A steam turbine operating under steady flow, conditions of steam at inletof steam turbine: p=13.8 bar, v=0.143 m3/kg, u=2590 kJ/kg, C= 30m/s. Thestate of the steam at the outlet leaving the turbine: p=0.35bar, v=4.37 m3/kg,u=2360 kJ/kg, C=90m/s. Heat is rejected to surroundings at the rate of 0.25kW and rate of steam flow through turbine is 0.38kg/s. Calculate powerdeveloped by the steam turbine.

To calculate the power developed by the steam turbine, we need to use the first law of thermodynamics for a steady-flow system, which states that the rate of energy entering the system minus the rate of energy leaving the system is equal to the rate of work done by the system.

The energy entering the system is the sum of the enthalpy (h), kinetic energy (KE), and potential energy (PE) of the steam at the inlet. The energy leaving the system is the sum of the enthalpy, kinetic energy, and potential energy of the steam at the outlet, plus the heat rejected to the surroundings.

The enthalpy of the steam can be calculated from the specific internal energy (u) and the pressure (p) and specific volume (v) of the steam, using the equation h = u + pv. The kinetic energy can be calculated from the velocity (C) of the steam, using the equation KE = 0.5*C^2. The potential energy is assumed to be zero, as there is no change in height.

First, calculate the enthalpy at the inlet and outlet:

h_inlet = u_inlet + p_inletv_inlet = 2590 kJ/kg + 13.8 bar0.143 m3/kg = 2590 kJ/kg + 1974 kJ/kg = 4564 kJ/kg

h_outlet = u_outlet + p_outletv_outlet = 2360 kJ/kg + 0.35 bar4.37 m3/kg = 2360 kJ/kg + 1530 kJ/kg = 3890 kJ/kg

Next, calculate the kinetic energy at the inlet and outlet:

KE_inlet = 0.5C_inlet^2 = 0.5(30 m/s)^2 = 450 J/kg

KE_outlet = 0.5C_outlet^2 = 0.5(90 m/s)^2 = 4050 J/kg

The rate of energy entering the system is the product of the mass flow rate (m_dot) and the sum of the enthalpy and kinetic energy at the inlet:

E_inlet = m_dot*(h_inlet + KE_inlet) = 0.38 kg/s*(4564 kJ/kg + 0.45 kJ/kg) = 1734.37 kW

The rate of energy leaving the system is the product of the mass flow rate and the sum of the enthalpy and kinetic energy at the outlet, plus the heat rejected to the surroundings:

E_outlet = m_dot*(h_outlet + KE_outlet) + Q_dot = 0.38 kg/s*(3890 kJ/kg + 4.05 kJ/kg) + 0.25 kW = 1477.69 kW

Finally, the power developed by the steam turbine is the difference between the rate of energy entering and leaving the system:

Power = E_inlet - E_outlet = 1734.37 kW - 1477.69 kW = 256.68 kW

So, the power developed by the steam turbine is approximately 256.68 kW.