MATHEMATICAL RELATIONSHIP BETWEEN DESORPTION AND SORPTION SOLUTIONS

This paper presents a theoretical equation relating the dimensionless times of Newman's solutions of the diffusion equation for desorption and sorption. The derived equation provides a theoretical proof of the linear relationship between dimensionless time and the inverse of transport ratio for a given fraction of diffusing substance as observed numerically in the literature. Following the same approach proposed earlier by the writer, a theoretical expression for the diffusion coefficient is derived as a function of time and the inverse of desorption rate when the fraction of diffusing substance is 0.5. Evaluation of the diffusion coefficient from an experimental desorption curve is discussed.     

Mathematical Relationship Between Surface Emission and Diffusion Coefficients

ABSTRACT

This naver analvzes the surface emission coefficient corresponding to anv diffusion coefficient expressed as an exponential function of the concentration diffusing substance in capillary porous solids. Theoretical equations for surface emission coefficient for both sorption and desorption are presented. Procedures to derive the diffusion coefficient and to verify the corresponding surface emission coefficient are made using experimental sorption data of aspen (Populur sp.) wood. It is of interest to note that the theoretical models for diffusion and surface emission coefficients can be established and/or evaluated using the same set of experimental dara.     

TWO-STAGE MOISTURE DIFFUSION IN WOOD WITH CONSTANT TRANSPORT COEFFICIENTS

ABSTRACT

To determine whether transport coefficients in desorption curves for northern red oak are constant, four sets of desorption data were compared against previously established mathematical conditions for infinite-series solution of the non-steady-state diffusion equation. For each data set, when moisture fraction in wood is above a certain value, designated as the first stage, these conditions are satisfied, with the diffusion and surface emission coefficients being positive and finite; below that value, designated as the second stage, these conditions are still satisfied, with the diffusion coefficient taking a smaller positive value but the surface emission coefficient becoming negative and finite. Mathematically, these two pairs of transport coefficients can be used to predict the whole diffusion curve that describes the variation of moisture fraction with time. However, the negative surface emission coefficient in the second stage of the desorption process implies that the moisture gradient has cut the surface at a point below the equilibrium moisture content, which is physically impossible. Alternatively, the second stage can be considered as a new stage with moisture fraction values normalized with respect to the lowest value in the first stage, which is also the starting point of the second stage. The transport coefficents are obtained in the same manner as in the first stage and are found to be positive and finite. The two pairs of transport coefficients can describe the diffusion curve with high accuracy. However, the assumption used in the second stage is that the initial moisture content in wood is uniform, which again is physically impossible. We therefore conclude that the transport coefficients for northern red oak are not constant. Although the two-stage approaches presented in this study can predict the diffusion curves accurately, their physical interaction is difficult to justify.     

TWO-STAGE MOISTURE DIFFUSION IN WOOD WITH CONSTANT TRANSPORT COEFFICIENTS

To determine whether transport coefficients in desorption curves for northern red oak are constant, four sets of desorption data were compared against previously established mathematical conditions for infinite-series solution of the non-steady-state diffusion equation. For each data set, when moisture fraction in wood is above a certain value, designated as the first stage, these conditions are satisfied, with the diffusion and surface emission coefficients being positive and finite; below that value, designated as the second stage, these conditions are still satisfied, with the diffusion coefficient taking a smaller positive value but the surface emission coefficient becoming negative and finite. Mathematically, these two pairs of transport coefficients can be used to predict the whole diffusion curve that describes the variation of moisture fraction with time. However, the negative surface emission coefficient in the second stage of the desorption process implies that the moisture gradient has cut the surface at a point below the equilibrium moisture content, which is physically impossible. Alternatively, the second stage can be considered as a new stage with moisture fraction values normalized with respect to the lowest value in the first stage, which is also the starting point of the second stage. The transport coefficents are obtained in the same manner as in the first stage and are found to be positive and finite. The two pairs of transport coefficients can describe the diffusion curve with high accuracy. However, the assumption used in the second stage is that the initial moisture content in wood is uniform, which again is physically impossible. We therefore conclude that the transport coefficients for northern red oak are not constant. Although the two-stage approaches presented in this study can predict the diffusion curves accurately, their physical interaction is difficult to justify.     

