Lecture 8 Aircraft Mission Text: Constraints analysis - - PowerPoint PPT Presentation

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.108

Lecture 8

Aircraft Mission

Text: Motori Aeronautici

• Mar. 22, 2016

Mauro Valorani Università La Sapienza

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Agenda

1

Constraints analysis Introduction Concept of Constraints Mathematical model

Aerodynamic Polar Throttle Lapse

Flight phases

2

Mission analysis Introduction Aircraft weights Cruise weight ratio

TSFC behavior

BCM/BCA Takeoff weight estimation

3

Conclusions

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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The difficulties of engine design Gas Turbine engines exert a dominant influence on aircraft performance and must be custom tailored for each specific application. ⇒ Engine Specifications come from Aircraft Specifications The design process is both started by and constrained by an identified need The process is inherently iterative

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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The need : Request for Proposal (RFP) It’s the mission specification that defines the desired engine performance. The aircraft customer describes the desired aircraft performance in a document such as a Request for Proposal Example:

1

Takeoff, field is at 2000 ft pressure altitude. Takeoff ground roll must be less than 2500 m at MTOW

2

Takeoff rate of climb greater than 1000ft/min

3

Subsonic cruise at Best Cruise Mach, maximum range 10000 km

4

Payload of 60000 kg

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.112

Design Process

Figure: Design process, schematic

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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A Roadmap DESIGN PROCESS ⇒ Constraint and Mission Analysis

Choice of (TSL/WTO) and (WTO/S) Estimation of WTO to obtain TSL

ENGINE SELECTION ⇒ Parametric Cycle Analysis and Performance ENGINE COMPONENTS ⇒ Components Sizing

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Design Process

MASS FLOW CONSTRAINT & MISSION ANALYSIS PARAMETRIC CYCLE ANALYSIS Mission Specs Efficiencies (1st attempt) Thrust Cycle parameters (βc, T4, BPR, …) Specific Thrust Ia Component sizing Assumed TSFC behavior with h, V, δT OFF-DESIGN Geometries Efficiencies (Actual) Cross-section, blade profiles, combustor, … Desired TSFC Reference flight condition Tech limitation

Figure: Design process, schematic

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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The concept of constraints The requirements of the RFP can be converted into a series of functional relationships between: the thrust-to-weight ratio at sea-level takeoff TSL/WTO the wing loading at takeoff WTO/S We are looking for equations of the kind: TSL/WTO = f (WTO/S) for each of the requirements (flight phases). These will represent constraints that have to be attained simultaneously. Of course, many legitimate solutions exist, and none can be identified as

• ptimal or unique.

The "best" solution is always given by judgment and compromise.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Constraints Diagram Each requirement gives life to a curve in the constraint diagram. The solution space is the region above all the curves For a given WTO, a low WTO/S means large wing area and increased drag, while a high TSL/WTO results in a large thrust requirement. One may prefer, therefore, relatively low thrust and high wing loadings.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Constraints Diagram Design points of actual passenger/cargo aircrafts. The selected design point is very sensitive to the application and the preferences of the designer.

1 lbf ft2 = 47.88 N m2

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Constraints Diagram Design points of actual fighter aircrafts.

1 lbf ft2 = 47.88 N m2

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Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Master equation The design process starts by considering the forces that act on the aircraft (modeled as a point mass): lift, drag, thrust, weight. Equation of motion in the velocity direction: T cos(AOA + ϕ) − D − W sin(θ) = W g0 dV dt (15) where AOA is the angle between Velocity and Wing Chord Line, ϕ is the angle between Wing Chord Line and Thrust axis. Multiplying by the velocity V, we obtain the energy conservation equation: (T cos(AOA + ϕ) − D)V = W

• V sin(θ) + d

dt

• V 2

2g0

• (16)

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Master equation Assuming small angles of attack (AOA ≈ 0) and small thrust vector misalignments with V (ϕ ≈ 0), and recalling that V Sinθ = dh

dt :

