Fick's Diffusion Equation Research Articles

Effect of hetero-structured nano-particulate coating on the oxygen surface exchange properties of La0.6Sr0.4Co0.2Fe0.8O3-δ

In this study, dense La0.6Sr0.4Co0.2Fe0.8O3-δ (LSCF) electrodes decorated with the novel hetero-structured ceramic oxide mixture in four different ratios of Ce0.8Gd0.2O2-δ (GDC) and La2Mo2O9 (LMO). The time-dependent conductivity transients were acquire using electrical conductivity relaxation (ECR) technique at a chosen conditions of temperature in the range of 650–850 °C and instantaneous pO2 step change between 0.2 and 0.8. Fitting of time-dependent conductivity to the appropriate non-equilibrium solutions of Fick's diffusion equation has yielded the chemical diffusion coefficient, Dchem, and oxygen surface exchange coefficient, kchem. As expected, the Dchem of the coated samples remained invariant, whilst the kchem is found to vary with the change in GDC-LMO coating mixture ratio. Substantial increase of a factor of 10 in the surface exchange coefficient is noticed for the LSCF coated with a 1:0.75 mixing ratio as compared to bare sample at 850 °C. The enhancement in kchem is attributed to the optimal triple-phase boundary (TPB) regions which promotes oxygen surface exchange kinetics. Thus, coating of selective ratio of hetero-structured oxide in a form of nano-particulate layer over the LSCF surface is considered to be a promising candidate for solid oxide fuel cell (SOFC) cathode.

Properties of nisin incorporated ZrO2/poly (vinyl alcohol)-wheat gluten antimicrobial barrier films

ABSTRACTThe properties of manufactured nisin-ZrO2/poly (vinyl alcohol)-wheat gluten (ZPWG) films were characterized. The formation of hydrogen bonds among the PVA, WG, and ZrO2 units were detected and caused a decrease in the film crystallinity. Also, the degradation rate was increased with WG content. As expected, the addition of WG and ZrO2 influenced the light transmission, tensile properties, gas permeability and water vapor permeability of the films. Nisin-loaded films presented a low initial burst release of the antimicrobial agent, followed by a gradual release. The initial diffusion of nisin (Mt/M∞<2/3) from the ZPWG film was formulated by the Fickian diffusion equation, showing a linear relationship (R2 > 0.998) between Mt/M∞ and t1/2, and the diffusion coefficient D increased with increasing WG content. Antimicrobial activity testing of nisin-ZPWG films with 40 wt% WG showed excellent inhibitory activity against Staphylococcus aureus, indicating its potential as a new active food packaging material.

Thin Layer Drying Kinetics Modelling of Curry Leaves (Murraya koenigii L.) in Cabinet Dryer

Curry leaves (Murraya koenigii L.) are the sweet smelling leaves of small tree of Rutaceae family native to Southwest Asia. In this study, the effect of temperatures (50, 55 and 60°C) on the drying of curry leaves was investigated. The experimental data were fitted to six thin layer mathematical models (Newton, Page, Handerson and Pabis, logarithmic, two-term exponential and Midilli et al). The models were evaluated in terms of coefficient of determination (R2), chi square (χ2) and root mean square error (RMSE). The Midilli et al model was best fitted to the experimental data of all the models evaluated. The effective diffusivity was calculated using Fick's diffusion equation, and the value varied from 2.07×10-12 m2/s to 2.643×10-12 m2/s. The activation energy and the diffusivity constant were found to be 21.808 kJ/mol and 4.667×10-8 m2/s respectively.

Non-Fickian Transport in Porous Media: Always Temporally Anomalous?

The methanol transport through the porous media, e.g., pelletized zeolite-based catalyst, was investigated. Standard Fickian diffusion equation was found to be inapplicable to a description of the experimental mass transfer kinetics. The diffusion equation with the fractional-order temporal and spatial derivatives was used in order to fit the experimental data. It is demonstrated that only the diffusion equation, which contains the time-fractional derivative or both, space–time-fractional derivatives, may thoroughly describe the experimental data. The measured time-fractional order is 0.85, which reveals the non-Fickian sub-diffusive transport through the pellet. The obtained fractional order allows the application of the continuous-time random walk model as a physical ground for the explanation of the measured non-Fickian transport. Contrary, using the space-fractional diffusion equation for the experimental data description leads to the inconsistent results similarly to the standard diffusion equation. The obtained findings reveal that the diffusate flux and the flux conservation law cannot be purely space fractional.

