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## Dispersion Equation (dispersion + equation)
## Selected Abstracts## Refractive index dispersion of relaxor ferroelectric 0.9Pb(Zn1/3Nb2/3)O3 -0.1PbTiO3 single crystal CRYSTAL RESEARCH AND TECHNOLOGY, Issue 2 2009Chongjun HeAbstract The refractive indices of 0.9Pb(Zn1/3Nb2/3)O3 -0.1PbTiO3 single crystal at different wavelengths have been measured by the minimum deviation method at room temperature, and their dispersion equations are obtained. The parameters connected to the energy band structure are obtained by fitting single-oscillator dispersion equation. Despersion energies are found to take on covalent crystal values. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The effect of surface,solute interactions on the transport of solutes through porous materials EUROPEAN JOURNAL OF SOIL SCIENCE, Issue 3 2009D. A. RoseSummary We have investigated the effect of differences in surface charge, valency of ion, solute concentration, solution flux and physical structure on the leaching and uptake of individual ions from simple solutions flowing through porous materials. We studied the miscible displacement of solutions of four salts (KBr, K2SO4, CaBr2 and CaSO4) having different cation : anion ratios separately through three inert materials (ballotini, pumice and ceramic) and two sizes of a reactive material (sepiolite) over several ranges of concentration (c) and at many pore-water velocities (v) under steady vertical saturated flow. Breakthrough curves of individual effluent ions (K+, Br,, Ca2+ and SO42,) were analysed by CXTFIT 2.0 to optimize transport parameters (retardation factor, R; dispersion coefficient, K) assuming that transport was governed by the convective,dispersion equation. In the inert materials, R did not differ significantly from 1 irrespective of c. In sepiolite, R was < 1 for anions and > 1 for cations, and became more extreme as c decreased. R varied with the valency of the anions, as predicted by diffuse double layer theory, and with that of the cations by a simple charge balance. Freundlich isotherms, reconstructed from R values, described the sorption of the cations and exclusion of the anions. For the inert materials, K did not depend on the ion or c and increased monotonically with v. For sepiolite, K also increased with v and with small but non-significant differences between ions and concentrations. The K(v) functions were consistent with Passioura's theory of dispersion in aggregated media, and the transport was reversible as R and K values did not depend on whether the media were being leached or resalinized. The effective dispersion coefficient of an ion is K* = K/R so, although K(v) appears to be unaffected by ion or concentration of solute in sepiolite, K*(v) will be affected. Thus, the controlling factor of these surface,solute interactions is R. [source] ## Pedotransfer functions for solute transport parameters of Portuguese soils EUROPEAN JOURNAL OF SOIL SCIENCE, Issue 4 2001M. C. Gonc, alvesSummary The purpose of this study is to quantify solute transport parameters of fine-textured soils in an irrigation district in southern Portugal and to investigate their prediction from basic soil properties and unsaturated hydraulic parameters. Solute displacement experiments were carried out on 24 undisturbed soil samples by applying a 0.05 m KCl pulse during steady flow. The chloride breakthrough curves (BTCs) were asymmetric, with early breakthrough and considerable tailing characteristic of non-equilibrium transport. The retardation factor (R), dispersion coefficient (D), partitioning coefficient (,), and mass transfer coefficient (,) were estimated by optimizing the solution of the non-equilibrium convection,dispersion equation (CDE) to the breakthrough data. The solution could adequately describe the observed data as proved by a median of 0.972 for the coefficient of determination (r2) and a median for the mean squared error (MSE) of 5.1 × 10,6. The median value for R of 0.587 suggests that Cl, was excluded from a substantial part of the liquid phase. The value for , was typically less than 0.5, but the non-equilibrium effects were mitigated by a large mass transfer coefficient (, > 1). Pedotransfer functions (PTFs) were developed with regression and neural network analyses to predict R, D, , and , from basic soil properties and unsaturated hydraulic parameters. Fairly accurate predictions could be obtained for logD (r2 , 0.9) and , (r2 , 0.8). Prediction for R and log, were relatively poor (r2 , 0.5). The artificial neural networks were all somewhat more accurate than the regression equations. The networks are also more suitable for predicting transport parameters because they require only three input variables, whereas the regression equations contain many predictor variables. [source] ## Influence of Small-Scale Heterogeneities on Contaminant Transport in Fractured Crystalline Rock GROUND WATER, Issue 5 2006Ralph MettierWe present a sequence of purely advective transport models that demonstrate the influence of