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Aircraft Performance

Module 5

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Climb and descent performance2

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 endurance7 : Payload-range, cost index

8 : Take-off performance

9 : Take-off performance

10 : Enroute and landing performance

11 : Wet and contaminated runways12 : Impact of performance requirements on aircraft design

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Climb

Introduction

Balance of forces

Rate of climb

Lift during climb

Climb gradient

Maximum rate of climb

Enroute climb speeds

Acceleration factor

Climb ceiling

Calculation of time, distance and fuel to climb

Certified climb performance data

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Climb - Introduction

In straight and level flight, T = D

If T > D, aircraft will accelerate and / or climb

Two flight regimes where climb performance is important

Obstacle clearance operations close to the ground

Enroute rate at which cruise conditions are achieved

Climb performance must be derived in terms of climb angle

or climb gradient for take-off or go-around climbout

Climb gradient (tangent of the climb angle) is used as the

reference to establish climb performance

For enroute performance, rate of climb is used as the

reference to establish climb performance

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Climb Balance ofForces

Summation of forces along flight path

T - D - W sin K - (W/g) dVg/dt = 0

Summation of forces normal to flight path

L + (W/g) dK/dt V W cos K = 0

K

W/g (dVg/dt)

K

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Climb Rate of Climb

From summation of forces along flight path:

sin K = (T-D)/W (1/g) dVg/dt

Rate of climb is defined as

r/c = dh/dt = V sin K

r/c = V(T-D)/W (V/g) dVg/dt

Geometric rate of climb, not pressure rate of climb

dVg/dt can be written as (dVg/dh)(dh/dt) and substituted in

previous equation

r/c = dh/dt = V(T-D)/W (V/g) (dVg/dh)(dh/dt)

dh/dt (1 + (V/g)(dVg/dh)) = V(T-D)/W

r/c = dh/dt = [V(T-D)/W] / [1 + (V/g)(dVg/dh)]

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Climb Rate of Climb (Contd)

The dimensionless term (V/g)(dVg/dh) is known as the

acceleration factor AFr/c = [V(T-D)/W] / (1 + AF)

AF will be defined later for different types of climb profiles

In summary:

When the aircraft is climbing at constant ground speed (no

accel.):

r/cunaccelerated = V(T-D)/W

When the aircraft is climbing and accelerating at the sametime:

r/caccelerated = V(T-D)/W (V/g) dVg/dt or

r/caccelerated = [V(T-D)/W] / (1 + AF)

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Climb Lift during Climb

Balance of forces normal to flight path leads to

L W cos K = - (W/g) dK/dt V

(W/g) dK/dt V is the centrifugal acceleration due to changing

the flight path at a rate dK/dt

dK/dt is nearly zero at any point in the climb

dK/dt is assumed to be equal to zero during climb

Balance of forces can be written as

L = W cos K

Lift during climb is less than W Drag during climb is lower than drag for level flight

cos K is essentially equal to 1 for most conditions on

commercial airplanes and we can assume that L = W

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Climb Climb Gradient

Climb gradient is defined as the tangent of the climb angle

Climb gradient = tan K

For small climb angles, tan K = sin K

Climb gradient = sin K = (r/c) / V

If K is expressed in radians, K = sin K

Climb gradient = K = (T-D)/W - (1/g) dVg/dt

K = (T/W CD/CL) - (1/g) dVg/dt

K = (T/W CD/CL)/(1 + AF)

Maximum K occurs at Vx when excess thrust (T-D) is

maximum

Speed typically slightly lower than VMD

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Climb Climb Gradient (Contd) Climb gradient is normally expressed as a percentage

Example: tan K = K = 0.05 is equivalent to a gradient of 5 %

A gradient of 5 % means that the aircraft climbs by 5 ft for every 100

ft traveled horizontally

If winds are present, the geometric climb gradient will be

different from the value ofK calculated with the previous

equations

Kgeometric = r/c / Vg = K (V/Vg) = K (V/(V-Vwind))

Where Vwind is the wind speed (headwind is positive)

Note : r/c is not affected by wind

Acceleration and climb gradient can be traded

Kno acceleration = (T-D)/W (referred to as total climb gradient available)

Kwith acceleration = (T-D)/W - (1/g) dVg/dt

Example : Kno acceleration = 5% can be traded for level flight acceleration

of 0.05 g

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Climb Maximum Rate of Climb

For a given weight and altitude condition, maximum rate of

climb is obtained at the speed VY

where V(T-D) is greatest

The speed for maximum rate of climb is slightly higher than the

speed for maximum excess thrust (T-D)

The true airspeed V for maximum rate of climb increaseswith altitude

The aircraft must accelerate along the flight path to maintain

the maximum rate of climb

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Climb Enroute climb speeds

In practice, it is desirable to choose an easily flown climb speed

Speed schedule independent of weight and temperature

Constant calibrated airspeed at low and medium altitudes

Climb at constant Mach number at higher altitudes wherecompressibility effects have a more important effect on climb

performance

Example of a climb speed schedule : 250 KCAS / M 0.70

Transition from climb at constant CAS to climb at constant Mach is

at a fixed altitude for a given CAS / M climb speed schedule

Altitude defined as the transition altitude

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Climb Enroute climb speeds (Contd)

