Consider a two$-$stage cascade refrigeration system operating between the pressure limits of $1\mathrm{.}6$ MPa and $140$ kPa with refrigerant$-134$a as the working fluid. Heat rejection from the lower cycle to the upper cycle, takes place in an adiabatic counterflow heat exchanger where the pressure in the upper and lower cycles are $0\mathrm{.}4$ and $0\mathrm{.}6$ MPa, respectively. In both cycles, the refrigerant is a saturated liquid at the condenser exit and a saturated vapour at the compressor inlet, If the mass flow rate of the refrigerant through the lower cycle is $0\mathrm{.}4$ kg$/$s$,$ determine the mass flow rate of the refrigerant through the upper cycle,