Transient sorption

Considerable progress has recently been made in developing the theoretical background necessary for the application of the above method of transient kinetic analysis. An important step in this direction was the use of WKB asymptotics to derive approximate analytical expressions for short- and long-time transient sorption and permeation in membranes characterized by concentration-independent continuous S(X) and Dt(X) functions 150-154). The earlier papers dealing with this subject152 154) are referred to in a recent review 9). The more recent articles 1S0 1S1) provide the correct asymptotic expressions applicable to all kinetic regimes listed above the usefulness... 

Some preliminary kinetic analyses of transient sorption and permeation data have been reported144,149. Examples from kinetic regimes where ideal kinetics is expected to be obeyed are shown in Fig. 17. The linearity of the relevant plots is satisfactory and the values of D3, D4 and D , Dd determined from these plots deviate from De in opposite directions, as expected l50-151 even though the detailed spatial dependence of S and DT in the membrane in question is two-dimensional145) (cf. previous subsection) and is, therefore, more complicated than envisaged in the theoretical... 

Transient sorption and permeation data may also be examined in the form of suitable moments. For example, in the case of sorption one may define the following moments171 ... 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

Information about the mass transport in zeolites can be obtained only from transient sorption or desorption experiments, and not from steady state permeation, because the materials are only available as small crystals. One can then either measure the spatial distribution of the... 

In both gravimetric and piezometric experiments it is generally desirable to make the measurement over a small differential concentration change in order to ensure that the assumption of system linearity is fulfilled. Under these conditions the transient sorption curve (expressed as fractional approach to equilibrium) should be independent of either the step size or direction (adsorption or desorption). Varying the step size and direction thus provides a simple and sensitive experimental test for system linearity. 

Figure 54 illustrates the way by which Eq. 16 allows the determination of the complete concentration dependence of the diffusivity from a single transient sorption profile [98]. 

Advection and diffusion in the cap materials are subject to retardation by transient sorption as is contaminant migration in groundwater. Under steady conditions, sorption does not influence the flux of contaminants through the cap materials. Thus, the steady-state fluxes (rates per unit area) of diffusion and advection in the cap are given simply by... 

A set of limiting cases of transport is developed, comprising of situations where diffusion through the void space, fiber space and external boundary layers, each contribute significantly to transport. By means of a scaling analysis, the conditions under which each limiting case is valid are identified. Finally, a comparison of the model predictions with experimental data indicates that the model is capable of describing transient sorption dynamics quite well. 

Obtained by comparing model predictions with experimental transient sorption data (for one ramp). 

Ottino J, Shah N. Analysis of transient sorption and permeation of small molecules in multiphase polymer systems. Polym Eng Soc 2004 24 153-62. 

Apart from the above mentioned redox type reactions, we like to consider (in connection with work to be published by us elsewhere) another type of relaxations, due to the possible reorganisations of sorption intermediates on the catalyst surface, as suggested by some investigations in our laboratory. This structuring on the catalyst surface is equivalent to a change in the entropy of the system catalyst surface / adsorbed intermediates and seems to be responsible e.g. for the selectivity change under transient conditions in the oxidation of hydrocarbons. Actually this structural organization of the surface intermediates is also a rate process which can be observed under transient conditions. 

If the step is only equilibrium controlled, a lumped analysis of both fluid and solid phase is possible and the corresponding nonlinear wavefront analysis is straight forward too. Analysis of propagation speed data yield information about the relevant equilibrium sorption mechanism (7). The different form of the transients when a reaction step has been stimulated, is discussed in the Appendix. 

Figure 2. Transient responses when stimulating pure sorption step that is kineti-cally controlled (5). Key left, loading and right, unloading the storage.

Furthermore for the relevant adsorption steps CO-adsorption, H O-adsorption, CC -desorption, H -desorption, one concludes from the form of the transients that tne first three are not kinetical-ly controlled, as they are shock fronts and that Hj has no measurable sorption capacitance in presence of CO. The last conclusion has been also confirmed by independent experiments, i.e. strip-ping-off of adsorbed H by CO, and the transients in this case suggest that also the desorption, at least under the selected experimental conditions, is not kinetically controlled. Independent transient experiments showing that the CO2 sorption transients can have the form of shock wavefronts or simple wavefronts (see (7)) support the conclusion that CO2 sorption is not kinetically controlled it could be also shown that 1 0 can strip off... 

Pr q/Pro = 5,0 are relevant sorption effects of CO- but not of H2 thus only the 1 wavefronts represent rather tne shift conversion). Therefore it seems conceivable that there are two different mechanisms which participate in the CO shift conversion which is also in agreement with the established two different sorption mechanisms for 1 0 and with the transient behavior, depicted on Figure 6. 

A successor to PESTANS has recently been developed which allows the user to vary transformation rate and with depth l.e.. It can describe nonhomogeneous (layered) systems (39,111). This successor actually consists of two models - one for transient water flow and one for solute transport. Consequently, much more Input data and CPU time are required to run this two-dimensional (vertical section), numerical solution. The model assumes Langmuir or Freundllch sorption and first-order kinetics referenced to liquid and/or solid phases, and has been evaluated with data from an aldlcarb-contamlnated site In Long Island. Additional verification Is In progress. Because of Its complexity, It would be more appropriate to use this model In a hl er level, rather than a screening level, of hazard assessment. 

Transient-transport measurements are a powerful tool for evaluating the validity of any sorption-transport model. The ability of a model to predict diffusion time lags is a test for its validity, as all the parameters are fixed by the equilibrium sorption and steady state transport, and because the time lag depends on the specific form of the concentration and diffusion gradients developed during the transient-state experiments. 

The transient contaminant transport from a dissolving DNAPL pool in a water saturated, three-dimensional, homogeneous porous medium under steady-state uniform flow, assuming that the dissolved organic sorption is linear and instantaneous, is governed by the following partial differential equation ... 

This form is actually used by Koros and Paul (1978). Eq. (18.38) can be solved for the boundary conditions of transient permeation. The set of equations then obtained is given in Table 18.12. These Eqs. (18.39)-(18.42) enable us to derive SCfj and b from experimental sorption isotherms, as presented by Eq. (18.36), and the values of Dd and DH from transient permeation experiments. The equations contain a number of dimensionless groups ... 

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