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Home , Propulsive efficiency, Turbofan

Section 6.1: The Propulsion Cycle

MAE 6530 - Propulsion Systems II 1 Overview

MAE 6530 - Propulsion Systems II 2 Overview (2) • Turbofan is the most modern variation of the basic gas engine. • As with other gas , there is a core engine, whose parts and operation are nearly identical to the operation. • In the turbofan engine, the core engine is surrounded by a in the front and an additional turbine at the rear.

MAE 6530 - Propulsion Systems II 3 Overview (3)

MAE 6530 - Propulsion Systems II 4 Overview(4) • Turbofan power now all civil transports flying at speeds up to Mach 0.9. • Several advantages to turbofan engines over both -driven and turbojet engines • By enclosing fan inside a duct or cowling, are better controlled. • Fan is not as large as propeller, so increase of speeds along blade is less, and there is less chance of tip stall and shock wave development. • Turbofan can suck in more total air massflow than a turbojet, thus offer potential for generating more . • Turbofan has low fuel consumption compared with turbojet.

MAE 6530 - Propulsion Systems II 5 Performance Efficiency (revisited) Ratio of Power Developed from Engine (desired beneficial output) Thrust to Change in Kinetic Energy of the Moving Airstream (cost of propulsion)

Propulsive power

Kinetic energy production rate

Maximum propulsive efficiency achieved by generating thrust moving as much air as possible with as little a change in velocity across the engine as possible. MAE 6530 - Propulsion Systems II 6 Jet Engine Performance Efficiency (2) • Maximum propulsive efficiency achieved by generating thrust moving as much air as possible with as little a change in velocity across the engine as possible. .. Fan provides that function

MAE 6530 - Propulsion Systems II 7 Jet Engine Performance Efficiency (3) • Recall that optimal thrust level of a turbo jet is characterized by

1 ⎛ 1 ⎞ ⎛ γ ⎞ τ = ⋅ ⎜1+ ⎟τ ⎜ γ−1 ⎟ ( c )opt ⎜ ⎟ λ ⎜ ⎟ τ ⎝⎜ f ⎠⎟ (Πc ) =⎜τc ⎟ = r opt ⎜ ⎟ ⎝ ⎠opt

• As supersonic flight Mach become larger, compression goes down until

1 ⎛ 1 ⎞ f 1= ⋅ ⎜1+ ⎟τ → τ = τ 2 ⎜ ⎟ λ λ r τr ⎝ f ⎠ f +1 …. optimal solution gets rid of all together, and best engine becomes a ! 8 MAE 6530 - Propulsion Systems II Jet Engine Performance Efficiency (4) • We are going to show that similar effect occurs as Mach drops significantly below Mach 1

• Trend emerges that replaces turbojet with Turbofan at lower Mach numbers

Decreasing

9 MAE 6530 - Propulsion Systems II Classification of Turbofan Engines

MAE 6530 - Propulsion Systems II 10 Classification of Turbofan Engines (2) • Turbofan engines may be classified based on fan location as either forward or aft fan. • Based on a number of spools, engine may be classified as single, double, or triple spools. •Based on a , engine may be categorized as either low- or high- bypass ratio. • Fan may be geared or ungeared to its driving low-pressure turbine. • Mixed-flow types (where flow merges in nozzle) may be fitted with .

MAE 6530 - Propulsion Systems II 11 Classification of Turbofan Engines (3) • High-bypass ratio turbofan (β > 5) achieves 75 % of thrust from bypass air - Ideal for subsonic transport . e.g. - Rolls-Royce Trent series (, A340, A350, A380), - Pratt & Whitney PW1000 G (geared) (Airbus A320neo, Bombardier CSeries, Embraer E2, Mitsubishi MC-21) - General Electric GE90 powering -300ER, 747.

• Low-bypass ratio turbofan β ~ 1 achieves approximately equal thrust distribution between gas generator and bypass duct - Well suited to high-speed military applications. e.g. - Rolls-Royce RB199 in the Tornado - Pratt & Whitney F100-PW-200 in F-16A/B and F-15 - EuroJet Turbo GmbH EJ200 powering the Typhoon Fighter

MAE 6530 - Propulsion Systems II 12 Classification of Turbofan Engines (4)

• Unmixed Bypass Flow Turbofan 2-Spool High-bypass turbofan engine with a unmixed exhaust, showing the fan (pink) and core (red) exhaust streams

• Mixed Bypass Flow Turbofan 2-spool, low-bypass turbofan engine with a mixed exhaust, showing the low-pressure (green) and high-pressure (purple) spools.

MAE 6530 - Propulsion Systems II 13 Classification of Turbofan Engines (5) High Bypass Ratio

• Modern high bypass ratio engine designed for long distance cruise at subsonic Mach numbers around 0.83. • Fan uses a single stage composed of a large diameter fan (rotor) with blades followed by a single nozzle stage (stator). • Bypass ratio is ~6 and the fan pressure ratio is ~2.

