CHAPTER 3 Gas Turbine Cycles for Aircraft Propulsion

CHAPTER 3 Gas Turbine Cycles for Aircraft Propulsion Chapter 2 Shaft Power Cycles

Gas Turbine Cycles for Aircraft Propulsion § Aircraft GT Cycles differ from Shaft Power Cycles in that the useful power output in the former is produced wholly or in part as a result of expansion in a propelling nozzle wholly in Turbojet and Turbofan engines and partly in Turboprop engines. § A second distinquishing feature is the need to consider the effect of forward speed and altitude on the performance of an aircraft engine. Chapter 2 Shaft Power Cycles 2

Gas Turbine Cycles for Aircraft Propulsion 3. 1 Criteria Of Performance For simplicity assume that mass flow ṁ is constant (i. e the fuel flow is negligible). The net thrust is then; ṁa Va Vj FIG. 3. 1 Schematic Diagram of a Propulsive Duct F = ṁj Vj - ṁa Va - ṁf Vf + Aj (Pj-Pa) F = ṁ a (Vj - Va ) + Aj (Pj-Pa) ( 3. 1 ) Chapter 2 Shaft Power Cycles 3

Criteria Of Performance § When the exhaust gasses are expanded completely to Pa in the propulsive duct i. e Pj = Pa ; then; § F = ṁ a (Vj-Va) ( 3. 2 ) § From this equation , it is clear that the required thrust can be obtained by designing the engine to poduce : § either a high velocity jet of small mass flow § or a low velocity jet of high mass flow. Chapter 2 Shaft Power Cycles 4

Criteria Of Performance The most efficient combination of these two variables is povided by the following analysis; Chapter 2 Shaft Power Cycles 5

Criteria Of Performance § From eqns. (3. 2) , (3. 3); § a) F is max when Va = 0 ηp = 0 § Vj =Va F=0 b) ηp is max when § We may conclude that; although Vj must be greater than Va, the difference (Vj-Va) should not be too great. Chapter 2 Shaft Power Cycles 6

Criteria Of Performance As a result a family of propulsion units are developed. § FAMILY of PROPULSION ENGINES § 1. Piston Engine § 2. Turboprop § 3. Turbofan § 4. Turbojet § 5. Ramjet § From 1 to 5 Vj increases and m decreases, for a fixed Va, F increases and ηp decreases Chapter 2 Shaft Power Cycles 7

Propulsion Engines Piston Engine Turboprop Turbofan Turbojet Ramjet Chapter 2 Shaft Power Cycles 8

Criteria Of Performance § Taken in the order shown : Propulsive jets of decreasing mass flowrate and increasing jet velocity § therefore suitable for aircraft of increasing design cruising speed. Chapter 2 Shaft Power Cycles 9

Fig 3. 2 Flight Regimes Ram-jet Turbojet Chapter 2 Shaft Power Cycles 10

Criteria Of Performance § Propulsion efficiency is a measure of the effectiveness with which the propulsive duct is being used for propelling the aircraft. § Efficiency of energy conversion ( 3. 4 ) Chapter 2 Shaft Power Cycles 11

Criteria Of Performance § From the above definitions; § ( 3. 6 ) § Efficiency of an aircraft power plant is inextricably linked to the aircraft speed. § For aircraft engines, Specific fuel consumption = sfc = Fuel Consumption/Thrust [kg/h. N] is a better concept than efficiency to define performance. Chapter 2 Shaft Power Cycles 12

Criteria Of Performance § § Since Qnet, p = const for a given fuel, then § for aircraft plants ; ηo = f (Va/sfc ). § for shaft power units ; ηo = f (1/sfc ). § Another important performance parameter is specific thrust, Fs ; § Fs thrust per unit mass flow of air [N. s/kg]. Chapter 2 Shaft Power Cycles 13

Criteria Of Performance § This ( Fs ) provides an indication of the relative size of engines producing the same thrust, because the dimensions of the engine are primarily determined by the airflow requirements. § Note that; § sfc = ηo / Fs Chapter 2 Shaft Power Cycles ( 3. 8 ) 14

