range of 30 to 40% of the processed raw material. A portion of these byproducts is irregularly discarded, violating Brazilian environmental legislation [4].

Some fruit processing industries in the Northeastern region, such as pineapple processing, often fail to recognize that they generate a significant volume of waste, which can be repurposed for various applications. These waste materials contain substantial nutritional value that can be utilized to create new products. Among these possibilities, one can mention their use in formulating supplements for human consumption and as animal feed components for nutritional purposes, thus fostering the development of novel products derived from this agro-industry [5]. In recent years, some authors have explored the feasibility of utilizing these waste materials to enhance human and/or animal nutrition [6].

To use these products, it is necessary to employ processing methods that ensure their quality and safety during storage. For this purpose, reducing biological activity by drying these waste materials is essential, which is the most conventional and cost-effective approach [7]. Once dried, the product can be stored for extended periods since microorganisms and enzymes that degrade and alter their chemical composition, such as fungi and yeasts, cannot function without water [8]. Dried pineapple waste holds significant potential for diverse applications, contributing to both human and animal nutrition as well as various industrial uses. For human consumption, it can be processed into nutritional supplements rich in dietary fiber, vitamins, and minerals, benefiting especially those in developing regions [9]. It also serves as a functional ingredient in foods like baked goods, cereals, and snack bars, and its bioactive compounds with antioxidant properties can be incorporated into functional foods to promote health benefits [5]. In animal feed, dried pineapple waste offers a cost-effective, nutritious alternative for livestock and poultry, enhancing their fiber intake, digestion, and overall health, and can also be used in pet food formulations [10]. Industrially, it is valuable for extracting bioactive compounds such as bromelain, producing compost and soil amendments, and developing biodegradable plastics and packaging materials [11]. Utilizing dried pineapple waste in these ways not only reduces environmental impact, but also supports food security and provides economic benefits, especially in regions with significant pineapple production and processing. As the processed product contains a high water content, it must be dried until it reaches a moisture content that does not facilitate rapid degradation.

Recognizing the significant waste generated in the pineapple industry during fruit pulp production presents an opportunity for sustainable waste management and valorization strategies. The critical research gap lies in the underexplored potential of converting this agro-industrial waste into valuable alternative food ingredients, necessitating a thorough understanding of the drying kinetics and the associated thermodynamic properties.

This study introduces a novel approach by integrating a comprehensive analysis of both constant and decreasing drying rate periods in the drying kinetics of industrial pineapple waste. By using detailed mathematical modeling and thermodynamic property determination, this approach offers a more precise representation of the drying process. Specifically, this study employed the Cavalcanti-Mata, modified Page, and modified Henderson–Pabis models to accurately capture the drying behavior. Additionally, the study considered the high moisture content and specific chemical composition of the pineapple waste, allowing for the development of tailored drying parameters that optimize the drying process for this particular type of agro-industrial residue. Additionally, this manuscript aims to illuminate the importance of studying these drying kinetics and obtaining precise values for effective diffusion parameters and thermodynamic properties, such as enthalpy, entropy, and Gibbs free energy, of pineapple waste. By analyzing the drying process at various temperatures and air velocities, and employing advanced mathematical models for parameter estimation, this study aims to unlock the potential of pineapple industry waste. Understanding these parameters is pivotal for optimizing drying processes, enhancing energy efficiency, and ultimately contributing to developing sustainable food production methods that effectively utilize waste products.

2. Materials and Methods

2.1. Materials

The raw material used in the experiments was pressed pineapple residue. The residue was collected from the COOPEAGRO (Cooperative of Small Organized Farmers) fruit processing industry located on AL 101 Norte Highway No. 382 Santa Tereza Verzeri District Maragogi-AL. The pressing of the residue was performed using a hydraulic press at a pressure of 100 kPa to ensure consistent porosity. The average porosity of the residue, determined by a method described elsewhere [12], was found to be 0.45. The chemical composition of the pineapple peels was 86% moisture, 5.8% crude protein (dry basis, d.b.), 5.6% crude fat d.b., 4.1% fiber d.b., 3.7% ash d.b., and 78.9% carbohydrate d.b. After pressing, the residues were placed in plastic bags and cooled to 5 °C in the company’s refrigeration chamber. They were then transported 25 km in Styrofoam boxes to the Food Laboratory of IFPE Campus-Barreiros, where they were stored in a freezer until the start of the experiments.

2.2. Pineapple Peel Treatment

The pineapple residues were washed and disinfected in a potable water solution at 5 °C with 200 mg/L of chlorine for 5 min. After this procedure, the material was rinsed again with running potable water to remove the chlorine. Subsequently, a heat treatment was performed in which the product was heated on an industrial stove at 100 °C for 30 s and then cooled by adding ice water until it reached a temperature between 35 and 40 °C to prevent the growth of filamentous fungi and yeast. After these treatments, the product was placed on a sieve to drain the remaining water and then distributed onto trays made of aluminum mesh with a 1 mm opening, measuring 130 × 70 × 50 mm, and placed in a dryer. Before being placed in the dryer, the initial moisture content was determined using the gravimetric method [13].