V (T − D) W = d

• V 2

2g0 + h

• dt

= dze dt = Ps (17) where ze represents the aircraft mechanical energy (kinetic + potential) and is often referred to as "energy height". Ps is the time rate of change of the energy height and is called weight specific excess power. Isolating the thrust-to-weight ratio at LHS: T W = D W + Ps V (18) Here, both T and W depend on the flight condition and mission phase.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Master equation: assumptions for T and W It is assumed that the installed thrust and the actual aircraft weight are given by (SL=Sea Level Static, TO=Take-Off): T = αTSL (19) W = βWTO (20) where α is the full throttle thrust lapse (dependent on altitude, speed and afterburner on/off) and β depends on how much fuel has been consumed. The equation becomes: TSL WTO = β α

• D

βWTO + Ps V

• (21)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Aerodynamic polar Recall that lift can be expressed through the lift coefficient as follows: L = nW = 1 2 ρV 2SCL = qSCL q := 1 2 ρV 2 (22) ⇒ CL = nW qS = nβ q WTO S (23) and that also drag has a similar expression: D = qCDS (24) where CD can be expressed through the aerodynamic lift-drag polar: CD = CD0 + K1C2

L + K2CL

(25)

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Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Aerodynamic polar Conventional form: CD = CDmin + K ′C2

L + K ′′(CL − CLmin)2

(26) where: K ′ is the induced drag (inviscid drag due to lift) K ′′ is the skin+pressure drag (viscous drag due to lift) Expanding: CD = (K ′ + K ′′)C2

L − (2K ′′CLmin)CL + (CDmin + K ′′C2 Lmin)

• r:

CD = CD0 + K1C2

L + K2CL

(27) where K1 = K ′ + K ′′ K2 = −2K ′′CLmin CD0 = CDmin + K ′′C2

Lmin

Assumptions widely used: K1 =

1 πARe

K2 ≃ 0

Aircraft Mission Constraints analysis

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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The equation: TSL WTO = β α

• D

βWTO + Ps V

• (28)

becomes, using the expression D = qCDS and the aerodynamic polar: TSL WTO = β α

• q

β(WTO/S)

• k1C2

L + k2CL + CD0

• + Ps

V

• (29)

and using the expression for CL = nβ

q WTO S

:

TSL WTO = β α

• q

β(WTO/S)

• k1
• nβ

q (WTO/S)

2 + k2

• nβ

q (WTO/S)

• + CD0
• + Ps

V

• (30)

This equation is the sought after TSL/WTO = f (WTO/S) which depends on flight conditions (α, β, V, ρ, n, PS) aircraft aerodynamic features (CD0, k1[AR, e], k2)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Throttle Lapse (α) The available thrust is expressed as: T = αTSL The meaning of α is NOT that of a throttle setting, but rather that of a

• ff-design engine behavior, affected by flight conditions.

It can be obtained from full off-design runs or from semi-empirical models. Flight conditions (altitude and Mach number) are often blended together into a single parameter: the Dimensionless Freestream Total Temperature θ0 θ0 := T0a TSL = Ta

• 1 + (γ−1)

2

M2

a

• TSL

(31) where 0 = total a = freestream SL = (Standard day) Sea Level ⇒ TSL = 15C

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Conclusions

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Constant θ0 contours note that: θ0 = 1 at sea level static conditions, θ0 can be greater or less than 1, θ0 depends only on Mach number above the tropopause (being T0 constant), the range of θ0 of today’s aircrafts is 0.8 /1.4

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Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Engine behavior The engine compressor pressure ratio πc depends upon the turbine temperature ratio τt the throttle setting T04 the flight conditions θ0 With the hypothesis of choked turbine (≡ constant temperature ratio τt), it can be demonstrated that πc varies only with the ratio

T04 θ0 .

From the TurboJet power balance [ −( ˙ ma + ˙ mf )LT = ˙ maLC ]: ηm(1 + f)(h04 − h05) = h03 − h02 (32) τc − 1 = ηm(1 + f) (1 − τt) 1 TSL (cpt T04) (cpc θ0) ; τ := h0dn h0up ; T02 ≈ T0a (33) ⇒ πc = [1 + ηc (τc − 1)]

γ γ−1 =

• 1 + C1

T04 θ0

• γ

γ−1

(34) where the constant C1 is: C1 = ηcηm(1 + f) (1 − τt) 1 TSL cpt cpc

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Conclusions

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Engine behavior (cont’d) We came up with the relationship between πc and the ratio T04

θ0

πc =

• 1 + C1

T04 θ0

• γ

γ−1

it follows that πc increases as the ratio T04

θ0 increases.

