Damkohler-number asymptotics

With simplified chemical kinetics, perturbation methods are attractive for improving understanding and also for seeking quantitative comparisons with experimental results. Two types of perturbation approaches have been developed, Damkohler-number asymptotics and activation-energy asymptotics. In the former the ratio of a diffusion time to a reaction time, one of the similarity groups introduced by Damkohler [174], is treated as a large parameter, and in the latter the ratio of the energy of activation to the thermal... 

The approach that has been described here is an example of a perturbation method for large Damkohler numbers and may be termed Damkohler-number asymptotics. It has been developed on the basis of an expansion that does not distinguish among special zones within the flame. It is possible that the Damkohler-number expansion will often be good in the hot reaction zone but poor elsewhere, while radical distributions away from the hot reaction zone have relatively little influence on the main characteristics of the flame. Under these circumstances, an approach based on matched asymptotic expansions, treating different zones differently, may be helpful. Sharper definitions of values of H consistent with the steady-state approximation (in the zone where it is applied) might thereby be developed. 

The catalytic reaction is simply a bimolecular reaction between B and R, with boundary conditions given by ce,m lz=0+ = cxBm, cj >m z=0+ = c] m. The yield of S increases monotonically as the Damkohler number of the catalytic reaction, Das, increases, and finally attains an asymptotic value when the catalytic reaction reaches its mass transfer limited asymptote. This feature is illustrated in Fig. 19, where the variation of Ys with Das is shown. It is interesting to note from Fig. 19, that the value of the mass transfer limited asymptote depends on the micromixing limitation of the homogeneous reaction. Larger is the micromixing limitation (rj) of the homogeneous reaction, more is the local... 

Figure 5 Influence of the Damkohler number. Da, and the monolith channel geometry on the asymptotic Sherwood number, Sh . = Circular channel = Square channel L = Equilateral triangular channel. (From Ref. 39.)...

Figure 7.4 Snapshots of the spatial distribution for the autocatalytic model (7.1) in the open blinking vortex-sink flow at time intervals equal to the flow period for a supercritical Damkohler number, Da = 7.0. Note, that after a transient time a time-periodic asymptotic state is reached where the autocatalytic growth, localized on the fractal unstable manifold, is balanced by the loss of product due to the outflow from the mixing region, in this case through the point sinks.

Effectiveness factor calculations summarized in Tables 19-1 to 19-5 are consistent with Langmuir-Hinshelwood kinetics, as discussed in this chapter. E is larger and approaches 1 asymptotically in the reaction-controlled regime where the intrapellet Damkohler number is small, and E decreases in the diffusion-controlled regime at large values of A a- These trends are verified by simulations provided in Table 19-1. 

Effect of the Damkohler Number on Conversion in Square Ducts. More conversion is predicted at higher Damkohler numbers because the rate of surface-catalyzed chemical reaction is larger. At a given axial position z, reactant conversion reaches an asymptotic limit in the diffusion-controlled regime, where oo. Actual simulations of I Abuik vs. f at /i = 20 are almost indistinguishable from those when p = 1000. The effect of p on bulk reactant molar density is illustrated in Table 23-5 for viscous flow in a square duct at = 0.20, first-order irreversible chemical reaction, and uniform catalyst deposition. These results in Table 23-5 for the parameter A, as a function of the Damkohler number p can be predicted via equations (23-80) and (23-81) when C = A and... 

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [55]. For a zero-order reaction, the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sh in circular tube reactors becomes Sh. = 4.36 [55]. The same value is obtained when reaction rates are low compared to the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the asymptotic value for Sh is = 3.66. As a consequence, the Sh in the reacting system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damkohler number defined in Equation 6.11. 

Fig. 11.1 Diffusion flame structure determined on basis of one-step kinetics in an asymptotic approximation to the kinetic activation I - reaction zone II - convection-diffusion zone Fp - fuel concentration Fq - oxidizer concentration Daoc - the Damkohler number corresponds to an infinitely fast reaction Daext - the Damkohler number when flame extinction takes place...

---