Equilibrium sorption models

FIGURE 12.3 Experimental and simulated breakthrough results of S04 effluent concentrations vs. pore volume (BTCs) from the E horizon (column E-I, input S04 of 0.005 M). The simulation is based on best fit of the data when a linear-equilibrium sorption model was assumed. 

FIGURE 12.4 Experimental and simulated (solid and dashed curves) BTCs of S04 effluent concentrations from the Bs horizon (column Bs-I, input S04 (CQ) of 0.005 M). Simulations are for a range of Kd values where a linear-equilibrium sorption model was assumed. 

Laboratory column experiments were used to identify potential rate-controlling mechanisms that could affect transport of molybdate in a natural-gradient tracer test conducted at Cape Cod, Mass. Column-breakthrough curves for molybdate were simulated by using a one-dimensional solute-transport model modified to include four different rate mechanisms equilibrium sorption, rate-controlled sorption, and two side-pore diffusion models. The equilibrium sorption model failed to simulate the experimental data, which indicated the presence of a ratecontrolling mechanism. The rate-controlled sorption model simulated results from one column reasonably well, but could not be applied to five other columns that had different input concentrations of molybdate without changing the reaction-rate constant. One side-pore diffusion model was based on an average side-pore concentration of molybdate (mixed side-pore diffusion) the other on a concentration profile for the overall side-pore depth (profile side-pore diffusion). 

Simulation of Mo(VI) breakthrough by the equilibrium sorption model is compared with the experimental data from the 0.043 mmol/1 column in Figure 3. Results for the 0.096-, 0.01- and 0.0016-mmol/l columns were similar and are not shown. The model simulates a very steep slope for the adsorption limb of the breakthrough curve and complete site saturation by the second pore volume. Experimental data from the column show that complete breakthrough did not occur until the sixth pore volume, which indicates the effects of a rate process. The equilibrium model also simulated complete rinse-out of Mo(VI) by the 9th pore volume whereas Mo(VI) in the column effluent did not reach zero until the 15th pore volume, which indicates that desorption also was affected by a rate process. 

Figure 3. Simulation of Mo(VI) experimental data from sewage-contaminated ground water, using the equilibrium sorption model.

Table 6.1 Frequently used equilibrium sorption models. ...

The function/(Q represents any sorption isotherm for a chemical species (linear or nonlinear). The most frequently encountered equilibrium sorption models are listed in Table 6.1 (Simimeket. al., 1999 Selim et. al. 1990)... 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

The First-Order Kinetic Model. Karickhoff (1, 68) has proposed a two-compartment equilibrium-kinetic model for describing the solute uptake or release by a sediment. This model is based on the assumption that two types of sorption sites exist labile sites, S, which are in equilibrium with bulk aqueous solution, and hindered sites, Sjj, which are controlled by a slow first-order rate process. Conceptually, sorption according to this model can be considered either as a two-stage process ... 

Of the various equilibrium and non-equilibrium sorption isotherms or sorption characteristics models, the most popular are the Langmuir and Freundlich models. The correct modeling of an adsorbate undergoing both transport and adsorption through a clay soil-solid system necessitates the selection of an adsorption isotherm or characteristic model which best suits the given system. The use of an improper or inappropriate adsorption model will greatly affect the... 

Most sorption/desorption kinetic models fit the data better by including an instantaneous, non-kinetic fraction described by an equilibrium sorption constant. 

Figure 19.19 Comparison of the solution of the linear sorption model with the radial diffusion model. Numbers on curves show y defined in Eq. 19-86. Y is the fraction of the chemical taken up by the sphere when equilibrium is reached. After Wu and Gschwend (1988).

The Loading Ratio Correlation. The equilibrium sorption therms for the pure components are correlated to the LRC model (1), which can be stated in the following manner ... 

Transient-transport measurements are a powerful tool for evaluating the validity of any sorption-transport model. The ability of a model to predict diffusion time lags is a test for its validity, as all the parameters are fixed by the equilibrium sorption and steady state transport, and because the time lag depends on the specific form of the concentration and diffusion gradients developed during the transient-state experiments. 

The assumption of equilibrium sorption has been supported by the long anticipated residence times in an installed barrier (e.g., days for a barrier thickness of 1-2 m), and by batch kinetic data reported by Cantrell (1996) that indicate near-equilibrium is achieved on the order of one day. Similar assumptions have been applied to the analysis of GAC barriers (e.g., Schad and Gratwohl, 1998). Cantrell (1996) also observed linear isotherms for Sr concentrations below approximately 0.1 mg/L, although this result should be viewed as particular to the specific experimental conditions. Although these results lend support to the simplified modeling approach, more data are clearly needed to better evaluate the key assumption of linear equilibrium sorption. 

