Multiple steady states in a catalyst

Example 3.2.5. Multiple Steady States in a Catalyst Pellet... 

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. 

They will also tell us that for certain types of reactions we can safely predict the performance of a packed bed reactor on the basis of a micro reactor, while for others we cannot. If the reaction Is highly exothermic and there Is a chance of multiple steady states, we might require experiments at flowrates similar to the large reactor. We often cannot accurately predict the occurrence of multiple steady states In a catalyst particle, nor do we want to. We want to avoid them, and to assure that we need reliable experiments. Theory provides a framework for them. It Is also useful In guiding the development of better catalysts. 

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. 

The catalyst has very low activity during warmup until the catalyst temperature attains -200°C. Since the oxidation reactions are exothermic, the catalyst provides heat, which further heats the catalyst and increases the rate. Therefore, the ACC can exhibit multiple steady states, where the catalyst has essentially no activity until a certain temperature where the catalyst ignites. This is described as lightojf, and a good catalyst system has a low lightoff temperature. Lightoff characteristics are sketched in Figure 7-18. 

Figure 7-18 Since tiie reactions in the ACC are sh ongly exothermic, hie catalyst can exhibit multiple steady states as hie catalyst suddenly lights off as the car is staled aid the terrqieratae rises to the ignition temperatiue, vdnch then causes a rapid increase in temperature.

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. 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

Figure 63 Multiple steady states in an autothermal reactor, with reaction rate limited by equilibrium heat production rates for fuUy active (a) and deactivated catalyst (a ) heat removal rates for normal (b), and increased heat transfer (b )-...

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 exciting issue of steady-state multiplicity has attracted the attention of many researchers. First the focus was on exothermic reactions in continuous stirred tanks, and later on catalyst pellets and dispersed flow reactors as well as on multiplicity originating from complex isothermal kinetics. Nonisothermal catalyst pellets can exhibit steady-state multiplicity for exothermic reactions, as was demonstrated by P.B. Weitz and J.S. Hicks in a classical paper in the Chemical Engineering Science in 1962. The topic of multiplicity and oscillations has been put forward by many researchers such as D. Luss, V. Balakotaiah, V. Hlavacek, M. Marek, M. Kubicek, and R. Schmitz. Bifurcation theory has proved to be very useful in the search for parametric domains where multiple steady states might appear. Moreover, steady-state multiplicity has been confirmed experimentally, one of the classical papers being that of A. Vejtassa and R.A. Schmitz in the AIChE Journal in 1970, where the multiple steady states of a CSTR with an exothermic reaction were elegantly illustrated. 

Figure 9-17 illustrates one possible outcome of a series of diagnostic experiments in an isothermal PFR in which an exothermic reaction takes place. The actual shape of a plot of outlet conversion versus linear or mass velocity through the PFR will depend on a number of factors, principally the values of the heat of reaction and the activation energy. If either of these values is very low, the maximum in die curve may not appear, and the plot may resemble Figure 9-16. However, if eidier AHr otE is high, individual catalyst particles in the reactor can have multiple steady states. In this case, a plot of outlet conversion versus linear velocity may exhibit discontinuities and will depend on how the experiments are conducted.

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]. 

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

The starting point of a number of theoretical studies of packed catalytic reactors, where an exothermic reaction is carried out, is an analysis of heat and mass transfer in a single porous catalyst since such system is obviously more conductive to reasonable, analytical or numerical treatment. As can be expected the mutual interaction of transport effects and chemical kinetics may give rise to multiple steady states and oscillatory behavior as well. Research on multiplicity in catalysis has been strongly influenced by the classic paper by Weisz and Hicks (5) predicting occurrence of multiple steady states caused by intrapellet heat and mass intrusions alone. The literature abounds with theoretical analysis of various aspects of this phenomenon however, there is a dearth of reported experiments in this area. Later the possiblity of oscillatory activity has been reported (6). 

Since the critical values of y(3 and Lw are y/3 = 4 and, Lw > 1 respectively, then referring to the results reported in Table II, it seems highly unrealistic to expect multiple steady states and periodic activity for a single catalyst particle resulting from intraparticle heat and mass transfer alone. 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

So far, there have been published only a few papers devoted to experimental investigation of multiplicity and oscillatory activity of a single catalyst particle. Observations of multiple steady states and/or oscillations for a single catalyst particle are reported in Table IV. Evidently three types of strong exothermic reactions have been investigated ... 

While we have for heat and mass transfer in a porous catalyst explicit relations for parameters giving rise to multiple steady states, there is nothing similar developed for the monolithic catalysts so far. Hence we are forced to investigate, for particular conditions, the region of multiple steady states numerically. 

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. 

Thus if the multiplicity of steady states for the catalyst surface manifesting itself in the multiplicity of steady-state catalytic reaction rates has been found experimentally and for its interpretation a three-step adsorption mechanism of type (4) and a hypothesis about the ideal adsorbed layer are used, the number of concrete admissible models is limited (there are four). It can be claimed that some types of adsorption mechanism have "feedbacks , but for the appearance of the multiplicity of steady states these "feedbacks must possess sufficient "strength . The analysis of these cases (mechanisms 4-7 in Table 2) shows that, to achieve multiplicity, the reaction conditions must "help the non-linear step. 

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