Damkoehler Number reactions

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 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. 

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... 

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.

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. 

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. 

Figure 7-5 shows the dependence of the isothermal conversion on the Damkoehler number for reactions of the first and second order performed in a batch reactor. If the expected conversion is equal to 90%, then the first order reaction will be four times faster than the second order reaction. 

Figure 4.36. Graphical representation of the concept of effectiveness factors (a) The effectiveness factor of reaction, rj, as a function of the Damkoehler number of the second kind, Dan, or of the Thiele modulus, (j) (cf. Equ. 4.74). (b) The effective reaction rate, r ff, as a function of the diameter of the particle, d. Part (b) can be used to obtain the value of for a given average diameter, of a population of floes with a distribution d. The range of validity of kinetic control (r js) and of diffusion control (Wfds) is indicated in part a.

A characteristic feature of reaction engineering is the introduction of dimensionless numbers to make reactions with different rates comparable. Here this is the Damkoehler number ... 

Figure4.3.4 Influence of the Damkoehler number Da on the conversion of reactant A for a zero-, first-, and second-order reaction [batch reactor reaction order n for definition of Do see Eqs. (4.3.13) and (4.3.19)].

If we compare this equation with the definition of the Damkoehler number [Eq. (4.3.19)], we see that Da can be regarded as the ratio of the reaction time and the characteristic reaction time ... 

The term kt is called the Damkoehler number Da (of a first-order reaction). 

Introducing the axial Peclet number for heat Peh,ax=UsCpPgdp/A,ax, the residence time T with regard to the empty tube and the Damkoehler number Da — km,efFPbf = k j Lfu leads to (if an exothermic reaction is taken as example) ... 

For ideal isothermal reactors, the conversion of a reactant A can be calculated by one parameter, the Damkoehler number. (For a cascade of CSTRs we also need the number of CSTRs.) For a reaction order n and a rate constant k, Da equals for a batch reactor (t = reaction time) and r (r = resi-... 

Da Damkoehler number Depends on reaction order and reactor type, for example, for batch reactor and first order Da — kt... 

Relative Rates of Mixing and Reaction The Damkoehler Number... 

Various Damkoehler numbers can be defined, preferably based on turbulence characteristics rather than on geometry-dependent variables. Regardless of the definition selected, there are definite limits of Da for the two limiting cases of fast and slow reactions. (Example 2-2)... 

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 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.)... 

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. 

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

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

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