Mixed 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. 

Mixed Side-Pore Diffusion Model Diffusion into immobile water in pores is the rate-limiting step. An average concentration is assumed throughout each side pore. 

The mixed side-pore diffusion model also reasonably simulated the experimental data (Figures 5a and 5b). This model was slightly more accurate than the reaction-rate model in simulating breakthrough curves for a range of input concentrations (Figure 5a-5c) however, significant discrepancies also were observed between experimental data and model simulations at concentrations of less than 0.01 mmol/1 Mo(VI). 

Figure 5. Simulation of experimental data from sewage-contaminated ground water, using the mixed side-pore diffusion model for four concentrations of Mo(VI).

The concept of Mo(VI) diffusion into and out of side pores that had an immobile-water phase resulted in a more accurate simulation of experimental breakthrough curves for a wider range of concentrations. The mixed side-pore diffusion model could be used to fit a particular experimental breakthrough curve with about the same degree of accuracy as the reaction rate model however, the mixed side-pore diffusion model was applicable for a wider range of concentrations. 

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 third model (mixed side-pore diffusion) (18) is based on the assumption of an effective average or mixed concentration of solute in the immobile-water phase. This model has no spatial dependence in the immobile-water phase. The balance equation for the immobile-water phase... 

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. 

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 transfer rate in the mixed side-pore model is proportional to the difference in concentration between the flowing-water and immobile-water phases. The transfer-rate constant kgA is a characteristic-rate parameter for diffusion in the immobile-water phase. Without the Freundlich sorption mechanism, this third model is the same as the dead-end pore model developed by Coats and Smith (19). The Freundlich sorption isotherm was included by van Genuchten and Wierenga (18) in their study, but they solved for the linear case only. Grove and Stollenwerk (20) described a similar model but included Langmuir sorption and a continuous immobile-water film phase. 

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