2.3. Drying Kinetics Determination

The drying kinetics of pineapple waste distributed on aluminum mesh trays were determined at temperatures of 40, 50, 60, and 70 °C with a drying air velocity of 1.5 m/s. The temperatures were selected to encompass a range that simulates typical industrial drying conditions, thereby providing comprehensive data across a broad temperature spectrum. During the kinetics, the trays were weighed at different time intervals to accurately capture the drying behavior at various stages. The initial intervals (every 5 min until 30 min had passed) were shorter to capture the rapid moisture loss during the initial drying phase. As the drying rate decreased, the intervals were extended (every 10 min until 60 min, every 15 min until 105 min, every 30 min until 180 min, and every 60 min until a constant weight was reached) to monitor the slower, steady-state drying phase. The weighing intervals were chosen to balance the need for detailed data collection and practical experimental constraints. Subsequently, triplicate samples of the material were taken to determine the dynamic equilibrium moisture content using the methodology described by [13]. The temperatures of 40, 50, 60, and 70 °C were chosen based on their relevance to industrial drying processes. These temperatures are commonly used in the drying of agricultural products, allowing for the study to be directly applicable to industrial settings. The range was selected to cover low to moderately high temperatures, thereby helping in understanding the drying behavior and energy requirements under different thermal conditions. The drying air velocity of 1.5 m/s was selected based on preliminary experiments and literature references that indicate this velocity is effective for convective drying of similar agricultural residues. The chosen velocity ensures sufficient air flow to carry away moisture without causing excessive turbulence or sample disturbance, thereby maintaining consistent drying conditions.

The sample weighing data during drying were used to calculate the moisture content ratios of pineapple waste, as shown in Equation (1):

$$\mathrm{R}\mathrm{X}=\frac{\mathrm{X}-{\mathrm{X}}_{\mathrm{e}}}{{\mathrm{X}}_0-{\mathrm{X}}_{\mathrm{e}}}$$

(1)

where RX = moisture content ratio of the product, dimensionless; X = moisture content, dry basis, decimal; X₀ = initial moisture content, dry basis, decimal; X_e = equilibrium moisture content, dry basis, decimal [14].

The drying of pineapple waste, due to its high initial moisture content, was divided into two periods: (a) constant drying rate; (b) decreasing drying rate. In the constant drying rate period, where water is removed in the absence of the diffusion phenomenon, the governing equation for the process is given by Equation (2), where it is assumed that there is no resistance from the product structure to the water removal [15]:

$$\frac{\mathbf{d}\left(\mathbf{X}\right)}{\mathbf{d}\mathbf{t}}=\frac{{\mathbf{h}}_{\mathbf{d}}{\mathbf{A}}_{\mathbf{s}}}{{{\mathbf{R}}_{\mathbf{v}}\mathbf{T}}_{\mathbf{a}\mathbf{b}\mathbf{s}}}({\mathbf{P}}_{\mathbf{v}\mathbf{s}}-{\mathbf{P}}_{\mathbf{v}})$$

(2)

A rectangular shape was considered due to its small thickness, as shown in Figure 1.

$${\mathrm{A}}_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r} \mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}=2\left[\mathrm{L}\mathrm{C}+\mathrm{C}\mathrm{h}+\mathrm{L}\mathrm{h}\right]$$

(3)

$${\mathrm{A}}_{\mathrm{s}\mathrm{f}}=\left[\sum {\mathrm{L}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{C}\mathrm{\prime }}_1^{\mathrm{n}}+{\mathrm{C}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{h}\mathrm{\prime }}_1^{\mathrm{n}}+{\mathrm{L}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{h}\mathrm{\prime }}_1^{\mathrm{n}}\right]$$

(4)

$${\mathrm{A}}_{\mathrm{s}}=2\left[\mathrm{L}\mathrm{C}+\mathrm{C}\mathrm{h}+\mathrm{L}\mathrm{h}\right]+\left[\sum {\mathrm{L}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{C}\mathrm{\prime }}_1^{\mathrm{n}}+{\mathrm{C}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{h}\mathrm{\prime }}_1^{\mathrm{n}}+{\mathrm{L}\mathrm{\prime }}_1^{\mathrm{n}}{\mathrm{h}\mathrm{\prime }}_1^{\mathrm{n}}\right]$$

(5)

where h_d = convective mass transfer coefficient, m s⁻¹; A_sf and A_s are the powder surface area and the total product surface area, respectively, m²; R_v = water vapor gas constant, 0.462 kJ kg⁻¹K⁻¹; T_abs = dry bulb temperature, K; L’, L = width of the powder and total pineapple waste rectangle, mm; C’, C = length of the powder and total pineapple waste rectangle, mm; h, h’ = height of the powder and total pineapple waste rectangle, mm; P_vs = saturation vapor pressure, Pa; P_v = vapor pressure, Pa.

2.4. Mathematical Models

The use of both constant and decreasing drying rate periods in this study allows for a comprehensive analysis of the drying kinetics of industrial pineapple waste. By examining these distinct phases, the study can provide detailed insights into the drying behavior, enabling the development of more accurate and effective drying models. This approach ensures that the drying process is optimized for efficiency and quality, contributing to sustainable waste management practices in the pineapple processing industry. The data obtained during the decreasing drying rate period were separated from the constant drying rate period, and these data were fitted to the three mathematical models derived from Fick’s equation, establishing for each of these models a new approach that allows the determination of their respective effective diffusion coefficients. The constant rate period is characterized by a relatively steady rate of moisture removal, where surface moisture evaporates quickly. The transition to the decreasing rate period occurs when the rate of moisture removal begins to decrease, indicating that surface moisture has been significantly reduced, and the drying process is now dominated by internal moisture diffusion. This changing point was identified by plotting the drying rate against time and observing the point where the slope of the curve starts to decline significantly.

