Volume-based effectiveness factor

The use of a surface-based effectiveness factor instead of the volume-base factors previously described = rj y) was suggested by Fujie et al. (1979) and defined as... 

IIt = First-order rate constant based on volume of catalyst m3/(m3 catalyst) s i) = Effectiveness factor... 

It is not easy to compare the activity of the V-W-Ti catalysts here tested with the lot of chromia, Pt and Pd based catalysts previously used because they have different shapes (monoliths and spheres) and because very different particle sizes arc involved (having thus very different effectiveness factors). For conqiarison purposes, all X-T curves were adjusted to a simple fust order kinetic model (with rate based on overall volume of catalyst, both for monoliths and for fixed beds). From the kinetic constants so obtained (see details of the method in ref 7), the preexponential factors (ko) of the Arrhenius law and the apparent energies of activation (E, p) were calculated for all catalysts. One example is shown in Figure 17. By the well Imown compensation effect between ko and E,pp, the kg values so obtained were recalculated for a given E.pp value of 44 kJ/mol. Such new ko value was used [7] as an activity index of the catalyst. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

Samples were usually transferred to the microfluidic system by electrokinetic injection and pressure injection. Electrokinetic sample introductimi is most commonly used for transferring samples to chips. This can be attributed to factors such as the simplicity in achieving elec-troosmotic flow (EOF), i.e., no moving parts and minimal back-pressure effects. The EOF in the microchannel acts as a pump and can easily be controlled by outside high voltages. The two commonly employed injection modes for microfluidic chips are time-based and discrete volume-based injection. In the case of the time-based (or gated) injection, the amount of sample introduced into the carrier stream can be controlled by adjusting the injectirMi time. 

In the literature are many articles on porous diffusion, especially in connection with carrier-bound enzymes or cells (for example. Pitcher, 1978). These are directly connected to the principles expressed in Sect. 4.5 concerning the influence of internal and external mass transport. The results are presented in the same graphical form as Fig. 4.36 in which the effectiveness factor of the reaction rj. is presented as a function of the Thiele modulus. For formulating an appropriate moduls one needs knowledge of the difficult to measure value. The following equation has shown itself useful in that the volume-based reaction rate is obtainable directly from the experimental measurements (Pitcher, 1978) (cf. Equ. 4.74)... 

They indicated that Even though specific surface area and total pore volume were important factors for increasing the capacity of hydrogen adsorption, the pore volume which has pore width (0.6-0.7 nm) was a much more effective factor than specific surface area and pore volume in PAN-based electrospun activated CNFs. 

Finally, the most noticeable result has been the understanding of the usually observed concentration effects on the elution volumes based not only on hydrodynamic aspects, but also on the important influence of thermodynamic factors such as the preferential solvation. In this sense, it has been quantitatively shown that, for a given solvent/polymer system and a given molar mass, the concentration effect is more acute in the packing that presents lower polymer-gel sorption and lower degree of cross-linking. 

The concept of an effectiveness factor of a porous slab can be extended to other geometries (solutions in Table 4.5.5). In contrast to some other textbooks is here always calculated based on the ratio of the particle volume to the external surface as characteristic length (L for a slab with thickness 21, dcyi/4 for long cylinders, and dp/6 for spheres) ... 

In this case, the effectiveness factor can be obtained simply by solving Eqn. (9-4), subject to the boundary conditions of Eqns. (9-4a) and (9-4b). For a first-order reaction, —Ra, = kv Ca. Here, kv is the rate constant based on geo/ncmc volume of catalyst. This rate constant is related to the one that we have used previously, k (moles/time-weight of catalyst), by kv = k/jp. 

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