Example: with fixed Mach number and throttle setting, πc increases if the aircraft climbs to higher altitudes (thus θ0 diminishes) πc diminishes if, for fixed altitude and throttle setting, the aircraft is accelerated to higher Mach numbers

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Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Theta break This image shows the behavior of πc in function of θ0 for fixed values of T04. The horizontal line corresponding to πcmax is due to the intervention of the Engine Control System

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Theta Break (cont’d)

Role of the Engine Control System

Prevent aircraft engines from operating outside their safety margins, such as: maximum compressor pressure ratio (preventing stall phenomena) maximum cycle temperature (preventing the turbine to overheat) These two limits are clearly visible in the previous slide. The ECS has two logics of operation: when πc is below its maximum allowable value, it limits the maximum allowable temperature reducing πc itself if πc reaches its maximum, it reduces T04 The value of θ0 at which the control logic switches from limiting πc to limiting T04 is known as: θ0break

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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The importance of Theta Break in the design phase At any flight condition different from θ0 = θ0break the engine cannot be

• perated at its maximum compressor pressure ratio and maximum cycle

temperature simultaneously. If θ0 < θ0break the engine can be operated at πcmax but T04 < T04max (obtaining less specific thrust than nominal) if θ0 < θ0break the engine can be operated at T04max but πc < πcmax (with greater fuel consumption than nominal) The engine designer would therefore have the engine operating as closer as possible to θ0 = θ0break and this can be done with a wise choice of θ0break The design with θ0break = 1 implies MAX THRUST AT SEA LEVEL STATIC A design with θ0break > 1 may be advisable for supersonic aircrafts

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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How to design Theta Break? Recalling from eq. (34) that the control system acts to maintain the ratio

Tt4 θ0

constant in order to keep πc = πcmax = constant, it follows immediately that, in the case of θ0break ≥ 1: T04max θ0break = T04 θ0

• πcmax

=

• T04SL

θ0SL

• πcmax

= T04SL (35) being θ0SL = 1. The engine has to be designed to have T04SL given by eq. (35), that is the ratio between the maximum allowable and the desired θ0break . Defining the Thottle Ratio as the ratio between the maximum allowable temperature and the sea-level-static temperature: TR := T04max T04SL = θ0break (36) it follows that TR is identical to θ0break .

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Throttle Lapse Example of engine behavior with varying Mach and altitude. It was obtained with semi-empirical models of the kind: α = f[h, M, TR] (TR fixed by design) (37)

Figure: Throttle Lapse of two Low-BPR-TurboFan (one with TR=1.0 and one with TR=1.1) as a function of Mach with varying altitude

Remember that T = αTSL

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Flight Phases The "master equation" (eq. 30) can be specialized for each flight phase, in

• rder to obtain the constraints that have to be attained to satisfy the RFP

. The following cases will be analyzed: Constant altitude/speed cruise Constant altitude/speed turn Takeoff Roll All the other flight phases can be obtained similarly and easily

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.135

Constant altitude/speed cruise Master equation (eqn. 30):

TSL WTO = β α

• q

β(WTO/S)

• k1
• nβ

q (WTO/S)

2 + k2

• nβ

q (WTO/S)

• + CD0
• + Ps

V

• Known quantities:

Constant altitude and speed ⇒

dh dt = 0 dV dt = 0 ⇒ PS = 0

Level flight ⇒ n = 1 (L = W) Aerodynamic polar (assigned): k1, k2, CD0 as functions of M, AR, e Assumptions: β = 0.7 ÷ 0.9 (an high enough value to simulate the beginning of the cruise phase, which is the more demanding scenario) The requirements are given in terms of: Desired cruise altitude h Desired cruise Mach number M Altitude and Mach number appear in: α(h, M) q = 1

2ρ∞(h)V 2 ∞ = 1 2γp∞(h)M2 ∞

Aerodynamic polar (CD depends on Mach)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.136

Constant altitude/speed cruise (cont’d)

Figure: Constant altitude/speed cruise constraint. h = 11000 m, M = 0.85

The minimum of the curve is found at:

WTO

S

• minT/W =

q β

• CD0

K1

⇒ TSL

WTO

• min = β

α

• 2
• CD0K1 + K2
• which is the condition of minimum thrust and drag (maximum range)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.137

