Variation Of Dispersion Coefficient Research Articles

Non-monotonic effect of compaction on longitudinal dispersion coefficient of porous media

Utilizing the discrete element method and the pore network model, we numerically investigate the impact of compaction on the longitudinal dispersion coefficient of porous media. Notably, the dispersion coefficient exhibits a non-monotonic dependence on the degree of compaction, which is distinguished by the presence of three distinct regimes in the variation of dispersion coefficient. The non-monotonic variation of dispersion coefficient is attributed to the disparate effect of compaction on dispersion mechanisms. Specifically, the porous medium tightens with an increasing pressure load, reducing the effect of molecular diffusion that primarily governs at small Péclet numbers. On the other hand, heightened pressure loads enhance the heterogeneity of pore structures, resulting in increased disorder and a higher proportion of stagnant zones within porous media flow. These enhancements further strengthen mechanical dispersion and hold-up dispersion, respectively, both acting at higher Péclet numbers. It is crucial to highlight that hold-up dispersion is induced by the low-velocity regions in porous media flow, which differ fundamentally from zero-velocity regions (such as dead-ends or the interior of permeable grains) as described by the classical theory of dispersion. The competition between weakened molecular diffusion and enhanced hold-up dispersion and mechanical dispersion, together with the shift in the dominance of dispersion mechanisms across various Péclet numbers, results in multiple regimes in the variation of dispersion coefficients. Our study provides unique insights into structural design and modulation of the dispersion coefficient of porous materials.

Spatiotemporal velocities, aggregation, and false dispersion: numerical considerations for the modeling of colonial and motile harmful algae

PurposeThe purpose of this study is to investigate the errors arising from the numerical treatment of model processes, paying particular attention to the impact of key system features including widely variable dispersion coefficients, spatiotemporal velocities of algal cells, and the aggregation of algae from single cells to large colonies. An advection–dispersion model has been presented to describe the vertical transport of colonial and motile harmful algae in a lake environment.Design/methodology/approachModel performance is examined for two different numerical treatments of the advective term: first-order upwind and quadratic upwind with a stability-preserving flux limiter (SMART). To determine how these schemes impact predictions, comparisons are made across a sequence of models with increasing complexity.FindingsUsing first-order upwinding for advection–dispersion calculations with a time oscillating velocity field leads to oscillatory numerical dispersion. Subjecting an initially uniform distribution of large-sized algal colonies to a spatiotemporal velocity creates a concentration pulse, which reaches a steady-state width at high-grid Peclet numbers when using the SMART scheme; the pulse exhibits contraction–expansion behavior throughout a velocity cycle at all Peclet numbers when using first-order upwinding. When aggregation dynamics are included with advection-dominated spatiotemporal transport, results indicate the SMART scheme predicts larger peak concentration values than those predicted by first-order upwind, but peak location and the time to large colony appearance remain largely unchanged between the two advective schemes.Originality/valueTo the best of the authors’ knowledge, this study is the first numerical investigation of a novel advection–dispersion model of vertical algal transport. In addition, a generalized expression for the effective dispersion coefficient of temporally variable flow fields is presented.

Dispersion of pollutants in a porous medium with finite thickness and variable dispersion coefficients

Groundwater is subject to the intrusion of pollutants of various types. These pollutants can have natural or anthropogenic sources. Their consumption can therefore affect human health, but also affects the development of vegetation. The objective of this article is to analyze the effect of the dispersion coefficient parameters on the spatio-temporal distribution of pollutants in a saturated porous medium with a finite thickness. This layer is subjected to two types of conditions at the outlet of the aquifer where the dispersion coefficient is a function of depth. The partial differential equations were solved using an implicit finite difference technique. The results of the analysis suggest that the behavior of the concentration profile was influenced by the different output boundary conditions. On the other hand, the parameter, b of the dispersion coefficient depending on the distance has a significant impact on the movement of the solute in a saturated porous medium compared to the parameter, a. In other words, it has more effect on the dispersion of pollutants in aquifers. This study highlights the need to bring insight on the transports parameters while modeling the transport of solute in a porous media.

General Backward Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water.

Pollutant source identification (PSI) has been conducted for four decades for tracking Fickian diffusive pollutants, while PSI for non-Fickian diffusion, well-documented in aquifers and rivers, requires novel, predictive models. To enable PSI for non-Fickian diffusive pollutants, this study derived a general backward model using the fractional-adjoint approach in sensitivity analysis for dissolved contaminants with transport governed by the spatiotemporal fractional advection-dispersion equation (fADE). The backward fADE contains a self-adjoint time-fractional term for subdiffusion and direction-dependent, non-self-adjoint space-fractional terms for superdiffusion. Field applications showed that the resultant backward location probability density function identified the point source location in all three test cases, one alluvial aquifer and two rivers. The backward model and boundary conditions derived in this study made it possible to reliably and efficiently backtrack pollutants (and may include other constituents, such as bedload) undergoing mixed sub- and superdiffusion in natural aquatic systems. The classical PSI model, however, underestimated the source location since it did not account for solute retention and preferential flow. In addition, the measured tracer snapshots (if available before PSI) can enhance the parameter predictability and improve the applicability of backward fADE PSI. Most importantly, a spatially variable dispersion coefficient is needed in the backward fADE since PSI is most likely scale dependent in natural hydrologic systems.

