Reaction Damkohler number

For a first-order reactioit the product xk is often referred to as the reaction Damkohler number. 

Da Second Damkohler number K l2 ID K = first-order reaction rate constant l = characteristic length D = diffusion coefficient... 

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. 

Where dissolution or precipitation is sufficiently rapid, the species concentration quickly approaches the equilibrium value as water migrates along the aquifer the system is said to be reaction controlled. Alternatively, given rapid enough flow, water passes along the aquifer too quickly for the species concentration to be affected significantly by chemical reaction. The system in this case is transport controlled. The relative importance of reaction and transport is described formally by the nondimensional Damkohler number, written Da. 

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... 

The relative importance of reaction with respect to diffusion can be described in terms of the nondimensional (second) Damkohler number [30-36] (also called Thiele modulus), in terms of the reaction layer thickness [37,38] or in terms of lability criteria [39,40]. 

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... 

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. 

In the reaction region the chemical source terms appear. The structure of this region is illustrated in Fig. 25.3, where it is seen that the limit 6 < p, < e is the one considered here. All three of these small parameters are related to appropriate Damkohler numbers [44]. The RRA analysis [44] results in predictions of peak temperature as a function of strain rate, shown in Fig. 25.4. The excellent agreement here is important for being able to calculate contaminant production with good accuracy. Figure 25.5 shows the sufficient agreement obtained for important radicals as well. 

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... 

Fig. 10. Fractional conversion versus Damkohler number for half,-, first- and second-order reactions taking place in a single ideal CSTR. Shaded areas represent possible conversion ranges lying between perfectly micromixed flow (M) and completely segregated flow (S). Data taken from reference 32. A = R —...

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]. 

Balakotaiah and Chakraborty introduce a four-mode hyperbolic model but with non-linear reactions, in three-dimensional geometry and with much bigger Damkohler number (Balakotaiah, 2004 Chakraborty and Balakotaiah, 2005). The effective model cannot be directly compared with our system (12)-(13). Nevertheless, in Section 3.1 we derive a four-mode hyperbolic model, analogous to the models from Balakotaiah (2004) and Chakraborty and Balakotaiah (2005). We show that it is formally equivalent to our model at the order This shows the relationship between the upscaled models... 

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. 

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-... 

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 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.

First, recall that the nondimensional Damkohler number, Da (Eq. 22-11 b), allows us to decide whether advection is relevant relative to the influence of diffusion and reaction. As summarized in Fig. 22.3, if Da 1, advection can be neglected (in vertical models this is often the case). Second, if advection is not relevant, we can decide whether mixing by diffusion is fast enough to eliminate all spatial concentration differences that may result from various reaction processes in the system (see the case of photolysis of phenanthrene in a lake sketched in Fig. 21.2). To this end, the relevant expression is L (kr / Ez)1 2, where L is the vertical extension of the system, Ez the vertical turbulent diffusivity, and A, the first-order reaction rate constant (Eq. 22-13). If this number is much smaller than 1, that is, if... 

Note that for the special case kTW= kr s, k is krw/fw. The extra factor (f yl compensates for the fact that in Eq. 25-46 k is multiplied only with the dissolved BC concentration, whereas here we assume that the hydrolysis affects the total (dissolved and sorbed) BC concentration. The Peclet Number (Eq. 25-13, with x0 = 30 m) is affected neither by sorption nor by reaction Pe = 10. The Damkohler Number is (Eq. 25-48) ... 

Develop and discuss a set of boundary conditions to solve the Graetz problem. Take particular care with the effects of surface reaction, balancing heterogeneous reaction with mass diffusion from the fluid. A second Damkohler number should emerge in the surface boundary condition,... 

Here rmix is a characteristic mixing time for the system, while rreac is a characteristic time for the chemical reaction. If the Damkohler number has a value much larger than unity, the process is diffusionally controlled a value much lower than 1 indicates a kinetically controlled process. The terms in the Damkohler number may be defined in different ways, according to the physical characteristics of the system of interest. 

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... 

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