Operational considerations

Aircraft speed is limited to 250 KCAS at altitudes up to 10,000 ft

(operational regulation)

Operational requirements normally dictate a climb speed greater than

250 kts above 10,000 ft

- Large jet aircraft typically use higher speeds

- ATC may not provide clearance to aircraft that use lower speeds

Aircraft is normally accelerated from 250 KCAS

to a higher climbspeed at 10,000 ft and the higher climb speed is maintained until

Mach reaches the climb Mach value

- Example : Climb speed schedule of250 kts / 290 kts / M 0.74

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Manufacturers typically define various climb speed schedules inorder to meet operational needs

Low speed climb speed schedule (e.g. 250 kts / M0.70) to minimize

fuel burn and maximize range capability

Also referred to long range climb speed schedule

High speed climb schedule (e.g. 250 kts / 320 kts / M 0.77) to

minimize flight time

Normal speed climb schedule (e.g. 250 / 290 / M0.74) provides a

compromise between fuel saving and flight time

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Best r/c is achieved at a speed close to constant calibrated

airspeed at altitudes up to about 30,000 ft and at a speed close toconstant Mach at altitudes above 30,000 ft

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Climb Acceleration factor

Acceleration factor (V/g)(dVg/dh) changes significantly during theclimb

Transition alt.

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Climb Acceleration factor (Contd)

When the acceleration factor (V/g)(dV/dh) is to be determined, the

incremental altitude, dh, is a change in true altitude and not

pressure altitude

Since all performance data and performance calculations are

based on pressure altitude, any pressure altitude increment can

be corrected to obtain the true altitude increment

(h = (hp (T/Tstd)

where :

T is the absolute average temperature over(hp

Tstd is the absolute average temperature over(hp under IS

Aconditions

It must be noted that the same concept applies when geometric rate

of climb is converted in pressure rate of climb (rate of change of

pressure altitude with time)

r/c pressure = r/c (Tstd/T)

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Climb Acceleration factor (Contd)

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Climb Climb Ceiling

Climb ceiling is the lowest pressure altitude at which the rate of

climb reaches a defined value

300 ft/min is typically used by airlines

Other values such as 100 f/min or 500 ft/min are sometimes used

Note that a fixed r/c can sometimes be achieved at two different

pressure altitudes

Why is the ceiling based on a fixed r/c capability?

Want to reach cruise altitude within a reasonable time

Want to have some excess thrust in order to be able to accelerateto cruise speed once the cruise altitude is reached

Example of climb ceiling chart is presented on the next slide

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Climb Calculation of Time, Distance, and Fuel

Manufacturers provide climb time, climb distance and climb

fuel data Time, distance and fuel data is used for flight planning

purposes

Data presented as a function of initial climb weight

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Climb Calculation of Time, Distance, and Fuel (Contd)

Calculation of time, distance and fuel to climb is calculated

using a step by step integration process using a timeincrement basis or a pressure altitude increment basis

Integration on a pressure altitude basis is more convenient

and is described on the next slide

Calculations are made for given values of

Climb speed schedule

Initial climb weight

Deviation from ISA

Engine bleed extraction

Climb thrust (T) and Fuel Flow (Wf) data is obtained from the

engine manufacturer

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Climb Calculation of Time, Distance, and Fuel (Contd)

1. Select hp1 and hp2

2. hpavg = (hp1 + hp2)/2

3. (hp = hp2 - hp1

4. (htrue = (hp (T/Tstd)

5. Wavg (at hpavg) is assumed

6. V, T, D and AF are evaluated at hpavg

7. r/cavg = [V(T-D)/Wavg] / (1 + AF)

8. (t = t2 t1 = (htrue/ r/cavg

9. t2 = t1 + (t

10. (dist = V (t

11. d2

= d1

+ (dist

12. Wfavg (Wfat hpavg) is evaluated from engine data

13. (fuel = Wfavg (t

14. fuel2 = fuel1 + (fuel

15. W2 = W1 - (fuel

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Climb Calculation of Time, Distance, and Fuel (Contd) Notes on the calculation process

Once the fuel burn (fuel has been calculated, the average weight

assumed for that step may be validated. Errors of up to 20 lb will notsignificantly affect the results

The size of(hp selected is a function of the rate of climb and is usually

in 1,000 ft increments or less

Altitudes where there are discontinuities in the rate of climb must beused as discrete points for calculation (i.e. (hp must be selected such

that hp2 = altitude with the discontinuity)

If a level acceleration segment is included in the climb profile, a similar

approach is used for the acceleration segment with the exception that:

