Nonlinear Sorption Isotherms Research Articles

Polanyi-Based Models for the Competitive Sorption of Low-Polarity Organic Contaminants on a Natural Sorbent

The development of appropriate equilibrium sorption relationships for anthropogenic organic contaminants with soils and sediments is essential to predicting the extents and rates of solid−water interactions in the environment. In this context, we previously reported results that showed how nonlinear sorption isotherms with nine low-polarity organic chemicals on a silty aquitard material could be modeled as a combined adsorption-partitioning process. The adsorption component was ascribed to presumed traces of a carbonaceous adsorbent, for which adsorption was well described by a Polanyi-based isotherm. In this work, we report competitive sorption behavior on the same material, as observed from batch experiments in binary solute systems that contain combinations of the previously studied sorbates (chorinated benzenes and polyaromatic hydrocarbons) over wide ranges of concentra tion. The results confirmed our expectation that partitioning is an increasingly dominating contribution to overall sorption when co...

Numerical Analysis of the Effect of the Lower Boundary Condition on Solute Transport in Lysimeters

Field lysimeters are often used to assess environmental behavior of agrochemicals. Most lysimeters used to date have a free‐draining lower boundary where leaching out of the lysimeter occurs by gravity alone. In this case, the lower boundary of a lysimeter is open to the atmosphere, and consequently, leachate is collected only if the bottom of the lysimeter becomes water saturated. In a field soil, such local water saturation does not occur. The objective of this study was to evaluate the effect of the lower boundary condition on chemical leaching. Numerical simulations were used to compare solute transport in field soils and in lysimeters. Simulations were carried out in homogeneous sandy and loamy soils under steady‐state, unsaturated water flow conditions. Water flow was described by the Richards equation and solute transport by the advection–dispersion equation. The effect of linear and nonlinear and instantaneous and kinetic sorption was investigated. The results showed that for a conservative solute the differences between field soil and lysimeter increase as the coarseness of the soil increases. Decreasing water flux increases the difference between field soil and lysimeter. In general, solute transport in the lysimeter is characterized by a slower mean velocity, a larger spreading, and smaller concentration values. For solutes subject to linear equilibrium sorption, the sorption mechanism compensates for the effects of the lower boundary condition. The larger the sorption coefficient, the less the difference between lysimeter and field soil. However, large differences are found in the case of strongly convex nonlinear sorption isotherms.

Diffusion of heterocyclic compounds from a complex mixture of coal–tar compounds in natural clayey till

The diffusion of coal–tar compounds in natural clayey till was studied experimentally. Cores were exposed to a solution with near constant concentration of coal–tar compounds in a multi-component mixture for 5 months, the cores were subsequently sub-sampled and analyzed. Diffusion profile data strongly indicates highly non-linear sorption isotherms. For dibenzofuran, dibenzothiophene, 2-methylquinoline, carbazole, phenanthrene, and fluorene, the profiles indicated significantly stronger sorption at high solute concentrations than at low solute concentrations, and than expected based on linear sorption isotherms. This corresponds with observed dramatic increase in sorption of these compounds at high surface density. For benzofuran, benzothiophene, quinoline, phenols, naphthalenes, and BTEXs the profiles indicated significantly weaker sorption at high solute concentrations than at low solute concentrations and than expected based on linear or Freundlich isotherms. The observed diffusion profiles have important implications with respect to transport of dissolved coal–tar compounds in multi-component mixtures of high concentrations as expected near immiscible phase coal–tar sources. Breakthrough times and concentration levels are affected significantly.

Analysis of Protein Purification Using Ion-Exchange Membranes

The performance of ion-exchange membranes for protein purification is analyzed using numerical solutions of different mathematical models. The models incorporate nonlinear sorption isotherms and mass-transfer coefficients based on either the overall or local solid-phase and liquid-phase driving forces. The numerical solutions are compared to analytical solutions which use overall mass-transfer coefficients only and, in general, are theoretically incorrect for nonlinear isotherms. The numerical solutions are fit to experimental breakthrough curves from the literature. The models allow the determination of the rate-controlling mass-transfer phenomena and solid-phase concentration, and prediction of the operating and membrane-design parameters needed to obtain sharp breakthrough curves.

