Damkoehler Number

Output Da the desired vector of values for the Damkoehler number ... 

Dastart starting value a for the Damkoehler number 10 a (exponent)... 

Figure 8 Analysis of the appropriate characteristic dimensions for specific heat transfer surface requirements in a chemical reactor exhibiting typical reaction rates. Da = Damkoehler number, NTU = number of thermal transfer units, Nu = Nusselt number, a = thermal diffusivity. (From Ref. 9.)... 

The parameter X contains the Damkoehler number, Da, the Stanton number for gas-to-liquid transfer, Stgj the external contact-... 

Lode (2002) and Fricke and Schmidt-Traub (2003) have discussed the influence of further parameters, e.g. Damkoehler number. Although, notably, the TMBR model only approximates an SMBR process, it can be applied to discuss the different influences on the SMBR. 

Damkoehler number feed solid D soUd = Cfeed ... 

The competition between reaction and mixing is given by the Damkoehler number Da), which is the ratio between the reaction rate and the local mixing rate, or conversely, the ratio of the characteristic local mixing time tm and the reaction time tr ... 

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. 

The importance of the feed location for chemical reactions has been clearly established by the work referenced above and many others. The literature contains less data on crystallization. However, undesired nucleation is potentially present for all crystallization systems, depending on the nucleation rate and the degree of local supersaturation. The analogy between reaction sensitivity and supersaturation sensitivity can be visualized through the concepts represented by the Damkoehler number, as discussed in Section 6.3.1.4 above. 

The Damkoehler number Da represents the ratio of a characteristic reaction time to the kinetic time constant of the reaction and is therefore a measure for the reaction time. The attribute characteristic refers to the individual definition necessary for each... 

A closer examination results in the recognition of the Damkoehler number as the term with which the dimensionless reaction rate is multiplied. This leads to a simplification for the general dimensionless mass balance ... 

The initial concentrations, which are required for the calculation of the initial reaction rate and the Damkoehler number, are preferably referenced to the final volume. This, on the one hand, allows for a straightforward calculation of the post-feed reaction phase as the final SBR concentrations directly correspond to the input concentrations of the concluding batch phase. On the other hand, this allows for a simple comparison to other ideal reactors without additional recalculations. 

When setting up the mass balance the Damkoehler number was referenced to the final reaction volume, this way enabling an easier comparison with other ideal reactors and a convenient calculation of SBR phase and post reaction batch phase. This shall be adopted for the definition of the Stanton number. 

Figures 4-10 and 4-11 can now be calculated by varying the steady state conversion for a fixed reaction order. The Damkoehler number calculated this way is subsequently used to determine flie corresponding steady state temperature using fixed values for the activation energy and Daoo. From the coupling equation, finally, the corresponding reference temperature is obtained. In order to simplify the later interpretation, two straight lines representing certain border line cases are added. The upper line is the border line for 100% conversion, the lower one for the special case that there is no conversion at all. The sigmoid curve presents all possible solutions. Each point on this curve corresponds to one steady state operating point.

In consequence of these high Damkoehler numbers the conversion achieved is also very high. An example is shown in 4-53a-b. 

If the Damkoehler number of isothermal semibatch processes is smaller than 10 the degree of accumulation becomes high. In such cases the reaction rate and thereby the heat production rate is not proportional to the feed rate alone. For such processes a synthesis optimization is recommended which in the last consequence will also benefit the overall process safety. As will be demonstrated in the context of the safety technical assessment of maloperations (c.f. Section 4.4), the reduction of the maximum accumulation is a very decisive factor. 

With the help of these parameters the given data about the reference temperature can be validated. In all cases the correct value can be determined. It will be required for the calculation of the Damkoehler number Da (To). 

In the following step the Damkoehler number can be calculated and evaluated in respect to its compliance with the safety criterion. 

The deviation in the pre-exponential factor, represented here by the Damkoehler number Da , seems to be quite severe. But this is a superficial judgement only. For numbers of this magnitude, deviations in percentages are always very large although the realistic assessment comes up with moderate differences. It becomes easier to evaluate the quality of the data determined, if they are used in a new simulation and if this result is compared with the original simulation. This is presented in Figures 4-76 to 4-78. 

One possible solution is to reduce the space/time-yield at fixed reactor size. If the reaction system is diluted and the residence time simultaneously prolonged, so that the product of Cbo "t remains constant, then the Damkoehler number and thereby the conversion are left unchanged, but the adiabatic temperature increase is lowered. Due to such a change in the manufacturing recipe, naturally, the amoimt of product manufactured per unit time is reduced. Such a measure and its effect is shown in Figure 4-81 for the example. 

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