Stability of Packed Tubular Reactors

In critical cases it may well be worthwhile to make a complete analysis of stability. In many cases, however, enough can be learned by studying what Bilous and Amundson (B7) called parametric sensitivity. These authors derived formulas for calculating the amplification or attenuation of disturbances imposed on an unpacked tubular reactor originally in a steady state, with the idea that if the disturbances grow unduly the performance of the reactor is too sensitive to the conditions imposed on it, that is, to the parameters of the system. The effect of feedback from a control system was not considered. As pointed out by the authors, it would be a much more complicated task to apply their procedure to a packed reactor, but it still would entail far less computation than a study of the transient response. 

A simple criterion for easy controllability was proposed by Wilson (W4). He pointed out that if 

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- 

With these variables, the relation between dimensionless temperature r and the fractional conversion Xf is 

This sharp curvature is particularly pronounced when is large, which is to say when the product of activation energy and heat of reaction is large. Barkelew proposed as a condition for sensitivity that the combination of N and be such that the value of N/S is above the point of contact with the envelope. Given this criterion, other families of curves were constructed, showing the critical relationship between N/S and for various combinations of the parameters a, /3, and t0, the initial dimensionless temperature. Several such families of curves are shown in the original paper. 

---