Dynamic Comparison of Alternative Tubular Reactor Systems Phisit Jaisathaporn † and William L. Luyben* Process Modeling and Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015 Direct quantitative comparisons of the dynamic controllability of four alternative tubular reactor systems are given in this paper. An exothermic, irreversible gas-phase reaction, A + B f C, is carried out in a packed-bed tubular reactor in a process that includes reaction, separation, feed preheating, and gas recycle. The four alternative tubular reactor designs are (1) a single-stage adiabatic reactor, (2) multistage adiabatic reactors with interstage cooling, (3) multistage adiabatic reactors with cold-shot cooling, and (4) a single-stage cooled reactor. All of the adiabatic systems are open-loop unstable (exhibit limit-cycle behavior) because of the sensitivity of the first reactor, which propagates thermal waves through the system. The cooled reactor system is open-loop stable for base-case parameters. The single-stage adiabatic reactor and the multistage adiabatic reactor system with intermediate cooling are closed-loop stable. However, the optimal seven-bed multistage adiabatic reactor system with cold-shot cooling is closed-loop unstable because of the severe disturbances generated by manipulating cold-shot flows. If the number of beds is reduced to three, the system is closed-loop stable. The cooled reactor is the most controllable. 1. Introduction A previous paper 1 provided a quantitative comparison of the steady-state economic optimal design of several alternative tubular reactor systems. This paper studies the dynamic controllability of four systems: (1) a single- stage adiabatic reactor; (2) multistage adiabatic reactors with interstage cooling; (3) multistage adiabatic reactors with cold-shot cooling; (4) a single-stage cooled reactor. An exothermic, irreversible gas-phase reaction, A + B f C, is carried out in packed-bed tubular reactors. The reactors are part of a plantwide system that includes reaction, separation, feed preheating, compres- sion, and gas recycle. Many papers have appeared in the literature that discuss the dynamics and control of tubular reactors in isolation. 2-10 Only a handful of papers have dealt with the dynamics and control of tubular reactors in a plantwide environment. Design and control of feed- effluent exchanger/reactor systems were reported by Douglas et al. 11 Tyreus and Luyben 12 discussed the inverse response, dead time, and open-loop instability of a reactor coupled with a preheater. The chaotic behavior of a similar system was reported by Bilden and Dimian. 13 Plantwide systems with single-stage adiabatic tubular reactors have been studied recently in several papers. 14-21 Stephens 22 investigated a methanol plant consisting of four adiabatic catalyst beds with cold-shot cooling and showed that the exit temperature control failed but the inlet temperature control was successful. Luyben 23 discussed the effect of design and kinetic parameters on the control of cooled tubular reactor systems. We compare the dynamics and control of the four systems considered in the previous paper. 1 The four optimal flowsheets of alternative tubular reactor sys- tems are shown in Figure 1. These alternative designs use different reactor configurations, but the separation, recycle, and preheating sections are essentially identi- cal. Steady-state conditions, equipment sizes, and design parameters are given in these figures for the optimal economic steady-state designs. The cold-shot design for three beds is shown, but a seven-bed design is optimal. As we will show, the seven-bed process is uncontrollable. 2. Dynamic Model The reactor is modeled by three partial differential equations: component balances on A and B (eqs 1 and 2) and an energy balance (eq 3 for an adiabatic reactor or eq 4 for a cooled reactor). The overall heat-transfer coefficient U in the cooled reactor in eq 4 is calculated by eq 5 and is a function of the Reynolds number Re (eq 6). Equation 7 is used for pressure drop in the reactor using the fiction factor f given in eq 8. The dynamics of the momentum balance in the reactor are neglected because they are much faster than the com- position and temperature dynamics. A constant mass flow through the reactor is assumed. The reaction rate r′ C is based on the volume of the reactor and is calculated by eq 9. * To whom correspondence should be addressed. Tel.: 610- 758-4256. Fax: 610-758-5057. E-mail: WLL0@Lehigh.edu. † Current address: Department of Chemical and Process Engineering, King Mongkut’s Institute of Technology North Bangkok, 1518 Piboolsongkram Road, Bangkok 10800, Thai- land. Tel.: 66-2-913-2500 ext 8230. Fax: 66-2-587-0024. E-mail: pjai@kmitnb.ac.th. ǫ ∂y A ∂t )-v ∂y A ∂z - (1 - y A )r′ C C (1) ǫ ∂y B ∂t )-v ∂y B ∂z - (1 - y B )r′ C C (2) F cat c cat ∂T ∂t )-cFv ∂T ∂z - λr′ C (3) F cat c cat ∂T ∂t )-cFv ∂T ∂z - λr′ C - 4U D tube (T - T st ) (4) 1003 Ind. Eng. Chem. Res. 2004, 43, 1003-1029 10.1021/ie030434d CCC: $27.50 © 2004 American Chemical Society Published on Web 01/20/2004