For Flight Dynamics

AER615: Aircraft Performance 1

Aircraft Performance The purpose of this course is to understand and predict how the airplane will actually perform in the air in order to achieve a desired performance or mission. z How fast can the a/c go? z How high can it go? z How much (weight) can it carry? z How far can it go without refueling? z How steep (or how quickly) can the a/c climb?

Key performance parameters z T/O z Climb z Cruise z Turning z Descent z Landing

Evaluation of a/c performance goes hand-in-hand with a/c design. We deal with atmospheric vehicle (always fly within the sensible atmosphere) : airplane, helicopter The flight performance of a/c is dependent upon such properties as p and of the air. Therefore (performance of any aerospace vehicle), the properties of the atmosphere must be taken into account. Flight manual for the performance of an a/c are always related to standard atmosphere. A standard atmosphere is defined in order to relate flight tests, wind tunnel results, and general airplane design and performance to a common reference. A standard atmosphere gives mean values of pressure, temperature, density and other properties as functions of altitude. The standard atmosphere reflects average atmosphere conditions. Its main function is to provide tables of common reference conditions.


Standard Atmosphere

SA reflects average atmosphere conditions Its main function is to provide tables of common reference conditions It is not an approximation of the actual atmosphere A reference performance flight tests and wind tunnel results Design purpose performance and comparisons Calibration of instruments

Elements of aircraft performance analysis

Thrust required (Drag) Thrust/weight, wing loading (W/S), Drag polar, L/D

Load factor (WLn = ) gives aerodynamic and structural limit for a given aircraft


Relationship with Aircraft Design Evaluation of aircraft performance goes hand-in-hand with aircraft design. It is generally the purpose of aircraft design to meet or exceed minimum performance requirements, including both aerodynamic performance and cost performance Design parameters related to aircraft performance Zero-lift drag coefficient Lift-dependent drag factor Maximum aerodynamic efficiency Maximum lift coefficient Wing planform area

Aerodynamics

Total weight of aircraft Maximum weight of fuel Maximum load factor

Structure

Maximum net thrust or power for engine Specific fuel consumption (thrust or power) Propeller efficiency

Powerplant

Performance Parameters Takeoff Performance The takeoff phase begins with the beginning of the ground roll and extends to the first moments of airborne flight until the aircraft clears a set value of screen height of usually 35' (FAR25:heavier a/c) or 50'(FAR23:small a/c). The parameters are:

Ground-roll distance and time Rotation distance and time Airborne distance and time (to screen height)

Climb Performance The climb phase immediately follows the takeoff phase and continues to level flight at some operational altitude. The parameters are:

Angle of climb Rate of climb Airspeed (or Mach number) of climb Time to climb Horizontal distance to climb Fuel consumed to climb Absolute and service ceilings


Cruise Performance The cruise phase involves the aircraft flying in a straight line. Typically, cruise will involve a constant speed or throttle setting and may involve a specified altitude profile. The parameters are:

Airspeed (or Mach number) of cruise Range Time to cover range Fuel consumed in cruise

Turning Performance This area studies steady level turns. The parameters are:

Airspeed (or Mach number) of turn Bank angle and associated load factor Rate of turn Turn radius

Descent Performance The descent phase begins with the reduction of altitude following cruise and continues until the aircraft is within the screen height (35 or 50) of the ground. The parameters are:

Rate of descent Angle of descent Airspeed of descent Time of descent Fuel consumed in descent

Landing Performance The landing phase begins with the moment of descent past the screen height and extends until the aircraft has come to a complete rest on the runway. The parameters are:

Airborne distance and time Rotation distance and time Ground-roll distance and time


