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Mathematical modelling of thin-layer drying of nance pulp (Byrsonima crassifolia (L.) Kunth)

This study mathematically modeled the thin-layer drying of nance pulp in a convective tunnel dryer at 60–80°C, identifying the Midilli et al. and Alonso et al. models as the best fits and determining an effective moisture diffusivity range of 2.4842 to 4.0409 × 10⁻⁹ m²/s with an activation energy of 23.62 kJ/mol.

Original authors: Apolinar Picado, Francisco Canelo

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Apolinar Picado, Francisco Canelo

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Preserving the bounty of the harvest is one of humanity's oldest challenges, particularly in tropical regions where fresh fruit spoils quickly under the sun. Drying remains one of the most reliable methods to extend the life of food, transforming perishable produce into stable ingredients that can be stored, transported, and enjoyed long after the season has passed. However, drying is not merely a matter of applying heat; it is a complex physical process where water moves from the inside of a fruit to its surface and then evaporates into the air. To make this process efficient, scientists study how different fruits behave under specific conditions, looking for mathematical patterns that describe exactly how fast they lose moisture. This knowledge allows engineers to design better dryers and predict how long a batch of fruit will take to process, ensuring that the final product retains its quality without wasting energy or time.

In this context, researchers in Nicaragua turned their attention to the nance, a small, yellow-orange fruit native to the tropics. Known for its creamy pulp and a distinctive sweet-sour flavor, the nance is rich in vitamin C and antioxidants, making it a valuable ingredient for jams, juices, and other food products. Despite its culinary potential, the fruit is often wasted when it becomes overripe. To find a better way to preserve it, a team of chemical engineers investigated the drying behavior of nance pulp. They wanted to understand the precise mechanics of how the pulp dries, identify the best mathematical tools to predict the process, and measure the energy required to remove the water. Their work focused on a specific type of drying setup: a laboratory-scale tunnel where hot air blows steadily over a thin layer of the fruit pulp.

The researchers conducted their experiments by placing the nance pulp into a shallow tray and exposing it to a stream of air moving at a steady speed of 2.0 meters per second. They tested three different temperatures: 60, 70, and 80 degrees Celsius. As the hot air passed over the pulp, the team carefully monitored the weight of the sample over time to track how much water was leaving the fruit. They observed that the pulp did not dry at a constant speed; instead, the rate of water loss slowed down continuously as the process went on. This indicated that the entire drying event happened during what is known as the falling rate period, where the movement of water from the center of the pulp to the surface becomes the limiting factor. They also noted that the pulp physically shrank as it lost water, a common but important detail that affects how the material behaves.

To make sense of these observations, the team compared their experimental data against six different mathematical models that scientists commonly use to describe drying. These models are essentially equations that attempt to predict how the moisture content changes over time based on temperature and other factors. After running the numbers, the researchers found that two specific models, developed by Midilli and colleagues and by Alonso and colleagues, matched the real-world data better than the others. These two models were able to accurately predict the drying curve, capturing the nuances of how the nance pulp lost moisture. The success of these models suggests that they account for the specific physical changes, such as shrinkage, that occur in the pulp during drying.

Beyond simply predicting how long drying takes, the team calculated how easily water moves through the nance pulp. This property, known as effective moisture diffusivity, measures how fast water molecules travel through the material. They found that this speed increased as the temperature rose. At the lowest temperature tested, the water moved at a rate of roughly 2.48 times 10 to the negative 9 square meters per second, while at the highest temperature, it sped up to about 4.04 times 10 to the negative 9 square meters per second. This increase happens because higher temperatures give the water molecules more energy, allowing them to move more freely and escape the fruit faster. The researchers also determined the activation energy for the process, which is the minimum amount of energy needed to start the drying reaction. They calculated this value to be 23.62 kilojoules per mole, a figure that helps engineers understand the energy efficiency of the drying process.

