Diffusion limitation effects

Goldstein, B. (1989). Diffusion limited effects of receptor clustering, Comm. Theor. Biol., 1, 109-127. 

Increase of the load of enzyme of a highly active enzyme shifts the kinetically controlled enzyme electrode to diffusional limitation. Since diffusion-limiting effects are associated with the immobilization, they may be manipulated by the procedure of coupling an enzyme to the electrode. 

These results hence show that grafting leads to the preservation of the zeolitic structure under acidic conditions, probably by means of both hydrophobic and diffusion limitation effects. 

This is confirmed also by the slightly lower mobility. In the thicker film a diffusion limited effect could explain the data. In other words, the observed films structure is particularly consistent with a picture in which the thicker is the film the larger is the structural interconnection, but the lower is the analyte amounts, reaching the two dimensional transport channel. 

The lifetime of many polymer products in use is limited by oxidative degradation. Exposed samples are usually non-uniformly oxidised. At the macroscopic level, the heterogeneities can result from oxygen-diffusion-limited effects. If the rate of oxygen consumption exceeds the rate of oxygen permeation, oxidation occurs in the surface layers, whereas the core remains practically unoxidised. The importance of this effect depends on several parameters. First, intrinsic parameters are linked to material geometry (e.g., sample thickness) coupled with the oxygen consumption rate, which depends on the reactivity of... 

Effective diffusion coefficient, in porous medium at bulk diffusion limit, 14... 

Examination of oven-aged samples has demonstrated that substantial degradation is limited to the outer surface (34), ie, the oxidation process is diffusion limited. Consistent with this conclusion is the observation that oxidation rates are dependent on sample thickness (32). Impact property measurements by high speed puncture tests have shown that the critical thickness of the degraded layer at which surface fracture changes from ductile to brittle is about 0.2 mm. Removal of the degraded layer restores ductiHty (34). Effects of embrittled surface thickness on impact have been studied using ABS coated with styrene—acrylonitrile copolymer (35). 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

The effectiveness of a porous catalyst T] is defined as the actual diffusion-limited reaction rate divided by the reaction rate that could have been achieved if all the internal surface had been at bulk concentration conditions. 

For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

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. 

Pore diffusion limitation was studied on a very porous catalyst at the operating conditions of a commercial reactor. The aim of the experiments was to measure the effective diffiisivity in the porous catalyst and the mass transfer coefficient at operating conditions. Few experimental results were published before 1970, but some important mathematical analyses had already been presented. Publications of Clements and Schnelle (1963) and Turner (1967) gave some advice. 

Changing the equilibrium conditions by having condensate in the sample due to water in the carrier gas or the diffusion limitation of the condensate in larger particles changes the reaction speed. Although the kinetics of the reaction and the diffusion of the condensate are not the process Imitating steps they have an effect on the overall reaction rate as described above. 

As shown in Ch. 2, the effect of pressure on the nature of the deposit is considerable. At high pressure (i.e., ca. atmospheric), the deposition is diffusion limited and, at low pressure, surface reaction is the determining factor. In practical terms, this means that low pressure generally provides deposits with greater uniformity, better step coverage, and improved quality. 

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

Torkelson and coworkers [274,275] have developed kinetic models to describe the formation of gels in free-radical pol5nnerization. They have incorporated diffusion limitations into the kinetic coefficient for radical termination and have compared their simulations to experimental results on methyl methacrylate polymerization. A basic kinetic model with initiation, propagation, and termination steps, including the diffusion hmitations, was found to describe the gelation effect, or time for gel formation, of several samples sets of experimental data. 

Ohshima, H Kondo, T, Electrophoretic Mobility and Donnan Potential of a Large Colloidal Particle with a Surface Charge Layer, Journal of Colloid and Interface Science 116, 305, 1987. O Neil, GA Torkelson, JM, Modeling Insight into the Diffusion-Limited Cause of the Gel Effect in Free Radical Polymerization, Macromolecules 32,411, 1999. 

Minimize the effects of transport phenomena If we are interested in the intrinsic kinetic performance of the catalyst it is important to eliminate transport limitations, as these will lead to erroneous data. We will discuss later in this chapter how diffusion limitations in the pores of the catalyst influence the overall activation energy. Determining the turnover frequency for different gas flow velocities and several catalyst particle sizes is a way to establish whether transport limitations are present. A good starting point for testing catalysts is therefore ... 

We define an effectiveness factor e as the ratio of the diffusion limited rate to the rate in the absence of diffusion limitation ... 

The presence of diffusion limitations has a strong effect on the apparent activation energy one measures. We can express both the rate constant, k, and the diffusion constant, Defr, in the Arrhenius form ... 

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