Airspeed and Its Measurement Airspeed (Actual and Estimated), also known as True Airspeed "True airspeed is the speed of the aircraft relative to the undisturbed air through which it is flying" V Actual (true) airspeed VTAS or TAS Estimated or measured true airspeed Equivalent Airspeed (Actual and Estimated) Equivalent airspeed is the airspeed at mean sea-level (MSL) in the Standard Atmosphere that has the same dynamic pressure as the true airspeed at local density. Ve Actual equivalent airspeed VEAS or EAS Estimated or measured equivalent airspeed The relationship between actual equivalent airspeed and (true) airspeed is

VVe = where is the local density ratio (relative to MSL). The equivalent airspeed is significant for performance since flying at a constant EAS is effectively flying at a constant dynamic pressure, where in certain regimes the aerodynamic forces (lift, drag, etc.) tend to remain constant. Measurement of Subsonic Airspeed Whether designed for low-subsonic or high-subsonic, a typical airspeed indicator (ASI) bases its measurement of airspeed on the difference ppp = 0 between total (stagnation) and static pressure as measured by a pitot-static tube or equivalent. Calibrated airspeed Calibrated airspeed is the airspeed at mean sea-level (MSL) in the Standard Atmosphere that would produce the same pressure difference p as the true airspeed in local conditions. VCAS or CAS Calibrated airspeed The CAS is what is read from the dial of the ASI(Air Speed Indicator). The actual reading from an uncorrected ASI is called IAS(indicated airspeed). In some cases manual correction of IAS to CAS may be possible. To obtain the true airspeed (TAS) from the CAS, corrections for local conditions must be applied.


The equivalent airspeed (EAS) can be obtained from the CAS if the local static pressure is known, which is generally the case since a standard altimeter measuring pressure altitude is available. The altitude is converted (based on the SA) back into its original pressure measurement form thus yielding the local static pressure p. To complete the measurement of TAS, the local air density is required. This quantity is generally not measured directly, and so the following options are available. 1. Measure temperature and calculate density from the ideal gas law, = p/(RT) , then calculate TAS from

/MSLEASTAS VV =

2. If temperature (or Mach number) is not available, estimate density as using the standard atmosphere at the current pressure altitude hp as read from the altimeter, then

hMSLEASTAS VV /

3. Measure Mach number (see next section), and temperature (using the latter to obtain the

local sound speed via RTa = ), then calculating MaVTAS =

Low-Subsonic Airspeed Measurement

Equivalent and calibrated airspeed are same at low speeds(below M=0.3). If Mach number is known then there is an equation to correct for higher values of Mach.

If an ASI is to always be used in low speed situations, the simpler incompressible flow relations (Bernoullis Law) can be applied to estimate the calibrated airspeed via

MSLCASCAS

pVV = 222

where we have defined CASV as the low-speed approximation to CAS with respect to the actual definition of CAS. As in the general case, the ASI has no knowledge of density, so the MSL-SA is used for CAS. For the incompressible flow assumption, it is observed that equivalent airspeed and calibrated airspeed are the same and one can simply write

CASEAS VV And finally, one can recover an estimate of the true airspeed by using either of options 1 or 2 from the previous section.


Generally, Mach numbers of 0.3 or less are considered appropriate for applying these simpler formulas without correction. At M = 0.3 the error in (equivalent) airspeed measurement will be less than 3%. If an ASI designed for low-speed (i.e., uses Bernoullis Law to convert pressure difference p to CAS) is used in a higher-speed flow it is possible to externally correct the estimated CAS to a proper EAS. For example, if Mach number is known,

1

)1(2

222 1

2)1(1

2

+=

MMVV CASEAS Measurement of Mach Number A Mach meter provides a direct measurement of Mach number based on the compressible flow relation equation

=

1)1(

2)1(

02

ppM

Velocities and Groundspeed Aircraft Velocity Relative to the Ground, V Aircraft Inertial Velocity For our performance simulations, we will generally assume a flat Earth as an inertial reference, so that the aircrafts inertial velocity is the same as the velocity relative to the ground. Keep in mind, however, that the Earths surface is in fact moving in excess of 1000 km/hr (in most places) and that for long distance flights, surface curvature and direction of flight relative to Earths rotation become a consideration. Wind Velocity (Relative to Ground), wV Aircraft Velocity Relative to Air, wVVV = ; and Airspeed, VV = It is this velocity that generates the aerodynamic forces on the aircraft. The direction of the vector with respect to the aircraft structure dictates the angles of attack and sideslip, and it is the magnitude of this vector that is the airspeed V , defining the dynamic pressure available to generate aerodynamic forces.


