A Review on Modeling the Drying Kinetics of Agricultural Bio Materials and Wastes

A Review on Modeling the Drying Kinetics of Agricultural Bio Materials and Wastes

Owoh I. P^1,3, Okonkwo W. I^1,2, Anyanwu C. N^1,2, Ojike O^1,2 and Nwagugu N. I^1,4

¹Department of Agricultural and Bioresources Engineering

²World Bank African Centre of Excellence for Power and Energy Development

³National Centre for Energy Research and Development, University of Nigeria, Nsukka, Nigeria

⁴Project Development Institute, Enugu, Nigeria

DOI: https://doi.org/10.51584/IJRIAS.2025.100500085

Received: 25 April 2025; Accepted: 04 May 2025; Published: 16 June 2025

ABSTRACT

Drying kinetics modeling is critical in optimizing drying processes for biomaterials and wastes, ensuring energy efficiency and product quality. This review provides a comprehensive synthesis of the major modeling approaches applied to drying kinetics, encompassing empirical, semi-theoretical, and theoretical models. Influencing factors such as moisture content, air velocity, temperature, and material structure are discussed. The review further examines modeling techniques specific to agricultural residues, food products, and animal wastes, highlighting the integration of traditional models with modern computational approaches, including artificial intelligence and computational fluid dynamics. Model selection criteria and current research gaps are analyzed, emphasizing the development of adaptive, material-specific models and the integration of real-time monitoring tools. The insights presented aim to guide future research and industrial applications in the valorization of organic wastes and sustainable drying system development.

Keywords: Drying kinetics, Biomaterials, Modeling, Waste valorization, Artificial intelligence

INTRODUCTION

Drying is an age-long practice of reducing or removal of moisture from a product in other to stop or retard the microbial reactions that may degrade, or decompose the product (Chauhan et al., 2015; Ahmad et al.,2022). Drying of agricultural products elongates their shelf life, enabling its storage and preservation for an extended time without decomposition. Drying is essential for preserving biomaterials and converting waste into value-added products. Effective drying reduces moisture content, and minimizes transportation costs. Modeling the drying kinetics of biomaterials is vital for optimizing drying equipment and enhancing process efficiency. Biomaterials, such as agricultural residues and food waste, display complex moisture diffusion behaviors due to their heterogeneous composition, necessitating robust and accurate models. This review outlines existing drying models, evaluates their applications, and highlights current research directions.

Mathematical modeling of drying Kinetics

Drying kinetic models are commonly used to estimate drying times of agricultural products during drying. Drying models simply mean a predictive mathematical relationship between the moisture content expressed as moisture ratio and time. Drying kinetics are affected by ambient temperature, air velocity, and material properties (Doymaz and Pala, 2003). Predicting the drying time is critical for boosting drier capacity and optimization or control of the operating conditions during drying (Inyang et al., 2018). In mathematical modelling of drying curve characteristics, the thin layer and equilibrium moisture content models are applied. Mathematical modelling of thin layer drying is important for optimization of operating parameters and performance improvements of the drying systems (Cihan et al., 2007). Thin layer drying models used for modelling the drying phenomenon of agricultural materials are classified into three categories, namely: theoretical, semi-theoretical, and empirical (Afzal and Abe, 2000, Panchariya et al., 2002; Akpinar and Bicer, 2005; Akpinar, 2006). The theoretical approach is concerned with diffusion or simultaneous heat and mass transfer equations. The semi-theoretical models approach is concerned with approximated theoretical equations (Afzal and Abe, 2000; Akpinar and Bicer, 2005). Simplifying the general series solution of Fick’s second law or the modification of simplified models generally derives semi-theoretical models. But they are only valid within the temperature, relative humidity, airflow velocity and moisture content range for which they were developed. They require small time compared to theoretical thin layer models and do not need assumptions of the geometry of a typical food, its mass diffusivity and conductivity (Parry, 1985). Nevertheless, the semi-theoretical equations have been successfully applied by many researchers to describe drying rates for various agricultural products. In this category, Henderson and Pabis model, Page model, and Lewis model is extensively utilized by researchers. Empirical models establish a direct relationship between average moisture content and drying time without regards to the fundamentals of the drying process and their parameters which have no physical meaning. Though it is easy to apply the empirical models in drying simulations but they cannot give clear accurate view of the important processes that takes place during drying although they may describe the drying curve for the conditions of the experiments (Afzal and Abe, 2000; Akpinar and Bicer, 2005). Thin layer equations describe the drying phenomena in a unified way regardless of the controlling mechanism. They have been used to estimate drying times of biological products and to generalize drying processes. Recent studies incorporate Computational Fluid Dynamics (CFD) for multiphase drying simulations (Norton & Sun, 2006) and Artificial Neural Networks (ANN) for nonlinear drying behavior prediction (Kumar et al., 2020) as advanced and hybrid drying models for the prediction of drying time in drying of both bio materials and non-bio materials as long as there are sufficient data collection for the machine learning algorithm

