Transport of direct dye into cellulose membrane

The dyeing process for a cellulose membrane–direct dye system is analyzed based on a parallel transport mechanism of surface and pore diffusion with Freundlich isotherm. Numerical solutions were obtained in order to clarify how the surface and pore diffusion resistances affect the uptake curve. The numerical solutions were also compared with an analytical solution for surface diffusion control to establish the range where the analytical solution can be considered as an acceptable approximation. The uptake curves in the cellulose membrane-chromophore (C.I. Direct Yellow 12) system in the presence of inorganic electrolyte were measured. The rate of adsorption and the maximum amount of adsorption increased with an increase in the concentration of those electrolytes. The rate of adsorption was approximately controlled by the surface diffusion rather than by pore diffusion. The surface diffusivities of the dye were little affected by either the concentration and or nature of the electrolyte.     

In-situ monitoring of axial particle mixing in a rotating drum using bulk density measurements

A device is described for measuring changes in the local composition of particulate materials in a rotary mixer by continuously monitoring changes in the bulk density. The bulk density is measured using a small cup mounted to the mixer wall that fills with powder during rotation through the bed of particles in the lower part of the mixer. The mass of material in the cup is measured using a load cell during rotation of the cup above the free surface of the particles, and the cup empties before re-entering the particle bed. For mixing of materials with a difference in either particle density, or packing density, the localised bulk density measurement gives a good measure of mixing progress. The measurement device is demonstrated in a 1 m diameter horizontal rotating drum in which two materials are mixed along the axis of the drum. Measurements of the rate of dispersion along the axis are consistent with other work in inclined rotary kilns and can be fitted with a simple diffusion model for the axial mixing of the two species.     

Critical review of the impact of tortuosity on diffusion

Most of the present formulations for the mass conservation of species do not correctly represent changes in the time scale of diffusion as a function of porosity in a system of porous media. In sediments, or in any porous system, the presence of solid particles causes the diffusion paths of species to deviate from straight lines. To represent the role of porosity on diffusion, the diffusion coefficient must be scaled with tortuosity. In this paper we present a review of the available formulations for the scaled diffusion coefficient with tortuosity, sensitivity tests with analytical solutions, and calibration of these formulations with respect to measured data.     

Mathematical modeling of diffusion and kinetics in amperometric immobilized enzyme electrodes

An analysis of the diffusion and kinetics in amperometric immobilized enzyme electrodes is presented for reactions of the enzyme and substrate. This analysis contains a non-linear term related to Michaelis-Menten kinetics. In this paper, we obtain approximate analytical solutions for the non-linear equations that describe diffusion and the reaction within the film by employing the homotopy perturbation method (HPM). The obtained analytical results are compared with the available limiting case results and found to be in satisfactory agreement.     

明胶层中的扩散和反应的动力学研究 (Ⅰ)纯扩散和扩散伴随一级反应的数值模型

设计了两个数值模型,分别描述明胶层中的反应物的纯扩散和扩散伴随一般反应的动力学过程。用这些模型可以给出反应物在明胶层中不同时间下的空间分布。比之通常的分析解法,计算机化的数值模型更易求解,而且更具有解决复杂动力学过程的能力。为了验证这些数值模型的适用性,还同时与相应的分析解法作了对照。     

溶剂浸渍树脂吸着螺旋霉素的动力学模型研究

本文根据颗粒扩散控制机制,认为溶剂萃取过程和固体内表面的吸附同时发生,建立了有限浴条件下溶剂浸溃树脂吸着螺旋霉素的动力学模型.浸渍树脂静态吸附螺旋霉素的计算值与实验值接近,得到螺旋霉素在浸渍树脂内的有效扩散系数为3.7×10~(-12)～29.4×10~(-12)m~2/s.线性吸附等温线条件下的分析解与非线性条件下的数值解比较,表明线性假设能较好地反映实际过程.     

