Liquid oxygen, which is extracted from the bottom of a lower pressure rectifier and compressed by a liquid oxygen pump to a predetermined supply pressure, is evaporated in a main heat exchanger to prepare an oxygen gas product, while oxygen gas is circulated in the main heat exchanger at a linear velocity which is equal to or higher than the terminal velocity, calculated depending on the supply pressure, of an oxygen droplet having a diameter of 200 &mgr;m. This process effectively prevents precipitation of heavy impurities in the heat exchanger and produces higher pressure oxygen gas at reduced operational costs.