Damkohler number critical values

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... 

Other criteria can be used to establish the extinction condition and that are partially equivalent to the critical Damkohler number. Such criteria are a critical mass transfer numbers (BCI) [21,32], critical mass flux of fuel [2,6,28] or critical temperatures (Ta) [2,5,29-31], The critical mass transfer number has a direct influence over the flame temperature, and thus, represents the link between the condensed phase (i.e., production of fuel) and the chemical time. The critical mass flux operates under the same principle, but assumes a consistent heat input. Combustion reactions generally have high activation energy, therefore, the reaction can be assumed to abruptly cease when the temperature reaches a critical value (Tcr). 

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. 

Again, there is a main transition at a critical Damkohler number, Da = Dac. For smaller values of Da the initial perturbation is quickly diluted and the activator decays to the C state, as in the bistable case. The same behavior is observed when the initial perturbation is not sufficiently large. For Da > Dac the perturbation grows as in the bistable case, forming a growing filament that eventually fills the whole system (in the closed flow case), or covers the unstable manifold of the chaotic saddle (in open flows). The filament consists now of a pulse of the C concentration, with a maximum close to the excited state, and accompanied by a smaller pulse of C2. In the closed... 

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,... 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

Notice that the molar density of reactant A does not decrease to zero at the center of the catalyst, where t] = 0, when the intrapellet Damkohler number is below its critical value. In fact. 

It is only necessary to solve this equation for values of tjcnticai between 0 and 1. Obviously, ciiticai is a function of the intrapellet Damkohler number, but an explicit analytical function for ( critical = /(A) is not possible. If A or A is incremented from its critical value to extremely large values in the diffusion-limited regime, then a Newton-Raphson root-finding approach can be implemented to find the reahstic root for ( critical at each value of the intrapellet Damkohler number (see Table 16-1). 

When the intrapellet Damkohler number is less than its critical value (i.e., /6), the critical dimensionless spatial coordinate Jjcnticai is negative, and boundary condition 2b must be employed instead of 2a. Under these conditions, the dimensionless molar density profile for reactant A within the catalytic pores is adopted from equation (16-24) by setting //criucai to zero. Hence,... 

REDEFINING THE INTRAPELLET DAMKOHLER NUMBER SO THAT ITS CRITICAL VALUE MIGHT BE THE SAME FOR ALL PELLET GEOMETRIES... 

This is an interesting challange from the standpoint of developing geometry-insensitive universal correlations for all catalyst shapes. As illustrated above, the critical value of the intrapellet Damkohler number is... 

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... 

A summary of the final results follows. When L = Vcataiyst/5 extemab critical values of the intrapellet Damkohler number are as follows ... 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

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. 

It is not necessary to introduce a critical spatial coordinate because the rate of disappearance of reactant A is extremely small when its molar density approaches zero in the central core of the catalyst at large values of the intrapellet Damkohler number. One-dimensional diffusion and first-order irreversible chemical reaction in rectangular coordinates is described mathematically by a frequently occurring... 

Problem. Consider zeroth-order chemical kinetics in pellets with rectangular, cylindrical and spherical symmetry. Dimensionless molar density profiles have been developed in Chapter 16 for each catalyst geometry. Calculate the effectiveness factor when the intrapellet Damkohler number is greater than its critical value by invoking mass transfer of reactant A into the pellet across the external surface. Compare your answers with those given by equations (20-50). 

Catalysts with Cylindrical Symmetry. This analysis is based on the mass transfer equation with diffusion and chemical reaction. Basic information has been obtained for the dimensionless molar density profile of reactant A. For zeroth-order kinetics, the molar density is equated to zero at the critical value of the dimensionless radial coordinate, criticai = /(A). The relation between the critical value of the dimensionless radial coordinate and the intrapeUet Damkohler number is obtained by solving the following nonlinear algebraic equation ... 

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 seven different values of the interpellet Damkohler number for which real and ideal packed catalytic tubular reactor performance is summarized in Table 22-2, it is possible to identify a critical value of the mass transfer Peclet number (Re Sc)cnticai, above which the effects of interpellet axial dispersion are insignificant for second-order irreversible chemical kinetics. For example, if ideal performance is justified when the outlet conversion of reactants under real and ideal conditions differs by less than 0.5%,... 

TABLE 22-3 Effect of the Interpellet Damkohler Number on the Critical Value of the Mass Transfer Peclet Number for Second-Order Irreversible Chemical Kinetics in Packed Catalytic Thbular Reactors"... 

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