NOVELTY - Extracting (M1) at least one insoluble oil from a liquid source involves feeding the liquid source comprising the insoluble oils; pumping at least one collection fluid through the membrane; contacting the insoluble oils in the liquid source in the contactor with collection fluids pumped through the membranes or membrane modules; removing a first stream comprising the liquid source without the insoluble oils from the contactor; and removing a second stream comprising the collection fluids and the extracted insoluble oils from the contactor.    USE - For extracting insoluble oil from liquid source useful for preparation of fatty acid methyl ester and biodiesel, and used as animal feed, and for energy generation (claimed); and also useful for recovery of released or secreted algae oil from an aqueous mixture, insoluble hydrocarbon and hydrocarbon-rich molecules from aqueous mixtures, omega fatty acids from an aqueous mixture, beta-carotene from an aqueous mixture, and removal of oil from produced water in petroleum exploration and production.    ADVANTAGE - The method extracts 45 (preferably 75, especially 100)% of the insoluble oils from the liquid source; allows more efficient separation and elimination of the solid-liquid-liquid-emulsion problem; and eliminates the need of a distillation system or a stripper to recover the solvent thus reducing the capital and operating cost of the overall oil recovery process.    DETAILED DESCRIPTION - Extracting (M1) at least one insoluble oil from a liquid source using at least one hydrophobic membrane or membrane module involves feeding the liquid source comprising the insoluble oils by pumping in a contactor or a vessel; pumping at least one collection fluid through the membrane or membrane module, where the collection fluids counterflow with the liquid source in the contactor or the vessel, where the collection fluid comprises solvents, a biodiesel, an algal oil, and/or a non-polar oil; contacting the insoluble oils in the liquid source in the contactor or the vessel with collection fluids pumped through the membranes or membrane modules; removing a first stream from the contactor or the vessel, where the first stream comprises the liquid source without the insoluble oils; and removing a second stream from the contactor or the vessel, where the second stream comprises the collection fluids and the extracted insoluble oils. An INDEPENDENT CLAIM is included for a contactor or vessel for extracting algal lipid components and/or algal oils from a lysed algal concentrate comprising an external metallic, polypropylene or other polymeric casing; hollow fiber membrane cartridges containing hollow fiber microporous membranes enclosed by the metal casing, where the membrane cartridges divide the casing into a shell-side and a fiber side; at least baffle on the shell-side of the metal casing; at least one distribution tube on the fiber-side of the metal casing; two inlet ports connected to the external metal casing, where the lysed algal concentrate is pumped to the shell-side through the first inlet port and a strip gas or a solvent is fed to the fiber side through the second inlet port; and two outlet ports connected to the metal casing, where the algal raffinate containing the algal biomass is removed from the first outlet port and a solvent/extracted lipid or oil mixture or the strip gas is removed from the second outlet port.