AA283 Homework 5, 2019 - 2020 Due Tuesday February 11 Reading: Chapter 4 lecture slides on gas turbine power generation, Chapter 5 Problem - In class we discussed the example of a power gas turbine equipped with a recuperator. In that example we assumed that the polytropic efficiency of the compressor was less than one and that the turbine was ideal with a polytropic efficiency equal to one. Analyze the more realistic case where the turbine can also have a polytropic efficiency less than one. Plot the maximum overall efficiency of the engine versus η = η pc η pe for several values of τ λ . Chapter 5 – Problem 4 Problem (from a previous midterm) – A turbojet engine is at rest, set for takeoff. The inlet, compressor, burner, turbine and nozzle operate ideally. Assume f << 1 . The ambient temperature is T 0 = 300 K and the turbine inlet temperature is T t 4 = 1250 K . Relevant area ratios are A 8 / A 4 * = 2 , A 4 * / A 2 = 1/8 . The convergent-divergent nozzle is fully expanded, P e = P 0 . The ambient pressure is 10 5 N / m 2 . Part I 1) Determine τ t and π t . 2) Determine τ c and π c . 3) Determine f M 2 ( ) . 4) Determine the dimensionless thrust, T /( ! m a a 0 ) . Part II - The same engine is modified with an aft fan similar to the design of the GE CJ700 shown in the figure where the low pressure turbine and fan are concentric blade rows of a single rotating disc. Assume f << 1 . The ambient temperature is T 0 = 300 K and the turbine inlet temperature is T t 4 = 1250 K . The turbine inlet to compressor entrance area ratio is the same, A 4 * / A 2 = 1/8 . The ambient pressure is 10 5 N / m 2 . The operating point of the engine up to station 5 remains the same as you have already determined in Part I items 1, 2 and 3. The area ratio A e / A 4 * , and the turbine of the aft fan are designed so that at the operating point shown, the engine core exit condition is P e = P 0 and M e = 1 . Both nozzles are simple convergent nozzles. Assume the aft fan and turbine are ideal. The bypass ratio of the aft fan is chosen to be β = 3.09 . Determine: 1) The pressure and temperature ratios across the aft turbine P t 6 / P t 5 and T t 6 / T t 5 and A e / A 4 * . 2) The pressure and temperature ratios across the aft fan, T t 16 / T 0 , P t 16 / P 0 and the fan exit Mach number M 1e and exit static pressure P 1e . 3) Determine the dimensionless thrust, T /( ! m a a 0 ) . Part III – To produce the CJ700 the engineers cleverly took the J85, a very reliable existing military turbojet designed for supersonic flight, and they replaced the back end with a fan creating a new engine suitable for efficient propulsion of several types of subsonic commercial business jets. The area ratios and turbine inlet temperature specified in this exam are very close to those of the actual engines, but the bypass ratio of the CJ700 is 2.0 not 3.09. Why?

AA283 Homework 5 2019 20 - web.stanford.educantwell/AA283_Course...Part III – To produce the CJ700 the engineers cleverly took the J85, a very reliable existing military turbojet

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Page 1: AA283 Homework 5 2019 20 - web.stanford.educantwell/AA283_Course...Part III – To produce the CJ700 the engineers cleverly took the J85, a very reliable existing military turbojet

Chapter 5 – Problem 4 Problem (from a previous midterm) – A turbojet engine is at rest, set for takeoff. The inlet, compressor, burner, turbine and nozzle operate ideally. Assume f <<1 . The ambient temperature is T0 = 300K and

the turbine inlet temperature isTt 4 = 1250K . Relevant area ratios are A8 / A4* = 2 , A4

* / A2 = 1 / 8 . The

convergent-divergent nozzle is fully expanded, Pe = P0 . The ambient pressure is 105N /m2 .

Part I 1) Determine τ t and π t . 2) Determine τ c and π c .

3) Determine f M 2( ) .

4) Determine the dimensionless thrust, T / ( !maa0 ) . Part II - The same engine is modified with an aft fan similar to the design of the GE CJ700 shown in the figure where the low pressure turbine and fan are concentric blade rows of a single rotating disc. Assume f <<1 . The ambient temperature is T0 = 300K and the turbine inlet temperature isTt 4 = 1250K . The

turbine inlet to compressor entrance area ratio is the same, A4* / A2 = 1 / 8 . The ambient pressure is

105N /m2 . The operating point of the engine up to station 5 remains the same as you have already determined in Part I items 1, 2 and 3.

The area ratio Ae / A4

* , and the turbine of the aft fan are designed so that at the operating point shown, the engine core exit condition is Pe = P0 and Me = 1 . Both nozzles are simple convergent nozzles. Assume the aft fan and turbine are ideal. The bypass ratio of the aft fan is chosen to be β = 3.09 . Determine:

1) The pressure and temperature ratios across the aft turbine Pt6 / Pt5 and Tt6 /Tt5 and Ae / A4* .

2) The pressure and temperature ratios across the aft fan, Tt16 /T0 , Pt16 / P0 and the fan exit Mach number M1e and exit static pressure P1e . 3) Determine the dimensionless thrust, T / ( !maa0 ) . Part III – To produce the CJ700 the engineers cleverly took the J85, a very reliable existing military turbojet designed for supersonic flight, and they replaced the back end with a fan creating a new engine suitable for efficient propulsion of several types of subsonic commercial business jets. The area ratios and turbine inlet temperature specified in this exam are very close to those of the actual engines, but the bypass ratio of the CJ700 is 2.0 not 3.09. Why?