Analytic solutions to the advective contaminant transport equation with non-linear sorption

This paper considers advective transport of a soluble contaminant through saturated soil with non-linear sorption of the contaminant onto a stationary porous media. The non-linear sorption isotherms considered in the transport analysis are the Langmuir and Freundlich sorption isotherms. A special case of the Freundlich sorption isotherm is the linear sorption isotherm, and it is shown that in this case transport through a homogeneous soil results in the initial concentration profile simply being translated in the direction of the groundwater flow. However, when the sorption isotherm is non-linear the initial concentration profile distorts as it is translated with the groundwater flow, leading to the development of concentration shock fronts and rarefactions. Analytic solutions to the non-linear first-order hyperbolic equations are developed for a number of contaminant transport problems of practical significance. It is shown that in the case of the Langmuir sorption isotherms, shock fronts develop at the leading edge of the concentration profile while for the Freundlich sorption isotherm shock fronts may develop at either the leading or trailing edge of the concentration profile. Copyright © 1999 John Wiley & Sons Ltd.

Two‐region linear/nonlinear sorption modeling: Batch and column experiments

AbstractBatch and column experiments were conducted to assist in elucidation of the mechanism and rates of slow diffusion of 1,2,4‐trichlorobenzene into Borden sand. Batch rate experiments were conducted at three concentration levels, and a full isotherm was developed over 550 d. The equilibrium isotherm showed significant nonlinearity (Freundlich exponent = 0.75). Three column experiments were conducted—two at low velocity (each at a different input concentration) and one at high velocity. The data were analyzed with four different models; the two most successful are described in detail here. These two models divided sorption between instantaneously equilibrating and diffusion‐limited domains, with either a linear or nonlinear sorption isotherm assumed for the instantaneously equilibrating fraction of sorption sites. Either model could simulate any single data set by adjustment of two parameters; however, no single parameter set could simulate all data sets. The model with a linearly sorbing instantaneous fraction described the batch data better than the model with a nonlinearly sorbing instantaneous fraction; however, a nonlinearly sorbing instantaneous fraction described the slow column experiments much better than a linearly sorbing instantaneous fraction. The high‐flow‐rate column experiment indicated that the assumed instantaneous fraction is actually influenced by rate limitations, suggesting that more complex models are needed in order to simulate this shorter time scale behavior. Overall, our work illustrates how one needs to include a wide variety of experimental conditions in order to fully test the complex set of sorption equilibrium and rate mechanisms that are present in natural solids.

Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media

Transport of a sorbing solute in a two-dimensional steady and uniform flow field is modeled using a particle tracking random walk method. The solute is initially introduced from an instantaneous point source. Cases of linear and nonlinear sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. The numerical simulation of solute particle transport yields the large scale behavior of the solute plume. Behavior of the plume is quantified in terms of the center-of-mass displacement distance, relative velocity of the center-of-mass, mass breakthrough curves, spread variance, and longitudinal skewness. The nonlinear sorption isotherm affects the plume behavior in the following way relative to the linear isotherm: (1) the plume velocity decreases exponentially with time; (2) the longitudinal variance increases nonlinearly with time; (3) the solute front is steepened and tailing is enhanced

Identification of nonlinear sorption isotherms by soil column breakthrough experiments

Identification of nonlinear sorption isotherms by soil column breakthrough experiments

Long-Term TNT Sorption and Bound Residue Formation in Soil

Soils surrounding former munitions production facilities are highly contaminated with 2,4,6-trinitrotoluene (TNT). Long-term availability and fate of TNT and its transformation products must be understood to predict environmental impact and develop appropriate remediation strategies. Sorption and transport in surface soil containing solidphase TNT are particularly critical, since nonlinear sorption isotherms indicate greater TNT availability for transport at high concentrations. Our objectives were to determine long-term sorption and bound residue formation in surface and subsurface Sharpsburg soil (Typic Argiudoll). Prolonged equilibration of ¹⁴C-TNT with the soil revealed a gradual increase in amount sorbed and formation of unextractable (bound) ¹⁴C residues. The presence of solid-phase TNT did not initially affect the amount of ¹⁴C sorbed during a 168-d equilibration. After 168 d, 93% of the added ¹⁴C was sorbed by uncontaminated soil, while 79% was sorbed by soil containing solid-phase TNT. In the absence of solid phase, pools of readily available (extractable with 3 mM CaCl₂) and potentially available (CH₃CN-extractable) sorbed TNT decreased rapidly with time and coincided with increased ¹⁴C in soil organic matter. More ¹⁴C was found in fulvic acid than in the humic acid fraction when no solid-phase TNT was present. After sequential extractions, including strong alkali and acid, 32 to 40% of the sorbed ¹⁴C was irreversibly bound (unextractable) in Sharpsburg surface and subsurface soil. Results provide strong evidence for humification of TNT in soil. This process may represent a significant route for detoxification in the soil-water environment. Publication no. 11551, Agric. Res. Div., Univ. of Nebraska — Lincoln, Lincoln, NE 68583-0915.

