Diffusion shrinking core model

Model II Pore Diffusion in the Particles is r.d.s. (Knudsen Diffusion, Shrinking Core Model] If pore diffusion (Knudsen diffusion at HV) in the silica particles limits the mass loss and we assume that an IL-free shell is formed (shrinking core model), the mass loss is given by... 

The shrinking core models described by Levenspiel cater for both reaction- and diffusion-controlled systems. Referring to the literature, how do these systems differ and which of these models do skeletal catalysts fit during their preparation by leaching ... 

Limitations of the Shrinking Core Model. The assumptions of this model may not match reality precisely. For example, reaction may occur along a diffuse front rather than along a sharp interface between ash and fresh solid, thus giving behavior intermediate between the shrinking core and the continuous reaction models. This problem is considered by Wen (1968), and Ishida and Wen (1971). 

Assuming that reaction proceeds by the shrinking-core model calculate the time needed for complete conversion of a particle and the relative resistance of ash layer diffusion during this operation. 

The shrinking-core model (SCM) is used in some cases to describe the kinetics of solid and semi-solids-extraction with a supercritical fluid [22,49,53] despite the facts that the seed geometry may be quite irregular, and that internal walls may strongly affect the diffusion. As will be seen with the SCM, the extraction depends on a few parameters. For plug-flow, the transport parameters are the solid-to-fluid mass-transfer coefficient and the intra-particle diffusivity. A third parameter appears when disperse-plug-flow is considered [39,53],... 

The nucleation model represents the other extreme. Here, the dissociation of hydrogen is the slow step. Once a nucleus of reduced metal exists, it acts as a catalyst for further reduction, as it provides a site where H2 is dissociated. Atomic hydrogen diffuses to adjacent sites on the surface or into the lattice and reduces the oxide. As a result, the nuclei grow in three dimensions until the whole surface is reduced, after which further reduction takes place, as in the shrinking core model. The extent of reduction (see Fig. 2.3a) shows an induction period, but then increases rapidly and slows down again when the reduction enters the shrinking core regime. 

In contrast, work published in journals of pure chemistry is usually concerned with reaction at the boundaries and commonly only initial rates are measured so that complications due to diffusion can be neglected. The results and conclusions in such studies are naturally different from case to case and will be enumerated later. First, however the shrinking core model will be reviewed. 

Kinetic effects of a different nature occur if the strong catalyst poisons diffuse so slowly that at first only the outer parts of the catalyst particles are poisoned (shrinking-core model, as discussed for dewaxing (10)) or that these poisons are converted in the outer layers of the catalyst, leaving a clean active core (11). 

To illustrate the principles of the shrinking core model, we shall consider the removal of carbon from the catalyst particle just discussed. In Figure 11-15 a core of unreacted carbon is contained between r = 0 and r = R. Carbon has been removed from the porous matrix between r = R arid r = R. Oxygen diffuses from the outer radius Ro to the radius R, where it reacts with carbon to form carbon dioxide, which then diffuses out of the porous matrix. The reaction... 

Where conditions eliminate diffusion resistance to reaction (12.1) [25], the rate is proportional to the surface area of the undecomposed CaCOj, consistent with the interface advance or shrinking core model. Calcite breakdown with greater than 90% efficiency is achieved in 2.5 s at 1270 K and the surface area of the product CaO is a maximum (about 100 m g ) after reaction at 1270 K. 

Determine whether the shrinking-core model will fit these data and evaluate the rate constant (the frequency factor in the rate equation) and an effective diffusivity Dg. The particles were granular and varied in size from 0.01 to 0.1 mm, but assume that a spherical particle with an average radius of 0.035 mm can represent the mixture. 

Closure. After completing this chapter, the reader should be able to define and describe molecular diffusion and how it varies with temperature and pressure, the molar flux, bulk flow, the mass transfer coefficient, the Sherwood and Schmidt numbers, and the correlations for the mass transfer coefficient. The reader should be able to choose the appropriate correlation and calculate the mass transfer coefficient, the molar flux, and the rate of reaction. The reader should be able to describe the regimes and conditions under which mass transfer-limited reactions occur and when reaction rate limited reactions occur and to make calculations of the rates of reaction and mass transfer for each case. One of die most imponant areas for the reader apply the knowledge of this (and other chapters) is in their ability to ask and answer "What if. , questions. Finally, the reader should be able to describe the shrinking core model and apply it to catalyst regeneration and pharmacokinetics. 

A very important particular case of gas-solid reactions is that in which a component of the solid phase reacts with the reactant gas phase very rapidly and the particle size remains constant. This establishes a reaction interface between the two zones in one zone there is diffusion (but no reaction) to the interface, and in the other there is only reaction at the interface. This is usually called the shrinking core model. If again we make the pseudo-steady-state assumption, then from equation (7-84)... 

The third limiting case of diemical reaction rate controlling is not consistent with the concept of a shrinking core model with a single diifu vity throughout the particle the existence of a sharp boundary implies transport by effective diffusion that is potentially slow with respect to the reaction. 

Comparing the last expression with Eq. 4.3-7, obtained for the shrinking-core model with diffusion rate controlling, shows that Pf — is nothing but the ratio of the time required to reach a given conversion to the time required to reach complete conversion. Sohn and Szekely showed that Eq. 4.4-8 leads to a remarkably accurate approximation to the results obtained by numerical integration. 

The main postulate of this delayed diffusion model is that phenyl anion II (designated as S), formed as the solid intermediate, reacts with the generated gas R after a certain minimum pressure is reached to give the final product Pj. Based on this hypothesis, plots have been prepared for the rate of outward advance (as against inward advance in the normal shrinking core model) of the second solid phase as a function of position within... 

CCC Shrinking core model Sulfur balance Plug-flow Partially backmixed Absent Diffusion controlled (29) ... 

Ruether (29) examined the case of oxydesulfurization for completely backmixed stirred tanks in series assuming the diffusion-controlled mechanism. The reaction in the particles was described by the shrinking core model. The results obtained on the conversion as a function of residence time were shown for various number of reactors in series. The procedure to calculate the conversion for a system having a distributed particle size... 

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