A membrane oxygenator includes an upper nested arrangement of a hard shell venous reservoir, a bowl-like defoamer within the reservoir and a heat exchanger coil vertically supported within the defoamer, and includes a membrane oxygenator unit mounted on the bottom of the upper arrangement. The oxygenator unit includes an inlet chamber with a tangential inlet providing swirling blood flow to dislodge air bubbles and centrifuge air to an upper center region of an inlet chamber where gas is exhausted through a vent. A bottom blood outlet chamber of the blood oxygenator unit has a relatively thin vertical cross-sectional area such that priming flow velocity therein discharges all air from the outlet chamber.