The theory of diffusion is based on Fick’s second law, which describes the diffusion of water in a solid through a concentration gradient [15]. This concept can be expressed by Equation (6):

$$\frac{\partial M}{\partial t}=\nabla (D\nabla M)$$

(6)

where ∇ f(x₁, …, xₙ) = vector of partial derivatives; D = diffusion coefficient, m²s⁻¹; t = drying time, s. Crank [16], employing Fick’s equation, developed solutions for various initial conditions and boundary conditions for materials with diverse geometric shapes. Equation (7) represents the equation in Cartesian coordinates:

$$\frac{\partial M}{\partial t}-D{\nabla }^2\mathrm{M}=\frac{\partial \mathrm{M}}{\partial t}-D\left(\frac{\partial M^2}{\partial x^2}+\frac{\partial M^2}{\partial y^2}+\frac{\partial M^2}{\partial z^2}\right)=0$$

(7)

Considering that there is a certain level of material homogeneity, the equation can be regarded as one-dimensional and simplified to Equation (8):

$$\frac{\partial \mathrm{M}}{\partial \mathrm{t}}=\frac{\partial }{\partial \mathrm{X}}\left(\mathrm{D}\frac{\partial \mathrm{M}}{\partial \mathrm{X}}\right)$$

(8)

The analytical solution provided by Crank [16] is expressed by Equation (9):

$$\mathrm{R}\mathrm{X}=\frac{8}{{\mathsf{\pi }}^2}{\sum }_{\mathrm{n}=0}^{\mathrm{\infty }}\frac{1}{{(2\mathrm{n}+1)}^2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{\left(2\mathrm{n}+1\right){\mathsf{\pi }}^2}{4{\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\mathrm{t}\right)$$

(9)

where D_ef = effective diffusion coefficient, m²s⁻¹; n = number of terms; h = characteristic dimension, half-thickness of the plate, mm.

$$\mathrm{R}\mathrm{X}=\frac{8}{{\mathsf{\pi }}^2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\mathsf{\pi }}^2}{4{\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\mathrm{t}\right)$$

(10)

In Equation (10), when considering the first term of the series at t = 0, RX equals RX =${8/\pi }^2$, or 0.810569 [17]. To achieve RX = 1.0, an indefinite number of terms from the series would be required. Therefore, by assuming that this value is determined through a regression equation, it could approach 1.0. In essence, it becomes possible to treat this term as a coefficient, denoted as A, to be determined via nonlinear regression, with an expected value close to 1.0. Consequently, this equation bears resemblance to the one originally proposed by Henderson and Pabis but with modifications introduced by the authors. This modified equation offers the potential to calculate the effective diffusivity of the product during the drying process. As a result, the Henderson–Pabis model can be expressed as Equations (11) and (12):

$$\mathrm{R}\mathrm{X}=\mathrm{A} \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\mathsf{\pi }}^2}{4{\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\mathrm{t}\right)$$

(11)

where:

$${\mathrm{K}}_1=\left(\frac{{\mathsf{\pi }}^2}{4{\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\right)$$

(12)

Continuing with the same approach, assuming A = 1 and introducing a potential time correction denoted as N, an equation like Page’s can be derived. This equation can be expressed as the authors’ modified Page equation, represented by Equation (13), which shares similarities with the description provided by [15]:

$$\mathrm{R}\mathrm{X}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\mathsf{\pi }}^2}{4{\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}{\mathrm{t}}^{\mathrm{N}}\right)$$

(13)

It is possible to modify Fick’s equation using two terms from the series, as shown in Equation (14):

$$\mathrm{R}\mathrm{X}=\frac{8}{{\mathsf{\pi }}^2}.\exp \left(-\frac{{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\mathrm{t}\right)+\frac{8}{{\mathsf{\pi }}^2}.\frac{1}{9}\exp \left(-\frac{3{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\mathrm{t}\right)$$

(14)

The modifications made to Equation (10) were as follows: in the 1st term of the series, instead of having fixed values like 8/π², a value of A₁ was assigned; in the 2nd term of the series, instead of 8/9π², a value of A₂ was assigned. The values of A₁ and A₂ should be determined by the nonlinear regression equation, as shown in Equation (15). The author also considered implementing a time correction of potential order, adding indices N₁ and N₂, respectively, to the 1st and 2nd terms of the series:

$$\mathrm{R}\mathrm{X}={\mathrm{A}}_1.\exp \left(-\frac{{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}{\mathrm{t}}^{{\mathrm{N}}_1}\right)+{\mathrm{A}}_2.\exp \left(-\frac{3{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}{\mathrm{t}}^{{\mathrm{N}}_2}\right)$$

(15)

where:

$${\mathrm{K}}_1=\left(\frac{{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\right)$$

(16)

$${\mathrm{K}}_2=\left(\frac{3{\mathsf{\pi }}^2}{{4\mathrm{h}}^2}{\mathrm{D}}_{\mathrm{e}\mathrm{f}}\right)$$

(17)

Considering the effective diffusion coefficients obtained, the activation energy was calculated by linearizing the Arrhenius equation, as illustrated in Equation (18):

$$\ln \mathrm{D}=\ln {\mathrm{D}}_0-\frac{{\mathrm{E}}_{\mathrm{A}}}{\mathrm{R}}.\frac{1}{\mathrm{T}}$$

(18)

where D₀ is the pre-exponential factor; E_A is the activation energy, kJ·mol⁻¹; R is the universal gas constant, 8.134 kJ·mol⁻¹ K⁻¹; T is the absolute temperature, K.

