Concentration Boundary Conditions Research Articles

Liquid solubilization dynamics: III. Use of a quiescent technique to determine the correct boundary condition for solubilization of oleic acid in sodium taurodeoxycholate

An unsteady state experimental technique has been employed to study interfacial mass transfer boundary conditions in quiescent liquid/liquid systems. This method is superior than steady state methods for identifying the specific nature of the interfacial step. Moreover, the technique lends mechanistic insight into those systems in which transient effects may dictate steady state behavior. Experimental data are compared to theoretical results derived by assuming specific boundary conditions at the interface. A constant concentration boundary condition correctly describes the intertransport of partially immiscible phases (benzaldehyde into water), and a constant flux boundary condition correctly describes the dissociation of solute dimer from one immiscible phase into another (benzoic acid from benzene to water). The method was extended to identify the underlying mechanism of the micellar solubilization of oleic acid by sodium taurodeoxycholate at 37°C, a model fatty acid digestion system. The phenomenon appears to be surfactant diffusion controlled. The effects of pH, added NaCl and surfactant concentration are also discussed.

Liquid solubilization dynamicsIII. Use of a quiescent technique to determine the correct boundary condition for solubilization of oleic acid in sodium taurodeoxycholate

Abstract An unsteady state experimental technique has been employed to study interfacial mass transfer boundary conditions in quiescent liquid/liquid systems. This method is superior than steady state methods for identifying the specific nature of the interfacial step. Moreover, the technique lends mechanistic insight into those systems in which transient effects may dictate steady state behavior. Experimental data are compared to theoretical results derived by assuming specific boundary conditions at the interface. A constant concentration boundary condition correctly describes the intertransport of partially immiscible phases (benzaldehyde into water), and a constant flux boundary condition correctly describes the dissociation of solute dimer from one immiscible phase into another (benzoic acid from benzene to water). The method was extended to identify the underlying mechanism of the micellar solubilization of oleic acid by sodium taurodeoxycholate at 37°C, a model fatty acid digestion system. The phenomenon appears to be surfactant diffusion controlled. The effects of pH, added NaCl and surfactant concentration are also discussed.

An approximate integral solution of vertical infiltration under changing boundary conditions

An approximate solution of the Richards equation was obtained with a spectral series solution for rain infiltration under changing boundary conditions. Prior to surface ponding a flux boundary condition was employed, which was followed by a concentration boundary condition. During this latter stage a flux matching boundary condition was used to describe the movement of the interface between the saturated and unsaturated zones. The solutions yield a general expression for ponding time, wetting front advance, and soil water profile. Infiltration rates beyond ponding include the effect of time dependent ponding height and saturation depth.

Effect of Mode of Growth on the Preferred Orientation in Bismuth Single Crystals

AbstractMelt growth involves mass transfer in a heated fluid, and an analysis of this problem needs a knowledge of the thermal, concentration and other boundary conditions. The geometry of the crystal growing apparatus also has significant influence (Pimpatkar, Ostrach). The factors affecting the orientation in bismuth single crystals have been studied (Bhatt; Shah). In this communication results of the effect of the mode of crystal growth on the preferred orientation are presented.

Rayleigh–Bénard instability in n-component reactive fluids

We study the hydrodynamic stability of n-component reactive fluids undergoing a single reversible reaction in the Bénard geometry and the Boussinesq approximation, and take the Soret, Dufour, and cross-diffusion effects to be negligible. The boundaries are assumed to be rigid and perfectly conducting; the dependence of the critical Rayleigh number on the assumed boundary conditions for concentrations is discussed in the limit of fast reactions. In this limit we show that, if the boundaries are impermeable to mass flow (the easiest to achieve experimentally), the critical Rayleigh number, R∞cr, is proportional to the one for nonreactive binary fluids. Apart from multiplicative factors, R∞cr depends only on a single dimensionless parameter δ which vanishes when all the diffusion coefficients are equal. For δ≳0 (<−1) stationary convection may set in only if R∞cr≳0(<0); for intermediate values of δ there exist both positive and negative solutions for R∞cr.

