Second-order reactions effectiveness factor

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

If the two competing reactions have the same concentration dependence, then the catalyst pore structure does not influence the selectivity because at each point within the pore structure the two reactions will proceed at the same relative rate, independent of the reactant concentration. However, if the two competing reactions differ in the concentration dependence of their rate expressions, the pore structure may have a significant effect on the product distribution. For example, if V is formed by a first-order reaction and IF by a second-order reaction, the observed yield of V will increase as the catalyst effectiveness factor decreases. At low effectiveness factors there will be a significant gradient in the reactant concentration as one moves radially inward. The lower reactant concentration within the pore structure would then... 

What equation would have to be solved for the catalyst effectiveness factor for a second-order reaction What is the apparent aetivation energy of a second-order reaction in the limit of... 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first order, whereas a second-order reaction has an apparent order of 3/2 when strongly influenced by diffusional effects. Because k and n are modified in the diffusion controlled region then, if the rate of the overall process is estimated by multiplying the chemical reaction rate by the effectiveness factor (as in equation 3.8), it is imperative to know the true rate of chemical reaction uninfluenced by diffusion effects. 

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

A different approach to the estimation of the overall effectiveness factor in porous catalysts was recently presented by Ho et al. (1994). These authors analyze the case of parallel bimolecular reactions, a case that is in general not one of uniform kinetics. Rather than trying to solve the coupled set of differential equations, Ho et al. chose to search directly for upper and lower bounds to the overall effectiveness factor, which are found by reducing the problem to that of finding the effectiveness factor for a single second-order reaction. The bounds can be es-... 

Discuss how your answer would change if you had used the effectiveness factor for a second-order reaction rather than a first-order reaction. Discuss what you learned from this problem and what you believe to be the point of the probiem. 

Effect of reaction order on diffusion factor y. Calculation of the characteristic function of y applicable to the case of an n order reaction yields similar functional relationships, in which the modulus

concentration term. For example, the case of second-order reaction involves the modulus... 

The effective interfacial areas for absorption with a chemical reaction [6] in packed columns are the same as those for physical absorption except that absorption is accompanied by rapid, second-order reactions. For absorption with a moderately fast first-order or pseudo first-order reaction, almost the entire interfacial area is effective, because the absorption rates are independent of kL as can be seen from Equation 6.24 for the enhancement factor for such cases. For a new system with an unknown reaction rate constant, an experimental determination of the enhancement factor by using an experimental absorber with a known interfacial area would serve as a guide. 

The choice of reaction model does make a difference in the modeling of HMX combustion. Different reaction models generate different burning surface structure, surface temperature and as a result, the burning rate of energetic materials. A critical concentration is introduced to the model to deal with the special problem that first-order and second-order reaction have. The effect of choosing this critical concentration turns out to be a very important factor to determine how well the model can simulate the combustion. 

Effectiveness factor versus Thiele, modulus for different values of the Biot number second-order reaction in a cylindrical pellet. 429... 

For the second-order reaction, Equation 7.101 show s that varies with the molar flow, which means and r vary along the length of the reactor as Na decreases. We are asked to estimate the catalyst mass needed to achieve a conversion of A equal to 7S%. So for this particular example, 4> decreases from 6.49 to 3.24. As shown in Figure 7,9, we can approximate the effectiveness factor for the second-order reaction using the analytical result for the first-order reaction, Equation 7.42,... 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

For this second-order reaction, the effectiveness factor changes substantially from the inlet of the catalyst bed to the outlet, because the Thiele modulus depends on the concentration of A at the external surface of the catalyst particle Ca,s-... 

For each reaction, a kinetic expression (r was formulated as a function of the product composition (y, kinetic constant, and effectiveness factor. Product compositions were determined from bench-scale mass balances and simulated distillation curves. The hydrocracking of vacuum residue was considered to follow second-order reaction, while the remaining reactions were assumed to be first-order as reported in the literature (Sanchez and Ancheyta, 2007). On the basis of these considerations, the reaction rates of the proposed model are as follows ... 

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