Kinetics external diffusion limitations

Fig. S.51. Eadie-Hofstee type plot showing departure from Michaelis-Menten kinetics due to external diffusion limitation...

As with the external diffusion limitation, a family of curves is obtained which shows that the overall rate of reaction decreases with increase in the Thiele modulus (as compared with the Damkdhler number for the external diffusion limitation). The rate is essentially under kinetic control at low values of 0(0< 1) that is, there is negligible diffusion limitation. In contrast to the case of external diffusion, it may be seen from the curvature of the lines in Fig. 5.53 that the rate of reaction is always a function of the kinetic parameters, even at the higher values of 0. 

In the kinetic studies of the adsorption process, the mass transport of the analyte to the binding sites is an important parameter to account for. Several theoretical descriptions of the chromatographic process are proposed to overcome this difficulty. Many complementary experiments are now needed to ascertain the kinetic measurements. Similar problems are found in the applications of the surface plasmon resonance technology (SPR) for association rate constant measurements. In both techniques the adsorption studies are carried out in a flow system, on surfaces with immobilized ligands. The role of the external diffusion limitations in the analysis of SPR assays has often been mentioned, and the technique is yet considered as giving an estimate of the adsorption rate constant. It is thus important to correlate the SPR data with results obtained from independent experiments, such as those from chromatographic measurements. 

Both studies show that at relatively low temperatures, i.e., during ignition of the catalyst, the rate-limiting step shifts from chemical kinetics to diffusion in the washcoat. This is clear from Fig. 7, computed using a one-dimensional model by Nakhjavan et al. [54]. Figure 7A shows the Thiele modulus and Fig. 7B an external diffusion limiting factor F versus dimensionless axial position in the reactor at various times on-stream for the catalytic combustion of propene in monolith reactors. The time is defined as the time after injection of the fuel in a preheated air flow. 

HDS activity of synthesized catalysts was studied in a tri-phasic slurry batch reactor (Parr 4575). The reaction mixture was prepared by adding 0.3 g of dibenzothiophene (99 mass %, from Aldrich) and 0,2 g of sieved catalyst (80-100 U.S. mesh) in 100 cm of -hexadecane (99 mass %, from Aldrich) Operating conditions, carefully chosen to avoid external diffusion limitations, were P= 5.59 + 0.03 MPa, T= 320 3°C and 1000 RPM. Samples taken periodically were analyzed by gas chromatography (Agilent 6890N, flame ionization detector and Econocap-5 capillary colunm (from Alltech). HDS kinetic constants were calculated assuming a pseudo-first order model referred to organo-S compound concentration and zero order with respect to excess H2. 

Prior to conducting the DOE (design of experiments) described in Table 3, it was established that no reaction took place in the absence of a catalyst and that the reactions were conducted in the region where chemical kinetics controlled the reaction rate. The results indicated that operating the reactor at 1000 rpm was sufficient to minimize the external mass-transfer limitations. Pore diffusion limitations were expected to be minimal as the median catalyst particle size is <25 pm. Further, experiments conducted under identical conditions to ensure repeatability and reproducibility in the two reactors yielded results that were within 5%. 

Low values of Dan corresponds to a situation where internal diffusion limitations, and hence internal concentration gradients, can be neglected, i.e. the observed rate agrees with that expected from the intrinsic reaction kinetics. Low values of Da lead to an effectiveness factor, r, close to 1. Large values of Dan correspond to strong internal diffusion limitations. In the limit the internal diffusion is potentially so slow that the reactants do not penetrate the pellet at all. In this case the reaction in limited to the external surface of the pellet. Large values of Dan lead to an effectiveness factor, T), close to 0. 

The observed rate will appear to be first-order with respect to the bulk reactant concentration, regardless of the intrinsic rate expression applicable to the surface reaction. This is a clear example of how external diffusion can mask the intrinsic kinetics of a catalytic reaction. In a catalytic reactor operating under mass transfer limitations, the conversion at the reactor outlet can be calculated by incorporating Equation (6.2.20) into the appropriate reactor model. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

We use the collocation method to solve the next example, which involves five species, two reactions with Hougen-Watson kinetics, both diffusion and external mass-transfer limitations, and nonconstant fluid temperature, pressure and volumetric flowrate. 

Diffusion, partition, and enzyme,reactions influence the sensor characteristics in a complex manner. The effect of enzyme immobilization on the reaction rate is described by the following terminology. Apparent or effective kinetics are observed when internal or external diffusion affects the overall rate. Inherent kinetics prevail when only partitioning (and not mass transfer) effects are present. Intrinsic kinetics describe the enzyme-catalyzed reaction when no partitioning effects or diffusion limitation are present. 

The coking and regeneration of a reforming catalyst was studied by physical characterization methods (pore volume, tortuosity, porosity, carbon distribution) as well as by kinetic investigations on the reaction rate of coke bum-off. For temperatures of industrial relevance for the Pt/Re-A Os catalyst, i.e. below 550°C (deactivation), the bum-off rate is determined by the interplay of chemical reaction and pore diffusion limitation by external mass transfer can be excluded. Based on the kinetic parameters, the process of the regeneration of a technical reactor is discussed. 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

Fig. 1 shows the typical kinetics of Pt adsorption on inels. The sorption attains equilibrium not less than in 4 days. The crashing of support granules and the circulation of impregnation solution do not change significantly the adsorption rate. Evidently, neither external nor inner diffusion limit the process. Pt is known... 

This mass transfer resistance and the reaction kinetics are usually compared using a dimensionless ratio, Damkohler number. Da, as in Equation 4.49. If Da is greater than unity, then the mass transfer resistance is significant and the reaction of the external diffusion is limited. On the other hand, when Da is less than unity, the reaction, in this case, is surface reaction limited. 

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