ISA (International. Standard Atm. ) § When estimating the cycle performance at altitude one needs to know the variation of ambient pressure and temperature with altitude. §ISA (International. Standard Atm. ) corresponds to middling lattitudes § Ta decrease by 3. 2 K per 500 m up to 11000 m. § After 11000 m Ta = const up to 20 000 m. § Then Ta starts increasing slowly Chapter 2 Shaft Power Cycles Z (m) Ta (K) 15

Criteria Of Performance § For high-subsonic or supersonic aircraft it is more appropriate to use Mach number rather than V (m/s) for aircraft speed, because "DRAG" is more a function of Ma. § Inrease in Mach number with altitude is experienced for a given Va Chapter 2 Shaft Power Cycles 16

Criteria Of Performance § Mach Number vs. Flight Velocity Va (m/s) Chapter 2 Shaft Power Cycles 17

3. 2 Intake & Propelling Nozzle Efficiencies § FIG 3. 4 Simple Turbojet Engine Chapter 2 Shaft Power Cycles 18

Intake & Propelling Nozzle Efficiencies § The turbine produces just sufficient work to drive the compressor and remaining part of the expansion is carried out in the propelling nozzle. § Because of the significant effect of forward speed, the intake must be considered as a seperate component. § In studying the performance of aircraft propulsion cycles it is necessary to describe the losses in the two additional components ; i. e. § INTAKE § PROPELLING NOZZLE Chapter 2 Shaft Power Cycles 19

3. 2. 1 Intakes § The intake is a simple adiabatic duct. § Since Q = W =0 , the stagnation temperature is constant, § although there will be a loss of stagnation pressure due friction and due to shock waves at supersonic flight speeds. Chapter 2 Shaft Power Cycles 20

Intakes § Under static conditions or at very low forward speeds ; intake acts as a nozzle in which the air accelerates from zero velocity or from low Va to V 1 at the compressor inlet. § At cruise speeds, however, the intake performs as a diffuser with the air decelerating from Va to V 1 and the static pressure rising from Pa to P 1. Chapter 2 Shaft Power Cycles 21

Intakes § Inlet isentropic efficiency, of temperature rise. “ηi “ defined in terms § The isentropic efficiency for the inlet ; ( 3. 9 ) here; § T 01' = Temperature which would have been reached after an isentropic Ram compression to P 01. Chapter 2 Shaft Power Cycles 22

V Intakes V V § Fig 3. 5 Intake Losses Chapter 2 Shaft Power Cycles 23

Intakes § P 01 - Pa = Ram Pressure Rise § RAM efficiency , ηr is defined in terms of pressure rise (pressure rise / inlet dynamic head ). Chapter 2 Shaft Power Cycles 24

Intakes § The isentropic efficiency for the INLET ; § ( 3. 9 ) Chapter 2 Shaft Power Cycles 25

Intakes § The intake presure ratio; § § ( 3. 10. a ) Chapter 2 Shaft Power Cycles 26

Intakes § Noting M= V / (g. RT )1/2 and R = Cp( g- 1 ) § § The stagnation temperature; ( 3. 10. b ) § § ( 3. 10. c ) Chapter 2 Shaft Power Cycles 27

Intakes § RAM efficiency is defined as; § § For supersonic inlets it is more usual to quote values of stagnation pressure ratio P 01 / P 0 a as a function of Mach number. § Chapter 2 Shaft Power Cycles 28

3. 2. 2 Propelling Nozzles Chapter 2 Shaft Power Cycles 29

3. 2. 2 Propelling Nozzles § Propelling nozzle is the remaining part of the engine after the last turbine stage. § The question is immediately arises, as to whether a simple convergent nozzle is adequate or whether a convergent - divergent nozzle should be employed. § It can be shown that for an isentropic expansion, the thrust produced is maximum when complete expansion to Pa occurs in the nozzle. Chapter 2 Shaft Power Cycles 30

