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.     

A STUDY ON THE SEPARATION OF DIFFUSION AND SURFACE EMISSION COEFFICIENTS IN WOOD

For a more accurate simulation of the moisture sorption process in wood, the precise separation of surface emission and diffusion coefficients is essential. In this paper, a non-linear simplex technique that was developed to obtain the optimum pair of the surface emission coefficient he, and the diffusion coefficient D from a single sorption curve is presented and tested with experimental data. An easy to use computer program was employed to carry out the evaluation. Data analysis showed that the new method resulted in a more statistically accurate calculation of b_c and D than currently existing methods.     

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.     

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.     

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.     

Water sorption kinetics in light‐cured poly‐HEMA and poly(HEMA‐co‐TEGDMA); determination of the self‐diffusion coefficient by new iterative methods

The present investigation is concerned with the determination of self‐diffusion coefficient (D) of water in methacrylate‐based biomaterials following Fickian sorption by two new methods: the Iterative and the Graphical methods. The D value is traditionally determined by means of the initial slope of the corresponding sorption curve and the so‐called Stefan's approximation. The proposed methods using equations without approximations and data resulting from the whole sorption range reach to accurate values of D, even when the sorption curve does not present an initial linear portion. In addition to D, the Graphical method allows the extrapolation of the mass of the sorbed water at equilibrium (M_∞), even when the equilibrium specimen's mass fluctuates around its limited value (m_∞). The test of the proposed procedures by means of ideal and Monte Carlo simulated data revealed that these methods are fairly applicable. The obtained D values compared with those determined by means of the Stephan's method revealed that the proposed methods provide more accurate results. Finally, the proposed methods were successfully applied to the experimental determination of the diffusion coefficient of water (50°C) in the homopolymer of 2‐hydroxyethyl methacrylate (HEMA) and in the copolymer of HEMA with triethylene glycol dimethacrylate (98/2 mol/mol). These polymers were prepared by light curing (λ = 470 nm) at room temperature in presence of camphorquinone and N,N‐dimethylaminoethyl methacrylate as initiator. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007     

MATHEMATICAL RELATIONSHIP BETWEEN DESORPTION AND SORPTION SOLUTIONS

ABSTRACT

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.     

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     

Arsenate and Dichromate Removal Efficiency of a New Calix[4]arene Impregnated Resin

This article highlights a newly synthesized 5,11,17,23-tetra-tert.butyl-25,26,27,28-tetrakis(hydrazidecarbonylmethoxy)-calix[4]arene (3) impregnated resin 4 and it's sorption ability for the removal of selected oxoanions, i.e., (arsenate and dichromate) from aqueous environment. Resin 4 was characterized by using FT-IR spectroscopy, scanning electron microscope (SEM), elemental, and thermogravimetic (TGA) analyses techniques. Sorption study was carried out through static and dynamic methods with continuous flow operation to evaluate the potency of resin 4 under various conditions. Several parameters, such as effect of sorbent dosage, contact time, pH, and initial concentration were optimized through batch experiments. Results justify that % sorption of these oxoanions is highly dependent on pH of the solution. Equilibrium was achieved in 45 min, while sorption efficiency of resin 4 can be better explained by Langmuir and Temkin isotherm models. Furthermore, kinetically both oxoanions follow pseudo second order model. Additionally, sorption of arsenate more appropriately could be explained by Moris-Weber diffusion kinetic model. For the dynamic method, Thomas model was used to calculate the kinetic coefficient (k_TH) and maximum sorption capacity (q_o) of resin 4 which are 3.84 and 5.94 ml mg^?1 min^?1 and 0.185, 0.160 mg g^?1 for As (V) and Cr (VI) respectively. Furthermore, resin 4 has been found as potential sorbent for removal of these oxoanions from contaminated surface water samples.     

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.     

