Study of the mechanism of pore diffusion in batch adsorption systems

In this study the film-pore diffusion model was applied to describe system transport kinetics of three basic dye-carbon systems, namely Basic Blue 69, Basic Red 22 and Basic Yellow 21. The mass transfer parameters evaluated were the external mass transfer coefficient k_f (cm s^?1) and the effective diffusivity D_eff (cm² s^?1). A single k_f value was sufficient to describe each dye system: these were 0.15 × 10^?2, 0.20 × 10^?2 and 0.50 × 10^?2 cm s^?1 for BB69, BR22 and BY21, respectively. The effective diffusivity was found to have values much larger than those of pore diffusivities calculated from liquid diffusivities and its value decreased with increasing initial dye concentration. This was attributed to the effect of surface diffusion, hence pore diffusivity was exchanged by the effective pore diffusivity in the model. The present model was solved by the exponential curve fit technique; results were expressed in the form of experimental and theoretical Sherwood Numbers compared in terms of the residual.     

Analytical solution using a pore diffusion model for a pseudoirreversible isotherm for the adsorption of basic dye on silica

An analytical solution for a two resistance mass transfer model explaining the adsorption of Astrazone Blue dye (Basic Blue 69) onto Sorbsil silica has been developed. The model includes a film mass transfer coefficient, k_f1 = 80 × 10⁻⁶cm·s−1, and an internal effective diffusivity, D_eff = 18×10⁻⁹cm²·s⁻¹ which controls the internal mass transport processes based on a pore diffusion mechanism.     

Effect of air temperature and velocity on moisture diffusivity in relation to physical and sensory quality of dried pumpkin seeds

Understanding the effect of drying process parameters on food quality is helpful in process optimization and control. The objective of this work is to understand the effect of mild and harsh effective moisture diffusivity (D_eff), varied by air temperature and velocity, of drying processes on the physical and sensory quality of flat food products. Pumpkin seeds were selected as a food representative. It was found that increments of air temperature and velocity resulted in increased D_eff and brown color on seed hull surfaces and embryos, but decreased hardness of seed embryos. Changes in taste and aroma of seed embryos were able to be sensed. Indicating that D_eff is related to seed physical quality. Similar phenomena occurred with both tray and fluidized bed drying. Air temperature, velocity, and D_eff should be controlled to ensure the best dried flat food products. Mild drying conditions are potentially preferred for good physical and sensory quality.     

Diffusion of semi‐flexible polyelectrolyte through nanochannels

The diffusion of sodium polystyrene sulfonate through polycarbonate nanochanels was studied in salt‐free dilute aqueous solution. A stronger molecular weight dependence of diffusion was observed compared to free diffusion in dilute solution. Scaling exponentials relating polymer size to diffusivity were between Flory's theory (D_eff ∝ N^?0.6) and Rouse's model (D_eff ∝ N^?1), revealing a crossover regime from 3‐D diffusion to 1‐D diffusion. Diffusion was less hindered for the polyelectrolyte (D_eff/D₀), than for a rigid sphere, when the polymer/channel size ratio exceeded 0.2. This is attributed to elongated chains with reduced frictional hindrance. Simulation of the confined diffusion based on an elongated cigar model gave D ∝ N^?1 ${R_{\rm t}^{2/3}}$ while the experimental results agree with D ∝ N^?0.94${R_{\rm t}^{ 2/3}}$ . For charged polyelectrolytes, the transition to 1‐D diffusion therefore begins before the polymer radius of gyration exceeds the channel size contrary to model assumptions. We attribute this to the charged nature of the polyelectrolytes causing extended chain conformations. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

Diffusion of electrolytes in dilute polymer solutions

The diffusion of sodium chloride in dilute solutions of carboxymethylcellulose (CMC) was studied using the porous frit technique. It was found that at relatively low CMC concentrations the diffusivity of sodium chloride (D) is higher than that in water (D₀); with increasing CMC concentration, the diffusivity decreases gradually and reaches a value lower than that in water. The relation between the diffusivity (D) and CMC concentration (C_p) was represented by the equation D = a – bC_p. The effect of temperature on the diffusivity of sodium chloride in CMC solutions was found to obey the Arrhenius equation with an activation energy of 2230 cal/mol.     