small-scale geometric inhomogeneities on contaminant transport in fractured crystalline rock. Special weight is placed on the role of statistically generated variable fracture apertures. The fracture network geometry and the aperture distribution are based on information from an in situ radionuclide retardation experiment performed at Grimsel test site (Swiss Alps). The obtained breakthrough curves are fitted with the advection dispersion equation and continuous-time random walks (CTRW). CTRW is found to provide superior fits to the late-arrival tailing and is also found to show a good correlation with the velocity distributions obtained from the hydraulic models. The impact of small-scale heterogeneities, both in fracture geometry and aperture, on transport is shown to be considerable. [source] ## Analytical power series solutions to the two-dimensional advection,dispersion equation with distance-dependent dispersivities HYDROLOGICAL PROCESSES, Issue 24 2008Jui-Sheng ChenAbstract As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection,dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection,dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio eL exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large eL values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Numerical study of boundary conditions for solute transport through a porous medium INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2001Glen P. PetersAbstract A transition region may be defined as a region of rapid change in medium properties about the interface between two porous media or at the interface between a porous medium and a reservoir. Modelling the transition region between different porous media can assist in the selection of the most appropriate boundary conditions for the standard advection,dispersion equation (ADE). An advantage of modelling the transition region is that it removes the need for explicitly defining boundary conditions, though boundary conditions may be recovered as limiting cases. As the width of a transition region is reduced, the solution of the transition region model (TR model) becomes equivalent to the solution of the standard ADE model with correct boundary conditions. In this paper numerical simulations using the TR model are employed to select the most appropriate boundary conditions for the standard ADE under a variety of configurations and conditions. It is shown that at the inlet boundary between a reservoir and porous medium, continuity of solute mass flux should be used as the boundary condition. At the boundary interface between two porous media both continuity of solute concentration and solute mass flux should be used. Finally, in a finite porous medium where the solute is allowed to advect freely from the exit point, both continuity of solute concentration and solute mass flux should be used as the outlet boundary condition. The findings made here are discussed with reference to a detailed review of previous relevant theoretical and experimental observations. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## A new approach to avoid excessive numerical diffusion in Eulerian,Lagrangian methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008A. YounesAbstract Lumping is often used to avoid non-physical oscillations for advection,dispersion equations but is known to add numerical diffusion. A new approach is detailed in order to avoid excessive numerical diffusion in Eulerian,Lagrangian methods when several time steps are used. The basic idea of this approach is to keep the same characteristics during all time steps and to interpolate only the concentration variations due to the dispersion process. In this way, numerical diffusion due to the lumping is removed at the end of each time step. The method is combined with the Eulerian,Lagrangian localized adjoint method (ELLAM) which is a mass conservative characteristic method for solving the advection,dispersion equation. Two test problems are modelled to compare the proposed method to the consistent, the full and the selective lumping approaches for linear and non-linear transport equations. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Modeling l-dopa purification by chiral ligand-exchange chromatography AICHE JOURNAL, Issue 3 2007Nooshafarin SanaieAbstract A model describing elution-band profiles that combines multiple chemical equilibria theory with the nonideal equilibrium,dispersion equation for solute transport is used to predict and characterize the separation of l,d-dopa by chiral ligand-exchange chromatography (CLEC). Formation constants and stoichiometries for all equilibrium complexes formed in the interstitial volume and pore liquid are taken from standard thermodynamic databases and independent potentiometric titration experiments. Formation constants for complexes formed with the stationary phase ligand (N-octyl-3-octylthio-d-valine) are determined from potentiometric titration data for a water-soluble analogue of the ligand. This set of pure thermodynamic parameters is used to calculate the spatially