Negligibility Criteria

Substitution of for CB in a kinetic expression is acceptable if, for the high rj region 

Substitution of the Aris numbers Ant and Ana into Equations 7.47 and 7.48 then yields 

Assuming that substitution of for CB is justifiable, provided that a relative error smaller than 10 % is introduced, the following criteria are obtained  

Criteria for other forms of kinetics are easily derived using the general formulae 7.51 and 7.52. For example, for Langmuir-Hinshelwood kinetics as in Equation 9.16, for the high 77 region the criterion becomes 

In summary, it can be said that calculating the effectiveness factor for bimolecular reactions does not pose any new problems as compared to simple reactions. Criteria for isothermal operation and criteria which determine whether or not CBj may be substituted for CB can easily be determined. From the last criteria it can be seen that the conclusion of Doraiswamy and Sharma [4], that can be substituted for CB if the key reactant 

---

Negligibility criteria are derived for both the high and low ij region. For the high ij region heat effects can be neglected if the number... 

Table 7.4 Comparison of the negligibility criteria for isothermal operation and nth-order kinetics...

Using the dusty gas model [5] analytical solutions are derived to describe the internal pressure gradients and the dependence of the effective diffusion coefficient on the gas composition. Use of the binary flow model (BFM, Chapter 3) would also have yielded almost similar results to those discussed below. After discussion of the dusty gas model, results are then implemented in the Aris numbers. Finally, negligibility criteria are derived, this time for intraparticle pressure gradients. Calculations are given in appendices here we focus on the results. 

Since we want to derive negligibility criteria, we can assume that the value of y is small. Then for nth-order kinetics Equation 7.91 yields... 

With the Equations 7.112 and 7.113 negligibility criteria can be derived in a similar way as before. Here only the results for (1,1) kinetics are given (Equation 7.40). A gas can be regarded as being diluted if the criterion... 

In Table 7.7 the most important dimensionless numbers introduced in this chapter are summarized (together with negligibility criteria). With these numbers, it is now possible to give generalized definitions for Aris numbers, for which two situations may be distinguished ... 

Table 7.7 Summary of the characteristic numbers, their ranges and the negligibility criteria for complex situations...

In the case of exothermic reactions, underestimating the transfer coefficients makes the real gradients less than the estimated ones. As such, this makes our estimates conservative, in the sense that if a criterion calls gradients negligible then they surely are. The intent here is to do most of the kinetic study and catalyst testing at gradientless conditions and this book will make use of the Colburn-type correlations as developed by Hougen (1951) and his associates. 

Equation 15.78 gives the criterion for heat transfer enhancement to cater to the new duty without increasing the heat transfer area. If it is assumed that the resistance to heat transfer across the tube wall is negligible (i.e. hw goes to infinity) and the difference between the inside and outside diameters is negligible (i.e. d0 = df), then Equation 15.78 simplifies to9 ... 

For most of the reactions frequently employed in limiting-current studies, the surface overpotential is not negligible. A criterion for assessing its magnitude is the exchange-current density i0, which is a measure of the reaction rate at the equilibrium potential of the electrode (i.e., when anodic and cathodic rates are equal). 

The Hatta criterion compares the rates of the mass transfer (diffusion) process and that of the chemical reaction. In gas-liquid reactions, a further complication arises because the chemical reaction can lead to an increase of the rate of mass transfer. Intuition provides an explanation for this. Some of the reaction will proceed within the liquid boundary layer, and consequently some hydrogen will be consumed already within the boundary layer. As a result, the molar transfer rate JH with reaction will be higher than that without reaction. One can now feel the impact of the rate of reaction not only on the transfer rate but also, as a second-order effect, on the enhancement of the transfer rate. In the case of a slow reaction (see case 2 in Fig. 45.2), the enhancement is negligible. For a faster reaction, however, a large part of the conversion occurs in the boundary layer, and this results in an overall increase of mass transfer (cases 3 and 4 in Fig. 45.2). 

The affinity constant of the test compound K can in turn be calculated from the IC50 value, the Kj of the marker and the concentration of M according to Cheng-Prusoff [see Eq. (3)] [22]. Since the concentration of free marker (M) is usually not determined in radioligand binding assays, it should be noted that the calculation of Ki according to Eq. (3) based on Mtot yields only reliable results if the depletion of the marker is negligible (i.e. <5% to 10% [7, 17, 21]). In competition experiments performed under typical conditions (Ttot Mtot this criterion is fulfilled. 

Saito (SI) showed that the prolate form is possible if the drop density is very large and the ratio of drop viscosity to field viscosity is so small as to become negligible. His criterion, obtained by considering the second-order velocity terms in Hadamard s model, is... 

The failure of Pauling s criterion for the fraction of ionic character of a bond (/ijer) in the case of alkali halides stems from the fact that the criterion fails to include the far from negligible polarization deformation of the ions in these completely ionic substances Rittner, Ref. 17, p. 1035). 

The short-time criterion is the more stringent except when H 1 and p/ 1. External resistance controls and the concentration within the particle is uniform when the inequalities in Eqs. (3-74) and (3-75) are reversed. Even if the external resistance is not negligible relative to the internal resistance, it may be possible to assume constant external resistance, i.e., quasi-steady behavior. Comparison... 

The internal resistance is always decreased substantially when a bubble or drop oscillates, but the external resistance may be unaffected if the Reynolds number is high enough. A rough criterion can be obtained from Eq. (11-63) for vibration of a particle in an axial stream. Oscillation has negligible effect on the external resistance if... 

For packed bed reactors, Carberry and Wendel (1963), Hlavacek and Marek (1966), and Carberry and Butt (1975) report that axial dispersion effects are negligible if the reactor length is sufficient. These and other researchers (Young and Finlayson, 1973 Mears, 1976) have developed criteria based on the reactor length for conditions where axial dispersion can safely be neglected. The criterion shown in Table V is a classic criterion for neglecting axial mass dispersion. The works by Young and Finlayson (1973) and Mears (1976) provide more detailed criteria to predict when axial dispersion is unimportant in nonisothermal packed bed reactors. 

The best mix of liability rules, information markets, and regulations varies across commodities even if efficiency is the only criterion. Because all are suboptimal in practice, the proper mix also depends on people s inefficiency preferences. Asserting that one values individual liberty eliminates only command-and-control regulation from the set of possibilities. Under the right circumstances, no liability, the negligence rule, strict liability, and strict liability with contributory negligence all promote individual liberty. On balance, 1 favor no liability or the negligence rule because 1 believe the adverse side effects of sfrict liability are too severe. 

As to this subject a few general remarks should suffice in the present context, as the decision was made at the beginning, to investigate only those systems in which the form birefringence is negligible. From this point of view, only a criterion for the absence of this effect would be needed. However, before this point is discussed, the reasons for the mentioned decision will be explained in more detail. 

Intraparticle Resistance. The Koros/Nowak (ref. 7) or Madon/Boudart (ref. 8) criterion states that, in the absence of mass transfer influences, the activity of a heterogeneous catalyst should be proportional to the number of active sites. In other words, the observed turn-over frequency (TOF) should be independent of the particle size if there is negligible intraparticle resistance since all active sites are fully effective. 

---