External diffusion limitations

Some simulation results for trilobic particles (citral hydrogenation) are provided by Fig. 2. As the figure reveals, the process is heavily diffusion-limited, not only by hydrogen diffusion but also that of the organic educts and products. The effectiviness factor is typically within the range 0.03-1. In case of lower stirrer rates, the role of external diffusion limitation becomes more profound. Furthermore, the quasi-stationary concentration fronts move inside the catalyst pellet, as the catalyst deactivation proceeds. 

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. 

The absence of external diffusion limitations (diffusion from the bulk to the surface of the support crystallite) was established by varying the space velocity with a constant catalyst volume. A twofold increase in the space velocity did not affect the TOF. This shows that external diffusion limitations are absent36. 

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. 

Because the conversion rate is independent of the external flow rate, external diffusion limitations can be neglected. Both from the proportionality of the conversion rate with the feed concentration and the independence of CjJC from CAj it follows that the reaction is first order, so that = 1/cosh, RA = (tanh0)/0 and DtA = L2, / 2. 

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. 

Figure 7 (A) Thiele modulus along the reactor. (B) External diffusion limiting factor F along the...

This figure is a schematic to show the various slopes of the resistance s and does not imply that internal diffusion limitations will always occur at particle sizes smaller than those shown for external diffusion limitations. 

The external diffusion limitation (mass transfer through a liquid-solid interface) is determined by the diffusion rate of the reactant to the external surface or the product out from catalyst particles surface. The flux to the external surface is defined by ... 

External diffusion limitation by mass transfer through layers in front of the enzyme membrane, eg, a semipermeable membrane or the boundary layer at the solution/biosensor membrane interface. 

Catalytic runs were carried out at atmospheric pressure in a quartz-made fixed-bed flow microreactor (10 mm i d.). All the stainless-steel equipment devices had been passivated by hot HNO3 treatment before the assembly. The catalyst was activated in situ (6 h at 773 K under COj-free air flow). 4-Methylpentan-2-ol was fed in with an N2 stream (partial pressure, Po.aicohoi = 19.3 kPa time factor, W/F = 0.54 gcat-h/gakohoi)- On-line capillary GC analysis conditions were Petrocol DH 50.2 column, oven temperature between 313 and 473 K, heating rate 5 K/min. Products identification was confirmed by GC-MS. For each catalyst a run in which several reactor temperatures were checked was carried out, in order to study the influence of the thermal history of the sample on its catalytic behaviour and to reach an appropriate conversion level (ca. 50%) at which the selectivities values of the different catalysts can be compared. Then, a new run with a fresh portion of the same sample was started at the desired temperature and carried out isothermally for 80 h. Further runs, where both the flow rate and the catalyst amount were considerably changed, while keeping the same W/F value, were also carried out no significant differences in conversion were observed, which rules out the occurrence of external diffusion limitations. 

The external diffusion limitation (EDL) indicates that substrate transport through diffusion (stagnant) layer is a rate-limiting process [10]. At internal diffusion limitation (IDL), the substrate diffusion through the external diffusion layer is fast, and process is limited by the diffusion inside an enzyme membrane. The disadvantage of these approximate solutions is an error at the boundaries between the different approximate treatments. It is helpful to illustrate this approach by reference to a trivial problem of the substrate conversion in the biocatalytical membrane of the biosensor and at the concentration of the substrate less than the Michaelis-Menten constant Km)- The calculated profile of substrate concentration at steady-state or stationary conditions is shown in Fig. 1. 

For Da < 0.1, external mass transfer can be neglected. Before this criterion can be used, the reaction rate must be known. To circumvent this issue, Berger and Kapteijn [48] proposed an easier to use criterion based on the conversion to check the absence of external diffusion limitations ... 

As long as the conversion is below the value of the right-hand side term, the external diffusion limitations can be neglected. On the other hand, the external mass transfer can be taken into account by a mass transfer coefficient for which Beretta et al. [45] derived the following correlation based on a functional form for the local Sherwood number to interpolate the exact solution ... 

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. 

However, the behavior of the catalysts measured in this work is different. At temperatures above 400 K the catalytic activity becomes limited, in agreem t with the Thiele theory. However, the apparent activation energy gradually decreases from 94 to 6 kJ/mol, rather than to 50 kJ/mol, which implies that the apparent activation energy of diffusion is exhibited. Nevertheless, the size of the wider pores in the pellet does appear to affect strongly the activity. Therefore, it is impossible that merely external diffusion limitation, that is, diffusion from the bulk of the gas flow to tiie external surface of the catalyst body, is rate-determining. Since the catalyst spheres had the same diameter, the activity of all catalysts should be equal if external transport is determining the activity. As the concentration of reactants inside the particle is nearly zero, the pore size should be of no importance. However, this is in contradiction with the measurements. 

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