Solute Concentration Profiles Research Articles

A general polynomial solution to convection–dispersion equation using boundary layer theory

A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute transport in semi-infinite systems such as soil column. However, previous studies have only proposed low order polynomial solutions such as parabolic and cubic polynomials. This paper presents a general polynomial boundary layer solution to CDE. Comparison with exact solution suggests the prediction accuracy of the boundary layer solution varies with the order of polynomial expression and soil transport parameters. The results show that prediction accuracy increases with increasing order up to parabolic or cubic polynomial function and with no distinct relationship between accuracy and order for higher order polynomials (\(n\geqslant 3\)). Comparison of two critical solute transport parameters (i.e., dispersion coefficient and retardation factor), estimated by the boundary layer solution and obtained by CXTFIT curve-fitting, shows a good agreement. The study shows that the general solution can determine the appropriate orders of polynomials for approximate CDE solutions that best describe solute concentration profiles and optimal solute transport parameters. Furthermore, the general polynomial solution to CDE provides a simple approach to solute transport problems, a criterion for choosing the right orders of polynomials for soils with different transport parameters. It is also a potential approach for estimating solute transport parameters of soils in the field.

ThermoCalc Application for the Assessment of Binary Alloys Non-Equilibrium Solidification

Abstract The paper presents the possibility of application of the developed computer script which allows the assessment of non-equilibrium solidification of binary alloys in the ThermoCalc program. The script makes use of databases and calculation procedures of the POLY-3 module. A solidification model including diffusion in the solid state, developed by Wołczyński, is used to describe the non-equilibrium solidification. The model takes into account the influence of the degree of solute segregation on the solidification process by applying the so-called back-diffusion parameter. The core of the script is the iteration procedure with implemented model equation. The possibility of application of the presented calculation method is illustrated on the example of the Cr-30% Ni alloy. Computer simulations carried out with use of the developed script allow to determine the influence of the back-diffusion parameter on the course of solidification curves, solidus temperature, phase composition of the alloy and the fraction of each phase after the solidification completion, the profile of solute concentration in liquid during solidification process, the average solute concentration in solid phase at the eutectic temperature and many other quantities which are usually calculated in the ThermoCalc program.

Double-Diffusion MHD Free Convective Flow along a Sphere in the Presence of a Homogeneous Chemical Reaction and Soret and Dufour Effects

A numerical solution is presented for the effects of chemical reaction, thermal radiation, Soret number, Dufour number and magnetic field on double-diffusion free convection flow along a sphere. The governing boundary-layer equations of the problem are formulated and transformed into non-similar form. The obtained equations are solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. The Roseland approximation is used to describe the radiative heat flux in the energy equation. Representative results for the fluid velocity, temperature and solute concentration profiles as well as the local heat and mass transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.

FINITE DIFFERENCE SOLUTION OF TWO-DIMENSIONAL SOLUTE TRANSPORT WITH PERIODIC FLOW IN HOMOGENOUS POROUS MEDIA

Two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finite-difference method for the transport of solutes through a homogeneous, finite, porous, two-dimensional, domain. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the first-order decay and zero-order production parameters proportional to the seepage velocity, periodic boundary conditions at the origin and the end of the domain. Results are compared to analytical solutions reported in the literature for special cases and a good agreement was found. The solute concentration profile is greatly influenced by the periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in a finite media, which is especially important when arbitrary initial and boundary conditions are required.

Consequences of phloem pathway unloading/reloading on equilibrium flows between source and sink: a modelling approach.

