Determination Of Diffusion Coefficients Research Articles

Determination of diffusion coefficients through gels with non-negligible finite-volume effects in the compartments of the diffusion cell

Diffusion cells are used to measure diffusion coefficients (DM) in gels. These measurements are of interest to understand and predict the availability of nutritive or toxic chemical species in waters, soils and sediments. When the diffusive flux from the donor to the acceptor compartment is constant (steady-state regime), DM is determined from the slope of the linear plot of the acceptor concentration vs time. However, at long enough times, there is a non-negligible concentration depletion in the donor compartment concomitant to a concentration increase in the acceptor compartment. Accordingly, the accumulation plot bends downwards preventing a linear fitting. This is the case of metals whose solubility (especially depending on pH values) limits the concentration in the donor compartment and the time required to reach concentrations above the limit of quantification in the acceptor compartment implies a non-negligible decrease of the concentration in the donor compartment.In this work, a simple linear regression is shown to provide the diffusion coefficient values from experiments exhibiting finite-volume effects. This expression is validated against rigorous numerical simulation as well as reported values in the literature. Diffusion coefficients of Zn, Ni and Pb in agarose cross-linked polyacrylamide (APA) gels (used in Diffusive Gradients in Thin-Film devices, DGT) are determined under finite-volume effects. The resulting values agree with those obtained under the standard linear regime.

Diode Laser Absorption Spectroscopy and DSMC Calculations for the Determination of Species-Specific Diffusion Coefficients of a CO2-N2O Gas Mixture in the Transition Gas Regime

Multicomponent gas mixture diffusion in a microscale confined flow in the transition gas regime at Knudsen numbers (Kn) above 0.1 has potential engineering applications in gas-phase microfluidics. Although the calculation of the diffusion coefficient accounts for the influence of the concentration of other species in a multicomponent gas mixture, the higher rate of gas-wall collision at 0.1 < Kn ≤ 10 introduces additional complications not predicted by conventional calculation methods. Thus, simultaneous measurement of diffusion coefficients for multiple gas species ensures accurate estimation of the diffusion coefficient of a particular species that includes the effect of interactions with other species and wall surface conditions in a multicomponent gas mixture at Kn > 0.1. However, most experimental methods for measuring the diffusion coefficient are not species-specific and therefore cannot directly differentiate between the species diffusing in a gas mixture. Thus, this paper demonstrates a new experiment methodology consisting of a two-bulb diffusion configuration accompanied by a tunable diode laser absorption spectroscopy detection technique for species-specific, in-situ, simultaneous measurement of the effective diffusion coefficient for a CO2-N2O gas mixture in the transition gas regime. The experimental results are compared against direct simulation Monte Carlo calculations and the Bosanquet approximation showing a deviation that has not been reported in the literature before.

Experimental and modeling approaches to determine drug diffusion coefficients in artificial mucus

Asthma, usually characterized by inflammation and mucus accumulation, causes restricted airflow and impaired lung function. The physiological and biochemical characteristics of mucus pose a strong barrier for drugs administered orally or via the pulmonary route for asthma treatment. In this study, two drugs commonly employed in the treatment of asthma, theophylline and albuterol, were placed in contact with an artificial mucus layer, measuring their interface concentrations, and modeling the concentration profiles to determine their diffusion coefficients. To monitor the diffusion process, the upper surface of a mucus layer was placed in contact with the drug solutions and the lower mucus surface was in contact with a zinc selenide crystal to allow for time-resolved Fourier transform infrared spectroscopy (FTIR) measurements. FTIR spectra were collected at constant time intervals and monitored for quantitative changes in spectral peaks corresponding to functional groups specific to each of these drugs. Changes in peak heights were correlated to concentration via Beer's Law. Fick's 2nd Law of Diffusion was used along with Crank's trigonometric series solution for a planar semi-infinite sheet to analyze the concentration data and determine diffusion coefficients. Using this method, fitting the experimental data resulted in diffusivity coefficients of D = 6.56 x 10−6 cm2/s for theophylline and D = 4.66 x 10−6 cm2/s for albuterol through artificial mucus. The drug diffusivity coefficients align closely with literature reports, wherein, diffusivity data was obtained experimentally using a rotating-disk apparatus and intrinsic dissolution technique. By coupling analytical and experimentally determined drug diffusion data, this approach provides a fast, non-invasive method for quickly assessing drug diffusion profiles through complex media.

