NOVELTY - The method involves operating a variable speed single-phase LNG expander (X1) to process nitrogen-rich feed gas. Rotational speed of a variable speed two-phase LNG expander (X2) contained within a surrounding vessel is set to decrease boil-off gas and optimize ratio between liquid LNG (LLNG) and vapor LNG (VLNG). The two-phase expander is operated to process nitrogen-rich LNG. Nitrogen from the LNG is separated by evaporation such that LNG sub-cooling process is optimized to produce greatest volume of sub-cooled liquid LNG to reduce boil-off losses.    USE - Method for performing optimum sub-cooling of LNG i.e. methane, to reduce boil-off losses at a natural gas liquefaction plant.    ADVANTAGE - The method enables reducing investment payback time of the LNG expanders, increasing overall plant profit by using the two-phase LNG expander in a LNG plant despite of gas well depletion, and optimizing output of the expanders in series by varying rotational speed of each expander independently so as to obtain coldest liquid LNG by optimizing ratio of temperature and mass flow rate, reduce energy costs and feed gas supply in the liquefaction plant by reducing LNG vaporization and LNG boil-off downstream of the expanders, thus extending lifetime of a gas well, and increasing production from the gas well. The two-phase expander expands the LNG for relieving full pressure, thus avoiding the need to utilize Joule Thompson valve. The method enables injecting nitrogen into natural gas at a liquefaction site, thus avoiding the need to pressurize well as the natural gas is under pressure in the well, and evaporating nitrogen from the LNG by thermodynamic process to remove evaporation heat from remaining LNG, thus sub-cooling remaining LNG to reduce boil-off losses. The variable speed single-phase expander, variable speed two-phase expander and a phase separator (PHS) are mounted together so as to avoid unnecessary losses in a piping system.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view illustrating liquefaction process for performing optimum sub-cooling of LNG by using single-phase and two-phase LNG expanders operated at variable rotational speed.    Liquid LNG (LLNG)    Main heat exchanger (MHE)    Outlet pressure (P2)    Vapor LNG (VLNG)    Single-phase expander (X1)    Two-phase liquid expander (X2)