Bifurcation Analysis of a Two-dimensional Thermally Coupled Homogeneous-Heterogeneous Combustion model
Imran Alam1, David H. West2 and Vemuri Balakotaiah1
1Chemical and Biomolecular Engineering Department,
University of Houston, 4800 Calhoun Road, Houston, TX 77004
2SABIC Technology Center, Sugarland, TX 77478
Catalytic combustion and catalytic partial oxidation systems have been analyzed by many investigators using theoretical, computational and experimental techniques over several decades. In models for such systems, coupling between the homogeneous and heterogeneous chemistries, low residence times and very high temperatures involved make the study difficult. Past analyses have used micro-kinetic models for homogeneous reactions or focused on a CFD approach to produce a direct numerical study of the system. However, bifurcation analyses of the system have been rare (except some work from Schmidt and coworkers for stagnation point flow, see for example, Song et al. ), and systematic studies of high-dimensional singularities are almost non-existent. Models with detailed chemistry or hydrodynamics are not suitable for computational bifurcation analysis. High adiabatic temperature rise leads to exponentially thin boundary layers in space and/or time, rendering the corresponding discretized systems ill-conditioned. Further, very exothermic reacting flow systems are not amenable to bifurcation analysis using standard CFD procedures because the ensuing exponentially thin boundary layers require extremely fine mesh resolutions. Instead of the usual CFD-type studies, we present in this work a bifurcation study for 2-D combustion systems, with a focus on qualitative features and underlying physics. This yields a clear picture of the essential features of the process as the operating parameters are changed.
Mathematical Models & Analysis
We present a mathematical model for the combustion of propane and methane in monolith reactors. For simplicity, we consider global kinetics with a single reaction occurring homogeneously as well as catalytically. Our focus here is on thermal rather than chemical coupling. We present a detailed bifurcation analysis of the model taking into account the interaction between transport phenomena and reactions in both gas and solid phases. For the homogeneous oxidation of propane, we used the rate expression from Westbrook and Dryer . For the catalytic reaction, we use the rate expression from Hiam, et al..
We use the inlet fluid temperature as the bifurcation variable and compute several bifurcation diagrams for the state variables (solid or gas temperature, conversion, etc.) vs. the inlet temperature . We use the notions of singularity theory with one distinguished parameter to analyze our models and classify the bifurcation diagrams.
For higher inlet concentrations, our bifurcation diagrams can have multiple ignitions and extinctions, and we interpret the first ignition-extinction pair as being due to the catalytic reaction and the subsequent ignition-extinction pairs being the result of thermal coupling. In practice, the intermediate stable state, between the first extinction and second ignition is important as it represents the lower temperature state and a reasonable conversion of the hydrocarbon. However, if the second ignition happens to the right of the first ignition, the system jumps directly to the high temperature ignited state, and the intermediate state is not observed. For stoichiometric feeds, it becomes clear from computations that the high temperature rise makes the search for intermediate branches impractical. We find that the bifurcation diagrams where the second ignition is to the right of the first occur typically at low residence times.
We observe that the first ignition temperature is not a monotonic function of channel hydraulic radius. For small values of the hydraulic radius, we have a higher catalytic reaction rate and good heat transfer between the solid and the gas phases. The bifurcation diagrams in this case are due to the catalytic chemistry alone. However, for larger values of the hydraulic radius, the ignition becomes controlled by the heat transfer between solid and gas phases as the temperature difference between the phases becomes significant. As can be expected, a second ignition occurs only when the catalytic chemistry goes into a mass transfer controlled regime, and therefore we observe that the extent of thermal coupling between heterogeneous-homogeneous systems can be controlled by tuning transport parameters such as the transverse Peclet number and other system parameters such as the hydraulic radius and the catalyst loading.
The bifurcation features such as ignition and extinction and hysteresis depend on several parameters, such as activation energies, adiabatic temperature rise (or the inlet concentration), residence time, inlet temperature, channel hydraulic radius, axial and radial Peclet numbers etc. We do a comprehensive bifurcation study of the partial differential equation model describing coupled homogeneous-heterogeneous combustion and classify the space of parameters into regions displaying different qualitative behaviors. The results of the analysis/computations will be presented.
. X. Song, W.R. Williams, L. D. Schmidt, R. Aris, Combustion and Flame 84(1991) 292-311.
. C. K. Wesbrook, F.L. Dryer, Combust. Sci. Tech. 27(1981), 31-43.
. L. Hiam, H. Wise, S. Chaikin, J. Catalysis 9 (1968), 272-276.