MAE 6530 - Propulsion Systems II 14 Classification of Turbofan Engines (6) Low Bypass Ratio

• Military turbofan designed for high performance at supersonic Mach numbers in the range of 1.1 to 1.5. • Fan has three stages with an of about 6 and a bypass ratio of only about 0.6. • Let’s investigate to understand why these engines look so different due to the the differences in the design flight condition. MAE 6530 - Propulsion Systems II 15 Classification of Turbofan Engines (7)

MAE 6530 - Propulsion Systems II 16 Thrust of a TurboFan Engine

m! a fan V m! a exit fan fan m! V m! a core exit core core

m! a V m! fan exit a fan fan

• 4-Primary Components of TurboFan Thrust o Bypass Momentum Thrust o Core Momentum Thrust o Fan Pressure Thrust Core Pressure Thrust o 17 MAE 6530 - Propulsion Systems II Thrust of a TurboFan Engine (2)

m! a fan V m! a exit fan fan m! V m! a core exit core core

m! a V m! fan exit a fan fan

⎛ ⎟⎞ ⎛ ⎟⎞ ⎛ ⎟⎞ ⎛ ⎟⎞ F = m! ⋅⎜V −V ⎟+ m! ⋅V + m! ⋅⎜V −V ⎟+⎜ p − p ⎟⋅ A +⎜ p − p ⎟⋅ A thrust acore exit ∞ ⎟ fuel exit a fan exit ∞ ⎟ exit ∞ ⎟ exit exit ∞ ⎟ exit ⎝⎜ core ⎠⎟ core ⎝⎜ fan ⎠⎟ ⎝⎜ core ⎠⎟ core ⎝⎜ fan ⎠⎟ fan Core Momentum Bypass Momentum Core Pressure Bypass Pressure Thrust Thrust Thrust Thrust

18 MAE 6530 - Propulsion Systems II Thrust of a TurboFan Engine (3) Air −to− fuel Ratio m! f = a m! fuel Bypass Ratio Total Air Massflow m! → a fan B m! a = m! a + m! a β = = core fan m! 1− B acore Bypass Fraction m! a β B = fan = m! a 1+ β

⎡ m! ⎤ m! a ⎛ ⎞ m! a ⎛ ⎞ F m! ⎢ core ⎜V V ⎟ fuel V fan ⎜V V ⎟⎥ p p A p p A thrust = a ⎢ ⋅⎜ exit − ∞ ⎟+ ⋅ exit + ⋅⎜ exit − ∞ ⎟⎥ +( exit − ∞ )⋅ exit +( 1exit − ∞ )⋅ exit m! ⎝⎜ core ⎠⎟ m! core m! ⎝⎜ fan ⎠⎟ fan ⎣⎢ a a a ⎦⎥

m! a m! a 1 1 core = core = = ! m! a m! a + m! a ma 1+ β ⎡ ⎤ core fan fan 1 ⎛ ⎞ 1 β ⎛ ⎞ 1+ F = m! ⎢ ⋅⎜V −V ⎟+ ⋅V + ⋅⎜V −V ⎟⎥ + m! thrust a ⎢ ⎜ exit ∞ ⎟ exit ⎜ exit ∞ ⎟⎥ a 1+ β ⎝⎜ core ⎠⎟ f core 1+ β ⎝⎜ fan ⎠⎟ core → ⎣ ⎦ m! m! m! a a − a ⎛ ⎞ ⎛ ⎞ core fan ⎜ p p ⎟ A ⎜ p p ⎟ A = =1− B ⎜ exit − ∞ ⎟⋅ exit +⎜ exit − ∞ ⎟⋅ exit ⎜ core ⎟ core ⎜ fan ⎟ fan m! a m! a ⎝ ⎠ ⎝ ⎠ m! a β m! 1 fan = B = , fuel = 19 ! ! MAEma 6530 -1Propulsion+ β ma Systemsf II Thrust of a TurboFan Engine (4)

• Similar to the Previous Discussion for the TurboJet, Normalized Thrust is

⎡ V V ⎤ ⎛ p A p A A + A ⎞ F m! V ⎢1+ f + exit exit ⎥ ⎜ exit exit exit exit exit exit ⎟ thrust a ∞ β core β fan ⎜ core core fan fan core fan ⎟ (T) = = ⎢ ⋅ + ⋅ −1⎥ +⎜ ⋅ + − ⎟= turbofan p A p A ⎢ f 1+ β V 1+ β V ⎥ ⎜ p A p A A ⎟ ∞ 0 ∞ 0 ⎢ ( ) ∞ ∞ ⎥ ⎜ ∞ 0 ∞ 0 0 ⎟ ⎣ ⎦ ⎝⎜ ⎠⎟