Isothermal Moisture Transport in Partially Saturated Porous Media

ABSTRACT

The prediction of a hydrodynamic model for the isothermal transport of liquid in partially saturated porous media is compared with experimentally obtained values of water transport in clay. The transport obeys the diffusion equation, The diffusion coefficient is described as a function of porosity, permeability and pore size distribution. The comparison indicates that the model needs some refinement.     

Adsorption and Diffusion of VO2+ and VO2 + across Cation Membrane for All‐Vanadium Redox Flow Battery

A method based on a selectivity coefficient and the Nernst‐Planck equation is proposed to determine diffusion coefficients of vanadium ions across a cation exchange membrane in VO²⁺/H⁺ and VO₂ ⁺/H⁺ systems. This simplified method can be applied to high concentrations of vanadium ions. Three cation exchange membranes were studied. The logarithmic value of the selectivity coefficient was linearly dependent on the molar fraction of vanadium ions in solution. The diffusion coefficient of vanadium ions decreased with decreasing water content. The membrane with the lowest diffusion coefficient was selected as a battery separator and showed the lowest capacity loss of the studied membranes.     

Drying of glassy polymer varnishes: A quartz resonator study

We report on desorption measurements on polymeric thin films coated onto quartz crystal resonators. Due to the high sensitivity of quartz crystal microbalances, the experiments can be performed on very thin films, which have small diffusion time constants even in the glassy state. When drying is performed slowly enough, diffusion equilibrium can be maintained through the whole process of desorption, including the glassy domain. From these quasi-stationary pressure ramps, we derived the solvent chemical potential as a function of polymer volume fraction μ(ϕ). The results fit well to a model recently proposed by Leibler and Sekimoto.¹ In addition, we have derived the mutual diffusion coefficient D(ϕ) from pressure step experiments. We observe a strong decrease of D(ϕ) for high polymer concentrations typical of hypodiffusive systems like polymers. We investigated the drying of an industrial varnish that is a blend of 2 copolymers as well as the drying of its components separately. Both the solvent chemical potential μ(ϕ) and the mutual diffusion coefficient D(ϕ) of the blend interpolate between the respective quantities of the components. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 2235–2246, 1998     

纸塑复合包装材料纸中化学物迁移预测模型

基于一维Fick扩散理论,考虑了纸和塑料涂层界面处的分配系数及纸和塑料涂层中不同的扩散系数,建立了迁移预测模型,得到了其解析解。分析了各参数如分配系数、扩散系数、纸中化学污染物初始浓度和量纲1时间对迁移行为的影响。结果表明,随纸塑界面处分配系数k_CP的增大,从纸中迁移入功能阻隔层塑料中的化学污染物量和通过功能阻隔层塑料迁移进入食品中的化学污染物量明显增多;塑料中扩散系数D_C的变化显著影响了塑料和食品中化学污染物的量,而纸中扩散系数D_P的改变对迁移的影响不显著。     

Kinetic parameters of isotopic exchange reaction in finite bath

A mathematical model has been proposed for an isotopic exchange reaction between the solid particles and ions in a solution of finite volume. A theoretical equation has been derived to predict the exchange fraction as a function of the dimensionless time T with the kinetic parameters ξ, ξ₁ and final fractional uptake U as parameters for the exchange rate controlled by combined film diffusion, surface chemical reaction and intraparticle diffusion. The exchange fraction increases as the values of T, U, ξ and ξ, increase. Experimental results of the isotopic exchange systems CaCO₃(s)Ca²⁺ (aq), CaC₂O₄(s)/ Ca²⁺ (aq) and Ca-form Dowex 50W-X8/Ca²⁺ (aq) agree well with the theoretical equation proposed in this study.     

