Multiple reactions effectiveness factor

Reactions in porous catalyst pellets are Invariably accompanied by thermal effects associated with the heat of reaction. Particularly In the case of exothermic reactions these may have a marked influence on the solutions, and hence on the effectiveness factor, leading to effectiveness factors greater than unity and, In certain circumstances, multiple steady state solutions with given boundary conditions [78]. These phenomena have attracted a great deal of interest and attention in recent years, and an excellent account of our present state of knowledge has been given by Arls [45]. 

The number of active sites is a multiplicative factor in the rate of the main reaction. See for example Equations (10.11) and (10.16). Thus, the decline in reaction rate can be modeled using a time-dependent effectiveness. A reasonable functional form for the time-dependent effectiveness factor is... 

Some of these effects have an enhancing influence on the overall rate of reaction. Others will have a detrimental effect. The relative magnitude of these effects will depend on the system in question. To make matters worse, if multiple reactions are being considered that are reversible and that also produce by products, all of the factors discussed in Chapter 6 regarding the influence of temperature also apply to gas-liquid reactions. 

Practical design problems may need to take into account many additional factors, including the recycle of some reactants (such as hydrogen), residence time distribution, inhomogeneity of the packing, multiple reactions, approach to equilibria, and so on. All of these problems have been encountered before, and professional simulator routines for solving them are versatile, effective and as reliable as the data provided to them. At least half a dozen such computer packages are commercially available. 

In practice, most industrial processes are staged with multiple reaction processes and separation units as sketched in Figure 4-15. A is the key raw material and is the key product, it is clear that many factors must be included in designing the process to maximize the yield of E. The effectiveness of the separations are obviously critical as well as the kinetics of the reactions and the choice of reactor type and conversion in each reactor. If separations are perfect, then the yields are equal to the selectivities, so that the overall... 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

Whether or not multiple steady states will appear, and how large the deviation of the effectiveness factors between both stable operating points will be, is determined by the values of the Prater and Arrhenius numbers. Effectiveness factors above unity generally occur when p > 0 (exothermal reactions). However, for the usual range of the Arrhenius number (y = 10-30), multiple steady states are possible only at larger Prater numbers (see Fig 13). For further details on multiple steady states, the interested reader may consult the monograph by Aris [6] or the works of Luss [69, 70]. 

This tutorial paper begins with a short introduction to multicomponent mass transport in porous media. A theoretical development for application to single and multiple reaction systems is presented. Two example problems are solved. The first example is an effectiveness factor calculation for the water-gas shift reaction over a chromia-promoted iron oxide catalyst. The methods applicable to multiple reaction problems are illustrated by solving a steam reformer problem. The need to develop asymptotic methods for application to multiple reaction problems is apparent in this example. 

Most textbook discussions of effectiveness factors in porous, heterogeneous catalysts are limited to the reaction A - Products where the effective diffusivity of A is independent of reactant concentration. On the other hand, it is widely recognized by researchers in the field that multicomponent single reaction systems can be handled in a near rigorous fashion with little added complexity, and recently methods have been developed for application to multiple reactions. Accordingly, it is the intent of the present communication to help promote the transfer of these methods from the realm of the chemical engineering scientist to that of the practitioner. This is not, however, intended to be a comprehensive review of the subject. The serious reader will want to consult the works of Jackson, et al. 

For exothermic reactions (fi > 0) a sufficient temperature rise due to heat transfer limitations may increase the rate constant Ay. and this increase may offset the diffusion limitation on the rate of reaction (the decrease in reactant concentrations CA), leading to a larger internal rate of reaction than at surface conditions CAs. This, eventually, leads to 17 > 1. As the heat of reaction is a strong function of temperature, Eq. (9.24) may lead to multiple solutions and three possible values of the effectiveness factor may be obtained for very large values of /I and a narrow range of catalytic reactions, (3 is usually <0.1, and therefore, we do not observe multiple values of the effectiveness factor. The criterion... 

With n = 2, we have a quadratic expression for the rate of reaction, which can display multiple steady states with high values of effectiveness factor for exothermic autocatalytic reactions. 

Note that the weighting factors correspond to the common H-isotopomer in the H/D and H/T isotope effects, and expressions for w are different for the serial and parallel cases. The equations in Eq. (11.11), developed by Northrop [49, 50], have proved to be useful in obtaining intrinsic kinetic isotope effects in enzymatic reactions with multiple rate-limiting steps that partially mask the full effect of an isotope-sensitive step. 

Multiplicity of the steady states has not been observed in the range of the normal operating conditions reported in the above results, however it was found in other reaction conditions at very low flow rate and higher o-Xylene concentration (50kg m. hr x -a = 0.02 Figure 5.25). Also it was noted that the phthalide negative effectiveness factor disappeared in these cases (Figure 5.26). 

Another interesting phenomenon can emerge under non-isothermal conditions for strongly exothermic reactions there will be multiple solutions to the coupled system of energy and mass balances even for the simplest first-order reaction. Such steady-state multiplicity results in the existance of several possible solutions for the steady state overall effectiveness factor, usually up to three with the middle point usually unstable. One should, however, note that the phenomenon is, in practice, rather rarely encountered, as can be understood from a comparison of real parameter values (Table 9.2). 

In the previous examples, we have exploited the idea of an effectiveness factor to reduce fixed-bed reactor models to the same form as plug-flow reactor models. This approach is useful and solves several important cases, but this approach is also limited and can take us only So far. In the general case, we must contend with multiple reactions that are not first order, nonconstant thermochemical properties, and nonisothermal behavior in the pellet and the fluid. For these cases, we have no alternative but to solve numerically for the temperature and species concentrations profiles in both the pellet and the bed. As a final example, we compute the numerical solution to a problem of this type. 

For the single-reaction, nonisothermal problem, we solved the so-called Weisz-Hicks problem, and determined the temperature and concentration profiles within the pellet. We showed the effectiveness factor can be greater than unity for this case. Multiple steady-state solutions also are possible for this problem, but for realistic values of the... 

The dimensionless scaling factor in the mass transfer equation for reactant A with diffusion and chemical reaction is written with subscript j for the jth chemical reaction in a multiple reaction sequence. Hence, A corresponds to the Damkohler number for reaction j. The only distinguishing factor between all of these Damkohler numbers for multiple reactions is that the nth-order kinetic rate constant in the 7th reaction (i.e., kj) changes from one reaction to another. The characteristic length, the molar density of key-limiting reactant A on the external surface of the catalyst, and the effective diffusion coefficient of reactant A are the same in all the Damkohler numbers that appear in the dimensionless mass balance for reactant A. In other words. 

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