Sorption rate model

The values of kj calculated by Bacon and Anderson (1982), and used in most models of Th scavenging, varied with particle concentration and ranged from 0.2 to 1.2 Such values are appreciably longer than expected from sorption rates onto particle surfaces. The discrepancy can be explained if dissolved Th is initially sorbed to surfaces of very small particles (colloids) that pass through the typical filters (0.1-0.4 im) used to separate dissolved from particulate fractions (Santschi et al. 1986). 

The data of Loukidou et al. (2004) for the equilibrium biosorption of chromium (VI) by Aeromonas caviae particles were well described by the Langmuir and Freundlich isotherms. Sorption rates estimated from pseudo second-order kinetics were in satisfactory agreement with experimental data. The results of XAFS study on the sorption of Cd by B. subtilis were generally in accord with existing surface complexation models (Boyanov et al. 2003). Intrinsic metal sorption constants were obtained by correcting the apparent sorption constants by the Boltzmann factor. A 1 2 metal-ligand stoichiometry provides the best fit to the experimental data with log K values of 6.0 0.2 for Sr(II) and 6.2 0.2 for Ba(II). 

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). 

Van Campen et al. [31] developed models describing the rate of moisture uptake above RH0 that consider both the mass transport of water to the solid substance and the heat transfer away from the surface. For the special case of an environment consisting of pure water vapor (i.e., initial vacuum conditions), the Van Campen et al. model is greatly simplified since vapor diffusion need not be considered. Here, only the rate at which heat is transported away from the surface is assumed to be an important factor in limiting the sorption rate, W. For this special case, an expression was derived to express the rate of moisture uptake solely as a function of RHj, the relative humidity of the environment, and RH0. 

Model Predictions. The rate for desorption of americium from the fissure surfaces into solution was assumed to equal the rate for the adsorption of americium from solution by the fissure surfaces. The sorption rate and the equilibrium fractionation of americium that were determined in the static experiments were used to determine input parameters to the ARDISC model. The ARDISC model predictions for the distributions of americium on the fissure surfaces in both sets of experiments are presented in Figures 5 through 10 along with the autoradiographs and the experimental histograms representing the various distributions of americium on the fissure surfaces. 

Another important experimental consideration is the potential for nonequilibrium sorption in short experimental columns. In particular, a drawback of using spatial data is that model calibration of a sorption rate constant is more difficult. For example, Khandelwal and Rabideau (in press) showed that multiple combinations of Kj and a can be used to generate equivalent predictions for a single-time contaminant spatial distribution. As discussed below, this concern can be addressed by using multiple columns operated for different durations. 

Calibrations performed using an equilibrium model indicated increasing Kd with time, which is consistent with kinetic effects (i.e., gradual approach to equilibrium). When the kinetic model was calibrated, good model fits were observed for all three columns using a calibrated Kd of 1.4 mL/g and first-order sorption rate constant of 0.15 day 1 (Figure 2). 

In the RT3D simulation, advective/dispersive transport of each contaminant is assumed. Sorption is modeled as a linear equilibrium process and biodegradation is modeled as a first-order process. Due to the assumed degradation reaction pathways (Fig. 2) transport of the different compounds is coupled. In the study, four reaction zones were delineated, based on observed geochemistry data. Each zone (two anaerobic zones, one transition zone, and one aerobic zone) has a different value for the biodegradation first-order rate constant for each contaminant. For example, since PCE is assumed to degrade only under... 

By combining thermodynamically-based monomer partitioning relationships for saturation [170] and partial swelling [172] with mass balance equations, Noel et al. [174] proposed a model for saturation and a model for partial swelling that could predict the mole fraction of a specific monomer i in the polymer particles. They showed that the batch emulsion copolymerization behavior predicted by the models presented in this article agreed adequately with experimental results for MA-VAc and MA-Inden (Ind) systems. Karlsson et al. [176] studied the monomer swelling kinetics at 80 °C in Interval III of the seeded emulsion polymerization of isoprene with carboxylated PSt latex particles as the seeds. The authors measured the variation of the isoprene sorption rate into the seed polymer particles with the volume fraction of polymer in the latex particles, and discussed the sorption process of isoprene into the seed polymer particles in Interval III in detail from a thermodynamic point of view. 

D. Gowanlock, R. Bailey, and F. F. Cantwell, Intra-particle sorption rate and liquid chromatographic bandbroadening in porous polymer packings I. Methodology and validation of the model, /. Chromatogr. A 726 (1996), 1-23. 