Dispersion of Pollutants in a Porous Medium with Finite Thickness and Variable Dispersion Coefficients

Dispersion of Pollutants in a Porous Medium with Finite Thickness and Variable Dispersion Coefficients

Achieving local mass conservation when using continuous Galerkin finite element methods to solve solute transport equations with spatially variable coefficients in a transient state

Mass balance checks are often used to evaluate the abilities of numerical simulators and assess the accuracy of the solutions but are often insufficient. In addition, there is a need to calculate the difference in mass fluxes, through downstream subdomain analysis, to determine whether remediation technologies are suitable for upstream contaminated sites. The Galerkin finite element method (GFEM) has been commonly considered locally nonconservative, whereas the finite difference method (FDM) is a local mass conservation method. In this study, we proved that the GFEM has local mass balance equations on “mass conserved elements” (constructed by connecting lines that perpendicularly cross element faces at midpoints) in solute transport problems with spatially variable velocities and dispersion coefficients, identical to discretized balance equations in the FDM. Furthermore, we found that locally conservative GFEMs with two types of approaches involving spatially variable coefficients (i.e., a linear combination approach and an average approach) give rise to two types of postprocessing techniques for calculating local mass fluxes. These are equivalent to an area-weighted method and first-order Taylor series method, respectively. Finally, to illustrate the practical applicability of these GFEMs, we used the proposed postprocessing approaches to resolve practical problems involving computation of local mass fluxes, based on information directly obtainable from finite element solutions of concentration distribution. Our findings revealed that the local mass balance errors were negligibly small, regardless of numerical concentration accuracy.

On the management fourth-order Schrödinger-Hartree equation

We consider the Cauchy problem associated to the fourth-order nonlinear Schrodinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in \begin{document}$ H^s $\end{document} -spaces. We also analyze the scaling limit of the fast dispersion management and the convergence to a model with averaged dispersions.

Преобразование собственных значений задачи Захарова–Шабата при столкновении солитонов

Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

Abundant excited optical breathers for a nonlinear Schrödinger equation with variable dispersion and nonlinearity terms in inhomogenous fiber optics

Under investigation is a nonlinear Schrödinger equation with variable dispersion and nonlinearity coefficients in inhomogenous fiber optics. Through extending Darboux transformation method, an analytic breather solution with two free functions is obtained for the first time. Furthermore, abundant optical breather structures can be excited by setting the variable coefficient functions. Linear, periodic, V-type, U-type and S-type breather structures are observed and demonstrated. These breathers are novel and interesting.

Periodic and N-kink-like optical solitons for a generalized Schrödinger equation with variable coefficients in an inhomogeneous fiber system

A generalized Schrödinger equation in an inhomogeneous optical fiber system is investigated, which is with variable gain coefficient, variable dispersion coefficient and variable nonlinearity coefficient. Analytical soliton solution and its compatible condition are derived by using the auxiliary equation method. Through setting the dispersion function and gain function, the periodic, doubly periodic, and N-kink-like solitons are excited and observed.

Interaction properties of solitonics in inhomogeneous optical fibers

In this paper, a generalized nonlinear Schrodinger equation with variable dispersion and nonlinear coefficients, which can be used to describe the pulse transmission in inhomogeneous optical fibers, is investigated analytically. By virtue of the Hirota method, analytic multiple soliton solutions are obtained. Interactions between solitonics are presented through choosing specific nonlinearity functions, and interaction properties of them are analyzed. Results obtained may potentially be useful in the area of optical communications.

Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Some analytical solutions of one-dimensional advection–diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green’s function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant’s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.

Vertical dispersion in vegetated shear flows

Canopy layers control momentum and solute transport to and from the overlying water surface layer. These transfer mechanisms strongly dependent on canopy geometry, affect the amount of solute in the river, the hydrological retention and availability of dissolved solutes to organisms located in the vegetated layers, and are critical to improve water quality. In this work, we consider steady state transport in a vegetated channel under fully developed flow conditions. Under the hypothesis that the canopy layer can be described as an effective porous medium with prescribed properties, i.e., porosity and permeability, we model solute transport above and within the vegetated layer with an advection-dispersion equation with a spatially variable dispersion coefficient (diffusivity). By means of the Generalized Integral Transform Technique, we derive a semianalytical solution for the concentration field in submerged vegetated aquatic systems. We show that canopy layer's permeability affects the asymmetry of the concentration profile, the effective vertical spreading behavior, and the magnitude of the peak concentration. Due to its analytical features, the model has a low computational cost. The proposed solution successfully reproduces previously published experimental data.

