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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