Damkohler number, equation

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. 

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. 

A notable aspect of this equation is that L appears within it as prominently as the rate constant k+ or the groundwater velocity vx, indicating the balance between the effects of reaction and transport depends on the scale at which it is observed. Transport might control fluid composition where unreacted water enters the aquifer, in the immediate vicinity of the inlet. The small scale of observation L would lead to a small Damkohler number, reflecting the lack of contact time there between fluid and aquifer. Observed in its entirety, on the other hand, the aquifer might be reaction controlled, if the fluid within it has sufficient time to react toward equilibrium. In this case, L and hence Da take on larger values than they do near the inlet. 

In dimensionless terms, there is a critical value for S (Damkohler number) that makes ignition possible. From Equation (4.23), this qualitatively means that the reaction time must be smaller than the time needed for the diffusion of heat. The pulse of the spark energy must at least be longer than the reaction time. Also, the time for autoignition at a given temperature T is directly related to the reaction time according to Semenov (as reported in Reference [5]) by... 

Applying Equations (5.21) to the adiabatic time corresponding to the critical Damkohler number, and realizing for a three-dimensional pile of effective radius, r,. A 3 (e.g. Sc = 3.32 for a sphere for Bi —> oo), then we estimate a typical ignition time at 6 = Sc 3 of... 

Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers... 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

The apparent reaction rate depends on the magnitude of Damkohler number (Da) as defined by Equation 7.14 - that is, the ratio of the maximum reaction rate to the maximum mass transfer rate. 

Figure 10.4 [1] shows the results for theoretical calculations [4] for the ratio n, the number of viable cells leaving the holding section of a continuous sterilizer, to Hq, the number of viable cells entering the section, as a function of the Peclet number (Pe), as defined by Equation 10.7, and the dimensionless Damkohler number (Da), as defined by Equation 10.8 ... 

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

However, if q is to be varied, Eq. (12) will not do, because q appears also in the dimensionless time r. Thus k must be used to make time dimensionless, for instance, r = kt. Da is still the Damkohler number, but the equation is now... 

The equations studied by Uppal, Ray, and Poore were written in terms of a dimensionless conversion, x, and a dimensionless temperature, x2, with origin at feed conditions and scaled by the dimensionless activation energy, y. This left three parameters Da, the Damkohler number, the ratio of reaction rate to flow rate B, a dimensionless heat of reaction /3, a dimensionless heat transfer coefficient and x2c, a dimensionless coolant temperature. The equations were... 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

We wish to see what the overall conversion of a continuous mixture will be, but, first, we have to ask which parameters will depend on jc, the index variable of the continuous mixture. Clearly k the rate constant in the Damkohler number will be a function of jc, and, if monotonic, can be put equal to Da.x. The parameter /3 is clearly hydrodynamic and so, for the most part, are the terms in the Davidson number. The only term in the equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity,... 

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