Propelling Nozzles § The pressure thrust arising from an incomplete expansion does not entirely compansate for the loss of momentum thrust due to smaller jet velocity. § But this is no longer true when friction is taken into account because theoretical jet velocity is not achieved. § For values of P 04/Pa ( nozzle pressure ratio ) up to 3 § Fconv-div thrust = F simple conv. § Converging-diverging nozzle at off-design condition Shock wave in the divergent section loss in stagnation pressure. Chapter 2 Shaft Power Cycles 31

Propelling Nozzles § With simple convergent nozzles; § a) It easy to employ a variable area nozzle, § b) It is easy to employ a thrust reverser, § c) It is easy to employ a noise suppressor ( i. e. increase the surface area of the jet stream ). § The thrust developed by a propulsive nozzle; § F = ṁ Vj + (Pj -Pa) Aj Chapter 2 Shaft Power Cycles 32

Propelling Nozzles § For a given m to determine the nozzle exit area that yields maximum thrust, differentiate the above eqn. § d. F = ṁ d. Vj + Ajd. Pj + Pj d. Aj - Pad. Aj § but § ṁ = AV = rj. Aj. Vj d. F = Aj (d. Pj + r j. Vj d. Vj) + (Pj -Pa) d. Aj Chapter 2 Shaft Power Cycles 33

Propelling Nozzles § since momentum equation; § d. P + r. Vd. V = 0 --> d. F = Aj [0] + (Pj -Pa) d. Aj § solve for d. F / d. Aj § d. F / d. Aj = Pj - Pa for max thrust (Pj -Pa) =0 § Therefore; the nozzle area ratio must be chosen so that the pressure ratio Pj / Po = Pa / Po § This design criterion is based on planar flow. Chapter 2 Shaft Power Cycles 34

Propelling Nozzles § This design criterion is based on planar flow. § If a similar equation is derived for a conical nozzle , it is seen that some under expansion is desirable. § Then the thrust gain is about 2% higher than Pe=Pa. § Variable exit / throat area ratio is essential to avoid shock losses over as much of the operating range as possible, and the additional mechanical complexity has to be accepted. Chapter 2 Shaft Power Cycles 35

Variable Area, Thrust Reversal and Noise Suppression Chapter 2 Shaft Power Cycles 36

Propelling Nozzles § The main limitations on the design of convergent divergent nozzles are : a) The exit diameter must be within the overall diameter of the engine, otherwise the additional thrust is offset by the increased external drag. b) In spite of the weight penalty; the included angle of divergence must be kept below about 30 o, because the loss in thrust associated with the divergence of the jet increases sharply at greater angles. Chapter 2 Shaft Power Cycles 37

Propelling Nozzles § In order to allow nozzle losses two approaches are commonly used; § i) Isentropic efficiency : § § § From definition of hj; ( 3. 12 ) Chapter 2 Shaft Power Cycles 38

Nozzle Loss for Unchoked Flows V Chapter 2 Shaft Power Cycles 39

Nozzle Loss for Choked Flows V V Chapter 2 Shaft Power Cycles 40

Propelling Nozzles § ii) Specific thrust coefficient : § Kf = actual thrust / Isentropic thrust § § For adiabatic flow with w =0 ; § § For critical pressure ratio M 5 = 1 § ( T 5 = Tc ); ( 3. 13 ) Chapter 2 Shaft Power Cycles 41

Propelling Nozzles § For adiabatic flow with w =0 ; § § For critical pressure ratio M 5 = 1 § ( T 5 = Tc ); ( 3. 13 ) Chapter 2 Shaft Power Cycles 42

Propelling Nozzles § For choked flow; § Then; Chapter 2 Shaft Power Cycles 43

Propelling Nozzles § thus critical pressure ratio is; § (3. 14) Chapter 2 Shaft Power Cycles 44

Propelling Nozzles § For a given mass flow m; ( 3. 15 ) Chapter 2 Shaft Power Cycles 45