Explanation Draw the T − s diagram for the Brayton cycle with regeneration as shown in Figure (1). Write the pressure ratio relation for the process 1-2. P r 2 = P 2 P 1 P r 1 (I) Here, relative pressure at state 1 is P r 1 and relative pressure at state 2 is P r 2 . Write the pressure ratio relation for the process 3-4. P r 4 = P 4 P 3 P r 3 (II) Here, relative pressure at state 3 is P r 3 and relative pressure at state 4 is P r 4 . Write the expression to calculate the net work output per unit mass w net . w net = W ˙ net m ˙ (III) Here, mass flow rate is m ˙ . Write the expression to calculate the thermal efficiency of the ( η t h ) . η t h = w net q in q in = w net η t h (IV) Write the expression for heat input per unit mass ( q in ) to calculate enthalpy at state 2 ( h 2 ) . q in = h 3 − h 2 h 2 = h 3 − q in (V) Here, enthalpy at state 3 is h 3 . Write the expression to calculate heat output per unit mass. q out = q in − w net (VI) Write the expression for heat output per unit mass during the process 4-1 to calculate enthalpy at state 4 ( h 4 ) . q out = h 4 − h 1 h 4 = q out + h 1 (VII) Write the expression for the efficiency of the compressor ( η C ) to calculate h 2 . η C = h 2 s − h 1 h 2 − h 1 (VIII) Here, enthalpy on isentropic condition 2 s is h 2 s . Write the expression for the efficiency of the turbine ( η T ) to calculate h 4 . η T = h 3 − h 4 h 3 − h 4 s (IX) Here, enthalpy on isentropic condition 4 s is h 4 s . Write the expression for the heat rejected by the regenerator ( q regen ) . q regen = ε ( h 4 − h 2 ) (X) Here, the effectiveness of the regenerator is ε and enthalpy at state 4 is h 4 . Write the expression for the new heat input to the gas turbine ( q in,new ) . q in,new = q in − q regen (XI) Write the expression to calculate the new thermal efficiency of gas turbine when regenerator is added with an effectiveness of 65 % percent ( η t h , n e w ) . η t h , n e w = w net,out q in (XII) Conclusion: From Table A-17, “Ideal-gas properties of air”, obtain the following properties at the temperature of 303 K . h 1 = 303.21 kJ / kg P r 1 = 1.4356 Here, enthalpy at state 1 is h 1 and relative pressure at state 1 is P r 1 . From Table A-17, “Ideal-gas properties of air”, obtain the following properties at the temperature of 1561 K . h 3 = 1710.0 kJ / kg P r 3 = 712.5 Here, enthalpy at state 3 is h 3 and relative pressure at state 3 is P r 3 . Substitute 14.7 for P 2 P 1 , and 1.4356 for P r 1 in Equation (I). P r 2 = ( 14.7 ) ( 1.4356 ) = 21.10 From the Table A-17, “Ideal-gas properties of air”. Obtain the value of enthalpy on isentropic state ( h 2 s ) at the relative pressure ( P r 2 ) of 21.10 by using interpolation method. Write the formula of interpolation method of two variables. y 2 = ( x 2 − x 1 ) ( y 3 − y 1 ) ( x 3 − x 1 ) + y 1 (XIII) Here, the variables denoted by x and y are relative pressure and enthalpy. Show relative pressure and enthalpy values from the Table A-17. Relative pressure ( P r 2 ) Enthalpy ( h 2 ) , in kJ / kg 20.64 649.22 21.10 ? 21.86 659.84 Substitute 20.64 for x 1 , 21.10 for x 2 , 21.86 for x 3 , 649.22 for y 1 , and 659.84 for y 3 in Equation (XIII). y 2 = ( 21.10 − 20.64 ) ( 659.84 − 649.22 ) ( 21.86 − 20.64 ) + 649.22 = 653.224 = 653.23 The value of enthalpy at the relative pressure of 21.10 by using interpolation method is 653.23 kJ / kg . Substitute 1 1.4356 for P 4 P 3 , and 712.5 for P r 3 in Equation (II)

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Author: CENGEL

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Chapter 9.12, Problem 109P

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The isentropic efficiencies of the turbine and the compressor and the thermal efficiency of gas turbine when regenerator is added with an effectiveness of 65% percent

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Chapter 9.12, Problem 108P

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The 7FA gas turbine manufactured by General Electric is reported to have an efficiency of 35.9 percent in the simple-cycle mode and to produce 159 MW of net power. The pressure ratio is 14.7 and the turbine inlet temperature is 1288°C. The mass flow rate through the turbine is 1,536,000 kg/h. Taking the ambient conditions to be 20°C and 100 kPa, determine: (a) the isentropic efficiency of the turbine and the compressor, (b) the thermal efficiency of this gas turbine if a regenerator with an effectiveness of 80 percent is added. Answers: (a) 0.849, 0.924 (b) 0.496

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A gas turbine operates in an ideal Brayton cycle of which Argon is used as the working substance. During the start of the compression, the temperature of the gas is at 340 K and pressure of 500kPa. The compression process is adiabatic with gas conditions at the outlet of 650 K and 1930 kPa. The working substance is heated and entered the turbine at 1400 K and 1930kPa and expands adiabatically to 900 K and 490kPa. Determine the compressor and turbine efficiencies, the actual compressor and turbine work, the net work of the cycle, the thermal efficiency, and the new thermal efficiency when assuming that a regenerator is installed with 85% effectiveness. Draw the schematic diagram.

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The isentropic efficiencies of the turbine and the compressor and the thermal efficiency of gas turbine when regenerator is added with an effectiveness of 65 % percent

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