Elucidating the separation role of C3N slit-pore on N2/CH4 in coalbed methane

A hybrid approach integrating molecular dynamics (MD) and Grand Canonical Monte Carlo (GCMC) simulations was employed to investigate the separation efficacy of N2/CH4 across various C3N slit apertures. MD simulations enabled the determination of separation selectivity and diffusion coefficients of N2/CH4, thereby aiding in the elucidation of the separation mechanism. On the other hand, GCMC simulations provided insights into the adsorption isotherms and isosteric heats of N2 and CH4 within the C3N slit. These findings reveal that the highest selectivity of N2 over CH4 occurs at the zigzag-edge C3N with a pore size of 6.2 Å. However, this selectivity is accompanied by the smallest diffusion coefficient, attributable to size sieving effects. As the pore diameter increases, more gas molecules can permeate into the pores, leading to a rise in both the maximum adsorption capacity and the isosteric heat of methane adsorption. With increasing pressure, gas molecules tend to reach saturation at higher pressure levels. Our study primarily centers on examining the separation of N2/CH4 across various C3N slit-pores, encompassing both adsorption and diffusion phenomena. These findings offer theoretical insights and comprehension for the experimental separation of coalbed methane utilizing cheaper carbon-based materials.

Determination of intracrystalline diffusion coefficient of organic solvents in metal organic framework adsorbents by zero-length column method

Determination of intracrystalline diffusion coefficient of organic solvents in metal organic framework adsorbents by zero-length column method

Preventing the impact of solute adsorption in Taylor dispersion analysis: Application to protein and lipid nanoparticle analysis

Taylor Dispersion Analysis (TDA) allows diffusion coefficient (D) or hydrodynamic radius (Rh) determination on a wide range of size between angstroms and about 300 nm. However, solute adsorption phenomena can affect the repeatability and reproducibility of TDA. Several numerical studies addressed the theoretical impact of solute adsorption in TDA, but very few experimental studies focus on this topic and no experimental methodologies were proposed so far to reduce the impact of adsorption in TDA. In this work, an experimental protocol, called plug-in-front TDA, consisting of adding the solute in the eluent at a lower concentration compared to the injected sample, was proposed to strongly limit the impact of adsorption on the Rh determination. This protocol was suggested based on the evidence that adsorption / desorption phenomena impacting narrow bore fused silica TDA in aqueous conditions are typically slow processes that can be counteracted by saturating the interaction sites during the experiments. Successful applications to proteins and mRNA lipid nanoparticles (LNP) in vaccine against Covid 19 and protein analysis were reported. TDA of proteins in conditions of strong interactions with the capillary surface was possible using the plug-in-front methodology. We anticipate that such experimental methodology will greatly help the experimentalist for implementing TDA in various applications.

Determination of Diffusion Coefficients of Bisphenol A (BPA) in Polyethylene Terephthalate (PET) to Estimate Migration of BPA from Recycled PET into Foods

