AE Aircraft Performance and Flight Mechanics

Transcription

1 AE 49 - Aircraft Performance and Flight Mechanics efinitions Range Total ground distance traversed on a full tank of fuel -- Anderson book istance an airplane can fly on a given amount of fuel -- alternative definition Endurance Total time an airplane stays in the air on a full tank of fuel -- Anderson book Time an airplane can fly on a given amount of fuel -- alternative definition Specific fuel consumption eight of fuel consumed per unit power per unit time 1

2 Propeller-riven Airplane Generalized endurance parameters Specific fuel consumption (SF or c) eight () lbs Time (t) secs or hrs Engine power (P) hp Airplane gross weight ( 0 ) lbs includes fuel and payload Airplane empty weight ( 1 ) lbs weight of airplane without fuel Endurance calculation w Fuel flow rate f - c P (lbs fuel/unit time) ifferential fuel burned c Pdt Solving for time and integrating E dt 0 cp f BS FUE BHP HR E f f f dt cp f c P Propeller-riven Airplane Generalized Range Equation Multiply V by dt to get differential air distance cp covered ds V dt V cp The total distance covered during a range flight is the integral of this expression R 1 V 0 V R ds 0 0 cp 1 cp

3 Propeller-riven Airplane Breguet Range Equation In level flight cruise (no acceleration), the throttle is set to maintain constant airspeed P A P R V P Shaft HP P A V η η Substituting into the general range equation for propeller airplanes 0 V η η 0 V 0 R 1 cp 1 cv 1 c Multiplying by / and using for equilibrium cruise flight (level and unaccelerated) R 0 1 η c 0 1 η c Propeller-riven Airplane Breguet Range Equation (continued) Assume h is constant throughout flight / is constant throughout flight c is constant throughout flight Then, we can integrate η R 0 η R ln c 1 c Because of the assumptions listed, the Breguet approximation is valid only for small Angle of attack changes over a weight change as large as the fuel load If weight changes significantly, / changes 0 1 3

4 Propeller-riven Airplane Breguet Endurance Equation Using the same assumptions ( and P A P R V ) 0 0 η 0 η E 1 cp 1 c V 1 c V Since and E If we assume 1 ρ S ρ V S, V 0 1 η ρ S c,, η, c, and ρ are all constant 3/ η 1 1 E ρ S c 1 0 in using this equation, E must be in seconds and c must be in ft -1 3/ 3/ Importance of the Ratios,, 1/ max Max Range for reciprocating engine/propeller airplanes 3/ max Max Endurance for reciprocating engine/propeller 4

5 Jet-powered airplane General endurance equation Specific fuel consumption is based on thrust, not power output lbs of fuel consumed TSF ct lb of thrust produced sec Again, let be the differential change in aircraft weight as fuel is burned in dt cttadt dt f c ctt t A TA E 1 0 Integrating: E dt 0 0 ctta 1 ctta Recalling that and T A T R 0 1 E 1 Assuming constant c t and / c t The Breguet approximation is: 1 E ln 0 ct 1 Jet-powered airplane To maximize endurance Reduce thrust specific fuel consumption, TSF Use a turbofan engine Fly at an altitude where the engine is efficient (high, but not too high) increase fuel fraction, f / 0, to increase 0 / 1 arry less payload Use drop tanks maintain /max oiter at velocity for /max (where,0 i ) ruise climb Notice that a jet airplane has best endurance at this velocity for /max, while a propeller-driven airplane gets best range at this true airspeed 5

6 Jet-powered airplane general range equation going back to the differential time expression and multiplying by V Vc ds V dt t T A of course, ds is the differential distance traveled by the jet during time dt s 0 when w w 0 s R when w w 1 integrating to get range R 1 V R ds 0 0 ctta for steady, unaccelerated level flight: T A T R 0 V R 1 ct Jet-powered airplane general range equation (continued) since, R becomes V ρ S Breguet approximation if we assume constant c t,,, and ρ for maximum range R ( 0 1 ) 1 R ρ S c t ρ S c t reduce TSF, c t fly at low density (high altitude) use a true airspeed to maximize 1/ 6

7 Jet-powered airplane ruise for Max range Fly at max The true air- Speed for This optimum Speed gives,0 3 d i 3/ Importance of the Ratios,, 1/ max Max Endurance for jet-propelled airplanes 1/ max Max Range for jet-propelled airplanes 7

8 Summary Relationship of,0 and i At / max Using,0 +,0 + ifferentiating with respect to and setting to zero d,0 + 0 d,0 + ear π 0,0 +,0 ear π,0,i FOR max Summary Relationship of,0 and i Similar developments show:,0 3,i FOR max Applies to maximum range for a jet airplane 3 3,0,i FOR max Applies to maximum endurance for a propeller-driven airplane 8

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