Damkohler first

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

This equation is a general result first derived by Damkohler (55). It is applicable for any form of the reaction rate expression since this quantity was eliminated through the require-... 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

The first expression here is very similar to the Damkohler result for A and B equal to 1. Since the turbulent exchange coefficient (eddy diffusivity) correlates well with IqU for tube flow and, indeed, /0 is essentially constant for the tube flow characteristically used for turbulent premixed flame studies, it follows that... 

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 —...

General discussions of several aspects concerning the treatment of chemical reactions with diffusion are given by Damkohler (D2), Horn and Kiichler (H12), Prager (P7), Schoenemann and Hofmann (SlO), and Trambouze (Til). Corrsin (C21) has discussed the effects of turbulence on chemical reactions from the fundamental point of view of turbulence theory. We will first discuss the application of each type of model to chemical reactors. Then a short comparison will be made between the different approaches. 

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... 

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,... 

If the reaction is first-order, what is the value of the Damkohler number Da in order to achieve a conversion of 0.75 at the exit of the reactor ... 

The ratio of reactor and reaction timescale is the (dimensionless, of course) Damkohler number of the first kind Da] = k-t (for a first-order reaction). In a... 

Fig. 12.1. Dependence of conversion on Damkohler number for a) a first-order reaction (Eq. (13)) b) a second-order reaction (Eq. (14)) and c) a second-order reaction with two reactants and a nonstoichiometric feed composition (Eqs. (15) and (16), here for X = 0.5).

The first Damkohler number quantifies the ratio between the heat transferred from the flame (conduction time) and the energy required to heat the reactants to the ignition temperature (residence time). Extinction will occur when heat cannot be transferred fast enough. Equation 3.53 is written in terms of the ratio between conduction and convection (residence time), but in a more general form, could include all forms of heat transfer like gas-phase radiation. The second Damkohler number indicates if the reaction has sufficient time to proceed. In general, extinction is attained when either of the Damkohler groups is reduced below unity. 

Here, we are concerned with intraparticle diffusion controlling a first-order readsorption reaction therefore, the Damkohler number (15), which reflects the ratio of reaction to diffusion rates within catalyst particles, is given by ... 

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. 

Little is known about the kinetics of the bioprocesses. A reasonable assumption is that the reaction rates are proportional to the amount of micro-organisms catalyzing the reactions. The influence of the other reactants is more complex, for example a nutrient in high concentration often has an inhibiting effect. Moreover, factors such as pH, salt concentrations, temperature, can have effects that are difficult to quantify. For this reason, we assume first-order kinetics and include all the other factors influencing the process rate in two Damkohler numbers. The following dimensionless reactor model is obtained ... 

A simple model of the bioreactor was used, assuming first-order kinetics with respect to FeEDTA species and lumping in two Damkohler numbers the effects of various factors such as micro-organisms concentration, pH, salt concentrations and temperature. 

Historical credit goes to Gerhard Damkohler (1908-1944) who was the first to use the theory of similarity [113] to investigate a chemical process in conjunction with mass and heat transfer. In a purely theoretical way he examined the conditions under which scale-up would be possible in the case of (inevitably) partial similarity and he... 

When the process pN- n-> pv in the mobile phase or stationary phase can be represented by first-order or pseudo-first-order interconversion kinetics and as a reversible binding event, the resolution of the interconverting species can be evaluated319 by treating the column as a chemical reactor with properties specified by the corresponding Damkohler number Da and the corresponding interconversion rate constants derived. Thus,... 

We first analyze the case where the mixing times of A and B are equal, i.e. rjA — riB — ri. Figure 11 illustrates how the conversion X(— 1 — CA>m(z — 0)/ CA m iC) varies with the Damkohler number Da for different values of the dimensionless mixing time r, for the case of stoichiometric feeding of reactants. 

The lack of attention to entropy relations in Brunauer s book is easy to understand. Indeed, it was only in 1943 that the first paper appeared in which an attempt was made to calculate and define entropies of adsorption It was only a rough attempt by Damkohler and Edse (3). The concepts, including a defini-... 

Chemical similarity in the diffuser section of the ACR can thus be defined in terms of the remaining (first) Damkohler number, Da/ =... 

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