Regions of multiplicity

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. 

Cutlip and Kenney (44) have observed isothermal limit cycles in the oxidation of CO over 0.5% Pt/Al203 in a gradientless reactor only in the presence of added 1-butene. Without butene there were no oscillations although regions of multiple steady states exist. Dwyer (22) has followed the surface CO infrared adsorption band and found that it was in phase with the gas-phase concentration. Kurtanjek et al. (45) have studied hydrogen oxidation over Ni and have also taken the logical step of following the surface concentration. Contact potential difference was used to follow the oxidation state of the nickel surface. Under some conditions, oscillations were observed on the surface when none were detected in the gas phase. Recently, Sheintuch (46) has made additional studies of CO oxidation over Pt foil. 

Using the qualitative stability analysis from the flow diagram as outlined in 1.7.3 and 6.1.5, we find the following pattern. For conditions where there is a single stationary state, it is always stable to perturbations in regions of multiple solutions, the uppermost and lowest are stable whilst the middle... 

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

Fig. 12.5. (a) The region of multiple stationary-state behaviour for the Takoudis-Schmidt-Aris model of surface reaction, with = 10 3 and k2 = 2 x 10-3 (b), (c), and (d) show how the stationary-state reaction rate varies with the gas-phase pressure of reactant R for different values of p, giving isola, mushroom, and single hysteresis loop respectively. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415, 363-87.)... 

If we choose a high value for the partial pressure of reactant P, p > 0.037 975, then the system does not cross the region of multiplicity as r is varied. The reaction rate varies monotonically with r. A similar situation occurs if the fixed partial pressure of P is too small, such that p < 0.020 133. However, with any value of p in the range 0.020 133 < p < 0.037975, there are multiple stationary-state reaction rates over some range or ranges of the partial pressure of R. 

With the values of the dimensionless desorption rate constants used above, k1 = 0.001 and k2 = 0.002, condition (12.64) describes two curves in the p-r parameter plane. These are shown in Fig. 12.6, which also gives their relative location with respect to the loci of turning points (i.e. where det(J) = 0) which mark the boundaries of the region of multiplicity. 

As an example, consider the horizontal cut obtained by varying r with p = 0.017. This cut does not traverse the region of multiplicity, so the reaction rate varies monotonically with r. Nor do we intersect the curve of Hopf points LK. We do, however, encounter the other Hopf curve, NM. The stationary... 

High electron density in the intemuclear region of multiple metal bonds leads to a large polarizability and thus the vibrational features are amenable to Raman studies. Resonance Raman effects are common owing to the presence of... 

Flo. 15. Typical temperature profiles in the bed. Region of multiplicity (Cu0/A1203, 4% CO, Re = 26, T = 166°C) (55). (Reprinted with permission from Advances in Chemistry Series. Copyright by the American Chemical Society.)... 

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. 

Fig. 27. Region of multiple steady states. Honeycomb structures, Pt catalyst, l%CO (58). (Copyright by Verlag Chemie GmbH. Reprinted with permission.)...

Above 70-80 °C, the deformation remains mixed, but crazing is dominant and corresponds to CDCs, as revealed by the behaviour of the strain to craze and the strain rate dependence this temperature of occurrence of CDCs is denoted T223. Furthermore, the SDZs become less widespread and tend to be accompanied at the tips of the main crazes by regions of multiple crazing (Fig. 56)... 

The properties of the rate oscillations in the oxidation of CO over Pt, Pd, and Ir were examined comprehensively in experiments by Turner et al. [85] who suggested that the reaction follows an adsorption mechanism. Studies of the reaction model written in accordance with the law of acting surfaces show the existence of regions of multiplicity for the curves WCq2 (T) at a fixed ratio of reactant partial pressures Pco/Po, and WCq2(PC0IP02) at a fixed temperature T, which was already known [166, 174-176], Experimental data indicate that self-oscillations take place between two stable branches of the kinetic curves in the region of hysteresis [85],... 

Steady-state kinetic curves W(PB) for the adsorption mechanism (8), taking into account the reversibility of adsorption steps, are illustrated in Fig. 4(a), (b). At a given value of k x with increasing k 2 [Fig. 4(a)], the region of multiplicity for steady states diminishes and at some value of k 2 it vanishes completely. With increasing k 2, the kinetic curve can achieve a maximum and finally take the form of that with saturation. A similar effect is observed with the reversibility of the first step at a given value oik 2 [Fig. 4(b)]. But the parametric sensitivity here is lower than in the former case. 

The region for the multiplicity of steady states in the coordinates T x P02 x Pco is shown in Fig. 2. The same region is represented in various d.s.s. (Fig. 3), characterizing its evolution by varying the respective parameter (T, P0i, Pco)- In all these diagrams the region of multiplicity is rather narrow but extended. 

A method of constructing the boundaries for the region of multiplicity of steady states [141] as applied to the CO oxidation over Pt is sure to be of a general character. Methodical strategy of this problem, consisting in obtaining analytical expressions for the boundaries of the multiplicity region, can also be useful for the analysis of the other systems and the other types of critical effect. 

As stated above, instead of plotting the effectiveness factor against the Thiele modulus which contains the unknown, intrinsic rate constant, it is often more convenient to relate it to the observable Weisz modulus i/f. This leads to the representation given in Fig. 14 for the same situation as depicted in Fig. 13. The dashed portions of the curves indicate the regions in which a unique solution of the effectiveness factor docs not exist, corresponding to the regions of multiple solutions in Fig. 13. 

It is demonstrated that volatility of the liquid reactant has a detrimental effect on the enhancement of gas absorption. It is also shown that failing to account for effects due to liquid evaporation in the modeling of gas-liquid CSTRs can lead to significantly different predictions of the number and region of multiple steady states. 

If the velocity and thus 0 are increased again, the pellet reenters the region of multiplicity for 0 = 0.0035. In this case the pellet remains at the higher stable operating point... 

DAE case. Even at a = 20, the steady-state multiplicity curves (Regenass and Aris, 1965) are not very close to those of a single component (a = < ). The region of multiplicity seems to increase as a increases, that is, in parameter space the single component case is the most likely to result in multiplicity. 

P8C-3 In an article on the kinetics of sucrose inversion by invertase with multiple steady sates in a CSTR [Chem. Eng. Common., 6, 151 (1980)], consider the following challenges Are the equations for Jf and K correct If not, what are the correct equations for these variables Can an analysis be applied to this. system to deduce regions of multiple steady statra ... 

In this problem bifurcation theory (CD-ROM Section 8.6,5) is used to determine the regions of multiple steady st es for the autocatalytic reaction... 

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