External Transport Limitations

With curve a of Fig. 4.29, the reaction is unimpeded (c - c ) here the reaction is under kinetic control and is in the kinetic regime. With curve c, transport is so slow in relation to the rate of reaction that the concentration in the L phase (or S phase) decreases to zero the reaction here is in the regime of diffusion control and transport limitation is paramount. 

Curve b is an intermediate case in which both phenomena are at work. The equation valid for a so-called effective reaction rate, r ff, is (for the case of a first-order reaction, Fitzer and Fritz, 1975) at steady state 

The reciprocal rate constants are additive in this case in which transport and reaction steps operate in series. 

One may take the ratio of the kinetic (/c ) to transport (/cjr, Equ. 2.2b) rate constants as a criterion for determining the boundary between the different types reflected in curves a and c. If kjkj ) 1, then the reaction is very slow in relationship to transport, and Equ. 4.61 may be reduced for the kinetic regime to 

To obtained undistorted kinetic parameters in cases such as these, transport processes must be arranged well enough so that Equ. 4.50 applies. This is accomplished mainly by good mixing in the fluid phase With good turbulence, a high fcL value is attained. If Equ. 4.62a is recast in the form, for example. 

---

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-... 

The kinetic modeling of the external transport limitation is successful with aid of a double-substrate-limitation function [61-63] ... 

Figure 6. Diagnostic tests for interphase (external) transport limitation.

COUPLED HEAT AND MASS TRANSFER IN PACKED CATALYTIC TUBULAR REACTORS THAT ACCOUNT FOR EXTERNAL TRANSPORT LIMITATIONS... 

Petersen [86] has demonstrated that with realistic values of the mass transfer and diffusion parameters, external transport limitations will never exist unless internal diffusion limitations are also present. This is most easily seen by comparing the reduction in reaction rate caused by internal limitations alone, tf, with that caused by the additional external transport limitations, (ticM)- Using Eq. 3.6.d-4... 

Figure 4.31. Plot of normalized effective reaction rate r ff/r ax against dimensionless substrate bulk concentration s/Xg for different values of modulus for external transport limitation in the case of Monod-type kinetics. (From Horvath and Engasser, 1974.)...

Figure 4.32. Plots of effectiveness factor of reaction f/r,ext versus modulus in case of external transport limitation as a function of substrate concentration s/Xg. The rate-determining steps (rds) are indicated with their range of validity by dotted lines, together with the limiting first-order effectiveness factor at low s values. (From Horvath and Engasser, 1974.)...

The influence of external transport on kinetic parameter estimation can also be illustrated in an Eadie-Hofstee plot, as shown in Fig. 4.33 (Hartmeier, 1972 Horvath and Engasser, 1974). Significant departures from linearity, however, are observed with increasing external transport limitation >0.1), particularly when a wide range of s is examined. 

This result is in agreement with similar situations under turbulent conditions. External transport limitation in this biofilm reactor was eliminated when the linear fluid velocity exceeded 0.8 m s L... 

When Bi 00, the external transport limitation becomes negligible (cf. Equ. 4.110) and hence approaches Comparing the magnitude of relative deviation from the corresponding true numerical values for film and floe geometries, Yamane showed that Equs. 4.81 and 4.82 can be recommended for a slab but never for a sphere. The variation of the relative error with the Thiele modulus is shown in Fig. 4.44. Equations 4.108 and 4.110 give quite similar degrees of accuracy. 

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... 

The permeability coefficient reflects the overall resistance of the composite barrier, and therefore includes external transport limitations (cf. Sect. 4.5.1), which can be quantified using the two-film theory (cf. Equ. 3.30) ... 

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. 

When data are taken under isothermal conditions at several temperatures in order to calculate the activation energy, the apparent rate constants evaluated will include the square root of (kyDe) at each temperature. Since D<, is not very sensitive to temperature changes, the Ea calculated from the Arrhenius equation will be equal to one-half the Ea of the surface reaction, provided that external transport limitations do not exist... 

The net rate of production of the intermediate R is derived along the same lines. Simplifying Carberry s equations by neglecting external transport limitations, the following equation for the selectivity for Ris obtained ... 

Petersen [1965a, b] has demonstrated that with realistic values of the mass transfer and diffusion parameters, external transport limitations will never exist unless internal diffusion limitations are also present. 

In kinetic studies it is advised to avoid experimentation in which external transport limitations are experienced. These can be easily avoided by adapting the flow rates. Yet, Koros and Nowak [1967] and Madon and Boudart [1982] have drawn attention to the lack of generality of the usual diagnostic test for external mass and heat transfer limitations. In this experimental test, the conversion (or the rate) is measured at constant space-time, W/Fao, but the flow velocity is varied by adapting simultaneously ITand Fao- If the conversion (or the rate) is not affected, the conclusion is drawn that there are no external transport limitations. At the low Reynolds numbers encountered in laboratory reactors, the mass and heat transfer coefficients are quite insensitive to changes in flow rates, so that these have to be varied over a very wide range. 

Assume that Pt was dispersed throughout the pore structure of the entire pellet in Problem 4.1 and apply the Weisz-Prater criterion to determine if mass transport limitations are expected. Do only one calculation using the lowest observed rate. Assume that the average pore diameter in the catalyst is 100 A (10 A = 1 nm), that Knudsen diffusion dominates, and that no external transport limitations occur (Cs = Co). 

---