Factors Affecting Drying Kinetics

A complex interplay of physical, environmental, and process-related factors influences the drying kinetics of biomaterials. One of the most critical parameters is the initial moisture content, which determines the duration of the constant-rate and falling-rate drying periods, depending on the extent to which water is free or bound within the material’s cellular matrix (Mayor & Sereno, 2004; Lewicki, 2006). Drying temperature also plays a pivotal role by enhancing the vapor pressure gradient and reducing water viscosity, thereby accelerating moisture migration; however, excessively high temperatures may lead to shrinkage, degradation, or case hardening (Kumar et al., 2014; Mujumdar, 2014). Environmental conditions such as relative humidity and air velocity directly influence the drying rate. Lower humidity levels improve the vapor pressure differential and increase the drying potential, while higher air velocity helps remove the saturated boundary layer from the material surface, promoting convective mass transfer (Sharma et al., 2009; Henderson et al., 2000). Additionally, the physical characteristics of the material, including thickness, porosity, cellular structure, and surface area-to-volume ratio, significantly impact internal moisture diffusion; denser or thicker materials tend to dry more slowly (Zogzas et al., 1996). Other parameters affecting the drying kinetics are pre-treatment techniques, such as blanching, ultrasonic treatment, and osmotic dehydration, which can modify the structural integrity of biomaterials, enhancing or impeding moisture diffusivity. For instance, ultrasound creates micro channels that facilitate water migration during drying (Nowacka et al., 2012). The drying method and energy source used (convective, solar, infrared, microwave, or freeze-drying etc) also impact the drying behavior by influencing heat transfer mechanisms and moisture removal efficiency (Ratti, 2001; Esper & Mühlbauer, 1998). The drying process is often described using semi-empirical or theoretical models, which rely on parameters such as effective diffusivity, drying rate constants, and activation energy—all of which are sensitive to the aforementioned variables (Midilli et al., 2002). A thorough understanding and integration of these factors are essential for optimizing drying processes and developing accurate predictive models.

Bio Materials and Corresponding Thin Layer Drying Kinetics  Models

Many bio materials have been dried, and their thin-layer drying kinetics modeled. Details of the thin layer models and their corresponding equations are presented in Table 1, while Table 2 displays the thin layer models used to predict moisture content and the related agricultural products.

Table .1: Table of thin layer drying models

Table 2. Thin layer models and their corresponding agricultural biomaterials

Advances in Drying Kinetics Modeling

AI and Machine Learning

Artificial Intelligence (AI) and Machine Learning (ML) have increasingly been applied to drying kinetics modeling to overcome the limitations of traditional physics-based methods, offering accurate predictions of moisture content, drying rates, and process optimization without explicitly solving complex differential equations (Shan et al., 2020; Tosun et al., 2022). Techniques such as artificial neural networks (ANN), support vector machines (SVM), and genetic algorithms (GA) have been widely used to model non-linear relationships in drying data, showing high predictive performance across various biomaterials (Erbay & Icier, 2010; Golpour et al., 2015). Furthermore, hybrid models that integrate ML with first-principles approaches are emerging, enhancing model generalization and interpretability (Chen et al., 2021). The use of deep learning and real-time sensor data also enables adaptive control and intelligent optimization of drying systems, paving the way for smart, automated drying technologies (Tosun et al., 2022). A compilation of the applications of AI and machine learning in modeling drying kinetics of agricultural products is shown in Table 3.

