Profile side-pore diffusion model

The mixed side-pore diffusion model gave a reasonable correlation with experimental data, and the parameters could be used for a variety of input concentrations. However, the profile side-pore diffusion model gave the most accurate simulations for the largest variety of input concentrations. 

The profile side-pore diffusion model is the most complex and, perhaps, the most realistic of the four models. Without the Freundlich sorption mechanism, the model is the same as that developed by Kipp ( ). The case of radial diffusion with linear sorption was considered by van (ienuchten et al. ( ) whereas, spherical diffusion that had linear sorption was described by... 

The four potential rate mechanisms were evaluated by calculating column-breakthrough curves for various parameter sets to obtain the most accurate correlation between observed column-breakthrough curves and calculated concentration data. The parameters pbf and pbs for the mixed side-pore and profile side-pore diffusion models were estimated from the 0.043 mmol/1 breakthrough curves. Simulations at other concentrations were made by changing only the solution concentration value in the Freundlich equation. Physical and chemical parameters common to all four models are listed in Table II. Results are for 0.096-, 0.043-, 0.01- and 0.0016-mmol/l columns. 

The breakthrough curve for nonreactive Br for one of the columns is shown in Figure 2 it is similar in shape to the Br-breakthrough curves for all columns. An immobile-water phase is indicated by the asymmetrical shape of the curve, and the profile side-pore diffusion model gave the best match to the experimental data. 

Figure 2. Simulation of Br experimental data, using the profile side-pore diffusion model.

Profile Side-Pore Diffusion Model Diffusion into immobile water in side pores is the rate-limiting step. A concentration gradient exists in each side pore. 

The profile side-pore diffusion model simulated the experimental data from the 0.043 mmol/1 column almost exactly and was within the accuracy of the breakthrough data (Figure 6b). Based on the best fit simulation of the Br-breakthrough curve (Figure 2), the immobile-water phase was calculated to be about 5 percent of the total porosity. Apparently, diffusion into and out of this volume of immobile water was responsible for the observed shoulder and tail of the curves of the experimental data. 

Experimentoi doto — Profile Side-pore diffusion model... 

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 most accurate results were achieved by using the more complex and conceptually realistic profile side-pore diffusion model. This model had a fit through all of the experimental data points from one column, could be used for a range of concentrations, and could simulate breakthrough curves for ground water of different compositions. 

Transferability of the results from this study to the Cape Cod natural gradient tracer test will provide information about the validity of laboratory experiments in providing information about onsite processes. Although actual values for some of the physical parameters determined in the laboratory may not apply to an aquifer because of scale differences between laboratory and field, conceptually realistic models such as the profile side-pore diffusion model may be able to simulate onsite transport conditions more accurately. 

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 fourth model (profile side-pore diffusion) is similar to the third but the assumption is made that a concentration profile exists throughout the thickness of the immobile-water phase. Molecular diffusion of solute is the major transport mechanism in the immobile-water phase. The transfer rate of solute from the flowing- to the immobile-water phase is assumed to be the dif-fusional flux at the interface between these phases. Therefore, Equations 7 and 9 are replaced by ... 

The concentration profile in the immobile-water phase is controlled by a diffusional-transport mechanism. The transfer rate from the immobile-water phase to the flowing-water phase is the diffusive flux, which depends on the concentration gradient in the immobile-water phase at the interface. Parameters V, 6, A, and Lg in the profile side-pore model are estimated from the shape of the breakthrough curve for a nonreactive tracer. Parameters Pbf and pbs are estimated from the shape of the breakthrough curve for a reactive solute. The effective molecular diffusivity Dm is estimated from values published in the literature. 

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