Observability estimates for convection dominated tubular reactors governed by the convection-(diffusion)-reaction pde

This paper investigates the state estimation problem for systems described by the convection-diffusion-reaction partial differential equation, with or without the diffusion term.The analysis of the observability leads to different results for the observability of the system e.g. a time-dependent approximate observability for the convection-reaction equation, this behavior does not occur for the convection-diffusion-reaction equation. To analyze these differences, a quantitative indicator of the unobservability is proposed based on the criterion of exact final observability. For the convection-reaction equation the indicator can be calculated analytically while for the convection-diffusion-reaction equation a numerical lower bound can be calculated using arbitrary precision calculation. For this indicator, the same qualitative behaviour as for the observability of the convection-reaction equation can be observed for the convection dominated convection-diffusion-reaction equation.

I. Introduction

Continuous manufacturing in the process industry is often performed in tubular reactors where the substances travel along the reactor with or without the removal of heat and chemical reactions or other processes as e.g. crystallization proceed such that at the outlet the desired products are obtained. The dynamics of tubular reactors are described by the convection-diffusion-reaction equation or, if diffusion is negligible by the convection-reaction equation. For process monitoring and control of such reactors, the knowledge of the states, i.e. the concentration and temperature profiles along the reactor is of crucial importance, but online measurements are expensive and difficult to install at multiple positions.Usually only concentration measurements at the outlet are available. Therefore, state estimation methods must be applied to get the desired information and to apply model based control techniques.