Bisphenol A (BPA) is a known substance that is found in food contact materials as an intentionally added as well as a non-intentionally added substance. Traces of BPA were found as a non-intentionally added substance in recycled PET (rPET). In 2023, the EFSA proposed a new TDI of 0.0002 µg/kg bw/d, which is lower than the previous (temporary) TDI of 4 µg/kg bw/d by a factor of 20,000. The TDI of 0.0002 µg/kg bw/d would translate for a default 60 kg person eating one kilogram of food into a migration limit of 0.012 µg/kg in the food. This very low migration limit is a challenge to measuring BPA levels in food. A solution is to use migration modeling to establish maximum concentrations in rPET for different food contact applications. Precise diffusion coefficients for BPA in PET were determined within this study by use of migration kinetics. In June 2024, the European Commission proposed a new migration threshold limit for BPA of 1 µg/kg, which should be understood as a detection limit. From the results of this study, it can be concluded that a BPA concentration in the PET bottle wall of 297 mg/kg (3% acetic acid), 255 mg/kg (10% ethanol), and 192 mg/kg (20% ethanol) after storage for 365 d at 25 °C is in compliance with the migration threshold limit of 1 µg/kg. These maximum concentrations are far above the measured BPA concentrations on rPET bottles in Europe between 2019 and 2023. Therefore, the new proposed migration threshold limit for BPA cannot be exceeded.

Influence of temperature and dry density coupled effects on HTO, 36Cl, 85Sr and 133Ba diffusion through compacted bentonite

This study investigated the combined impact of temperature (from 25 to 80 °C) and dry density of bentonite (from 1.2 to 1.65 g·cm−3) on the apparent diffusion coefficient, Da, of four radionuclides (HTO, 36Cl, 85Sr and 133Ba) in a Ca-Mg bentonite. In this study, the different porewater chemistry, present under the specific experimental conditions -- a factor often overlooked in diffusion studies -- was explicitly considered. As a support to diffusion studies, batch sorption tests were carried out at different temperatures (25, 40, 60, and 80 °C) with the sorbing elements (85Sr and 133Ba).The planar source method was used for the determination of apparent diffusion coefficients, Da, which demonstrated its efficacy for both conservative (HTO and Cl) and sorbing elements. For all the investigated radionuclides, Da values decreased with increasing the clay dry density and increased with increasing temperature. Notably, the highest Da value (1·10−9 m2 s−1) was attained for HTO diffusion in the clay at 1.2 g cm−3 and 80 °C, while the lowest Da (3.6·10−12 m2 s−1) value was determined for Ba diffusion at 1.65 g cm−3 and 25 °C.The experimental Da values were analysed employing the Arrhenius law and the Stokes-Einstein equations. Results revealed that diffusion data are in a reasonable agreement with Arrhenius behaviour, but deviations from the Stokes-Einstein equation were observed for compaction densities higher than 1.4 g cm−3.

Electrode reactions on the aluminum electrode in 1-butyl-3-methylimidazolium chloride chloroaluminate ionic liquid: A comprehensive study

Mechanisms of Al electrochemical deposition/dissolution in acidic AlCl3–1-butyl-3-methylimidazolium chloride (BMImCl) ionic liquids are studied. Stationary polarization curves (SPCs) are obtained in the anodic range of potentials. The anodic SPCs are characteristic of electrode processes, which are complicated by passivation phenomena. The peaks intensity of AlCl4−, Al2Cl7− ions were obtained by in-situ Raman spectroscopy during the aluminum electrode polarization and correlated with the behavior of the SPCs. Based on the results obtained, a mechanism of electrochemical oxidation of aluminum in AlCl3–BMImCl IL is proposed. According to the mechanism in question, the anodic process is complicated by the presence of a following chemical reaction involving AlCl3. Chronopotentiometric studies were carried out to determine diffusion coefficients of Al2Cl7− anion, which are unaffected by the change in the molar ratio of AlCl3 to BMImCl (N). The activation energy (Ea) for the Al2Cl7− diffusion coefficient is independent of N and equals 14.5 ± 0.4 kJ mol−1. Exchange current density (i0) at the Al | AlCl3–BMImCl interface was determined using impedance spectroscopy for the compositions with 1.1 ≤ N ≤ 2.0 at temperatures from 303 to 386 K. The i0 increases from 0.54 ± 0.13 to 2.73 ± 0.22 mA cm−2 with the concentration of the electrochemically active Al2Cl7− anion in the concentration range indicated above at 303 K. The data on i0 is used to calculate the transfer coefficient of cathodic reaction (α = 0.18 ± 0.02), the electrochemical reaction rate constant (ks= (1.1 ± 0.3)∙10−6 cm s−1 (at T = 303 K)) and the Ea for the ks (Eaks = 36.5 ± 0.9 kJ mol−1), which do not depend on the IL composition.

