Kinetic model breakthrough curve

Sorbed pesticides are not available for transport, but if water having lower pesticide concentration moves through the soil layer, pesticide is desorbed from the soil surface until a new equiUbrium is reached. Thus, the kinetics of sorption and desorption relative to the water conductivity rates determine the actual rate of pesticide transport. At high rates of water flow, chances are greater that sorption and desorption reactions may not reach equihbrium (64). NonequiUbrium models may describe sorption and desorption better under these circumstances. The prediction of herbicide concentration in the soil solution is further compHcated by hysteresis in the sorption—desorption isotherms. Both sorption and dispersion contribute to the substantial retention of herbicide found behind the initial front in typical breakthrough curves and to the depth distribution of residues. 

Jardine et al. (1985b) employed a two-site nonequilibrium transport model to study Al sorption kinetics on kaolinite. They used the transport model of Selim et al. (1976b) and Cameron and Klute (1977). Based on the above model, Jardine et al. (1985a) concluded that there were at least two mechanisms for Al adsorption on Ca-kaolinite. It appeared that there were equilibrium (type-1) reactions on kaolinite that involved instantaneous Ca-Al exchange and rate-limited reaction sites (type-2) involving Al polymerization on kaolinite. The experimental breakthrough curves (BTC) conformed well to the two-site model. 

Several theoretical models were constructed to describe the chromatographic process in the frontal 116.191 and the zonal elution mode 20. The conventional method of obtaining the kinetic parameters consists in fitting the model to the experimental breakthrough curves. Another method based on the split-peak effect is a direct measurement of the apparent association rate constant (7,211. Because of the slow adsorption process, a fraction of the solute injected as a pulse into the immunochromatographic column is eluted as a nonretained peak. This behavior is observed at high flow rates, with very short or low-capacity columns 121—251. 

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. 

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. 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

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. 

Numerical Solutions of the Kinetic Model for a Breakthrough Curve.671... 

Thus, we can conclude that, as long as the mass transfer kinetics is reasonably fast, the equilibrium-dispersive model can be used as a first approximation to predict shock layer profiles. As a consequence, the results of calculations of band profiles, breakthrough curves, or displacement chromatograms made with this model can be expected to agree well vsdth experimental results. Conclusions based on the s) stematic use of such calculations have good predictive value in preparative chromatography. 

This result means that even if the kinetics follows the solid film linear driving force model the breakthrough curve can be fitted successfully to the Thomas model, provided that the apparent desorption rate corrstant, k, given by Eq. 14.82b is used. 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

The comparability of the two methods was further tested by attempting to use kinetic-parameter values determined by gas purge to predict breakthrough curves (BTC) obtained from the same soil/solute pairs. A BTC obtained from miscible displacement of PCE through the Eustis soil column is presented in Fig. 11-5. The values for ki and F (fraction of sorbent for which sorption is instantaneous) determined from the Eustis/PCE gas-purge experiment were used to calculate values for /3 (fraction of instantaneous retardation) and w (ratio of hydrodynamic residence time to reaction time). The simulation produced by a first-order bicontinuum model (Brusseau and Rao, 1989a) with these independently determined kinetic parameters is shown in Fig. 11-5. The model simulation compares extremely well with the experimental BTC. Such predictions for organic chemicals, where values for... 

If the equilibrium isotherm is linear, analytic expressions for the concentration front and the breakthrough curve may, in principle, be derived, however complex the kinetic model, but except when the boundary conditions are simple, the solutions may not be obtainable in closed form. With the widespread availability of fast digital computers the advantages of an analytic solution are less marked than they once were. Nevertheless, analytic solutions generally provide greater insight into the behavior of the system and have played a key role in the development of our understanding of the dynamics of adsorption columns. 

The performance of fixed-bed adsorbers is governed by equilibrium, kinetics of mass transfer and hydrodynamics. The objective of this part of the work is the prediction of breakthrough curves of mixtures of n/iso-paraffins and propane /propylene in a fixed-bed adsorber including parameters independently measured. The mathematical model for the fixed bed adsorption process is based on the following assumptions ... 

---