Rate-controlled sorption model

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). 

Finite-difference techniques were used to compute numerical solutions as column-breakthrough curves because of the nonlinear Freundlich isotherm in each transport model. Along the column, 100 nodes were used, and 10 nodes were used in the side-pore direction for the profile model. A predictor-corrector calculation was used at each time step to account for nonlinearity. An iterative solver was used for the profile model whereas, a direct solution was used for the mixed side-pore and the rate-controlled sorption models. 

Rate-Controlled Sorption Model Accounts for the possibility that the rate of sorption reaction may be too slow for equilibrium to be achieved. 

Figure 4. Simulation of experimental data from sewage-contaminated ground water, using the rate-controlled sorption model for four concentrations of Mo(VI).

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.

The sorption of ethane from dilute mixtures with helium by 4A sieve crystal powder and pellets made without binder has been studied with a microbalance in a flow system at temperatures between 25° and 117°C. Results show clearly that intracrystalline diffusion is the rate-controlling process and that it is represented well by a Pick s law diffusion model. Transient adsorption and desorption are characterized by the same effective diffusivity with an activation energy of 5660 cal/gram mole. 

Sorption Kinetics. The adsorption and desorption data were analyzed in terms of a model based on the following main assumptions. Micropore diffusion within the sieve crystals is the rate-controlling process. Diffusion may be described by Fick s law for spherical particle geometry with a constant micropore diffusivity. The helium present in the system is inert and plays no direct role in the sorption or diffusion process. Sorption occurs under isothermal conditions. Sorption equilibrium is maintained at the crystal surface, which is subjected to a step change in gas composition. These assumptions lead to the following relation for the amount of ethane adsorbed or desorbed by a single particle as a function of time (Crank, 4). 

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). 

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 ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Attempts to model chemical weathering of catchments have used a variety of approaches and were originally designed to understand acidification processes. The BIRKENES code (Christophersen et al., 1982) was one of the first developed to model catchment stream chemistry. It used cation-anion charge balance, a gibbsite equilibrium solubility control for aluminum concentrations, a Gapon ion exchange for metals sorption, and rates for sulfate adsorption/ desorption in a two-reservoir model. The model was calibrated by input mass fluxes and output mass fluxes for the Birkenes catchment in Norway to provide the water flux information and to fit empirical parameters. 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

Transport models employing the bicontinuum-sorption formulation, with one domain equilibrium controlled, were presented by Selim et al. (1976) and Cameron and Klute (1977), while Selim et al. (1976) also presented a model where both domains were rate limited. The one-site model mentioned previously is a special case of the two-site model, where all sorption sites are assumed to be of the time-dependent class (Selim et al., 1976 van Genuchten, 1981). The bicontinuum-based model has generally been able to represent nonequilibrium data much better than has the one-site model. 

Many one-dimensional solute-transport models have been developed and used to analyze column data. For a recent review, see Grove and Stollenwerk (13). Four different models were used in the study discussed in this article to simulate the shape of the column-breakthrough curves. All four models contain a one-dimensional solute-transport equation and use the Freundlich equation to describe sorption. They differ in the rate mechanism that is assumed to control transport of Mo(VI) from flowing phase to solid surface. The essential features of each model are summarized in Table III. 

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