Experimental and fundamental critical analysis of diffusion model of airflow drying

Scientific literature of agromaterial drying present contradictory conclusions in terms of the kinetic effect of airflow velocity. Some authors confirmed that it does not trigger any modification of drying, while some articles tried to establish empirical models of the effective diffusivity D_eff versus the airflow velocity, what is fundamentally erroneous. By analyzing internal and external transfer phenomena, this research aimed at recognizing that once air velocity is higher than a critical airflow velocity (CAV), the internal transfers become the limiting phenomenon. CAV depends on the effective diffusivity and the product size. It was calculated in the cases of two studied raw materials (apple and carrot), differently textured by instant controlled pressure drop (DIC). Values of CAV greatly depend on diffusivity of water within the matrix. At temperature T?=?40°C, they were 1?m/s for untreated carrot and 2.1?m/s for DIC-textured carrot, whose D_eff values were 1.31 and about 3?×?10^?10?m²/s, respectively. Also, at temperature T?=?40°C, they were 2.1?m/s for untreated apple and 3?m/s for DIC-textured apple, whose D_eff were 1.4 and about 10.4?×?10^?10?m²/s, respectively.     

Influence of morphology on the transport properties of polystyrene/polybutadiene blends: 2. Modelling results

Unsteady-state and steady-state models are used to predict the effective diffusion coefficients, D_eff, of small molecules in randomly interspersed polystyrene-polybutadiene blends and comparisons are made with experimental values obtained by sorption measurements. It is found that the polybutadiene and polystyrene phases are not topologically equivalent. This fact is responsible for the discrepancies between the experimental values of D_eff and the predictions from standard steady-state models.Simulations of sorption experiments based on an unsteady-state model have shown that values of D_eff obtained by different methods from the same sorption experiment differ consistently from each other. This supports the hypothesis that the experimentally observed differences between the various values of D_eff, for a given blend-permeant pair, can be justified solely on the basis of morphological arguments.     

Estimation of surface diffusion coefficient in liquid phase adsorption

An estimation procedure of surface diffusion coefficient, D_s, in liquid phase adsorption was proposed. The procedure is based on a restricted diffusion model, in which D_s is correlated with molecular diffusivity by considering a restriction energy due to an adsorptive interaction between adsorbates and adsorbents. In some adsorption systems, D_s of different adsorbates could be calculated with an error less than about 50% from only one datum of each adsorption equilibrium constant. Irrespective of temperature, the procedure, can be applied for the estimation of D_s even in a wide range of D_s of about 4 orders of magnitude.     

Measurement of lyophilisation primary drying rates by freeze-drying microscopy

The mass transfer processes that influence the rate of primary drying during lyophilisation are characterised by freeze-drying microscopy. A simple diffusion model is shown to describe the observed progress of drying fronts through isothermal, one-dimensional systems of slab and cylindrical geometry. Measured drying rates are modelled by an effective diffusion coefficient, D_eff, which describes the diffusion of water vapour from the inner frozen core through the outer dried cake of the lyophilising mass. In this way, the relationship between the instantaneous thickness of the dried cake and the drying time is determined for different drying temperatures. Values of D_eff determined by analysis of freeze-drying microscopy data for two simple model systems are shown to be in satisfactory agreement with those predicted from theory. Furthermore, the lyophilisation behaviour of a citrate and a tris-HCl buffer are well described by the model and values of D_eff are constant at temperatures below the collapse temperatures of the dried cakes. However, values of D_eff increase approximately four- and six-fold, respectively as the collapse temperatures are approached. Microscopic examination of the citrate buffer dried cake shows its structure to be homogeneous at temperatures well below the collapse temperature, as required by the model, but indicates significant cracking of the cake as the collapse temperature for this buffer is approached. As a result, channeling of the water vapour through the dried cake causes enhanced drying rates. Finally, we show that the total time required to sublime water from aqueous slurries of glass beads in a conventional laboratory lyophiliser are in reasonable agreement with those times estimated using the values of D_eff determined by freeze-drying miscroscopy.     

Separation of surface and bulk phase diffusivities: A kinetic theory analysis

A kinetic model for the effective (e.g. surface plus bulk) diffusivity in a porous bed composed of rectangular channels is presented. The merit of this approach is that no distinction is made between the bulk and adsorbed phase and the effective axial flux can therefore be compared directly with experimental measurements. We utilize the elements of the BGK kinetic equation and express D_eff as an explicit function of the surface potential and collision frequency. Details concerning gas-gas collision dynamics and gas-solid interactions are contained in the relaxation time, τ, and the external force, F_s, respectively. For definiteness, the analysis is limited to hard sphere gas-gas collisions, although no special significance is implied by this choice. The assumption of a discontinuous gas-solid interaction potential results in a separation of the form D_eff = D_g + K_eqD_s, with interactive modification of both D_g and D_s. Preliminary calculations indicate that only when K_eq, is large will surface diffusion become a significant mode of transport.     