m! V ρ ⋅ A ⋅V ⋅V γV 2 a ∞ = ∞ 0 ∞ ∞ = ∞ = γ ⋅ M 2 p A p A γ ⋅ R ⋅T ∞ ∞ 0 ∞ 0 g ∞ Isp ⋅g0 f (I) = =(T) ⋅ turbofan c turbofan γ ⋅ M Aexit + Aexit = Aexit → ∞ ∞ core fan ⎡ 1 ⎤ ⎢1 1 V V ⎥ ⎛ A p A ⎞ + ( + β) exit exit ⎛ ⎞⎜ exit exit exit ⎟ ⎢ f core β fan ⎥ A ⎜ p core fan fan ⎟ 2 ⎜ exit ⎟⎜ exit ⎟ (T) = γ ⋅ M∞ ⎢ ⋅ + ⋅ −1⎥ +⎜ ⎟⎜ ⋅ + −1⎟ turbofan ⎢ ⎥ ⎜ ⎟⎜ ⎟ (1+ β) V∞ 1+ β V∞ ⎝ A0 ⎠⎜ p∞ Aexit p∞ Aexit ⎟ 20 ⎢ ⎥ ⎝⎜ ⎠⎟ MAE 6530 - Propulsion⎣⎢ Systems II ⎦⎥ Thrust of a TurboFan Engine (5) • Fully expanded nozzle & f >> 1 • Inlet, fan, compressor, turbine, and fan /core nozzles are isentropic • heat addition is as constant pressure and Low Mach p = p = p → π = π = π = π =1 exit exit ∞ d b ncore n fan core fan γ γ ⎛ ⎞γ−1 ⎛ ⎞γ−1 γ ⎜ ⎟ , ⎜ ⎟ , γ−1 . πc =⎜τc ⎟ πc =⎜τc ⎟ πt =(τt ) core ⎝⎜ core ⎠⎟ fan ⎝⎜ fan ⎠⎟ ⎡ V V ⎤ ⎢⎛ 1 ⎞ exit ⎛ ⎞ exit ⎥ 2 ⎢⎜ ⎟ core ⎜ β ⎟ fan ⎥ (Toptimal ) = γ ⋅ M∞ ⎜ ⎟⋅ +⎜ ⎟⋅ −1 = turbofan ⎢⎝⎜1+ β ⎠⎟ V ⎝⎜1+ β ⎠⎟ V ⎥ ⎢ ∞ ∞ ⎥ ⎣ ⎦ ⎡ ⎛ ⎞ ⎛ ⎞⎤ Vexit ⎟ Vexit ⎟ ⎢⎛ 1 ⎞ ⎜ ⎟ ⎛ β ⎞ ⎜ ⎟⎥ M 2 ⎢⎜ ⎟ ⎜ core 1⎟ ⎜ ⎟ ⎜ fan 1⎟⎥ = γ ⋅ ∞ ⎜ ⎟⋅⎜ − ⎟+⎜ ⎟⋅⎜ − ⎟ ⎢⎝⎜1+ β ⎠⎟ ⎜ V ⎟ ⎝⎜1+ β ⎠⎟ ⎜ V ⎟⎥ ⎢ ⎜ ∞ ⎟ ⎜ ∞ ⎟⎥ ⎣⎢ ⎝ ⎠ ⎝ ⎠⎦⎥ f = ⋅ (Ioptimal )turbofan (Toptimal )turbofan γ ⋅ M∞ 21 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle

• Too lengthy to analyze all types of turbofan engines. So, only a simple single spool fan, compressor, turbine cycle will be analyzed • Look at Fan Bypass Flow Stream

V ∞ m! a fan Vexit fan V M T • Look at by pass velocity ratio exit exit exit fan = fan ⋅ fan V∞ M∞ T∞ • First calculate bypass exit stagnation pressure

P0 P0 P0 P0 exit P0 P0 → P = ∞ ⋅ 1 ⋅ 2 ⋅ fan ⋅ p → ideal cycle → π = 1 ⋅ 2 =1 0exit ∞ d fan p P P P P P ∞ 0∞ 01 02 0∞ 01 γ γ−1 ⎛ γ +1 2 ⎞ ⎜ ⎟ 22 → P0 = πr ⋅πc ⋅ p∞ = pexit ⋅⎜1+ Mexit ⎟ exit ⎜ ⎟ MAE 6530 - Propulsionfan Systemsfan II fan ⎝⎜ 2 fan ⎠⎟ The Ideal TurboFan Cycle (2) • Look at Fan Bypass Flow Stream

V ∞ m! Vexit a fan fan

γ ⎧ ⎫ ⎛ ⎞γ−1 ⎪ ⎪ γ +1 2 ⎟ fully expanded nozzle → ⎨⎪p = p ⎬⎪→ π ⋅π =⎜1+ M ⎟ exit ∞ r c ⎜ exit ⎟ ⎩⎪ fan ⎭⎪ fan ⎝ 2 fan ⎠ γ ⎛ ⎞γ−1 ⎛ γ +1 ⎞ Isentropic fan ⎜ ⎟ ⎜1 M 2 ⎟ → πc =⎜τc ⎟ → τr ⋅τc =⎜ + exit ⎟ fan ⎝⎜ fan ⎠⎟ fan ⎝⎜ 2 fan ⎠⎟

T0 ⎛ γ +1 ⎞ γ +1 τ = ∞ ⎜1+ M 2 ⎟→ M 2 = τ −1 r ⎜ ∞ ⎟ ∞ r T∞ ⎝ 2 ⎠ 2

γ +1 2 2 Mexit τr ⋅τc −1 Mexit τr ⋅τc −1 γ +1 2 2 fan fan fan fan → Mexit = τr ⋅τc −1→ = → 2 = 2 fan fan γ +1 2 τ −1 M τ −1 M r ∞ r 2 ∞ 23 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (3) • Look at Fan Bypass Flow Stream

Substitute

γ−1 V M T ⎛τ ⋅τ −1⎞ T γ−1 exit exit exit ⎜ r c ⎟ exit ⎛ ⎞ γ γ fan fan fan ⎜ fan ⎟ fan ⎟ = ⋅ = ⎜ ⎟⋅ → isentropic fan →⎜ p ⎟ = p → T = T ⎜ ⎟ ⎜ exit ⎟ ( ∞ ) exit ∞ V∞ M∞ T∞ ⎜ τr −1 ⎟ T∞ ⎝ fan ⎠ fan ⎝⎜ ⎠⎟