According to the dual-sorption model, gas sorption in a polymer (cm) occurs in two types of sites. The first type is filled by gas molecules dissolved in the equilibrium free volume portion of material (concentration cH). In rubbery polymers this is the only population of dissolved gas molecules, but in glassy polymers a second type of site exists. This population of dissolved molecules (concentration cD) is dissolved in the excess free volume of the glassy polymer. The total sorption in a glassy polymer is then... 

Sorption of organic contaminants onto aquifer solids is frequently described as a partitioning process, where the hydrophobic organic compound partitions into natural organic material associated with the aquifer solids [8]. Sorption can be characterized as either an equilibrium or rate-limited phenomenon. Equilibrium sorption can be modeled as either a linear or non-linear process. Equilibrium sorption may be assumed when the flow of groundwater and other processes affecting contaminant transport are slow compared to the rate of sorption. In this event the sorption of the contaminant can be considered instantaneous. If we assume equilibrium sorption, the relationship between sorbed and aqueous contaminant concentrations may be described by a sorption isotherm. 

Most field-scale modeling studies of contaminant plumes make the local equilibrium assumption (LEA) [18,19]. The LEA is based on the premise that the interactions between the contaminant and the aquifer material are so rapid compared to advective residence times that it can be assumed that the interactions are instantaneous [3]. Linear equilibrium sorption assumes that the binding of contaminants to aquifer solids is instantaneous and that the concentration of sorbed contaminant is directly proportional to the concentration of the dissolved contaminant. This can be modeled by a linear sorption isotherm [2] ... 

BIOPLUME III is a public domain transport code that is based on the MOC (and, therefore, is 2-D). The code was developed to simulate the natural attenuation of a hydrocarbon contaminant under both aerobic and anaerobic conditions. Hydrocarbon degradation is assumed due to biologically mediated redox reactions, with the hydrocarbon as the electron donor, and oxygen, nitrate, ferric iron, sulfate, and carbon dioxide, sequentially, as the electron acceptors. Biodegradation kinetics can be modeled as either a first-order, instantaneous, or Monod process. Like the MOC upon which it is based, BIOPLUME III also models advection, dispersion, and linear equilibrium sorption [67]. 

In Figures 12.10 and 12.11, we illustrate a comparison of simulated (curve-fitted) versus predicted BTCs for the BC-II column. Clearly, regardless of whether a kinetic or equilibrium model was used, the predictions overestimated the extent of sorption and resulted in much-delayed BTCs. These predictions were obtained with independently derived parameters from the BC-I column for the equilibrium linear model as well as the kinetic model. It is obvious that the independently measured parameters for the high concentration were inadequate in describing BTC for the low concentration. Therefore, the reactivity or retention of S04 during transport is concentration dependent. 

If the addition of pentane occurs at 60-70% conversion, two influences result an increase in pressure due to the arranged loss of soluble styrene with increasing conversion, and a decrease in pressure because of increasing diffusion of pentane into the beads. The equilibrium pressure for the quaternary system styrene-polystyrene-isopentane-n-pentane has been calculated by Wolfahrt [27] for different conversions, temperatures and //-/isopentane ratios using a thermodynamic sorption model based on chain-of-rotators equation-of-state. 

Equilibrium sorption isotherms heats of sorption, and diffusivities have been calculated for carbon dioxide in type A zeolites, using an idealized model of the molecular potential field. Good agreement with practical results are obtained for heats of sorption, but results are poor for equilibrium isotherms, and the simple approximation used for diffusivity is quite inadequate. Nevertheless, useful insight is obtained on the basic assumptions. 

Equation 6 can be shown to correspond in mathematical form to a model predicated on a continuous spectrum of sorption interaction energies. If this interpretation is imposed on equation 6, the variable n can be said to reflect both the level and distribution of sorption energies, and KF the sorption capacity. For most natural solids, n generally ranges in value between 0.5 and 1.0, the upper limit characterizing a linear isotherm. As defined, KF would logically incorporate the specific reactive surface area, SH, of the sorbent, which can be abstracted to yield a capacity term, KFh, expressed per unit surface area (KFh = KF/SH). A logarithmic transform of equation 6 can be used to facilitate evaluation of both KVu and n from observed equilibrium sorption data. 

If transport occurs much faster than sorption, sorption processes may not reach equilibrium conditions. Nonequilibrium sorption may result from physical causes such as intraparticle rate-limited diffusion, chemical causes such as rate-limiting reaction kinetics, or a combination of the two. One approach used to model rate-limited sorption is bi-continuum models consisting of one region where transport is described by the advection-dispersion equation with equilibrium sorption, and another region where transport is diffusion limited with equilibrium sorption, or another region where sorption is chemically rate limited. 

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