Stagnant Film Research Articles

Fate and Transport of Combined Residual Chlorine in Small Stream

Concentration profiles of combined residual chlorine (CRC) are measured in a small stream just below a municipal wastewater outfall. Loss of CRC is found to be due primarily to volatilization. Vertical and longitudinal concentration profiles are successfully modeled with a 2‐dimensional diffusive transport model, not incorporating a stagnant film. To calibrate this model, a Henry's Law constant for CRC is determined in the laboratory. The model proves superior in the case studied to the classical two‐film model for the prediction of depth‐integrated combined residual chlorine. The diffusive transport model is nearly identical to the two‐film model under conditions of imposed gas‐phase limitation. However, the two‐film theory predicts liquid‐phase limitation for the data in this study, and as a result, significant differences exist between the two sets of model predictions. Field data clearly indicates gas‐phase limitation, which was predicted by the 2‐dimensional diffusive transport model.

Kinetics and mechanism of metal extraction with acidic organophosphorus extractants (I): Extraction rate limited by diffusion process

The extraction mechanism of metal ion by acidic organophosphorus extractant was discussed by taking into account the physicochemical properties of extractant. Since the acidity of the extractant is higher than that of chelating agent, the rate of complex formation of metal ion with the extractants is rapid compared to that with the chelating agents. It is found that the rate determining step of the extraction such as Cu(II) and Co(II) by 2-ethylhexyl phosphonic acid mono-2-ethylhexyl ester is not the complex formation at the interface, but the diffusion of the metal complex in the organic stagnant film, which is instantaneously formed at the interface. This result was also confirmed by the stirring speed dependence of the initial extraction rate. Furthermore, the time course of Co(II) extraction from the initial stage to the equilibrium state was quantitatively interpreted by considering only the diffusion rates of reactants and products in both stagnant films.

The role of water in the low-temperature hydrogen reduction of hematite into iron, and the role of the surface oxygen chemical potential in double reactions

The striking role played by water vapour during the low temperature reduction of hematite into magnetite by carbon monoxide has been reported but the possibility that this effect was not due to hydrogen produced from the H 2O+CO shift reaction had to be eliminated. A very similar effect by water vapour during the reaction of hematite with hydrogen is reported, the nucleation of magnetite is more regularly distributed and the magnetite layer is much more regular, but the rate of reaction is lower. As the reduction proceeds to give more iron, another striking effect is observed: in the presence of water, there is no overlap of the steps hematite → magnetite and magnetite → iron; the reduction of magnetite starts once the hematite has disappeared. It is shown that rather than being a problem of transfer through a stagnant film, the clearly different steps may be attributed to the fact that when adsorption-desorption is not fast enough, the surface oxygen chemical potential differs from that of the gas.

GENERAL LINEARIZED MATRIX THEORY OF ELECTROCHEMICAL MASS TRANSFER WITH APPLICATIONS TO ELECTRODE PROCESSES AND ION EXCHANGE

Abstract A general linearized matrix theory for electrochemical mass transfer is presented. The accuracy of the theory when applied to cases where electric current exists is tested by comparing results with exact calculations of the effect of ionic migration on limiting currents at a rotating disk electrode with laminar flow and at a flat plate electrode adjacent to a stagnant fluid film. Application of the theory is further illustrated by using it to calculate the effect of ionic migration on limiting currents in a turbulent, law-of-the-wall boundary layer. Finally, the theory is used to evaluate commonly used “film-model” methods for generalizing binary mass-transfer correlations to multicomponent nonideal mixtures. For this last purpose, a film-model generalization method was developed for nonideal electrolyte mixtures. The method is equivalent to that used by Krishnamurthy and Taylor (1985) for mass-transfer in nonideal nonelectrolyte mixtures. Results using the film-model method are compared to matri...

Kinetic mechanism of metal extraction with hydroxyoximes.

