With pore diffusion

Here, as in Section 8.5.4, we treat the isothermal case for ijo, and relate tj0 to 17. may then be interpreted as the ratio of the (observed) rate of reaction with pore diffusion and external mass transfer resistance to the rate with neither of these present. 

Cooper, R.S. and Liberman, D.A. (1970) Fixed-bed adsorption kinetics with pore diffusion control. Ind. Chem. Eng. Fund., 9, 620. 

In some cases, surface diffusion - that is, the diffusion of adsorbate molecules along the interface in the pores - may contribute substantially to the mass transfer of the adsorbate, and in such cases, the effective diffusivity may become much larger than the case with pore diffusion only. 

Extension of the equilibrium model to column or field conditions requires coupling the ion-exchange equations with the transport equations for the 5 aqueous species (Eq. 1). To accomplish this coupling, we have adopted the split-operator approach (e.g., Miller and Rabideau, 1993), which provides considerable flexibility in adjusting the sorption submodel. In addition to the above conceptual model, we are pursuing more complex formulations that couple cation exchange with pore diffusion, surface diffusion, or combined pore/surface diffusion (e.g., Robinson et al., 1994 DePaoli and Perona, 1996 Ma et al., 1996). However, the currently available data are inadequate to parameterize such models, and the need for a kinetic formulation for the low-flow conditions expected for sorbing barriers has not been established. These issues will be addressed in a future publication. 

A branched pore leaching model as applied to release of water-soluble carbon from soil incorporates reaction to soluble compounds coupled with pore diffusion within the solids and leaching into the bulk solution. Application of such a model appears to describe hemicellulose hydrolysis reasonably well but not significantly better than chemical reaction only or simple leaching models. 

Thus the surface temperature is fixed at 65.6°C with pore diffusion the rate determining step. At all other surface temperatures, pore diffusion gives the largest t value for the mass transfer steps, and boimdary layer heat transfer gives the largest t value for the heat transfer steps. 

Solutions of the General Rate Model with Pore Diffusion Model. 757... 

The effectiveness factor, rj, is defined as the ratio of the reaction rate with pore diffusion resistance to the reaction rate without pore diffusion resistance (i.e., all of the active catalytic sites are restricted to the external surface of the particle). In mathematical terms... 

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

If the (r,.)ob, does not vary with (p Sj), external mass transfer is dominant and if varies with pore diffusion is limiting. If there are no mass transfer... 

Now with pore diffusion, the two independent rates are each merely multiplied by their own effectiveness fiictor to give... 

The selectivity ratio with pore diffusion limitations is foutul by solving the diffusion-reaction equation ... 

Surface reaction with pore diffusion dp, w, [ ], [5](, temperature Pore structure Type of impeller and stirring speed (mechanically agitated contactor), gas velocity (sparged contactor)... 

Surface reaction with pore diffusion dp, [,4], [ ](, temperature, replacement of active by inactive catalyst particles Superficial liquid velocity (above certain minimum) Superficial gas velocity ... 

In the design of a fixed bed reactor, it is necessary to know the rate of reaction encompassing mass and diffusion effects. These effects on the reaction rate can be represented by the effectiveness factor r), with pore diffusion, besides the effects of mass. This will be represented by an overall rate r" (mol mass h ) or r (mol/area h ). 

We first discuss the various simple models, and start with linear models, favoured for the possibility of analytical solution which allows us to study the system behaviour in a more explicit way. Next we will discuss nonlinear models, and under special conditions such as the case of rectangular isotherm with pore diffusion analytical solution is also possible. Nonisothermal conditions are also dealt with by simply adding an energy balance equation to mass balance equations. We then discuss adsorption behaviour of multicomponent systems. 

What has been presented so far is the analysis of the irreversible case with pore diffusion mechanism only. The concept of travelling concentration front is valid as long as there is no surface diffusion because if surface diffusion is present, which is unlikely for very strong adsorption, it will distort the sharp concentration wave front and hence the concept of two distinct regions will no longer be applicable. 

Fig 5 7b Diagrams of C/Co versus r for irreversible adsorption with pore diffusion kinetics... 

In recent years considerable advances have been made in our understanding of gas-solid reaction systems. These advances are due in part to the development of more sophisticated mathematical models in which account is taken of such structural effects as pore size, grain size, and pore diffusion. Another important contributory factor has been the use of more sophisticated experimental techniques such as electron microscopy. X-ray diffraction, and porosimetry, which together with pore diffusion measurements provide information on the key structural parameters and make possible the critical assessment of this new generation of models. 

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