Constant altitude/speed turn Master equation (eqn. 30):

TSL WTO = β α

• q

β(WTO/S)

• k1
• nβ

q (WTO/S)

2 + k2

• nβ

q (WTO/S)

• + CD0
• + Ps

V

• Known quantities:

Constant altitude and speed ⇒

dh dt = 0 dV dt = 0 ⇒ PS = 0

Aerodynamic polar (assigned): k1, k2, CD0 as functions of M, AR, e Assumptions: β = 0.7 ÷ 0.9 The requirements are given in terms of: Desired cruise altitude h Desired cruise Mach number M Desired turn rate n > 1 Remember that during a turn L = nW where n =

1 cosφ =

• 1 + V 2

g0r (r is the

radius of curvature)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Constant altitude/speed turn (cont’d)

Figure: Constant altitude/speed turn constraint. h = 11000 m, M = 0.85, n = 1.1

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.139

Takeoff Ground Roll Master equation (eqn. 18): V(T − D) W = d

• V 2

2g0 + h

• dt

the takeoff ground roll distance is obtained with the assumption of constant altitude dh/dt = 0 and TSL >> D: TSL WTO = β αg0 dV dt which can be rearranged to (dt = ds/V): ds = β αg0 WTO TSL VdV and integrated to yield: sG = β αg0 WTO TSL V 2

TO

2g0 Being: 1 2 ρSV 2

STALLCLmax = βWTO

where VTO = kTOVSTALL and kTO ≈ 1.2, the equation can be recast to give: sG = β2 α k2

TO

(TSL/WTO) ρ g0CLmax WTO S

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Takeoff Ground Roll (cont’d) The ground roll distance equation: sG = β2 α k2

TO

(TSL/WTO) ρ g0CLmax WTO S

• tells that the ground roll distance is inversely proportional to:

density ρ, which is function of altitude and temperature. High elevation airports and hot days deteriorate take-off performance. aerodynamic performance CLmax , whose value is increased by the use

• f high lift devices (slats, flaps, ...)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Solving the previous equation for TSL/WTO, the take-off constraint as a function of WTO/S can be obtained: TSL WTO = β2 α k2

TO

sG ρ g0CLmax WTO S

• (38)

Known quantities: CLmax (≈ 1.5 ÷ 2.7 depending on installed H-L-devices) Assumptions: β = 1 (MTOW) The requirements are given in terms of: Desired field elevation and temperature ⇒ ρ[h, T] Desired Takeoff speed (VTO = kTO VSTALL, es: kTO = 1.2) Field length available sG Note that airport elevation and temperature affect also the engine performance through α(h, M0)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Takeoff Roll (cont’d) The relationship is linear

Figure: Takeoff Roll. h = 1500 m, CLmax = 2.4, sG = 2000 m, HOT (ISA +30) vs COLD (ISA +0) day

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Constraint diagram Cases analyzed: constant altitude/speed cruise, constant altitude/speed turn, takeoff roll, horizontal acceleration, service ceiling

Figure: All constraints

The solution space is the white region over the constraint curves. We can choose: TSL WTO and WTO S

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Aim of the mission analysis Constraint analysis ⇒

TSL WTO

and

WTO S

Next step ⇒ establish the size of the aircraft via the estimation of the maximum takeoff weight WTO (o MTOW) With the MTOW in hand, the Sea level Max Thrust TSL and the wing area S are immediately obtained: TSL WTO · WTO = TSL WTO S −1 · WTO = S

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Who is WTO? WTO is the sum of 3 main contributions: empty weight WE

aircraft structures equipments (engines, avionics, seats, etc.)

payload weight WP fuel weight WF WTO = WE + WP + WF (39) The sum of Empty weight and Payload weight is the Maximum Zero Fuel Weight (MZFW) it follows that: WTO = WP 1 − WF

WTO − WE WTO

(40) The fuel weight WF represents the fuel gradually consumed during the mission: the aircraft weight decreases at exactly the same rate at the which the fuel is consumed: dW dt = − dWF dt = −TSFC · T (41)

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Cruise weight ratio We need WF /WTO, given desired range and payload (from RFP). For the cruise flight phase, the required thrust is not known, being it throttled down so that: T = D(h, M, AOA) Equation (41) becomes: dW W = −TSFC D W dt = −TSFC D W ds V (42) The integration requires the knowledge of the behavior of the term −TSFC D