Aircraft Motion in Performance Analysis (sign convention) Aircraft motion exists in 3 dimensions with 6 degrees-of-freedom required to describe general motion: up, forward, sideways, roll rotation, pitch rotation and yaw rotation. Traditionally, these motions are described with respect to the aircraft body-axes and divided into two groups: the longitudinal group consisting of up, forward and pitch; and the lateral group consisting of sideways, roll, and yaw Right-handed coordinate system defining positive directions for the forces & moments acting on an airplane as well as its linear and angular velocity vectors at the CG

x,y,z Coordinate X,Y,Z Resultant aerodynamic forces along axes u,v,w Linear velocities along axes p,q,r Angular velocities about axes L,M,N Aerodynamic moments about axes ,, Angular displacements about axes Angular rates about the x,y, and z are rolling, pitching, and yawing, respectively. Motion in the airplane's line of symmetry = longitudinal motion (linear motion along x and z axes, rotation about y axis) Motion of the plane of symmetry = lateral-directional motion (linear motion along the y axis, rotation about x and z axes) In aircraft performance, we mainly concerned with longitudinal motion (x-z plane). The aircraft motion through the atmosphere is governed by the equation of motion. We will use our knowledge of the lift, drag, and thrust of an aircraft to analyze how a given airplane responds to the four forces of flight, and how this1 response determines its performance.


The Aircraft Model and Equations of Longitudinal Motion Simplified Equations of Longitudinal Motion In this course, we will usually restrict ourselves to longitudinal motion of symmetrical aircraft during wings-level flight. This means that we study motion where the wings remain level to the ground (the aircrafts weight always acts in the z-x (symmetry) plane of the aircraft). For this situation to come about, the following must be true

: The net side force and yawing moments are zero. In simple terms, this is accomplished by the pilots manipulation of the aircrafts rudder together with a small amount of constant sideslip. The sideslip will be generally small enough so as to neglect its effect on the longitudinal situation.

== 0,0 NY

: The net rolling moment is zero. Moreover the net roll angle will be zero the wings remain level. = 0L

Stability and Control Assumptions Our study of aircraft performance will focus on the four fundamental forces of flight: lift, drag, thrust and weight; however, longitudinal motion is also dependent on pitch moment and the balance thereof. The basic assumption we will use in this course is that the aircraft is at all times in a state of pitch equilibrium (or quasi-equilibrium if we are changing our pitch or angle-of-attack) The fact of the matter is that the pilot cannot arbitrarily select the speed and angle-of-attack, hence lift coefficient. Rather the pilot adjusts the elevators and throttle simultaneously to establish any particular condition of trimmed flight. In simple terms, elevator control directly affects the angle-of-attack at which pitch balance occurs for a given flight speed and climb angle.


Longitudinal Forces in the Aircraft Body-Frame An aircraft body-fixed frame of reference is useful for describing the key angles and forces that affect the longitudinal motion. This is true since the direction of aircraft thrust is essentially a body-fixed quantity and the aircraft angle-of-attack is tied to the aircraft body orientation.