Properties of exact and approximate traveling wave solutions for transport with nonlinear and nonequilibrium sorption

Nonlinear sorption leads to the existence of moving concentration fronts of substances transported in porous media that do not change shape. The cause for the existence of such traveling wave fronts is a balance between the self‐sharpening effect of nonlinear sorption and the spreading effects of dispersion and sorption kinetics. While analytical solutions for the transport of nonlinearly sorbing substances are usually not available, the asymptotic front speed and the shape of traveling waves can in some situations be determined analytically. This is done by deriving a traveling wave equation in a moving coordinate system. In the case of simultaneous linear equilibrium and nonlinear nonequilibrium sorption, this traveling wave equation is a second‐order differential equation. Because this equation cannot be solved analytically for typical nonlinear sorption isotherms, the second‐order term is usually neglected. In this paper the significance of the second‐order term is discussed for a simple piecewise linear sorption isotherm that allows the analytical solution of both the full and the simplified differential equations. These analytical solutions make it possible to calculate the approximation error as a function of flow and isotherm parameters and thus to localize domains where the error may be significant. In addition, the analytical solutions are used to discuss the identifiability of model parameters from asymptotic front shape data.

An analytical model for nonlinear adsorptive transport through layered soils

Solute transport with nonlinear adsorption occurs in many situations involving inorganic chemical and metal contamination in soil and groundwater systems. The resulting isotherms can be highly nonlinear, and numerical solutions of the transport equation can encounter severe convergence difficulties. An analytical solution is developed for simulating one‐dimensional transport in the layered soil column with nonlinear adsorption effects. The analytical solution is an exact solution for nondispersive transport and becomes an approximate one when dispersion effects are included. The accuracy of the solution has been examined using a numerical model under advection‐dominated conditions. The derived analytical solution can be used for any form of nonlinear sorption isotherms given as closed‐form functions, or tabulated data, with a general time‐dependent source boundary condition. The heterogeneity of soils is handled by treating the soil column as a layered system. The resulting analytical model is suitable for estimating the leaching potential of metals or other solutes in the subsurface and for determining travel times of contaminants. Our analysis shows that the shape and nonlinearity of the sorption isotherm may have a controlling influence on transport behavior. Evaluation and application examples are provided to demonstrate applicability of the solution to solute and metal transport with both functional and tabular nonlinear isotherms.

Sorption of phenylthioacetates on natural soil: Application of partition-adsorption mechanism and model

Sorption of eight phenylthioacetates from water by natural soil is studied at very low to high relative concentrations (ratios of equilibrium solute concentration to solute water solubilities). The Freundlich isotherm is used to fit the experimental data, where nonlinear sorption isotherms are observed. The partition-adsorption mechanism and model is used to explain the nonlinear isotherms, which show that sorption of phenylthioacetates from water by natural soils consists of two contributions: a linear partition in SOM and a nonlinear residual adsorption on soil minerals and onto SOM because of the specific interactions between functional groups of phenylthioacetatesand OH and COOH groups of SOM Partition dominates the sorption process at moderate to high relative concentrations, and both partition and the residual adsorption dominates over partition at low relative concentration.

Competitive Sorption and Displacement of Hydrophobic Organic Contaminants in Saturated Subsurface Soil Systems

The sorption of tetrachloroethylene and trichlorobenzene by a subsurface soil that exhibits moderately nonlinear sorption isotherms for each solute was found to be strongly competitive, while that by a soil for which both solutes exhibit more linear sorption isotherms was essentially additive. The isotherm nonlinearity and competitive sorption observed in the former case suggest that a greater fraction of the sorption by that soil occurred on or within tightly ordered or condensed organic matrices rather than within more amorphous soil organic matter. Ideal adsorbed solution theory was used in conjunction with a Freundlich isotherm in a dual‐domain model to characterize mathematically the sorption behavior of these two soils in column‐type reactor systems. Model parameters obtained independently from breakthrough curves for single‐solute systems were used to predict breakthrough curves for bisolute systems. The results demonstrate that (1) even moderate isotherm nonlinearity can affect breakthrough curve behavior in single‐solute systems, and (2) the effects of isotherm nonlinearity on solute breakthrough are significantly increased by the additional manifestation of competitive effects in bisolute systems.

