Steady States in a Catalyst Pellet

Example 3.2.5. Multiple Steady States in a Catalyst Pellet 

The catalyst pellet problem solved in example 3.2.2 is solved here using the shooting technique. The Maple program is given below  

The governing equation is entered here (after substituting the parameter values)  

The boundary value problem has multiple solutions. The solution obtained depends on the initial guess provided for a. An initial guess of 0.9 is given  

For this example, the error is calculated based on the boundary condition at x = 1. 

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Example 3.2.13. Multiple Steady States in a Catalyst Pellet - T] vs. 3>... 

In this chapter, nonlinear boundary value problems were solved numerically. In section 3.2.2, series solutions were derived for nonlinear boundary value problems. This is a powerful technique and is even capable of predicting multiple steady states in a catalyst pellet. However, these series solutions should be used cautiously. The convergence of the solution is not guaranteed and should be verified. This can be done by increasing the number of terms in the series and plotting the profiles. 

In section 3.2.7, boundary value problems were solved as initial value problems. This methodology is especially useful for predicting the performances in chemical reactors. Maple s stop condition was used in this section to obtain t] vs. O curves. This is very useful because, it is generally easier to solve an initial value problem than a boundary value problem. This technique was then used in section 3.2.8 to predict multiple steady states in a catalyst pellet in section 3.2.8. This methodology is extremely useful for predicting the hysteresis curves in multiple steady state problems. 

Problem 9-1 (Level 1) The reaction A B is taking place at steady state in a catalyst particle that can be represented as an infinite flat plate of thickness 2L. Internal temperature gradients are negligible. The effective diffusivities of A and B are equal. The concentrations of A and B at the pellet surface are Ca,s and Cb,s, respectively. The reaction rate is strongly influenced by pore diffusion, such that 0 is large. 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

Multiple steady states in a rectangular catalyst pellet were analyzed in this example. This problem will be revisited later in this chapter. 

Problems with multiple steady states are interesting to solve numerically. Computational effort required for solving these problems can be highly demanding. Multiple steady states in a rectangular catalyst pellet were analyzed in example 3.2.2, 3.2.5 and 3.2.9. One has to provide an approximate solution or a guess value to predict the three multiple solutions. It is difficult to predict the effectiveness factor of the pellet as a function of O or y using the numerical approaches described earlier in this chapter. In the next example, this boundary value problem will be solved as an initial value problem. 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

Coupling may also occur via boundary conditions, e.g. the reaction rate in a catalyst pellet depends on the concentration and the temperature of the fluid surrounding the pellet. At steady state, when coupling between equations occnrs throngh boundary conditions, an exact or approximate analytical solution can be calculated with boundary conditions as variables, e.g. the effectiveness factor for a catalyst particle can be formulated as an algebraic fimction of surface concentrations and temperature. The reaction rate in the catalyst can then be calculated using the effectiveness factor when solving the reactor model. However, this is not possible for transient problems. The transport in and out of the catalyst also depends on the accumulation within the catalyst, and the actual reaction rate depends on the previous history of the particle. 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

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]. 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

Determining the Limiting Step Because at any point in the column the overall rate of transport is at steady state, the rate of transport from the bubble is equal to the rate of transport to the catalyst smface, which in turn is equal to the rate of reaction in the catalyst pellet. Consequently, for a reactor that is perfectly mixed, or one in which the catalyst, fluid, and bubbles all flow upward together in plug flow, we find that... 

The steady states which are unstable using the static analysis discussed above are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. That is to say simply that branch 2 in Figure 4.8 is always unstable, while branches 1,3 in Figure 4.8 and branch 4 in Figure 4.8 can be stable or unstable depending upon the dynamic stability analysis of the system. As mentioned earlier, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped parameter models or a distributed parameter model made discrete by a technique such as the orthogonal collocation technique. However, in the latter case, the system dimensionality will increase considerably, with n dimensions for each state variable, where n is the number of internal collocation points. 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

The influence of capillary condensation upon catalyst effectiveness factor has been assessed both by approximate calculations and by pore network simulations. It was found that catalyst effectiveness could be affected by the presence of capillary condensation, depending on the ratio of reaction rates in the gas and liquid phases. The effectiveness factor under conditions of capillary condensation is sensitive to operating conditions of the reactor, such as pressure, and to properties of the catalyst pore structure like pore-size distribution and connectivity. Once the catalyst pellet contains some pores filled with liquid, the kinetics of the process become dependent upon the phase equilibria of the system. This can lead to multiple steady states in the reaction rate as a function of temperature or pressure, because the current state of the catalyst pellet depends on the history of temperature and pressure profiles to which it has been subjected. 

The data given below, on diffusion of nitrogen (A) and helium B in porous catalyst pellets, have been provided by Henry, Cunningham, and Geankoplis [1967], who utilized the steady-state Wicke-Kallenback-Wesz technique. An alumina pellet with the following properties was used ... 

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