Effective diffusivity limiting cases

When the rate of diffusion is very slow relative to the rate of reaction, all substrate will be consumed in the thin layer near the exterior surface of the spherical particle. Derive the equation for the effectiveness of an immobilized enzyme for this diffusion limited case by employing the same assumptions as for the distributed model. The rate of substrate consumption can be expressed by the Michaelis-Menten equation. 

Also Senkan et al. (6) and Schehl et al. (3) have shown that for methanation, the material balance equation can be solved independently of the energy balance equation in diffusion-limited cases, because the effects of temperature variation on gas properties essentially cancel each other. It is therefore justified to consider the isothermal model for the purpose of yield optimization. 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

To illustrate, consider the limiting case in which the feed stream and the two liquid takeoff streams of Fig. 20-44 are each zero, thus resulting in batch operation. At steady state the rate of adsorbed cariy-up will equal the rate of downward dispersion, or afF = DAdC/ah. Here a is me surface area of a bubble,/is the frequenity of bubble formation. D is the dispersion (effective diffusion) coefficient based on the column cross-sectional area A, and C is the concentration at height h within the column. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

The effect of catalyst particle size was investigated by two different catalyst particle size fractions 63-93 pm and 150-250 pm, respectively. The effect of the particle size is very clear as demonstrated by Figure 47.2. The overall hydrogenation rate was for smaller particles 0.17 mol/min/gNi while it was 0.06 mol/min/gNi, for the larger particles, showing the presence of diffusion limitation. This kind of studies can be used to determine the effectiveness factors. The conversion levels after 70 min time-on-stream were 21% and 3%, respectively, for these two cases. 

The in vitro measurements of permeability by the cultured-cell or PAMPA model underestimate true membrane permeability, because of the UWL, which ranges in thickness from 1500 to 2500 pm. The corresponding in vivo value is 30-100 pm in the GIT and nil in the BBB (Table 7.22). The consequence of this is that highly permeable molecules are (aqueous) diffusion limited in the in vitro assays, whereas the membrane-limited permeation is operative in the in vivo case. Correcting the in vitro data for the UWL effect is important for both GIT and BBB absorption modeling. 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

The analysis of the literature data shows that zeolites modified with nobel metals are among perspective catalysts for this process. The main drawbacks related to these catalysts are rather low efficiency and selectivity. The low efficiency is connected with intracrystalline diffusion limitations in zeolitic porous system. Thus, the effectiveness factor for transformation of n-alkanes over mordenite calculated basing on Thiele model pointed that only 30% of zeolitic pore system are involved in the catalytic reaction [1], On the other hand, lower selectivity in the case of longer alkanes is due to their easier cracking in comparison to shorter alkanes. 

The studies of Thomas and Raja [28] showed a remarkable effect of pore size on enantioselectivity (Table 42.3). The immobilized catalysts were more active than the homogeneous ones, but their enantioselectivity increased dramatically on supports which had smaller-diameter pores. This effect was ascribed to more steric confinement of the catalyst-substrate complex in the narrower pores. This confinement will lead to a larger influence of the chiral directing group on the orientation of the substrate. Although pore diffusion limitation can lead to lower hydrogen concentrations in narrow pores with a possible effect on enantioselectivity (see Section 42.2), this seems not to be the case here, because the immobilized catalyst with the smallest pores is the most active one. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

In the case that the chemical reaction proceeds much faster than the diffusion of educts to the surface and into the pore system a starvation with regard to the mass transport of the educt is the result, diffusion through the surface layer and the pore system then become the rate limiting steps for the catalytic conversion. They generally lead to a different result in the activity compared to the catalytic materials measured under non-diffusion-limited conditions. Before solutions for overcoming this phenomenon are presented, two more additional terms shall be introduced the Thiele modulus and the effectiveness factor. 

The details of these short-time calculations, made for the case that U(r) = 0, are given elsewhere (28). Searching for the inverted effect in unimolecular systems (reactants linked to each other) would also be very desirable since their rates would not be diffusion limited. 

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