Nonlinear Sorption Isotherms Research Articles

Performance expectation of coal waste in permeable reactive barrier for removal of cadmium considering contamination level and pore water velocity

The effectiveness and longevity of permeable reactive barriers (PRBs) depend on the performance of the reactive materials and the subsurface environment. The relationship of the groundwater velocity on performance of coal waste for the heavy metal removal was reported in our previous study. In this study, we investigated the performance and longevity of coal waste as a PRB material for the removal of Cd considering subsurface environmental conditions such as contamination level and groundwater velocity. The artificial groundwater contaminated by Cd were prepared with various concentrations ranging from 10 to 100 mg L−1. Lab-scale column experiments were conducted using coal waste filled columns by injecting the artificial groundwater. The breakthrough curves were analyzed advection dispersion equation coupled with equilibrium sorption model to determine the retardation factor. The Cd breakthrough curves exhibited different retardation with respect to the contamination levels. The Cd transport was more retarded as the contamination level lowered. The relationship between the retardation factor and the contamination levels could be explained with empirical equations based on non-linear sorption isotherms. By adopting the velocity dependency of sorbent performance in our previous study, transport of Cd within coal waste was simulated under various subsurface environmental conditions to construct the longevity function. The function could be used for the longevity prediction of coal waste as a PRB material considering groundwater velocity and contamination level in subsurface environment.

Contaminant transport analysis under non-linear sorption in a heterogeneous groundwater system

ABSTRACT In this study, a one-dimensional non-linear advection–dispersion equation subject to spatial–temporal dependent advection and dispersion coefficients is solved for a heterogeneous groundwater system. The non-linearity of the governing equation is based on the Freundlich and Langmuir sorption isotherms. The groundwater flow is considered to vary exponentially with time. Also, a generalized theory of the dispersion coefficient is used for extensive study of the model problem. The approximate solutions of the model problem are obtained in a semi-infinite and finite heterogeneous media by employing the Crank–Nicolson scheme. The exact solutions are obtained in both domains by the Laplace transform technique subject to linear sorption isotherm and non-transient flow conditions. Further, various graphical solutions are obtained using MATLAB scripts to examine the contaminant transport behaviour. For quantitative evaluation of the proposed model, a root mean square (RMS) error is computed. Overall, the results show that RMS error of the approximate solutions with respect to the exact solutions is within acceptable limits (less than 5%) for different combinations of discretization parameters. The robustness of the proposed model suggests its better suitability for modelling groundwater transport phenomena under the consideration of a non-linear sorption isotherm.

Ensemble Smoother with Multiple Data Assimilation as a Tool for Curve Fitting and Parameter Uncertainty Characterization: Example Applications to Fit Nonlinear Sorption Isotherms

The ensemble smoother with multiple data assimilation (ES-MDA) coupled to a normal-score transformation is used to fit a Langmuir isotherm curve to estimate its parameters ( $$S_{m}$$ and b) and their uncertainty. The highlights of this work are threefold: (1) the ES-MDA can be used as a curve fitting procedure, (2) the ES-MDA provides also a full uncertainty quantification about the fitted parameters, and (3) for the specific case of the Langmuir isotherm, parameter $$S_m$$ is well identified with little uncertainty, while parameter b is well identified with a larger uncertainty, indicating that solute concentrations are more sensitive to $$S_m$$ than to b. As a by-product, the number of samples required to characterize the joint uncertainty of Langmuir isotherm parameters is also investigated; it can be concluded that the minimum number of samples to use is six, with best results obtained with eight samples, a value larger than the number recommended in the literature.

Utilization of factorial design methodology to optimize Pr Red Hegxl dye uptake and prediction of removal efficiency via artificial neural network: comparison of linear vs non-linear sorption isotherm and kinetic parameters

Utilization of factorial design methodology to optimize Pr Red Hegxl dye uptake and prediction of removal efficiency via artificial neural network: comparison of linear vs non-linear sorption isotherm and kinetic parameters

The Influence of Salinity on the Removal of Ni and Zn by Sorption onto Iron Oxide- and Manganese Oxide-Coated Sand