The kinetic mechanism was elucidated for metal extraction with hydroxyoximes. The reaction of the 1:2 metal–chelating agent complex formation determines the rate of copper extraction with the chelating agents, which are surface-active and less soluble in the aqueous phase. It is found from the results about the diluent’s effect on the rate that this complex is formed at the interface through the reactions between the adsorbed 1:1 complex and the chelating agent dissolved in the aqueous phase, and also between the adsorbed chelating agent and the 1:1 complex in the aqueous phase. The overall rate constant of this interfacial reaction can be related to the pKa value of the chelating agent. The 1:1 complex formation in the stagnant film of the aqueous phase controls the rate of copper extraction with the chelating agent, which is surface-inactive and soluble in the aqueous phase. The catalysis of α-hydroxyoxime in the copper extraction with β-hydroxyoxime is caused by the fast reaction between the 1:1 copper–β-hydroxyoxime complex adsorbed at the interface and the α-hydroxyoxime dissolved in the aqueous phase. The question of increasing the rate of extraction is examined by focusing on the physical properties, the partition coefficient of the chelating agent between the aqueous and the organic phases, and the surface activity of the chelating agent. Modification of the substitutional group in the hydroxyoximes is also examined as a means of increasing the rate.

PARAMETER ESTIMATION IN CAKE WASHING

Abstract The uniform-film model for cake washing considers the cake to be composed of channels in which a stagnant film of solute-rich filtrate adheres to the surface of the channel. Two independent methods are presented for an unbiased evaluation of the uniform-film model parameters. The first is an iterative procedure which requires adjustment of only one model parameter, and the second is a direct analytical procedure based on the method of moments. Both require experimental data to identify the model parameters. Application of the two methods give wet-base porosities greater than unity which suggests that the experimental data used to determine the model parameters is beyond the domain of applicability of the uniform-film model.

Membrane formation through the phase‐inversion of Poly(vinylidene)/Dimethylformamide solutions. Experimental and theoretical study of evaporative flux

AbstractThe phase inversion of a polymeric solution is modeled assuming the validity of the diffusion equation and the existence of a mass‐cumulative stagnant film of vapor on the cast membrane. Unlike previous analyses, the model predicts a maximum in the evaporative flux leaving the polymeric solution. A comparison of theory with experiments indicates that the solvent flux is predicted only qualitatively. The discrepancy is probably due to some assumptions introduced, for mathematical tractability, in the model.

Dissolution of ionizable drugs in buffered and unbuffered solutions.

The dissolution kinetics of ionizable drugs (weak acids or bases) are analyzed with a mathematical model derived from the theory of mass transfer with chemical reaction. The model assumes that the overall process is diffusion limited, that all the reactions are reversible and instantaneous, and that dissolution and reaction are limited to the stagnant fluid film adjacent to the solid phase. Dissolution into buffered and unbuffered aqueous solutions are considered separately, with convenient analytical solutions obtained in both cases. In addition, equations for the time to partial and complete dissolution are derived. The dissolution rate is shown to be dependent on the pKa and intrinsic solubility and the medium properties, i.e., pH, buffer capacity, and mass transfer coefficient. Equations of a form analogous to the nonionized case are derived to account explicitly for all these factors, with dissolution rates expressed in terms of the product of a driving force (concentration difference) and resistance (inverse of mass transfer coefficient). The solutions are in an accessible analytical form to calculate the surface pH and subsequently the surface concentrations driving the drug dissolution. Numerical examples to illustrate dissolution into unbuffered and buffered media are presented and the results are shown to be in accord with experimental data taken from the literature.

Kinetics of release from enteric-coated tablets.

Controlled and localized release of drugs in the intestine can be achieved by enteric coating. The design of enteric-coated tablets has so far remained empirical, in part because of the lack of a quantitative description of the drug release kinetics. In this paper, a mathematical model is presented that describes the dissolution of the polymer coating and release kinetics of weakly acidic drugs from enteric-coated tablets in buffered media. This model can also be used to predict the time of onset of core disintegration. The model assumes that the release rate is limited by diffusion, and furthermore, all the reactions are considered as reversible and instantaneous. Dissolution and reaction are assumed to take place in the polymer layer and a hypothetical stagnant liquid film adjacent to the polymer layer (the classical film theory approach). The dissolution of the enteric coating is found to depend on the intrinsic solubilities and pKa's of the drug and polymer and the medium properties. The release rate of the drug is found to depend on the intrinsic solubilities and pKa's of drug and polymer, the medium properties, i.e., pH and buffer capacity, and a mass transfer coefficient. Explicit relationships between the release rates and all these factors are derived. Successful prediction of experimental data indicates that the model provides an adequate description of release from enteric coated tablets. Limitations of the model and its potential application to the design of appropriate in vitro testing conditions and to the formulation of enteric coated tablets are also discussed.