Predicting Effective Transport Properties of Randomly Oriented Polycrystalline Li-Ion Battery Electrodes

Widely practiced Li-ion battery models, such as pseudo-two-dimensional (P2D) models [1-3], use representative spherical electrode particles with assumed isotropic transport properties. Electrode particles, such as NMC, are typically composites that are comprised of numerous randomly oriented crystallites. Each of the single-crystal primary particles that compose secondary particle can have strongly anisotropic transport properties that affect the intercalation process. The empirically approximated effective properties (e.g., diffusion coefficients) as reported in literature vary by as much as six orders of magnitude. In many cases, the effective diffusion coefficients are determined as a part of a parameter-fitting procedure to represent battery polarization measurements.The present approach develops both analytical and computational homogenization methods that derive effective properties based upon the intrinsic single-crystal properties of the crystallites. Homogenization techniques are computationally effective methods to predict macroscale behavior based on microscale properties.As illustrated in Fig. 1, the computational approach begins by assembling over 1000 randomly oriented single-crystal primary particles, with each crystallite having anisotropic properties associated with the crystal lattice. The dodecahedron-shaped primary particles are tightly assembled into a secondary particle that is approximately spherical. A three-dimensional finite-element model predicts transient Li-intercalation fractions throughout a secondary particle during charge or discharge processes. In graphite, the transverse and normal diffusion coefficients differ by approximately six orders of magnitude. Based on the cut plane in Fig. 1, and despite extreme anisotropy within the crystallites, the homogenized macroscopic behavior is nearly isotropic.The analytical approach extends a self-consistent method (SCM) for polycrystalline materials first presented by Ponte-Castañeda and Willis [4], which mathematically combines tensors describing diffusion pathways for every primary particle. The present study considers the concentration-dependent and anisotropic diffusion parameters collected by Persson et al. [5] and Zhou et al. [6] for graphite and NMC, respectively. Another homogenization method is the use of an orientation distribution function (ODF) [7], which predicts the effective diffusion parameters based on the statistical average of every possible primary particle orientation. Fig. 2 shows the predicted effective diffusion coefficients bounded by Hashin-Shtrikman bounds for anisotropic composites proposed by Willis [8] and the experimentally determined diffusion coefficients. The predicted effective diffusion coefficients are lower than the experimentally determined values in transverse direction but are significantly larger than the values in the normal direction. Additionally, the in-plane diffusion dominates the curvature of the predicted effective properties.[1] M. Doyle, T. F. Fuller, and J. Newman. Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J. Electrochem. Soc., 140:1526–1533, 1993.[2] T. F. Fuller, M. Doyle, and J. Newman. Simulation and optimization of the dual lithium ion insertion cell. J. Electrochem. Soc., 141:1–10, 1994.[3] M. Doyle, J. Newman, A. S. Gozdz, C. N. Schmutz, and J.-M. Tarascon. Comparison of modeling predictions with experimental data from plastic lithium ion cells. J. Electrochem. Soc., 143:1890–1903, 1996[4] P.P.- Castañeda and J.R. Willis. The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids, 43:1919–1951, 1995[5] K. Persson, V. A. Sethuraman, L. J. Hardwick, Y. Hinuma, Y. S. Meng, A. van der Ven, V. Srinivasan, R. Kostecki, and G. Ceder. Lithium diffusion in graphitic carbon. J. Phys. Chem. Lett., 1:1176–1180, 2010[6] H. Zhou, F. Xin, B. Pei, and M. S. Whittingham. What limits the capacity of layered oxide cathodes in lithium batteries? ACS Energy Lett., 4:1902–1906, 2019[7] S. Nemat-Nasser, M. Hori, and J. D. Achenbach. Micromechanics: Overall properties of heterogeneous materials. In Micromechanics, volume 37. Elsevier Science I& Technology, The Netherlands, 1993[8] J.R. Willis. Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids, 25:185–202, 1977 Figure 1

