The Damkohler Number

We have seen that, if the dense phase is well-mixed and only the rate constant is a function of x, the performance of the fluid bed depends only on the modified Damkohler number, Da. If the Davidson number, Tr, depends on x, the linearity may still be exploited but we have to go back to equation (21), which may be rearranged to give 

Diffusion within the catalyst particle can be accounted for by using an effectiveness factor, tj, but x should no longer be defined to make k(x) linear in x. Rather, it would be sensible to make tjk linear in x. Of course, straightening out the monotone dependence on x of one parameter, distorts the distribution, as in equation (4), and it may be better to think of the Damkohler number as a function of x. We can alway write it as Da.u (x), where a (x) has a mean value of 1. 

If the catalyst decays as a known function of the length of time it spends in the reactive environment, the catalyst must be drawn off from the dense phase, reactivated and returned, so that the burden is constant. A steady state is quickly built up in which the fraction of catalyst of age in the interval (a, a + da) is (w/W).exp - (walW)da, where W is the weight of catalyst in the bed and w the rate of replenishment. If the decay is accounted for by saying that the rate constant should be multiplied by a factor, for example exp 

If the decay cannot be expressed solely as a function of age, the linearity is lost and the generalization to continuous mixtures is no longer possible. 

If the catalyst were not to decay but for some other reason, perhaps temperature control, the particles were taken out and recycled, then each might be supposed to be in pristine condition on entering the bed. Each particle would then undergo a transition during which the steady state profile of reactant within the particle would be built up. The analysis of Amundson and Aris (1962 this part is not tainted with the error mentioned above in fn. 13) may be used. We assume spherical particles of radius R, and call the profile of concentration at time a, c r, a). If D is the diffusivity of the reactant and k the rate constant per unit volume of catalyst, 

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Furthermore, analytical solutions of premixed edges were studied and the dependence of edge speed with the Damkohler number was reported [61], in which the edge speed ranged from positive to negative values, in a similar fashion as nonpremixed edge flames. 

With all three programs, vary the Damkohler number. Da over a range from 1 to 500 and explain the results. Note that when Da no longer has any influence on the rate then mixing must be controlling... 

In order to implement the PDF equations into a LES context, a filtered version of the PDF equation is required, usually denoted as filtered density function (FDF). Although the LES filtering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damkohler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re = 4,000. 

Now, if the system is observed over a distance scale L, the characteristic time frrans associated with advective transport is simply the time L/vx required for groundwater to traverse distance L. The Damkohler number, the ratio of the two characteristic times,... 

The time for spontaneous ignition to occur will decrease as the Damkohler number... 

The parameter term (kt), which is called the Damkohler Number Da, is dimensionless and is now the single governing parameter in the model. This results in a considerable model simplification because originally the three parameters, r, k and CA0, all appeared in the model equation. 

The chemical time scales ra and the mixing time scale can be used to define the Damkohler number(s) Da, = /x . Note that fast reactions correspond to large Da, and... 

To determine the dominant combustion mechanism for a given set of conditions, the Damkohler number (Da) is given as... 

The flame lift-off height, which is related to the ignition distance, was inversely affected by the excitation frequency. Since the flow time scale decreased with increasing frequency, the data were plotted as a function of the Damkohler number in Fig. 29.14, where the characteristic flow time scale was estimated by large-eddy turnover time as 1/17 and the characteristic chemical reaction time was computed using an ignition delay model [21] for ethylene jet. While the results did not show any evidence of critical Damkohler number, the range... 

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... 

The only two parameters appearing in eqn. (65) are the dispersion number, DjiiL, or inverse Peclet number, and the Damkohler number, or dimensionless rate group, t/jCa," - Solutions to eqn. (65) are therefore functions only of these two groups. If term (4) in eqn. (65) is absent, then... 

The Peclet number, uLjD, when written in the form (L /D)l(Llu) is seen to be a ratio of characteristic dispersion time to characteristic residence time and the Damkohler number can, in similar manner, be considered as a ratio of characteristic residence time, L/u, to characteristic reaction time, l/feCA " [59]. 

Perry and Lee [28,29] offer an enhancement of QPA, based upon use of dual heat flux sensors and additional thermocouples in autoclave curing. This enhancement entails determining heat transfer properties during the cure, then using these properties in conjunction with PID regulatory control of autoclave temperature. Using the additional sensors, Perry and Lee employ an on-line Damkohler number in lieu of the second time-derivative of temperature to avoid exothermic thermal runaway within the prepreg stack thermoset resin. The Damkohler number is defined as ... 

Moreover, the Damkohler number (Da) can be used to simplify the relations in Table 5.4 ... 

A high Damkohler number means that the global rate is controlled by mass transfer phenomena. So, the process rate can be rewritten in terms of the Damkohler number and the external effectiveness factor for each reaction order can be deduced, as shown in Table 5.5. In Figure 5.3, the external effectiveness factor versus the Damkohler number is depicted for various reaction orders. It is clear that the higher the reaction order, the more obvious the external mass transfer limitation. For Damkohler numbers higher than 0.10, external mass transfer phenomena control the global rate. In the case of n = 1, the external effec-... 

Different kinetic dimensionless numbers These can also be used depending on the conditions of the process. The most important is the Damkohler number. 

In a similar way the Damkohler Number (Eq. 22-1 lb) measures the relative pace of diffusion versus advection in the time period tT = k which corresponds to the mean life of the reactive chemical. In fact ... 

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