Explanation Draw the T − s diagram for regenerative Brayton cycle as shown in Figure (1). Write the isentropic relation to obtain the temperature at the exit of the compressor. T 2 = T 1 ( P 2 P 1 ) ( k − 1 ) / k = T 1 ( r p ) ( k − 1 ) / k Here, the pressure ratio for the cycle is r p , and the specific heat ratio is k . Write the isentropic relation to obtain the temperature at the exit of the turbine. T 4 = T 3 ( P 4 P 3 ) ( k − 1 ) / k = T 3 ( 1 r p ) ( k − 1 ) / k = T 3 r p ( 1 − k ) / k Write the expression to obtain the heat input for the ideal regenerative Brayton cycle ( q in ) . q in = h 3 − h 5 = c p ( T 3 − T 5 ) (I) Here, the specific heat of the working fluid at constant pressure is c p . Write the expression to obtain the heat rejection for the ideal regenerative Brayton cycle ( q out ) . q out = h 6 − h 1 = c p ( T 6 − T 1 ) (II) Write the expression to obtain the net work output for the ideal regenerative Brayton cycle ( w net ) . w net = q in − q out (III) Write the expression to obtain the thermal efficiency of the ideal regenerative Brayton cycle ( η th ) . η th = 1 − q out q in (IV) Conclusion: Substitute T 4 for T 5 , T 3 r p ( 1 − k ) / k for T 4 in Equation (I). q in = c p ( T 3 − T 3 r p ( 1 − k ) / k ) = c p T 3 ( 1 − r p ( 1 − k ) / k ) Substitute T 2 for T 6 , T 1 r p ( k − 1 ) / k for T 1 in Equation (II). q out = c p ( T 1 r p ( k − 1 ) / k − T 1 ) = c p T 1 ( r p ( k − 1 ) / k − 1 ) Substitute c p T 1 ( r p ( k − 1 ) / k − 1 ) for q out , and c p T 3 ( 1 − r p ( 1 − k ) / k ) for q in in Equation (III). w net = c p T 3 ( 1 − r p ( 1 − k ) / k ) − c p T 1 ( r p ( k − 1 ) / k − 1 ) = c p ( T 3 − T 3 r p ( 1 − k ) / k − T 1 r p ( k − 1 ) / k + T 1 ) Write the relation to maximize the net work output of the cycle. ∂ w net ∂ r p = 0 . (VI) Substitute c p ( T 3 − T 3 r p ( 1 − k ) / k − T 1 r p ( k − 1 ) / k + T 1 ) for w net in Equation (VI). ∂ ∂ r p [ c p ( T 3 − T 3 r p ( 1 − k ) / k − T 1 r p ( k − 1 ) / k + T 1 ) ] = 0 Differentiate w net partially with respect to r p

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Chapter 9.12, Problem 174RP

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The pressure ratio that maximizes the thermal efficiency of the cycle.

Compare the pressure ratio for maximum thermal efficiency with the pressure ratio that maximizes the cycle net work.

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Chapter 9.12, Problem 173RP

Chapter 9.12, Problem 184FEP

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The pressure ratio that maximizes the thermal efficiency of the cycle. Compare the pressure ratio for maximum thermal efficiency with the pressure ratio that maximizes the cycle net work.

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The pressure ratio that maximizes the thermal efficiency of the cycle.

Compare the pressure ratio for maximum thermal efficiency with the pressure ratio that maximizes the cycle net work.