Preparation and Characterization of Porous Core-Shell Fibers for Slow Release of Tea Polyphenols.

This study focused on the fabrication, characterization, and release properties of electrospun tea polyphenol (TPP) loaded porous core-shell structured fibers. The morphology, structure and properties of the electrospun TPP loaded porous core-shell fibers were investigated by a combination of Fourier transformation infrared spectroscopy (FTIR), scanning electron microscopy (SEM), contact angle (CA) measurements, transmission electron microscopy (TEM), etc. In addition, the cumulative drug release rate of TPP loaded porous core-shell fibers was determined by ultraviolet (UV) spectrophotometer, and the release mechanism was investigated using Fickian diffusion equation, which would provide the theoretical basis for future study. The results showed TPP loaded porous core-shell structured fibers were successfully prepared by coaxial electrospinning, and the porous structure of the core-shell fibers could further enlarge the specific surface area, enhance the hydrophobic properties, and improve the drug release properties.

Water sorption equilibria and kinetics of henna leaves

In this work, firstly the sorption isotherms of henna leaves were determined using a dynamic vapor sorption (DVS) device at 3 temperatures (30, 40, 50 °C). The equilibrium data were well fitted by the GAB model. Secondly, drying kinetics were measured using a pilot convective dryer for 3 air temperatures (same as above), 3 velocities (0.5, 1, 1.42 m/s) and 4 relative humidities (20, 30, 35, 40%). The drying kinetic coefficients were identified by fitting the DVS and pilot dryer data by Lewis semi-empirical model. In order to compare the obtained kinetic parameters with literature, the water diffusivities were also identified by fitting the data by the simplified solution of fickian diffusion equation. The identified kinetic coefficient was mainly dependent on air temperature and velocity what proved that it represented rather the external transfer and not the internal one.

Kinetics of Gelatinized White Yam (Dioscorea Rotundata, Poir) During Convective Drying

Gelatinized white yam cubes, having a moisture content of 196% dry basis were dried in a convective dryer under different conditions of air temperature (40, 50, 60 and 70°C) and relative humidity (20 - 50%). There was no constant rate period throughout the entire drying period as drying took place entirely during a falling rate period. The effect of temperature was more pronounced than that of relative humidity. The drying data were fitted to five thin-layer drying models. The goodness of fit of the models were evaluated by comparing the percent mean relative deviation modulus (E%), RMSE, χ2 and R2 between their observed and predicted moisture ratio. The Binomial approximation of Fick's diffusion equation gave the best fit to the drying data as the highest values of R2 and the lowest values of χ2 and RMSE were consistently obtained with the Binomial model equation.

A formal description of middle ear pressure-regulation

IntroductionMiddle ear (ME) pressure-regulation (MEPR) is a homeostatic mechanism that maintains the ME-environment pressure-gradient (MEEPG) within a range optimized for “normal” hearing. ObjectiveDescribe MEPR using equations applicable to passive, inter-compartmental gas-exchange and determine if the predictions of that description include the increasing ME pressure observed under certain conditions and interpreted by some as evidencing gas-production by the ME mucosa. MethodsMEPR was modeled as the combined effect of passive gas-exchanges between the ME and: perilymph via the round window membrane, the ambient environment via the tympanic membrane, and the local blood via the ME mucosa and of gas flow between the ME and nasopharynx during Eustachian tube openings. The first 3 of these exchanges are described at the species level using the Fick's diffusion equation and the last as a bulk gas transfer governed by Poiseuille's equation. The model structure is a time-iteration of the equation: PMEg(t=(i+1)Δt) = ∑s(PMEs(t=iΔt)+(1/(βMEsVME)∑P(ҚPs(PCs(t=(iΔt)-PMEs(t=(iΔt))). There, PMEg(t=iΔt) and PMEs(t=iΔt) are the ME total and species-pressures at the indexed times, PCs(t=iΔt) is the species-pressure for each exchange-compartment, βMEsVME is the product of the ME species-capacitance and volume, ҚPs is the pathway species-conductance, and ∑S and ∑P are operators for summing the expression over all species or exchange pathways. ResultsWhen calibrated to known values, the model predicts the empirically measured ME species-pressures and the observed time-trajectories for total ME pressure and the MEEPG under a wide variety of physiologic, pathologic and non-physiologic conditions. ConclusionsPassive inter-compartmental gas exchange is sole and sufficient to describe MEPR.

