Single generalized Thiele modulus

Figture 16 Effectiveness factor of single-file reaction plotted against the generalized Thiele modulus as defined by Eq. (42). (From Ref. 162.)... 

Fig. 5 Concentration profiles of the molecules of species B within the single-file system under stationary conditions and comparison with the dependence to be expected for ordinary diffusion (broken line, Eq. 29). The quantity 2L(klDy (the Thiele modulus 0) in Eq. 29 has been chosen to coincide with the generalized Thiele modulus (cf. Eq. 30) of the single-file reaction for k = 1.27 x 10 (0 = 2.77). z denotes the distance from the middle of the file and L = NX its length. From [1] with permission...

This dependence is represented by the dotted line in Fig. 6. In a first attempt to systematize the simulation results of molecular reaction and diffusion in single-file systems, a generalized Thiele modulus has been introduced [1]. Combining Eq. 25 and Eq. 23, the Thiele modulus may be expressed in the alternative notation... 

Fig. 6 Effectiveness factor rj = of single-file reaction plotted as a function of the generalized Thiele modulus. From [1] with permission...

These numbers reveal that the (intrinsic) alkylation reaction is exceedingly fast. Using the value of 10- cm /s for the diffusivity, it was found that the generalized Thiele modulus will be unity at 100 for a particle with a 3.1 pm diameter. This diameter is approximately the same as that of the USHY single crystals Thus it can be seen that, practically, the reaction can not be run at 100 °C in the liquid phase without diffusion limitations masking the intrinsic kinetics. 

Generalizati on. It has been observed that all plots of effectiveness against Thiele modulus are similar and that a single plot can represent the nine main cases fairly adequately by defining a generalized modulus as... 

Biofilters are chemically enhanced absorbers, and therefore mass transfer limited (see Absorption with Chemical Reaction in Sec. 14). The magnitudes for the Hatta [= Damkohler II = (Thiele modulus)2] numbers are quite low, perhaps below 5. Nevertheless, for design simplicity, mass-transfer limitation is generally assumed to be in the liquid phase (the biofilm). For a single-component biofilter, the simplified biofilter model and design equation is... 

This literature survey leaves the impression of an enormous but chaotic quantity of publications, because all the models and correlations focus on a single rather specific problem. A number of attempts have been made to integrate some of the models and correlation s, but without success. Also, the authors tend to use their own tailor-made definitions rather than generalized ones. As a result a large variety of definitions can be found for the same number (Doraiswamy and Sharma [4] use 20 different definitions for the Thiele modulus in their survey). This makes the results of different correlations hard to compare. Finally, some of the formulae which were given contradict each other, which is of course very confusing. 

With this more general equation, representation of the effectiveness factor in terms of the Thiele modulus also becomes possible for single-file diffusion. As an example. Fig. 16 shows the result of a computer simulation of diffusion and reaction within a single-file system consisting of A = 100 sites for different occupation numbers in comparison with the dependence... 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

Because the equations for the three shapes produce three different curves, it is desirable to formulate a single Thiele modulus which will be applicable to these and other shapes. It is also desirable to generalize the modulus to include any... 

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