Step—Variable Diffusivity

In the preceding analysis we considered the diffusivity as constant but if the uptake curve is measured over a large concentration step this may be a poor approximation. In many zeolitic systems the concentration dependence of the intracrystalline diffusivity is given approximately by Eq. (5.6), with Dq independent of concentration. If the adsorption equilibrium isotherm obeys the Langmuir equation this gives as the expression for the concentration dependence of the diffusivity 

FIGURE 6.4. ( i) Theoretical uptake curves for a Langmuir system with micropore diffusion control and b) variation of effective diffusivity with sorbate concentration for a Langmuir system and a Volmer system calculated from the theoretical solution of Garg and Ruthven. [Reprinted with permission from Chem. Eng, Set. 27, Garg and Ruthven (ref. 2). Copyright 1972, Pergamon Press, Ltd.] 

The shape of the uptake curve does not differ greatly from the shape of the constant diffusivity curve but the uptake rate is modified. The effective diffusivity DJ thus becomes dependent on the step size, lying within the range 

ISOTHERMAL SINGLE-COMPONENT SORPTION MACROPORE DIFFUSION CONTROL 

The appropriate form of the diffusion equation for a macropore-controlled system may be obtained from a differential mass balance on a spherical shell element  

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The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

This is the dispersion relation for a two-variable reaction-diffusion system with a step-function diffusivity for V. It is the analog of (10.26) for homogeneous reaction-diffusion systems and relates the growth rates X of spatial perturbations to the parameter values of the system. In contrast to the homogeneous case, the dispersion relation (11.47) is a complicated expression that cannot be solved analytically if D D. A diffusion-driven instability of the uniform steady state of the system occurs if the stability condition (10.23) is satisfied and (11.47) has solutions with a positive real part. 

For the alkylation of isobutane and aromatics, the phenomena in the reactors are complicated involving numerous consecutive and simultaneous reaction steps variable and unknown kinetics for the different reactions and numerous mass transfer or diffusion steps between phases or in a specific phase (2,5). In this article, comparative processes are evaluated. Methods to improve current processes are proposed. 

For a constant diffusion coefficient and boundary conditions of constant current (galvanostatic operation) or constant surface concentration (je.g., for a potential step experiment), this equation can be integrated directly [58]. For nonconstant boundary conditions but constant diffusion coefficient, the equation can be solved using DuhameTs superposition integral [59]. With an arbitrarily variable diffusion coefficient, the equation must be solved numerically. 

These equations describe the full oxidation of a conducting polymer Submitted to a potential step under electrochemically stimulated confer-mational relaxation control as a function of electrochemical and structural variables. The initial term of /(f) includes the evolution of the current consumed to relax the structure. The second term indicates an interdependence between counter-ion diffusion and conformational changes, which are responsible for the overall oxidation and swelling of the polymer under diffusion control. 

Transient computations of methane, ethane, and propane gas-jet diffusion flames in Ig and Oy have been performed using the numerical code developed by Katta [30,46], with a detailed reaction mechanism [47,48] (33 species and 112 elementary steps) for these fuels and a simple radiation heat-loss model [49], for the high fuel-flow condition. The results for methane and ethane can be obtained from earlier studies [44,45]. For propane. Figure 8.1.5 shows the calculated flame structure in Ig and Og. The variables on the right half include, velocity vectors (v), isotherms (T), total heat-release rate ( j), and the local equivalence ratio (( locai) while on the left half the total molar flux vectors of atomic hydrogen (M ), oxygen mole fraction oxygen consumption rate... 

Table IV includes theoretical transition times (C2, R14, SI7c) in laminar flow between parallel plates, following a concentration step at the wall (Leveque mass transfer). Clearly, in laminar flow (Re 100 or lower), transition times are comparable to those in laminar free convection. Here, however, the dependence on concentration (through the diffusivity) is weak. The dimensionless time variable in unsteady-state mass transfer of the Leveque type is...

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

Reaction 5.45 is at least partly hypothetical. Evidence that the Cl does react with the Na component of the alanate to form NaCl was found by means of X-ray diffraction (XRD), but the final form of the Ti catalyst is not clear [68]. Ti is probably metallic in the form of an alloy or intermetallic compound (e.g. with Al) rather than elemental. Another possibility is that the transition metal dopant (e.g. Ti) actually does not act as a classic surface catalyst on NaAlH4, but rather enters the entire Na sublattice as a variable valence species to produce vacancies and lattice distortions, thus aiding the necessary short-range diffusion of Na and Al atoms [69]. Ti, derived from the decomposition of TiCU during ball-milling, seems to also promote the decomposition of LiAlH4 and the release of H2 [70]. In order to understand the role of the catalyst, Sandrock et al. performed detailed desorption kinetics studies (forward reactions, both steps, of the reaction) as a function of temperature and catalyst level [71] (Figure 5.39). 

Although MC does not have a proper time variable, the time equivalent of MC moves can be estimated, for example, by computing rotational correlation functions or molecular diffusion coefficients in well characterized liquids these are time-dependent quantities also experimentally accessible, and the comparison between the number of MC moves and the corresponding experimental data provides the required time equivalents. An order of magnitude estimate with typical MC translational or rotational steps of 0.2 A and 4° gives 1 or 2 ps per 10 translational or rotational MC moves, respectively. 

Central differences are typically used when the flux is predominately due to a diffusive term, where the gradients on both sides of the control volume are important. An exphcit equation is one where the unknown variable can be isolated on one side of the equation. To get an explicit equation in a computational routine, we must use the flux and source/sink quantities of the previous time step to predict the concentration of the next time step. 

As the first step in introducing some basic electro-diffusion notions, let us define the following dimensionless variables... 

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