Internal diffusion limitations

It is therefore important to examine under what conditions the above criterion is met (i.e. fast ion backspillover relative to its desorption or consumption) for otherwise the promotional process will be internally diffusion limited not due to slow diffusion of the reactants but due to slow diffusion (backspillover) of the promoting species. 

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]. 

Modeling of Internal Diffusion Limitations in a Fischer-Tropsch Catalyst... 

Zhan, X., Davis, B. H. 2002. Assessment of internal diffusion limitation on Fischer-Tropsch product distribution. Applied Catalysis A General 236 149-61. 

The book focuses on three main themes catalyst preparation and activation, reaction mechanism, and process-related topics. A panel of expert contributors discusses synthesis of catalysts, carbon nanomaterials, nitric oxide calcinations, the influence of carbon, catalytic performance issues, chelating agents, and Cu and alkali promoters. They also explore Co/silica catalysts, thermodynamic control, the Two Alpha model, co-feeding experiments, internal diffusion limitations. Fe-LTFT selectivity, and the effect of co-fed water. Lastly, the book examines cross-flow filtration, kinetic studies, reduction of CO emissions, syncrude, and low-temperature water-gas shift. 

For lipase, initial activity corresponds to the amount of protein that was adsorbed. Specific activity is constant at 1 mmoFs gE for this carrier-enzyme system, which compares to 27% of the free enzyme activity. The trypsin system shows a lower specific activity that is only 10% of the free enzyme. The reason for the lower recovered activity of this system is not known. To rule out possible internal diffusion limitations, the Wheeler-Weisz modulus was estimated, assuming a carrier layer thickness of 0.1 mm for all carriers. Using the data of the experiments performed at 150 rpm, one finds ... 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

In the first stage of the investigation the catalyst can be considered in the form of powder in order to derive intrinsic transient kinetics of all the relevant reactive processes. To this purpose, dynamic reactive experiments can be performed in a simple tubular fixed-bed microreactor over small quantities (50-200 mg) of finely powdered catalyst in principle, this guarantees negligible transport limitations and more controlled conditions (e.g. isothermal catalyst bed), hence enabling a direct estimation of intrinsic rate parameters by kinetic fit. Internal diffusion limitations are particularly relevant to the case of bulk (extruded) monolith catalysts, such as vanadium-based systems for NH3/urea SCR however, they... 

Fig. S.S4. Eadie-Hofstee type plot showing departure from Michaelis-Menten kinetics due to internal diffusion limitation...

The structural differences between the various sulfur-containing molecules make it impractical to have a single rate expression applicable to all reactions in hydrodesulfurization. Each sulfur-containing molecule has its own hydrogenolysis kinetics that is usually complex because several successive equilibrium stages are involved and these are often controlled by internal diffusion limitations during refining. 

Catalysts pre-treatment (calcination and reduction) was performed in the same testing system or in a parallel automatic activation system prior to reaction test Calcination is carried out at 600 °C under airflow for 8 h and reduction at 250 °C for 2 h under hydrogen flow. Catalytic tests were carried out at 30 bar total pressure, temperature range 200-240°C, and 2.26h-1 WHSV, H2/hydrocarbons molar ratio of 2.93. Each fixed bed microreactor contained 500 mg of catalyst (particle size 0.4—0.6 mm, for which there are no internal diffusion limitations). Reaction products distribution are analysed using a gas chromatograph (Varian 3380GC) equipped with a Plot Alumina capillary column. 

Establishing the presence of internal diffusion limitations is less trivial. An easy check would be to decrease the crystallite size. If internal diffusion limitations are absent, this would not affect the activity. However, the internal diffusion is related to the pore-size. The catalyst particles of the catalyst with the smallest pores, the LTL zeolite, are constructed of several... 