W

during the mission. It is often found that this term remains relatively unaltered over the flight leg, and the integral can be approximated as: Wf Wi = exp

• −TSFC

D W

• ∆t
• (43)

where ∆t is the cruise flight time.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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TSFC behavior We now seek the behavior of TSFC with varying flight conditions. TSFC is a complex function of the combination of instantaneous altitude, speed and throttle setting A satisfactory approximation for this design stage is the following: TSFC ≈ (C1 + C2M) (44) where C1 and C2 are constants that are known in advance for each type of engine cycle.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

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Best Subsonic Cruise Subsonic cruise is usually the most important portion of any mission because it uses the largest amount of onboard fuel.

• Eqn. (42) can be simplified to yield:

dW W = − C1/M + C2 astd CD CL

• ds

(45) where astd is the speed of sound for ISA conditions. Best Cruise condition ⇒ minimum fuel consumption ⇒ min

• (C1/M + C2)

CD CL

• = (C1/M⋆ + C2)
• C⋆

D

C⋆

L

• The best cruise is identified by the values M⋆, C⋆

L , C⋆ D

Next task: find their expressions

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Best Subsonic Cruise Mach number Below the critical drag rise Mach number MCRIT (approximately 0.8), neither CL nor CD depend on M. In this range: CD CL = CD0 + K1C2

L + K2CL

CL (46) whose minimum can be found differentiating with respect to CL: CD CL ∗ =

• 4K1
• CD0
• + K2

at C∗

L =

• CD0

K1 (47) which is the maximum efficiency CL, as expected. Next, because the lowest achievable value of

• CD

CL

• is constant below MCRIT , it

follows that: (C1/M + C2)

• CD

CL

• decreases as M increases

Further increases past MCRIT cause

• CD

CL

• to increase again

⇒ M⋆ = MCRIT

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Best Subsonic Cruise Weight Fraction The final equation is: dW W = −

• C1

MCRIT + C2 4(CD0)K1 + K2 ds astd (48) whose integral is exact and yields: ΠCRZ = Wf Wi = exp

• −
• C1

MCRIT + C2 4(CD0)K1 + K2 ∆s astd (49) Knowing: the desired range ∆s the values for the TSFC constants C1 and C2 the aerodynamic polar we can estimate the cruise fuel depletion.

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Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.151

Best Subsonic Cruise Altitude The altitude may not be arbitrarily chosen It is the altitude that allows the condition of self-sustained flight with those values of M⋆, C⋆

L and C⋆

• D. Being:

L = βWTO ⇒ 1 2 γM2pSC⋆

L = βWTO

Recalling that C⋆

L =

• CD0/K1, it follows that:

δ := p pstd = 1 pstd 2β γM2

CRIT

1

• (CD0)/K1

WTO S

• (50)

Remember that β decreases during cruise, so: δ (⇒ altitude) must gradually increase being M fixed to M⋆, the speed must gradually decrease until the tropopause is reached Usually steps of 2000 ft are employed, due to air traffic control concerns.

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.152

Takeoff weight estimation Recalling eqn. 39 WTO = WE + WP + WF (51) it follows that: WTO = WP 1 − WF

WTO − WE WTO

(52) WP is the desired payload weight

WF WTO is the product of the weight ratios of all the phases of flight (we

examined best cruise only)

WE WTO can be estimated with empirical models as a function of WTO

N.B. The calculations of WTO requires an iterative procedure because of the dependence of

WE WTO on WTO

Aircraft Mission Constraints analysis

Introduction Concept of Constraints Mathematical model Aerodynamic Polar Throttle Lapse Flight phases

Mission analysis

Introduction Aircraft weights Cruise weight ratio TSFC behavior BCM/BCA Takeoff weight estimation

Conclusions

8.153

What we have done so far and what’s next Done: RFP ⇒ Constraints on TSL/WTO and WTO/S First attempt choice of TSL/WTO and WTO/S Mission Analysis ⇒ WTO ⇒ TSL Next: How to achieve this TSL? We have performance to meet (TSFC) We have design limitations (T4max , βCmax , etc.) We can play with design parameters (BPR, βF , βC, etc.) ⇒ Parametric Cycle Analysis and Performance estimation