Following the determination of the activation energy, it became feasible to calculate the enthalpy (∆H), entropy (∆S), and Gibbs free energy using Equations (19)–(21):

$$∆\mathrm{H}=\mathrm{E}-\mathrm{R}\mathrm{T}$$

(19)

$$∆\mathrm{S}=\mathrm{R}\left(\ln \mathrm{A}-\ln \frac{{\mathrm{k}}_{\mathrm{B}}}{{\mathrm{h}}_{\mathrm{p}}}-\ln \mathrm{T}\right)$$

(20)

$$∆\mathrm{G}=∆\mathrm{H}-\mathrm{T}∆\mathrm{S}$$

(21)

where ln(A) represents the intercept obtained from the regression analysis applied to the graph obtained in the activation energy calculation; k_B is the Boltzmann constant, $1.38\times {10}^{-23}J{K}^{-1}$; h_p is the Planck constant, $6.626\times {10}^{-34}Js$.

The mathematical models were adjusted to the pineapple residue drying kinetics curves using linear regression for constant drying rates. For decreasing drying rates, nonlinear regression using the quasi-Newton method was employed. The coefficients of the regressions were determined using the Statistica 7.0 computer program. To assess the quality of the model fits to the experimental data, coefficients of determination (R²), relative mean error (P), and estimated mean error (SE) were calculated:

$$\mathrm{S}\mathrm{E}=\sqrt{\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[{\left({\mathrm{R}\mathrm{X}}_{{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{i}}}-{\mathrm{R}\mathrm{X}}_{{\mathrm{e}\mathrm{s}\mathrm{t}}_{\mathrm{i}}}\right)}^2\right]}{\mathrm{G}\mathrm{L}\mathrm{R}}}$$

(22)

$$\mathrm{P}=\frac{100}{\mathrm{n}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[\frac{\left|{\mathrm{R}\mathrm{X}}_{{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{i}}}-{\mathrm{R}\mathrm{X}}_{{\mathrm{e}\mathrm{s}\mathrm{t}}_{\mathrm{i}}}\right|}{{\mathrm{R}\mathrm{X}}_{{\mathrm{e}\mathrm{x}\mathrm{p}}_{\mathrm{i}}}}\right]$$

(23)

For the comparative analysis between the models, the theoretical Fick’s equation with 4 terms of the series was used.

3. Results and Discussion

This study introduces a novel approach by analyzing both constant and decreasing drying rate periods in the drying kinetics of industrial pineapple waste. Using the Cavalcanti-Mata, modified Page, and modified Henderson–Pabis models, it accurately captures the drying behavior, considering the high moisture content and chemical composition of the waste. Implementing this approach in industrial applications enhances drying efficiency and effectiveness. It allows for the optimization of drying parameters, leading to faster and more uniform drying, reduced energy consumption, and lower costs. Additionally, it preserves the nutritional and functional properties of the dried waste, making it more suitable for use in food, animal feed, and industrial products. This method supports sustainable waste management and adds value to byproducts, promoting a circular economy in the agro-industrial sector.

Figure 2 displays the experimental data for the drying kinetics of pineapple industrial waste at temperatures of 40, 50, 60, and 70 °C, expressed as a function of the moisture content ratio over time.

From the analysis of Figure 2, there is a noticeable rapid decrease in the initial moisture content over time, indicating the presence of a constant drying rate period. Upon examining the experimental data, this constant drying rate occurs from the initial moisture content of 88% on a wet basis (733% on a dry basis) to approximately 75% on a wet basis (300% on a dry basis). This corresponds to drying times of 20,500, 15,300, 11,000, and 8100 s, for the respective temperatures of 40, 50, 60, and 70 °C.

In Figure 3, the periods of a constant drying rate for industrial pineapple waste can be observed, where there is a variation in the range from 7.21 × 10⁻⁵ to 2.97 10⁻⁵ g s⁻¹ (grams per second) of water removal as the drying temperature varies from 40 to 70 °C, respectively. Equations (24)–(27) are the linear regression applied exclusively to the constant rate period.

RX(40 °C) =1.0047 − 2.9699 × 10 ⁻⁵ t, R² = 0.9987; p = 0.00001;

(24)

RX(50 °C) = 0.9902 − 4.1172 × 10⁻⁵ t, R² = 0.9962: p = 0.00001;

(25)

RX(60 °C) = 1.0024 − 5.4697 × 10⁻⁵ t, R² = 0.9982; p = 0.00001;

(26)

RX(70 °C)= 0.9931 − 7.2180 × 10⁻⁵ t, R² = 0.9869; p= 0.00001;

(27)

Table 1. Technical data for the constant drying rate period of chickpeas, determining the convective mass transfer coefficient, m/s (h_d).

