The unified thermal and chain branching model of hydrocarbon oxidation in a well stirred continuous flow reactor

Abstract
The Gray-Yang ‘skeleton’ model of hydrocarbon oxidation, which unified the thermal and chain branching theories of hydrocarbon oxidation, is analysed in detail in order to probe its ability to account for the recent experimental observations in continuously stirred tank reactors (c. s. t. r.). Using rigorous mathematical analysis, the nature of the thermo-kinetic steady states is investigated, and particular attention is paid to the existence of stable limit cycles originating from supercritical Hopf bifurcations. These oscillations representing the ‘cool flames’ disappear at the low temperature limit via limit cycle-saddle point bifurcations. The origin of ‘two-stage’ ignitions is traced to these bifurcations; the computed trajectories exhibit ‘shoulders’ characteristic of such ignitions. When the ambient temperature is traversed slowly upward or downward, rapid transitions between quasi-steady states occur. The system shows hysteresis in the transitions between oscillatory and non-oscillatory steady states. Simple ignitions may take place as a result of the classical tangency condition when the ambient temperature is ascending. More­over, at sufficiently high pressures, two-stage ignitions occur on lowering the ambient temperature as a result of a limit cycle-high temperature saddle point bifurcation. This type of bifurcation extends the general theory of explosions which previously described only explosions originating from non-oscillatory steady states. The analysis shows very good agreement with many qualitative features of hydrocarbon oxidation in c. s. t. r., particularly the combustion of acetaldehyde. It is suggested that a more detailed investigation of the experimental temperature traces during the transitions between the steady states may assist in ascertaining the nature of these transitions and the bifurcations they represent.