The influence of salinity on the single and binary sorption of Ni and Zn onto iron oxide- and manganese oxide-coated sand (IOCS and MOCS) was investigated at pH = 5. The single sorption experimental data were fitted to Freundlich, Langmuir, Dubinin–Radushkevich, and Sips models, and a nonlinear sorption isotherm was observed (NF = 0.309–0.567). The higher Brunauer–Emmett–Teller (BET) surface area (ABET) and cation exchange capacity (CEC) of MOCS contributed to the higher maximum sorption capacities (qmL) of Ni and Zn than that of IOCS. The Ni sorption capacities in the single sorption were higher than that in the binary sorption, while the Zn sorption capacities in the single sorption were less than that in the binary sorption. The single and binary sorptions onto both IOCS and MOCS were affected by the salinity, as indicated by the decrease in sorption capacities. Satisfactory predictions were shown by the binary sorption model fitting including P-factor, ideal adsorbed solution theory (IAST)–Freundlich, IAST–Langmuir, and IAST–Sips; among these, the P-factor model showed the best fitting results in predicting the influence of salinity of Ni and Zn in the binary sorption system onto IOCS and MOCS. IOCS and MOCS offer a sustainable reactive media in a permeable reactive barrier (PRB) for removing Ni and Zn in the presence of salinity.

Supercritical CO2 extraction of pelletized oilseeds: Representation using a linear driving force model with a nonlinear sorption isotherm

This work develops a general, predictive model for the SuperCritical (SC) CO2 extraction of pelletized oilseeds. Cumulative SC-CO2 extraction curves of small (S, dp = 2.5 mm) and large (L, dp = 4.0 mm) cylindrical evening primrose seed pellets were modeled simultaneously using a Linear Driving Force (LDF) approximation modified to consider the nonlinear oil partition between CO2 and the substrate and extrapolated using the actual solubility of oil in CO2 at extraction conditions (40 °C and 45 MPa). Model best-fitting parameter was a single interparticle porosity (εp) of pellets considering equilibrium conditions after static extraction, and mass transfer kinetics during dynamic extraction. The adaptation of general model to bisized mixtures containing 25–75% S-pellets, predicted experimental results better than a monosized version (same Sauter mean diameters) because of the negative impact on overall cumulative extraction curves of slow-extracting L-pellets. Finally, a simplified version of the general model was developed that neglected the intraparticle porosity (εp = 0) and used a microstructural factor (FM∝(εp)−2) in the general model as the best-fitting parameter. This simplified model saved 33% of computational time at the expense of overestimating extraction rates, particularly at the beginning of the diffusion-controlled period, which diminished as εp decreased and FM increased.

Sorption Behavior of Hexabromocyclododecanes (HBCDs) on Weihe River Sediment.

The sorption of hexabromocyclododecanes (HBCDs) on sediment affects the fate and transport of HBCDs in rivers. The sorption of HBCDs on sediment from the Weihe River was investigated by performing batch equilibration experiments, and the effects of changing the pH ionic, strength, and humic acid concentration (HA) on sorption were evaluated. The obtained results indicated that fast rather than slow sorption was the dominant process. Nonlinear sorption isotherms were acquired, and the Freundlich (R2 0.94–0.98) and Langmuir (R2 0.95–0.99) models both described the sorption of HBCDs well. The adsorption capacity for α-HBCD, β-HBCD, and γ-HBCD were calculated using the Langmuir model, and were 443.56, 614.29 and 1146.37 mg/kg, respectively. Thermodynamic analysis shows that HBCDs sorption on sediment is a spontaneous exothermic process. HBCDs sorption was affected by the HA concentration and ionic strength. The amounts of HBCDs sorbed to the sediment decreased as the ionic strength increased, and first increased and then decreased as the HA concentration increased. Changes in pH did not clearly affect the sorption of HBCDs. Synchrotron radiation Fourier-transform infrared spectra (SR-FTIR) was used to characterize the adsorption mechanism, and the obtained result indicated that hydrophobic interactions dominated the mechanism involved in HBCDs sorption on sediment.

Sorption and desorption behaviors of triphenyl phosphate (TPhP) and its degradation intermediates on aquatic sediments

As triphenyl phosphate (TPhP) can biodegrade extensively in sediments, researches should further the understanding of the fate and transport of TPhP and its degradation intermediates in the environment. Therefore, the sorption/desorption behaviors of TPhP, diphenyl phosphate (DPhP) and phenyl phosphate (PhP) on sediments were investigated. The kinetic process was well-fitted by pseudo-second-order model, suggesting that chemisorption was involved. And the Langmuir model could describe the sorption isotherms of TPhP and DPhP well except for PhP. The redundancy analysis revealed that the sorption amount had a positive correlation with sediment organic matter, zeta potential and C/H of sediments. Besides the sorption/desorption behaviors were greatly influenced by the physicochemical properties of the sorbates. PhP with high molecular electrostatic potential (0.132 e0) was prone to protonation and formed hydrogen bonds, leading to higher sorption. Furthermore, hydrophobicity, π-π interactions, Lewis acid-base interaction and hydrogen bonding were involved in the sorption process and resulted in nonlinear sorption isotherms. TPhP, DPhP and PhP exhibited apparent desorption hysteresis on the sediments. Sediments with organic matter removed, which have complex pore distributions, exhibited more hysteresis. These results may contribute to the risk assessment and fate modeling of TPhP and its degradation products in sediments.

