Comparison of analytical and numerical simulations of cowpea (Vigna unguiculata) grain drying using Fick's law and the finite element method

 Corresponding author: Marco Aurélio Amarante Ribeiro Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Comparison of analytical and numerical simulations of cowpea (Vigna unguiculata) grain drying using Fick’s law and the finite element method Marco Aurélio Amarante Ribeiro * Doctoral Student, Federal University of Viçosa (UFV), Viçosa, MG, Brazil. Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 Publication history: Received on 17 March 2025; revised on 27 April 2025; accepted on 30 April 2025 Article DOI: https://doi.org/10.30574/gjeta.2025.23.2.0139 Abstract This article presents a comprehensive analysis of the drying process of a single cowpea bean (Vigna unguiculata) using both analytical and numerical approaches. The process, governed by coupled heat and mass transfer (Bird et al., 2002) mechanisms, was simulated by modeling the bean as an ellipsoidal body under isotropic conditions. The analytical solution was based on a truncated series representation of Fick’s second law, while the numerical solution employed the Finite Element Method (FEM) with convective boundary conditions. Simulations were implemented in Python, incorporating thermophysical properties of air and grain derived from empirical models in the literature. Drying behavior at different temperatures was evaluated, and the results from both methods were compared using relative error and standard error of the estimates. The two solutions exhibited strong agreement, particularly during the active drying phase, confirming the accuracy and robustness of both modeling strategies. This comparative study contributes to the development of accurate thin-layer drying models (Madamba, 1996) and supports the optimization of postharvest processing techniques. Keywords: Drying Process Modeling; Finite Element Method; Fick’s Second Law; Moisture Diffusion; Cowpea 1. Introduction Cowpea (Vigna unguiculata (L.) Walp.), commonly known in Brazil as feijão-de-corda, feijão-macassar, or feijãofradinho, is a legume of significant socioeconomic importance, particularly in the North and Northeast regions of the country. Adapted to semi-arid climates and low-fertility soils, cowpea is notable for its resilience, short growth cycle, and high drought tolerance. These characteristics make it a strategic crop for food security in areas with challenging edaphoclimatic conditions. In addition to serving as a vital source of plant-based protein in local diets, cowpea cultivation sustains the livelihoods of smallholder farmers and is widely used for both fresh consumption and industrial processing, such as flour production and traditional regional dishes. Brazil ranks among the world’s leading cowpea producers, with significant advancements in breeding programs aimed at increasing productivity, resistance to pests and diseases, and the nutritional quality of the grains (Freire Filho et al., 2012). Grain drying, regardless of type, is a critical stage in post-harvest crop handling, as it directly impacts the shelf life, quality, and usability of the final product. A solid understanding of the fundamental principles underlying the drying process is essential for optimizing drying technologies and improving operational efficiency. At its core, grain drying involves the transfer of heat and water from the grain to the surrounding air, governed by a series of physical and thermodynamic mechanisms (Chen & Pan, 2023). These principles have been extensively documented in the literature (Brooker et al., 1974; Pabis et al., 1991; Brooker et al., 1992; Bengtsson & Ahrné, 2005; Jian & Jayas, 2022). Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 2 During the initial drying phase, surface water evaporates rapidly, maintaining a constant drying rate. This stage is primarily influenced by the availability of heat and the efficiency of mass transfer between the grain and the drying medium, typically air (Bird et al., 2002; Li et al., 2023). As drying progresses, the process transitions into a falling-rate period, during which water migrates from the interior to the surface. The drying rate gradually declines due to reduced water gradients and increasing internal resistance to water movement (Jayas et al., 2023). Eventually, the grain reaches its equilibrium water content, at which point further drying becomes negligible. The drying process encompasses both heat and mass transfer phenomena. Heat may be delivered by conduction, convection, or radiation, depending on the drying technique employed. This thermal energy increases the grain’s temperature, facilitating internal water movement toward the surface. However, excessive or uneven heating can degrade grain quality, leading to structural damage or uneven drying (Jibril et al., 2024). Mass transfer, defined as the movement of water from within the grain to the surface for evaporation, is influenced by factors such as grain temperature, relative humidity, and airflow conditions (Zhang et al., 2006). Several mathematical models have been developed to describe drying kinetics and predict the rate of water removal. Thin-layer drying models—such as the Page (Page, 1949), Midilli, and Henderson and Pabis (Henderson & Pabis, 1961) models—are widely applied to represent the drying behavior of grains, including cowpea (Jayas et al., 2023). These models typically assume a single-layer configuration and express the drying process in terms of water content ratio and time. A key parameter in these models is the effective water diffusivity, which quantifies the ease of internal water migration and is influenced by temperature, initial water content, and the physical properties of the grain (Li et al., 2023). Energy consumption is another important consideration, as drying is one of the most energy-intensive stages in postharvest processing. Assessing the energy balance during drying is fundamental to developing sustainable and energyefficient technologies (Jimoh et al., 2023; Jibril et al., 2024). Environmental conditions also play a significant role in drying performance. High air temperatures typically accelerate drying but may compromise product integrity if not carefully controlled. Similarly, high ambient humidity levels can reduce the moisture gradient between the grain and the air, slowing the process (Chen & Pan, 2023). Airflow velocity is also crucial, as sufficient air circulation promotes uniform moisture removal (Sahin & Sumnu, 2006). Various drying technologies have been developed, each with specific advantages and limitations. Hot air drying is the most commonly used method, valued for its simplicity and cost-effectiveness, although it may result in uneven drying if not properly controlled (Jayas et al., 2023). More advanced systems, such as fluidized bed dryers, promote uniformity through constant grain agitation, while freeze and vacuum drying offer superior preservation of nutritional content at higher energy costs (Ho, 1992). Given the complexity of drying—which involves coupled heat and mass transfer mechanisms—advanced simulation tools are essential. Modeling the drying of a single grain allows for precise analysis of internal water diffusion and surface interactions. Both analytical and numerical methods have been employed for this purpose. Analytical solutions, based on Fick’s law, provide valuable theoretical insight, while numerical approaches such as the Finite Element Method (FEM) offer greater versatility and accuracy for grains with complex geometries. This work aims to compare these two modeling approaches—analytical and numerical (FEM)—in simulating the drying of a single cowpea grain. The grain is modeled as an ellipsoid with isotropic properties. The simulation incorporates convective boundary conditions and uses thermophysical data derived from empirical models and published literature. FEM implementation was carried out using Python-based algorithms to solve the diffusion equation under realistic drying scenarios, with the goal of evaluating the strengths and limitations of each modeling strategy. 