Modeling breakthrough and elution curves in fixed bed of inert core adsorbents: analytical and approximate solutions

A mathematical model is developed for fixed-beds packed with inert core adsorbents. New analytical solutions to predict breakthrough and elution curves are derived for linear adsorption systems coupled with axial dispersion, film mass transfer and intraparticle mass transfer. New approximate solutions are also obtained based on the assumptions of parabolic concentration profile in the adsorbent shell and the quasi-lognormal distribution for the impulse response in order to predict breakthrough and elution curves. The applicability of these approximate solutions is suggested by comparison with the new analytical solutions. The effects of the size of inert core, sample input mode, axial dispersion, film mass transfer resistance and intraparticle diffusion resistance, on the breakthrough and elution curves are discussed. The decrease of the intraparticle mass transfer resistance by using inert core adsorbents is quantitatively analyzed by introducing the parameter 1/Θ. Furthermore, an analytical expression for the resolution of two components is derived based on the quasi-lognormal distribution approximate solution; the resolution of two components is improved with the inert core adsorbent when compared with the conventional adsorbent, especially for biomacromolecules where the intraparticle diffusion rate is slow.     

Treatment of size-dependent aerosol transport processes using quadrature based moment methods

The use of moment methods for simulation of aerosol settling and diffusion phenomena in which the settling velocity and diffusion coefficient are functions of the size of the particles leads to difficult computational problems, especially if the moment equations need to be closed. In this study, a simple one dimensional problem of aerosol diffusion and gravitational settling is carried out using quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). Analytical solutions can be obtained for the number density function, and issues related to the integration of the solutions to get the moments are discussed. Comparison of the solutions of the moment equations to the moments obtained from the analytical solutions reveals that solutions depend on the initial choice of moments. Results also indicate that the proper choice of moments of the initial number density function may be a significant factor in obtaining more accurate solutions from QMOM or DQMOM.     

Determination of the aerosol particle size distribution by means of the diffusion battery: Analytical inversion

The algorithm of the analytical inversion of aerosol size distribution is proposed in this work. As the diffusion battery separates particles into several fractions according to their diffusivity, the total spectrum can be represented as the sum of spectra of fractions. Analytical formulas are derived to calculate mean diameters for particles in different fractions using diffusion battery penetrations as input parameters. The spectra of fractions are approximated by lognormal functions. Two analytical solutions for the aerosol size distribution inversion problem are discussed. The sizing accuracy of analytical solutions is investigated, comparing them with the measurements through transmission electron microscopy using the laboratory-generated NaCl aerosol. The agreement is demonstrated to be within 10% accuracy. It is shown that in case of two-mode size distribution, the spectrum components are well resolved for rather distant peaks (modal diameters of 10 and 300?nm) and poorly resolved for nearby modes (50 and 300?nm). To improve the peak resolution, the procedure of spectrum correction is applied demonstrating an excellent peak separation. Finally, the peak resolution is experimentally verified for the laboratory-generated two-mode spectra of tungsten oxide–NaCl aerosol with the modal diameters of 10 and 60?nm, respectively. Both analytical solutions demonstrated good peak resolution.

Copyright © 2018 American Association for Aerosol Research     

System of coupled non-linear reaction diffusion processes at conducting polymer-modified ultramicroelectrodes

The theoretical analysis of the steady-state amperometric response for conducting polymer-modified ultramicroelectrodes is discussed. The effect of substrate diffusion in the solution adjacent to the polymer film on both the concentration profile and current response is also examined. Simple analytical expressions for substrate and mediator concentrations and current responses for all values of reaction/diffusion parameters are presented. The model is based on non-stationary system of coupled reaction/diffusion equations containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reactions. He's variational iteration method is used to give approximate analytical solutions of coupled non-linear reaction diffusion equations. A good agreement with available limiting case results is noticed.     

明胶层中的扩散和反应动力学研究Ⅵ．球型扩散及扩散伴随一级反应过程的数值解

本文采用有限差分法导出了扩散物从空间一点向周围作三维球型扩散和边扩散边进行一级反应的动力学方程的数值解，并描述了其求解方法。在相同的参数下，获得的数值解与相近边界条件的分析解结果对照，一致性良好，为进而研究染料云的形成过程打下基础。     

Analysis of Field-Assisted Binary Ion Exchange

A differential equation was derived for ionic migration in a system with two mobile ions. Stationary solutions were obtained for the case where the slower ion follows the faster, which reduce to those obtained when diffusion is ignored. When the faster ion is "behind" the slower no stationary solutions are possible, but the diffusion term is not important for this case and appropriate solutions are readily obtained. However, by taking account of diffusion at the boundaries, where it is important, infinite gradients are avoided and concentration equilibrium is permitted at the glass-salt interface. The concentration distributions predicted from this approach are compared with those measured on glass that has been ion exchanged with electric field assistance. Although the agreement is reasonable, there is some discrepancy, due in part to the assumption of concentration-independent species mobility.     

