Parametric sensitivity criteria

A clear understandig of the characteristics of the various parametric sensitivity criteria reported in the literature can be best achieved by considering the following simple model, describing the temperature - conversion phase plane behavior of a pseudohomogeneous plug-flow reactor, where an exothermic irreversible n-th order reaction occurs [c.f. Morbidelli and Varma (1988)] ... 

A more comprehensive approach consists of studying the variation of the Semenov criterion as a function of the reaction energy. Such an approach is presented in [12], where the reciprocal Semenov criterion is studied as a function of the dimensionless adiabatic temperature rise. This leads to a stability diagram similar to those presented in Figure 5.2 [11, 13]. The lines separating the area of parametric sensitivity, where runaway may occur, from the area of stability is not a sharp border line it depends on the models used by the different authors. For safe behavior, the ratio of cooling rate over heat release rate must be higher than the potential of the reaction, evaluated as the dimensionless adiabatic temperature rise. 

The Villermaux criterion and the Da/Si criterion are dynamic stability criteria, meaning that with a cooling medium temperature above the limit level, 20 resp. 30 °C, the reactor will be operated in the instable region and present the phenomenon of parametric sensitivity. If instead of B12, B is used, both criteria lead to the same result. This should not be surprising since they derive from the same heat balance considerations, that is, the heat release rate of the reaction increases faster with temperature than the heat removal does. 

This criterion expresses the fact that the reaction rate must be high enough to avoid the accumulation of reactant, even if the reaction is performed at the temperature of the coolant, and the cooling capacity must be sufficient to control the temperature. If this criterion is fulfilled, the reactor will not be parametric sensitive. The set temperature of the reactor can be chosen using the following criterion ... 

Thus, the equations describing the thermal stability of batch reactors are written, and the relevant dimensionless groups are singled out. These equations have been used in different forms to discuss different stability criteria proposed in the literature for adiabatic and isoperibolic reactors. The Semenov criterion is valid for zero-order kinetics, i.e., under the simplifying assumption that the explosion occurs with a negligible consumption of reactants. Other classical approaches remove this simplifying assumption and are based on some geometric features of the temperature-time or temperature-concentration curves, such as the existence of points of inflection and/or of maximum, or on the parametric sensitivity of these curves. 

Bilous and Amundson [1] were the first to describe the phenomenon of parametric sensitivity in cooled tubular reactors. This parametric sensitivity was used by Barkelew [2] to develop design criteria for cooled tubular reactors in which first order, second order and product- inhibited reactions take place. He presented diagrams from which for a certain tube diameter dt the required combination of CAO and Tc can be derived to avoid runaway or vice versa. Later van Welsenaere and Froment [3] did the same for first order reactions, but they also used the inflexion points in the reactor temperature T versus relative conversion XA trajectories, which describe the course of the reaction in the tubular reactor. With these trajectories they derived a less conservative criterion. Morbidelli and Varma [4] recently devised a method for single order reactions based on the isoclines in a temperature conversion plot as proposed by Oroskar and Stern [5]. 

Wilson s criterion is based on the one-dimensional reactor. A more useful criterion of parametric sensitivity based on this model was given by Barkelew (B2). Barkelew proceeded by assuming that only one reac-... 

A very simpie criterion for estimating the region of stabie reactor behavior is based on a maximum temperature peak. If the maximum temperature difference is fixed to AT = 1.2, thefoiiowing correiation is obtained for separating the region of stabie operation from that of high parametric sensitivity (see Figure 5.22) ... 

Figure 6. Comparison of predicted region of parametric sensitivity with experimental data runaway condition, O nonrunaway condition data of Emig et al. (1980), [from Morbidelli and Varma (1987)]. (a) — Generalized criterion = 0 — Generalized criterion, = 0.1 (b) —...

As discussed by Chemburkar et al. (1986), none of the previous criteria based on the presence of some temperature profile can be applied in this case, since there is neither a temperature profile nor a hot spot. Therefore, only the generalized criterion for parametric sensitivity developed above can be used. 

Morbidelli, M. and Varma, A. (1988) A Generalized Criterion for Parametric Sensitivity Application to Thermal Explosion Theory , Chem. Engng. Sci. 43, 91-102. 

First, the cis-peak assignment for band III is primarily based on the theoretically estimated energy (54), which appears to be very sensitive to the choice of parameters in the calculation (48,93). In fact, recent theoretical considerations (93) show that changes in parametrization markedly affect the energy but not the relative intensity of the cis peak. The intensity criterion, which... 

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