KINETICS OF CADMIUM ION SORPTION ON ION EXCHANGE AND CHELATING RESINS

ABSTRACT

Cadmium sorption from aqueous solutions on sulfonic (C-150) and iminodiacetic (S-930) Purolite macroporous resins was investigated. The influence of operating variables such as initial pH, Cd(H) concentration, time and temperature on the equilibrium parameters was measured. The ion preference and sorption ability of resins, i.e. binding constant (b) and saturation capacity {x_m), derived from sorption isotherm, depend on the functional group structure. The Cd(II) uptake is only particle diffusion controlled. The kinetic parameters, i.e. t_1/2 values for 50 % attainment of equilibrium sorption, rate constant (K¯ ) and diffusion coefficient (D¯) are higher on the sulfonic resin. The “moving boundary” particle diffusion model fits the entire ion sorption process on chelating resin, but only the initial sorption on the sulfonic resin, confirming the difference in chemistry between chelation and ion exchange.     

Diffusion coefficient and equilibrium solubility of water molecules in biodegradable polymers

The diffusion coefficient and solubility of water molecules were measured in polyglycolide (PGA), poly(L ‐lactide) (PLLA), poly[(R)‐3‐hydroxybutyrate] (PHB), poly(ϵ‐caprolactone) (PCL), and Skygreen^R (SG). The diffusion coefficient and equilibrium solubility decreased in the order SG > PCL > PLLA > PHB > PGA and PGA > SG > PLLA > PHB > PCL, respectively. The diffusion coefficient and solubility of water at low sorption temperature in PHB varied according to the initial crystallinity of the matrix polymer even though crystallization of PHB molecules took place during the sorption experiment. In contrast, the amorphous PLLA and the crystalline PLLA showed an almost identical diffusion coefficient and solubility of water, in spite of the fact that the amorphous PLLA remained practically amorphous during the whole sorption procedure. A strong correlation existed between the water solubility and the surface tension or contact angle of the polymer matrix. The water diffusivity in PGA was almost 2 orders of magnitude lower while water was more soluble in PGA with a lower heat of sorption than that corresponding to the other more hydrophobic polymers, indicating that the transport of water molecules in PGA followed the solution–diffusion model. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 1716–1722, 2000     

建材VOC散发过程模拟与传质参数测定新方法

建材中挥发性有机化合物(VOC)的散发是一个复杂传质过程。为准确把握传质特性,首先建立了一套描述干建材散发行为的显性完全解析模型,适用于模拟对人体最不利的无换气情况;代入有关文献中的传质参数预测了环境舱浓度,与文献中对应的实验数据及数值算法预测值吻合良好。然后基于对模型的分析提出一套简便快捷的实验方法,能够利用不同VOC背景值下干建材在密闭舱中散发的平衡浓度或逐时浓度,求取预测散发过程的4个重要的传质参数:可散发浓度C₀、扩散系数D、分配系数K和对流传质系数h_m;实验部分测算了两类密度板中甲醛散发的C₀、D、K、h_m,代入数值算法预测了密闭舱和直流舱的环境舱浓度,与实验数据吻合良好。该套模型和测定方法能够应用于建材散发的模拟研究。     

Mathematical Relationship Between Surface Emission and Diffusion Coefficients

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.     

On the use of internal mass transfer coefficients in modeling of diffusion and reaction in catalytic monoliths

We utilize the recently developed concept of internal or intraphase mass transfer coefficient to simplify the problem of diffusion and reaction in more than one spatial dimension for a washcoated monolith of arbitrary shape. We determine the dependence of the dimensionless internal mass transfer coefficient (Sh_i) on washcoat and channel geometric shapes, reaction kinetics, catalyst loading and activity profile. It is also reasoned that the concept of intraphase transfer coefficient is more useful and fundamental than the classical effectiveness factor concept. The intraphase transfer coefficient can be combined with the traditional external mass transfer coefficient (Sh_e) to obtain an overall mass transfer coefficient (Sh_app) which is an experimentally measurable quantity depending on various geometric and transport properties as well as kinetics. We present examples demonstrating the use of Sh_app in obtaining accurate macro-scale low-dimensional models of catalytic reactors by solving the full 3-D convection-diffusion-reaction problem for a washcoated monolith and comparing the solution with that of the simplified model using the internal mass transfer coefficient concept.     

