Fick's Law Research Articles

Microencapsulation of Grape Pomace Extracts with Alginate-Based Coatings by Freeze-Drying: Release Kinetics and In Vitro Bioaccessibility Assessment of Phenolic Compounds.

The phenols from grape pomace have remarkable beneficial effects on health prevention due to their biological activity, but these are often limited by their bioaccessibility in the gastrointestinal tract. Encapsulation could protect the phenolics during digestion and influence the controlled release in such an intestine where their potential absorption occurs. The influence of freeze-drying encapsulation with sodium alginate (SA) and its combination with gum Arabic (SA-GA) and gelatin (SA-GEL) on the encapsulation efficiency (EE) of phenol-rich grape pomace extract and the bioaccessibility index (BI) of phenolics during simulated digestion in vitro was investigated. The addition of a second coating to SA improved the EE, and the highest EE was obtained with SA-GEL (97.02-98.30%). The release of phenolics followed Fick's law of diffusion and the Korsmeyer-Peppas model best fitted the experimental data. The highest BI was found for the total phenolics (66.2-123.2%) and individual phenolics (epicatechin gallate 958.9%, gallocatechin gallate 987.3%) using the SA-GEL coating were used. This study shows that freeze-dried encapsulated extracts have the potential to be used for the preparation of various formulations containing natural phenolic compounds with the aim of increasing their bioaccessibility compared to formulations containing non-encapsulated extracts.

Mathematical modeling of the drying and oil extraction processes of passion fruit seeds using fractional calculus

AbstractBACKGROUNDThis article aims to fit kinetic models of the passion fruit seed drying process, in addition to mathematical modeling using traditional models and fractional calculus of the extraction of lipids from the seeds.RESULTSRegarding the drying process, it was found that the model that best suited the two drying temperatures employed was the fractional order one.CONCLUSIONAnalysis showed that for both drying conditions the value of parameter α was greater than 1, indicating that the profile that occurs for the seeds is superdiffusion. Regarding the mathematical fits of oil extraction, it was found that the two models that presented the best fits for the lipid extraction kinetics were hyperbolic and fractional order. Regarding the fractional order model, as the value of parameter α was lower than 1, it can be seen that the process again does not follow Fick's law, and in this case a subdiffusion process occurs. © 2024 Society of Chemical Industry (SCI).

A numerical validation between the neutron transport and diffusion theories for a spatial kinetics problem

In this paper, a comparative analysis of numerical results of the neutron transport and diffusion theories for steady-state and transient multigroup problems is presented. The neutron transport equation is known as the one that best describes the behavior of the neutron population in a nuclear reactor. However, due to the difficulty of working with its complete form, other models are considered as approximations to this equation. One such approximation is the neutron diffusion equation, which uses the Fick's Law. It is well known, however, that the diffusion model may not work well under specific conditions. The objective of this work is to establish a quantitative comparison of numerical results obtained for the K dominant eigenvalue and for the scalar fluxes from the two theories and to analyze the influence of the model on the results.

Effect of process parameters on the extraction kinetics of Leptocarpha rivularis DC. in a packed bed extractor using supercritical carbon dioxide

The kinetics of supercritical extraction from L. rivularis stalks using CO2 were studied in terms of temperature (40−60ºC), pressure (18–30 MPa), specific CO2 consumption (30−50 kg/kg d.s.), and particle diameter (0.5−1.1 mm). The extraction yield was found to range from 17.30 to 27.23 g/kg d.s. The highest extraction yield was obtained at 40°C, 30 MPa, 30 kg CO2/kg d.s., and mean particle size of 0.5 mm. The yield increased with pressure and decreased with temperature and particle size (p≤0.05). The diffusion model based on Fick's 2nd law adequately described the cumulative extraction curves, using the effective diffusion coefficient (De) as the adjusted parameter, which ranged between 3.50 and 19.26×10−12 m2/s. The Biot number was estimated, indicating a prevalence of internal control to mass transfer resistance. Additionally, the “apparent solubility” was obtained from the initial slope of the cumulative extraction curves. The extraction yield correlated positively with apparent solubility (p≤0.05) and De coefficient (p≤0.05).

