Consider a cryogenic energy storage $($CES$)$ system in which nitrogen is liquefied during off$-$peak hours using surplus electricity generated by wind turbines and stored in a $515$m$3515-$m$3$ cryogenic tank at T$1=200$C$$and$$P$1=0\mathrm{.}12$MPa$.$T$1=200$C$$and$$P$1=0\mathrm{.}12$MPa$.$ During peak times, the entire liquid nitrogen is pumped to P$2=15$MPaP$2=15$MPa and is heated by ambient air to T$3=20$CT$3=20$C at that pressure before it is routed to a nitrogen turbine where it expands to a pressure of P$4=100$kPaP$4=100$kPa $.$ The properties of nitrogen are h$1=440\mathrm{.}1$kJ$/$kg$,$$1=825\mathrm{.}9$kg$/$m$3,$h$3=33\mathrm{.}36$kJ$/$kg$,$and$$h$4$s$=245\mathrm{.}5$kJ$/$kgh$1=440\mathrm{.}1$kJ$/$kg$,$$1=825\mathrm{.}9$kg$/$m$3,$h$3=33\mathrm{.}36$kJ$/$kg$,$and$$h$4$s$=245\mathrm{.}5$kJ$/$kg $($enthalpy at turbine exit for the case of isentropic expansion$).$

a$)$ Determine the work consumed by the pump and the work produced by the turbine if the isentropic efficiency of the pump is $73$ percent and that of the turbine is $82$ percent. $($Round the final answers to the nearest whole numbers.$)$

b$)$ What fraction of work consumed during the liquefaction process is recovered when the stored liquid nitrogen is used to generate work? Assume that it takes $0\mathrm{.}8$ kWh of electricity to liquefy a unit mass of nitrogen. $($Round the final answer to four decimal places.$)$