A novel methodology for the diffusion coefficient determination in porous media by X-ray high-resolution radiographies

The first step in diagnosing a patient with cancer is the biopsy and the study of the tissue sample taken, suitably processed, is essential for a correct diagnosis. Tissue processors are machines that automatically perform the entire processing protocol, often relying on empirical timing related to chemical diffusions in the biological samples. The determination of the diffusion coefficient in porous materials, such as histological tissue, is still under study with multiple methods often resulting in inaccurate predictions and with few amounts of literature data for biological samples. The authors present an experimental technique, based on a multitude of X-ray radiographs acquired over time, where the intensity of the attenuation to the ionising beam is proportional to the concentration of the solvent that diffuses in the histological preparation. The paper so describes a proof-of-concept of a novel method to estimate the diffusion coefficient in a histological sample, based on the acquisition of time-dependent X-ray radiographs and the solution of an inverse diffusion problem.

Potential of Benchtop NMR for the Determination of Polymer Molar Masses, Molar Mass Distributions, and Chemical Composition Profiles by Means of Diffusion-Ordered Spectroscopy, DOSY.

The determination of molar masses and their distributions is crucial in polymer synthesis and design. This work presents the current performance and limitations of diffusion-ordered spectroscopy (DOSY) on a low-field (benchtop) NMR spectrometer (at 90MHz) as an alternative to size exclusion chromatography (SEC) for determining diffusion coefficient distributions (DCDs) and molar mass distributions (MMDs). After optimization for narrowly distributed homopolymers, MMDs obtained with inverse Laplace transformation (ILT) and log-normal distribution are compared with average molar masses obtained with mono- and bi-exponential fits, as well as MMDs obtained from SEC. This approach enables ILT to determine DCDs and MMDs even for bimodal homopolymers with fully spectrally overlapping signals and block copolymers with various chemical compositions, for which chemical composition profiles are determined. The feasibility of low-field diffusion NMR with samples dissolved in non-deuterated solvents is further demonstrated and methods for solvent suppression are discussed.

A novel approach for determining the chloride ion diffusion coefficient in seawater sand concrete: The extravasation method

The significance of the chloride ion diffusion rate cannot be overstated when evaluating the durability of concrete structures. Nonetheless, conventional techniques employed for determining the chloride ion diffusion coefficient in concrete, particularly in the case of seawater and sand concrete (SSC), are prone to inaccuracies due to the inherently elevated levels of inherent chloride ions present within SSC. This research introduces a fresh approach for precisely ascertaining the chloride ion diffusion rate within SSC, addressing this issue. Based on the ion concentration gradient conversion mechanism, an exfiltration method was established, and a corresponding apparatus was developed. The study also investigates the effects of W/C, fly ash content, and LC2 content on chloride ion leaching behavior of SSC. Additionally, the determination of the chloride ion diffusion coefficient and the evaluation of SSC's resistance to chloride ingress were carried out by employing Fick's second law in conjunction with analyzing chloride ion profiles following sample submersion.

Temporal Evolution of Defects and Related Electric Properties in He-Irradiated YBa2Cu3O7-δ Thin Films.

Thin films of the superconductor YBa2Cu3O7-δ (YBCO) were modified by low-energy light-ion irradiation employing collimated or focused He+ beams, and the long-term stability of irradiation-induced defects was investigated. For films irradiated with collimated beams, the resistance was measured in situ during and after irradiation and analyzed using a phenomenological model. The formation and stability of irradiation-induced defects are highly influenced by temperature. Thermal annealing experiments conducted in an Ar atmosphere at various temperatures demonstrated a decrease in resistivity and allowed us to determine diffusion coefficients and the activation energy ΔE=(0.31±0.03) eV for diffusive oxygen rearrangement within the YBCO unit cell basal plane. Additionally, thin YBCO films, nanostructured by focused He+-beam irradiation into vortex pinning arrays, displayed significant commensurability effects in magnetic fields. Despite the strong modulation of defect densities in these pinning arrays, oxygen diffusion during room-temperature annealing over almost six years did not compromise the signatures of vortex matching, which remained precisely at their magnetic fields predicted by the pattern geometry. Moreover, the critical current increased substantially within the entire magnetic field range after long-term storage in dry air. These findings underscore the potential of ion irradiation in tailoring the superconducting properties of thin YBCO films.

