Transport limited pattern formation in catalytic monoliths

We consider the problem of flow in a tube with an exothermic surface reaction and show that the azimuthally symmetric steady-state can lose stability giving rise to patterned states with nonuniform concentration and temperature profiles. The primary cause for this transport limited pattern formation is the slow relative rate of heat and mass diffusion compared to surface reaction. Patterned states in which the temperature and concentration profiles vary in the azimuthal direction, can exist (or coexist with symmetric states) for all values of the fluid Lewis number (Le_f) though the patterned states are more pronounced and exist in a wider range of parameter space when Le_f<1. We analyze a three-dimensional model of catalytic monolith and develop analytical criteria for identifying the parameter regions in which patterned states exist. These criteria indicate that patterned states are formed whenever the local balance equations have multiple solutions and the characteristic reaction time is much smaller compared to the heat/mass diffusion time. Examination of the numerical values of the various parameters shows that most catalytic monoliths and combustors may operate in the region in which a large number of patterned states may exist. It is also found that when nonuniform and three-dimensional temperature and concentration fields exist, there can be hot spots in which the temperature exceeds the adiabatic temperature even when Le_f=1.