Multiple steady states in coupled flow tank reactors

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Abstract
Coupling between continuous‐flow, stirred tank reactors (CSTR’s), each having multiple steady states, can produce new steady states with different concentrations of the chemical species in each of the coupled tanks. In this work, we identify a kinetic potential ψ that governs the deterministic time evolution of coupled tank reactors, when the reaction mechanism permits a single‐variable description of the states of the individual tanks; examples include the iodate‐arsenous acidreaction, a cubic model suggested by Noyes, and two quintic models. Stable steady states correspond to minima of ψ, and unstable steady states to maxima or saddle points; marginally stable states typically correspond to saddle‐node points. We illustrate the variation in ψ due to changes in the rate constant for external material intake (k 0) and for exchange between tanks (k x ). For fixed k 0 values, we analyze the changes in numbers and types of steady states as k x increases from zero. We show that steady states disappear by pairwise coalescence; we also show that new steady states may appear with increasing k x , when the reaction mechanism is sufficiently complex. For fixed initial conditions, the steady state ultimately reached in a mixing experiment may depend on the exchange rate constant as a function of time, k x (t) : Adiabatic mixing is obtained in the limit of slow changes in k x (t) and instantaneous mixing in the limit as k x (t)→∞ while t remains small. Analyses based on the potential ψ predict the outcome of mixing experiments for arbitrary k x (t). We show by explicit counterexamples that a prior theory developed by Noyes does not correctly predict the instability points or the transitions between steady states of coupled tanks, to be expected in mixing experiments. We further show that the outcome of such experiments is not connected to the relative stability of steady states in individual tank reactors. We find that coupling may effectively stabilize the tanks. We provide examples in which coupled CSTR’s can be operated stably with one of the tanks at or beyond the single‐tank marginal stability point.

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