Transients in Adiabatic Tubular Reactors. Axial Dispersion Models with Well-mixed Appended Sections

Using boundary conditions more general than those due to Danckwerts, the authors recently showed that significant variations may be encountered in the global stability characteristics of an axially dispersed adiabatic tubular reactor. The numerical results presented in this communication demonstrate that for a fixed initial state of the reactor, steady states very different from those predicted with Danckwerts conditions are reached depending on the initial state ofthe reactor appendages. Thecases where the reactor is preceded or succeeded by well-mixed (non-reactive) sections are considered. These findings will be very useful in regulation of the dynamics of tubular reactors. 1. lNTEODUCTION Recently, Parulekar and Ramkrishna [l] had investigated the stability of an axially dispersed tubular reactor with boundary conditions more general than those due to Danckwerts [2]. The general boundary conditions arose as a result of appending either semiinfinite (axially dispersed) sections without chemical reaction or perfectly mixed sections of finite volume. Regardless of the nature of the appended sections, the Danckwerts boundary conditions are always implied at steady state. As a result, under this more general framework, the steady state picture remains the same as that encountered with Danckwerts boundary conditions. Thus the locations (and obviously the multiplicity) of the steady states are unaltered. The analysis of Parulekar and Ramkrishna [l] showed that the local, asymptotic stability conditions remain also the same as those obtained by Amundson with Danckwerts boundary conditions [3]. However, the authors [l] have pointed out that significant variations may be encountered in the global stability characteristics of the steady states. Thus, for a given initial condition in the reactor section, rather widely varying steady states may be reached within the framework of the authors depending on the start-up conditions outside the reactor. (With Danckwerts boundary conditions, the initial state of the system outside the reactor is irrelevant.) Transient calculations were not presented [l] earlier because of the tremendous computational effort required when semi-infinite sections are considered. In this communication we present numerical calculations on reactor transients for the cases where well-mixed appendages of finite volume are included at either end of the reactor. In particular, we consider here the example of a first order, exothermic reaction. The reaction is considered to be catalytic in nature. Since only the reactor section contains the catalyst, no reaction occurs in the reactor appendages. The calculations are compared with those done using Danckwerts boundary conditions at both reactor ends. The initial conditions used assume non-zero residual enthalpy and are such that the steady state eventually attained cannot be predicted by the maximum principle [I]. 2. MODEL The various reactor assemblies, used in the computations, are shown in Fig. 1. The model equations in terms of dimensionless variables are reproduced below from our earlier work [l]. ax -= -~+Pe~-G(~,B),--IO (1) dr ae __= dr -~+F+G(~,@,