Diffusional limitations

Figure 6.3.2 shows the feed-forward design, in which acrolein and water were included, since previous studies had indicated some inhibition of the catalytic rates by these two substances. Inert gas pressure was kept as a variable to check for pore diffusion limitations. Since no large diffusional limitation was shown, the inert gas pressure was dropped as an independent variable in the second study of feed-back design, and replaced by total pressure. For smaller difftisional effects later tests were recommended, due to the extreme urgency of this project. 

When the products are partially or totally miscible in the ionic phase, separation is much more complicated (Table 5.3-2, cases c-e). One advantageous option can be to perform the reaction in one single phase, thus avoiding diffusional limitation, and to separate the products in a further step by extraction. Such technology has already been demonstrated for aqueous biphasic systems. This is the case for the palladium-catalyzed telomerization of butadiene with water, developed by Kuraray, which uses a sulfolane/water mixture as the solvent [17]. The products are soluble in water, which is also the nucleophile. The high-boiling by-products are extracted with a solvent (such as hexane) that is immiscible in the polar phase. This method... 

The effect of temperature on the rate of ethanol production is markedly different for free and immobilised systems. Thus while a constant increase in rate is observed with free S. cerevisiae as temperature is increased from 25 to 42 °C, a maximum occurs at 30 °C with cells immobilised in sodium alginate. The lower temperature optimum for immobilised systems may result from diffusional limitations of ethanol within the support matrix. At higher temperatures, ethanol production exceeds its rate of diffusion so that accumulation occurs within the beads. The achievement of inhibitory levels then causes the declines observed in the ethanol production rate. 

However, in most cases enzymes show lower activity in organic media than in water. This behavior has been ascribed to different causes such as diffusional limitations, high saturating substrate concentrations, restricted protein flexibility, low stabilization of the enzyme-substrate intermediate, partial enzyme denaturation by lyophilization that becomes irreversible in anhydrous organic media, and, last but not least, nonoptimal hydration of the biocatalyst [12d]. Numerous methods have been developed to activate enzymes for optimal use in organic media [13]. 

In ICC 1 there were only a few references to diffusional limitations, but they may have been present in a number of papers. Despite improved attention, problems may still exist particularly in systems involving transport from the gas to the liquid phase. Absent a demonstration that the rate of a hydrogenation was proportional to the amount of catalyst one may suspect that C(H2)(liq.) was not in equilibrium with P(H2)(gas). 

In order for diffusional limitations to be negligible, the effectiveness factor must be close to 1, i.e. nearly complete catalyst utilization, which requires that the Thiele modulus is suffieiently small (< ca. 0.5), see Figure 3.32. Therefore, the surface-over-volume ratio must be as large as possible (particle size as small as possible) from a diffusion (and heat-transfer) point of view. There are many different catalyst shapes that have different SA/V ratios for a given size. 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

At low temperatures diffusion will be rapid compared to chemical reaction and diffusional limitations on the reaction rate will not be observed. In this temperature regime, one will observe the intrinsic activation energy of the reaction. However, since chemical reaction rates increase much more rapidly with increasing temperature than do diffusional processes, at... 

This equation gives the differential yield of V for a porous catalyst at a point in a reactor. For equal combined diffusivities and the case where hT approaches zero (no diffusional limitations on the reaction rate), this equation reduces to equation 9.3.8, since the ratio of the hyperbolic tangent terms becomes y/k2 A/ki v As hT increases from about 0.3 to about 2.0, the selectivity of the catalyst falls off continuously. The selectivity remains essentially constant when both hyperbolic tangent terms approach unity. This situation corresponds td low effectiveness factors and, in tliis case, equation 12.3.149 becomes... 

Generally, under either isothermal or noniso-thermal conditions, intrapartiole diffusional limitations are undesirable because they reduce the selectivity below that which can be achieved in their absence. The exception to this generalization is a set of endothermic reactions that take place in nonisothermal pellets where the second reaction has an activation energy that is greater than that of the first. 

The intrinsic rate expressions for these reactions are both first-order in hydrogen and zero-order in acetylene or ethylene. If there are diffusional limitations on the acetylene hydrogenation reaction, the acetylene concentration will go to zero at some point within the core of the catalyst pellet. Beyond this point within the central core of the catalyst, the undesired hydrogenation of ethylene takes place to the exclusion of the acetylene hydrogenation reaction. 

The activity of the Pt-exchanged catalyst for n-C f, transformation increases when the crystallites size increases, which was totally unexpected. External diffusional limitations cannot be invoked since the size of the grains of catalyst is the same. Moreover, this would lead to the opposite result. Other experiments showed that the activity of zeolite-... 

In the model, the kinetic constants for propagation and termination are allowed to vary as a function of free volume, as suggested by Marten and Hamielec (16) and Anseth and Bowman (17). To account for diffusional limitations and still predict the non-diffusion controlled kinetics, the functional forms for the propagation and carbon-carbon termination kinetic constants are ... 

Figure 5 shows similar experimental rate data for the DEGDMA/DMPA/TED polymerization. As seen in the case of HEMA, TED addition decreases both the initial rate and the maximum rate of polymerization of DEGDMA. As described earlier, polymerization of DEGDMA results in a highly crosslinked polymer. The autoacceleration effect is characterisitc of highly crosslinked systems as the diffusional limitations reduce the carbon-carbon radical termination kinetic constant... 

In a detailed rotating-disk electrode study of the characteristic currents were found to be under mixed control, showing kinetic as well as diffusional limitations [Ha3]. While for low HF concentrations (<1 M) kinetic limitations dominate, the regime of high HF concentrations (> 1 M) the currents become mainly diffusion controlled. However, none of the relevant currents (J1 to J4) obeys the Levich equation for any values of cF and pH studied [Etl, Ha3]. According to the Levich equation the electrochemical current at a rotating disk electrode is proportional to the square root of the rotation speed [Le6], Only for HF concentrations below 1 mol 1 1 and a fixed anodic potential of 2.2 V versus SCE the traditional Levich behavior has been reported [Cal 3]. 

For homogeneously doped silicon samples free of metals the identification of cathodic and anodic sites is difficult. In the frame of the quantum size formation model for micro PS, as discussed in Section 7.1, it can be speculated that hole injection by an oxidizing species, according to Eq. (2.2), predominantly occurs into the bulk silicon, because a quantum-confined feature shows an increased VB energy. As a result, hole injection is expected to occur predominantly at the bulk-porous interface and into the bulk Si. The divalent dissolution reaction according to Eq. (4.4) then consumes these holes under formation of micro PS. In this model the limited thickness of stain films can be explained by a reduced rate of hole injection caused by a diffusional limitation for the oxidizing species with increasing film thickness. 

Although examples of the methodology will utilize entirely reaction rates or reactant concentrations, the procedures are equally valid for other model responses. They have been used, for example, with responses associated with catalyst deactivation and diffusional limitations as well as with copolymer reactivity ratios and average polymer molecular weights. 

Hence, combining Eqs. (7)-(10) gives the diffusional limited rate of burning ... 

---