It is now accepted that the transport phloem, linking major sources and sinks, is leaky, and this leakage can be considerable. Hence for phloem transport to function over the long distances observed, a large fraction of this unloaded photosynthate must be reloaded. A fraction of this unloaded solute is used to maintain tissues surrounding the phloem, as well as being stored. Also, pathway unloading/reloading acts as a short-term buffer to source and sink changes. In this work we present the first attempt to include both pathway unloading and reloading of carbohydrate in the modelling of pressure driven flow to determine if this has any significant effect upon source-sink dynamics. Our results indicated that the flow does not follow Poiseuille dynamics, and that pathway unloading alters the solute concentration and hydrostatic pressure profiles. Hence, measurement of either of these without considerable other detail tells us very little about the flow mechanisms. With adequate reloading along the pathway, the effects of pathway unloading can completely compensate for, making the entire system look like one with no pathway unloading.

Simulation of the Dendritic Growth Velocity for Binary Alloy Al-Cu in the Undercooled System

The phase-field model was applied to simulate the solidification kinetics to undercooled Al-Cu alloy. The relationships between material properties and model parameters are presented. The diffusivity of solute in the solid region and liquid and liquidus temperature are calculated during the simulation of solidification process. As an example, the two-dimensional computations for the dendritic growth in Al–Cu binary alloy have been performed. The dendritic morphology calculated by phase-field model showed features that are commonly found in experiments on the solidification. The concentration profiles of solute calculated in the solid region and liquid are not completely horizontal, showing evidence of microsegregation. The velocity of the dendrite tip and solute concentration at the interface front are calculated. It is found that the tip velocity is greatly concentration dependent around interface. In order to validate the growth kinetics predicted by this model tests have been performed for comparison with Stefanescu’s model. The present work based results show good agreement with those obtained by Stefanescu. The dependence of growth velocity on the initial concentration and super-cooling are also demonstrated.

Boundary-induced autophoresis of isotropic colloids: anomalous repulsion in the lubrication limit

When suspended in a liquid solution, chemically active colloids may self-propel due to an asymmetry in either particle shape or the interfacial distribution of solute absorption. We here consider a chemically homogeneous spherical particle which undergoes self-diffusiophoresis due to the presence of nearby inert wall. In particular, we focus upon the near-contact limit where it was recently observed (Yariv,Phys. Rev. Fluids, vol. 1 (3), 2016, 032101) that the solute-concentration profile within the narrow gap separating the particle and the wall cannot be uniquely determined by a gap-scale analysis. We here revisit this near-contact limit using matched asymptotic expansions, the inner region being the gap domain and the outer region being on the particle scale. Asymptotic matching with the Hankel-transform representation of the outer distribution of solute concentration serves to determine both the scaling and magnitude of the corresponding inner profile. The ensuing gap-scale pressure field, set by a lubrication mechanism, gives rise to an anomalous particle–wall interaction, scaling as an irrational power of the gap clearance.

Prandtl model for concentration polarization and osmotic counter-effects in a 2-D membrane channel

An accurate 2-D numerical model that accounts for concentration polarization and osmotic effects is developed for the cross-flow filtration in a membrane channel. Focused on the coupling between laminar hydrodynamics and mass transfer, the numerical approach solves the solute conservation equation together with the steady Navier-Stokes equations under the Prandtl approximation, which offers a simplified framework to enforce the non-linear coupling between filtration and concentration polarization at the membrane surface. The present approach is first validated thanks to the comparison with classical exact analytical solutions for hydrodynamics and/or mass transfer, as well as with approximated analytical solutions that attempted at coupling the various phenomena. The effects of the main parameters in cross-flow reverse osmosis (RO) or nanofiltration (NF) (feed concentration, axial flow rate, operating pressure and membrane permeability) on streamlines, velocity profile, longitudinal pressure drop, local permeate flux and solute concentration profile are predicted with the present numerical model, and discussed. With the use of data reported from NF and RO experiments, the Prandtl approximation model is shown to accurately correlate both average permeate flux and local solute concentration over a wide range of operating conditions.

