Runaway criteria

In the above example it was shown how hot spots develop in fixed bed reactors for exothermic reactions. An imp ortant problem associated with this is how to limit the hot sp ot in the reactor and how to avoid excessive sensitivity to variations in the parameters. Several approaches have been attempted to derive simple criteria that would permit a selection of operating conditions and reactor dimensions prior to any calculation on the computer. Such criteria are represented in Fig. 

In Fig. 11.5.C-2 the locus of the partial pressure and temperature in the maximum of the temperature profile and the locus of the inflection points before the hot spot are shown as p and (pj), respectively. Two criteria were derived from this. The first criterion is based on the observation that extreme sensitivity is found for trajectories—the p-T relations in the reactor—intersecting the maxima curve p beyond its maximum. Therefore, the trajectory going through the maximum of the p -curve is considered as critical. This is a criterion for runaway based on an intrinsic property of the system, not on an arbitrarily limited temperature increase. The second criterion states that runaway will occur when a trajectory intersects (Pi)i, which is the locus of inflection points arising before the maximum. Therefore, the critical trajectory is tangent to the (pi)i-curve. A more convenient version of this criterion is based on an approximation for this locus represented by p in 

The formulas used in the first criterion are easily derived as follows. Considering again the case of a pseudo-first-order reaction treated under Sec. 11.5.b and dividing Eq. 11.5.b-2 by Eq. ll.5.b-l leads to 

This curve is called the maxima curve. It can be seen from Fig. 11.5.C-2 that it has a maximum. The temperature corresponding to this maximum, T, is obtained by differentiating Eq. 11.5.C-2 with respect to and setting the result equal to zero  

Notice the slightly different definition of / in this formula, compared to that of /, used in conjunction with Fig. 1 l.S.c-l. 

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There are many studies in the literature on the derivation of runaway criteria. Theoretically derived criteria are described briefly in Table 5.4-40, while those based on experiments are listed in Table 5.4-41. 

Runaway criteria developed for plug-flow tubular reactors, which are mathematically isomorphic with batch reactors with a constant coolant temperature, are also included in the tables. They can be considered conservative criteria for batch reactors, which can be operated safer due to manipulation of the coolant temperature. Balakotaiah et al. (1995) showed that in practice safe and runaway regions overlap for the three types of reactors for homogeneous reactions (1) batch reactor (BR), and, equivalently, plug-flow reactor (PFR), (2) CSTR, and (3) continuously operated bubble column reactor (BCR). 

Less conservative criteria for runaway can be found by removing the assumption of negligible reactant consumption. Along this line, a class of runaway criteria has been devised by linking the reactor behavior to suitable geometric features of the temperature-time history. 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

To deal with the modifications to the runaway criteria determined by the existence of an upper temperature limit, let us denote by 9ma the value of the generic parameter 9 that generates a temperature profile with a maximum equal to Tma. In the same way, let us denote by 9C the critical value of the parameter corresponding to the runaway boundaries obtained with any of the criteria discussed in the previous sections and generating a temperature profile with maximum value Tc. Two possible cases can be distinguished, namely... 

When condition (4.44) is satisfied, the existence of a maximum allowable temperature does not affect the critical parameter value 9C obtained with any of the classical runaway criteria. On the contrary, when condition (4.45) is satisfied, the actual limit value of the parameter to define safe operation must be replaced by 9ma. 

Fig. 4.11 Safety boundaries for the phenol-formaldehyde reaction according to the runaway criteria of Morbidelli and Varma (7), Thomas and Bowes (2), and for an imposed maximum allowable reactor temperature Tr,ma=98°C(5)...

Moreover, in some practical cases, as highlighted by the case study, the observance of a maximum allowable temperature may even nullify the results deriving from the direct application of the runaway criteria. In detail, this happens when operation that is close, but not necessarily beyond, the runaway boundaries already produces a maximum temperature that, for some reason, the system cannot comply with. Hence, warnings are given about the necessity to include this aspect when investigating safe conditions at which exothermic reactions are to be carried out. 

In the literature there are numerous runaway criteria with which operating ranges of high parametric sensitivity can be precalculated for known reaction kinetics [16, 59, 60]. In practice, however, these parameters are of only limited importance because they rarely take into account the peculiarities of individual cases. Sensitive reactions such as partial oxidation and partial hydrogenation are therefore generally tested in singletube reactors of the same dimensions as those in the subsequent multitubular reactor. This allows the range of parametric sensitivity to be determined directly. Recalculation of the results for other tube diameters is only possible to a limited extent due to the uncertainties in the quantification of the heat transfer parameters (see Section 10.1.2.4). 

For a zero-order reaction (rA = /c in mol m s ), the heat released by the reaction does not depend on the concentration and conversion. Thus the simplification of a negligible influence of the conversion on the runaway (to derive simple runaway criteria as shown above) is now not needed. Instead of Eqs. (4.10.53) and (4.10.55) we obtain for the first and second condition of a stable operation of a cooled batch reactor ... 

Figure 1. Comparison between approximate analytic (curves with numbers as in Table 1) and exact numerical runaway criteria -, Barkelew (1959) A, Dente and Collina (1964) 0, Morbidelli and Varma (1982) [from Morbidelli and Varma (1985)].

Morbidelli, M. and Varma, A. (1985) On Parametric Sensitivity and Runaway Criteria of Pseudohomogeneous Tubular Reactors , Chem. Engng. Sci. 40, 2165-2168. 

Figure 9.1 Runaway criteria for first-order reactions obtained by various authors. (1 Barke-lew 1959 2 Dente and Collina 1964 3 Hlavecek et al. 1969 4 Van Welsenaere and Froment 1970.) (After Froment and Bischoff, Chemical Reactor Analysis and Design, 1979, Wiley. Reprinted by permission of John Wiley Sons, Inc.)...

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