Fractional derivatives in the diffusion process in heterogeneous systems: The case of transdermal patches

In this note, we present a simple mathematical model of drug delivery through transdermal patches by introducing a memory formalism in the classical Fick diffusion equation based on the fractional derivative. This approach is developed in the case of a medicated adhesive patch placed on the skin to deliver a time released dose of medication through the skin towards the bloodstream.The main resistance to drug transport across the skin resides in the diffusion through its outermost layer (the stratum corneum). Due to the complicated architecture of this region, a model based on a constant diffusivity in a steady-state condition results in too simplistic assumptions and more refined models are required.The introduction of a memory formalism in the diffusion process, where diffusion parameters depend at a certain time or position on what happens at preceeding times, meets this requirement and allows a significantly better description of the experimental results.The present model may be useful not only for analyzing the rate of skin permeation but also for predicting the drug concentration after transdermal drug delivery depending on the diffusion characteristics of the patch (its thickness and pseudo-diffusion coefficient).

Diffusion of a mono-energetic implanted species with a Gaussian profile

The implanted profile in an isotropic substrate of a mono-energetic ion species is usually very near a Gaussian profile. An exact solution to the time-dependent Fick diffusion equation of an initially Gaussian profile is presented. This solution is a general one also covering the diffusion within the two limiting cases usually considered in solutions to the Fick equation, viz. a perfect sink at the surface and a perfectly reflecting surface plane at the surface. An analysis of the solutions for these two cases shows that at small diffusion times the main effect of annealing is a nearly symmetric broadening of the implanted profile. At the origin and for longer diffusion times the profile deviates significantly from Gaussian. A review is also given of past attempts to extract diffusion coefficients by fitting experimental data to approximate equations based on simplified initial profiles.

Mathematical model for dehydration and shrinkage: Prediction of eggplant's MRI spatial profiles

In the preceding work, Brasiello et al. (2013) developed a mathematical model for eggplant's dehydration taking into account the shrinkage effects. Although the model provided good results in predicting the total water weight loss of slices, at that time, no information was given about the capability of predicting water content profiles inside the material. This is a very important issue in order to catch some fundamental aspects of dehydration with shrinkage. In this paper, we analyze the evolution of the water content profiles inside cylindrical samples of eggplant during dehydration using a mathematical model which takes into account shrinkage effects. In particular, we show the good agreement between numerical derived spatial profiles and the Magnetic Resonance Imaging (MRI) data available in Literature. Model's parameters are calculated through a suitable simplified procedure, which uses experimental data from samples of a different shape. The numerical results are compared to those derived from the same procedure applied to Fick's diffusion equation.

Water Diffusion from a Bacterial Cell in Low-Moisture Foods.

We used a Fick's unsteady state diffusion equation to estimate the time required for a single spherical shaped bacterium (assuming Enterococcus faecium as the target microorganism) in low-moisture foods to equilibrate with the environment. We generated water sorption isotherms of freeze-dried E. faecium. The water activity of bacterial cells at given water content increased considerably as temperature increased from 20 to 80 °C, as observed in the sorption isotherms of bacterial cells. When the water vapor diffusion coefficient was assumed as between 10(-12) and 10(-10) m(2) /s for bacterial cells, the predicted equilibration times (teq ) ranged from 8.24×10(-4) to 8.24×10(-2) s. Considering a cell membrane barrier with a lower water diffusion coefficient (10(-15) m(2) /s) around the bacterial cell with a water diffusion coefficient of 10(-12) m(2) /s, the teq predicted using COMSOL Multiphysics program was 3.8×10(-1) s. This result suggests that a single bacterium equilibrates rapidly (within seconds) with change in environmental humidity and temperature.