MASS TRANSFER COEFFICIENTS FOR REACTIVE FLUIDS FLOWING IN CONDUITS WITH SEMI-PERMEABLE WALLS

The advancing front theory is an approximate solution for mass transfer into a reactive fluid when the reaction can be assumed to be very fast. The theory has had considerable use in predicting mass transfer characteristics for reactive fluids flowing in conduits. In this paper, the mass transfer coefficient, in the form of the local, fluid-side Sherwood number, is derived for reactive flow in conduits with semi-permeable walls. The local, fluid-side Sherwood number is given as a function of the Graetz number, the wall Sherwood number, and a dimensionless reaction strength parameter. The limiting cases of both the constant wall concentration boundary condition (Shw−∞) and the constant wall flux boundary condition (Shw−∞)are investigated. Comparisons of the results with the classical Graetz and Leveque theories give conclusions about the accuracy of the advancing front theory for the worst possible case.

A generalized solution for solute flow in soils with mobile and immobile water

An analytical solution is presented for the movement of a solute through a porous medium which is leached at a constant rate. The solution takes into account transfer into an immobile water fraction. The generalized solution is valid for both finite or semi‐infinite media and for concentration or flux type boundary conditions. The solution consists of a convolution integral of two functions. The first function is the derivative with respect to injected pore volume of the corresponding analytical solution of the solute transfer equation without transfer into immobile water. The second function is the J function, for which series approximations are presented. The influence of the boundary conditions on the solution is discussed.

Mass transfer in porous media with immobile water

An analytical solution is presented for the movement of a solute through a porous medium, which is leached at a constant rate. The solution takes into account diffusion into an immobile water fraction. The generalized solution is valid for both finite or semi-infinite media and for concentration or flux-type boundary conditions. The solution consists of a convolution integral of two functions. The first function is the derivative vs. time of the corresponding solutions of the solute transfer equation without lateral diffusion. The second function is the J-function, for which a series approximation was developed. The use of the solution is illustrated with displacement studies in a 30 cm long column, filled with glass beads. It is shown that effluent concentration distributions from, and solute concentration distributions inside the column, when unsaturated, cannot be explained without taking into account an immobile water fraction.

Excess vacancy generation by E-center dissociation in the case of phosphorus diffusion in silicon

Phosphorus diffusion equations in silicon are proposed taking into account the following: Phosphorus diffuses as E centers; an E center is formed by pairing a negatively ionized vacancy with a positively ionized phosphorus atom; with increasing phosphorus concentration, the equilibrium concentration of negatively ionized vacancies increases and the ionization rate of phosphorus decreases. The equations are solved numerically under the boundary conditions of constant surface concentrations of total phosphorus atoms and negatively ionized vacancies. The phosphorus diffusion coefficient depends not only on the vacancy concentration but also on its gradient. The excess vacancies are generated by the E-center dissociation. The concentration and the distribution of the excess vacancies are good enough to explain qualitatively the anomalous diffusion of phosphorus, except the kink in the phosphorus concentration profile.

Examples of the simplest boundary conditions for electron and ion concentrations and electron and heavy-particle temperatures

Examples of boundary conditions in certain particular cases are considered. The conditions proposed in [1, 2] can be used for any plasma component The boundary conditions for the same wall will generally be distinct for each of the components.

Regular expansion solutions for heat or mass transfer in concentrated two-phase particulate systems at small peclet and reynolds numbers

General solutions for small Peclet number heat or mass transfer in concentrated two-phase particulate systems submerged in arbitrary Stokesian flows with arbitrary temperature or concentration boundary conditions were obtained using regular perturbation expansions of the dependent variable in powers of the Peclet number. Uniformly valid solutions for all orders of approximation were obtained within the domain of interest, subject to a restriction imposed on the value of the Peclet number by the dimensions of that domain. The solutions were specialized to cases of practical interest. For a particulate system submerged in a uniform flow with a linear temperature gradient in the direction of flow, the results reduced to those of Maxwell for effective electrical conductivity of a composite medium. Specific relations were also obtained for local and overall interfacial fluxes and Nusselt numbers. The results reflect the effects of particle volume fraction and retardation of internal circulation by surfactant impurities.

