A gas turbine operates with an inlet temperature of $1200$C and an inlet pressure of $4$ MPa. The gas expands isentropically to a pressure of $1$ MPa. Calculate the work done by the turbine per kilogram of gas, the change in enthalpy, and the exit temperature assuming air behaves as an ideal gas with a specific heat ratio of $1\mathrm{.}4.$Air flows through a converging nozzle with an inlet pressure of $300$ kPa, an inlet temperature of $400$ K$,$ and an inlet velocity of $200$ m$/$s$.$ The exit pressure is $100$ kPa. Calculate the exit temperature, exit velocity, and exit area assuming isentropic flow.A steel beam with a length of $6$ meters and a cross$-$sectional area of $0\mathrm{.}05$ m$^2$ is subjected to a uniformly distributed load of $10$ kN$/$m$.$ Calculate the maximum bending stress in the beam, the maximum deflection if the modulus of elasticity for steel is $200$ GPa, and the shear force at the ends of the beam.A heat exchanger transfers heat from hot water entering at $70$C and leaving at $50$C to cold water entering at $20$C and leaving at $40$C$.$ The hot water flow rate is $1$ kg$/$s$,$ and the cold water flow rate is $0\mathrm{.}5$ kg$/$s$.$ Calculate the rate of heat transfer in the exchanger and the effectiveness of the heat exchanger.A gear train consists of two gears with $24$ and $36$ teeth, respectively. If the smaller gear rotates at $1200$ rpm$,$ calculate the speed of the larger gear and the torque transmitted if the power input is $8$ kW$.$

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