Explanation Draw the T − s diagram for pure jet engine as shown in Figure (1). Consider that the aircraft is stationary, and the velocity of air moving towards the aircraft is V 1 = 1200 ft / s , the air will leave the diffuser with a negligible velocity ( V 2 ≅ 0 ) . Diffuser : Write the expression for the energy balance equation for the diffuser. E ˙ in − E ˙ out = Δ E ˙ system (I) Here, the rate of energy entering the system is E ˙ in , rate of energy leaving the system is E ˙ out , and the rate of change in the energy of the system is Δ E ˙ system . Write the temperature and pressure relation for the process 1-2. P 2 = P 1 ( T 2 T 1 ) k / ( k − 1 ) (II) Here, the specific heat ratio of air is k , pressure at state 1 is P 1 , pressure at state 2 is P 2 , temperature at state 1 is T 1 and temperature at state 2 is T 2 . Compressor: Write the pressure relation using the pressure ratio for the process 2-3. P 3 = P 4 = ( r p ) ( P 2 ) (III) Here, the pressure ratio is r p , pressure at state 3 is P 3 and pressure at state 4 is P 4 . Write the temperature and pressure relation for the process 2-3. T 3 = T 2 ( P 3 P 2 ) ( k − 1 ) / k T 3 = T 2 ( r p ( k − 1 ) / k ) (IV) Here, temperate at state 3 is T 3 . Turbine: Write the temperature relation for the compressor and turbine. w c o m p , i n = w t u r b , o u t h 3 − h 2 = h 4 − h 5 c p ( T 3 − T 2 ) = c p ( T 4 − T 5 ) T 3 − T 2 = T 4 − T 5 T 5 = T 4 − T 3 + T 2 (V) Here, the specific heat at constant pressure is c p , enthalpy at state 2 is h 2 , enthalpy at state 3 is h 3 , enthalpy at state 4 is h 4 , enthalpy at state 5 is h 5 , work input to the compressor is w c o m p , i n , work output from turbine is w t u r b , o u t , temperature at state 4 is T 4 and temperature at state 5 is T 5 . Nozzle: Write the temperature and pressure relation for the isentropic process 4-6. T 6 = T 4 ( P 6 P 4 ) ( k − 1 ) / k (VI) Here, pressure at state 6 is P 6 and temperature at state 6 is T 6 . Write the energy balance equation for the nozzle. E ˙ in − E ˙ out = Δ E ˙ system (VII) Write the expression to calculate the specific impulse of the jet engine. F m ˙ = ( V e x i t − V i n l e t ) (VIII) Here, the thrust force produced by engine is F , mass flow rate of air is m ˙ , inlet velocity of air is V inlet , and outlet velocity of air is V exit . Conclusion. From Table A-1E, “Molar mass, gas constant, and critical-point properties”, obtain the value of gas constant ( R ) for air substance as 0.3704 psia ⋅ ft 3 / lbm ⋅ R . From Table A-2Ea, “Ideal-gas specific heats of various common gases”, obtain the following values for air at room temperature. k = 1.4 c p = 0.24 Btu / lbm ⋅ R The rate of change in the energy of the system ( Δ E ˙ system ) is zero for the steady state system. Substitute 0 for Δ E ˙ system Equation (I). E ˙ in − E ˙ out = 0 E ˙ in = E ˙ out h 1 + V 1 2 2 = h 2 + V 2 2 2 0 = h 2 − h 1 + V 2 2 − V 1 2 2 0 = c p ( T 2 − T 1 ) − V 2 2 − V 1 2 2 (IX) Here, inlet velocity is V 1 or V i n l e t and velocity at state 2 is V 2 . Substitute 0 for V 2 , 30 ° F for T 1 , 1200 ft / s for V 1 , and 0.24 Btu / lbm ⋅ R for c p in Equation (IX). 0 = 0.24 Btu / lbm ⋅ R ( T 2 − ( 30 ° F ) ) − V 2 2 − ( 1200 ft / s ) 2 2 T 2 = ( 30 + 460 ) R + ( 1200 ft / s ) 2 ( 2 ) ( 0

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Author: CENGEL

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Chapter 9.12, Problem 177RP

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Solution Summary: The author explains the specific impulse of the jet engine and the energy balance equation for the diffuser.

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