NOVELTY - The system for trans-esterifying a triglyceride, comprises a reaction tank (20) inputted with the triglyceride and a monohydric alcohol for undergoing a trans-esterification reaction to produce a mixture, a separation tank (24) inputted with the mixture for a phase separation to generate an ester phase solution and a glycerol phase solution having first and second parts, a first pipe connecting the reaction tank and the separation tank and transporting the mixture from the reaction tank to the separation tank, and a second pipe (26) connecting the reaction tank and the separation tank.    USE - The system is useful for trans-esterifying a triglyceride to produce biodiesel (claimed), which is useful for diesel engine of the vehicles.    ADVANTAGE - The system is capable of economically and rapidly trans-esterifying the triglyceride with high reaction rate and less energy consumption, achieves the trans-esterification mole conversion rate of 99%, can be economically constructed, operated and maintained, can effectively reduce the quantity of the compounds needed to be separated and purified, and can shrink the scale of the separation and purification apparatus.    DETAILED DESCRIPTION - The system for trans-esterifying a triglyceride, comprises a reaction tank (20) inputted with the triglyceride and a monohydric alcohol for undergoing a trans-esterification reaction to produce a mixture, a separation tank (24) inputted with the mixture for a phase separation to generate an ester phase solution and a glycerol phase solution having first and second parts, a first pipe connecting the reaction tank and the separation tank and transporting the mixture from the reaction tank to the separation tank, a second pipe (26) connecting the reaction tank and the separation tank and transporting the first part of the glycerol phase solution from the separation tank to the reaction tank for undergoing the trans-esterification reaction, a purification apparatus (22) inputted with the ester phase solution and the second part of the glycerol solution to produce a purified monohydric alcohol, a purified fatty acid alkyl ester and a purified glycerol, a third pipe connecting the separation tank and the purification apparatus and transporting the ester phase solution from the separation tank to the purification apparatus, a fourth pipe (28) connecting the separation tank and the purification apparatus and transporting the second part of the glycerol phase solution from the separation tank to the purification apparatus, and a fifth pipe connecting the purification apparatus and the reaction tank and transporting the purified monohydric alcohol from the purification apparatus to the reactor for undergoing the trans-esterification. The triglyceride and the monohydric alcohol are heated and stirred in the reaction tank so as to promote the trans-esterification reaction. The triglyceride is purified from natural plant or edible oil. An INDEPENDENT CLAIM is included for a method for trans-esterifying a triglyceride.    DESCRIPTION OF DRAWING(S) - The diagram shows a schematic view of a trans-esterification reaction system.    Reaction tank (20)    Purification apparatus (22)    Separation tank (24)    Second pipe (26)    Fourth pipe. (28)