A chloride diffusion model for cementitious material with pore-solution viscosity enhancement

A promising approach to retard chloride attack has been proposed by increasing the pore solution viscosity in contrast to the traditional methods, but the dependence of chloride diffusion coefficient on viscosity has not been included in the current transport models. This paper presents a model for the diffusion of chloride in cementitious materials based on the viscosity-enhancing behavior of pore solution using viscosity modifying admixtures (VMAs). The parameters affecting the diffusion of chloride in the viscosity-enhancing solution are proposed to be the VMA molecular shape factor and VMA volume fraction based on the Maxwell approximation of spherical inclusions in the effective medium approximation theory. VMA shape factor is modeled with VMA molecular mass as a key variable. VMA volume fraction is calculated as a function of water-cement ratio, hydration age, initial VMA volume fraction, and VMA leaching residual fraction. This chloride diffusion model is developed and validated, which is based on the modification of Fick's second law and takes into account the pore solution viscosity-enhancing behavior and VMA-induced pore coarsening. This model differs from traditional models based on ion transport paths and provides a new reference for concrete durability design in terms of the intrinsic properties of the pore solution.

Diffusion in multicomponent mixed solvent electrolyte systems

Thermodynamic ideality is often assumed in diffusion calculation, implying concentrations are assumed equal to the activities with activity coefficients of one. A general set of diffusion flux and molar flux equations are developed, incorporating equilibrium thermodynamic models for non-ideal conditions in well-known non-equilibrium diffusion theory. Derivations are based on the Maxwell-Stefan friction theory with the chemical potential as the driving force. Isothermal and isobaric conditions are assumed, and the theory does not include the Soret coefficient or the Onsager formalism.In this work, Fick's law is derived from the general diffusion theory, also as a function of activity gradients. The results illustrate that concentration cannot just be substituted by activity. Using a similar approach demonstrates that the Nernst-Planck (NP) equation may be derived from the general theory incorporating terms for migration and convection. An extended NP equation is derived including the thermodynamic correction factors (TCF) to account for the non-ideality of the liquid. The derived equation shows the improvement of the self-diffusion terms and friction coefficients in the NP equation and that the TCF enhances the observed effective diffusivities.Examples of TCF are presented using the extended UNIQUAC thermodynamic model. This is done for the CO2-NaOH-monoethyleneglycol-water system, which is typically found in wet natural gas pipeline CO2 corrosion systems. pH-stabilization is classically used in the prevention of corrosion, but the results show that glycol has a noticeable effect of the diffusion on key components in the corrosion mechanism. A TCF of 0.4 to 1.3 may be expected. This indicates that the flux calculation deviates up to 60 % from the real-life scenario if ideality is assumed. The thermodynamic model may therefore improve the diffusion calculations considerably.

Experimental and Numerical Study of Superheated Steam Drying for Sliced Avocado

AbstractSuperheated steam drying (SSD) of sliced avocado is studied by both experiment and theory. An experimental system is built to evaluate the effects of drying temperature and gas velocity on the evaporation rate and color change of the product. Experiments are conducted for ripened Booth avocado from Daklak province of Vietnam. Experimental results show that evaporation is faster at higher temperatures and velocities, but the effect of temperature decreases at higher velocities. At temperatures below 130 °C, the product is not brown, but at a temperature in the range of 130–150 °C, the dried sample is burned. The experimental model is fitted with high accuracy to predict the moisture rate as a function of time. The evaporation processes of SSD are also compared with hot air drying at the same drying conditions. To extend the application ability of the result, the theoretical model is developed on the basis of Fick's law in which the spatial distributions of temperature and moisture are taken into account. The effective diffusivity is determined by an optimization tool with good agreement. This parameter is constant for all drying conditions, so the model can be applied to a huge range of drying conditions. The numerical results show that the temperature is almost uniform in the sample because of high thermal conductivity, but the moisture gradient is significant.