Mechanism of iodine vapor sorption in polyacetylene

Simultaneous measurements of weight uptake and conductivity variation during iodine vapor sorption of polyacetylenes (PA) at 20–25°C show that the sorption is a two-stages process. For the first stage, iodine diffusion to the fibrilar surface of the cis-rich and trans-rich PA is of Knudsen type pore diffusion as supported by the sorption measurements of hexane for the cis-rich PA, and has a diffusion coefficient in the order of 10⁷ cm²/sec. Conductivity of the PA rises rapidly and reaches a maximum at end of the stage. For the second stage, the diffusion is more restrictive and has a diffusion coefficient lower than the first stage by a factor of about 10 due to multilayer sorption of iodine, which leads to a decrease in the pore diameter and therefor the diffusion rate. In addition to the restricted diffusion in the second stage, diffusion of the iodine molecules adsorped on the fibrilar surface into the interior of the fibrils is appreciable for cis-rich PA (leading to a conductivity drop) and is negliglible for tran-rich PA (leading to insignificant variation in conductivity).     

Sorption Properties of Ni2+ Loaded Iron Hydroxide Toward Phosphate and Cd2+ Ions

Abstract

Ni²⁺ ion loaded iron hydroxide was used for sorption of both Phosphate and Cd²⁺ ions from aqueous solution at 303 K and pH range 3–7 in 0.1 M KNO₃ as a background electrolyte. The surface charge density of the loaded surface was computed from potentiometric titrations both in the presence of 0.1 M KNO₃ and 0.01 M Phosphate. It was noted that the presence of phosphate alters the surface and decreases the point of zero charge toward lower pH. The sorption of phosphate was observed to increase with concentration and decrease with increase in pH of the system. However, the sorption of Cd²⁺ was observed to increase both with increase in concentration and pH of the system. The Freundlich equation was successfully applied to the sorption data which gives a straight line with the R² (regression coefficient) values in between 0.94 to 0.99.     

Solubility and diffusion coefficient of supercritical-CO2 in polycarbonate and CO2 induced crystallization of polycarbonate

The solubility and diffusion coefficient of supercritical CO₂ in polycarbonate (PC) were measured using a magnetic suspension balance at sorption temperatures that ranged from 75 to 175 °C and at sorption pressures as high as 20 MPa. Above certain threshold pressures, the solubility of CO₂ decreased with time after showing a maximum value at a constant sorption temperature and pressure. This phenomenon indicated the crystallization of PC due to the plasticization effect of dissolved CO₂. A thorough investigation into the dependence of sorption temperature and pressure on the crystallinity of PC showed that the crystallization of PC occurred when the difference between the sorption temperature and the depressed glass transition temperature exceeded 40 °C (T − T_g ≥ 40 °C). Furthermore, the crystallization rate of PC was determined according to Avrami's equation. The crystallization rate increased with the sorption pressure and was at its maximum at a certain temperature under a constant pressure.     

UNSTEADY-STATE SORPTION MEASUREMENTS A TEST OF DIFFERENT METHODS TO EVALUATE THE DIFFUSIVITY FROM UNSTEADY-STATE SORPTION MEASUREMENTS

ABSTRACT

Different simple methods for evaluating the diffusivity from a sorption measurement are presented and tested when there is a surface resistance (a limited mass transfer coefficient), a non-perfect step in relative humidity or non-Fickian effects. It is concluded that the different methods have different qualities and it is impossible to give a general recommendation on which method to use. The best tactic is probably to use more than one of the methods. A knowledge of the experimental conditions will help in choosing the best method.     

Transport of water and methanol vapors in alkyl substituted poly(norbornene)

This report examines the ability of aliphatic-substituted polynorbornenes to separate methanol vapor from water-wet air streams. Single component transport properties of water and methanol were measured for each candidate polymer at 30 °C. The diffusion coefficient, sorption coefficient, and permeability were determined by measuring the rate of mass gain or loss of a sample due to sorption or desorption. The ratio of the permeabilities of the pure components yielded the ideal selectivity. The materials studied in this were methyl-, nbutyl-, nhexyl-, and ndecyl-polynorbornene. These polymers were glassy, with T_gs of 150-380 °C, and have relatively high fractional free volumes (16-19%). The results of the study showed polynorbornenes were methanol selectivity with ideal selectivities of 1.2-9. This permeation selectivity is due primarily to the solubility selectivity of the materials.