Module 6

Aircraft Performance

Module 6

Where are we? 1 : Introduction to aircraft performance, atmosphere 2 : Aerodynamics, air data measurements 3 : Weights / CG, engine performance, level flight 4 : Turning flight, flight envelope 5 : Climb and descent performance 6 : Cruise and endurance 7 : Payload-range, cost index 8 : Take-off performance 9 : Take-off performance 10 : Enroute and landing performance 11 : Wet and contaminated runways 12 : Impact of performance requirements on aircraft design

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Cruise         

Introduction Definition of specific air range (SAR) Calculation of SAR Typical SAR chart Cruise range calculation Types of cruise Temperature effects Altitude effects Wind effects

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Cruise - Introduction  Range is defined as the distance that an aircraft can travel with a given fuel quantity and a given payload • For example, the Bombardier Global Express has a (non-stop) range capability of 6,500 nm at a cruise speed of Mach 0.80 and with a payload of 8 passengers (1,600 lb)

 Range is the sum of climb, cruise and descent distances  Efficient cruise performance is required in order to maximize range and minimize operating costs  The concepts associated with cruise performance will be discussed in the next slides

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Cruise – Definition of Specific Air Range (SAR)  SAR is defined as the air distance traveled per unit of mass of fuel during steady state and level flight cruise conditions  SAR units • Nautical air miles (nam) per lb of fuel (nam/lb)  SAR = true airspeed / fuel flow = V / Wf  SAR is independent of wind speed (if wind is constant)  Cruise air distance is proportional to SAR for a given fuel quantity

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Cruise – Definition of SAR (Cont’d)  Specific Range (SR) is defined similarly but in terms of ground distance and speed • SR = ground speed / fuel flow = Vg / Wf • Vg = V + Vwind • SR = SAR * (V + Vwind ) / V • For mission performance analysis, headwinds are negative (negative impact on performance) and tailwinds are positive (positive impact on performance)

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Cruise – Calculation of SAR  SAR can be calculated with an exact numerical method or with a theoretical method  Exact method : • Specific cruise conditions are assumed : weight, altitude, Mach number, deviation from ISA , engine bleed extraction and number of engines operating (neng) • V = ao θ 0.5 • q is calculated (q = 1481.3 δ M2 , q in lb/ft2) • CL= W/(qS) • From the high speed drag model of the airplane, CD is determined knowing CL and M Cruise and endurance

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Cruise – Calculation of SAR (Cont’d) • Total thrust required = Treq = D = CDqS • T required per engine = Treq / neng • Data supplied by the engine manufacturer is used to determine Wf per engine knowing altitude, Mach number, deviation from ISA, engine bleed extraction and Trequired per engine

•

Wf = Wf per engine * neng

• SAR = V / Wf

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Cruise – Calculation of SAR (Cont’d)  Theoretical method is useful to understand the parameters that affect SAR • SAR = V / Wf • SAR = V / (Treq SFC) • Knowing that Treq = W / (L/D) : • SAR = (V/SFC) (L/D) (1/W) • Knowing that V = ao θ 0.5 M, • SAR = (ao θ 0.5 / SFC) (M L/D) (1/W)  Theoretical method shows that for a given weight and altitude combination, SAR is maximum when M L/D is maximized • assumes that SFC is independent of the thrust level • Maximum M L/D is equivalent to maximum CL0.5/CD

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Cruise – Calculation of SAR (Cont’d)  Tangent lines on CL – CD graph defines maximum M L/D  CL for maximum M L/D reduces as M increases  Maximum M L/D occurs when compressibility effects are becoming important

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Cruise – Typical SAR chart

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Cruise – Cruise range calculation 

Cruise range is defined as the cruise air distance traveled while burning a given quantity of fuel (given fuel burn)

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Cruise range can be obtained by using SAR data or from a theoretical equation

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For flight at constant altitude, cruise range can easily be determined from SAR data : • R = SARavg * (W1-W2) Where R = Cruise range (nam) (still air distance) W1 = weight at beginning of cruise segment (lb) W2 = weight at end of cruise segment (lb) W1-W2 = fuel burn during cruise segment (lb) SARavg = average SAR during cruise segment (nam/lb) Cruise and endurance