Chitosan/teff flour active films incorporated with citric acid and beetroot leaf extract: Physicochemical properties and mathematical modeling of phenolic release

Active compounds are integrated into food packaging films to enhance their food protection capabilities. Understanding the release of these components in films, particularly in crosslinking scenarios, is crucial. This study aimed to mathematically model the release of phenolic compounds from chitosan/teff flour films to understand how active compounds gradually release. Moreover, it was aimed to study the effects of incorporation of beetroot leaf extract and citric acid crosslinking. The collective observations, encompassing increased density and thermal stability, alongside concurrent reductions in moisture content, water solubility, water vapor permeability and swelling index following citric acid addition, strongly suggested the presence of crosslinking. Applying Fick's law and the finite element method revealed a substantial influence of the crosslinking agent on diffusion coefficients. The model exhibited strong agreement with experimental data, as reflected in low root mean square error values ranging from 3.02 to 8.50 mmol/m3 for films. Furthermore, the influence of citric acid crosslinking on the release of TPC was evident, as indicated by a decrease in average diffusion coefficient values from 3.499 × 10−13 m2 s−1 to 1.770 × 10−13 m2 s−1 with the formula with 1.5 % citric acid and 0.5 % beetroot leaf extract. This showcases the impact of various parameters on controlled release in food packaging.

Revisiting the bleeding effect in historical cobalt porcelain pigments: Mechanism, influence and technical responses

The bleeding phenomenon, a persistent and widespread issue in the application of cobalt-bearing pigment during porcelain decoration, has spurred different civilisations to develop various response strategies to alleviate this problem. In this study, we challenge the prevailing hypotheses concerning the role of composition and viscosity in determining the bleeding effect on blue-and-white wares, proposing a novel physical model framing it as a diffusion process that occurs within vitreous silicate, where the severity can be qualitatively expressed using the diffusion distance. Drawing upon the phenomenological Fick's law and microscopic diffusion mechanism, we quantitatively discuss the primary physical parameters that influence the diffusion behaviour for the first time, clarifying that the diffusion of cobalt ions is not related to the medium viscosity. Moreover, compared with the microstructural features of blue decors with and without the bleeding effect, we reveal that domestic cobalt particles are all encapsulated by anorthite crystals, acting as a passivation shell to hinder the dissolution of cobalt particles. Significantly, our reinterpretation has broader archaeological implications for the bleeding effect associated with cobalt pigment in ceramics, elucidating the historical trajectories of responsive practices and the multifaceted interplay between resource form, artistic expression and technological advancement across varying environmental and cultural contexts. Overall, these responses to the bleeding effect exemplify the complexities of technological evolution, highlighting that technology is not merely an extension of technical knowledge but also functions as a form of social construction deeply intertwined with its local context. This comprehensive understanding contributes to our understanding of historical narratives in ceramics and the diversity of human ingenuity in ancient societies, with potential implications for contemporary pigment manufacture.

Experimental‐Assisted Approach to Develop a Numerical Model for Simulating the Reaction Propagation in Reactive Multilayers

One outstanding feature of self‐propagating reactions is their ability to release heat of reaction over both temporal and spatial scales, enabling the sustained progression of the reaction after a local ignition. They propagate in the form of a continuous reaction front through the mixture of the starting materials. Previous research on reactive materials has predominantly focused on unraveling the microstructure property relationships influencing released energy in reacting multilayers. This involved considering coupled differential equations, including the heat conduction equation and Fick's law. In this study, the introduction of a purely thermal numerical macroscale model is made, incorporating two states of material properties that differentiate between the thermal characteristics before and after phase formation. The homogenization of material properties before the phase formation is accomplished through the consideration of directional‐temperature‐dependent thermal conductivity and temperature‐dependent‐specific heat capacity. The energy‐release function is derived using experimental data for the reaction velocity depending on bilayer thickness. This model allows for the exploration of reaction motion and temperature profiles, achieving qualitative conformity with experimental measurements for freestanding foil, and necessitating reasonable computational effort.

Hydraulic permeation-induced water concentration gradients in ion exchange membranes