A new photoacoustic method to study diffusion in soft matter

We describe an improvement to the photoacoustic method for determining diffusion coefficients of dyes in soft matter, such as agar hydrogels. The proposed method is based on the properties of thermal wave formed due to a heat transfer at the interface of the two media: the gel and polyethylene film. It is inherently simple, requires a small number of reagents and relatively fast, since samples are small. To assess the obtained results, we independently determined diffusion coefficients of the dyes using an optical image analysis method and found both sets of results to be in a good agreement.

Mathematical Models for Estimating Diffusion Coefficients in Concentrated Polymer Solutions from Experimental Data

Diffusion processes in operations involving polymeric materials are of significant interest. Determining experimental values for diffusion coefficients is often challenging. Estimating these coefficients in concentrated polymer solution, polymer films, and membranes relies on experimental tests where the polymer is brought into contact with certain components/solvents. The diffusion coefficient values depend on the diffusion type, which is affected mainly by the nature of the polymer, concentration, and temperature. The literature presents an extensive amount of information regarding the diffusion phenomenon. This paper makes a particular contribution by showing how experimental data obtained from different applications can be processed to determine diffusion coefficients. The manuscript addresses some aspects regarding solvent diffusion in polymers, and illustrates how to determine the diffusion coefficients from experimental data. For specific cases of diffusion, several models for the predictive estimation of diffusion coefficients are also presented. Polymer–solvent systems such as polyvinyl alcohol (PVA)–water, cellulose acetate (CA)–tetrahydrofuran (THF) and cellulose triacetate (CTA)–dichloromethane (DCM) are investigated, with their diffusion mechanisms influenced by changes in structure caused by variations in concentration and temperature. The experimental data obtained through a gravitational technique allow for the highlighting of the diffusion mechanism and the selection of an appropriate mathematical model. A change in the structure of the polymer during the experiment leads to diffusion anomalies. Modeling the experimental data yielded diffusion coefficient values that vary based on the type of system investigated, composition and temperature. Thus, in the case of the CTA-DCM system, the diffusion coefficient at 303 K, at various concentration values, is in the range of 4.5 and 8·10−11 m2/s; for the PVA-H2O system, D = 4.1·10−12 m2/s at 303 K, and D = 6.5·10−12 m2/s at 333 K; while for the CA-THF system, the solvent–polymer diffusion coefficient has a value of 2.5∙10−12 m2/s at 303 K, and D = 1.75∙10−11 m2/s at 323 K. Mathematical models can be useful in studies regarding the drying of polymer films with complex structures, providing knowledge for designing or selecting suitable equipment.

Hydrogen diffusivity, uptake and embrittlement of solution annealed and cold deformed austenitic stainless steel 1.4435 at hydrogen pressures up to 1000 bar and 200 °C

Austenitic stainless steel 1.4435, a candidate material for high-pressure tubing for hydrogen applications, was investigated in two different stages of cold deformation (0 and 27 %) regarding its hydrogen embrittlement tendency. The determination of effective diffusion coefficients revealed slightly increased diffusion kinetics at 60°C for the cold-reduced material (8.0 · 10−15 m²/s) compared to the solution annealed one (5.0 · 10−15 m²/s). High-pressure gaseous hydrogen charging shows no differences regarding the degree of cold deformation with a maximum hydrogen concentration of 112 wt.-ppm. Furthermore, a significant ductility loss expressed by the ratio of the reduction of area was observed.