Nonlinear Sorption Isotherm of Zeolites by Frequency Response Analysis

Frequency response analysis was used to study the physical adsorption and diffusion of ethene on 13X molecular sieves, a system for which the adsorption isotherm has a relatively large curvature. Transient experiments were carried out in a Berty reactor with effluent sample analysis every 0.5 s. Mathematical models describing sorption−diffusion were solved by applying perturbation theory and Laplace transform. Fast Fourier transform was used to separate the responses into the fundamental and first harmonic frequencies. The slope and curvature of the sorption isotherm and effective diffusivity were determined by applying the Hooke and Jeeves algorithm. The sorption isotherm constructed with data from dynamic experiments was comparable to the isotherm from steady-state experiments.

Modeling Transport of Multiple Organic Compounds: Segregated Transport‐Sorption/Solubilization Numerical Technique

The transport of multiple organic contaminants in the subsurface may be affected by their mutual interactions. Conventional modeling approaches cannot simulate the transport of multiple organic species whose sorption and/or solubilization is interdependent. This paper describes a numerical modeling approach for characterizing interactive solute sorption and solubilization reactions coupled with one‐dimensional advective‐dispersive transport in porous media. This numerical approach, referred to as segregated transport‐sorption/solubilization (STSS), allows for interphase mass transfer at each time step after solute advection and dispersion have occurred. The overall capabilities of the STSS technique include modeling equilibrium or nonequilibrium transport of individual and multiple solutes with linear or nonlinear sorption isotherms. Application of this modeling approach is illustrated for the transport of a conservative tracer, a surfactant, a hydrophobic organic compound (HOC), and a polychlorinated biphenyl (PCB) mixture. The coupling of sorption modules for phenanthrene and surfactant is used to reasonably describe surfactant‐enhanced flushing of sorbed phenanthrene from a sand column.

Numerical Simulation of Solute Transport Instantaneously Injected in a Single Rock Fracture System: Inclusion of a Nonlinear Sorption Term

AbstractThe effect of nonlinear Freundlich sorption isotherm on the transport of sorptive solute in a system of fracture and porous matrix was investigated through numerical simulation. To solve a set of partial differential equations of solute transport with nonlinear term, use of the Laplace transform Galerkin (LTG) technique was investigated. It was shown that the LTG method could be applied successfully to solve quasi-linearized transport equation. Sensitivity analysis showed that nonlinear sorption in porous matrix with order less than 1 increased the skewness of the breakthrough curve. The fitting of experimental effluent data using a transport model with nonlinear isotherms was also conducted.

Modelling the complex problem of intracrystalline diffusion and first-order phenomena in microporous solid particles under constant-volume/variable-concentration conditions

Microporous solids with pore diameters comparable to effective molecular cross-sections are mainly used as both stereo-selective absorbents and shape-selective catalysts. Often complex phenomena govern the overall rate of processes of both physical adsorption and catalytic reaction. Transport and reaction rates are usually determined independently of each other. In contrast to this procedure, this paper describes a way to model such phenomena comprising internal diffusion in microporous particles coupled with any first-order rate process inherent both in the physical system, i.e. sorption system with nonlinear sorption isotherm and any particle size distribution, and the experimental apparatus characteristics. Modelling and complete solution of the models for both constant and variable boundary conditions were carried out by means of non-linear Volterra integral equations. It becomes possible to determine both the diffusion coefficients and rate constants of constituents of a complex process using only one experimental arrangement. The approach is incorporated into the software ZEUS (zeolite uptake simulator).

Competition and enhancement of VOC sorption at low sorbent moisture contents

Alumina coated with humic acid was used to evaluate the sorption of mixtures of organic vapors under partially saturated conditions. Vapor interactions resulted in both enhanced and suppressed sorption relative to the uptake of single vapors. The interactions observed for VOCs may result from interaction between adjacent molecules at the surface, solvophobic effects, or nonlinear sorption isotherms.

Modeling of a Diffusion-Sorption Experiment by Linear and Nonlinear Sorption Isotherms

The results of a diffusion-sorption experiment, using simulated groundwater spiked with a mixture of I25I, 85Sr, and 137Cs, are modeled by a one-dimensional porous-medium approach in which sorption...

Exact solutions of the problem of adsorption-desorption dynamics with a nonlinear sorption isotherm

Analytic solutions of the problem of adsorption-desorption dynamics with a nonlinear Sorption isotherm are obtained for cases of practical importance in which the Danckwerts condition is satisfied at the inlet to the porous medium and the adsorbate concentration in the mobile phase at the surface of the porous medium is given.