Although substantial literature has detailed transport through swollen membranes, there are still conflicting reports regarding the fundamental physics of hydraulic water transport in such materials, with a long-standing debate centered on the applicability of the pore-flow and solution-diffusion mechanisms. The critical thermodynamic distinction between these models is the presence of a membrane-phase concentration gradient. For pure solvent transport, the pore-flow model presumes that a membrane-phase pressure gradient drives transport in the absence of a concentration gradient. In contrast, the solution-diffusion model proposes that a membrane-phase concentration gradient drives transport and that there is a negligible pressure gradient through the membrane. To address this topic for ion exchange membranes, this study reports measurements of water concentration profiles in pressurized films of Nafion and a commercial membrane based on sulfonated poly (styrene-co-divinylbenzene). Several films were stacked together in a custom-built permeation cell, with the feed side pressurized to between 68.9 and 137.9 bar, and the permeate side held at atmospheric conditions. At steady state, the experiment was stopped, the films were separated, and the water content of each film was measured, revealing a water concentration gradient through the stack thickness that increased with increasing feed pressure. The experimental data were in good agreement with theoretical predictions based on Fick's law and polymer-solvent swelling theories (i.e., Flory-Huggins and Flory-Rehner). These results are consistent with the solution-diffusion model and cannot be explained by a pore-flow mechanism.

Study on the Optimization, Extraction Kinetics and Thermodynamics of the Ultrasound-Assisted Enzymatic Extraction of Tremella fuciformis Polysaccharides.

In this study, Tremella fuciformis polysaccharides (TFPs) were extracted by ultrasound-assisted enzymatic extraction (UAE) at different extraction parameters in order to explore the potential of ultrasound in intensifying the extraction yield. The effects of experimental conditions on the extraction yields were optimized using response surface methodology, with the optimal ultrasonic power of 700 W, temperature of 45 °C and time of 50 min. The kinetic analysis revealed that UAE significantly promoted the dissolution, diffusion and migration with the maximum yield of 26.39%, which was enhanced by 40.45% and 156.96% compared with individual ultrasonic extraction (UE) and enzymatic extraction (EE). According to the modified Fick's second law of diffusion, the extraction process of TFPs illustrated a good linear correlation (R2 ≥ 0.9), and the rate constant gradually elevated as the temperature increased from 25 to 45 °C, while the presence of ultrasound exerted a vital role in extracting TFPs. Regarding to the thermodynamic results, the positive values of ΔH and ΔG demonstrated that UAE, UE and EE were endothermic and unspontaneous processes. This study provides a theoretical basis for polysaccharide extraction processing.

Interactions between methyl octabromo ether flame retardant and expanded polystyrene microplastics in the photoaging process

Expanded polystyrene (EPS) plastic is widely used because of its low density and lightweight properties, enabling it to float on water and increase its exposure to sunlight. In this study, we simulated the photoaging process of flame retardant-added EPS (FR-EPS) and common original EPS (OR-EPS) microplastic (MP) particles with and without methyl octabromoether flame retardant (MOBE) in the laboratory to explore the effect of MOBE on the photodegradation of EPS. Results showed that MOBE accelerated size reduction and surface hole formation on the particles, hastening the shedding and replacement of particle surfaces. FR-EPS particles exhibited a weight loss exceeding that of OR-EPS, reaching 40.85 ± 3.72% after 36 days of irradiation. Moreover, rapid physical peeling of the FR-EPS surface was accompanied by continuous chemical oxidation and fluctuations of the carbonyl index and O/C ratio. A diffusion model based on Fick's second law fitted well for the concentration of MOBE remaining in FR-EPS particles. MOBE's sensitivity to direct photochemical reactions inhibited the early-stage photoaging of EPS MP particles by competing for photons. However, MOBE as chromophores could absorb photons and produce •OH to promote the aging of EPS. Moreover, the capacity of EPS to absorb light energy also accelerated MOBE degradation. These findings suggested that the photoaging behavior of commercial EPS products containing flame retardants in the environment is quite different from that of pure EPS, indicating that additive-plastic interactions significantly alter MP fate and environmental risks.

Chitosan/zein-based sustained-release composite films: Fabrication, physicochemical properties and release kinetics of tea polyphenols from polymer matrix

This study investigated the properties of chitosan/zein/tea polyphenols (C/Z/T) films and analyzed the release kinetics of tea polyphenols (TP) in various food simulants to enhance the sustainability and functionality of food packaging. The results revealed that TP addition enhanced the hydrophilicity, opacity and mechanical properties of film, and improved the compatibility between film matrix. 1.5 % TP film showed the lowest lightness (76.4) and the highest chroma (29.1), while 2 % TP film had the highest hue angle (1.5). However, the excessive TP (above 1 % concentration) led to a decrease in compatibility and mechanical properties of film. The TP concentration (2 %) resulted in the highest swelling degree in aqueous (750.6 %), alcoholic (451.1 %), and fatty (6.4 %) food simulants. The cumulative release of TP decreased to 16.32 %, 47.13 %, and 5.87 % with the increase of TP load in the aqueous, alcoholic, and fatty food simulants, respectively. The Peleg model best described TP release kinetics. The 2 % TP-loaded film showed the highest DPPH (97.13 %) and ABTS (97.86 %) free radical scavenging activity. The results showed TP release influenced by many factors and obeyed Fick's law of diffusion. This study offered valuable insights and theoretical support for the practical application of active films.