Moisture Diffusion in Unsaturated Self-Desiccating Concrete with Humidity-Dependent Permeability and Nonlinear Sorption Isotherm

AbstractA nonlinear diffusion model for the drying of concrete, previously developed at Northwestern University and embedded in some design codes, was improved and calibrated on the basis of recent...

Cs+ sorption onto Kutch clays: Influence of competing ions

The sorption of Cs+ onto raw Kutch clay (India) over a wide range of concentrations and pH levels was studied and compared with its sorption onto the well-studied FEBEX clay (Spain). The principal clay mineral in both samples is montmorillonite. Non-linear sorption isotherms were measured for both clays. The shapes of the isotherms prompted the use of a two-site exchange thermodynamic model to describe the experimental data. However, differences in the highly selective sorption sites (Type 1) were observed for the two studied clays. It was found that the presence of different cations (H+, Ca2+, Mg2+, etc.) in solution affected the sorption of Cs+. A model accounting for the competition process was developed for predicting the behavior of the raw clay in complex solutions.

A Review on Water Vapor Pressure Model for Moisture Permeable Materials Subjected to Rapid Heating

This paper presents a comprehensive review and comparison of different theories and models for water vapor pressure under rapid heating in moisture permeable materials, such as polymers or polymer composites. Numerous studies have been conducted, predominately in microelectronics packaging community, to obtain the understanding of vapor pressure evolution during soldering reflow for encapsulated moisture. Henry's law-based models are introduced first. We have shown that various models can be unified to a general form of solution. Two key parameters are identified for determining vapor pressure: the initial relative humidity and the net heat of solution. For materials with nonlinear sorption isotherm, the analytical solutions for maximum vapor pressure are presented. The predicted vapor pressure, using either linear sorption isotherm (Henry's law) or nonlinear sorption isotherm, can be greater than the saturated water vapor pressure. Such an “unphysical” pressure solution needs to be further studied. The predicted maximum vapor pressure is proportional to the initial relative humidity, implying the history dependence. Furthermore, a micromechanics-based vapor pressure model is introduced, in which the vapor pressure depends on the state of moisture in voids. It is found that the maximum vapor pressure stays at the saturated vapor pressure provided that the moisture is in the mixed liquid/vapor phase in voids. And, the vapor pressure depends only on the current state of moisture condition. These results are contradictory to the model predictions with sorption isotherm theories. The capillary effects are taken into consideration for the vapor pressure model using micromechanics approach.

A new characterization method of membranes with nonlinear sorption isotherm systems based on continuous upstream and downstream time-lag measurements

A theoretical foundation for a new characterization method for dual-mode sorption membranes is presented. The method relies on simultaneous monitoring of pressure decay and pressure rise and continuous evaluation of the respective instantaneous upstream and downstream time lags as well as their ratio. Assuming the simplest gas transport model in dual-mode sorption membranes, i.e. complete immobilization and instantaneous equilibrium between Henry's and Langmuir's sites, the expression for the upstream time lag and the ratio of the upstream and downstream time lags for dual-mode sorption membranes has been derived using Frisch's asymptotic solution. The analytically derived ratio between the upstream and downstream time lags is used as an extra equation that reduces the number of permeation experiments required to fully recover the kinetic and sorption parameters of the membrane. The proposed characterization method, developed for the ideal boundary conditions, is extended to the real boundary conditions, in which gas concentrations at the upstream and downstream interfaces of the membrane vary with time, by recognizing a set of unique relationships between the maximum downstream time lag and the time at which the ratio of the upstream and downstream time lags is equal to the unique value that would exist under the ideal boundary conditions. Finite difference numerical modeling was utilized to examine the proposed method under the real boundary conditions using some case studies from the literature. The accuracy of the recovered kinetic and sorption parameters is promising and allows the elimination of challenging sorption measurement involving dense membranes.

Sorption of Cationic Surfactants to Artificial Cell Membranes: Comparing Phospholipid Bilayers with Monolayer Coatings and Molecular Simulations.