Aerosol diffusion battery: The retrieval of particle size distribution with the help of analytical formulas

A new algorithm is proposed for the determination of aerosol particle size distribution from a set of screen diffusion battery penetrations. The idea is to determine the size spectra of the fractions of particles separated by the sections of diffusion battery, so the total size distribution is the sum of the spectra of fractions. The spectrum of each fraction is approximated by the lognormal function, which is defined by two parameters: the standard geometric deviation (SGD) and geometric mean diameter. The SGD value is chosen to be 1.35 for each fraction. The geometric mean diameters of fractions are calculated from the diffusion battery penetrations. For this purpose, analytical formulas are derived to link the mean single-fiber collection efficiency for each fraction with the experimentally measured penetrations. Then the mean diameters of fractions are calculated from the collection efficiencies using the fan model filtration theory. To achieve a better size resolution, numerical approach is proposed to calculate the particle size spectrum using the analytical solution as an initial approximation. The validity of the analytical and numerical solutions is investigated by comparing them with the spectra determined by means of transmission electron microscopy and gravity settling. For this purpose, the aerosol is generated using the evaporation-nucleation technique, Collison-type nebulizer, and hot-wire bulb generator. It is found that the analytical solution demonstrates a good sizing accuracy but relatively poor size resolution, while the numerical approach results in both good sizing accuracy and good size resolution for the two-mode aerosol.

Copyright © 2018 American Association for Aerosol Research     

Modeling separation of proteins by inert core adsorbent in a batch adsorber

Adsorption/desorption kinetics of protein on the binding ligand of inert core adsorbent in a batch adsorber is analyzed theoretically for Langmuir isotherm coupled with the intraparticle diffusion and film mass transfer resistances. For the two limiting cases of Langmuir isotherm, there are analytical solutions. New analytical solutions are derived for Henry isotherm, and the analytical solution of shrinking core model is recommended for rectangular isotherm. The effects of the inert core radius, equilibrium constant, intraparticle diffusion and film mass transfer resistances on the time evolution of bulk concentration and particle radial profiles were investigated. The applicable range of the analytical solution with rectangular isotherm is given. A new method to estimate both film mass transfer coefficient, k_f, and effective pore diffusivity, D_pe, from a single bulk concentration-time curve in batch adsorber is given and tested with literature data for the adsorption of BSA on CB-6AS inert core adsorbent.     

Radial diffusion model for fragrance materials: Prediction and validation

A predictive model based on Fick's second law for radial diffusion is proposed and validated for modeling the diffusion of fragrance materials. A pure component, two binary systems, and a ternary system were used for validation. The model combines the prediction model to represent the liquid phase nonidealities, using the UNIFAC group contribution method, with the Fickian radial diffusion approach. The experimental headspace concentrations were measured in a diffusion chamber using the solid-phase microextraction technique and quantified using gas chromatography with a flame ionization detector. The numerical solutions were obtained along with an analytical model considering constant surface concentration. The odor intensities of the studied systems were calculated using Stevens' power law and the strongest component model, respectively. The numerical simulation presented good adherence to the experimental gas concentration data. The proposed methodology is an efficient and validated tool to assess the radial diffusion of fragrance and volatile systems.     