Table:3 Applications of AI and Machine Learning in Modeling Drying Kinetics of Agricultural Products

Computational Fluid Dynamics (CFD)

CFD simulation models are based on heat and mass transfer for optimization of dryer geometry and the drying processes using the Navier–Stokes equations as the governing equations, Energy, continuity, and momentum equations (Ahmad et al., 2023; Mellalou et al.,2021; Rouissi et al., 2021)

Computational Fluid Dynamics (CFD) has become a vital tool in drying kinetics modeling, enabling detailed simulation of heat and mass transfer, airflow distribution, and moisture evolution in drying systems with spatial and temporal resolution that traditional models often lack (Younis et al., 2017; Ratti, 2001). By solving the Navier–Stokes, energy, and species transport equations, CFD helps in analyzing complex geometries, optimizing dryer design, and improving energy efficiency and product quality (Nathakaranakule et al., 2007; Karim & Hawlader, 2005). CFD also facilitates the study of coupled phenomena such as shrinkage, phase change, and turbulence effects during drying, especially in porous media (Kumar & Prasad, 2007). Integrating CFD with experimental data and advanced modeling approaches like multiphysics and AI further enhances its predictive power and practical applicability (Tosun et al., 2022).

Multiscale and Multi-Physics Modeling

Multiscale and multi-physics modeling provides a comprehensive framework for understanding drying kinetics by integrating phenomena from cellular-scale moisture transport to bulk-level heat and mass transfer (Waananen & Okos, 1996; Zhang et al., 2014). These models couple thermal, mass, and structural dynamics to simulate the complex interactions that occur during drying, especially in porous biomaterials (Cai & Chen, 2008; Shan et al., 2020). Numerical techniques such as finite element analysis and computational fluid dynamics, often implemented in platforms like COMSOL and ANSYS, enable flexible, multi-physics simulation environments (Younis et al., 2017). Despite their promise, challenges remain regarding computational load, multiscale parameter estimation, and experimental validation, though recent advances in image-based modeling and machine learning are helping to address these issues (Tosun et al., 2022). Table 4 compiles some examples of multiscale and multi-physics modeling approaches used to analyze the drying kinetics of agricultural products.

Table 4. Table of multiscale and multi-physics modeling approaches used to analyze the drying kinetics of agricultural products,

Model Selection and Evaluation Criteria

Graphical and statistical analyses are always used to validate and select the best-fitted models in kinetics modeling. The key performance metrics for determining the model of best fit (goodness of fit) are;

Mean Relative Error. (MRE)

The models with Mean relative error (MRE) values below or equal to 10% are usually considered as a good fit (Simal et al., 2005). It is given as equation (1)

\[
\text{MRE}(\%) = \frac{100}{N} \sum_{i=1}^{N} \left\lfloor \frac{X_{\text{exp},i} – X_{\text{pred},i}}{X_{\text{exp},i}} \right\rfloor
\tag{1}
\]

Coefficient of Determination (R²)

The coefficient of determination (R²) is the square of the correlation coefficient, quantifying the proportion of variance in the dependent variable that can be explained by the independent variable. R² values range from 0 to 1; a value of 0 indicates no explained variance, while a value of 1 signifies that the model accounts for all variance in the dependent variable. Typically, the model with the highest coefficient of determination (R²) and a low root mean square error (RMSE) (Demir et al., 2004) is selected as the best-fitting model.

It is given as R² and expressed as equation (2)

\[
R^2 = 1 – \frac{\sum \left( X_{\text{exp}} – X_{\text{pred}} \right)^2}{\sum \left( X_{\text{exp}} – X_{\text{avg exp}} \right)^2}
\tag{2}
\]

where

Y_obv. is the experimental value

Y_pred is the predicted value

Y_avg  is the average of the observed values

Average Absolute Difference (AAD),

The Average Absolute Difference (AAD), also known as Mean Absolute Error (MAE), provides a measure of how far off the predictions were from the actual values, on average. This metric quantifies and compares the average magnitude of errors between the values predicted by the models and the actual values. It is given as equation (3)

\[
\text{AAD} = \frac{1}{N} \sum_{i=1}^{N} \left\lfloor X_{\text{exp},i} – X_{\text{pred},i} \right\rfloor
\tag{3}
\]