Determination of the infinite dilution diffusion and activity coefficients of alkanes in polypropylene by inverse gas chromatography

Infinite dilution diffusion and activity coefficients of n‐hexane, n‐heptane, and n‐decane in polypropylene were measured from 373.15 to 393.15 K by inverse gas chromatography. The influences of small molecule solvent and temperature on the infinite dilution diffusion and activity coefficients were investigated. The results showed that the infinite dilution diffusion coefficient decreased but the infinite dilution activity coefficient increased with an increasing number of CH₂ group in the aliphatic solvents. The temperature increase resulted in the increase in the infinite dilution diffusion coefficient and the decrease in the infinite dilution activity coefficient. The graphs plotted according to the results of the infinite dilution diffusion coefficient versus temperature were in agreement with the Arrhenius equation. Diffusion constant and activation energy were obtained from the Arrhenius equation. Also, the interdependence on the infinite dilute activity coefficient and temperature accorded with Gibbs‐Helmholtz equation. From the temperature dependence of the limiting activity coefficients partial molar excess enthalpy at infinite dilution H^E,∞ was obtained. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 101: 1925–1930, 2006     

离子扩散型抗菌陶瓷中银离子的扩散

研究了离子扩散法制造抗菌陶瓷过程中Ag+在釉层中的扩散行为.用扫描电镜和电子探针分析了扩散过程中Ag的分布.用Fick定律分析扩散过程,提出用表层Ag+质量分数和表观扩散系数来描述实际的扩散过程.用余误差函数近似法确定扩散温度下不同成分釉层中Ag+的表观扩散系数.对于所研究的釉层,表观扩散系数为1.6×10-11～1.3×10-10 cm2/s.确定了上述2个参数与釉层成分、熔盐比例、扩散温度及扩散时间的关系.利用这些关系建立了优化制造条件的基础.     

Determination of effective diffusion coefficient and interfacial mass transfer coefficient of bovine serum albumin (BSA) adsorption into porous polyethylene membrane by microscope FTIR-mapping study

Here, a new method for simultaneous determination of diffusion coefficient D and interfacial mass transfer coefficient (or convective mass transfer coefficient) k was proposed for bovine serum albumin (BSA) adsorption into porous polymeric membranes. The experimental data for BSA concentration at different membrane depth and different time were determined from FTIR-mapping measurements. Then the diffusion coefficient D and interfacial mass transfer coefficient k were estimated from the calculated dimensionless concentration data at different time and membrane depth by a trial-and-error method based on the diffusion equation initiated in this paper. The diffusion coefficient D and interfacial mass transfer coefficient k evaluated in this manner are respectively: and . The theoretical concentration values calculated from the determined parameters were compared with experimental reading from FTIR mappings, which showed a good agreement between them, especially for the case of a relatively long-time adsorption.     

MODELING OF ROUGH RICE DRYING IN THIN VERTICAL COLUMNS

ABSTRACT

The solution of classical diffusion equation based on the assumption of average moisture diffusion coefficient did not adequately represent natural convection drying of rough rice in thin vertical columns exposed on both sides to hot air. Instantaneous moisture diffusivity coefficients determined from experimental drying curves decreased continuously with an increase in exposure duration and were linearly related to moisture ratio. The proponionality constant which was called apparent moisture diffusion coefficient was distinctly related to air temperature, relative humidity, and initial moisture content of rough rice. The modified moisture diffusion model using the instantaneous moisture diffusion coefficient was found to best represent the moisture removal from bulk rough rice.     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

ABSTRACT

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally oblained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

Mass Transfer in the Celery Slice: Effects of Temperature,Moisture Content,and Density on Water Diffusivity

Abstract

Statistical tests were applied to determine the effects of temperature, moisture content, density, and porosity of material on the effective moisture diffusion coefficient during convective drying of root celery. In biological materials with colloidal capillary-porous structure (like root celery), which shrink considerably during drying and show high heterogeneity, the effective water diffusion coefficient depends not only on material temperature and moisture content, but also on its density. It was found that statistical tests can be applied to predict which independent variables should describe the water diffusivity in colloidal capillary-porous materials. A mathematical model of the effective water diffusion coefficient in root celery was formulated as Arhenius-type equation with moisture content of the raw material, its temperature and density as independent variables.     