Effect logs of double diffusion on MHD Prandtl nano fluid adjacent to stretching surface by way of numerical approach

The current communication is carried to contemplate the unique and novel characteristics of nanofluids by constructing formulation of Prandtl fluid model. The fascinating aspects of thermo diffusion effects are also accounted in this communication. Mathematical modelling is performed by employing boundary layer approach. Afterwards, similarity variables are selected to convert dimensional non-linear system into dimensionless expressions. The solution of governing dimensionless problem is executed by shooting method (SM). Graphical evaluation is displayed to depict the intrinsic behavior of embedded parameters on dimensionless velocity, temperature, solutal concentration and nanoparticle concentration profiles. Furthermore, the numerical variation for skin friction coefficient, local Nusselt number, Sherwood number and nano Sherwood number is scrutinized through tables. The assurance of current analysis is affirmed by developing comparison with previous findings available in literature, which sets a benchmark for implementation of computational approach. It is inferred from the computation that concentration profile increases whereas Sherwood number decreases for progressive values of Dufour solutal number.

Thermophoresis on boundary layer heat and mass transfer flow of Walters-B fluid past a radiate plate with heat sink/source

The Walters-B liquid model is employed to simulate medical creams and other rheological liquids encountered in biotechnology and chemical engineering. This rheological model introduces supplementary terms into the momentum conservation equation. The combined effects of thermal radiation and heat sink/source on transient free convective, laminar flow and mass transfer in a viscoelastic fluid past a vertical plate are presented by taking thermophoresis effect into account. The transformed conservation equations are solved using a stable, robust finite difference method. A parametric study illustrating the influence of viscoelasticity parameter (Γ), thermophoretic parameter (τ), thermal radiation parameter (F), heat sink/source (ϕ), Prandtl number (Pr), Schmidt number (Sc), thermal Grashof number (Gr), solutal Grashof number (Gm), temperature and concentration profiles as well as local skin-friction, Nusselt and Sherwood number is conducted. The results of this parametric study are shown graphically and inform of table. The study has applications in polymer materials processing.

Effects of temperature gradient in the growth of Si0.5Ge0.5 crystals by the traveling liquidus-zone method on board the International Space Station

Si0.5Ge0.5 crystals were grown at two different temperature gradients on board the International Space Station (ISS) using the traveling liquidus-zone (TLZ) method and effects of temperature gradient on crystal quality were investigated. Although average axial Ge concentration profile was not affected by the temperature gradient, crystal quality was affected greatly. Single crystal length was shortened and constitutional supercooling (CS) is shown to occur more easily at higher temperature gradient. The calculated degree of CS based on the solute concentration profile in the melt and phase diagram data is about 4 times larger when the temperature gradient is twice, which supports the experimental results. Instability at high temperature gradient is unique to the TLZ method and is not common to other crystal growth methods such as the directional solidification method and Czochralski method.

Solute source depletion control of forward and back diffusion through low-permeability zones

Solute diffusive exchange between low-permeability aquitards and high-permeability aquifers acts as a significant mediator of long-term contaminant fate. Aquifer contaminants diffuse into aquitards, but as contaminant sources are depleted, aquifer concentrations decline, triggering back diffusion from aquitards. The dynamics of the contaminant source depletion, or the source strength function, controls the timing of the transition of aquitards from sinks to sources. Here, we experimentally evaluate three archetypical transient source depletion models (step-change, linear, and exponential), and we use novel analytical solutions to accurately account for dynamic aquitard-aquifer diffusive transfer. Laboratory diffusion experiments were conducted using a well-controlled flow chamber to assess solute exchange between sand aquifer and kaolinite aquitard layers. Solute concentration profiles in the aquitard were measured in situ using electrical conductivity. Back diffusion was shown to begin earlier and produce larger mass flux for rapidly depleting sources. The analytical models showed very good correspondence with measured aquifer breakthrough curves and aquitard concentration profiles. The modeling approach links source dissolution and back diffusion, enabling assessment of human exposure risk and calculation of the back diffusion initiation time, as well as the resulting plume persistence.

