The Bureau of Mines made time-dependent calculations to determine the size of a stationary-source fire within a ventilated duct (or passageway) necessary to ignite a combustible duct liner. The objective of this work was to gain more understanding of what occurs in duct fires and to determine critical conditions for ignition and propagation of mine fires. Heat transfer to the combustible surface includes both convective and radiated components, while the heat is conducted into the combustible and reradiated to the surroundings. The combustible is assumed to ignite when the net heat transfer to its surface is sufficient to raise the surface temperature to some minimum temperature, defined as the ignition temperature. A nondimensional parameter, which characterizes the source-fire intensity in the presence of ventilation flow, emerges as a parameter for a criterion of the critical conditions. The results of these calculations indicated that the minimum fire size necessary for ignition increases with the ventilation rate and duct cross section. These results are compared with experimental results obtained from fires in a 0.8- By 0.8-By 11-m-long duct and a full- scale gallery (2.4- By 2.4-By 65-m). Agreement between the theory and the experiment is good.