An asymptotic theory for the propagation of a surface-catalysed flame in a tube

Experiments have shown that when a mixture of fuel and oxygen is passed through a zirconia tube whose inner surface is coated with a catalyst, and then ignited at the end of the tube, a reaction front, or flame, propagates back along the tube towards the fuel inlet. The reaction front is visible as a red hot region moving at a speed of a few millimetres per second. In this paper we study a model of the flow, which takes into account diffusion, advection and chemical reaction at the inner surface of the tube. By assuming that the flame propagates at a constant speed without change of form, we can formulate a steady problem in a frame of reference moving with the reaction front. This is solved using the method of matched asymptotic expansions, assuming that the Reynolds and Damköhler numbers are large. We present numerical and, where possible, analytical results, first when the change in fluid density is small (a simplistic but informative limit) and secondly in the variable-density case. The speed of the travelling wave decreases as the critical temperature of the surface reaction increases and as the mass flow rate of fuel increases. We also make a comparison between our results and some preliminary experiments.