Damkohler I number

Damkohler I number. A further dimensionless number is the Damkohler I number (Boucher and Alves 1959 Lassey and Blattner 1988 Bowman and Willett 1991 Lasaga... 

Here the rate, of the k fluid-mineral exchange reaction is compared to the pore velocity of the fluid, Vp. A high Damkohler I number implies fast reaction kinetics... 

The summation included for completeness is over all mineral species k. It reduces to the last term for the case of a single fluid and mineral species containing oxygen. Figure 13 shows the solutions of the equation for different Peclet and Damkohler I numbers calculated for a fluid composed of water and a one-dimensional rock column. Bowman et al. (1994) presented many more examples. The curves shown in Figure 13a are solutions for an infinite Damkohler I number at a Peclet number of 100. In this case of rapid... 

From the dimensionless form of this equation, the Damkohler I number... 

In case of nonuniform residence times, a constant Damkohler I number can still be assured by also changing the reaction constant through temperature. 

Although Damkohler traced numbers I to IV back to the above combinations of named dimensionless numbers known at that time, numbers I to IV have come to be known as the four Damkohler numbers Da to DaIV in chemical literature. We will not identify them in this way, instead we will only refer to the new, genuine reaction kinetic pi-number... 

Figure 13. Normalized stable isotope composition of a rock column infiltrated by a reactive fluid, (a) The solution assumes local equilibrium, with a Peclet number (Npe) of 100 and infinite Damkohler 1 number, (b) solution calculated for the case of a Peclet number of 100 and a Damkohler 1 number (No) of 1. The dimensionless parameters are given by normalized concentration c = (5, - >j )/(8j -5y ) distance z=xlL dimensionless time T =i( I 3jL) (after Bowman and Willet 1991—note that captions for their Figures 1 and 3 are switched).

A plot of the conversion as a function of the number of reactors in series for a first-order reaction is shown in Figure 5-6 for various values of the Damkohler Economics number tA. Observe from Figure 5-6 that when the prcnluct of the space lime and the specific reaction rate is relatively large, say. Da I, approximately 90% conversion is achieved in two or three reactors thus the cost of adding subsequent reactors might not be justified. When the product tA is small. Da 0.1, the conversion continues to increase significantly with each reactor added. 

In any circumstances, it can be expected that and (5x are algebraic functions of turbulence length scale and kinetic energy, as well as chemical and molecular quantities of the mixture. Of course, it is expedient to determine these in terms of relevant dimensionless quantities. The simplest possible formula, in the case of very fast chemistry, i.e., large Damkohler number Da = (Sl li)/ SiU ) and large Reynolds Re = ( Ij)/ (<5l Sl) and Peclet numbers, i.e., small Karlovitz number Ka = sjRej/Da will be Sj/Sl =f(u / Sl), but other ratios are also quite likely to play a role in the general case. 

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

For large Damkohler numbers, the chemistry is fast (i.e., reaction time is short) and reaction sheets of various wrinkled types may occur. For small Da numbers, the chemistry is slow and well-stirred flames may occur. 

Dfc Damkohler number characterizing Kolmogorov-scale fluctuations D Damkohler number characterizing large-scale fluctuations I integral length scale... 

Many practical industrial processes are diffusion limited (i.e., have a high Damkohler number), and the assumption that the chemistry is fast is often sufficient to predict the overall characteristics of the process. For instance, in turbulent diffusion flames, the rates of fuel oxidation and heat release are often governed by the turbulent transport and mixing. 

Mathematicians have their names attached to theorems, lemmas and even conjectures naturalists have species and genera called after them physicists have their principles and chemists their reactions and reagents. Engineers can claim the dimensionless parameters, for they, who, in their function as designers, have often to work with units, appreciate more than anyone the conceptual beauty of the dimensionless number, invested as it is with full contextual meaning and magnitude. I submit that it would be more than appropriate to call this transfer parameter the Davidson number, for who has done more to elucidate the mechanism of this transfer process than John Davidson Damkohler has preempted the initial letters Da, but that is no matter, for transfer gives us Tr and we immediately think of Trinity and its present Vice-Master. 

Calculations were carried out for different values of the Damkohler Number (i.e., for various reactor lengths) and various concentrations of the activator [A], which determines the number of active centers and therefore the final degree of polymerization. The ratio Da/Da was used, as a... 

This Damkohler criterion is the Damkohler number of type I (Dof). Other Damkohler numbers were defined [12] type II used to characterize the material transport at the surface of a solid catalyst, type III used to characterize the convective heat transport at the catalyst surface, and type IV used to characterize the temperature profile in a solid catalyst. For batch reactions, the reaction time xr is defined at a reference temperature, the cooling medium temperature, as... 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

Damkohler s original analysis [2] resulted in four dimensionless numbers which today are referred to as Damkohler numbers I through to IV and are given by... 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

In Fig. 4.7, the PSPS is depicted together with the kinetic surfaces at four selected reboiler Damkohler numbers. At nonreactive conditions (i.e., Dar= 0 Fig. 4.7(a)),... 

Fig. 4.8(b)). At Damkohler numbers Dac> 0.085 and Dar> 0.166, pure isobutene and pure MeOH are feasible top and bottom products, respectively. At Dar< 0.166, both pure MeOH and a kinetic azeotrope (i.e., the mixture on the branch from MTBE to the pinch point) are possible bottom products, while another kinetic azeotrope (i.e., the mixture on the branch between isobutene and the nonreactive azeotrope isobutene-MeOH) is the possible top product. 

Other criteria can be used to establish the extinction condition and that are partially equivalent to the critical Damkohler number. Such criteria are a critical mass transfer numbers (BCI) [21,32], critical mass flux of fuel [2,6,28] or critical temperatures (Ta) [2,5,29-31], The critical mass transfer number has a direct influence over the flame temperature, and thus, represents the link between the condensed phase (i.e., production of fuel) and the chemical time. The critical mass flux operates under the same principle, but assumes a consistent heat input. Combustion reactions generally have high activation energy, therefore, the reaction can be assumed to abruptly cease when the temperature reaches a critical value (Tcr). 

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