Aircraft Pitch (body angle relative to horizontal) (Body) Angle-of-attack (relative to air velocity) Thrust Angle (relative to body a constant) 0 Angle of the Zero-Lift Line (ZLL) (relative to the aircraft body) The equations of motion can be expressed in this body-fixed frame as follows...

sinsincoscos

sinsincoscos

,

,

DTLWVm

WLDTVm

Tz

Tx

=+=

&&

Before proceeding, we note the following basic relationships that allow one to calculate angle-of-attack once the lift-coefficient is known:


LLL CCC += 0 The lift coefficient at zero angle-of attack, CL0 , is related to the angle of the ZLL by the relationship

00LL CC =

where both CL0 and 0 are generally both positive for a typical aircraft. Longitudinal Forces in the Ground (Earth) Frame While expressing the longitudinal motion in the aircraft body-frame is in many ways more natural and direct, the measures of aircraft performance that we are most interested in (rate of climb, range, etc.) are more directly related to a ground-fixed reference frame.

a Air-Climb Angle (angle of air-relative velocity to horizontal) Flight-Path Angle (angle of ground-relative velocity to horizontal) Angle of thrust vector (relative to horizontal)


The equations of motion can be expressed in this ground-frame are then

aTaz

aaTx

DTWLVm

LDTVm

sinsincos

sincoscos

,

,

+==

&&

One should note the following about this setup:

The flight-path angle and air-climb angle a are generally different unless the wind velocity is zero or not considered. (Most simplified performance formulas implicitly assume these two angles are the same or equivalently that motion is to be evaluated with respect to the moving air rather than to the ground.)

The thrust angle T is generally unknown unless the aircraft pitch is known. In principal this is straightforward since LLL CCC += 0 , TT += and the difference between and a can be determined

however, the approximation that thrust direction is collinear with the flight path is frequently used, i.e., T = a, for simplicity in low angle-of-attack situations.


Summary of Aerodynamics

Source of any aerodynamic force and moment on a body

pressure distribution skin friction distribution

Dimensionless coefficients are used to quantify these forces and moments

),,(,, eDML RMfCCC = Aerodynamic center on a body

moments do not vary with Total lift and drag of an aircraft are not equal to the sum of the parts due to aerodynamic interference effects Wing aerodynamics is a function of the wing shape

high AR straight wing low AR straight wing high AR swept wing

The drag polar CL vs. CD contains almost all the necessary aerodynamics for an aircraft performance analysis


Summary of Propulsion Systems The variations of thrust, power, and specific fuel consumption with flight velocity and altitude. Different types of engine

Reciprocating engine/propeller combination : up to approximately M = 0.3 Turbojet engine: above M =2 + Turbofan engine: up to approximately M = 1 Turboprop engine: up to approximately M = 0.6

All of our four main engine types work by processing air to increase its momentum. By Newtons third law (action-reaction) the air pushes forward on the engine as the engine pushes the air back. The thrust is equal to the rate-of-change of momentum of the air processed by the engine:

)( VVmT j = & where T is the thrust, m is the mass flow-rate of the processed air (not fuel), V is the forward speed of the aircraft relative to the surrounding air, and V

&j is the velocity of the

processed air leaving the engine (relative to the surrounding air). Please note that this is a simplified analysis that neglects changes in pressure and the additional mass-flow due to spent fuel applicable to gas turbine engines. Energy is imparted both to the aircraft as it is driven forward, and to the air which is blown back. The effort to do both must be provided by the engine. In terms of power (energy per unit time), we have

Available Power TVPA = : the power imparted to the aircraft Lost Power 2)(

21 VVmP jL = & : the power imparted to the air

So that the total power P (for example, the power provided by a propeller shaft, hence shaft power) is

