NOVELTY - The system comprises a reaction vessel comprising an upper section having a fatty acid feedstream introduced to the reaction vessel, a lower section having an alcohol vapor introduced to the reaction vessel and an esterification zone maintained under esterification conditions, a conversion unit, a fat splitter, and a pre-esterification unit for pre-esterification of the fatty acid feedstream before introduction of the fatty acid feedstream into the reaction vessel. The esterification zone comprises esterification trays mounted one above the other.    USE - The system is useful for preparing biodiesel from a fatty acid feedstream.    ADVANTAGE - The system is capable of simply, continuously and effectively preparing the biodiesel from the fatty acid feedstream with high quality and low glycerin, water and sulfur content.    DETAILED DESCRIPTION - The system comprises a reaction vessel comprising an upper section having a fatty acid feedstream introduced to the reaction vessel, a lower section having an alcohol vapor introduced to the reaction vessel and an esterification zone maintained under esterification conditions, a conversion unit, a fat splitter, and a pre-esterification unit for pre-esterification of the fatty acid feedstream before introduction of the fatty acid feedstream into the reaction vessel. The esterification zone comprises esterification trays mounted one above the other, and includes an ion exchange resin catalyst with sulfonic acid or carboxyl functional groups. Each tray has a predetermined liquid volume capacity, and is charged with a heterogeneous ion exchange resin catalyst. The alcohol vapor and fatty acid feedstream are catalytically esterified in the esterification zone to produce a fatty acid alkyl ester product stream (10) and water. The water is removed by unreacted alcohol vapor. The unreacted alcohol vapor is recycled to the lower section of the reaction vessel. The conversion unit removes impurities from the fatty acid alkyl ester product stream by distillation, reboiled stripping, inert gas stripping, physical adsorption and gravity separation. The fatty acid feedstream is a product of introducing a triglyceride feedstock (1) into the fat splitter to produce a fatty-acid rich feedstream. The catalyst is present in structured packing. A catalyst concentration on the trays is 2-20 wt./vol.%. The biodiesel comprises a water content of less than 0.050 vol.%, a kinematic viscosity of 1.9-6 mm2/s, a cetane number of greater than 47, and a cloud point of less than -40 degrees C.    DESCRIPTION OF DRAWING(S) - The diagram shows a schematic view of a system for preparing biodiesel from a fatty acid feedstream.    Fatty acid feedstock (1)    Alcohol (3)    Column (5)    Reaction zone (6)    Alcohol/water separation unit (8)    Product stream separation unit (11)    Product stream. (10)