Provided is a stirred tank polymerization reactor system having a reactor tank, a stirring assembly including a rotatable shaft that extends through a wall of the tank, and a triple barrier mechanical seal. The mechanical seal includes an outer cylinder mounted in the wall of the tank, an inner cylinder that is rotatable relative to the outer cylinder and connected to the shaft, and first, second, and third fluid barrier seals mounted between the outer and inner cylinders at different positions along the axis of rotation of the inner cylinder. First and second sources of pressurized barrier fluid are connected between the first and second pressure fluid barrier seals and the second and third barrier seals, respectively. The three barrier seals advantageously “step down” the differential pressure experienced by the uppermost barrier seals, substantially reducing mechanical stresses experienced by the annular sealing rings and seats. The mechanical seal can further include a closed loop cooling channel in its outer cylinder that is connected to a circulating source of cooling fluid.