This study reports the distribution coefficient between phospholipid bilayer membranes and phosphate buffered saline (PBS) medium (DMW,PBS) for 19 cationic surfactants. The method used a sorbent dilution series with solid supported lipid membranes (SSLMs). The existing SSLM protocol, applying a 96 well plate setup, was adapted to use 1.5 mL glass autosampler vials instead, which facilitated sampling and circumvented several confounding loss processes for some of the cationic surfactants. About 1% of the phospholipids were found to be detached from the SSLM beads, resulting in nonlinear sorption isotherms for compounds with log DMW values above 4. Renewal of the medium resulted in linear sorption isotherms. DMW values determined at pH 5.4 demonstrated that cationic surfactant species account for the observed DMW,PBS. Log DMW,PBS values above 5.5 are only experimentally feasible with lower LC-MS/MS detection limits and/or concentrated extracts of the aqueous samples. Based on the number of carbon atoms, dialkylamines showed a considerably lower sorption affinity than linear alkylamine analogues. These SSLM results closely overlapped with measurements on a chromatographic tool based on immobilized artificial membranes (IAM-HPLC) and with quantum-chemistry based calculations with COSMOmic. The SSLM data suggest that IAM-HPLC underestimates the DMW of ionized primary and secondary alkylamines by 0.8 and 0.5 log units, respectively.

Mathematical simulation of atomic hydrogen diffusion transfer through a multilayer metal membrane at finite pressures

Diffusion transfer of atomic hydrogen through multilayer metal membranes has been studied within the lattice model of an ideal gas, with the transfer being described by a set of nonlinear algebraic equations. It has been shown that, for multilayer membranes composed of less than four layers, an analytical expression describing a diffusion flux can be derived. Atomic hydrogen transfer through a membrane consisting of a vanadium layer, the surfaces of which are coated with palladium films, has been analyzed in detail. It has been found that the value of the flux may depend on the transfer direction. The effect of diffusion asymmetry arises at finite pressures of hydrogen on the outer membrane surfaces, when its dissolution in metals is described by nonlinear sorption isotherms. The degree of the diffusion asymmetry increases with a rise in hydrogen pressure and depends on the arrangement of the layers composing a membrane.

Characteristics and influencing factors of tetrachloroethylene sorption-desorption on soil and its components.

To investigate the effects of soil structure, soil organic carbon (SOC), minerals, initial tetrachloroethylene (PCE) concentration (C0), and ionic strength (Ci) on PCE sorption-desorption, six types of soil were adopted as adsorbents, including two types of natural soil and four types of soil with most of the "soft carbon" pre-treated by H2O2 or with all SOC removed from the original soil by 600 °C ignition. The results showed that all of the sorption-desorption isotherms of PCE were non-linear within the experimental range, and the H2O2-treated samples exhibited higher non-linear sorption isotherms than those of the original soils. The hysteresis index of PCE sorption to original soil is less pronounced than that of the H2O2-treated and 600 °C-heated samples due to the entrapment of sorbate molecules in the "hard carbon" domain, together with the meso- and microporous structures within the 600 °C-heated samples. Both SOC and minerals have impacts on the sorption-desorption of PCE, and the sorption-desorption contribution rate of minerals increased with decreasing SOC content. C0 has almost no influence on the sorption to minerals of the soils, but the contribution rate of minerals decreased with increasing C0 in the desorption stage. As a result of the salting-out effect, PCE sorption capacity was increased by increasing Ci, especially when Ci ≥ 0.1 M. Moreover, desorption increased and hysteresis weakened with increasing Ci, except for the 600 °C-heated samples. In addition, no significant effect of Ci on desorption of PCE and no hysteresis was observed in this experimental range for the 600 °C-heated samples.

Transmembrane gas transfer through a two-layer membrane

Gas diffusion through a two-layer membrane is studied within the lattice model of an ideal gas. Analytical expressions are derived for transmembrane diffusion gas flux and permeability of the membrane. It is revealed that the value of the transmembrane gas flux may depend on the membrane orientation. An expression is obtained for the asymmetry coefficient of diffusion gas transfer. It is shown that the revealed effect of asymmetry of membrane diffusion permeability arises in the case of nonlinear sorption isotherms. The results obtained can be used to predict the separating characteristics of composite membranes.