LA PPP += and we can define the power efficiency as

)/(12

VVPP

j

A

+==


From this it can been seen that propeller that increases the speed of a large volume of air by a relatively small amount is inherently a much more efficient device than a jet engine that increases the speed of a smaller volume of air by a much larger amount. However, more efficiency does not mean more thrust, and large aircraft that travels at high speed with lots of drag requires more thrust. The thrust produced by a propeller is limited by a couple of factors: (a) As propeller rotation speed increases to meet the demands of faster flight, increased drag on the blade increases the torque load on the driving engine and saps away the total power P ; and, (b) the shaft power produced by the engine will be limited, so that even at 100% efficiency the thrust at a given flight speed must drop off with increasing speed. Engine Power/Thrust Performance Modelling For the purpose of this course we can build simplified models of the various powerplant types. This will allow us to focus on performance calculations while not getting mired in detailed engine specifications. In addition our simplified models will be able to be imported directly into our simulation capability. The prop-types, piston-prop and turboprop, will be treated from a power viewpoint, with a thrust conversion added on, while the jet-types, turbojet and turbofan, will be treated directly from a thrust viewpoint. In all cases, the bottom-line will be: the thrust that drives our aircraft, propulsive power if required, and the fuel-consumption. We will utilize the concept of a virtual throttle, based on MSL-SA conditions that can vary the shaft-power or thrust linearly from zero to maximum depending on the engine type. Following the virtual throttle control we will continue to scale the output according to variations with: airspeed, relevant atmospheric conditions, and Mach number as applicable. Key Specifiers (at MSL-SA conditions)

Rated Maximum Engine Power PmaxEffective Operating Propeller Efficiency p(Power) Specific Fuel Consumption, SFC c

Piston -Prop

Static Thrust T0Rated Maximum (Takeoff) Thrust TmaxTurbo-Jet (Thrust) Specific Fuel Consumption, TSFC ct

More details will be discussed in AER710(Propulsion)


Takeoff Performance Aircraft Configuration for Takeoff and Landing Drag polar is a key encapsulation of aircraft aerodynamic properties affecting performance:

2

min LDDKCCC +=

As presented, the drag polar implicitly refers to the cruise or normal flying configuration of the aircraft. During takeoff and landing operations, however, the aircraft is usually in a different configuration that leads to somewhat different aerodynamic properties:

Retractable landing-gear is deployed, affecting mainly drag Wing flaps are used to enhance lift at low speed, and add some drag

Landing-Gear Effect on Lift and Drag To first-order, landing gear affects only the profile drag of the aircraft, thus [ ] 2,min LgDDD KCCCC ++=

This increase in drag coefficient can be treated as a constant for a particular situation. For the aircraft we will study in this course that have permanently deployed landing gear (Cessna for example), the additional landing gear drag will already be included in .

minDC

Wing-Flap Effect on Lift and Drag Various high-lift devices acting at the trailing and/or leading edge (flaps, slats, etc.) of the main wing are used to modify the lift characteristics to produce enhanced lift for the low airspeed associated with takeoff and landing operations. The amount of flap deployment is generally adjustable. In simple terms the deployment of high-lift devices

Increases the lift at all angles of attack by some amount fLC , whose value depends on the flap setting. The LC relationship becomes

[ ] LfLLL CCCC ++= ,0


Increases the maximum value of (though not necessarily by and amount

) maxL

C

fLC , Adds some profile drag so that fDC , [ ] 2,min LfDDD KCCCC ++=

The use of high-lift devices for takeoff and landing has two specific benefits: (1) the higher lift coefficient means a lower stall speed; and (2) noting that while on the ground, angle-of-attack must be achieved by rotating the aircraft about its wheels, the upward shift of the lift-curve (i.e., greater ) means more lift can be generated with less rotation - the latter being limited by ground clearance.

0LC

Ground-Effect Consideration When and aircraft is close to the ground (In Ground Effect, or IGE) the airflow around it is different than when away from the ground (Out-of Ground Effect, or OGE). In simple terms, the ground constrains the normal downwash from the aircraft generally increasing the upward pressure on the aircraft enhancing lift. The aerodynamic calculations will be discussed later. Aerodynamic Stall and Stall-Speed The stall speed, Vs , is the speed below which an aircraft cannot generate enough lift to support its own weight occurring when max , CL = CL,max beyond which flow separation from the wing will occur, generally resulting in a significant loss of lift (lower CL ) and drag increase. Its value is defined with respect to level flight where L = W, then

max,

12

Ls CS

WV

= The stall speed of an aircraft is a characteristic of a particular configuration with an additional dependence on altitude (due to density). For example, flap position will significantly affect stall speed.


