Taylor-diffusion-controlled combustion in ducts

An analysis is presented for the Burke–Schumann flame established when a fuel tank discharges with mean velocity U along a circular duct of radius a filled initially with air. Attention is focused on effects of interactions of shear with transverse diffusion resulting in enhanced longitudinal dispersion. The analysis accounts for preferential-diffusion effects arising for non-unity values of the fuel Lewis number 𝐿F, with the Peclet number 𝑃⁢𝑒=𝑈⁢𝑎/𝐷𝑜 based on the thermal diffusivity 𝐷𝑜 taken to be of order unity for generality. The solution to the associated Taylor-dispersion problem is described for times 𝑡′ much larger than the characteristic diffusion time across the pipe 𝑎2/𝐷𝑜, when the flame is embedded in a mixing region of increasing longitudinal extent moving with the mean velocity. At leading order in the limit 𝑡′≫𝑎2/𝐷𝑜, the longitudinal flame location, the burning rate, and the peak temperature are found to be a function of the effective Lewis number 𝐿e⁢f⁢f=𝐿F⁢(1+𝑃⁢𝑒2/48)/(1+𝐿2
F⁢𝑃⁢𝑒2/48), whose value changes from 𝐿e⁢f⁢f=𝐿F for 𝑃⁢𝑒≪1 to 𝐿e⁢f⁢f=1/𝐿F for 𝑃⁢𝑒≫1. As a result of this variation, the flame exhibits preferential-diffusion effects that depend fundamentally on 𝑃⁢𝑒, with important implications in designs of microcombustion devices employing narrow channels and pipes.