CORRELATIONS FOR ESTIMATING THE DIFFUSIVITIES OF NITROUS OXIDE IN AQUEOUS SOLUTIONS OF DIETHANOLAMINE AND N-METHYLDIETHANOLAMINE AND THE SOLUTION DENSITIES AND VISCOSITIES

Experimentally measured diffusion coefficients of nitrous oxide in water, aqueous solutions of diethanolamine, aqueous solutions of N-methyldiethanolamine, and aqueous blends of diethanolamine and N-methyldiethanolamine have been used to arrive at a modified Stokes-Einstein correlation by using nonlinear least-squares filling. The average deviation of this correlation from the experimentally measured diffusion coefficients is about 13%. Comparisons are also made to the estimated diffusivities of nitrous oxide in the amine solutions which are obtained from the Wilke-Chang and Hayduk-Minhas correlations. The average deviations of the Wilke-Chang and Hayduk-Minhas correlations from the measured values are about 13% and 19%, respectively. Correlations were also presented for estimating the solution densities and viscosities. Predicted values for the solution densities and viscosities from these correlations deviate from experimentally measured values on average by 1.2% and 4.8% respectively.     

Insight into dynamics of polyelectrolyte chains in salt-free solutions by laser light scattering and analytical ultracentrifugation

We have investigated the dynamics of xanthan aqueous solutions with and without added salt (NaCl) by using laser light scattering (LLS) and analytical ultracentrifugation (AUC) via sedimentation velocity (SV). The fast and slow modes are observed in salt-free and low-salt xanthan solutions by dynamic light scattering (DLS). The scattering ratio (KC/R_s(q)) and apparent diffusion coefficient (D_s,app(q)) of the slow mode is linearly related to scattering vector (q²), indicating that it is related to the diffusion of scattering objects. The intensity contribution (α_s) of the slow mode is independent of scattering angles, indicating that the slow mode is not related to some scattering objects larger than the LLS observation length. However, the slow mode disappears in SV experiments, indicating that it arises from the temporal aggregates due to long range electrostatic interactions between chains, which can be destroyed in centrifugal field. The diffusion coefficient measured by SV is close to that of the fast mode in DLS measurements, indicating that it is the coupling diffusion of macroions and counterions. The present studies also demonstrate that the chain stiffness does not change the characteristics of the dynamics of polyelectrolyte in solutions.     

Diffusion of Moisture in Adhesively Bonded Joints

In this article, diffusion of moisture in adhesively bonded composite joints is discussed and analysed experimentally, analytically and numerically. The experimental studies concentrate on moisture diffusion in adhesive films and in unidirectional and multidirectional composite substrates exposed to two different conditioning environments, namely 45°C/85% RH and 90°C/97% RH for the absorption studies and 90°C/ambient for the desorption studies. The coefficients of diffusion are determined from the water uptake plots. The analytical solutions for diffusion in joints with impermeable adherends are based on the classical theory of diffusion and are used to derive equations in two-dimensions for different adhesive fillet shapes, namely radiused fillet, triangular fillet and rectangular fillet. In the finite element analysis, the diffusion of moisture from the composite substrates into lap-strap joints is also taken into account. Both unidirectional and multidirectional composites are considered, as well as two different fillet shapes, i.e., rectangular and triangular fillet. A comparison between the results obtained using FEA and those obtained using the analytical solution is made. Finally, fatigue test data for lap-strap joints aged and tested in different environments is presented and a tentative link between fatigue threshold and water concentration at the site of failure initiation is made, indicating a semi-empirical method of predicting the strength of joints subjected to moisture-induced degradation.     

The Proper Use of Mass Diffusion Equations in Drying Modeling: Introducing the Drying Intensity Number

This article intends to clearly define the possibilities and limitations offered by a simple diffusion approach of drying. Actually, many works use a simple diffusion equation to model mass transfer during drying, probably because a simple analytical solution of this equation does exist in the case of simple boundary conditions. However, one has to be aware of the limitations of this approach. Using a comprehensive formulation and a relevant computational solution, the most frequent assumptions of the diffusion approach were rigorously tested. It is concluded that analytical solutions must be discarded for several reasons: analytical solutions, either using Dirichlet or third kind boundary conditions, are often misleading and should be avoided; in the drying process, the coupling between heat and mass transfer is mandatory; nonlinearity (variation of diffusivity with moisture content) can hardly be avoided for mass transfer. In order to reach a verdict, a dimensionless number, the Drying Intensity Number (N_DI), is introduced. It allows the level of coupling between heat and mass transfer to be easily assessed. Thanks to this number, a guide is proposed for choosing the right level of modeling, depending on the drying configuration.