Sorption–desorption of imidacloprid onto a lacustrine Egyptian soil and its clay and humic acid fractions

Sorption–desorption of the insecticide imidacloprid 1-[(6-chloro-3-pyridinyl)-methyl]-N-nitro-2-imidazolidinimine onto a lacustrine sandy clay loam Egyptian soil and its clay and humic acid (HA) fractions was investigated in 24-h batch equilibrium experiments. Imidacloprid (IMDA) sorption–desorption isotherms onto the three sorbents were found to belong to a non-linear L-type and were best described by the Freundlich model. The value of the IMDA adsorption distribution coefficient, Kdads, varied according to its initial concentration and was ranged 40–84 for HA, 14–58 for clay and 1.85–4.15 for bulk soil. Freundlich sorption coefficient, Kfads, values were 63.0, 39.7 and 4.0 for HA, clay and bulk soil, respectively. The normalized soil Koc value for imidacloprid sorption was ∼800 indicating its slight mobility in soils. Nonlinear sorption isotherms were indicated by 1/nads values <1 for all sorbents. Values of the hysteresis index (H) were <1, indicating the irreversibility of imidacloprid sorption process with all tested sorbents. Gibbs free energy (ΔG) values indicated a spontaneous and physicosorption process for IMDA and a more favorable sorption to HA than clay and soil. In conclusion, although the humic acid fraction showed the highest capacity and affinity for imidacloprid sorption, the clay fraction contributed to approximately 95% of soil-sorbed insecticide. Clay and humic acid fractions were found to be the major two factors controlling IMDA sorption in soils. The slight mobility of IMDA in soils and the hysteresis phenomenon associated with the irreversibility of its sorption onto, mainly, clay and organic matter of soils make its leachability unlikely to occur.

Effect of nonlinear sorption on multispecies radionuclide transport in a coupled fracture-matrix system with variable fracture aperture: a numerical study

A one-dimensional numerical analysis on multispecies radionuclide transport in a single-horizontal coupled fracture-matrix system has been performed. Analysis considering linear and nonlinear adsorption cases with linear, Langmuir, and Freundlich adsorption models has been carried out. The results indicate that there is a significant change in the spatial distribution of radionuclides in a coupled fracture-matrix system when a nonlinear adsorption isotherm is considered as compared to the simplified linear sorption isotherm. Sensitivity analysis of Langmuir constant and Freundlich exponent has been performed, and the numerical results indicate that the behavior of radionuclide transport is strongly influenced by these nonlinear sorption isotherm parameters. In addition, an attempt has been made to consider the variation of fracture aperture thickness along its flow direction by introducing logarithmic distribution. A clear distinction in the spatial distribution of radionuclide concentration was observed when variable fracture aperture is considered as opposed to the conventional parallel plate model with a constant fracture aperture thickness. Further, the results suggest that there is an enhanced retardation of radionuclides within the high permeable fracture with varying fracture aperture.

Application of nonequilibrium fracture matrix model in simulating reactive contaminant transport through fractured porous media

Nonequilibrium and nonlinear sorption of the contaminants in the fractured porous media could significantly influence the shape of the breakthrough curve (BTC). For the fracture-matrix system, there are very few studies which consider these processes. In this study, the nonequilibrium fracture-matrix model with two different nonlinear sorption isotherms, namely nonlinear Freundlich and Langmuir sorption isotherms were developed. The effect of sorption nonlinearity and nonequilibrium conditions on the shape of the BTC was studied using the temporal moments. The developed models along with the linear equilibrium, linear nonequilibrium fracture matrix models, and the multirate mass transfer model were used to simulate the BTC, which were compared with the experimental data available in the literature. Both sorption nonequilibrium and nonlinearity were found to significantly influence the shape of the BTC. Presence of sorption nonlinearity reduces the solute spreading, whereas presence of nonequilibrium conditions increases the solute spreading. Considering the sorption nonequilibrium along with the sorption nonlinearity leads to an improved simulation of the BTC. The nonequilibrium nonlinear sorption models could simulate the extended BTC tailing resulting from sorption nonlinearity and rate-limited interaction in the fracture-matrix system.

Numerical studies on kinetics of sorption and dissolution and their interactions for estimating mass removal of toluene from entrapped soil pores

When the non-aqueous phase liquid (NAPL) contaminant is entrapped in soil pores, their release to the subsurface environment is mainly limited by interphase mass transfer such as dissolution and sorption. Sorption onto the aquifer material plays an important role as a retardation mechanism in subsurface contaminant transport processes. Considering the heterogeneity associated with aquifer properties as well as contaminant source distribution, long-term contamination is an inevitable consequence of mass transfer limitation of residual mass from soil micropores. A comprehensive numerical model is presented in this study for understanding the kinetic nature of dissolution and sorption of hydrophobic hydrocarbons and their interactions in order to estimate the rate and extent of mass removal from different phases. The results showed that sorption by soils and sediments having different physicochemical properties generally follow non-linear behavior. It is observed from the study that sorption non-linearity can be effectively incorporated by the two-site kinetic model rather than combination of linear and non-linear sorption isotherms. Even though initial phase of mass transfer is dissolution-dominated, the extended tailing of concentration at higher pore volumes is controlled by non-equilibrium sorption, which is better explained by the two-site kinetic model.