The mathematical model of the ignition process of a drop of water-coal fuel (WCF) developed by the authors in the flow of a high-temperature oxidizer (air) has been presented. According to the results of the mathematical modeling, it has been established that the zone of ignition of the WCF particle with an increase in the flow velocity shifts to the area of the aerodynamic trace of the drop. A prognostic modeling of the ignition processes of the water-coal fuel drops under conditions corresponding to the furnace spaces (intensive radiation-convective heating, change in the dynamics of the aerodynamic spectrum of a fuel particle) of typical boiler units has been carried out. A comparative analysis of the ignition delay times (t(ign)), obtained theoretically, and published earlier experimental values of (t(i)(gn)) has showed their good conformance. The results of the mathematical modeling have shown that the best option for stable ignition and burning of the WCF drop can be a combustion space consisting of two successive combustion chambers: in the first one, the thermal preparation and ignition of the WCF particle is carried out, and the second one is direct combustion. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.