A data-driven model for damage evolution of bridge stay cable

Chloride ion concentration is crucial for the reliability of stay cables, particularly when their protective covers are compromised. Previous studies have focused on the corrosion of single wires or cables over time, proposing estimation models that primarily consider corrosion duration while neglecting other significant factors. Addressing this gap, this paper establishes a corrosion damage model for stay cables considering time, temperature, humidity and inclination angle based on the diffusion characteristics of chloride ions. Firstly, We conducted accelerated corrosion experiments on stay cables in different environments to assess the characteristics of the spatial distribution of chloride ion concentrations within the cables and the mass loss characteristics of the wires. Secondly, a three-dimensional diffusion model for chloride ions in stay cables is proposed based on Fick's second law. A pivotal innovation of our research is the application of diverse machine learning techniques to explore how the diffusion coefficient of chloride ions is influenced by factors like time and environmental temperature, leading to the formulation of a sophisticated chloride ion diffusion model. Finally, by amalgamating this diffusion model with the established corrosion law for steel reinforcement, we devised a comprehensive corrosion damage evolution model for the steel wires of stay cables. The result demonstrates the anisotropic diffusion of chloride ions and its substantial correlation with the corrosion damage observed in steel wires. Notably, the PSO-BPNN method emerged as a highly accurate predictor for the diffusion coefficient of chloride ions. The model established in this paper considers a variety of influencing factors and accurately predicts the corrosion damage characteristics of stay cables, which makes an important contribution to improving the reliability of stay cables.

Mechanical properties: Lifespan and retention forecast for jute fiber woven fabric reinforced epoxy matrix composite

AbstractPolymer composites bonded with natural fibers are extensively used for a range of engineering purposes. Preserving their mechanical strength and estimating their maximum service life are thus essential for efficient usage. The primary goals of this effort are to analyze and forecast the longest possible life span of composites. To do this, Fick's law and the Arrhenius principle are employed to analyze the diffusion coefficient and activation energy. Compression molding technology and a manual technique for layup were employed to make the composites reinforced with epoxy laminate and woven mats of jute fiber (JRC). Three layering patterns were used to prepare the laminated composite: 45° angle‐ply laminate [0°/+45°/0°/−45°/0°], 0° balanced laminate [0°/0°/0°/0°/0°/0°], and 30° angle‐ply laminate [0°/+30°/0°/−30°/0°]. In order to examine how aging affected the composites' mechanical properties, the materials were immersed in water for 10, 20, 30, and 40 days throughout the study. The study's conclusions showed that the composite samples' weight rose with age. It was possible to make precise predictions about strength retention by comparing the mechanical characteristics of aged (wet) and unaged composites. It was discovered that the strongest composites were those with a 45‐degree stacking pattern. The activation energy of composites with a 45° angle‐ply laminate layering pattern is maximum, according to the Arrhenius principle. Additionally, using an electron microscope with scanning capabilities (SEM), it was possible to determine the fiber matrix and the tensile cracked surface contact connection.Highlights The widespread usage of polymer composites with natural fibers in engineering applications. Preservation of mechanical strength and predicting the maximum service life of these composites show how important these materials are for engineering efficiency. Fick's rule and the Arrhenius principle are used to analyze diffusion coefficient and activation energy. Laminated composites are made using 45° angle‐ply, 0° balanced laminate, and 30° angle‐ply layering patterns. The impact of simulated aging on composite mechanical characteristics by water immersion for varied periods.

Predicting Diffusion Coefficients in Nafion Membranes during the Soaking Process Using a Machine Learning Approach.