Water absorption by swelling seeds. I. Constant surface boundary condition

Recent analyses of water absorption by seeds have used a non-swelling model with constant diffusivity and constant surface concentration boundary condition. A physically realistic modification of such analyses is developed, where the spherical seed has the properties of normal swelling, moisture content dependent diffusively, and a moisture characteristic which is described by the double layer theory relevant to colloids under mechanical restraint. Analysis of this model by a finite difference approximation produces a relationship between linearized moisture content and dimensionless time which is appropriate to all spherical swelling materials. A good match was obtained between this relationship and experimental absorption data for wheat seed. The relationship given graphically in dimensionless form is applicable to any swelling system, e.g. soil aggregates, which satisfy normal swelling and constrained double layer boundary conditions. The analysis gave values for the diffusivity of the seed similar to, but larger than, those calculated from non-swelling models. These are much smaller than most values of diffusivity of soils in the available water range. The magnitude of the maximum rate at which a seed can absorb water relative to that which the soil can supply, predicts that for seeds embedded in most soils, the constant surface concentration boundary condition used by most authors is inappropriate. In terms of the swelling behaviour of most soils, the swelling model, unlike the non-swelling model, predicts that imbibing seed embedded in soil will be subject to a mechanical constraint.

Laminar heat or mass transfer in rectangular channels and in cylindrical tubes for fully developed flow: comparison of solutions obtained for various boundary conditions

A comparison is made of results obtained with various boundary conditions at the wall of the channels. By comparing blood Sherwood numbers for finite wall resistance with those for constant concentration boundary condition only a small variation is found for various points along the blood channel.

An Improved Model for Capillary Diffusion with No External Stirring

Average capillary concentration as a function of time has been determined by computer simulation of the diffusion process from a capillary into an infinite stationary medium. These results may be used as the basis for experimental determination of diffusivity to eliminate the usual errors associated with stirring, and to correct results obtained without stirring but using the zero concentration boundary condition solution.

Analysis of Radial, Steady‐State Solution, and Solute Flow

AbstractRadial, steady‐state solution and solute flow to a root can be described as flow in a hollow cylinder. The analysis was broken into two parts: the solution flow and solute flow. The equation for the solution flow was solved analytically to obtain the soil solution tension distribution subject to fixed tensions at the inner and outer boundaries. From the tension distribution, the solution content distribution and solution flux were derived. The solution flux approaches a finite limit as the tension across the cylinder increases. The equation of flow for the solute was solved seminumerically, subject to fixed inner and outer concentration boundary conditions. The solute diffusion coefficient was assumed to be dependent upon the solution content. The convective part of the solute flow equation was dependent upon the solution flux. For most cases a good approximation would be the use of a constant solute diffusion coefficient, only dependent upon a mean solution content. As the solution flux increased, the solute concentration gradient increased at the root for both plant uptake or salt sieving of the solute.

Unsteady diffusion with first‐order reaction

AbstractOutline below is a method for simplification of the solution of diffusion problems involving irreversible first‐order reactions. It permits writing the concentration profiles for reactive systems in terms of those for nonreactive systems of the same geometry and similar boundary conditions. The method is valid where fluid flow patterns and concentration boundary conditions are time independent and where the assumptions of constant density and mass diffusivity are acceptable. The useful results of this development are summarized in Equations (31) through (33) and represent a generalization of an earlier development by P. V. Danckwerts (1).

38—A Method of Calculating Diffusion in Fibres Coupled with Irreversible Adsorption or Rapid Reaction

The problem of calculating diffusion constants in a fibre as the diffusing substance is irreversibly adsorbed or reacts with the fibre phase is discussed. An iteration procedure is applied for the boundary condition of constancy of the total amount of active substance and this is applied to the periodate consumption by cellulose. From the results, a direct method of calculation is developed for the alternative boundary conditions of constant amount and constant external concentration of reactive substance. The method is compared with previous theoretical work regarding applicability and the principal results.