$2.(25$ points$)$ In steam power plants, open feedwater heaters are frequently utilized to heat the feedwater by mixing it with steam bled off the turbine at some intermediate stage. Consider an open feedwater heater that operates at a pressure of $1000$ kPa $.$ Feed water at $\backslash (50^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and $1000$ kPa is to be heated with superheated steam at $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and $1000$ kPa $.$ In an ideal feedwater heater, the mixture leaves the heater as saturated liquid at the feedwater pressure. Determine the ratio of the mass flow rates of the feedwater $(\backslash (\backslash$dot$\{$m$\}\_\{1\}\backslash ))$ and the superheated vapor $(\backslash (\backslash$dot$\{$m$\}\_\{2\}\backslash ))$ for this case. State your assumptions clearly.