Nonlinear state estimation of packed‐bed tubular reactors

Abstract

Packed-bed tubular reactors are characterized by nonlinear behavior, limited on-line measurements, and stiff dynamics. Because of the limited number of temperature and composition sensors normally available, state estimation algorithms would be a desirable part of the reactor control system. However, to complicate matters, the thermal transients are usually several orders of magnitude slower than the concentration transients, thus leading to very stiff distributed parameter modeling equations. In spite of a number of reported state estimation studies of this class of problems, a good understanding of the structure of the problem has not conie forth, and the previously published results have important practical limitations. In this note we shall illustrate how recently developed, general, theoretical results ( Soliman and Ray, 1978) can be profitably applied to this problem and help elucidate the structure of these state estimation problems.To illustrate the essential features of the problem, we shall choose to study a simple reactor in which a firstorder, irreversible, exothermic reaction is taking place. We shall assume a pseudohcmogeneous axial dispersion model for the reactor 2. There is the danger that essential features of the problem will be lost through brute force discretization. A theoretical analysis of the full partial differential equations is safer and is to be preferred. 3.There is a similar danger in just setting c = 0 and applying the state estimation theory to the degenerate equations, This procedure usually gives incomplete and sometimes even incorrect results (for example, Haddad and Kokotovic, 1971). In particular, the estimator resulting from setting € = 0 does not allow dynamic estimates of the fast variables ( x i here) or the proper treatment of composition measurements.TWO papers (Ajinkya et al., 1974;Ramirez, and Clough, 1976) have appeared in which the full distributed nature of the problem is treated. In the first paper, only the catalyst activity profile was estimated, while the gas phase concentrations were not estimated but measured. Thus, the problem of two time scales did not arise. In the second paper, state estimation of concentration was attempted using an incorrect algorithm, and rather poor filter performance was reported. A third recent paper describes studies of distributed parameter observers applied to the packed-led estimation problem (Zeitz, 1977) and reports reasonable estimates with sufficient temperature measurements.In contrast to earlier estimation algorithms, our recent results, reported in more generality and mathematical detail elsewhere (Soliman and Ray, 1978), allow the estimation prollem to be decomposed into two time scales with two state estimators, one for the fast variables alone and one for the slow variables. In addition, our results allow, for the first time, the proper inclusion of concentration measurements into the filtering equations for c + 0. To illustrate these results, the resulting state estimator for the packed-led reactor system ( 1) to (6) ...

Optimal continuous-time state estimation for linear finite and infinite-dimensional chemical process systems with state constraints

“…More recently, Zavala and Biegler [16] presented the application of the MHE for multi-zone low-density polyethylene tubular reactors. In 1981, Ray [11] summarized and applied the works of [15] in both lumped parameter and distributed parameter systems (see [17,18]). The optimal state estimation technique developed by Thomas and Ray was formulated by utilizing the variational method for continuous-time systems and the resulting estimation formulations have analytical expression.…”