A numerical solution to a two-dimensional model of flow and transient heat transfer involving solidification in a pipe has been established. Where the temperature of pipe wall is below the freezing point of fluid, phase change of flowing fluid and the influence of different boundary condition, such as pipe wall temperature, initial temperature and inlet velocity has been taken into account. Also it has been investigated to elicit proper non-dimensional numbers to show the solidification proceeding results. Additionally comparing the two acceptable inlet conditions, show distinctions between velocity inlet and pressure inlet boundary condition in such problems, which affect the whole freezing process.