The multi-component adsorption equilibrium was predicted by the LRC model and the sorption rate into an adsorbent pellet was described by the following modified LDF model. 

SO2 Sorber. In applying of the Westvaco process to boilers, Claus units, smelters, etc, a variety of waste gases will be encountered. Therefore, to estimate reactor size differential, sorption rate studies were made with simulated flue gases. Stepwise regression techniques were then applied to obtain a rate expression (Equation 5). A more detailed discussion of the rate models considered and the quality flt of the model... 

The net acceptor reaction Reap is included in the continuity equation since the CO2 mass captured is removed from the interstitial gas phase. The pseudo-homogeneous continuity equation was used for the heterogeneous models as well due to numerical convergency problems. A truely heterogeneous continuity equation should not contain the sorption reaction rate but the sum of all the species mass balance interfacial mass transfer terms. The sorption rate should then be included in the pellet continuity equation instead. 

Modeling the influence of rate-limited sorption on the transport of organic chemicals has been a topic of interest for some time. Initial attempts incorporated the one-site sorption kinetics model into the advective-dispersive transport equation (cf., Oddson et al., 1970). This approach was based on that taken by researchers in chemical engineering (i.e., Lapidus and Amund-... 

Loss of Flavor Compounds in Model Systems. Preliminary work conducted in our laboratory showed sorption of aldehydes by LDPE used in aseptic packages (Hansen, A. P. Arora, D. K., North Carolina State University, Raleigh, unpublished data). The sorption rate was related to concentration of aldehydes and to time, as both parameters increased so did the amount of flavor compounds sorbed into the film. 

The sigmoidal sorption-rate curves were analyzed using the model of Joshi and Astarita (9), which is based on a combination of a... 

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). 

The second transport model (rate-controlled sorption) is based on the assumption that the sorption mechanism does not have time to reach equilibrium at each point along the column. Mansell et al. (16) used this model to simulate phosphorus transport through sandy soils. Therefore, Equation 1 is replaced by the sorption rate expression ... 

Figure 7. Simulation of Mo(VI) experimental data from uncontaminated ground water, using the equilibrium sorption, rate-controlled sorption, mixed side-pore diffusion, and profile side-pore diffusion models.

It seems to be the exception rather than the rule that the rate of sorption in zeolites is controlled by a simple diffusion process in the solid, characterized by a constant diffusion coefficient. This result is not surprising in view of the complexity of the structures of zeolites and related to the fact that the isotherms can in most cases not be explained by an ideal model. Sorption rates depend in many cases considerably on the type of cation in the solid and the pretreatment (degree of dehydration). 

Adsorption of vapors on test chamber walls has been previously described by means of models including two or three rate constants for adsorption/desorption processes in the ease of dynamic experiments (Dunn et al., 1988 Colombo et al., 1993) and with three adsorption/desorption constants in the case of static experiments (Colombo et al., 1993). Two rate constants describe a reversible sink whereas three rate constants describe a reversible and an irreversible (i.e. leak type) sink. However, these models did not adequately describe the sorption process(es), especially in the case of long-term tests, as resulted from two observations (Colombo et al., 1993) (a) the model with three sorption rate constants (reversible + irreversible sink) provided a better description of the experimental data than the one-sink model and (b) desorption experiments following adsorption gave strong indications that the irreversible sink was in fact slowly rever-... 

Fast sorption rates were assumed, thereby closely approaching the instantaneous equilibrium case. Mass transfer is described by the linear driving force model. 

The unstirred layer adsorption model can be generalized by the introduction of surface concentration dependent sorption rate constants k and This subject is currently being studied as well as the existence of a second, irreversible, surface reaction following reversible initial adsorption for fibrinogen and prothrombin on a 60% DOPS/40% DOPC mixture. 

The apparent simplicity of this approach is, however, deceptive. For measurement of intracrystalline diffusion the method works well when diffusion is relatively slow (large crystals and/or low diffusivity), but when sorption rates are rapid the uptake rate may be controlled by extracrystalline diffusion (through the interstices of the adsorbent bed) and/or by heat transfer. The intrusion of such effects is not always obvious from the shape of the uptake curve, but it may generally be detected by changing the sample quantity and/or the sample configuration. It is in principle possible to allow for such effects in the mathematical model used to interpret the uptake curves (Fig. 2), and indeed the modeling of nonisothermal systems has been studied in considerable detail [8-12]. However, any such intrusion will obviously diminish the accuracy and confidence with which the intracrystalline diffusivities can be determined. 

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