2 Fan Thrust ⎡ ⎤ ⎛V ⎞ ⎛τ ⋅τ −1⎞ ⎡ ⎛V ⎞⎤ ⎛ τ ⋅τ −1 ⎟⎞ ⎜ exit ⎟ ⎜ r c ⎟ ⎢⎛ ⎞ ⎜ exit ⎟⎥ ⎢⎛ ⎞ ⎜ r c ⎟⎥ ⎜ fan ⎟ ⎜ fan ⎟ 2 β ⎟ ⎜ fan ⎟ 2 ⎢ β ⎟ ⎜ fan ⎟⎥ ⎜ ⎟ =⎜ ⎟ →(T) = γ ⋅ M ⎢⎜ ⎟⋅⎜ −1⎟⎥ = γ ⋅ M ⎜ ⎟⋅⎜ −1⎟ ⎜ ⎟ ⎜ ⎟ fan ∞ ⎢⎜ ⎟ ⎜ ⎟⎥ ∞ ⎢⎜ ⎟ ⎜ ⎟⎥ ⎜ V∞ ⎟ ⎜ τr −1 ⎟ ⎝1+ β ⎠ ⎜ V∞ ⎟ ⎢⎝1+ β ⎠ ⎜ τr −1 ⎟⎥ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎢ ⎝⎜ ⎠⎟⎥ ⎜ ⎟ ⎣⎢ ⎦⎥ ⎣⎢ ⎝ ⎠⎦⎥

24 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (4) • Look at Fan Bypass Flow Stream

Substitute

1 ⎛ ⎞ γ− γ−1 Vexit Mexit Texit τr ⋅τc −1⎟ Texit ⎜ ⎟ ⎛ ⎞ γ γ fan fan fan ⎜ fan ⎟ fan isentropic fan ⎜ p ⎟ p T T = ⋅ = ⎜ ⎟⋅ → →⎜ exit ⎟ =( ∞ ) → exit = ∞ V M T ⎜ τ −1 ⎟ T ⎝⎜ fan ⎠⎟ fan ∞ ∞ ∞ ⎜ r ⎟ ∞ ⎝ • Fan⎠ Thrust 2 ⎡ ⎤ ⎛V ⎞ ⎛τ ⋅τ −1⎞ ⎡ ⎛V ⎞⎤ ⎛ τ ⋅τ −1 ⎟⎞ ⎜ exit ⎟ ⎜ r c ⎟ ⎢⎛ ⎞ ⎜ exit ⎟⎥ ⎢⎛ ⎞ ⎜ r c ⎟⎥ ⎜ fan ⎟ ⎜ fan ⎟ 2 ⎜ β ⎟ ⎜ fan ⎟ 2 ⎢⎜ β ⎟ ⎜ fan ⎟⎥ ⎜ ⎟ =⎜ ⎟ →(T) = γ ⋅ M ⎢⎜ ⎟⋅⎜ −1⎟⎥ = γ ⋅ M ⎜ ⎟⋅⎜ −1⎟ ⎜ ⎟ ⎜ ⎟ fan ∞ ⎢⎜ ⎟ ⎜ ⎟⎥ ∞ ⎢⎜ ⎟ ⎜ ⎟⎥ ⎜ V∞ ⎟ ⎜ τr −1 ⎟ ⎝1+ β ⎠ ⎜ V∞ ⎟ ⎢⎝1+ β ⎠ ⎜ τr −1 ⎟⎥ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎢ ⎝⎜ ⎠⎟⎥ ⎜ ⎟ ⎣⎢ ⎦⎥ ⎣⎢ ⎝ ⎠⎦⎥

25 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (5) • Now look at Core Flow Stream

V m! core exit core

• Look at by core velocity ratio

Vexit Mexit Texit core = core ⋅ core V∞ M∞ T∞ • First calculate core exit stagnation pressure P P 01 02 πd = ⋅ =1 P0 P0 P0 P0 P0 ∞ 1 P0 P0 P0 3 4 5 → P = ∞ ⋅ 1 ⋅ 2 ⋅ core ⋅ core ⋅ core p → ideal cycle → π =1 0exit ∞ b core p P P P P P ∞ 0∞ 01 02 03 04 core core pexit = p∞ core 26 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (6) • Now look at Core Flow Stream

Vexit core

Substitute γ ⎛ ⎞γ−1 ⎜ γ +1 2 ⎟ → P0 = πr ⋅πc ⋅πt ⋅ p∞ = pexit ⋅⎜1+ Mexit ⎟ exit ⎜ ⎟ core core ⎝⎜ 2 core ⎠⎟ γ ⎛ γ +1 ⎞ Isentropic compressor γ−1 ⎜1 M 2 ⎟ → πc =(τc ) → τr ⋅τc ⋅τt =⎜ + exit ⎟ ⎝⎜ 2 core ⎠⎟

T0 ⎛ γ +1 ⎞ γ +1 τ = ∞ ⎜1+ M 2 ⎟→ M 2 = τ −1 r ⎜ ∞ ⎟ ∞ r T∞ ⎝ 2 ⎠ 2 27 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (7) • Now look at Core Flow Stream

Vexit core

Solve for exit Mach

γ +1 2 2 Mexit Mexit γ +1 2 2 core τr ⋅τc ⋅τt −1 core τr ⋅τc ⋅τt −1 → Mexit = τr ⋅τc ⋅τt −1→ = → 2 = 2 core γ +1 2 τ −1 M τ −1 M r ∞ r 2 ∞ Substitute V M T T exit exit exit ⎛τ ⋅τ ⋅τ −1⎞ exit core = core ⋅ core = ⎜ r c t ⎟⋅ core ⎜ ⎟ V∞ M∞ T∞ ⎝⎜ τr −1 ⎠ T∞ 28 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (8) • Now look at Core Flow Stream