2. Fundamental Considerations Single grain drying can be described as a water removal process in which water is transferred from the grain surface to the atmosphere, usually through evaporation. This process is influenced by several factors, such as air temperature, air velocity, relative humidity, and physical properties of the grain. Characteristics such as porosity, shape, size, and thermal conductivity play a crucial role in the drying rate, and these factors can vary significantly according to the grain type, which makes modeling single grain drying a considerable technical challenge (Dincer & Dost, 2019; Iqbal et al., 2020). Grain drying modeling has been an important field of study due to its relevance in several industrial processes, such as agriculture and food production. Modeling the grain drying process fundamentally depends on the analysis of heat and Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 3 mass transfer (Albini, Freire, & Freire, 2019; Bird et al., 2002) phenomena in a thin layer of grains and their interactions with the drying medium. Understanding these phenomena is obtained through the analysis of a single grain, since a thin layer of grains is usually formed by a mesh with the thickness of a single grain, uniformly arranged on a tray with a perforated bottom. Thus, modeling a single grain allows greater precision in modeling the drying of thin layers of grains. Numerous researchers have effectively employed the Finite Element Method (FEM) to simulate the drying behavior of individual grains (Haghighi et al., 1990; Irudayaraj, 1993; Jia et al., 2002). The drying of a single grain involves complex challenges arising from its geometry, thermophysical properties, and the dynamic nature of moisture transport within the material. To better understand and optimize this process, mathematical modeling and numerical simulations are essential tools. Among the available approaches, this study focuses on comparing the analytical solution of the diffusion equation with the numerical solution provided by FEM, considering the bean grain as an ellipsoidal body. 3. Methodology This study focused on modeling the drying process of a single cowpea bean grain, idealized as an ellipsoidal body with isotropic properties, through a comparative analysis of two simulation approaches: the analytical solution of Fick’s second law of diffusion (Goneli et al., 2010) and the Finite Element Method (FEM). This comparative modeling is essential for understanding thin-layer drying phenomena and extends to applications in thick-layer drying systems, offering insights that contribute to process optimization and equipment design (Crank, 1975; Dincer & Dost, 2019; Kudra & Mujumdar, 2002). The modeling was performed by integrating the numerical solution of the diffusion equation using FEM under convective (Robin-type) boundary conditions, and comparing the results with those obtained via the classical analytical series solution. The numerical approach was implemented in Python (Hunter, 2007; Harris et al., 2020; Virtanen et al., 2020), and it relied on empirical models and thermophysical correlations to define the physical properties of the grain and air under drying conditions, supplemented by data from established literature sources (Jamaleddine & Ray, 2010; Iqbal, Haider, & Sattar, 2020; Liu, Zhang, & Wang, 2017). The FEM framework enabled the solution of the diffusion equation in a three-dimensional domain, making it particularly appropriate for grains with non-spherical geometries. Within this framework, the governing partial differential equations were discretized through the assembly of mass matrices, stiffness matrices (representing the diffusion term), and boundary matrices (modeling the convective flux), in line with standard finite element procedures (Bird, Stewart, & Lightfoot, 2002; COMSOL, 2017). This methodology offers flexibility in the definition of material properties, meshing strategies, and boundary conditions—features often limited in purely analytical approaches (Ramachandran et al., 2018; Arsène, Hénault, & Lefebvre, 2021). Additionally, the weak form of Fick’s second law used in FEM enables precise temporal and spatial resolution, especially when implicit time discretization schemes are employed. The implementation conducted in this study utilized Pythonbased algorithms, which are particularly effective for the numerical simulation of mass transfer processes in complex geometries (Fish & Belytschko, 2007). 