色谱法测定葡萄糖果糖在制备柱上的传质和扩散系数

色谱模型对研究色谱分离过程中的传质动力学具有重要的意义。根据物料守恒原理,结合传质扩散模型推导色谱分离连续方程。假设吸附过程相平衡是线性,连续性方程推导出理论板数与流速和色谱模型参数有关。在多组不同流速条件下测定色谱柱参数,计算得到两组分的传质系数和扩散系数。分别用测定模型参数模拟脉冲实验,结果表明拟合较好。     

Sorption and desorption parameters of water or ethanol in light‐cured dental dimethacrylate resins

Two cycles of sorption/desorption of water or ethanol by light‐cured dental resins of bisphenol A glycol dimethacrylate (Bis‐GMA), bisphenol A ethoxylated dimethacrylate (Bis‐EMA) urethane dimethacrylate (UDMA) triethylene glycol dimethacrylate and decanediol dimethacrylate (D₃MA) were studied. The experimental curves m_t = f(t) taken for the first water sorption by poly‐Bis‐GMA, poly‐Bis‐EMA and poly‐UDMA showed a maximum. A maximum was also observed in the curve obtained for first sorption of ethanol by poly‐Bis‐GMA. In all other cases, the curves for sorption or desorption of water or ethanol showed Fickian behavior. The experimental data obtained for first sorption of water or ethanol were perfectly fitted to a new proposed equation, which predicts water or ethanol sorption with simultaneous extraction of unreacted the monomer. This equation gave us the possibility for the determination of the diffusion coefficient of the extraction of the unreacted Bis‐GMA during the water and ethanol sorption, as well as the diffusion coefficient of the extraction of the unreacted Bis‐EMA and UDMA during the water sorption. The maximum water or ethanol absorbed at equilibrium and the diffusion coefficient are determined from the second sorption/desorption cycle during which the extraction of the monomer is negligible. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008     

Isothermal Vapour and Liquid Transport Inside Clay During Drying

ABSTRACT

A model is presented to describe the moisture transport inside a partially saturated porous material. The transport is caused by vapour diffusion and liquid diffusion. The evaporation inside the porous material is described with a mass transfer coefficient and a specific evaporating surface. Predictions of the model for moisture profiles are compared to experimentally obtained profiles found in the literature. The model needs further extension in the form of incorporating sorption isotherms.     

SOLUTIONS OF DIFFUSION EQUATION WITH CONSTANT DIFFUSION AND SURFACE EMISSION COEFFICIENTS

ABSTRACT

Closed-form solutions of the non-steady state diffusion equation with constant transport coefficients are presented. The diffusion coefficient is assumed to be finite, but the surface emission coefficient can be either finite or infinite. Mathematical conditions are established for the transport coefficients to be constant. When these conditions are met, the transport coefficients can then be easily evaluated. Diffusion test data can be compared against these conditions to determine whether or not the transport coefficients are constant. Desorption test data of northern red oak indicate that initial moisture content in wood and equilibrium moisture content in the environment are closely related to the constancy of the transport coefficients.     

Drying Simulation of a Solid Slab with Three Dimensional Shrinkage

ABSTRACT

A mathematical model is developed to simulate the drying of a hygroscopic porous solid. The model, based on the gradient of moisture concentration per unit volume as driving force, takes into account the migration of water within the solid by diffusion and the evaporation at the interface. A mathematical equation for diffusion in a slab with three dimensional shrinkage has been derived, assuming that the magnitude of shrinkage is equal to the volume of water evaporated. The resulting diffusion equation and the heat balance eauation for infinite thermal conductivitv were solved n;merically with temperature dependent diffusion coefficient and convective boundary conditions. The deDendence of the desorption isotherm with temperature is-also considered. corndination of all these factors in a single model provides a tool that is effective in predictinq dryinq behavior and also useful in exploring and understanding the impact of important variables on the drying process.