Vexit core

Calculate exit Stagnation Temperature ⎛ ⎞ ⎜ T0 T0 T0 T0 T0 ⎟ ⎜T0 2 3 4 5 exit ⎟ ⎜ ∞ core core core core core ⎟ Calculate → T0 = T∞ ⋅⎜ ⋅ ⋅ ⋅ ⋅ ⋅ ⎟= T∞ ⋅(τr ⋅τd ⋅τc ⋅τt ⋅τn ) exit ⎜ ⎟ core ⎜ T∞ T0 T0 T0 T0 T0 ⎟ ∞ 2 3 4 5 ⎟ ⎝⎜ core core core core ⎠⎟ ⎛ ⎞ ⎜ γ +1 2 ⎟ Isentropic nozzle → T0 = Texit ⋅⎜1+ Mexit ⎟ exit ⎜ ⎟ core core ⎝⎜ 2 core ⎠⎟ → T0 = Texit ⋅(τr ⋅τc ⋅τt ) exit ⎛ γ +1 ⎞ core core From previous →⎜1+ M 2 ⎟= τ ⋅τ ⋅τ ⎜ exit ⎟ r c t ⎝⎜ 2 core ⎠⎟ 29 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (9) • Now look at Core Flow Stream

Vexit core

Write in terms of maximum temperature ratio ⎛ γ +1 ⎞ T ⎜1+ M 2 ⎟ Texit 0 ⎜ 4 ⎟ Texit T0 T0 T0 T0 4 ⎜ ⎟ 4 2 4 2 core core ⎝ 2 core ⎠ core core core T∞ core core T∞ τλ = → M4 ≈ M3 ≈ 0 → = ⋅ ⋅ = ⋅ ⋅ = T ⎛ γ +1 2 ⎞ core core T T T T T T T τ ⋅τ ∞ ⎜ ⎟ ∞ 03 0∞ ∞ ∞ 03 0∞ c r T0 ⎜1+ M3 ⎟ core core 3 ⎜ ⎟ core ⎝⎜ 2 core ⎠⎟ 2 T V ⎛V ⎞ exit exit ⎛ 1⎞ ⎜ exit ⎟ ⎛ 1⎞ ⎛ ⎞ core τλ core ⎜τr ⋅τc ⋅τt − ⎟ τλ ⎜ core ⎟ ⎜τr ⋅τc ⋅τt − ⎟ ⎜ τλ ⎟ = → = ⎜ ⎟⋅ →⎜ ⎟ =⎜ ⎟⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ T∞ τc ⋅τr V∞ ⎝ τr −1 ⎠ τc ⋅τr ⎜ V∞ ⎟ ⎝ τr −1 ⎠ ⎝τc ⋅τr ⎠ ⎝⎜ ⎠⎟

30 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (10) • Now look at Core Flow Stream ⎡ V V ⎤ ⎛ p A p A A + A ⎞ F m! V ⎢1+ f + exit exit ⎥ ⎜ exit exit exit exit exit exit ⎟ thrust a ∞ β core β fan ⎜ core core fan fan core fan ⎟ (T) = = ⎢ ⋅ + ⋅ −1⎥ +⎜ ⋅ + − ⎟= turbofan p A p A ⎢ f 1+ β V 1+ β V ⎥ ⎜ p A p A A ⎟ ∞ 0 ∞ 0 ⎢ ( ) ∞ ∞ ⎥ ⎜ ∞ 0 ∞ 0 0 ⎟ ⎣ ⎦ ⎝⎜ ⎠⎟ Vexit core

2 m! aV∞ ρ∞ ⋅ A0 ⋅V∞ ⋅V∞ γV∞ 2 = = = γ ⋅ M∞ p∞ A0 p∞ A0 γ ⋅ Rg ⋅T∞ • Core Thrust Aexit + Aexit = Aexit → core fan ⎡⎡ 1 ⎤ ⎤ ⎢1+ 1+ ⎛V ⎛ V ⎥ ⎞ ⎛⎛ A ⎞ p ⎞ A ⎞ ⎢⎛ 1( β⎞) ⎜ exit τ ⋅τ ⋅exiτt −1⎛ ⎞⎜ τ exit ex⎟it ⎥ exit ⎟ 22 ⎢ f core⎜ r β c fan t ⎥ ⎟A ⎜⎜ p λ co⎟re fa⎟n fan ⎟ MM ⎜ ⎟ ⎜ 1 ⎜ ⎟exit ⎟⎜ exit ⎟ 1⎟ 1⎟ (T)= γ=⋅γ ⋅ ∞ ⎢⎢⎜ ⎟⋅⋅⎜ ⎜+ ⋅ − ⎥ +⎜ ⋅⎜⎟⎜ ⋅ +− ⎥ − ⎟ (T)core turbofan ∞ ⎢ 1+ ⎟ V ⎜ 1+ β V ⎥ ⎜ ⎟A ⎜⎟⎜ p A ⎟ p ⎟ A ⎟ ⎜1(+ββ) ⎟ ⎜ ∞⎜ τ −∞ 1 ⎝ ⎟ 0⎜⎠⎜τ ∞⋅τ ex⎟it ∞⎟ exit ⎟ ⎢⎢⎝ ⎠ ⎝⎜ ⎝ r ⎥ ⎠ ⎝⎝⎜ c r ⎠ ⎠⎟⎥ ⎠⎟ ⎣⎢⎣⎢ ⎦⎥ ⎦⎥