The takeoff phase begins with the beginning of the ground roll and extends to the first moments of airborne flight until the aircraft clears a set value of screen height of usually 35 or 50. The parameters are:

Ground-roll distance and time Rotation distance and time Airborne distance and time (to screen height)

. Takeoff Fundamentals The takeoff phase is subdivided into three portions:

Ground-roll (all wheels down) Rotation (nose wheel lifted off ground) Climb to screen height.

Consider an airplane standing motionless at the end of runway

Where Sg (Ground Roll): distance covered over the ground before airplane lifts into the air Sa (Airborne distance): extra distance covered over the ground after the airplane is airborne but before it clears an obstacle of a specific height Screen height 50 ft for small A/C, military A/C (FAR23)


35 ft for heavy A/C (FAR25) The ground roll Sg is divided into intermediate segments. These segments are defined by various velocities

Vstall Stall speed, takeoff configuration (partial flaps typical) Vmcg Minimum control speed on the ground at which enough aerodynamic forces can

be generated on the vertical fin with rudder deflection while the airplane is still rolling along the ground to produce a yawing moment sufficient counteract when there is an engine failure for a multiengine aircraft

Vmca Minimum control speed in the air without the landing gear in contact with the ground; Min. speed required for yaw control in case of engine failure. This is essentially a reference speed (the airplane is still on the ground when this speed is reached)

V1 Decision speed (critical engine failure speed).(should be > Vmcg in order to maintain control of the airplane). If an engine fails before V1 is reached, the takeoff must be stopped. If an engine fails after V1 is reached, the takeoff can still be achieved.

VR Takeoff rotation speed (> 1.05Vmca ) Pilot begins to rotation of a/c to desired angle of attack in preparation for optimal liftoff (CL increase). Nose gear may be retracted at this time

Vmu Minimum unstick speed; with critical engine out, a/c could liftoff under adequate control. @Vmu, a/c can liftoff but in order to provide an additional margin of safety, the a/c continues to accelerate. Assume that the angle of attack is achieved during rotation is the max. allowable by the tail clearance.

VLO Liftoff speed. (> 1.1Vmu with all engine operative, > 1.05Vmu with critical engine operative) The a/c becomes airborne. Total distance covered along the ground to this point = ground roll, Sg.

V2 Takeoff climb speed (> 1.2Vstall in takeoff configuration and > Vmca )


Usually, stallLO VV 1.1Relative Velocity (Head wind, Tail wind)

Ground Speed, V

The velocity of the object measured relative to the ground is called the ground speed. Again, this is a vector quantity.

Airspeed, V

The important quantity in the generation of lift is the relative velocity between the object and the air, which is called the airspeed. Airspeed cannot be directly measured from a ground position, but must be computed from the ground speed and the wind speed. Airspeed is the vector difference between the ground speed and the wind speed. On a perfectly still day, the airspeed is equal to the ground speed. But if the wind is blowing in the same direction that the aircraft is moving, the airspeed will be less than the ground speed.

Examples

Suppose we had an airplane that could take off on a windless day at 100 mph (liftoff airspeed is 100 mph). Now suppose we had a day in which the wind was blowing 20 mph towards the west. If the airplane takes off going east, it experiences a 20 mph headwind (wind in your face). Since we have defined a positive velocity to be in the direction of the aircraft's motion, a headwind is a negative velocity. While the plane is sitting still on the runway, it has a ground speed of 0 and an airspeed of 20 mph [airspeed = ground speed (0) - wind speed (-20) ]. At liftoff, the airspeed is 100 mph, the wind speed is -20 mph and the ground speed will be 80 mph [airspeed (100) = ground speed (80) - wind speed (-20) ].


