Mixing Damkoehler number

The competition between reaction and mixing is well represented by a mixing Damkoehler number, Dum, which is the ratio between the reaction rate and the local mixing rate, or conversely, the ratio of the characteristic mixing time, tm, and the reaction time, tr ... 

Relating Mixing and Reaction Time Scales The Mixing Damkoehler Number... 

The mixing Damkoehler number the ratio of rates of the first or second reaction and the local mixing rate. 

Mixing Damkoehler number the ratio of mixing time to reaction time, DaM = Tm/tr. The mixing Damkoehler number may be referred to simply as the Damkoehler number. (Note that the traditional Damkoehler number is the vessel residence time divided by the reaction time.)... 

These values were combined into a mixing Damkoehler number as the ratio of mixing time constant to reaction time constant ... 

The mixing Damkoehler numbers obtained were 0.050, 0.032, and 0.023, respectively. These are reasonably close for such radically different experimental conditions vessel size, impeller rotation rate, feed location, and concentration. 

The concepts embodied in the mixing Damkoehler number (Dum) are extremely useful for initial evaluation of reaction conditions in which mixing effects must... 

An adaptation of the Damkochicr number (Da) is a useful concept for evaluation of mixing effects in crystallization. It is the ratio of the characteristic mixing time to its corresponding process time (nucleation induction time, crystal growth/supersaturation release time, or reaction time). Studies of these times and the resulting predicted Damkoehler number in a laboratory setting can provide evidence of possible scale-up problems. 

An effectiveness factor has been presented by Garside (1971) as the ratio of the actual growth rate to the one that would occur if the interface were exposed to the bulk conditions. This is also presented as a Damkoehler number, but it is different from the Damkoehler numbers described in Chapter 6, which relate to mixing time/process time ratios. 

It is helpful to visualize the relationship between mixing and nucleation rates through an analogy with the reaction Damkoehler number (Da). The Da number for reaction is defined as Da for reaction = mixing time/reaction time... 

Sections 6.3.1.4 and 6.4 on the use of the Damkoehler number concept, the need for effective location of a feed line may be evaluated from a comparison of nucleation and mixing rates. 

Thus the value of r that is needed to ensure that axial dispersion of mass affects the reactor length by less than 5%, depends on the Damkoehler number Da. For Da = l this criterion is fidfilled if the residence time r equals 5% of the dispersion time Td.sx- For Da higher or lower than unity, the criterion is tightened (tD,ax<0.05TD,ax) Or diminished (rD,ax>0.05 ro.ax)- This effect is easy to understand if we consider that deviation of the behavior of a back-mixed reactor compared to an ideal PFR increases with increasing conversion and value of Da -simply think of the case of a CSTR compared to a PFR - and thus the criterion for the minimal residence time compared to the axial dispersion time is stricter. 

It would be more convenient to nse a mixing time that is not geometry specific. A number of such times and Damkoehler numbers were compared by Brodkey and Kresta (1999) nsing varions local turbulence scales in the Toor reactor. All of these times nse local tnrbnlence parameters or characteristic times. The position at which these are evalnated for the two multitube reactors is at the point of coalescence of the feed jets. It tnms out that this is very close to Mao and Toor s (1971) characteristic half mixing length. 

Table 2-2 Damkoehler Numbers Based on Different Mixing Times...

Figure 13-2 Normalized yield, Y/Ye p, as a function of Damkoehler number based on ki. This is a qualitative conceptualization of the interaction between mixing rate as expressed by a local mixing time, tm, reaction rate, kiCBo, and reaction yield. As the mixing improves (smaller Da i), the yield increases. As the second reaction gets faster (increasing k2), the mixing time must also drop, to maintain yield.

Figure 13-3 By-product selectivity, Xj, as a function of Damkoehler number based on k2. These data of Bourne in Sharratt (1997) show the increased by-product formation with increasing mixing time based on the engulfment model, Te- As the reaction rate for the second reaction, kiCe, increases, the mixing time must decrease to maintain yield.

The Damkoehler number requires characteristic time scales for both mixing and reaction. Calculation of the reaction time scale is relatively straightforward, although the necessary data may be difficult to obtain. Many choices for the mixing time have been proposed, and data are available for many common semibatch geometries. 

A step-by-step recommended procedure for determining mixing sensitivity based on evaluation of the Damkoehler number may be found in Section 13-4. 

The defining number for this discnssion is the Damkoehler nnmber (Da), the ratio of mixing time to reaction time ... 

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