Damkohler number, definition

The maximum effective reaction rate is obtained for the limiting value of cs = 0. This means that the product of the effectiveness factor and the second Damkohler number can never exceed unity. A comparison of the definition of the Weisz modulus (eq 56) with the definition of Dan (eq 78) gives the equivalence... 

The approach that has been described here is an example of a perturbation method for large Damkohler numbers and may be termed Damkohler-mimber asymptotics. It has been developed on the basis of an expansion that does not distinguish among special zones within the flame. It is possible that the Damkohler-number expansion will often be good in the hot reaction zone but poor elsewhere, while radical distributions away from the hot reaction zone have relatively little influence on the main characteristics of the flame. Under these circumstances, an approach based on matched asymptotic expansions, treating different zones differently, may be helpful. Sharper definitions of values of Q consistent with the steady-state approximation (in the zone where it is applied) might thereby be developed. 

Ratios of fluid-dynamical or transport times to chemical times are Damkohler numbers (see Sections 3.4 and 5.4) that are of relevance in identifying regimes of reacting flows. One definition of a Damkohler number... 

Flow rate (f 1) and composition (ci) of the reactor-inlet depend on the flow rate (F3) and composition (C3) of the recycle, and therefore cannot be used as references. For this reason, the dimensionless quantities are defined using the flow rate (Fo) and concentration (cq) at the plant inlet. The plant Damkohler number obtained in this way is different from the classical definition that makes use of reactor inlet as reference. 

The dimensionless spatial coordinate rj is measured in the thinnest dimension of rectangular catalysts. For cylindrical and spherical catalysts, r] is measured in the radial direction. The characteristic length L which appears in the intrapellet Damkohler number and is required to make the spatial coordinate dimensionless (i.e., rj = spatial coordinate/L) is one-half the thickness of catalysts with rectangular symmetry, measured in the thinnest dimension the radius of long cylindrical catalysts or the radius of spherical catalysts. q A is the molar density of reactant A divided by its value in the vicinity of the external surface of the catalyst, CAsurf- Hence, by definition, q A = 1 at r = 1. 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

Calculate the intrapellet Damkohler number when ijcnticai = 0, which corresponds to the largest value of A that is consistent with the presence of reactant A throughout the catalyst. This is the definition of the critical intrapellet Damkohler number, Acnticai- At higher values of A, reactant A... 

What is the critical value of the intrapeUet Damkohler number for onedimensional diffusion and zeroth-order irreversible chemical reaction in catalytic pellets with spherical symmetry The radius of the sphere is used as the characteristic length in flie definition of the Damkohler number. 

Figure 20-1 Dimensionless correlations between the effectiveness factor and the intrapellet Damkohler number for radial diffusion and wth-order irreversible chemical kinetics in long porous cylindrical catalysts (i.e., n = 0,1). The quantity on the horizontal axis is A, not A. The cylindrical radius R is the characteristic length in the definition of A.

Using log-log coordinates, graphs are provided in Figure 20-3 through Figure 20-6 which illustrate the effectiveness factor versus intrapellet Damkohler number (i.e.. A) for nth-order irreversible chemical reactions, where n = 0,1,2. Each graph corresponds to catalysts with different symmetry, and contains data for three different reaction orders. The characteristic length L in die definition of the Damkohler number is ... 

Obtain an analytical expression for the effectiveness factor (i.e., E vs. tjcriticai) in Spherical catalysts when the chemical kinetics are zeroth-order and the intrapeUet Damkohler number is greater than its critical value. Use the definition of the effectiveness factor that is based on mass transfer via diffusion across the external surface of the catalyst. 

At high-mass-transfer Peclet numbers, sketch the relation between average residence time divided by the chemical reaction time constant (i.e., r/co) for a packed catalytic tubular reactor versus the intrapeUet Damkohler number Aa, intrapeiiet for zeroth-, first-, and second-order irreversible chemical kinetics within spherical catalytic pellets. The characteristic length L in the definition of Aa, intrapeiiet is the sphere radius R. The overall objective is to achieve the same conversion in the exit stream for all three kinetic rate laws. Put all three curves on the same set of axes and identify quantitative values for the intrapeiiet Damkohler number on the horizontal axis. 

For the design of RD processes, besides information on the reaction, information on phase equUibria is of prime importance, especially on vapor-liquid equilibria and in some cases also on liquid-liquid equilibria (see above). The systematic investigation of phase equUibria for the design of RD processes will generally involve also studies of reactive systems (see examples above). Studies of phase equUibria in reactive systems generally pose no problem if the reaction is either very fast or very slow as compared with the time constant of the phase equilibrium experiment (high or low Damkohler number Da). In the first case, the solution will always be in chemical equUibrium, in the second case, no reaction will take place. The definition of the time constant of the phase equilibrium experiment win depend on the type of apparatus used. If the RD process is catalyzed and the catalyst does not substantially influence the phase equilibrium, the phase equilibrium experiments can often be performed without catalyst and again no or only little conversion will take place. 

According to this definition, Damkohler number gives the ratio of the characteristic process time to the characteristic reaction time (Doherty and Malone, 2001) and captures effectively the major dependence of conversion, production rate and product purities on the feed flowrate, catalyst level and holdup (Chen et al., 2002). For an open... 

The definition of Damkohler number implies that no reaction takes place if Da << 1 and chemical equilibrium limit is reached if Da >> 1. 

B. If the reaction is second order and the numerical value of Da (Damkohler number) is the same as in part 1 (although its definition is slightly different), find the exit conversion using the Fox s iterative method (explain your formulation of adjoint equations for the iterative solution of the nonlinear two-point boundary-value differential equation). 

Here v and d represent dimensionless convection velocity and dimensionless thermal dif-fusivity, resp. (axial mass dispersion is neglected) B is dimensionless reaction enthalpy. Da is the Damkohler number and Le is the Lewis number. We have avoided the conventional introduction of the Peclet number since we would like to examine the effects of convection and thermal diffusion separately. Otherwise, the scaling of the variables and definitions of the dimensionless quantities are conventional, see e.g. (Nekhamkina et al. (2000)). 

The designer should be aware that there is a critical reactor volume, which generally corresponds to a bifurcation point of the mass balance equations. For stable operation the reactor should be larger than this critical value. As example, for essentially first-order reaction with pure product and recycle, the feasibility condition is simply Da>. The definition of the plant Damkohler number includes reactor volume, reaction kinetics and fresh reactant feed flow rate. Similar expressions hold for more complex stoichiometry. 

We will use these characteristic times to define some dimensionless numbers, such as Damkohler number, Thiele modulus, and Hatta modulus. Here, we will give the broad definitions, and the true meanings will be clear as we move along the chapter ... 

However, observing the definition of Damkohler and Peclet numbers both have the same order of magnitude. The Da number relates reaction and diffusion rates, while Pe relates convective and diffusion rates. 

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