Oxygen diffusion capacity scales as oxygen consumption

Metabolic scaling is one of the central enigmas of comparative biology. Oxygen consumption, as a proxy for aerobic metabolism, does not scale proportionally with body mass. Expressed as an exponential function, oxygen consumption scales as mass to the 2/3–8/10 power. This fundamental principle explains why the appropriate drug dose per kilogram of pet cat is higher than that of an average adult human, and why the cat's resting heart rate is over 140 beats min−1 when my own is closer to 70 beats min−1, and a large whale's is closer to 15 beats min−1. The scaling pattern holds in endotherms and ectotherms alike. Yet, in spite of the tremendous utility of metabolic scaling, the cause of this phenomenon remains elusive.James Gillooly and colleagues at the University of Florida, Gainesville, used another fundamental principle of comparative physiology to look into one possible driver of metabolic scaling: Fick's law of diffusion. Passive diffusion of oxygen into the body may be a rate-limiting step in aerobic metabolism and, as the first step in the process, seemed a likely candidate. According to Fick's law, the rate of passive diffusion across a membrane is proportional to the surface area of that membrane, and inversely proportional to the thickness of the membrane. Given this relationship, if you know how membrane surface area and thickness relate to mass, you can relate passive diffusion capacity to mass directly.Gillooly and his team compiled data on respiratory membrane surface area and thickness for a range of vertebrates, including animals from all major vertebrate classes, with representatives from all ecosystems and all forms of locomotion. This is particularly remarkable for the amphibians included in the study, as they obtain oxygen through their mouths and skin in addition to their lungs. Once the team had this information, they found that respiratory surface area scales with mass strongly, whereas membrane thickness changes little with body size.Using the scaling exponents calculated from the published data, and estimates of the physiological properties of respiratory membranes, the researchers applied Fick's diffusion equation to model how passive oxygen diffusion capacity varies with mass. Doing so revealed that the relationship between oxygen diffusion capacity and body size was statistically indistinguishable from the relationship between resting oxygen consumption and metabolic rate for both endotherms and ectotherms. In other words, the body's capacity for passive oxygen diffusion precisely matches its resting metabolic oxygen requirements. This supports the idea of symmorphosis, a hypothesis suggesting that no one bodily system is over-engineered relative to the others.The researchers noted that the match between resting oxygen consumption rate and diffusion capacity is inconsistent with the idea that respiratory systems evolved to optimize performance at maximum activity levels: it would appear instead that the respiratory surface is acting at capacity during resting activity and that the jump between resting and active metabolic rate could be made by modifying the oxygen partial pressure gradient over the respiratory surface, consistent with their model.It may be a jump to suggest that oxygen diffusion capacity is the rate-limiting step in oxygen consumption, but Gillooly and colleagues present intriguing evidence of a structure–function match: a step in the right direction on the path to explaining the existence of metabolic scaling. Of course, only time will tell whether the symmorphosis hypothesis will hold for other components of the metabolic pathway.

Effect of drying temperature on the rheological characteristics of dried seedless grapes

The grape is the second most widely cultivated fruit in the world after the orange; it is of great commercial interest. The Sultana variety is grown in large quantities in Tunisia and in the Mediterranean countries. The product is easily deformable; the thermo physical properties depend on the water content. By using the static gravimetric method, the sorption isotherms for sultana grapes were determined. By using the experimental drying kinetics and adopting a numerical procedure for solving the Fick's diffusion equation, the mass diffusion coefficient was determined as a function of the temperature and water content of the product. Using the constraint relaxation test, with a texture analysis machine, the mechanical properties of grains of dried grapes were measured. The relaxation function and the Young's modulus were then deduced. These characteristics are key parameters in running simulation models addressing the thermo hydro mechanical behavior and allow the appropriate process and drying time to be subsequently determined in order to avoid unnecessary energy consumption.

Isothermal Langevin dynamics in systems with power-law spatially dependent friction.

We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially dependent diffusion coefficient of the form D(x)∼|x|^{c}, at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At long times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for c<0 and superdiffusion for 0<c<1. For c>1, the diffusion equation predicts that the particles accelerate. Here we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit (〈x^{2}〉∼t^{2}). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin and is correctly captured by Langevin's equation.

Three-Dimensional Simulations of the Electrothermal and Terahertz Emission Properties ofBi2Sr2CaCu2O8Intrinsic Josephson Junction Stacks

Stacks of intrinsic Josephson junctions (JJs) in cuprate superconductors are promising emitters of terahertz light, which is sought for many applications in imaging, spectroscopy, and communication. The authors simulate JJ stacks in three dimensions, simultaneously solving a Fickian diffusion equation for heat and coupled sine-Gordon equations for current. They predict, and confirm experimentally, that applying a modest magnetic field to a stack's long side enhances emission significantly---one more step toward efficient devices.

