Damkohler’s number

A fundamental aspect in the MR design is the eflect of Damkohler s number -the ratio between reaction rate at the reactor entrance and feed flow rate - on the MR performances ... 

Figure 12.6 CO conversion and H2 recovery profiles as a function of reactor length at different Damkohler s numbers. Furnace temperature = 280 °C. Feed pressure = 1000 kPa. Permeate pressure = 100 kPa.

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... 

In Damkohler s analysis, which applied to a continuous chemical reaction process in a tubular reactor, he solved these dilemmas by completely abandoning geometric similarity and fluid dynamic similarity. In other words, L/D idem and assuming that the Reynolds number is irrelevant in the scaling. Hence, his scale-up depends exclusively on thermal and reaction similarity. In our case it is even easier to see that the Reynolds number is very small and does not play a role in the process. By allowing to adjust L/D accordingly, there is more flexibility in the scaling problem. 

Fig. 80 External catalyst effectiveness factor tie as a function of the Damkohler-ll number DaM s k/ kGa and the reaction order m from [118].

Note. V is the molar volume, JVyi is Advogadro s number, is the equilibrium concentration, D is the diffusion coefficient, sub-s surface, hHfiB the heat of fusion, t) is the Damkohler number. Ah is the thermal conductivity, i die area shape factor for surface nuclei , y, is the distance between steps, n is the equilibrium surface concentration, p = 1 - o-JS is one minus the maximum surface supersaturation divided by the solution supersaturation, and p is the density. ihG - pl- fPMpAkBT In S)... 

S-shaped flame temperature response with Damkohler number exhibiting edge propagation characteristics. (From Kim, J. and Kim, J.S., Combust. Theory Model, 10, 21,2006.)... 

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 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... 

Barlow, R. S., R. W. Dibble, J.-Y. Chen, and R. P. Lucht (1990). Effect of Damkohler number on superequilibrium OH concentration in turbulent nonpremixed jet flames. Combustion and Flame 82, 235-251. 

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 —...

Table 7 presents Hilder s data for the residual concentration in a PFR and in a laminar flow reactor, as predicted by both eqns. (56) and (57) over a wide range of Damkohler numbers. 

The center manifold approach of Mercer and Roberts (see the article Mercer and Roberts, 1990 and the subsequent article by Rosencrans, 1997) allowed to calculate approximations at any order for the original Taylor s model. Even if the error estimate was not obtained, it gives a very plausible argument for the validity of the effective model. This approach was applied to reactive flows in the article by Balakotaiah and Chang (1995). A number of effective models for different Damkohler numbers were obtained. Some generalizations to reactive flows through porous media are in Mauri (1991) and the preliminary results on their mathematical justification are in Allaire and Raphael (2007). 

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. 

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) ... 

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... 

The first inequality characterizes recycle systems with reactant inventory control based on self-regulation. It occurs because the separation section does not allow the reactant to leave the process. Consequently, for given reactant feed flow rate F0, large reactor volume V or fast kinetics k are necessary to consume the whole amount of reactant fed into the process, thus avoiding reactant accumulation. The above variables are grouped in the Damkohler number, which must exceed a critical value. Note that the factor z3 accounts for the degradation of the reactor s performance due to impure reactant recycle, while the factor (zo — z4) accounts for the reactant leaving the plant with the product stream. 

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