3.1. Classic Drying Models It is necessary to report on empirical drying models because they are frequently used to represent grain drying, especially for modeling grain drying in thin layers with a thickness of one grain, to obtain the drying curve under isothermal conditions under controlled conditions. These equations are fundamental for applications in modeling thick layer drying on a commercial scale. These empirical models are built from adjustments to experimental data, when there is little experimental data or the complexity of the system does not justify the use of advanced numerical methods, generally with simple expressions and few parameters, while semi-empirical models try to incorporate a simplified physical basis, incorporating better adjustment capacity, although they continue to maintain a degree of empiricism. Some simpler classical models are presented below, although a variety of these models are found in the literature, but in some ways, they are all modifications of these classical models. 3.1.1. Lewis model This model is the simplest one used to describe the drying rate according to the difference between the grain moisture content and the equilibrium moisture content, derived directly from the analytical solution of the first-order Fick's law, Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 4 assuming one-dimensional and isothermal diffusion. This model is effective when drying occurs uniformly, usually in the constant-rate period, typical of products with high initial moisture content, as expressed by Equation (1): 𝑀𝑅=𝑀(𝑡)−𝑀𝑒 𝑀0−𝑀𝑒=𝑒−𝑘∙𝑡……………..(1) where 𝑀𝑅 − Water content ratio. 𝑀(𝑡) − Grain water content over time, [kg waterkg ⁄ dry air]. 𝑀0 − Initial water content of the grain, [kg waterkg ⁄ dry air]. 𝑀𝑒 − Equilibrium water content, [kg waterkg ⁄ dry air]. 𝑡 − Time, [s] 𝑘 − Drying rate constant, [s−1] 3.1.2. Page model Another widely used model is known as the Page Model, due to its simplicity and ability to describe drying at various stages of the process, such as flash drying, using an exponential function (Page, 1949): 𝑀𝑅=𝑒𝑥𝑝(−𝑘∙𝑡𝑛)………………(2) where 𝑛 is an exponent that depends on the drying conditions. 3.1.3. Henderson and Pabis model This model is a simplified version of the Lewis model, but with a multiplicative coefficient that improves the fit, assuming an exponential loss of water from the grain over time, without taking into account the complexity of non-linear variations in the drying rate (Henderson & Pabis, 1961): 𝑀𝑅=𝑎∙𝑒𝑥𝑝(−𝑘∙𝑡)…………….(3) where 𝑎 is an adjustment coefficient that depends on the drying conditions. These models are widely used due to their simplicity and excellent performance in nonlinear regression with experimental data, although they disregard the effects of internal gradients, resistances and variations in physical properties. 3.2. Models based on the Three-Dimensional Diffusion Equation The drying process of a single grain involves heat transfer due to the hot air surrounding it and also the migration of water from the interior to the surface of the grain, where evaporation occurs. Therefore, it involves a process of simultaneous heat and mass transfer (Bird et al., 2002) described by the equations of conservation of mass and conservation of energy. The fundamental equations for simulating the drying of a grain are presented below. 3.2.1. Physical Basis of the Diffusion Model The internal diffusion of water in capillary-porous media can be described by Fick's Second Law in three dimensions, which is a partial differential equation (PDE) that models the propagation of water in a diffusive medium: 𝜕𝑢 𝜕𝑡=𝐷𝑒𝑓∙(𝜕2𝑢 𝜕𝑥2+𝜕2𝑢 𝜕𝑦2+𝜕2𝑢 𝜕𝑧2)…………….(4) or in vector form 𝜕𝑢 𝜕𝑡=𝐷𝑒𝑓∙∇2𝑢 ; 𝑢=𝑢(𝑥,𝑦,𝑧,𝑡)…………….(5) Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 5 were u − Water concentration (or water content, dry basis), [kg/m³]. 𝑡 − Time, independent variable that represents the temporal evolution of drying, [s] 𝐷𝑒𝑓 − Effective diffusion coefficient, [m2s ⁄] 𝜕𝑢 𝜕𝑡 − Partial derivative of 𝑢 with respect to 𝑡, representing the temporal variation of water content, [kg water(kg dry air)∙s ⁄]. 𝛻2𝑢 − Laplacian operator, which represents the sum of the second derivatives in each spatial direction. Equation (4) is used to describe the migration of water from the interior of the grain to its surface. For grains with regular geometry—such as spheres or ellipsoids—it can be solved analytically using an infinite Fourier series, and numerically for any geometry using the Finite Element Method-FEM (Fish & Belytschko, 2007) or Finite Volume Method-FVM. 3.2.2. Initial conditions 𝑢(𝑥,𝑦,𝑧,0)=𝑢0(𝑥,𝑦,𝑧)……………..(5) It indicates that at the initial instant 𝑡=0, the distribution of water content 𝑢 inside the grain is known and given by a function 𝑢0. 