First-order kinetics effectiveness factors

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Consider a nonporous catalyst particle where the active surface is all external. There is obviously no pore resistance, but a film resistance to mass transfer can still exist. Determine the isothermal effectiveness factor for first-order kinetics. 

Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets. 

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

The reaction will then appear to follow first-order kinetics, regardless of the functional form of the intrinsic rate expression and of the effectiveness factor. This first-order dependence is characteristic of reactions that are mass transfer limited. The term diffusion controlled is often applied to reactions that occur under these conditions. 

If the effectiveness factor of the catalyst is known to be 0.42, estimate the tortuosity factor of the catalyst assuming that the reaction obeys first-order kinetics and that Knudsen diffusion is the dominant mode of molecular transport. 

Detailed kinetics investigations have shown that the reaction follows pseudo-first-order kinetics. A linear relationship exists between pH and log k (the log of the rate constant), such that log k decreases by 1.7 (i.e., tm increases by a factor of ca. 50) with each increase of one pH unit. For example, the tu2 value of diphenhydramine (11.24, R = R = H, Fig. 11.2) and orphenadrine (11.24, R = 2-Me, R = H) at pH 0 and 25° were found to be 550 and 460 min, respectively, from which f1/2 values of ca. 460 and 360 h could be calculated for pH 1. Each increase of 10° in temperature led to a decrease in the f/2 value of 3 - 4 h. Hence the tV2 value of diphenhydramine and orphenadrine in the stomach at 37°, assuming pH 1 and neglecting any effect caused by ionic strength, should be ca. 4 d. This is clearly too slow for any significant nonenzymatic formation of benzhydrol in the body (see below). 

Few reports are available on the potential effect of chemical concentration on the BAF in an aquatic organism (e.g., Mayer, 1976). Yet, a key assumption of EP theory is the independency of BAF relative to exposure concentration. To our knowledge, there is only one report (Huckins et al., 2004) in the peer-reviewed literature, where the effect of chemical exposure level on concentration factors (CFs) or BAFs has been tested in side-by-side BMO and passive sampler exposures. Huckins et al. (2004) defined CF as the ratio of the concentration in a sample matrix (whole body [soft tissues in the case of bivalves] or whole SPMDs) relative to the concentration in the ambient exposure medium at any moment in time, whereas the A sw and BAF (includes biomagnilication) represent the maximal CF. Similar to ATs s and BAFs, CFs are expected to be independent of exposure concentrations, when residue exchange follows first-order kinetics. 

Find the effectiveness factor for a spherical catalyst particle with first-order kinetics. 

Introduced as a factor in the last term of Equation 20.52 is the dimensionless first-order kinetics parameter kttk. Variations in this parameter will allow the effect of any first-order reaction to be simulated. 

Fig. 3.5 Effect of Thiele s modulus on the effectiveness factor for spherical immobilized particles for first-order kinetics and Michaelis-Menten kinetics with p = 5. Thiele s modulus for first-order kinetics is...

Davidson et al. (1978a) also used first-order kinetics for nitrogen transformations, but they considered that some of the transformation rate coefficients were dependent on several factors including environmental ones The rate for nitrification was empirically adjusted for water suction. Overall, most of the nitrogen models assume first-order kinetics. Some of them also consider the effects of temperature on rate coefficients. 

Effectiveness factors calculated in this way are of the order of 0.02-0.04 for both reactions. Van Hook (1 9) and Rostrup-Nielsen (20) published values which are of the same order of magnitude, but the latter were obtained assuming first order kinetics for the methane conversion and equilibrium for the watergas shift. The corresponding reaction layer amounts to 0.65-1.3 10 A, and to a surface area still exceeding Sp by a factor IOOO. 

Little is known about the kinetics of the bioprocesses. A reasonable assumption is that the reaction rates are proportional to the amount of micro-organisms catalyzing the reactions. The influence of the other reactants is more complex, for example a nutrient in high concentration often has an inhibiting effect. Moreover, factors such as pH, salt concentrations, temperature, can have effects that are difficult to quantify. For this reason, we assume first-order kinetics and include all the other factors influencing the process rate in two Damkohler numbers. The following dimensionless reactor model is obtained ... 

A simple model of the bioreactor was used, assuming first-order kinetics with respect to FeEDTA species and lumping in two Damkohler numbers the effects of various factors such as micro-organisms concentration, pH, salt concentrations and temperature. 

Figure 6.2. Effectiveness factor i) versus the Thiele modulus

Table 6.1 Effectiveness factor 17 as a function of the zeroth Aris number An, for first-order kinetics an infinitely long slab, an infinitely long cylinder and a sphere...

Equations 6.57 and 6.58 are illustrated in Figure 6.15 where A according to Equation 6.56 is plotted versus 17 for first-order kinetics in a slab. The effectiveness factor 17 and the approximation rj were calculated from Tables 6.1 and 6.2 and Equations 6.57 and... 

The value of Equation 6.59 is demonstrated in Figure 6.16 for first-order kinetics in a slab. It can be seen that Equation 6.59 predicts the effectiveness factor with an error of... 

Wall Effects. In the above discnssion, we have assnmed that the reaction is homogeneous (i.e., no catalytic reaction at the walls of the reaction bnlb). The fact that the data give first-order kinetics is not a proof that wall effects are absent. This point can be checked by packing a reaction bnlb with glass spheres or thin-walled tnbes and repeating the mea-snrements under conditions where the surface-to-volume ratio is increased by a factor of 10 to 100. This will not be done in this experiment, but the system chosen for study must be free from serious wall effects or it may not be possible to discnss the experimental results in terms of the theory of nnimolecular reactions. 

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