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Cruise – Cruise range calculation (Cont’d) • Note : if SAR does not vary linearly with weight, it is necessary to analyze smaller cruise segments and to add them up in order to obtain the cruise range  Cruise range can also be estimated with theoretical methods if certain assumptions are made: • SFC is constant • CL is constant  Cruise range is obtained by integrating the SAR equation over a weight range  Two cases are analyzed in the next slides • Flight at constant altitude • Flight at constant speed Cruise and endurance

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Cruise – Cruise range calculation (Cont’d) Case 1 : flight at constant altitude  SAR equation : SAR = (1/ SFC) (V L/D) (1/W) 

Knowing that V = (2 W / (ρ S CL))0.5 , we can rewrite SAR as

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SAR = (1/SFC) (2 / (ρ S))0.5 (CL0.5/CD) (1/W0.5)

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Integration of the SAR equation with respect to weight and conversion to nam gives: R = [ 1.676 / (SFC (ρ S)0.5) ] (CL0.5/CD) (W10.5- W20.5) (nam) Where R = Cruise range (nam) (still air distance) W1 = weight at beginning of cruise segment (lb) W2 = weight at end of cruise segment (lb)

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Range is maximized by minimizing ρ (i.e. high altitude) and by maximizing CL0.5/CD (i.e. CL = (CD0 / (3K) )0.5) if compressibility effects are neglected) Cruise and endurance

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Cruise – Cruise range calculation (Cont’d) Case 2 : flight at constant speed  

SAR equation : SAR = (1/ SFC) (V L/D) (1/W) Integration of the SAR equation with respect to weight and conversion to nam gives: R = (V/ SFC) (CL/CD) (ln (W1/W2)

(nam)

or R = ( (661.5 θ0.5) / SFC) (ML/D) (ln (W1/W2) (nam) Where R = Cruise range (nam) (still air distance) V = TAS (knots) W1 = weight at beginning of cruise segment (lb) W2 = weight at end of cruise segment (lb) 

Range is maximized by maximizing ML/D

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Cruise – Cruise range calculation (Cont’d)  Constant speed and constant CL imply that the aircraft must climb as its weight reduces : • CL = W / (1481.3 δ M2 S) • W / δ must be constant • As weight reduces due to fuel burn, the aircraft must climb • Resulting climb angle is small (order of 0.02o) and the basic assumption that T = D is still essentially valid  If SFC, M or L/D are not constant, the integration can be done numerically

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Cruise – Cruise range calculation (Cont’d)  Range for the two cases can be compared with the following example: -

S = 450 ft2 W1 = 45,000 lb and W2 = 42,000 lb FL370 / ISA / Mach 0.8 L / D = 15 SFC = 0.6 (lb/hr) / lb fuel

 Flight at constant altitude : • R = [ 1.676 / (SFC (ρ S)0.5) ] (CL0.5/CD) (W10.5- W20.5) ∀ ρ = 0.0006759 slugs / ft3 • q = 1481.3 δ M2 = 202.7 lb/ft2 • CL = W/ (qS) = 0.493 • CD = CL / (L/D) = 0.0329 • R = 778 nam

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Cruise – Cruise range calculation (Cont’d)  Flight at constant speed : • R = ( (661.5 θ 0.5) / SFC) (ML/D) (ln (W1/W2) ∀ θ = 0.7519 • R = 791 nam

 Flight at constant speed and constant CL provides better range  Verify climb angle for flight at constant speed • W/ δ = 45000 / 0.2138 = 210,477 lb = constant ∀ δ at end of cruise = W / (W/ δ) = 42,000 / 210,477 = 0.1996 • Altitude at end of cruise = 38,437 ft • Climb angle = atan ( 1437 / (791*6077)) = 0.017o

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Cruise – Types of cruise  Cruise at constant speed and constant CL implies increasing altitude • Not possible operationally • Is approximated operationally by cruise segments at constant altitude followed by step climbs

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Cruise – Types of cruise (Cont’d) 

