Effectiveness factor multiple steady states

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]. 

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]. 

To give an example, Fig. 20 shows a diagram for E = 10 and various selected values of Kp fi. The dashed lines indicate the range over which multiple steady states of t/(ip) occur. Here, by means of numerical methods it is not possible to determine a unique solution of the effectiveness factor of the pellet for given conditions at the external pellet surface [91]. Which operating point will be observed in a real situation again depends upon the direction from which the stationary state is approached [91]. 

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. 

Stewart, W. E., and J. V. Villadsen, Graphical calculation of multiple steady states and effectiveness factors for porous catalysts, AIChE J.,15, 28-34, 951 (1969). 

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. 

For a given value of , equation (3.63) can be solved for the boundary conditions ((3.65) and (3.66)). Once the numerical solution is obtained the effectiveness factor can be calculated using equation (3.67). It is of interest to plot the effectiveness factor as a function of O. For this purpose, one can solve the boundary value problem for different values of O and predict the effectiveness factor. However, since multiple steady states occur different initial guesses have to be used to capture all of the steady states. This boundary value problem is difficult to solve because for every value of O, there can 3 different qs and, hence. 

This plot captures the multiple steady state part. However, to predict the effectiveness factor at higher values of O, we choose initial values ranging from le - 40 to le - 10. For these low values, one has to perform highly accurate simulations. For this purpose, absolute error (abserr) is set to le - 41 and the relative error (relerr) is set to le - 12. 

At the multiple steady state region the phthalide effectiveness factor is negative for all steady states and takes very high negative values which exceed -2x 10" ) as shown in Figure 5.33. 

The numerical method developed shows very fast convergence characteristics for all steady states even when there are multiple steady states, although the difference in temperature between the bulk phase and surface of the pellet is sometimes quite high. In the following the effectiveness factors are computed for the two kinetic schemes and the effect of different parameters on the effectiveness factor is discu.ssed. 

A temperature gradient would also be expected. For an isothermal case, with rj set equal to 1, multiple steady-state solutions may be found (see Figure 10), and the concentration gradient is very significant at temperatures above 427°C (800°F). The non-isothermal catalytic effectiveness factors for positive order kinetics under external and internal diffusion effects were studied by Carberry and Kulkarni (8) they also considered negative order kinetics. 

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... 

Intraparticle diffusion can have a significant effect on the kinetic behavior of enzymes immobilized on solid carriers or entrapped in gels. In their basic analysis of this problem. Moo-Young and Kobayashi (1972) derived a general modulus and effectiveness factor. The results also predicted possible multiple steady-states as well as unstable situations for certain systems. While these results are very interesting it should be remembered that they are primarily mathematical and await extensive experimental support data. 

Figure 11.9.a-l shows the relation between the effectiveness factors rj and tjo and the modulus [104, 108]. This relation can only be obtained by numerical integration of the system, Eqs. 11.9.a-l to 11.9.a-8, except for the cases already mentioned. With isothermal situations ri tends to a limit of 1 as 0 increases, with nonisothermal situations, however, r/ or tic, may exceed 1. Curve 1 corresponds to the t] concept, curves 2, 3, and 4 to r/o. The dotted part of curve 4 corresponds to a region of conditions within which multiple steady states inside the catalyst are... 

This whole field of oniquotess and stability has been reviewed recently by Aris [100]. As previously mentioned the possibility of multiple steady states seriously complicates the design of the reactor. Indeed, transient computations have to be performed in order to make sure that the correct steady-state profile throughout the reactor is predicted. Another way would be to check the possibility of multiple steady states on the effectiveness factor chart for every point in the reactor. This... 

The plots of Weisz and Hicks (1962) are reproduced in Figure 7.6. The nature of the curves at high values of /3m suggests multiple solutions. In other words, the reaction can occur at three steady states, two stable and one unstable. We shall not be concerned with this aspect of effectiveness factors, but it is instructive to note that e given by one of the solutions in the multiple steady-state region can be orders of magnitude higher than unity. Instabilities of this kind are essentially local in nature, and are briefly considered in Chapter 12. The reactor as a whole can also exhibit multiple steady states, a feature that is briefly treated in Chapter 13. 

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. 

One unique but normally undesirable feature of continuous emulsion polymerization carried out in a stirred tank reactor is reactor dynamics. For example, sustained oscillations (limit cycles) in the number of latex particles per unit volume of water, monomer conversion, and concentration of free surfactant have been observed in continuous emulsion polymerization systems operated at isothermal conditions [52-55], as illustrated in Figure 7.4a. Particle nucleation phenomena and gel effect are primarily responsible for the observed reactor instabilities. Several mathematical models that quantitatively predict the reaction kinetics (including the reactor dynamics) involved in continuous emulsion polymerization can be found in references 56-58. Tauer and Muller [59] developed a kinetic model for the emulsion polymerization of vinyl chloride in a continuous stirred tank reactor. The results show that the sustained oscillations depend on the rates of particle growth and coalescence. Furthermore, multiple steady states have been experienced in continuous emulsion polymerization carried out in a stirred tank reactor, and this phenomenon is attributed to the gel effect [60,61]. All these factors inevitably result in severe problems of process control and product quality. 

Shown in Figure 4.6 is this relationship obtained by Carberry and Kulkarini (1973). Similar behavior as in Figure 4.5 can be observed. However, multiple steady states do not exist. It is obvious that the external effectiveness factor by itself is mainly of academic interest except for those unusual cases where the catalyst has only an external surface. 

Multiphase reactors, 431. See also Slurry reactor Trickle-bed Multiple steady state for effectiveness factor, 109 in interfacial transport (reactor point effectiveness), 123... 

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

In real cells, multiple transmembrane pumps and channels maintain and regulate the transmembrane potential. Furthermore, those processes are at best only in a quasi-steady state, not truly at equilibrium. Thus, electrophoresis of an ionic solute across a membrane may be a passive equilibrative diffusion process in itself, but is effectively an active and concentra-tive process when the cell is considered as a whole. Other factors that influence transport across membranes include pH gradients, differences in binding, and coupled reactions that convert the transported substrate into another chemical form. In each case, transport is governed by the concentration of free and permeable substrate available in each compartment. The effect of pH on transport will depend on whether the permeant species is the protonated form (e.g., acids) or the unprotonated form (e.g., bases), on the pfQ of the compound, and on the pH in each compartment. The effects can be predicted with reference to the Henderson-Hasselbach equation (Equation 14.2), which states that the ratio of acid and base forms changes by a factor of 10 for each unit change in either pH or pfCt ... 

For = 0.1 the results are shown in Figure 5.56. A narrow multiplicity range exists (or/iy=0. However, the multiplicity range for Py=0A is quite large and a narrow range of the values of over which five steady states exist is evident. The maximum effectiveness factor reaches a value of 500 (Elnashaie and Mahfouz, 1978). 

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