Diffusion of water-soluble sorptive drugs in HEMA/MAA hydrogels

We measure and, for the first time, theoretically predict four prototypical aqueous-drug diffusion coefficients in five soft-contact-lens material hydrogels where solute-specific adsorption is pronounced. Two-photon fluorescence confocal microscopy and UV/Vis-absorption spectrophotometry assess transient solute concentration profiles and concentration histories, respectively. Diffusion coefficients are obtained for acetazolamide, riboflavin, sodium fluorescein, and theophylline in 2-hydroxyethyl methacrylate/methacrylic acid (HEMA/MAA) copolymer hydrogels as functions of composition, equilibrium water content (30–90%), and aqueous pH (2 and 7.4). At pH2, MAA chains are nonionic, whereas at pH7.4, MAA chains are anionic (pKa≈5.2). All studied prototypical drugs specifically interact with HEMA and nonionic MAA (at pH2) moieties. Conversely, none of the prototypical drugs adsorb specifically to anionic MAA (at pH7.4) chains. As expected, diffusivities of adsorbing solutes are significantly diminished by specific interactions with hydrogel strands. Despite similar solute size, relative diffusion coefficients in the hydrogels span several orders of magnitude because of varying degrees of solute interactions with hydrogel-polymer chains. To provide a theoretical framework for the new diffusion data, we apply an effective-medium model extended for solute-specific interactions with hydrogel copolymer strands. Sorptive-diffusion kinetics is successfully described by local equilibrium and Henry's law. All necessary parameters are determined independently. Predicted diffusivities are in good agreement with experiment.

Numerical Study of Contaminant Transport in Fractured Porous Rock with Distance Dependent Dispersivity

This paper presents solute concentration profiles with constant, linear and exponential distance-dependent dispersion models for solute transport through fractured porous rock matrices. A numerical analysis has been developed for solution of the equations governing the dispersion model using the hybrid finite volume method. The results of the numerical analysis have been used to investigate the effect of the matrix diffusion coefficient, fracture velocity, and matrix and fracture retardation factors on concentration profiles. It is found that in the presence of higher matrix diffusion coefficients, the behaviour of concentration profiles is different for constant and distance-dependent dispersion models. However, the behaviour of concentration profiles is observed to be nonlinear during the small transport time period, and the matrix diffusion is found to control the spreading of solute in the fracture in the presence of constant and distance-dependent dispersion models.

Effect of Nb Content on Solidification Characteristics and Microsegregation in Cast Ti–48Al–xNb Alloys

Microstructural evolution in nonequilibrium solidification of Ti–48Al–xNb alloys with Nb contents ranging from 2 to 8 at% has been studied by containerless electromagnetic levitation. Levitated drops of controlled undercooling were quenched onto chill copper substrates and subjected to phase and microstructure analysis. With increasing Nb content, the solidification path changes gradually from hyperperitectic solidification to hypoperitectic solidification and both solidification segregation (S-segregation) and β-solidification gradually increase. A transition from typical hypoperitectic solidification to a sole solidification of the β phase beyond a critical undercooling is revealed for the Ti–48Al–8Nb hypoperitectic alloy. For the Ti–48Al–2Nb alloy, the morphologies of the primary β dendrites are not observed. With increasing undercooling, the coarsening of the lamellar colonies occurs, which can be attributed to the transition of the primary β dendritic morphology. Furthermore, the solute concentration profiles for the final solidification microstructure are obtained to examine the segregation behaviors of alloying elements. With increasing Nb content, the undercooling eliminating S-segregation gradually increases.

