A gas-turbine power plant operates on the regenerative Brayton cycle between the pressure limits of 100 and 700 kPa. Air enters the compressor at 30°C at a rate of 12.6 kg/s and leaves at 260°C. It is then heated in a regenerator to 400°C by the hot combustion gases leaving the turbine. A diesel fuel with a heating value of 42,000 kJ/kg is burned in the combustion chamber with a combustion efficiency of 97 percent. The combustion gases leave the combustion chamber at 871°C and enter the turbine, whose isentropic efficiency is 85 percent. Treating combustion gases as air and using constant specific heats at 500°C, determine (a) the isentropic efficiency of the compressor, (b) the effectiveness of the regenerator, (c) the air–fuel ratio in the combustion chamber, (d) the net power output and the back work ratio, (e) the thermal efficiency, and (f) the second-law efficiency of the plant. Also determine (g) the second-law efficiencies of the compressor, the turbine, and the regenerator, and (h) the rate of the exergy flow with the combustion gases at the regenerator exit.
a)
The isentropic efficiency of the compressor.
Answer to Problem 152P
The isentropic efficiency of the compressor is
Explanation of Solution
Draw the layout of the gas-turbine plant functioning on the regenerative Brayton cycle as shown in Figure (1).
Consider, the pressure is
Write the expression to calculate the temperature and pressure relation ratio for the isentropic compression process 1-2s.
Here, the specific heat ratio is k.
Write the expression to calculate the isentropic efficiency of the compressor
Write the expression to calculate the temperature and pressure relation ratio for the expansion process 3-4s.
Write the expression for the isentropic efficiency of the turbine
Conclusion:
From Table A-2b, “Ideal-gas specific heats of various common gases”, obtain the following values of air at
Substitute 303 K for
Substitute 303 K for
Thus, the isentropic efficiency of the compressor is
Substitute 1144 K for
Substitute 1144 K for
b)
The effectiveness of the regenerator for regenerative Brayton cycle.
Answer to Problem 152P
The effectiveness of the regenerator for regenerative Brayton cycle is
Explanation of Solution
Write the expression to calculate the effectiveness of the regenerator
Conclusion:
Substitute 673 K for
Thus, the effectiveness of the regenerator for regenerative Brayton cycle is
c)
The air-fuel ratio in the combustion chamber.
Answer to Problem 152P
The air-fuel ratio in the combustion chamber is
Explanation of Solution
Write the expression for the heat input for the regenerative Brayton cycle
Here, the specific heat at constant pressure is
Write the expression to calculate the air-fuel ratio in the combustion chamber (AF).
Write the expression to calculate the total mass of the air-fuel mixture
Write the expression to calculate the heat input for the regenerative cycle
Conclusion:
Substitute
Substitute
Thus, the air-fuel ratio in the combustion chamber is
Substitute
Substitute
d)
The net power developed by the gas-turbine plant and the back work ratio for the gas-turbine plant.
Answer to Problem 152P
The net power developed by the gas-turbine plant is
The back work ratio for the gas-turbine plant is
Explanation of Solution
Write the expression to calculate the power given to the compressor
Write the expression to calculate the power developed by the turbine
Write the expression to calculate the net power developed by the gas-turbine plant
Write the expression to calculate the back work ratio for the gas-turbine plant
Conclusion:
Substitute
Substitute
Substitute 3168 kW for
Thus, the net power developed by the gas-turbine plant is
Substitute 3168 kW for
Thus, the back work ratio for the gas-turbine plant is
e)
The thermal efficiency of the gas-turbine plant.
Answer to Problem 152P
The thermal efficiency of the gas-turbine plant is
Explanation of Solution
Write the expression to calculate the thermal efficiency of the gas-turbine plant
Conclusion:
Substitute 2266 kW for
Thus, the thermal efficiency of the gas-turbine plant is
f)
The second-law efficiency of the gas-turbine plant.
Answer to Problem 152P
The second-law efficiency of the gas-turbine plant is
Explanation of Solution
Write the expression to calculate the second-law efficiency of the gas-turbine plant
Here, the maximum possible efficiency of the gas-turbine plant is
Write the expression to calculate the maximum possible efficiency of the gas-turbine plant.
Conclusion:
Substitute 303 K for
Substitute 0.735 for
Thus, the second-law efficiency of the gas-turbine plant is
g)
The exergy efficiency for compressor , turbine and regenerator.
Answer to Problem 152P
The exergy efficiency for the compressor is
The exergy efficiency for the turbine is
The exergy efficiency for the regenerator is
Explanation of Solution
Write the expression to calculate the stream exergy difference between the inlet and exit of the compressor
Here, the temperature of the surroundings is
Write the expression to calculate the exergy efficiency for the compressor
Write the expression to calculate the stream exergy difference between the inlet and exit of the turbine
Write the expression to calculate the exergy efficiency for the turbine
Applying energy balance for the regenerator process.
Write the expression to calculate the exergy increase of the cold fluid for the regenerator
Write the expression to calculate the exergy decrease of the cold fluid for the regenerator
Write the expression to calculate the exergy efficiency for the regenerator
Conclusion:
Substitute
Substitute
Thus, the exergy efficiency for the compressor is
Substitute
Substitute
Thus, the exergy efficiency for the turbine is
substitute
Substitute
Substitute
Substitute
Thus, the exergy efficiency for the regenerator is
h)
The rate of exergy of the combustion gases at the regenerator exit.
Answer to Problem 152P
The rate of exergy of the combustion gases at the regenerator exit is
Explanation of Solution
Write the expression to calculate the rate of exergy of the combustion gases at the regenerator exit
Conclusion:
Substitute
Thus, the rate of exergy of the combustion gases at the regenerator exit is
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