Nafion, a versatile polymer used in electrochemistry and membrane technologies, exhibits complex behaviors in saline environments. This study explores Nafion membrane's IR spectra during soaking and subsequent drying processes in salt solutions at various concentrations. Utilizing the principles of Fick's second law, diffusion coefficients for these processes are derived via exponential approximation. By harnessing machine learning (ML) techniques, including the optimization of neural network hyperparameters via a genetic algorithm (GA) and leveraging various regressors, we effectively pinpointed the optimal model for predicting diffusion coefficients. Notably, for the prediction of soaking coefficients, our model is composed of layers with 64, 64, 32, and 16 neurons, employing ReLU, ELU, sigmoid, and ELU activation functions, respectively. Conversely, for drying coefficients, our model features two hidden layers with 16 and 12 neurons, utilizing sigmoid and ELU activation functions, respectively.

PEA GRAINS IN DRYING: UNRAVELING THE KINETICS OF HOT-AIR DRYING AND EXPLORING MATHEMATICAL MODELS FOR MOISTURE DIFFUSION

Pea drying studies were assessed to learn more about the kinetics and properties of drying in a hot-air dryer. Research was done on impact of temperatures and pre-treatments on drying behaviours. The drying rate graphs demonstrated that the entire drying procedure took place when rates were declining. To properly understand the experimental data, four mathematical models (Henderson & Pabis, Page, Wang & Singh, and Aghbashlo et al.) were used. The Page model was discovered to be the ideal one to depict peas' curves of drying. The identification of the Page model as the most suitable for depicting pea drying curves underscored the applicability in modeling drying behaviors in similar agricultural products. With Fick's second law of diffusion, effective moisture diffusivity (Deff) sorted from 2.45x10-10 to 6.55x10-10 m2/s at given temperature. Deff was expressed as a function of temperature with an Arrhenius type equation. For samples from Potas, Blanch, and Control codes, the activation energy for moisture diffusion was computed as 21.48, 22.82, and 22.32 kJ/mol, respectively. The computation of activation energy for moisture diffusion for different samples offered practical information for optimizing drying processes under various conditions. The results showed the importance of pea drying kinetics and practical implications for industry on drying efficiency and product quality.

Electrochemical Characterization of Diffusion in Polymeric vs. Monomeric Solvents.

Polymer electrolyte was used as a medium for testing the performance of microband electrodes under conditions of linear diffusion. Cyclic voltammetry (CV) and chronoamperometry (CA) experiments were performed in a highly viscous medium, where diffusion rates are much slower than in fluid solutions. The log i vs. log v (CV) or log i vs. log t (CA) relationships with the current equation confirmed the existence of such conditions, yielding slope values that were lower than the expected 0.5. This could indicate an impure linear diffusion profile, i.e., some contribution from radial diffusion (edge effects). However, the desired value of 0.5 was obtained when performing these tests in monomeric solvents of similar viscosities, such as glycerol or propylene glycol. These results led to the conclusion that the current equations, which are based on Fick's laws, may not be applicable for polymer electrolytes, where various obstructions to free diffusion result in a more complicated process than for monomeric solvents.

Temperature-time dependence and mechanisms of redox reaction in Cr-coated Zr alloy cladding during steam oxidation at 900–1250 ℃

The structural integrity of the Cr coating can be significantly affected with increasing temperature during steam oxidation. Herein, the elemental diffusion, layered structure, and kinetics evolution of Cr coating in steam environment of 900–1250 ℃ for 0.5 to 3 h were studied. The oxidation behavior of Cr-coated Zr claddings associated with redox reaction between Cr2O3 and Zr were investigated in detail. Cr2O3 layer forms on Cr coating for steam oxidation at 900 ℃ and Cr-Zr diffusion layer appears at interface between Cr coating and Zr substrate when the samples are subjected to steam oxidation at 1000–1150 ℃ for 3 h. The starting time for redox reaction decreases dramatically when the temperature is higher than 1150 ℃. A prediction modal based on Fick's Law was put forward to predict the starting time of redox reaction for 1100 to 1250 ℃. The prediction is consistent with the experimental results. Furthermore, the initial process of redox reaction was investigated and analyzed in detail. Predominated by ion transportation within residual Cr layer, O2- from Cr2O3 could diffuse along the ZrO2 particles at Cr coating boundaries to the Cr-Zr layer, resulting in the reduction of Cr2O3 and partial oxidation of diffusion layer.