Damkohler numbers second-order reaction

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 —...

Figure 11. Equivalence between the droplet diffusion model (81) and the IEM model for a zero-order reaction and a second-order reaction in a CSTR. The Damkohler numbers are such that f = 0.5 for perfect micromixing. The agreement is excellent for the second-order reaction, more approximate for the zero-order one.

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).

Figure 13 shows how the steady-state exit conversion X[— 1 — Cnm(z — 1)/C 5jii m] varies with the Damkohler number Da for different values of the dimensionless mixing time t](= tmix/r). The figure shows how non-uniform feeding could significantly reduce the conversion as compared to premixed feed for the case of a bimolecular second-order reaction (e.g. by a factor of 2 for the case of f] — 0.1), when mixing limitations are present in the system. 

Fig. 15. Variation of conversion with Damkohler number for a bimolecular second-order reaction for uniform and distributed feeding in a CSTR.

The accuracy of low-dimensional models derived using the L S method has been tested for isothermal tubular reactors for specific kinetics by comparing the solution of the full CDR equation [Eq. (117)] with that of the averaged models (Chakraborty and Balakotaiah, 2002a). For example, for the case of a single second order reaction, the two-mode model predicts the exit conversion to three decimal accuracy when for (j>2(— pDa) 1, and the maximum error is below 6% for 4>2 20, where 2(= pDd) is the local Damkohler number of the reaction. Such accuracy tests have also been performed for competitive-consecutive reaction schemes and the truncated two-mode models have been found to be very accurate within their region of convergence (discussed below). 

Figure 11.3 Isothermal external effectiveness factor as a function of the second Damkohler number and different reaction orders, n.

A minimum conversion ofX = 0.75 is required. For a second order reaction the conversion in an ideal plug flow reactor depends on the first Damkohler number. Dal, as shown in Equation 5.52 (see Chapter 2) ... 

The above expressions are inserted into the appropriate balance equations, for example, for tanks-in-series, segregated tanks-in-series, and maximum-mixed tanks-in-series models. The models are solved numerically [3], and the results are illustrated in the diagrams presented in Figure 4.29, which displays the differences between the above models for second-order reactions. The figure shows that the differences between the models are the most prominent in moderate Damkohler numbers (Figure 4.29). For very rapid and very slow reactions, it does not matter in practice which tanks-in-series model is used. For the extreme cases, it is natural to use the simplest one, that is, the ordinary tanks-in-series model. 

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

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... 

If the reaction is second-order and the numerical value of the Damkohler number Da is the same as in part 1, find the exit conversion using the solution of the nonlinear two point boundary value differential equation. 

Fig. 18. Variation of conversion (X) with the Damkohler number, Daa, for a bimolecular second-order wall-catalyzed reaction occurring in a tubular reactor.

The Damkohler is a dimensionless number that can give us a quick estimate of the degree of conversion that can be achieved in continuous-flow reactions. The Damkdhler number is the ratio of the rate of reaction of A to the rate of convective transport of A at the entrance to the reactor. For first- and second-order ineversible reactions the Damkohler numbers are... 

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,... 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

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. 

Quantitative results in Table 30-1 reveal that one achieves maximum conversion of reactants to products in ideal (i.e., 30%) and non-ideal (i.e., 25%) packed catalytic tubular reactors when the mass transfer Peclet number is approximately 6 for second-order irreversible chemical kinetics with an interpeUet porosity of 50%. Specific values for PeMT and the corresponding maximum conversion are sensitive to the simple mass transfer Peclet number and the chemical reaction coefficient, where the latter is defined by the product of the effectiveness factor, the interpeUet Damkohler number, and the catalyst filling factor. For example, when Pesimpie is 50 and the chemical reaction coefficient is 5 for second-order irreversible chemical kinetics, the critical value of PeMT [i e., (Re Sc)criacai] is approximately 30, whereas maximum conversion is obtained when PeMT is only 6. Hence, one concludes that the ideal simulations in Table 30-1 with a 0,... 

The external effectiveness factors as function of the second Damkohler number are obtained by solving Equation 2.141. This is done for reaction orders = 1, 2, i/2> and -1 and displayed in Figure 2.20 [27]. 

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [55]. For a zero-order reaction, the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sh in circular tube reactors becomes Sh. = 4.36 [55]. The same value is obtained when reaction rates are low compared to the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the asymptotic value for Sh is = 3.66. As a consequence, the Sh in the reacting system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damkohler number defined in Equation 6.11. 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

For a second-order irreversible reaction, the Damkohler number is... 

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