3.2.3. Dirichlet boundary conditions It considers the fixed water content at the surface and defines the imposed surface water content, 𝑢𝑠|Γ, directly at the boundary Γ: 𝑢|Γ=𝑢𝑠(𝑡)……………..(7) 3.2.4. Neumann boundary conditions Maintains fixed flow (impermeable surface), indicating that there is no water flow in the normal direction 𝑛 to the outer surface: 𝜕𝑢 𝜕𝑛|Γ=0………………(8) 3.2.5. Robin type boundary condition Defines the mass exchange by convection between the grain and the external air, where ℎ𝑚 is the mass transfer coefficient (Incropera et al., 2007) and 𝑢𝑒 is the equilibrium water content of the grain in contact with the air (dry basis). Diffusivity is also used in the convective boundary condition in the numerical solution of the diffusion equation by the finite element method, expressed by Equation (9): −𝐷𝑒𝑓𝜕𝑢 𝜕𝑛=ℎ𝑚(𝑢−𝑢𝑒)…………….(9) Here comes the mass transfer coefficient (Incropera et al., 2007), ℎ𝑚, which can be estimated with correlations of the Sherwood number type, as a function of the Reynolds and Schmidt numbers: 𝑆ℎ=0,2+0,6∙𝑅𝑒12 ⁄∙𝑆𝑐13 ⁄…………..(10) 3.2.6. Reynolds number 𝑅𝑒=𝜌𝑎∙𝑣∙𝑑𝑒𝑞 𝜇……………(11) Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 6 3.2.7. Schmidt number 𝑆𝑐=𝜌𝑎 𝜌𝑎∙𝐷𝑎…………….(12) 3.2.8. Sherwood number 𝑆ℎ=ℎ𝑚∙𝑑𝑒𝑞 𝐷𝑎……………….(13) 3.2.9. Analytical Solution of the Diffusion Model For an ellipsoidal grain with equivalent radius, 𝑟𝑒𝑞, assuming symmetry and constant diffusivity, the water content ratio can be expressed by a series of sines and exponentials: 𝑀𝑅(𝑡)=∑( 6 𝑛2∙𝜋2) ∞ 𝑛=1 ∙𝑒𝑥𝑝[(−𝑛2∙𝜋2)(𝐷𝑒𝑓∙𝑡 𝑟𝑒𝑞 2)]………….(14) where the equivalent radius of the ellipsoid with axes 𝑎,𝑏,𝑐 is given by: 𝑟𝑒𝑞=√𝑎∙𝑏∙𝑐 3……………(15) This analytical expression considers the Fick solution with uniform Dirichlet boundary condition on the surface. Therefore, the greater the number of terms in the series, the greater the accuracy. The thermal dependence of diffusivity can be expressed by the Arrhenius equation (Dincer & Dost, 2019): 𝐷𝑒𝑓(𝑇)=𝐷0∙𝑒𝑥𝑝(−𝐸𝑎 𝑅∙𝑇)……………(16) The sensitivity of diffusivity to temperature varies non-linearly according to the variation of the activation energy, 𝐸𝑎. 3.3. Numerical Formulation by the Finite Element Method The Finite Element (FEM) formulation solves the weak equation of Fick's Second Law, with a Robin (convective) boundary condition. This formulation allows solving the diffusion equation considering complex geometries, heterogeneities in the medium and varied boundary conditions. The weak formulation of the Fick equation with a Robin (convective) boundary condition: ∫ 𝑣𝜕𝑢 𝜕𝑡 ⬚ Ω𝑑Ω+∫𝐷𝑒𝑓 ⬚ 𝛺∇𝑢∙ ∇𝑣 𝑑Ω=∫ℎ𝑚(𝑢−𝑢𝑒) ⬚ Γ𝑣 𝑑Γ…………..(17) The discretized equations for implementation in finite elements are presented below. 3.3.1. Mass matrix The mass matrix (𝑴) is a fundamental component in the solution of transient models by the finite element method: 𝑴𝒊𝒋=∫𝜙𝑖𝜙𝑗 ⬚ Ω𝑑Ω………….(18) Where; 𝑴𝒊𝒋 − Term representing the water content coupling between nodes i and j. Ω − Body domain (bean grain). 𝜙𝑖,𝜙𝑗 − Shape functions associated with element nodes. Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 7 The mass matrix represents the inertia of the drying system in accumulating or releasing water (water vapor) and, by analogy with heat transfer, is equivalent to thermal capacitance. This matrix reflects the distribution of water contained in the grain and its evolution over time. 3.3.2. Stiffness matrix (diffusion) The stiffness matrix (𝑲) represents the internal flux of water (vapor), caused by spatial differences in concentration, and models the coupling of the domain with the external environment through surface convection. This matrix represents the resistance or ease with which the material loses water to the external environment, acting analogously to a surface conductance in heat transfer. This matrix represents the diffusion of water within the grain domain and originates from the term containing the water content gradient in the weak formulation: 𝑲𝒊𝒋=∫𝐷𝑒𝑓∇𝜙𝑖∙∇𝜙𝑗 ⬚ Ω𝑑𝛺……………(19) Where; 𝐊𝐢𝐣 − It represents the diffusion of water within the grain domain. 𝐷𝑒𝑓 − Effective diffusion coefficient. ∇𝜙𝑖 − Gradient of the shape function associated with the node. 𝑖 The matrix 𝑲 is only assembled on the boundary elements (elements with faces on the grain surface), is additive to the global system and appears together with the stiffness matrix in the time advance equation. 