On the effective viscosity of a suspension of drops partially coated by a stagnant film

A simple model is presented for determining the effective viscosity of a dilute suspension of small drops or bubbles partially coated by a stagnant surface which is due to the presence of a surfactant. The boundary value problem for a single drop or bubble having a stagnant film is solved approximately using a proven minimization technique. The overall mean suspension viscosity is found as a function of the fraction of the drop surface area that is immobile or contaminated.

Kinetics of the formation of nickel aluminide coatings on pure nickel by chemical vapour deposition

A chemical vapour deposition process was used to deposit aluminium onto a pure nickel substrate which was not in contact with the powder mixture. After deposition for about 1 h the aluminium content at the surface reached a high value and then remained almost constant with time. The kinetics of the process is governed by a combination of solid and gas diffusion rates, and a qausi-steady-state appears to exist at the gas-coating interface. The experimental results can be explained satisfactorily by a combination of a stagnant film model for gaseous diffusion and solid diffusion rates.

Dissolution of Acidic and Basic Compounds from the Rotating Disk: Influence of Convective Diffusion and Reaction

A mass transfer model was developed to describe the dissolution and reaction of acidic and basic compounds from a rotating disk in unbuffered water. Dissolution of two carboxylic acids, 2-naphthoic acid (1) and naproxen [(+)-6-methoxy-α-methyl-2-naphthaleneacetic acid, 2], and the free base, papaverine (6,7-dimethoxy-1-veratryliso-quinoline, 3), in aqueous solutions (μ=0.1 with KCl) at 25°C were investigated. An automated dissolution apparatus, which consisted of microcomputer-controlled autoburets, was constructed to monitor and adjust the pH of the aqueous solutions during the experiments. Unique features of the mass transfer model include treatment of mass transfer as a convective diffusion process rather than a stagnant film diffusion only process; treatment of ionization and acid-base reactions as heterogeneous reactions; use of experimental diffusion coefficients for all species, particularly H+ and OH−; and application of boundary conditions that specify flux for surface ionization produced species. The model accurately predicted the dissolution rate assuming the solubility, pKa, and diffusion coefficient of the compound were independently known. The model also predicted pH at the solid-liquid surface, the flux of H+ from the surface, and the contribution of A− to the total acid flux as a function of bulk pH of the aqueous solution.

Response of Radioactive Trace Metals to Acid–Base Titrations in Controlled Experimental Ecosystems: Evaluation of Transport Parameters for Application to Whole-Lake Radiotracer Experiments

Radiotracer experiments were carried out in 20 enclosures located in two lakes at the Experimental Lakes Area (ELA), northwestern Ontario, to study pathways of trace metal removal from the water column of shallow lakes. Two removal mechanisms were characterized: (1) sorption to and subsequent transport with falling particles and (2) direct adsorption to surface sediments. Our approach was to measure independently the kinetics of radiotracer sorption, fluxes and concentrations for particles, particle settling velocities, and the "equivalent stagnant boundary film." Our radiotracer results enabled us to test the sensitivity of the tracer removal rates on these rate-determining processes using a numerical transport model. Acid titrations of whole ecosystems revealed that some trace metals (e.g. Mn, Co, and Zn) can diffuse back to the water column as the pH is lowered from 6.5 to 4.8 after 18 d, while others remain tightly bound (e.g. Sn, Fe, Se, Cr, Ag, and Hg isotopes). Subsequent CaCO3 additions to bring back the pH to its original value restored the initial removal conditions for acid-sensitive radiotracers, indicating that the pH sensitivity is reversible. Transport parameters for particle-related pathways or diffusive pathways across the sediment–water interface obtained from our enclosure experiments were used to predict the removal rates of "particle-reactive" 60Co and the "diffusive" pathway tracer 134Cs observed in earlier experiments where radiotracers were added to whole lakes or to larger enclosures.

Design calculations for countercurrent gas-liquid reactors

A mathematical model has been developed for a countercurrent differential contactor in which a gas-liquid reaction occurs. The local mass transfer rates near the gas-liquid interface are computed from a stagnant film model that includes a second-order reaction term. The contactor model accounts for enhancement of the gas absorption rate as well as the contributions to liquid reactant conversion from both the boundary layer region and the bulk liquid. Orthogonal collocation based on Lobatto polynomials has been used to solve the nonlinear equations for both the local mass transfer rates and the macroscopic differential mass balances. Parametric studies indicate that the number of transfer units, the chemical reaction kinetics and the absorption factor all play important roles in determining reactor performance. Collocation techniques are also employed to solve the design equations that include liquid-phase axial dispersion in the countercurrent gas-liquid reactor. These techniques have been found to be sufficiently efficient and robust that they should be useful for reactor design and parameter estimation applications.

