Internal Transport Limitations

GP 9[ [R 16]The extent of internal transport limits was analysed for the wide fixed-bed reactor, using experimental data on carbon monoxide conversion and matter and process parameter data for the reactants [78]. The analysis was based on the Weisz modulus and the Anderson criterion for judging possible differences between observed and actual reaction rates. As a result, it was found that the small particles eliminate internal transport limitations. 

GP 9] [R 16[ The extent of external transport limits was made in an approximate manner as for the internal transport limits (see above), as literature data on heat and mass transfer coefficients at low Peclet numbers are lacking [78]. Using a Pick s law analysis, negligible concentration differences from the bulk to the catalyst sur-... 

Heterogeneously catalyzed reactions. Macroscopic fluid models are combined with microscopic transport models in the catalyst particles to describe how concentration changes with time and position in a catalytic reactor. Special considerations must be given to the selection of experimental temperature and catalyst particle size to minimize (and hopefully eliminate) internal transport limitations on the catalytic reaction rate. The next requirement is that the flow pattern in the reactor Is accurately represented by the well-mixed or plug-flow assumption. The subsequent discussion applies to gas-phase reactants. 

The effects of catalyst decay and transport were qualitatively similar to those discussed for the simulation of la-aft lignin. Intrinsic kinetics resulted in the highest yields of monomeric products, and the presence of both deactivation and internal transport limitations resulted in the lowest. 

Another application of the principle of separation is the case of truly heterogeneous processes (see Sects. 4.5, 6.6, and 6.7). Here, cases of external and internal transport limitation of a bioprocess must be distinguished from the case where the physical transport may be enhanced by the biological reaction. A simplified treatment in all these cases is possible through introduction of the concept of efficiency defined by... 

In principle, internal transport limitation, can be checked by measuring the characteristic dimension of the biocatalytic mass and comparing it with... 

Atkinson and Ur-Rahman (1979) used an approach based on Equ. 4.62a, replacing the rate constant by an equation for which also takes into account internal transport limitation rjr,iml see next section). 

The concept of Equs. 4.49 resp. 4.51 is also applicable in the case of internal transport limitation here the ratio of rate constants is treated as a dimensionless number, the so-called Thiele modulus (j) (Thiele, 1939) or the Damkoeh-ler number of the second degree, Da . 

Figure 4.35. Plots of normalized overall reaction rate Teff/r x versus dimensionless substrate concentration s/K for different values of the modulus in case of internal transport limitation with saturation-type kinetics. (From Horvath and Engasser, 1974.)...

Figure 4.37. Effect of internal transport limitation on Monod-type kinetics demonstrated on a Lineweaver-Burk plot (see Fig. 4.24c) with varied values of a modulus according to Shieh (1980a). The limiting case of no transport limitation is indicated...

More complex cases, such as cases with S- and P-inhibition kinetics, have been solved numerically by Moo-Young and Kobayashi (1972). Recently criteria have been developed specifically for Monod-, S-inhibited-, Teissier-, and maintenance-type kinetics to quantify and predict diffusional control within whole cells and cell floes (Webster, 1981). Further details concerning the use of the effectiveness factor concept for the quantification of biological processes will be given in Sect. 5.8, presenting simple formal kinetics in the case of internal transport limitation. 

Figure 5.74. Normalized kinetic plot of the Powell approach to internal transport limitation (cf. Equ. 5.43) with variation of the ratio Kd/Ks- Monod-type kinetics appear when this ratio is zero (after Humphrey, 1978).

A unique benefit of biofilm reactors for research purposes is the fact that due to the distinct separation between the L and the S phase (difference in density — Pl) high relative velocities can be realized. As a result, external transport limitation can be excluded or easily studied simultaneously with internal transport limitations in the case of uniform and controlled biofilm thickness. In this respect, biofilm reactors are superior to the conventional... 

Ensuring the Absence of Transport Limitations at the Catalyst Pellet Scale. Several criteria have been developed for verifying that concentration and temperature gradients, internal or external from the catalyst pellet, can be neglected. Section 2.2.1 deals with external transport limitations and, hence, with steps 1 and 7 from the catalytic cycle discussed above, whereas Section 2.2.2 focuses on steps 2 and 6, that is, on internal transport limitations. Apart from the calculation of these criteria, also a few experimental tests are at hand to verify the absence of transport limitations (see Section 2.2.3). It must be noted, however, that the result of such experimental tests may depend on the reaction order and not necessarily lead to a conclusive interpretation. As a result, it is recommended to double check the outcome of the experimental verification by calculating the corresponding criteria. 

Experimental Verification of the Absence of Transport Limitations. Also experimental techniques have been established to verify the intrinsic character of the measured chemical kinetics. Different diagnostic tests for external as well as internal transport limitations have been developed (2). 

A second test addresses the potential occurrence of internal transport limitations and consists of performing experiments with catalyst pellets of varying dimensions. For larger pellet diameters, diffusional effects are more likely to affect the observed kinetics. Hence, for positive reaction orders, a decrease in the observed reaction rate is expected when internal mass transport limitations become important (see Fig. 5). Although rather unlikely at the laboratory scale. 

Impact of internal transport limitations on kinetic studies... 

The first set of preliminary experiments in a kinetic study carried out in laboratory PBRs should establish the flow conditions at which external transport effects are negligible. The second set of diagnostic experiments conducted should verify the particle size requirements under which intraparticle transport effects are eliminated. A study of intrinsic kinetics must make use of the flow rates and particle sizes that exclude both external and internal transport limitations, respectively. 

In many cases, it is necessary to rapidly estimate whether internal transport limitations are present or not, without having to use the reaction rate constant. The Weisz-Prater criterion is... 

It should also be emphasized that all reactions used have to be tested with respect to external and internal transport limitations, and the results should be documented in the additional material accompanying the paper. The problem is even more severe if molecular sieves with different pore sizes and... 

Note of caution In all catalytic testing experiments the absence of any external and internal transport limitations within the catalyst particle and at its outer surface has to be ascertained otherwise, results regarding activity and selectivity, by which catalytic performance is to be characterized, may not be of any significance. 

Diagnosing Internal Transport Limitations in Experimental Studies... 

On the other hand, if /3 > x, r /rc will decline as Ca declines. Since Ca is less than Ca,s throughout the cataly st particle, an internal transport limitation will cause to be less than its... 

If B < X, / c will increase as Ca declines. An internal transport limitation will cause... 

In the presence of an internal transport limitation, the net rate of formation of t-CeHis is reduced because Cn inside the catalyst particle is less tiian Cn,s> and it is further reduced because Cj inside the particle is greater than Therefore, the net forward rate of the isomerization reaction is reduced more than the rate of the hydrogenolysis reaction, which is reduced in proportion to Cn. The actual selectivity to i-CgHM should be less than the intrinsic selectivity when the Thiele modulus is high. 

In this equation, ija is the effectiveness factor for the first reaction and t)b is the effectiveness factor for the second. Let A° denote the intrinsic ratio of the two reaction rates, i.e., the ratio when internal transport limitations are insignificant and both effectiveness factors are equal to 1. For the present example. 

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