Temperature		C	L	h	Asf	As	Rv	Pvs	Pv	dX/dt	hd
°C	K	mm	Mm	mm	mm2	mm2	kJ/kg K	kPa	kPa	g/s	m s−1
40	313	130	70	50	903 × 103	1358 × 103	0.462	7.38	1.826	2.96 × 10−5	5.69 × 10−7
50	323	130	70	50	899 × 103	1354 × 103	0.462	12.34	1.966	4.12 × 10−5	4.37 × 10−7
60	333	130	70	50	934 × 103	1389 × 103	0.462	19.93	1.995	5.47 × 10−5	3.38 × 10−7
70	343	130	70	50	954 × 103	1409 × 103	0.462	31.18	2.051	7.22 × 10−5	2.79 × 10−7

presents the parameters for determining the convective mass transfer coefficient during the constant drying rate period. It can be observed that this coefficient varies from 5.69 × 10⁻⁷ to 2.79 × 10⁻⁷ m/s as the temperature changes from 40 to 70 °C. This implies that at higher temperatures, the mass transfer coefficient is lower. Therefore, the drying efficiency is higher, as it requires less time to complete the drying during the constant drying rate period. Additionally, this table shows that the thickness (h) of the tray used for drying industrial pineapple waste is 50.0 mm, which is an average value obtained from 100 observations.

The high moisture content in industrial pineapple waste is a characteristic of the process. Pineapples have a natural water content ranging from 85% to 88%. Even when processed for juice extraction or the production of jams and other derivatives, their water content undergoes minimal changes. This is why drying is essential to provide stability to the product, as the waste from pineapple processing can degrade or ferment within a few hours [18]

Based on the experimental data presented in Figure 2, it was observed that during the decreasing drying rate process at temperatures of 40, 50, 60, and 70 °C, with corresponding relative humidities of 25%, 15%, 9%, and 6%, the pineapple waste reached its equilibrium moisture content at moisture levels of 4.6%, 2.6%, 2.2%, and 1.6%, respectively, on a dry basis (d.b.). This observation highlights the direct influence of temperature and relative humidity on the drying process. During this phase, the drying kinetics were investigated, focusing on removing water (mass) from the product, and the effective mass diffusion coefficient was determined in mm²s⁻¹.

Another factor directly influenced by temperature is the drying time, which decreases as the drying temperature increases during this phase. It was observed that at a temperature of 40 °C, the industrial pineapple waste took 36,000 s to reach equilibrium. At 50 °C, this time was reduced to 32,400 s; at 60 °C, it decreased further to 27,000 s. The shortest drying time was observed at 70 °C, with a duration of 21,600 s.

Table 2. Effective diffusion coefficients for the Fick Model with the first four terms of the series.

Temp.	First Term				Second Term				Third Term				Fourth Term
(°C)	Def	R2 (%)	SE	P	Def	R2 (%)	SE	P	Def	R2 (%)	SE	P	Def	R2 (%)	SE	P
	mm2/s				mm2/s				mm2/s				mm2/s
40	0.017368	91.23	0.0092	19.60	0.017361	93.51	0.00683	18.80	0.017355	94.31	0.0060	18.35	0.017346	97.24	0.00573	18.18
50	0.019950	91.38	0.0964	26.49	0.019960	93.79	0.0818	25.63	0.019948	94.57	0.0765	25.26	0.019937	97.38	0.0747	25.12
60	0.029992	92.51	0.0983	21.38	0.029981	94.75	0.0822	20.39	0.029960	95.57	0.0755	19.88	0.029937	97.91	0.0732	19.68
70	0.039956	92.29	0.1020	17.61	0.039934	94.71	0.0845	16.50	0.039924	95.44	0.0785	16.18	0.039916	97.81	0.0764	16.02

contain the effective diffusivity coefficients obtained from nonlinear regression fits for the Fick model, considering the drying of a flat plate geometry product during the decreasing drying rate period. In these tables, it can be observed that as the number of terms in the Fick model series (from 1 to 4) increases, the coefficients of determination (R²), estimated mean error (SE), and relative mean error (P) improve.

However, a few more terms from the series should still be used to better fit this model to the experimental data. Some other observations, however, can be noted, such as the lack of a significant increase in effective diffusion coefficients with an increase in the number of series terms. This is observed up to the fourth term of the Fick equation series, contrary to other authors, including [15], who observed significant differences in the effective diffusion coefficient when increasing the number of terms in the Fick equation series.

When using only the first term of the Fick equation series, there is a discrepancy in the equation. For t = 0, it implies RX = 0.810566, which is far from the actual experimental conditions where, at t = 0, RX = 1. Even when using all four terms of the series, this premise still deviates from reality because, for t = 0, RX = 0.9166, which is still distant from the ideal boundary condition for the real process. Thus, there is a need to use many terms in the series so that at the initial time of zero, the initial moisture content ratio is close to 1. However, for the Fick model to be used with fewer terms in the series, there should have been a rapid water loss at the beginning of the drying process, meaning that the values would be below RX = 0.81 in the subsequent seconds.

When analyzing the table containing the modifications proposed by Cavalcanti-Mata [19] to the Page and Henderson–Pabis models, using the first term of the Fick model series as a premise, along with some considerations, it becomes evident that the coefficients of determination (R²) are higher and range from 0.9855 to 0.9921. This signifies an improved relationship between the experimental data and the proposed models.

However, when analyzing the Page model modified by Cavalcanti-Mata and the Cavalcanti-Mata model, it can be observed that the coefficients of determination (R²) are greater than 0.997, indicating that these equations should be recommended to describe the drying of industrial pineapple waste. The estimated mean error (SE) is also low, which is desirable. However, in terms of the mean relative error (P), most values are above 10%.

It can be observed in

Table 3. Data from the fittings of mathematical models to the experimental drying data.

