  The problem of spontaneous acceleration of premixed flames propagating in
open horizontal tubes with smooth walls is revisited. It is proved that in long
tubes, this process can be considered quasi-steady, and an equation for the
flame front position is derived using the on-shell description. Numerical
solutions of this equation are found which show that as in the case of uniform
flame movement, there are two essentially different regimes of flame
propagation. In the type I regime, the flame speed and its acceleration are
comparatively low, whereas the type II regime is characterized by significant
flame acceleration that rapidly increases as the flame travels along the tube.
A detailed comparison of the obtained results with the experimental data on
flame acceleration in methane-air mixtures is given. In particular, it is
confirmed that flames propagating in near-stoichiometric mixtures and mixtures
near the limits of inflammability belong to the types II and I, respectively,
whereas flames in transient mixtures undergo transitions between the two
regimes during their travel.