Phase transfer catalysis with rapid reaction

We consider the process of solid-liquid phase-transfer catalysis with rapid homogeneous reaction. The mathematical problem based on a simple stagnant film model (Melville and Goddard, 1985, Chem. Engng Sci. 40, 2207–2215) is analysed. The case of rapid and reversible, homogeneous reaction is first discussed. It is shown that a boundary layer in the concentration gradient but not in the concentration develops at the surface of the solid reactant. A method for the explicit calculation of the performance of the process for quite general conditions is presented. The case of rapid and irreversible, homogeneous reaction is next considered. The structure of the thin reaction zone that may develop as the reaction tends towards irreversibility is analysed. It is found that, in this case, the width of the reaction zone is a function of both the bulk and the surface Damköhler numbers.

The role of the ocean in a regional sulfur cycle

Dimethylsulfide (DMS) concentrations were measured in ocean waters along the West Coast of the United States on three cruises during 1983 and 1984. Concentrations in surface waters ranged from 13 to 380 ng S/L with a summer average of 60 and a winter average of 20 ng S/L. The flux of sulfur from the ocean to the atmosphere was calculated using the stagnant film boundary layer model to be 28 mg S/m2/a. On the basis of a non‐sea salt sulfate residence time of 5 days, the calculated net flux of biogenic sulfur to the West Coast of the United States is 0.045 Tg/a. This is 4–13% of the combined total anthropogenic emissions from the western United States.

Physical, chemical and biological characteristics* of CO2 gas flux across the air‐water interface

Abstract Carbon dioxide gas flux across the air‐water interface is most often treated as a ‘simple’ physical process, primarily responding to wind speed and water temperature. Available experimental data yield an exponential regression equation relating wind speed to the thickness of a stagnant boundary film through which gas diffuses to or from the water. Flux of CO2 is influenced by CO2 hydration reactions in the stagnant boundary layer. High pH and a thick stagnant boundary layer favour chemical enhancement of the CO2 gas flux.The rate of CO2 flux reflects the sum of net organic metabolism plus CaCO3 reactions. Some interesting gas‐flux constraints on the rate of net organic carbon production and on global geochemical cycling of CaCO3 emerge. At high pH (circa 10), the maximum net organic carbon production which can be supported by CO2 flux across the air‐water interface is about 0.06 mol C m&2 d&1. On a global scale, organic C, not atmospheric C, appears to account for the ‘CO2’ term in the classical CaCO3 dissolution‐precipitation reaction.

Mathematical simulation of a biofilm process

A mathematical model has been developed for a fixed-film biological process (film flow over a flat plate) that describes bulk liquid transport, diffusional transport of oxygen and organics across a stagnant film, diffusional transport of oxygen and organics into the biofilm, biochemical reactions by the individual cells within the biofilm, biofilm growth, and cell density changes within the biofilm due to cellular decay. Simulation studies are presented to show how contact time and diffusion layer thickness affect process performance.

Influence of convection on lamellar spacing of eutectics

The influence of convection on the concentration field in front of a growing lamellar eutectic was determined numerically for 50% and 33% volume fraction eutectics. The results were used to calculate the change in lamellar spacing λ, assuming the eutectic grows at the spacing giving the minimum interfacial undercooling. The value of λ increases with the square of the stirring intensity, especially at low growth rate V. Large increases in λ can occur for typical crystal growth situations. The fictitious stagnant film model yields erroneous results that predict little effect of convection on λ and an incorrect dependence of λ on position along the interface.

A dual equilibrium diffusion model for epitaxial growth of gallium arsenide layers from the gas phase and an a priori computation of growth rates

A model is proposed and quantitatively treated of epitaxial growth of gallium arsenide layers, where the rate controlling step consists in the diffusion of reactants through a stagnant gas film adhering to the substrate, and where chemical equilibria are established between the reactants in the main gas stream and at the surface of substrate. The boundary layer theory is applied to the hydrodynamic part of the model which is simplified by introducing a mean effective film thickness, and the system of Ga-As-Cl-H is reduced to six molecular species and to three chemical reactions. With this basis and using estimated values of diffusion coefficients, the growth rates of epitaxial gallium arsenide layers have been a priori computed in dependence of the feed rate, its composition and on temperature. The predicted three dependences are discussed from the view-point of their courses and of the significance of computed results.