Temperature	Parameters—Henderson–Pabis Model Modified by Cavalcanti-Mata
	h mm	A	K	${\mathit{D}}_{\mathit{e}\mathit{f}}$ m2/s		R²	SE	P
40 °C	25	1.04817	0.000087	0.882 × 10−9		0.9868	0.00139	10.09
50 °C	25	1.04746	0.000099	1.003 × 10−9		0.9874	0.0369	22.79
60 °C	25	1.03908	0.000150	1.520 × 10−9		0.9921	0.0083	13.65
70 °C	25	1.05893	0.000220	2.229 × 10−9		0.9855	0.0176	11.75
Temperature	Parameters—Page model modified by Cavalcanti-Mata
	h mm	K	N	${\mathit{D}}_{\mathit{e}\mathit{f}}$ m2/s		R²	SE	P
40 °C	25	5.4 × 10−6	1.2842	5.47 × 10−9		0.9985	0.00015	11.61
50 °C	25	6.5 × 10−6	1.28504	6.59 × 10−9		0.9984	0.0131	5.74
60 °C	25	1.3 × 10−5	1.26474	13.2 × 10−9		0.9995	0.0083	13.65
70 °C	25	1.5 × 10−5	1.29495	15.2 × 10−9		0.9977	0.0176	11.75
Temperature	Parameters—Cavalcanti-Mata Model
	A1	N1	A2	N2	${\mathit{D}}_{\mathit{e}\mathit{f}}$ m2/s	R²	SE	P
40 °C	1.698121	1.118394	−0.700480	0.976141	3.620 × 10−9	0.9996	0.00005	13.07
50 °C	2.292974	1.114945	−1.29365	0.997683	4.34 × 10−9	0.9985	0.0132	8.15
60 °C	1.688064	1.128602	−0.688994	0.989407	4.62 × 10−9	0.9996	0.0083	13.65
70 °C	1.252784	1.209190	−0.262371	1.157325	8.849 × 10−9	0.9951	0.0176	11.75

that in the Henderson–Pabis model modified by Cavalcanti-Mata, only at 40 °C, a relative mean error (P) of 10% was found; in the Page model modified by Cavalcanti-Mata and the Cavalcanti-Mata model, a relative mean error (P) below 10% was obtained only at 50 °C. Although this may indicate that the Cavalcanti-Mata model has the highest coefficient of determination, and the difference between this model and the modified Page model only occurs at the third or fourth decimal place, they can be considered equivalent.

In all four models, the effective diffusivity (D_ef) shows an increase in its value with temperature, implying that higher temperatures result in faster drying. However, it is observed that this phenomenon does not occur in the Cavalcanti-Mata model, where the effective diffusivity at 70 °C did not increase compared to the temperature of 60 °C.

Several authors have confirmed this increase in effective diffusivity with temperature, including [4], who studied flour products from agro-industrial residues and found that higher temperatures lead to higher drying rates, reaching the equilibrium moisture content in less time. Similar findings were also reported by Santos et al. (2018), who investigated the drying of passionfruit seeds [20]. The parameter “N” in these equations is a temporal conformity effect and serves to correct potential errors resulting from neglecting to consider the impact of internal resistance on water transfer.

The effective diffusion coefficient is an index that allows evaluation of the drying rate and its temperature dependence [21]. Table 2 contain the values of the effective diffusion coefficients according to each model. It can be observed that for the Fick model with four terms of the series, the variation in the effective diffusivity of pineapple waste ranged from 0.017346 to 0.039916 m²s⁻¹, corresponding to a variation from 1.735 × 10⁻⁸ to 3.991 × 10⁻⁸ m²s⁻¹, as the temperature changed from 40 to 70 °C. In the Henderson–Pabis model modified by Cavalcanti-Mata, the effective diffusivity of industrial pineapple waste ranged from 0.882 × 10⁻⁹ to 2.229 × 10⁻⁹ m²s⁻¹ for the same temperature variation. In the Page model modified by Cavalcanti-Mata, there was a variation from 5.47 × 10⁻⁹ to 15.2 × 10⁻⁹ m²s⁻¹, and in the Cavalcanti-Mata model, the variation was from 5.470 × 10⁻⁹ to 9.174 × 10−9 m²s⁻¹.

To validate the drying model, we compared the experimental drying curves with those predicted by our model. Figure 4 shows the evaporation kinetics for both the experimental data and the simulation results. The comparison reveals a good agreement between the experimental and simulated data, indicating the model’s accuracy in predicting the drying behavior of pineapple waste. Moreover, we acknowledge that the effective diffusivity may vary with moisture content or temperature. Hence, we propose an extended model where the effective diffusivity is a function of both variables. This approach aligns with the findings of several studies that report variable diffusivity during drying processes.

The dependence of diffusion on temperature elevation is attributed to the level of vibration of water molecules, which promotes the diffusion of water vapor during the drying process within the product. Various researchers have frequently reported this phenomenon in different studies. For instance, [22] investigated the drying of 5 mm thick apple slices at a pressure of 0.02 bar and found effective diffusivity values ranging from 1.50 × 10⁻⁹ to 2.60 × 10⁻⁹ m²s⁻¹ for temperatures between 50 and 70 °C. They also observed effective diffusivity values between 0.42 × 10⁻⁹ and 1.49 × 10−9 m²s⁻¹ for 4 mm thick apple slices.