In Table 7 the effectiveness and corresponding Thiele modulus for the different support materials is given. The particle size for the ASA, SiC>2 and HT supports was taken equal to the sieve fraction. This is a worst-case scenario, since it is far more likely that the particles in the sieve fraction are constructed of several crystallites which contain the relevant pores and Pt particles. Between those crystallites, the pore radii will be very large compared to the pore radius in the support material. Even in this worst case scenario, the effectiveness is still high, close to unity, for all catalysts. This demonstrates that the observed reaction kinetics reflect the intrinsic catalyst properties, since internal diffusion limitations are absent. 

For a comparison of catalyst activities and in kinetic studies, one needs data that are not disguised by concentration gradients. For a 5% tolerance level the criterion for the effectiveness factor for the absence of internal diffusion limitations reads... 

If internal diffusion limitations dominate, tj — /f and, for an nth order reaction,... 

When the diffusion of a reactant inside the pellet is not fast enough to compensate for its disappearance by reaction a decreasing concentration profile is established in the pellet. For positive partial reaction orders with respect to the reactant this leads to lower reaction rates at positions away from the external surface and hence to a lower reaction rate when averaged over the complete pellet volume. A measure for the degree of internal diffusion limitations is given by the internal effectiveness factor, t, defined as ... 

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 flux that would be established in the absence of internal diffusion limitations is given by ... 

At the centre plane and for strong internal diffusion limitations the concentration of A either vanishes or equals the equilibrium concentration, leading to ... 

Note that criterion 7.186 requires a knowledge of the apparent activation energy for the reaction. Criterion 7.186 holds whether internal diffusion limitations exist or not. When the criterion concerning external heat transfer is compared to criterion 7.167 concerning radial heat transport limitations through the bed, it can been seen that the latter are more critical unless ... 

Internal diffusion limitations can generally be easily avoided in laboratory slurry reactors as the minimum of the pellet diameter is determined by the maximum of the pore size of the membrane filter used to maintain the catalyst in the reactor. Membrane filters allowing the use of pellets with a diameter of 1 pm are commercially available. 

In this case the experimental determination of the rate and the knowledge of the reaction order is sufficient to assess the importance of internal diffusion limitations. Note that only n > -1 is meaningful. 

An experimental test to verify the absence of significant concentration gradients inside the catalyst pellet is based on the inverse proportional relation between the effectiveness factor and the pellet diameter for strong internal diffusion limitations. Hence, a measured rate which is independent of the pellet size indicates that internal diffusion limitations can be neglected. Care should be taken to avoid artifacts. External heat transfer effects also depend on pellet size and for exothermic reactions might compensate the internal diffusion limitations. If the catalyst pellet consists of a support with an non-uniformly distributed active phase, crushing and sieving to obtain smaller pellets is hazardous. 

By varying the speed of the stirrer from 200 to 500 rpm no effct on the reaction velocities is observed in SC CO2 or in n-hexane. However, without stirring, the reaction velocities decrease 6 % and 13 % respectively. All further SC C02 studies are performed at 300 rpm. So, no external diffusion limitation can be assumed however, internal diffusion limitations may happen, as discussed in the following section. 

Monoliths that were anodized extensively (72) had an anodization thickness of up to 25 pm with a BET surface area of 40 m /g, which is sufficient for many applications. However, because this layer contained only mesopores (pore diameters up to 20 nm) and no macropores, internal diffusion limitations can easily be a problem. An extensive report on the anodization of aluminum monoliths, with the aim of using the anodization layer as catalyst support, was provided by Burgos et al. (73). 

The dynamic adsorption capacity of activated carbon containing monoliths has been shown to be equivalent to the micropore volume. However, this condition can only be met when the external area is above c. 100 m g" and the threshold diameter wide. In systems with no micropore volume or poor internal diffusion due to a low external surface area and narrow threshold diameter the breakthrough point is reached when c. 9% of the external area is covered. Future work will concentrate on using higher linear velocities and adsorption temperatures md different monolith geometries (wall thickness and channel width) in order to study the internal diffusion limitations of these types of adsorption units. 

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