Integration is based on a step in speed

Acceleration is calculated (instead of r/c) at the average speed

If winds are present, r/c is not affected but the distance increments

must be be calculated with the ground speed Vg ((dist = Vg (t)

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Climb Certified Climb Performance Data

Items covered in previous pages fell in the category of

operational performance, i.e. not certified performance

Certified Climb Performance data will be reviewed in another

module and will include :

Take-off weight limited by climb requirements (WAT limits)

Take-off weight limited by obstacle clearance considerations

Improved take-off climb

Engine-out enroute climb performance (driftdown)

Landing weight limited by climb requirements (WAT limits)

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Descent

Introduction

Rate of descent Descent speed schedules

Cabin pressurization considerations

Emergency descent

Gliding flight

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Descent - Introduction

Descent analysis is analogous to climb analysis except that

appropriate sign corrections are required

The methodology used for climb time, distance and fuel can

also be used for descent time, distance and fuel

Descents are normally carried out with engines at idle

Net thrust is very low and sometimes negative

General considerations

Minimum glide angle descents are flown at the speed for the

best L/D, and that speed increases with weight

Maximum rate of descent is obtained at maximum speeds andwithmaximum use of available drag devices

For a given Mach / CAS descent speed schedule, heavier

aircraft have lower rates of descent and lighter aircraft have

higher rates of descent

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Descent Rate of Descent

Rate of descent (r/d) is calculated as follows:

r/d =- (dh/dt) = V(D-T)/W + (V/g) dVg/dt

r/d = [V(D-T)/W] / [1 + (V/g)(dVg/dh)]

r/d = [V(D-T)/W] / [1 + AF]

Descent gradient Kd

Kd = (r/d)/V = (D-T)/W + (1/g) dVg/dt

Kd

= [(D-T)/W] / [1 + (V/g)(dVg/dh)]

Kd = [(D-T)/W] / [1 + AF]

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Descent Descent Speed Schedules

Same approach as for climb speeds

Fixed Mach number at higher altitudes

Fixed CAS at medium and lower altitudes

Manufacturer may provide descent data for more than one

descent speed schedule in order to enhance operational

flexibility

Restriction of250 KCAS at altitudes below 10,000 ft

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Descent Cabin Pressurization Considerations

Cabin pressure altitude is typically equal to 8,000 ft when the

aircraft is at the maximum certified altitude

During descent, cabin pressure altitude will be increased

progressively

A maximum rate of change of cabin pressure equivalent to a rate of

descent of 300 ft/min at sea level is normally selected for

passenger comfort

Rate of change of pressure = 22.9 (lb/ft2)/min For a descent from the maximum certified altitude to 1,500 ft, the

time required for the cabin altitude to reduce from 8,000 ft to 1,500

ft pressure altitude is calculated as follows

Pressure for cabin at 8000 ft = 1572 lb/ft2

Pressure at 1,500 ft = 2,004.5 lb/ft2

Minimum descent time = (2004.5 1,572) / 22.9 = 18.9 minutes

An idle descent may sometimes result in smaller descent time

Partial power may have to be used during the initial part of the

descent to increase descent time

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Descent Cabin Pressurization Considerations

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Descent Emergency Descent

May be carried out when it is necessary to descend to a

lower altitude very quickly

Loss of cabin pressurization for example

Typically carried at Vmo/Mmo with idle thrust and spoilers

extended

Minimum thrust

High drag

High speed

Very high descent rates can be achieved

r/d can reach 10-15,000 ft/min

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Descent Gliding flight

Gliding flight can be analyzed by setting T = 0 in the descent

equations

Kd = (r/d)/V = D/W + (1/g) dVg/dt

Kd = (D/W) / [1 + (V/g)(dVg/dh)]

Kd = (CD/CL) / [1 + AF]

Descent gradient is minimized during flight at maximum L/D or at

VMD

For an aircraft with a drag polar defined as CD = CDO + KCL2 :

CD/CL = CD0/CL+ KCL

d (CD/CL)/ dCL = -CD0/CL2+ K = 0 at maximum L/D

CL = (CD0/K)0.5 results in minimum descent gradient

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Descent Gliding flight (Contd)

Rate of descent (r/d) is calculated as follows:

r/d =- (dh/dt) = VD/W + (V/g) dVg/dtr/d = (VD/W) / [1 + (V/g)(dVg/dh)]

r/d = (V CD/CL) / [1 + AF]

Rate of descent is minimized when V CD/CL is minimized

From CL = W / (0.5 V2 S) V = (W/(0.5 S CL))

0.5

For a given weight and altitude, V is proportional to 1/CL0.5

R/D is minimized when CD/CL1.5 is minimized

For an aircraft with a drag polar defined as CD = CDO + KCL

2

:

CD/CL1.5= CD0/CL

1.5+ KCL0.5

d (CD/CL1.5)/ dCL = -1.5CD0/CL

2.5+ 0.5 KCL-0.5= 0

CL = (3 CD0/K)0.5 results in minimum rate of descent

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