TECHNOLOGY FOCUS - BIOLOGY - Preferred Components: The oil is derived from: biomass, algae, animal tallow and/or waste oil. The fatty acid alkyl ester is a fatty acid methyl ester.    TECHNOLOGY FOCUS - INSTRUMENTATION AND TESTING - Preferred Method: The step of inputting the microwave energy is carried out for a first period of time, and the step of inputting the ultrasonic energies carried out for a second period of time, where the first and second periods of: time at least partially overlap; or time do not overlap. The reaction is carried out in a glass or glass-lined vessel. In the method, a stream of the reaction mixture is flowed through a reactor in which the microwave energy and the ultrasonic energy are input to the stream, where the reaction is carried out in a continuous process. The inputting of the ultrasonic energy produces an emulsion and the inputting of microwave energy further comprises inputting microwave energy into the emulsion: to produce one or more fatty acid alkyl esters; and to allow the transesterification to proceed at a temperature below the critical temperature of the alcohol. The reaction is carried out at temperature of at least 200, 190, 180, 170, 160, 150, 140, 13, 120, 110, 90, 80, 70, 60, 50, or 40 degrees C, preferably less than 125 degrees C or 40-150, 50-140, 60-130, 65 to 145, 70-120, or 60-110 degrees C below the supercritical temperature of the alcohol. The two immiscible reactants (at least 90, 95, 96, 97, 98, or 99%) are emulsified to an emulsion by the ultrasonic energy, and microwave energy is inputted into the emulsion. The apparatus further comprises a controller for controlling the operation of one or more of the source of microwave energy and the source of ultrasonic energy. The reaction vessel is configured and operable to have a reaction mixture circulated in it. The reactor further comprises a pump for circulating the reaction mixture through the reaction vessel. The second period of time is commenced before the first period of time is commenced. The inputting of ultrasound energy is commenced before the inputting of microwave energy. Preferred Components: The inputting of the ultrasonic energy produces an emulsion that has an average particle size of less than 5 micrometers, where 90% of particles in the emulsion have a particle size of less than 10, 9, 8, 7, 6, or 5 micrometers. The ultrasonic energy has a frequency comprising a frequency of at least 22 KHz., preferably 22, 23, 24, 25 or 44 KHz. The microwave energy has a frequency of 2.45 GHz or 915 MHz.    TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Method: In the method, provided that when the reaction contains an esterification reaction between n-butanol and stearic acid, where the reaction mixture does not comprise k catalyst. The method further comprises detecting one or more fatty acid alkyl esters, where the fatty acid alkyl esters are detectably produced in less than 5, 4, 3, 2, or 1 hour; in less than 59, 50, 40, 30, or 20 minutes; or in less than 10, 9,5 8, 7, 6, 5, 4, 3, 2, or 1 minute. The reaction comprises an esterification reaction and/or a transesterification reaction. The immiscible reactants are an alcohol and an oil, and the reaction mixture further optionally comprises a catalyst, where the ratio of the alcohol to the oil is 9:1-1:1, or 8:1-1:1. Preferred Components: The oil is a vegetable oil, where the alcohol comprises 1-5 carbon alkyl alcohol.