Study of impedance model and water transport behavior of modified solvent-free epoxy anticorrosion coating by EIS

This work studied the evolution of an impedance model of 60 μm modified solvent-free epoxy anti-corrosion coating on a Q235 steel surface in 3.5% NaCl solution using electrochemical impedance spectroscopy. The electrochemical process of the system was divided into five stages. During the early stage of immersion, water absorption occurred mainly at the coating, coating resistance decreased, and coating impedance deviated from purely capacitive characteristics. The corrosion reaction started after water permeated into the metal/coating interface. During the mid-immersion stage, the electrochemical reaction at the coating/metal interface was controlled by semi-infinite diffusion of corrosion products due to the barrier effect of the coating. The types of corrosion product diffusion gradually became finite layer diffusion and barrier layer diffusion with the clogging of coating pores. Logarithm of coating capacitance and the square root of time showed a linear relationship in the early immersion stage, which was a typical characteristic of Fick’s diffusion. Afterwards, increase in the coating capacitance slowed down, and a non-Fickian diffusion process occurred, which presented the two-stage absorption characteristics. Water diffusion coefficient in coating was calculated to be 2.95 × 10−10 cm2/s, while volume fraction and total water absorption at saturation of coating were 2.3% and 185 μg, respectively, indicating good water resistance and protective properties of the coating. The water kinetic equation in the coating which can reflect the whole Fick diffusion process of water and containing time and location variables was obtained via the Fick diffusion equation and three-dimensional images of water distribution in the coating were drawn by transforming the dynamic equation into programs.

Ethylene production, respiration and gas exchange modelling in modified atmosphere packaging for banana fruits

SummaryA mathematical model was developed from experimental measurements to describe the evolution of the O2, CO2 and ethylene in a modified atmosphere packaging system for Cavendish bananas. The respiration and ethylene production in the fruits were experimentally obtained from a closed system method and then represented by Michaelis–Menten equations of enzyme kinetics. The gas transfer through the packaging was described by a Fick's diffusion equation, and the temperature dependence was represented based on the Arrhenius law. The model was validated by packaging the fruit in perforated bags of polypropylene and low density polyethylene at 12 °C for a period of 8 days. With the developed model it was possible to satisfactorily describe the experimental evolution of the gas content in the headspace of the packages, obtaining coefficients of determination (R2) of 0.93 for the O2 levels, 0.90–0.91 for the CO2 levels, and 0.89–0.93 for the ethylene levels.

The development of a new LDF mass transfer correlation for adsorption in fixed beds

In this study, a general LDF model has been introduced to predict mass transfer rate through adsorbents with the macropore diffusion as the controlling step. Using this relation eliminates the need for solving the time-consuming diffusion equation to find mass transfer rate through the porous adsorbent. The proposed relation was successfully applied in the general mathematical model for an adsorption fixed bed. This correlation was adjusted to be capable of predicting the mass transfer rate in a wide range of gas adsorption systems reported in the literature. This correlation was used in 21 different adsorbent and adsorbate systems. The results demonstrated an excellent agreement between the correlation results and those obtained using Fickian diffusion equation. By applying the developed LDF model instead of diffusion model, a great deal of CPU time can be saved. The latter characteristic will be very important when this model is employed in commercial software such as Aspen Adsorption or Prosim Dynamic Adsorption Column.

A Fast Gradient Projection Method for a Constrained Fractional Optimal Control

Optimal control problems governed by a fractional diffusion equation tends to provide a better description than one by a classical second-order Fickian diffusion equation in the context of transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significantly increased computational complexity and storage requirement than the corresponding classical control problem, due to the nonlocal nature of fractional differential operators. We develop a fast gradient projection method for a pointwise constrained optimal control problem governed by a time-dependent space-fractional diffusion equation, which requires the computational cost from $$O(M N^3)$$O(MN3) of a conventional solver to $$O(M N\log N)$$O(MNlogN) and memory requirement from $$O(N^2)$$O(N2) to O(N) for a problem of size N and of M time steps. Numerical experiments show the utility of the method.