A kinetic Monte Carlo approach to diffusion in disordered nanoporous carbons

The efficiency of different technological processes where nanoporous carbons are used depends on their storage capacity and an appropriate gas transport. Different experimental and theoretical works that relate storage to global structural parameters of the solid such as specific surface area (SSA) or total pore volume (V_t) can be found in literature. The structure–transport relationships have been less studied. The combined use of the truncated pore network model (TPNM) and the kinetic Monte Carlo (KMC) method is proposed in this work to find hydrogen and methane effective self-diffusivities (D_eff) and to delve into those relationships. It was found that for Knudsen and free molecular diffusion in the simulated materials, the D_eff/V_t vs. SSA graphic nearly follows a power law. The KMC/TPNM approach was also used to predict the self-diffusion coefficients of hydrogen in Vulcan XC-72 and of methane in a carbon aerogel. The obtained values are within the expected range. KCM/TPNM is computationally fast and it allows a study of the diffusion synchronously and globally in the network, avoiding thus its fractionation in single pores and the use of just one geometric model to describe the porous spaces.     

Color removal from dye wastewaters by adsorption using powdered activated carbon: Mass transfer studies

Color removal from synthetic dye wastewater which typically emanates from the Taiwan textile industry has been studied using powdered activated carbon (PAC) as an adsorbent. The CIE colorimetric system has been used in the measurement of color for the treatment of disperse-red-60 dye wastewater. The effect of contact time, dye concentrations and PAC dosage on color and color removal has been investigated. A film-pore double resistance diffusion model for mass transfer has also been used in this study to determine the effective diffusivity, D_eff, for the adsorption of disperse-red-60 dye wastewater to PAC.     

Free volume model and diffusion of organic solvents in natural rubber

The concentration dependence of the diffusion coefficients and the equilibrium isotherms of benzene, o-xylene, ethylbenzene, and chloroform in natural rubber membranes at 303 K were experimentally determined. The data were used to critically test the predictive capability of the Vrentas–Duda free volume model. It was found that although the model works well for some polymer–solvent systems such as toluene-polystyrene, the use of some of the parameters from pure component properties yields unacceptably low diffusion rates for the rubber–solvent systems studied. The parameters D_o1 and ξ obtained from experimental zero-concentration diffusivity data, and parameter V?₁^* calculated from the solvent molecular geometries are needed to achieve good predictions. The diffusion coefficients described by the revised model can be used to predict quite well the breakthrough times of the rubber–solvent systems that were also experimentally measured by a permeation method. © 1995 John Wiley & Sons, Inc.     

Gas and water transport properties of epoxy–amine networks: Influence of crosslink density

The gas and water transport properties were studied on seven epoxy–amine networks that differ by their di/mono epoxy composition and their cure cycle. The characteristic parameters of transport M_∞ and D were determined in each case, and they show that the solubility and diffusion coefficients are influenced by their composition. The solubility increases with the extend of cure and with the decrease of dangling chains, and in both cases the evolution of the diffusivity is inverse. The variations of the solubility and diffusivity have been discussed as a function of two parameters: polar groups and unrelaxed holes of the glassy networks. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 80: 2058–2066, 2001     

Dispersion approximations to the multicomponent convective diffusion equation for chemically active systems

We show that expressions which approximate the asymptotic values of the coefficients K_eff, V_eff, D_eff in the dispersion formulation of the vector convective diffusion equation for fully developed lammar flow in a tube with catalytic walls can be derived very simply using an extension of the intuitive procedure of Taylor[1].     

The apparent and effective chloride migration coefficients obtained in migration tests

The apparent (D_app) and effective (D_eff) migration coefficients obtained in chloride migration tests are investigated in this study. The presented D_app profiles in concrete show that the apparent migration coefficient is strongly concentration-dependent. As demonstrated, the binding of chlorides during the migration tests is very low at low free-chloride concentrations and therefore the chloride penetration front progresses throughout the concrete only slightly retarded by the binding. The diffusion flux during migration tests is shown to be insignificant compared to the migration flux. The D_RCM obtained from the Rapid Chloride Migration (RCM) tests are found to be equal to the computed D_app at the locations of the chloride penetration fronts, which gives an indication that the D_RCM represents only the migration coefficient at the front. A linear correlation is found between the D_RCM obtained from the traditional RCM model and the D_eff obtained from the chloride transport model which includes non-linear chloride binding and concentrations in non-equilibrium.     

