Unusual dynamics of a reactor/preheater process with deadtime, inverse response and openloop instability Bjorn D. Tyreus” and William L. Luyben+ “Engineering, E. 1. Du Pont De Nemours Co, Newark, DE 19714-6090, USA tDepartment of Chemical Engineering, lacocca Hall, Lehigh University 11 I, Bethlehem, PA 18015, USA Received 25 June 1993; revised 7 September 1993 This paper presents a mathematical analysis of the unusual dynamics found in a coupled reactor/preheater process. The outlet temperature of the reactor exhibits inverse response (wrong-way behaviour) for a change in the inlet reactor temperature and a large deadtime. The coupling of the preheater with the reactor produces positive feedback, which makes the coupled process openloop unstable. Both theoretical linear analysis and simulation studies reveal some unusual dynamics and some interesting frequency response plots. The Nyquist plot of the openloop characteristic equation of the reactor itself is used to determine the number of poles in the right half of the s-plane (RHP) of the openloop characteristic equation of the coupled reactor/preheater system. For small values of reactor gain and small values of the positive-zero time constant, there is one pole; so the closedloop system is stable if the total openloop system (process plus controller) Nyquist plot encircles negatively the (- 1,O)point. As the gain and positive-zero time constant are increased, a point of discontinuity is reached. This occurs when the Nyquist plot of the openloop reactor indicates two poles instead of one in the RHP. Now there must be two negative encirclements of the (- 1,O) point for closedloop stability. For small values of the positive-zero time constant and reactor gain, the Nyquist plot of the coupled system does show two encirclements. However, for a given deadtime, there are maximum values for the positive-zero time constant and the reactor gain beyond which the closedloop system cannot be stabilized. One of the most unusual dynamic features of this system is the effect that reset (integral action) in the controller has on closedloop stability. Conventional wisdom says that adding integral action in a feedback controller always degrades dynamic stability and performance. However, in this coupled reactor/preheater process the addition of integral action can improve closedloop stability. Keywords: unusual dynamics; reactor/preheater process; mathematical analysis Tyreusr recently discussed a process that contains a fixed-bed adiabatic tubular reactor coupled with a feed preheater. The inertial effect of the heat capacity of the packing coupled with a large adiabatic temperature rise results in ‘wrong-way’ behaviou? as it is called in the reactor-design community, or ‘inverse response’ as it is called in the control community: the initial response to a decrease in reactor ,inlet temperature is a temporary increase in the reactor exit temperature, followed by an eventual decrease in exit temperature. The steadystate gain between the inlet and outlet temperatures can be quite large. In addition, the reactor response contains appreciable deadtime due to the slow propagation speed of the thermal front in the packing. The exit stream from the reactor is used in a feed preheater to warm up the reactor feed. This positive *Present address: PROSIM Inc., PO Box 1455, Hockessin, DE 19707, USA feedback loop makes the coupled system openloop un- stable if the loop gain is greater than unity. This type of system has been discussed in the literature by several workers over the years3-19. Most of the work has con- sidered issues of steadystate design, multiple steady- states, stability or mathematical modelling. The study of Douglas zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON et al.’ used analogue simulation methods to explore the effects of sensing element location, loop time constant and controller gain in a feed-effluent, exchanger-reactor process. The reactor was not packed and no wrong-way effect was reported. The controller used was proportional only. Apparently no in-depth fundamental dynamic analysis has been published. That is the purpose of this paper. Process zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML studied Figure I gives a sketch of the physical system, and Figure 0959-1524/93/040241-11 0 1993 Butterworth-Heinemann Ltd J. Proc. Cont. 1993, Volume 3, Number 4 241