Measurement of Diffusion Coefficients and Assessment of Atomic Mobilities in HCP Mg-Ag-Sn Alloys

Reliable thermodynamic and kinetic databases are essential for designing and developing novel magnesium (Mg) alloys for elevated-temperature applications in the automotive and aerospace industries. This study investigated diffusion behaviors of HCP Mg-Ag-Sn alloys by four sets of three diffusion couples annealed at 450, 500, and 550 °C, respectively. Interdiffusion and impurity diffusion coefficients at the three annealing temperatures were extracted using the forward-simulation method. The determined diffusion coefficients, combined with prior results in the literature, were then used to assess atomic mobilities for the HCP phase of the Mg-Ag-Sn system. This assessment was conducted using the DICTRA and the TCMG5 thermodynamic database within the Thermo-Calc software. Comprehensive comparisons between calculated and experimental diffusion coefficients yielded an excellent agreement. The atomic mobilities were also further validated by appropriate predictions of concentration profiles and diffusion paths observed in independent diffusion couple experiments performed at 525 °C for 18 hrs. It has been found that the addition of Sn and Ag both increase the diffusivity of Ag in HCP Mg and Sn in HCP Mg in the stable temperature range of HCP Mg, respectively. This improved understanding of the kinetics in the Mg-Ag-Sn ternary system is essential for controlling diffusion-dependent properties such as coarsening and, therefore, the mechanical properties of Mg-Ag-Sn alloys at elevated temperatures.

Mathematical description of protein extraction from muscle tissue of hydrobionts and determination of the effective molecular diffusion coefficient

The purpose of this work is to obtain an array of experimental data, kinetic dependences of the protein extracting from the particles of dispersed muscle tissue of hydrobionts (Alaska pollack), its mathematical description, and to find the molecular diffusion coefficients of proteins in the system of equations using solving the inverse problem of the modeling. This work’s novelty is to study swelling and protein extraction from animal tissue and its mathematical description. This study has established that the diffusion of proteins from the muscle tissue of hydrobionts during stirring suspension (raw material-extractant) occurs at a significantly higher rate than diffusion from raw materials of plant origin. It indicates the contribution of a convective component to the swelling process and dissolution of particles due to intensive blending and easy differentiability of muscular tissue particles of hydrobionts. Based on the solution of the mathematical model, the molecular diffusion coefficient of proteins from raw materials of animal origin (muscle tissue of hydrobionts) (3÷8)×10−9 m2/s, which exceeds the diffusion coefficients of carbohydrates from raw materials of plant origin (e.g., sugar 0.3×10−9 m2/s) by more than an order of magnitude. Researchers can use the obtained molecular diffusion coefficient to calculate the parameters of the diffusion process of proteins during its extraction from dispersed particles of hydrobionts.

Determination of effective liquid-phase diffusion coefficients by tracer method in continuous packed columns

Many industrially relevant chemical processes are affected by the liquid-phase diffusion resistance in porous materials, for example in heterogeneous catalysts. Therefore, reliable experimental methods for the determination of effective diffusion coefficients in porous structures are highly desirable. In this work, a method was developed to study the internal diffusion effects in porous media by step response experiments with tracers injected to fixed beds filled with porous particles and inert material. The tracer responses were recorded at the outlet of the bed by UV–vis spectrometry. Ethanol was used as the tracer and water as the solvent. Porous aluminum oxide particles were selected as the model material. The step responses were described with a dynamic axial dispersion model, i.e. partial differential equations for the inert and catalyst sections of the bed. The partial differential equations describing the model were discretized with finite differences and the resulting ordinary differential equations were solved numerically with a solver for stiff ordinary differential equations. From the experimental data, the dispersion coefficients of the bed and the effective diffusion coefficient in the porous aluminum oxide particles were successfully determined by non-linear regression analysis. The values of the effective diffusion coefficents had a good agreement with the liquid-phase diffusion coefficients estimated from correlations. The method is applicable for any porous material.