Pore diffusion breakthrough curve

Fig. 15. Theoretical breakthrough curves for a nonlinear (Langmuir) system showing the comparison between the linear driving force (—), pore diffusion (--------------------), and intracrystalline diffusion (-) models based on the Glueckauf approximation (eqs. 40—45). From Ref. 7.

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

A plot of C/C0 versus pore volume (q/V0) produces what is known as a breakthrough curve. Idealized breakthrough curves are given in Figure 10.6. These data show that the solute spreads owing to velocity distribution and molecular diffusion only. In other words, there is no interaction between the solute, solvent, and solid (Nielsen and Biggar, 1962). For this reason, the following equation is obeyed ... 

McCoy and Liapis [36] used two different kinetic models to represent the column affinity process. In both models the transport of the adsorbate in the adsorbent particle is considered to be governed by the diffusion into the pores. In model I the adsorption is assumed to be completely reversible with no interaction between the adsorbed molecules, In model 2, it is assumed that the biomolecule may change conformation after adsorption. Although these two models represent different overall adsorption mechanisms, the differences between the simulated breakthrough curves is very small. 

The rate of contaminant adsorption onto activated carixm particles is controlled by two parallel diffusion mechanisms of pore and surface diffusion, which operate in different manners and extents depending upon adsorption temperature and adsorbate concentration. The present study showed that two mechanisms are separated successfully using a stepwise linearization technique incorporated with adsorption diffusion model. Surface and pore diffiisivities were obtained based on kinetic data in two types of adsorbers and isothermal data attained from batch bottle technique. Furthermore, intraparticle diffiisivities onto activated carbon particles were estimated by traditional breakthrough curve method and final results were compared with those obtained by more rigorous stepwise linearization technique. 

In this paper, we will present a procedure to separate pore diffusion and surface diffusion in GAC adsorption by use of a stepwise linearization. Furdiermore, a simplified method to estimate concentration dependency of apparent diffiisivities from breakthrough curves will be proposed. 

Breakthrough curves from column experiments have been used to provide evidence for diffusion of As to adsorption sites as a rate-controlling mechanism. Darland and Inskeep (1997b) found that adsorption rate constants for As(V) determined under batch conditions were smaller than those necessary to model breakthrough curves for As(V) from columns packed with iron oxide coated sand the rate constants needed to model the breakthrough curves increased with pore water velocity. For example, at the slowest velocity of 1 cm/h, the batch condition rate constant was 4 times smaller than the rate constant needed to model As adsorption in the column experiment. For a velocity of 90 cm/h, the batch rate constant was 35 times smaller. These results are consistent with adsorption limited by diffusion of As(V) from the flowing phase to sites within mineral aggregates. Puls and Powell (1992) also measured more retardation and smaller rate constants for As(V) at slower flow velocities where there was sufficient time for diffusion to adsorption sites. 

Ma et al. (1996) and Whitley et al. (1993) have provided methods to decide if effects such as pore and surface diffusion or adsorption kinetics have to be considered in a model. Their approach is based on the qualitative assessment of breakthrough curves, which are the result of a step input for different feed concentrations and flow rates. When the physical parameters are known or can be estimated, the value of dimensionless parameters defined in these publications may be used to select a model. 

Pellet [35] and Rasmuson and Neretnieks [36] extended the solution of Rosen by including axial dispersion, but still assuming that the kinetics of adsorption-desorption is infinitely fast. Later, Rasmuson [37] extended the earUer solution and calculated the profile of a breakthrough curve (step bormdary condition, or frontal analysis) in the framework of the general rate model (Eqs. 6.58 to 6.64a), which includes axial dispersion, the film mass transfer resistance, the pore diffusion, and a first-order slow kinetics of adsorption-desorption. 

Van Den Broeke and Krishna [56] compared the calculated and the experimental breakthrough curves of single components and of mixtures containing methane, carbon dioxide, propane, and propene on microporous activated carbon and on carbon molecular sieves. They ignored the external mass transfer kinetics and assumed that there is local equilibrium for each component between the pore surface and the stagnant fluid phase in the macropores. They also assumed that the surface-diffusion contribution is much larger than that of pore diffusion and they neglected pore diffusion. They used in their calculations three different... 

C. 1 Breakthrough curves in ion-exchange columns. Experiments are proposed for a column packed with H+/Na" ion-exchange resin. The resin has been totally regenerated with acid. A step function of sodium-chloride solution is injected into the column at time t = 0. Sketch output curves of concentration vs time for the Cl, Na+, and H+ ions. It is proposed that only a pH meter at the outlet is needed to get information about the breakthrough curves and of pore diffusion and kinetics. Discuss this possibility and its significance. 

The breakthrough curve for the irreversible case with pore diffusion is (7)... 

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). 

Diffusion of a solute through immobile water to a reaction site also is affected by interstitial water velocity. If the diffusion rate is slow compared to the interstitial velocity, physical nonequilibrium occurs (5-7). The immobile water can be a layer on the grain surface (film diffusion), in dead-end pores between tightly packed grains (pore diffusion), or within crevices or pits on the grain surfaces (particle diffusion). Calcium and chloride breakthrough curves from column experiments done by James and Rubin (8) indicate that nonequilibrium transport occurs unless interstitial velocities are decreased so that the hydrodynamic-dispersion coefficient is of the same order of magnitude as the molecular-diffusion coefficient. 

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. 

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. 

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). 

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. 

As the concentration of Mo(VI) in the influent decreased, the shoulder and tail parts of the experimental breakthrough curves became progressively less prominent (compare Figures 6a and 6d). This trend may be explained by a slower rate of Mo(VI) diffusion into side pores caused by a decrease in the concentration gradient between flowing and immobile phases. 

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. 

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. 

If pore diffusion controls the rate of adsorption, the breakthrough curve has the opposite shape from that for external-film control. The corresponding line in Fig. 25.10 was taken from the work of Hall et al., who presented breakthrough curves for several cases of irreversible adsorption. For pore diffusion control the initial slope of the curve is high, because the solid near the front of the mass-transfer zone has almost no adsorbate, and the average diffusion distance is a very small fraction of the particle radius. The curve has a long tail because the final molecules adsorbed have to diffuse almost to the center of the particle. 

When a cake of porous particles is washed, solute in the pores must first diffuse to the surface of the particles. This is a slow process compared to displacement from the external channels in the cake. The curves of concentration vs. time are similar to the breakthrough curves for adsorption or other fixed-bed processes, and the equations in Chap. 25 can be applied to predict the effects of particle size, fluid velocity, and other variables. 

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