31 MAE 6530 - Propulsion Systems II The Ideal TurboFan Cycle (11) Substitute Fan/Core/Pressure Thrust Summary γ−1 V M T ⎛τ ⋅τ −1⎞ T γ−1 exit exit exit ⎜ r c ⎟ exit 2 ⎛ ⎞ γ γ fan fan fan ⎜ fan ⎟ fan ⎛V ⎞ ⎛τ ⋅τ −⎟ 1⎞ = ⋅ = • ⎜Fan ⎟⋅ → isentr⎜opexiict ⎟ fan⎜→r ⎜ pc ⎟ ⎟ = p → T = T ⎜ ⎟ ⎜ fan ⎟ ⎜ ⎜ efaxnit ⎟ ⎟ ( ∞ ) exit ∞ V M T ⎜ τ −1 ⎟ T ⎜ ⎟ =⎜ ⎝⎜ fan ⎠⎟ ⎟ fan ∞ ∞ ∞ ⎜ r ⎟ ∞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎜ V∞ ⎟ ⎜ τr −1 ⎟ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ 2 ⎡ ⎤ ⎛ ⎞ Vexit ⎡ Vexit ⎛ pexit Aexit pexit Ae⎞xi⎤t Aexit + Aexit ⎟ ⎛V ⎞ ⎛τ ⋅τ −1⎞ F⎡ m! V⎛V⎢1+ f + ⎞β⎤ β ⎥ ⎜ τ ⋅τ −1 ⎟ ⎟ ⎜ exit ⎟ ⎜ r c ⎟ ⎢thrust a ⎜∞ ⎢ exit ⎟⎥ core ⎢ fan ⎥ ⎜⎜ corre ccore fan f⎟an⎥ core fan ⎟ ⎜ fan ⎟ ⎜ fan ⎟ (T) =2 ⎛ β= ⎞ ⎜ fan ⎟ ⋅ + 2 ⎛⋅ β−1⎞+⎜⎜ ⋅ fan + ⎟ − ⎟= ⎜ ⎟ ⎜ ⎟ turbofan ⎢⎜ ⎟ ⎜ ⎢ ⎟⎥ ⎢⎜ ⎟⎥ ⎜⎜ ⎟⎥ ⎟ ⎜ ⎟ =⎜ ⎟ →(T) = γ ⋅ M∞p∞⎜A0 p⎟∞⋅A⎜0 f (1−+1β⎟) =V∞γ ⋅ M1+∞β⎜ V∞ ⎟⋅⎜⎜ p∞ A0 p∞−1A⎟0 A0 ⎟ ⎜ ⎟ ⎜ ⎟ fan ⎢⎜ ⎟ ⎜ ⎢ ⎟⎥ ⎢⎜ ⎟⎥ ⎜⎝⎜ ⎟⎥ ⎠⎟ ⎜ V∞ ⎟ ⎜ τr −1 ⎟ ⎝1+ β ⎠ ⎜ V⎣ ∞ ⎟ ⎢⎝1+ β ⎠⎦ ⎜ τr −1 ⎟⎥ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎢ ⎝⎜ ⎠⎟⎥ ⎜ ⎟ ⎣⎢ ⎦⎥ ⎣⎢ ⎝ ⎠⎦⎥ m! V ρ ⋅ A ⋅V ⋅V γV 2 a ∞ = ∞ 0 ∞ ∞2= ∞ = γ ⋅ M 2 • Core p A ⎛pVA ⎞ γ ⋅ R ⋅T ∞ ∞ 0 ⎜ ∞ ex0 it ⎟ ⎛τg ∞⋅τ ⋅τ −1⎞ ⎛ τ ⎞ ⎜ core ⎟ ⎜ r c t ⎟ ⎜ λ ⎟ ⎜ ⎟ =⎜ ⎟⋅⎜ ⎟ A A ⎜AV ⎟ ⎜ τ −1 ⎟ ⎜τ ⋅τ ⎟ exit + exit =⎜ exit ∞→ ⎟ ⎝ r ⎠ ⎝ c r ⎠ core fan ⎝⎜ ⎠⎟ ⎡ 1 ⎤ ⎡⎢1 1 ⎛V V ⎥ ⎛ A p ⎞A⎤ ⎞ ⎛+ 1( + β⎞) ⎜ exit⎛τ ⋅τ e⋅xitτ −⎛1⎞ ⎞⎛⎜ τ exit ⎞ exit ⎟ exit ⎟ 2 2 ⎢⎢ f core⎜ rβ c fan t ⎥ A⎟ ⎜⎜ p λ core ⎟ fan ⎟⎥fan ⎟ M ⎜ ⎟ ⎜ ⎜ ⎟exit ⎟⎜ exit ⎟ 1⎟ ⎟ (T) (T=) γ ⋅= γ ⋅ M∞⎢⎢⎜ ⋅⋅⎜ ⎜+ ⋅ −1⎥ +⎜ ⋅⎟⎜⎜ ⋅ +− ⎥ −1 core turbofan ∞ ⎢⎜ ⎟ ⎜V ⎜ 1 V ⎥ ⎜ A⎟ ⎟⎜⎜ p A ⎟ p ⎟A ⎟ ⎢⎝1(+1+ β) ⎠⎟ ∞⎜ + βτ −∞ 1 ⎝ ⎟0 ⎠⎜⎜τ∞ ⋅τexit ⎟ ∞ ⎟⎥exit ⎟ ⎢ ⎝⎜ ⎝ r ⎥ ⎠ ⎝⎝⎜ c r ⎠ ⎠⎟ ⎠⎟ ⎣⎢⎣⎢ ⎦⎥ ⎦⎥