In another study, [23] examined the drying of 7 mm thick apple slices and obtained effective diffusivity values ranging from 2.36 × 10⁻⁹ to 3.92 × 10⁻⁹ m²s⁻¹ for the same temperature variation. Additionally, Wanderley et al. (2023) investigated the drying kinetics of pomegranate peels and seeds. They reported effective diffusion coefficients of 3.758 × 10⁻¹² to 4.680 × 10⁻¹² m²s⁻¹ and 1.3106 × 10⁻⁹ to 2.591 × 10⁻⁹ m²s⁻¹ for temperatures ranging from 50 to 70 °C [24]. The Fick diffusion model for a flat plate geometry was used to determine effective diffusivity in these studies.

Therefore, the results obtained for industrial pineapple waste fall within the range mentioned in the literature, which reports that effective diffusion coefficients for agricultural products are in the order of 10⁻⁹ to 10⁻¹¹ m²s⁻¹. However, it is necessary to qualify this statement, as it was made by Madamba et al. (1996), dating back to the last century, and various authors with different products have conducted many studies [25]. For example, Lisbôa et al. (2015) found values between 2.396 to 2.428 × 10⁻¹³ m²s⁻¹ for canary seed grains at temperatures ranging from 10 to 40 °C. Similarly, Gilago et al. (2023) studied solar drying under two configurations, where in configuration A, the average temperature was 61.2 °C, and in configuration B, the average temperature was 53.1 °C, with effective diffusivity values of 6.7 × 10⁻⁹ m²s⁻¹ and 7.35 × 10⁻⁹ m²s⁻¹, respectively [26].

However, what may appear as a contradiction, with a higher effective diffusivity (7.35 × 10⁻⁹ m²s⁻¹) at the lower drying temperature of 53.1 °C, is due to the drying process. In configuration A, drying is achieved through natural convection, while in configuration B, it is undertaken using forced convection. As a result, the drying of carrot slices has a heat rate of 705.64 W in configuration A and 789.55 W in configuration B, making it acceptable for drying to be faster in configuration B, where the average collector efficiencies are 56.84% and 68.74%, respectively.

The determination of activation energy and thermodynamic properties was obtained using the proposed models through nonlinear regression for temperatures ranging from 40 to 70 °C. The activation energy determined by linearizing the Arrhenius equation is presented in Figure 5 for the different models.

In(D_eff) = 5.8148 − 3108.9819 (1/T), R²= 0.9633; p = 0.0185

(28)

In(D_eff) = 3.7881 − 3413.5844 (1/T), R²= 0.9499; p = 0.0254

(29)

In(D_eff) = 3.0325 − 4037.5339 (1/T), R²= 0.9259; p = 0.0378

(30)

In(D_eff) = 3.6025 − 2913.8247 (1/T), R²= 0.8076; p = 0.1013

(31)

Activation energy can be described as the minimum energy required to initiate the mass transfer process of water from the interior of the solid to the surface. Zogzas et al. reported that the activation energy for agricultural products varies between 12.7 and 110 kJ·mol⁻¹ [27]. In this context, numerous studies can be cited to support this narrative. For instance, the work of Gomes et al. (2018) examined the drying kinetics of crushed jambu leaves at different temperatures (60, 70, and 80 °C) and layer thicknesses (5 and 10 cm) [28].

Drying was conducted in an air-circulating oven with an air velocity of 1.5 m s⁻¹. The activation energy of the material was 16.61 kJ·mol⁻¹ for a thickness of 0.005 m and 16.97 kJ·mol⁻¹ for a thickness of 0.010 m, as reported by Moura et al. (2021), who studied the drying of trapia residues (peel and seeds) at temperatures ranging from 50 to 80 °C, finding an activation energy (Ea) of 18 kJ·mol⁻¹ for trapia peel and 24.2 kJ·mol⁻¹ for the seeds [14]. Gilago et al. (2023) dried carrot slices using indirect solar dryers with natural convection (configuration-A) and forced convection (configuration-B) [26]. In configuration-A, the average temperature was 61.2 °C, while in configuration-B, the average temperature was 53.1 °C.

The authors obtained an activation energy of 42.71 kJ·mol⁻¹ and 37.85 kJ·mol⁻¹ for the respective configurations (A and B) in their study. Wanderley et al. (2023) investigated the drying of pomegranate peels and seeds in an air-circulating oven at temperatures of 50, 60, and 70 °C [24]. They determined the effective diffusivities and thermodynamic properties of the process, finding activation energies of 10.60 kJ·mol⁻¹ for pomegranate peels and 31.39 kJ·mol⁻¹ for their seeds.

In this research, the activation energy for industrial pineapple waste was determined to be 25.3 kJ mol⁻¹ for the Fick model using four terms of the series, 27.8 kJ mol⁻¹ for the Henderson–Pabis model modified by Cavalcanti-Mata, 32.8 kJ mol⁻¹ for the Page model modified by Cavalcanti-Mata, and 23.7 kJ mol⁻¹ for the Cavalcanti-Mata model, in the temperature range of 40 to 70 °C. These values fall within the range reported in the literature for agricultural products.

Table 4. Thermodynamic property values.