3.3.3. Contour matrix (Robin condition) The boundary matrix appears in the finite element formulation when a Robin-type boundary condition is used, generating an additional term in the weak formulation of the Fick’s equation and contributing an additional matrix called the boundary matrix 𝑹: 𝑹𝒊𝒋=∫ℎ𝑚𝜙𝑖𝜙𝑗 ⬚ Γ𝑑Γ…………..(20) Where Γ − Outer boundary of the domain (grain surface). 𝜙𝑖,𝜙𝑗 − Shape functions associated with surface nodes. ℎ𝑚 − Mass transfer coefficient (m/s). 𝑹𝒊𝒋 − Stiffness term - resistance to water diffusion or heat conduction. 3.3.4. Source font vector The surface source vector (𝑭) represents the effect of mass exchanges between the grain surface and the environment (air), derived from the Robin (or convection) boundary condition: 𝑭𝒊=∫ℎ𝑚𝑢𝑒 𝜙𝑖 ⬚ Γ𝑑Γ………….(21) Where 𝑭𝒊 − Surface source vector of element i. 𝑢𝑒 − Equilibrium water content at the surface. 𝜙𝑖 − Shape function associated with node i on the surface. Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 8 This vector is responsible for quantifying the gain or loss of water (water vapor) across the grain boundary. 3.3.5. Matrix system after implicit temporal discretization After discretizing Equation (17), the matrix system represented in Equation (22) appears: [𝑴+Δ𝑡(𝑲+𝑹)]∙𝒖𝑛+1=𝑀∙𝒖𝑛+∆𝑡∙𝑭…………..(𝟐𝟐) Where; 𝑴 − Mass matrix. 𝑲 − Stiffness matrix (diffusivity). 𝑹 − Boundary matrix (convection). 𝒖𝑛 − Water content vector in time 𝑡𝑛. 3.3.6. Comparison of the Efficiency of Analytical Solution and Finite Element Solution A comparison of the efficiency of the analytical series solution of the diffusion equation (with a large number of terms), in relation to the finite element solution, is found in Table 1. Table 1 Comparison between Analytical and Finite Element Solutions. Feature Analytical Solution Finite Element Solution Geometry Requires ideal symmetry Any geometry Boundary condition Dirichlet (usually) Dirichlet, Neumann or Robin Precision High (with many terms) High (depending on mesh and method) Flexibility Limited High Computational cost Low Moderate to high 3.4. Method for Comparing Results The results of the simulations using the two solution methods were compared using two statistical parameters: Relative Error 𝐸𝑟=|𝑀𝑅𝐹𝐸𝑀(𝑡)−𝑀𝑅𝐴𝑛𝑎𝑙(𝑡) 𝑀𝑅𝐴𝑛𝑎𝑙(𝑡) |∙100…………….(23) Where 𝐸𝑅 − Relative error, [%] 𝑀𝑅𝐹𝐸𝑀 − Water content ratio value simulated using finite elements. 𝑀𝑅𝐴𝑛𝑎𝑙 − Water content ratio value simulated using analytical solution of diffusion equation with thirty terms. 3.4.1. Standard Error of Estimates 𝑆𝐸𝐸=√1𝑛∑(𝑦𝑖−𝑦𝑖 )2 𝑛 𝑖−1 ………..(24) Where 𝑆𝐸𝐸 − Standard deviation between predicted and observed values. Global Journal of Engineering and Technology Advances, 2025, 23(02), 001-015 9 𝑦𝑖 − Observed or experimental value. 𝑦𝑖  − Value estimated or predicted by the model. 𝑛 − Number of observations. 3.5. Diffusion Model Solution Two solutions were presented: (i) analytical solution using 30 terms of the Fick series, Equation (14), and (ii) numerical solution implemented in a Python algorithm, using the finite element method, considering a bean as an isotropic ellipsoid (Figure 1). The mathematical formulation was implemented in an algorithm coded in Python language to perform the simulations, prepare the data tables and graphs of the drying curves. For the numerical solution, the ellipsoidal geometry of a cowpea grain, as illustrated in Figure 1, with axes a=9 mm,b=6 mm,c=5 mm, was discretized into a mesh 20×20×20, resulting in approximately 10,000 elements. An illustration of the discretization mesh is shown in crosssection in Figure 2. The drying conditions established for the simulations, with a drying air flow with a speed equal to 1 m/s are in Table 2 and the physical properties of the bean grain (Goneli et al., 2010) are in Table 3. (A) (B) Figure 1 (a) Ellipsoidal geometry assumed for a cowpea grain, and (b) Illustration of the finite element mesh (A) (B) Figure 2 (a) Illustrative image of a cowpea grain, and (b) mesh with, approximately, 10,000 elements of a quarter of the ellipsoid representing the cowpea grain 3.5.1. Physical properties of dry air Table 2 Physical properties of dry air as a function of temperature in the range from 𝟎℃ 𝒕𝒐 𝟏𝟎𝟎℃. Property Unit Equation Density: kgm3 ⁄ 𝜌𝑎(𝑇)= 101325 287,05∙(𝑇+273,15) Dynamic viscosity: Pa∙s 𝜇(𝑇)=𝜇0∙(𝑇0+𝐶 𝑇+𝐶)∙(𝑇 𝑇0)32 ⁄