MAE 6530 - Propulsion Systems II 32 Turbine-Compressor-Fan Work Matching

• Work taken from flow by high and low pressure turbine drives both compressor and fan,

W! = m! + m! ⋅ h −h = m! ⋅ h −h + m! ⋅ h −h turbine ( acore fuel ) ( 04 05 ) ( acore ) ( 03 02 ) ( a fan ) 03 02 ( fan ) Approximating Cp ≈ const, factoring out m! C T • a p ∞ ⎡ ⎛ ⎞ ⎤ T0 −T0 ⎛ ⎞ ⎛T T ⎞ ⎢⎛m! m! ⎞ ⎛T −T ⎞ ⎛m! ⎞ ⎜ 3 2 ⎟ m! 0 − 0 ⎟⎥ ⎜ a fuel ⎟ ⎜ 0 0 ⎟ ⎜ a ⎟ ⎜ core ⎟ ⎜ a fan ⎟ ⎜ 3 fan 2 ⎢⎜ core ⎟ ⎜ 4 5 ⎟ ⎜ core ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎥ m! aCpT + ⎟⋅ ⎟= ⎟⋅⎜ ⎟+⎜ ⎟⋅⎜ ⎟ ∞ ⎢⎜ m! m! ⎟ ⎜ T ⎟ ⎜ m! ⎟ ⎜ T ⎟ ⎜ m! ⎟ ⎜ T ⎟⎥ ⎝⎜ a a ⎠⎟ ⎝⎜ ∞ ⎠⎟ ⎝⎜ a ⎠⎟ ⎜ ∞ ⎟ ⎜ a ⎟ ⎜ ∞ ⎟ ⎢ ⎝⎜ ⎠⎟ ⎝ ⎠ ⎝⎜ ⎠⎟⎥ ⎣⎢ ⎦⎥ 33 MAE 6530 - Propulsion Systems II Turbine-Compressor-Fan Work Matching (2) → Simplify ⎛ T ⎞ ⎛ T ⎞ ⎛ ⎞ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎛ 1 ⎞ T0 T0 T0 ⎟ ⎜T0 3 T0 ⎟ ⎛ ⎞ ⎜T0 3 T0 ⎟ ⎜ ⎟ ⎜ 4 5 4 ⎟ 2 core 2 ⎟ ⎜ β ⎟ 2 fan 2 ⎟ ⎜1− B+ ⎟⋅⎜ − ⎟=(1− B)⋅⎜ − ⎟+⎜ ⎟⋅⎜ − ⎟ ⎜ f ⎟ ⎜T T T ⎟ ⎜T T T ⎟ ⎜1 ⎟ ⎜T T T ⎟ ⎝ ⎠ ⎝⎜ ∞ 0 ∞ ⎠⎟ ⎜ ∞ 0 ∞ ⎟ ⎝ + β ⎠ ⎜ ∞ 0 ∞ ⎟ 4 ⎝⎜ 2 ⎠⎟ ⎝⎜ 2 ⎠⎟ → Substitute ⎡ ⎤ ⎢ T0 T0 T0 T0 ⎥ ⎢ τ = 4 τ = 4 τ = ∞ = 2 ⎥ ⎢ λ T t T r T T ⎥ ∞ ∞ ∞ ∞ ⎛ 1 ⎞ ⎛ ⎞ ⎛ β ⎞ ⎛ ⎞ ⎢ ⎥ ⎜1 B ⎟ 1 B ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ →⎜ − + ⎟⋅(τλ −τtτλ )=( − )⋅⎜τrτc −τr ⎟+⎜ ⎟⋅⎜τrτc −τr ⎟ T0 T0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ 3 3 ⎥ ⎝ f ⎠ ⎝ core ⎠ ⎝1+ β ⎠ ⎝ fan ⎠ ⎢ core = τ fan = τ ⎥ ⎢ c c ⎥ T0 core T0 fan ⎣⎢ 2 2 ⎦⎥

→ Solve for τt τ ⎛ ⎞ ⎛ β ⎞ τ ⎛ ⎞ 1− B ⋅ r ⋅⎜τ −1⎟+⎜ ⎟⋅ r ⋅⎜τ −1⎟ ( ) ⎜ c ⎟ ⎜ ⎟ ⎜ c ⎟ τλ ⎝ core ⎠ ⎝1+ β ⎠ τλ ⎝ fan ⎠ τt =1− ⎛ 1 ⎟⎞ ⎜1− B+ ⎟ ⎝⎜ f ⎠⎟

34 MAE 6530 - Propulsion Systems II Turbine-Compressor-Fan Work Matching (3)

⎛ β ⎞ ⎛ ⎞ ⎛ β ⎞ ⎛ ⎞ ⎜1− ⎟⋅⎜τ −1⎟+⎜ ⎟⋅⎜τ −τ ⎟ ⎛ ⎞⎜ ⎟ ⎜ c ⎟ ⎜ ⎟ ⎜ c r ⎟ β τ ⎝⎜ 1+ β ⎠⎟ ⎝⎜ core ⎠⎟ ⎝⎜1+ β ⎠⎟ ⎝⎜ fan ⎠⎟ ⎜ r ⎟ → let B = → τt =1−⎜ ⎟ 1 ⎜ ⎟ ⎛ ⎞ + β ⎝τλ ⎠ β 1 ⎟ ⎜1− + ⎟ ⎝⎜ 1+ β f ⎠⎟ → Simplify ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 ⎟ β τr ⎟ ⎟ ⎟ ⎜ ⎟⋅⎜τ −1⎟+⎜ ⎟⋅ ⋅⎜τ −1⎟ ⎜τ −1⎟+ β ⋅⎜τ −1⎟ ⎛ ⎞⎜ ⎟ ⎜ c ⎟ ⎜ ⎟ ⎜ c ⎟ ⎛ ⎞⎜ c ⎟ ⎜ c ⎟ τr ⎟⎝1+ β ⎠ ⎝ core ⎠ ⎝1+ β ⎠ τ ⎝ fan ⎠ τr ⎟⎝ core ⎠ ⎝ fan ⎠ τ =1−⎜ ⎟ λ =1−⎜ ⎟ t ⎜ ⎟ ⎜ ⎟ ⎝⎜τ ⎠⎟ ⎛ 1 1 ⎞ ⎝⎜τ ⎠⎟ ⎛ 1+ β ⎞ λ ⎜ + ⎟ λ ⎜1+ ⎟ ⎝⎜1+ β f ⎠⎟ ⎝⎜ f ⎠⎟