Characterization of Structure and Transport in Porous Media by Pulsed Field Gradient (PFG) NMR Technique. Part I: Master Curve and Characteristic Inner Length

Systematic experimental studies focusing on the practical application of observation time dependent pulsed‐field‐gradient (PFG) NMR were performed. The objective was to provide engineering scientists with a reliable experimental tool for characterizing the structure and transport in fluid‐filled porous media. Observation time dependent self‐diffusion in glass bead packs as model systems was investigated, where the diffusing species (molecules of the solvent or dissolved particles) served as probes for the confining geometry in the porous medium. First, the basic question whether the obtainable structure information is independent of the actual mobility of the diffusing probe particles was examined experimentally. It could be demonstrated that plotting the normalized time‐dependent diffusion coefficient D(t)/D₀ versus the actual migration length l_D(t) during a given observation time t indeed yields a characteristic “master curve” that is independent of the mobility of the diffusing species, thus reflecting, as desired for a reliable method, solely the effects of the confining geometry of the porous system of interest. It was further shown that from the master curve a new quantity, i.e., a “characteristic inner length” or “correlation length” ξ_D can be derived that corresponds to a path length in the porous medium, after which particles in the pore fluid experience an averaged restricting geometry and diffuse with an effective diffusion coefficient D_eff. It turned out that ξ_D is surprisingly short, that is, in the order of the bead radius. Moreover, it was demonstrated that normalization of this migration length with the bead radius yields a common master curve for all monodisperse bead packs used and thus, it is obviously possible to derive and record master curves for different kinds of packs, beds or other porous media as references that can be used to characterize or certify the kind of the porous matrix of interest.     

Binder removal studies in ceramic thick shapes made by laminated object manufacturing

Binder removal was studied in a body of alumina made by Laminated Object Manufacturing (LOM) with polymethyl methacrylate as binder, benzyl butyl pthalate and polyethelene glycol as plasticisers and glyceryl trioleate as dispersant with an optimized solids loading of 58% by volume. The burnout of the organics from laminated cuboids of different thickness was carried out in nitrogen atmosphere at different temperatures between 200 and 300 °C. The variation of the organic content from the core to the surface was also determined by thermogravimetric analysis (TGA) and FTIR studies. The organic content of the body at a particular temperature exposed for a particular time was modeled based on the parameter, S (∞ D_eff/l²) where D_eff is the effective diffusivity and l is the diffusion path length. The variation of S as a function of exposed surface area per unit volume of the cuboids is linear. Since the dependence of S on temperature follows an Arrhenius behaviour, the value of S can be predicted at any temperature and for any exposed surface area per volume of the cuboids. Thus the model can be used to estimate the organic content of the body exposed for a certain duration at a particular temperature.     

Sorption of dissolved organics from aqueous solution by polystyrene resins—II. External and internal mass transfer

External and internal mass transfer were characterized for the sorption of 2,4-dichlorophenol and p-nitrophenol on polystyrene resins. An evaluation method is presented which estimated the external mass-transfer coefficient, κ_L, from the concentration uptake in an integral fixed-bed reactor. The experimental κ_L values are compared with values predicted using four different empirical correlations. The effect of axial dispersion on the values of κ_L was shown to be small under the given conditions.To quantify the intraparticle mass transfer, an effective diffusion coefficient, D_eff, was estimated using the pore diffusion model interpretation of data from a differential column reactor within a recirculated batch system. The D_eff values depend on both resin structure and solute properties. Values of D_eff between 0.3 × 10⁻⁹ and 1.8 × 10⁻⁹ m²/s were observed for macroporous resins, which exceed values that are characteristic of pore diffusion, and a value of 0.6 × 10⁻¹¹ m²/s was estimated for a microporous resin.     

Drying kinetics and microstructure evolution of nano-zirconia under microwave pretreatment

The effects of microwave power and sample quality on microwave drying kinetics and characteristics of zirconia were studied. It is found that by increasing the microwave power and decreasing the sample mass, the surface diffusion coefficient (D_eff) appears to an upward tendency. The corresponding value D_eff at a sample mass of 10, 20, 30, and 40g are 1.849E-14, 2.443E-14, 3.210E-14, and 3.278E-14 m²/s, respectively. The corresponding value D_eff at a microwave power of 300, 400, 500, 600, and 700W are 1.270E-14, 1.784E-14, 2.619E-14, 3.392E-14, and 4.497E-14 m²/s, respectively. Besides, the materials were characterised by X-ray diffraction (XRD), scanning electron microscopy (SEM), Fourier transforms infrared spectroscopy (FT-IR) to evaluate the changes of materials before and after drying. The results show that microwave accelerates the drying of zirconia and increases its dispersibility. The heat conduction direction of microwave drying is the same as that of moisture diffusion, which avoids being affected by heat inertia and heat transfer loss. The drying process is fast and efficient, and the microwave directly penetrates the product, avoiding the disadvantage of slow evaporation caused by the temperature gradient.