discretized composition of each column volume element as a function of time. The model includes a temperature-dependent pure-component parameter, determined by regression to a single elution band for the pure component, that corrects for subtle effects associated with immobilizing the N-octyl-3-octylthio-d-valine ligand onto the stationary phase. The model is shown to accurately predict elution chromatograms and separation performance as a function of key column operating variables. The model is then used to better understand the connection between chemical equilibria within the system and changes in band profiles and band separation resulting from changes in column operating conditions. © 2007 American Institute of Chemical Engineers AIChE J, 2007 [source] ## Variational formulation for the stationary fractional advection dispersion equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2006Vincent J. ErvinAbstract In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] ## Simulation of pesticide leaching in a cracking clay soil with the PEARL model PEST MANAGEMENT SCIENCE (FORMERLY: PESTICIDE SCIENCE), Issue 5 2005Rômulo P Scorza JúniorAbstract Testing of pesticide leaching models is important to increase confidence in their use in pesticide registration procedures world-wide. The chromatographic PEARL model was tested against the results of a field leaching study on a cracking clay soil with a tracer (bromide), a mobile pesticide (bentazone) and a moderately sorbing, persistent pesticide (imidacloprid). Input parameters for water flow and solute transport were obtained from site-specific measurements and from literature. The model was tested using a stepwise approach in which each sub-model was sequentially and separately tested. Uncalibrated simulations for the water flow resulted in moisture profiles that were too wet. Calibration of the hydraulic relationships resulted in a good description of the moisture profiles. Calibration of the dispersion length was necessary to obtain a good description of bromide leaching. The calibrated dispersion length was 61 cm, which is very long and indicates a large non-uniformity of solute transport. The half-life of bentazone had to be calibrated to obtain a good description of its field persistence. The calibrated half-life was 2.5 times shorter than the half-life derived from the laboratory studies. Concentrations of bentazone in drain water and groundwater were described reasonably well by PEARL. Although measured and simulated persistence of imidacloprid in soil corresponded well, the bulk of the imidacloprid movement was overestimated by PEARL. However, imidacloprid concentrations in drain water were underestimated. In spite of the extensive calibration of water flow and tracer movement, the behaviour of the moderately sorbing pesticide imidacloprid could not be simulated. This indicates that the convection,dispersion equation cannot be used for accurate simulation of pesticide transport in cracking clay soils (even if extremely long dispersion length is used). Comparison of the model results from a poorly sorbed chemical (bentazone) and a moderately sorbed chemical (imidacloprid) were useful in defining the limitations of using a chromatographic flow model to simulate the effects of preferential flow. Copyright © 2005 Society of Chemical Industry [source] ## Refractive index dispersion of relaxor ferroelectric 0.9Pb(Zn1/3Nb2/3)O3 -0.1PbTiO3 single crystal CRYSTAL RESEARCH AND TECHNOLOGY, Issue 2 2009Chongjun HeAbstract The refractive indices of 0.9Pb(Zn1/3Nb2/3)O3 -0.1PbTiO3 single crystal at different wavelengths have been measured by the minimum deviation method at room temperature, and their dispersion equations are obtained. The parameters connected to the energy band structure are obtained by fitting single-oscillator dispersion equation. Despersion energies are found to take on covalent crystal values. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A new approach to avoid excessive numerical diffusion in Eulerian,Lagrangian methods INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2008A. YounesAbstract Lumping is often used to avoid non-physical oscillations for advection,dispersion equations but is known to add numerical diffusion. A new approach is detailed in order to avoid excessive numerical diffusion in Eulerian,Lagrangian methods when several time steps are used. The basic idea of this approach is to keep the same characteristics during all time steps and to interpolate only the concentration variations due to the dispersion process. In this way, numerical diffusion due to the lumping is removed at the end of each time step. The method is combined with the Eulerian,Lagrangian localized adjoint method (ELLAM) which is a mass conservative characteristic method for solving the advection,dispersion equation. Two test problems are modelled to compare the proposed method to the consistent, the full and the selective lumping approaches for linear and non-linear transport equations. Copyright © 2007 John Wiley & Sons, Ltd. [source] |