⎛ ⎟⎞ ⎛ ⎟⎞ ⎜τ −1⎟+ β ⋅⎜τ −1⎟ ⎛ ⎞⎜ c ⎟ ⎜ c ⎟ τr ⎟⎝ core ⎠ ⎝ fan ⎠ τ =1−⎜ ⎟ t ⎜ ⎟ ⎛ ⎞ ⎝τλ ⎠ 1+ β ⎟ ⎜1+ ⎟ ⎝⎜ f ⎠⎟

⎛⎛ ⎞⎞⎡⎛⎡ ⎞ ⎛ ⎞⎤⎤ 1 ⎜ττr ⎟ ⎛ ⎟ ⎞ ⎛ ⎞⎟ → for m! >> m! → = 0 → τ =11−−⎜⎜ r⎟⎟⋅⋅⎢⎜⎢⎜ττ −−1⎟1+⎟+β ⋅β⎜τ⋅⎜ττ −−τ1⎟⎟⎥⎥ air fuel t ⎜⎜ ⎟⎟⎢⎜⎢⎜ c c ⎟ ⎟ ⎜ ⎜r cc r⎟⎟⎥⎥ f ⎜⎜ττ ⎟⎟ ⎝⎜⎜ cocroere ⎠⎟ ⎟ ⎝⎜ ⎜ ffaann ⎟⎠⎟ ⎝⎝ λλ⎠⎠⎣ ⎣⎝ ⎠ ⎝ ⎠⎦⎦ 35 MAE 6530 - Propulsion Systems II Air to Fuel Ratio → Rearrange

m! fuel ⋅ hfuel −h0 = m! a ⋅ h0 −h0 ( 4 ) core ( 4 3 ) core

→ factor out m! aCp ⋅T∞ ⎡ ⎛ ⎞⎤ m! ⎛ h T ⎞ m! ⎜T T0 T0 ⎟ ⎢ fuel ⎜ fuel 04 ⎟ acore 04 2core 3core ⎟⎥ m! C ⋅T ⎢ ⋅⎜ − ⎟= ⋅⎜ − ⋅ ⎟⎥ a p ∞ m! ⎜C T T ⎟ m! ⎜T T T ⎟ ⎢ a ⎝⎜ p ⋅ ∞ ∞ ⎠⎟ a ⎜ ∞ ∞ 0 ⎟⎥ ⎣⎢ ⎝ 2core ⎠⎦⎥ → Simplify ⎛ ⎞ τλ −τr ⋅τc ⎟ 1 1 ⎜ ⎟ = ⋅⎜ core ⎟ ⎜ ⎟ f 1+ β ⎜ τ fuel −τλ ⎟ 1 1 ⎛ ⎞ ⎝⎜ ⎠⎟ ⎜ ⎟ ⋅(τ fuel −τλ )= ⋅⎜τλ −τr ⋅τc ⎟→ f 1+ β ⎝⎜ core ⎠⎟ ⎛ ⎞ ⎜ τ −τ ⎟ f = 1+ β ⋅⎜ fuel λ ⎟ ( ) ⎜ ⎟ ⎜τλ −τr ⋅τc ⎟ ⎝⎜ core ⎠⎟ 36 MAE 6530 - Propulsion Systems II Collected TurboFan Matching Equations

⎛ ⎟⎞ ⎛ ⎟⎞ ⎜τ −1⎟+ β ⋅⎜τ −1⎟ ⎛ ⎞⎜ c ⎟ ⎜ c ⎟ τr ⎟⎝ core ⎠ ⎝ fan ⎠ • Turbine Work τ =1−⎜ ⎟ t ⎜ ⎟ ⎛ ⎞ ⎝τλ ⎠ 1+ β ⎟ ⎜1+ ⎟ ⎝⎜ f ⎠⎟

⎛ ⎞ τλ −τr ⋅τc ⎟ 1 1 ⎜ ⎟ = ⋅⎜ core ⎟ ⎜ ⎟ f 1+ β ⎜ τ fuel −τλ ⎟ ⎝⎜ ⎠⎟ m! • Fuel/Air Flow f = a ⎛ ⎞ ⎟ m! fuel ⎜ τ −τ ⎟ f = 1+ β ⋅⎜ fuel λ ⎟ ( ) ⎜ ⎟ ⎜τλ −τr ⋅τc ⎟ ⎝⎜ core ⎠⎟ If the bypass ratio goes to zero the matching condition reduces to the usual turbojet formula. MAE 6530 - Propulsion Systems II 37 Questions??

MAE 6530 - Propulsion Systems II 38