During flight at constant altitude, range can be maximized (or fuel burn minimized for a given range) by flying at the Mach number for maximum SAR – referred to as Maximum Range Cruise (MRC) Mach number • MRC is not used very much operationally as it normally results in unacceptably low cruise speeds (long flight time) • Flight at MRC implies that M and thrust are reduced as weight reduces • MRC speed schedule can be derived from SAR chart

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Cruise – Types of cruise (Cont’d) 

Flight at Long Range Cruise (LRC) cruise speed or Mach number provides a good compromise between fuel efficiency and flight time

 LRC speed is the speed that provides 99 % of max. SAR • Implies that Mach and thrust are reduced as weight reduces • LRC can be derived from SAR chart

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Cruise – Types of cruise (Cont’d)  Flight at constant Mach number • Commonly used operationally, specially on short range missions where flight at varying speed schedule (e.g. LRC) would only result in a small benefit in operating costs  Flight at maximum cruise speed • Speed limited by maximum level flight speed capability (i.e. D = MCR or maximum cruise thrust) or Vmo/Mmo • MCR limit varies as a function of temperature (MCR is lower at higher deviations from ISA) • Results in low SAR • Used when flight time is more important than fuel cost

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Cruise – Types of cruise (Cont’d)  Effect of different types of cruise conditions on cruise range

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Cruise – Temperature effects 

Unless maximum cruise speed is used, temperature has a negligible effect on SAR and range • SAR = V / Wf • In order to maintain a constant thrust level at a higher temperature, fuel flow increases • The fuel flow increase is essentially compensated by the higher true airspeed V at the higher temperature • Effect on SAR is typically less than 0.1 % for every degree of deviation from ISA conditions

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Significant impact when max. cruise speed is limited by MCR

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Cruise – Altitude effects • Cruise altitude can have a significant impact on fuel required Example : CRJ200, 500 nm mission, 30 passengers Climb: 250 kts/M 0.70 - Cruise: M 0.74 16 14 12 10

% increase in block fuel for cruise below FL390

8 6 4 2 0

FL FL FL FL FL FL 390 370 350 330 310 290 Cruise and endurance

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Cruise – Wind effects  When winds are present, it is desired to maximize nm/lb of fuel  SR (nm/lb) = SAR (V + Vwind)/V

(tailwind positive)

 With a tailwind : • SR is greater than SAR • Mach number for MRC and LRC are lower than in zero wind conditions  With a headwind : • SR is lower than SAR • Mach number for MRC and LRC are higher than in zero wind conditions  Range (still air distance) is corrected similarly in order to obtain ground distance Cruise and endurance

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Endurance  Introduction  Conditions for best endurance

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Endurance - Introduction  Endurance is defined as the length of time that an aircraft can remain airborne  Some aircraft are used for missions where it is required to maximize the time that the airplane remains airborne • Ex. : surveillance mission  In addition, air traffic controllers may require that an aircraft stays in holding mode before proceeding with the planned mission  For such cases, it is desirable to fly at a condition where fuel flow is minimized • Maximum endurance is obtained when the aircraft is operated at a flight condition where fuel flow is minimized

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Endurance – Conditions for best endurance  For practical considerations, level flight is assumed  Basic SFC definition leads to : dW = - SFC T dt dt/dW = -1 / (SFC T)  Knowing that T = D = W D/L in level flight : dt = - (1/SFC) L/D 1/W dW  Integrating between beginning and end of flight segment : E = (1/SFC) L/D ln (W1/W2) Where E = Endurance (hours) W1 = weight at beginning of flight segment (lb) W2 = weight at end of flight segment (lb)  For the case where SFC is constant with varying thrust levels, maximum endurance is obtained during flight at maximum L/D or VMD Cruise and endurance

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Endurance – Conditions for best endurance (Cont’d)  Typical data shows that minimum fuel flow occurs at a specific pressure altitude

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Endurance – Conditions for best endurance (Cont’d)  In practice, the speed for best endurance or holding is normally defined by VMD  Other constraints may force a further increase of the holding speed • Maneuvering margin prior to stall warning • Maneuvering margin prior to buffet

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