The study confirms that drying nance pulp is a process governed by the falling rate period, where internal diffusion controls the speed of water removal. By identifying the most accurate mathematical models and quantifying the movement of water, the researchers have provided a clear roadmap for processing this fruit. Their findings show that while the pulp shrinks and the drying slows down over time, the process can be reliably predicted using specific equations. This work offers a practical foundation for industries looking to turn fresh nance pulp into stable, long-lasting food products, ensuring that the unique flavor and nutritional value of the fruit can be preserved with greater efficiency.

Technical Summary: Mathematical Modelling of Thin-Layer Drying of Nance Pulp

Problem Statement
The study addresses the need to characterize and mathematically model the drying kinetics of nance pulp (Byrsonima crassifolia (L.) Kunth). While nance fruit is valued for its nutritional content and potential in various food products, its preservation requires effective drying techniques to extend shelf life and reduce waste. Although previous studies have examined the drying of whole nance fruits (with and without peels) and foam-mat drying of the pulp, there is a specific need to identify the most accurate thin-layer drying models for nance pulp under convective conditions and to determine its effective moisture diffusivity and activation energy.

Methodology
The research utilized experimental data obtained from a prior study by Ruiz and Blandón (2021), conducted using a lab-scale convective tunnel dryer. The experimental conditions involved drying nance pulp at three distinct temperatures (60, 70, and 80 °C) with a constant air velocity of 2.0 m/s. The pulp was placed in a sample holder (45 mm × 75 mm cross-section, 5 mm depth) and weighed at specific intervals to monitor moisture loss.

The core of the methodology involved:

Data Conversion: Experimental moisture content was converted into the moisture ratio (MR) over time.
Model Fitting: The experimental MR data were fitted to six (6) thin-layer drying models commonly used for fruits and vegetables using non-linear regression analysis (MATLAB's Curve Fitting Toolbox).
Statistical Evaluation: Model performance was assessed using the coefficient of correlation ( $r$ ), root-mean-square error (RMSE), and chi-square ( $\chi^2$ ). The best models were selected based on the highest $r$ and lowest $\chi^2$ and RMSE values.
Diffusivity and Activation Energy: The effective moisture diffusivity ( $D_{eff}$ ) was calculated using Fick's second law of diffusion (simplified for an infinite slab). The temperature dependence of $D_{eff}$ was analyzed using the Arrhenius equation to determine the activation energy ( $E_a$ ).

Key Results

Drying Characteristics: The drying process occurred entirely within the falling rate period; no constant rate period was observed. This indicates that internal diffusion is the primary mechanism for moisture removal. Shrinkage of the pulp was observed throughout the drying process.
Model Performance: Among the six models evaluated, the models proposed by Midilli et al. and Alonso et al. provided the best fit for the experimental data across all tested temperatures. These models demonstrated the highest correlation coefficients and the lowest error values. The study notes that models with a greater number of constants tended to fit the data better, potentially due to the influence of shrinkage on the drying curve pattern.
Effective Moisture Diffusivity: The calculated $D_{eff}$ values ranged from 2.4842 to 4.0409 ( $\times 10^{-9}$ ) m²/s. As expected, diffusivity increased with higher drying temperatures due to enhanced molecular activity.
Activation Energy: The activation energy for the drying process was calculated to be 23.62 kJ/mol, derived from the slope of the Arrhenius plot ( $\ln(D_{eff})$ vs. $1/T$ ).

Significance and Claims
The paper claims to expand upon previous findings by Ruiz and Blandón (2021) by rigorously identifying the thin-layer models that most accurately describe the drying kinetics of nance pulp. The primary contribution is the validation of the Midilli et al. and Alonso et al. models as the most suitable for predicting the drying behavior of this specific material under the tested conditions. Additionally, the study provides specific thermodynamic parameters (diffusivity range and activation energy) essential for the design and optimization of industrial drying processes for nance pulp. The authors emphasize that these findings offer a mathematical basis for understanding the drying mechanism, which is dominated by the falling rate period and influenced by material shrinkage.

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