The application of singularity theory to isothermal autocatalytic open systems: the elementary scheme A +mB = (m+ 1)B

Abstract

The stationary-state behaviour of the simple autocatalytic models: quadratic autocatalysis A + B → 2B; rate =k1ab; cubic autocatalysis A + 2B → 3B; rate =k1ab2, in a well stirred, open system (isothermal c. s. t. r.) is determined via singularity theory methods. These methods allow all of the possible patterns for the dependence of the stationary-state extent of conversionxon the residence timeDa(i. e. thex —Dabifurcation diagrams) to be identified and located. The cubic rate law has a cusp singularity, separating diagrams with a unique dependence ofxonDafrom those which display a simple S-shaped hysterisis loop with multiple stationary states. This behaviour is qualitatively similar to that shown by a simple exothermic reaction in an adiabatic c. s. t. r. (i. e. the two systems are contact equivalent). If the autocatalyst is not infinitely stable but instead undergoes a simple decay B → C; rate =k2b, a wider range of bifurcation diagrams is possible, with isolas or mushroom patterns. These arise as the system is ‘unfolded’ from its winged cusp singularity by varying parameters such as the catalyst lifetime and inflow concentration. It is shown that these are also the only patterns possible for a generalized order of autocatalysis, i. e. for a rate proportional toabm, withmtaking any value greater than one, integral or non-integral. The ranges of the above parameters over which the different responses are found are also given analytically for the generalm. These parameters cannot, however, give a complete unfolding, so certain additional bifurcation diagrams that are found for the exothermic reaction in a non-isothermal, non-adiabatic reactor (which also has a winged-cusp singularity) are not found in the autocatalytic system.