A method and system is provided for combusting a fuel having application to a heat consuming device such as a boiler or furnace or a reactor. An oxygen-containing stream is introduced into one or more oxygen transport membranes subjected to a reactive purge or a sweep gas. The oxygen transport membrane(s) can advantageously be subjected to a reactive purge or a sweep gas passing in a cross-flow direction with respect to the membranes to facilitate separation of the oxygen. In case of a reactive purge, temperature control of the oxygen transport membrane(s) is effectuated by the use of a suitable heat sink. Further, the oxygen transport membranes can be arranged in a row and be connected in series such that retentate streams of ever lower oxygen concentrations are passed to successive oxygen transport membranes in the row. The fuel or sweep gas can be introduced in a direction counter-current to the bulk flow of the retentate streams.