Soret and MHD effects on bioconvection wall jet flow of nanofluid containing gyrotactic microorganisms

Various applications of bioconvection phenomena in the field of medicine and biotechnology boost us to present the study of laminar wall jet flow in this specific direction. For the purpose, we have considered nanofluid containing gyrotactic microorganisms in the presence of normally applied magnetohydrodynamic forces along with Soret effects. Boundary layer approximation and similarity transformation are utilized to convert governing equations into ordinary differential equations. Influence of different emerging parameters on velocity, temperature and concentration profiles of solute, nanoparticle and motile microorganisms has been investigated. The role of physical quantities like Nusselt number, Sherwood number and density number of microorganisms is also highlighted. Increase in Nusselt number and density number of motile microorganism is observed for incremental values of bioconvection Peclet number. Soret number reflects increasing effect on Nusselt number and decreasing effect on Sherwood number because solute diffusion faces resistance due to higher values of Soret number and in return decreases rate of mass transfer. Also bioconvection Rayleigh number imposes decreasing effect on density number of the motile microorganisms.

Modeling solute segregation during the solidification of γ-phase U-Mo alloys

Using first principles calculations, it is demonstrated that solute segregation during U-Mo solidification can be modeled using the classic Brody-Fleming limited diffusion framework. The necessary supporting equations specific to the U-Mo alloy, along with careful verification of the assumptions underpinning the Brody-Fleming model are developed, allowing for concentration profile predictions as a function of alloy composition and cooling rate. The resulting model is compared to experimental solute concentration profiles, showing excellent agreement. Combined with complementary modeling of dendritic feature sizes, the solute segregation model can be used to predict the complete microstructural state of individual U-Mo volume elements based upon cooling rates, informing ideal processing routes.

Effect of Chemical Reaction on Convective Heat Transfer of Boundary Layer Flow in Nanofluid over a Wedge with Heat Generation/Absorption and Suction

The aim of the present study is to examine the convective heat transfer of nanofluid past a wedge subject to first-order chemical reaction, heat generation/absorption and suction effects. The influence of wedge angle parameter, thermophoresis, Dufour and Soret type diffusivity are included. The local similarity transformation is applied to convert the governing nonlinear partial differential equations into ordinary differential equations. Shooting method integrated with fourth-order Runge-Kutta method is used to solve the ordinary differential equations. The skin friction, heat and mass transfer rates as well as the effects of various parameters on velocity, temperature and solutal concentration profiles are analyzed. The results indicate that when the chemical reaction parameter increases, the heat transfer coefficient increases while the mass transfer coefficient decreases. The effect of chemical reaction parameter is very important in solutal concentration field compared to velocity and temperature profiles since it decreases the solutal concentration of the nanoparticle.

Convective heat transfer and thermo-diffusion effects on flow of nanofluid towards a permeable stretching sheet saturated by a porous medium

Boundary layer flow of a nanofluid towards a permeable stretching surface saturated by porous medium with a convective boundary condition is considered. After employing viable similarity transforms, equations governing the flow are reduced to a system of nonlinear ordinary differential equations. Well known numerical technique Runge–Kutta–Fehlberg method is applied to solve the system. Concrete graphical analysis is carried out to study the effects of different emerging parameters on velocity, temperature, solute and nanoparticle concentration profiles coupled with a comprehensive discussion. Tables are also constructed to compare current study with already existing results in the literature. Numerical effects of local Nusselt number, local Sherwood number and nanofluid Sherwood number are also discussed.

Continuum‐scale characterization of solute transport based on pore‐scale velocity distributions

AbstractWe present a methodology to characterize a continuum‐scale model of transport in porous media on the basis of pore‐scale distributions of velocities computed in three‐dimensional pore‐space images. The methodology is tested against pore‐scale simulations of flow and transport for a bead pack and a sandstone sample. We employ a double‐continuum approach to describe transport in mobile and immobile regions. Model parameters are characterized through inputs resulting from the micron‐scale reconstruction of the pore space geometry and the related velocity field. We employ the outputs of pore‐scale analysis to (i) quantify the proportion of mobile and immobile fluid regions and (ii) assign the velocity distribution in an effective representation of the medium internal structure. Our results (1) show that this simple conceptual model reproduces the spatial profiles of solute concentration rendered by pore‐scale simulation without resorting to model calibration and (2) highlight the critical role of pore‐scale velocities in the characterization of the model parameters.