Root Mean Square Error

Root mean square error (RMSE) or sometimes referred to as root mean square deviation (RMSD) is derived by squaring the differences between the sum of the experimental value of the moisture ratio and the predicted value, dividing that by the number of test points, and then taking the square root of that result.

\[
\text{RMSE} = \sqrt{\frac{\sum_{i=1}^{N} \left( X_{\text{exp},i} – X_{\text{pred},i} \right)^2}{N}}
\tag{4}
\]

RMSE = root mean square error

I = variables

N = Number of data points

X exp,i = mean experimental moisture ratio

X pred,i = mean predicted moisture ratio

Mean Absolute Error (MAE)

The mean absolute error is defined as the ratio of the absolute error of the predicted to the actual value. Using this method, one can determine the magnitude of the absolute error in terms of the actual size of the observations. Mean absolute error, MAE, is a quantity used to measure how close forecasts or predictions are to the eventual outcomes. The mean absolute error is given by (Tripathy and Kumar, 2008; Mota et al., 2010)

The mean absolute error (MAE) is given as equation (5)

\[
\text{MAE} = \frac{1}{n} \sum_{i=1}^{n} \left[ \text{exp}_i – \text{pred} \right]
\tag{5}
\]

Where:

n is the number of data points or observations.

Σ denotes the summation symbol.

exp,i is the actual experimental value

pred,i is the predicted value

|Actual – Predicted| represents the absolute difference between the actual value and the predicted value for each data point.

Chi-Square (χ²) test

The Chi-Square (χ²) test measures how close your model’s predictions are to the experimental (actual) data points. It does this by squaring the difference between predicted and actual values and summing them up. The smaller the result, the better your model is at matching the real data. It is given as equation (6) by Ertekin, C., & Yaldiz, O. (2004).

\[
\chi^2 = \sum_{i=1}^{N} \frac{\left( X_{\text{exp},i} – X_{\text{pred},i} \right)^2}{X_{\text{pred},i}}
\tag{6}
\]

It can be normalized depending on the parameters being evaluated (goodness of fit or relative error per degree of freedom) and the type of data being analyzed. The normalized chi-square is given as equation (7)

\[
\chi^2 = \frac{1}{N – n} \sum_{i=1}^{N} \left( X_{\text{exp},i} – X_{\text{pred},i} \right)^2
\tag{7}
\]

Despite the evaluations of all the model testing criteria, model robustness should be evaluated under varying conditions such as temperature, humidity, and material thickness to ensure generalizability. A sensitivity analysis of model parameters further aids in understanding the influence of each variable on drying behavior (Henderson et al., 1997).

Research Gaps and Future Directions

Need for Material-Specific Models

Generic models often fail to capture the unique drying behaviors of specific biomaterials. There is a need for customized models that integrate moisture-binding mechanisms and internal structure changes during drying.

Integration with Real-Time Monitoring Systems

The use of sensors and data acquisition systems allows for real-time monitoring of drying parameters. Future models should incorporate feedback loops and predictive control systems for dynamic adaptation (Ratti, 2001).

Climate-Resilient and Energy-Efficient Drying

Future drying models must support energy-efficient systems that adapt to fluctuating climatic conditions, particularly in solar and hybrid drying.

Open Access Databases and Modeling Platforms

Collaborative platforms that house experimental data and model parameters are necessary for cross-laboratory comparison and development.

Coupled Modeling Approaches

Integrating mechanical, thermal, chemical, and biological phenomena into a unified modeling framework—e.g., drying and microbial inactivation—offers a complete process understanding (Chen et al., 2020).

CONCLUSION

Modeling drying kinetics is key for process optimization in bio-material drying. While empirical and semi-theoretical models are widely used because of their ease of application, advanced computational tools offer improved accuracy and adaptability. Emphasis should be placed on creating smart, adaptive systems informed by real-time data and tailored to specific biomaterial properties. Further research should be focused on advanced computational tools for improved accuracy and adaptability

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