$(28$ points$)$ In a closed heat exchanger, water is supplied to the state$-3$ at $5$ bars with the quality $1$ and leave at $T_4=180C($state$-4),$ and $T_5=320C($state$-5).$ As shown in the figure, wasted hot Nitrogen is supplied to the state$-1$ with the temperature of $550K.$ This $N_2$ leaves at state$-2.$ The $92\%$ rate of heat transfer from the Nitrogen stream are transferred to the water stream. There is no pressure drop on water or nitrogen line. The summary of each state as follows:

State$-1$: $T_1=550K,N_2,$ State$-2$: $N_2$

State$-3$: Water, with quality $1,P_3=5$ bars, State $-4$ : Water, $T_4=180C,P_4=5$ bars

State$-5$: Water, $T_4=320C,P_5=5$ bars, Mass flow rate of the Nitrogen at the State$-1=$ the mass flow rate of the water at State$-3,$ The water supplied into the State$-3$ will be divided equally to the State$-4$ and State$-5,$ The mass flow rate at the State$-4=$ the mass flow rate of the water at State$-5.$ Calculate:

$($a$)$ The rate of heat transfer per unit mass into water, in $k\frac{J}{k}g$

$($b$)$ The enthalpy at State$-2,$ in $k\frac{J}{k}g$