A method includes directing a refrigerant fluid mixture and a flow of natural gas through a first heat exchanger for exchanging heat between a natural gas flow path and a first refrigerant flow path. The method also includes expanding the flow of natural gas exiting from the first heat exchanger via a first throttle valve. Further, the method also includes directing a generated cold natural gas vapor and a slurry having a liquefied natural gas and solidified carbon dioxide through a filter sub-assembly. Moreover, the method also includes separating the solidified carbon dioxide by the filter sub-assembly to form a purified liquefied natural gas. Finally, the method includes directing a pulse of a cleaning fluid having at least one of methane and carbon dioxide through the filter sub-assembly to remove the solidified carbon dioxide therefrom and storing the purified liquefied natural gas in a storage tank assembly.