To compute steady weak flames in MFR, we use a steady 1-D reactive diffusive flow model with heat convection between the gas and the reactor wall. More specifically for simplicity, we use a modified steady 1-D flame code such as PREMIX in CHEMKIN [a,b] or Cantera [c] with additional heat convection term to the gas-phase energy equation as follows [41]:
The last term of LHS is the heat convection term added and the other terms are the same as those of the original energy equations in the codes. The temperature of the inner surface of the reactor tube in flow direction is measured and its profile is used as T_w of the heat convection term for computations. This approach can be considered as a reasonable compromise between computational quality and cost since weak flame is formed at very low flow velocity in MFR, mostly less than a couple of centimeters/second. By this approach, we are able to limit computational costs of weak flames within spatially 1-D modeling approach. This is beneficial for kinetics study using weak flames in MFR. Validity of the present 1-D approach has been confirmed through the comparison with 2-D model of the reactive flow in a cylindrical tube with detailed chemistry in Appendix of [111] where difference between the 1-D and 2-D models is found to be negligibly small.

We frequently get a comment like "a plug flow model is inappropriate in low flow velocity conditions of weak flames." but it is indeed a misunderstanding. The plug flow model is well accepted for many flow reactor systems, which arises this frequent comment. However, we use the different model.