		Enthalpy (J mol−1)
Temperature (K)	Fick Four Terms	Modified Henderson–Pabis	Page	Modified Cavalcanti-Mata
313.15	22,725.03	25,218.94	30,294.14	21,153.89
323.15	22,643.69	25,137.60	30,212.80	21,072.55
333.15	22,562.35	25,056.26	30,131.46	20,991.21
343.15	22,481.01	24,974.92	30,050.12	20,909.87
Entropy (Jmol−1K−1)
313.15	−192.71	−209.21	−215.34	−223.58
323.15	−192.96	−209.46	−215.59	−223.84
333.15	−193.21	−209.71	−215.84	−224.08
343.15	−193.45	−209.95	−216.08	−224.32
Gibbs Free Energy (Jmol−1)
313.15	83,070.87	90,732.20	97,726.94	91,167.99
323.15	84,999.21	92,825.56	99,881.60	93,405.07
333.15	86,930.07	94,921.44	102,038.77	95,644.68
343.15	88,863.37	97,019.75	104,198.39	97,886.72

presents the thermodynamic parameters based on the Fick model using four terms of the series, the Henderson–Pabis model modified by Cavalcanti-Mata, the Page model modified by Cavalcanti-Mata, and the Cavalcanti-Mata model.

It is observed that enthalpy decreases with increasing temperature. This behavior is related to the increase in the partial vapor pressure of water in the product with the rise in the drying air temperature, indicating that higher temperatures require less energy to remove free water from the product, as noted by Oliveira et al. (2015) [29]. From this perspective, there is an increase in the water diffusion rate from the interior to the surface of the product, resulting in water loss through desorption.

In Table 4, it is observed that entropy also decreases with an increase in the drying temperature. This happens because as the drying air temperature rises, there is an increase in the molecular excitement, resulting in an enhanced diffusion rate of water, which requires a lower entropy value in the process. With respect to the models, it is noted that the Cavalcanti-Mata model has the highest entropy value (−223.58 to −224.32 J mol⁻¹ K⁻¹) within the temperature range (313.15 to 343.15 K), while the Fick model with four terms of the series has the lowest entropy value (−192.71 to −193.45 J mol⁻¹ K⁻¹) within the same temperature range. This implies that the Cavalcanti-Mata model exhibits a higher degree of molecular excitement and, therefore, a greater mass diffusion process with increasing temperature. According to Martins (2015), negative entropy values are attributed to the presence of chemical adsorption and/or structural modifications of the adsorbent [30].

Gibbs free energy measures the overall energy associated with a thermodynamic system. It can be observed in Table 4 that the highest values are associated with the Page equation modified by Cavalcanti-Mata (97.73 to 104·2 KJ·mol⁻¹), while the lowest values are found in the Fick model with four terms of the series (83.07 to 88.86 KJ·mol⁻¹). It is noted that within this range, Gibbs free energy increased with the rise in temperature, with positive values for the entire temperature range studied. According to Oliveira et al. (2015), the positive value of Gibbs free energy characterizes an endergonic reaction, which requires an addition of energy from the surroundings for the reaction involving the product to occur. However, in the case of drying, there is no chemical reaction, making this operation non-spontaneous, and is explained by the addition of energy required for the transition from the liquid phase to the gas phase.

The results of this study are in line with the findings of Araújo et al. (2017) for peanut drying [31], Guimarães et al. (2018) for okara drying [32], Quequeto et al. (2019) for noni seed drying [33], Almeida et al. (2020) for azuki bean drying [34], Gilago et al. (2023) for carrot slice drying [26], and Wanderley et al. (2023) for pomegranate peel and seed drying [24]. These authors also observed an increase in Gibbs free energy with an increase in the drying air temperature.

One notable limitation of this study is the selection of mathematical models used to represent the drying kinetics of pineapple waste. While the Cavalcanti-Mata, modified Page, and modified Henderson–Pabis models were effective in fitting the experimental data, the choice of these specific models inherently limits the scope of the analysis. There are numerous other models available in the literature that could potentially offer different insights or more accurate predictions. Additionally, the assumption of constant effective diffusivity throughout the drying process is a simplification that may not fully capture the complexities of moisture movement within the material, as effective diffusivity is known to vary with moisture content and temperature. This limitation could affect the accuracy of the models used. Furthermore, the study was conducted under specific experimental conditions—fixed temperatures and air velocity—that may not encompass the full range of possible industrial scenarios. Future research should explore a broader range of models and experimental conditions to validate and enhance the robustness and generalizability of the findings.

4. Conclusions

This study successfully introduced a novel approach to the mathematical modeling and determination of thermodynamic properties during the drying process of industrial pineapple waste. The investigation revealed a two-phase drying process: an initial constant rate drying phase followed by a decreasing drying rate phase. The convective mass transfer coefficients and effective diffusion coefficients were determined for various temperatures, providing a comprehensive understanding of the drying kinetics.

The study demonstrated that the Cavalcanti-Mata and the modified Page models best fit the experimental data, offering robust tools for predicting drying behavior. The Henderson–Pabis model modified by Cavalcanti-Mata was identified as a practical option for engineering applications due to its simplicity and effectiveness.

The effective diffusivity was found to vary with temperature, aligning with the trends reported in the literature. The activation energy and thermodynamic properties, such as enthalpy, entropy, and Gibbs free energy, were thoroughly analyzed, showing that the drying process is non-spontaneous and requires energy input.

Overall, this research provides significant insights into the drying process of pineapple waste, contributing to the optimization of drying operations in the pineapple processing industry. The findings highlight the potential for developing efficient and sustainable methods for managing and utilizing agro-industrial waste, thereby promoting environmental sustainability and resource efficiency.