Developing semianalytical solutions for multispecies transport coupled with a sequential first‐order reaction network under variable flow velocities and dispersion coefficients

Key Points semianalytical method for solving a multispecies reactive‐transport equation sequential first‐order reaction under spatially or temporally varying flow GITT and general linear‐transformation method by Clement (2001)

Spatially Variable Dispersion Coefficients in Meandering Channels

To analyze the spatial variability of two-dimensional dispersion coefficients in meandering channels, an observed dispersion tensor was calculated by applying a theoretical equation to the measured velocity profiles of two laboratory channels. From the observed dispersion tensor, local longitudinal and transverse dispersion coefficients were estimated by the least squares method. In both channels, the spatial variation in the dispersion coefficients was found to be strongly dependent on the specific location within the channel, because alternating bends in channels induces the periodic generation of secondary currents that alter the magnitude of transverse dispersion. Dimensionless transverse dispersion coefficients of each measurement section vary from 0.33 to 1.14 for the M1 channel and from 0.63 to 3.40 in the M2 channel around the entrance and apex of the second bend. The maximum negative correlation occurred near the apex of the channels because of the separation of the point of maximum primary velocity and the center of the circulation cells by the complex flow structure in the meandering channel.

Modeling of nitrate removal for ion exchange resin in batch and fixed bed experiments

In this study, nitrate removal from aqueous solutions was studied using a nitrate selective ion exchange resin in both batch and fixed bed experiments. Nitrate retention experiments were carried out using nitrate-rich groundwater and several synthetic waters with different initial nitrate concentrations and the presence of competitive ions, such as sulfate. The equilibrium data were modeled and compared using the Langmuir adsorption isotherm and the mass action law and both models described the experimental data well. However, different values of Langmuir coefficients were obtained for different operating conditions, whereas it is possible to state all equilibrium data with a simple general equation obtained by applying the mass action law. A mathematical model (advection–dispersion equation) was solved numerically to predict the dynamic behavior of the columns. The experimental breakthrough curves from column experiments were in good agreement with the model predictions. The mass action isotherm parameters obtained from these column experiments were confirmed by the results of the batch experiments. Consequently, column behavior can be predicted from the batch equilibrium data by using the mass action isotherm. Finally, sensitivity analysis showed that the variation of dispersion coefficient between two times higher and two times lower than the base value was negligible.

Indirect Method of Measuring Dispersion Coefficients for Granular Flow in a Column of Dihedrons

In the chemical, pharmaceutical and food-processing industries, the drying operation is a major intermediate step in the production of finished or semi-finished products. There are a great many drying processes and types of equipment, such as dryers with immersed channels; the main advantage of this type of dryer is the homogeneous distribution of the drying air, made possible by dihedrons, a special device for introducing the air. The basis tool for dimension, a drier with immersed channels, is the control of the dehydration of the product, and the distribution function residence time of the particles inside the column. In the present study, a model and thus an indirect measurement of dispersion coefficients of solids were proposed in contrast with previous studies where the variation of dispersion coefficient has often been disregarded in the granular flow models. The comparison of calculations with measurements gave an indirect measurement of the longitudinal and transversal variable coefficients as a function of the position in the column. It is concluded that it is necessary to account for the variation of dispersion coefficients during the granular flow in the column to obtain a correct description of their flow. It is worth noting that this indirect measurement proves that the transversal dispersion coefficient varied much more than the longitudinal one.

Optimization of PSA process for producing enriched hydrogen from plasma reactor gas

Hydrogen purification with optimum recovery and energy consumption from various process streams represents one of the major commercial uses of pressure swing adsorption (PSA) technology. In the plasma process for hydrogen production, the effluent stream from a reactor has to be processed to obtain the high purity hydrogen before being fed to fuel cells for power generation or stored in tanks. The major impurity in the effluent stream is unreacted methane. A rigorous PSA model for simulation and optimization purposes has been developed. The non-isothermal, bulk separation with variable superficial velocity and dispersion coefficient, linear driving force approximation for particle uptake, and Langmuir isotherm to represent adsorption equilibrium were applied in the PSA modeling. The model was solved using gPROMS software. The extensive simulation results involving parametric studies of PSA separation performance were well matched with the experimental results using an activated carbon made from coconut shell as an adsorbent. The optimal conditions for separation of 50%H 2/50%CH 4 mixture in a laboratory-scale PSA unit, and separation of 25%H 2/75%CH 4 mixture, representing exit gas from plasma reactor, in a pilot-scale PSA unit were obtained by carrying out optimization routines in gPROMS.

Stochastic modeling of a fluid batch mixed system with variable dispersion coefficient

This contribution presents the numerical solution of the one-dimensional stochastic differential equation with a diffusion coefficient which is a function of the spatial coordinate, as a model of a fluid batch system stirred by a rotating mechanical impeller. It is demonstrated that neither the Ito nor the Stratonovich methods of solution, which are widely used to solve equations of this kind, lead to a uniform distribution of concentration of the blended species component at steady-state. A method of solution is suggested, with the aid of the so-called transport integral, which converges to the uniform steady-state distribution, consistent with the physical conception of the course of the mixing process.

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.