Damkohler numbers series

General design charts can be drawn which display the relationships between the Damkohler number, rkC/, the number of tanks in series, N, the achieved fractional conversion, and the total volume of the N... 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

F re 4-4 Conversion as a function of the number of tanks in series for different Damkohler numbers for a first-order reaction. 

A simplified mathematical description of this three-zone model proved [332], however, that the quantity of material converted in the dispersion zone could be ignored, if the original concentration ratio of A and B was less than 0.1 and the action of micro- and macro-mixing is described by the Damkohler number Da = fczCBo/s and n = 27twD/ qs) (where w is the flow rate D is the dispersion coefficient q is the liquid throughput and s [s ] is the shrinkage rate of the volume element). The model was evaluated with competitive reactions in series and parallel reactions, and was found to be adequately applicable. 

In Figure 3.23 the unreacted fraction of the key reactant (/j = 1 - X) is plotted against the Damkohler number for different stirred tanks in series. For N oo the final conversion corresponds to that of an ideal PFR. 

The above expressions are inserted into the appropriate balance equations, for example, for tanks-in-series, segregated tanks-in-series, and maximum-mixed tanks-in-series models. The models are solved numerically [3], and the results are illustrated in the diagrams presented in Figure 4.29, which displays the differences between the above models for second-order reactions. The figure shows that the differences between the models are the most prominent in moderate Damkohler numbers (Figure 4.29). For very rapid and very slow reactions, it does not matter in practice which tanks-in-series model is used. For the extreme cases, it is natural to use the simplest one, that is, the ordinary tanks-in-series model. 

FIGURE 4.29 yj (tanks in series), yur (maximum-mixed tanks in series), and ysT (segregated tanks in series) at M = 1.00, M = vaCob/(vbCoa)> R = kAtcoA h/ AyR = DamkOhler number. 

An additional simple approximate analytical solution is available in the so-called thin film" theory near the diffusion limit when the Damkohler number (kb m/DA) representing the ratio of diffusion time to reaction time approaches zero. Smith et al. (1973) have obtained the following estimate of the facilitation ratio by a power series expansion ... 

A plot of the conversion as a fanction of the number of reactors in series for a first-order reaction is shown in Figure 4-4 for various values of the Damkohler... 

A plot of the conversion as a function of the number of reactors in series for a first-order reaction is shown in Figure 5-6 for various values of the Damkohler Economics number tA. Observe from Figure 5-6 that when the prcnluct of the space lime and the specific reaction rate is relatively large, say. Da I, approximately 90% conversion is achieved in two or three reactors thus the cost of adding subsequent reactors might not be justified. When the product tA is small. Da 0.1, the conversion continues to increase significantly with each reactor added. 

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