This work examines reaction-induced flow maldistributions in adiabatic

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).

The coupling between the momentum and energy balances and the change of physical properties with temperature can give rise to situations in which the pressure drop vs. flow rate curve is nonmonotonic. This can lead to thermoflow multiplicity-the existence of different flow rates in a tube under the same overall pressure drop. A two-dimensional model is used to analyze the conditions leading to thermoflow multiplicity for an incompressible non-lrlewtonian fluid flowing in a cooled tube. First, the multiplicity features of various limiting models are determined. These results are later used to gain an understanding of the asymptotic multiplicity features of the general model. The results show that the temperature sensitivity of the viscosity necessary for thermoflow multiplicity to occur decreases with increasing Brinkman number or p (modified Stanton number), or with decreasing cooling temperature, Biot number, or power law parameter. Multiple flow rates for a prescribed pressure drop are unlikely to occur in heat exchangers in which the Brinkman numbers are usually low and Biot numbers are high but may be found in polymer processing applications.