On the number of steady states of the nonadiabatic tubular reactor

Abstract

Tubular reactors have been the subject of many theoretical and experimental studies. A summary of these results can be found in review articles by Varma and Aris (1977) and Razon and Schmitz (1987). One main issue was to find criteria for the uniqueness of the steady state and the number of steady state solutions. Using computational and asymptotic methods, it is now established that the maximum number of steady states for an adiabatic tubular reactor in which a single exothermic reaction occurs is three. For the nonadiabatic tubular reactor, Varma and Amundson (1973) found five steady state solutions. Kapila and Poore (1982) extended this maximum number from five to seven using large activation energy asymptotics. In a recent study, Alexander (1990) showed using the method of averaging that the nonadiabatic tubular reactor may have an arbitrarily large number of steady states, provided the activation energy is sufficiently large.We show here that the last restriction is not necessary. For any activation energy that lies above the boundary for uniqueness of a steady state, an arbitrarily large number of steady states can exist. We analyze the limiting Neumann model (Balakotaiah, 1989) using bifurcation theory and find an infinite number of bifurcation points on the unstable branch. This leads to the conclusion that there are always parameter regions that give rise to arbitrarily many steady states. The results for the limiting model are compared with those of the complete model by numerical calculations.We consider a cooled tubular reactor in which an irreversible first-order reaction occurs. For simplicity, we assume that the cooling temperature is equal to the inlet temperature. The dimensionless energy and species balances arewhere 0 is the dimensionless temperature, x is the dimensionless concentration, and z is the axial coordinate. Pe is the Peclet number (heat and mass), y is the activation energy, B is the dimensionless heat of reaction, St is the Stanton number, and A is a Damkoehler number. The corresponding boundary conditions areIf B is much larger than A, then the reactant consumption is negligible and we may drop the species balance. The energy balance may then be written as (Balakotaiah, 1989).

Chemical Reactor Dynamics: Stability of Steady States

“…The condition, c > eY6, in Proposition 3 is equivalent to the strong cooling condition of [16]. The result obtained may be interpreted as follows.…”