If the plane took off to the west, it would have a 20 mph tailwind (wind at your back). Since the wind and aircraft direction are the same, we assign a "+" to the wind speed. At liftoff, the airspeed is still 100 mph, the wind speed is 20 mph and the ground speed will now be 120 mph [airspeed (100) = ground speed (120) - wind speed (20) ]. So the aircraft will have to travel faster (and farther) along the ground to achieve liftoff conditions with the wind at it's back.

"A head wind Vw in takeoff reduces the a/c's ground speed while maintaining specified air speed (V=Vg+Vw), resulting in a shorter takeoff distance. Tailwind will result in a longer t/o distance"


EXAMPLE #1

The below a/c is to perform a t/o at the given indicated conditions. The available runway length is 750 m. During the t/o roll, the runway conditions change from a dry surface to a heavily puddled wet surface, when the a/c's airspeed reaches 70 % of lift off airspeed. Using mean kinetic energy estimation, determine if the pilot will have enough runway to lift off the main landing gear cleanly from the runway surface, or if there will be some tire damage by overrunning the runway. Note that any pilot reaction delays are implicit in the airspeeds provided for your calculations (i.e., don't add any extra delay distances) Aircraft Twin turbofan business jet b = 16.1 m, S = 37 m2, e (@ t/o) = 0.78, CD0 = 0.027 (@ t/o), CLmax (@t/o) = 1.7, CLg (t/o ground roll) = 0.85, W = 135 kN, VLO = 1.08 VS t/o Engines Two medium-bypass turbofan engines; for a single engine, max. Forward thrust given by Teng S/L = 34 - 0.11V + 2x10-4 V2 kN (@ S/L, V in m/s), Teng (h)/Teng S/L = 0.7 (variation of thrust as function of altitude), ratio of air density = Airfield ISA conditions at 900 m altitude, 5 m/s tailwind Mean height of wing over ground in roll , h = 2.2 m Runway upslope: R = 1o , = 0.02 (in t/o roll on dry runway), = 0.07 (in t/o roll on wet paved runway)

Ground effect factor:

)4.705.1(

)32.11(1

bh

bh

+

=


Climb segments #2 &3 (35' to 1500') Once we enter climb segment #2, regulated performance will now be in terms of minimum allowable gross flight slope angles. Flight angle is simplified via % gradient in climb, as 100 tan (%). During segment #2 from 35' to 400', the aircraft may fly at or exceed V2. Typically one would expect the aircraft to accelerate to a satisfactory holding speed for climbing to the desired cruise altitude, at max. climb thrust.

VLO

V2

35' 400' 1500'

#1 #2 #3

Referring to FAR25, Min. allowed climb gradients are as follows

Total No. of Engines Segment Configuration 2 3 4 #1 T/O flaps, gear down, critical engine inoperative

0 % (0o)

0.3 % (0.2o)

0.5 % (0.3o)

#2 T/O flaps, gear coming up, critical engine inoperative

2.4 % (1.4o)

2.7 % (1.55o)

3 % (1.7o)

#3 T/O flaps, gear up, critical engine inoperative

1.2 % (0.7o)

1.4 % (0.8o)

1.5 % (0.9o)


Performance of aircraft depends on temperature and altitude of airfield. WAT(Weight Altitude Temperature) charts or table must be checked such that at t/o weight, local airfield altitude and temperature, the a/c's climb performance through the various segments will meet or exceed the specified minimums. The a/c's t/o weight must be at or below the max. t/o weight indicated on the WAT chart.

h

max. t/o weight

ISA+ 30degISA+20 deg

ISA+10 deg

ISAISA-10 deg

Trend: Max. t/o weight drops with increasing altitude, temperature, and humidity (since water vapor is less dense than dry air)


Assuming a specified constant flight angle , or one under gradual transition so that we can neglect aircraft rotation,

then we can apply the force balance along the flight path:

dtdV

gWWDT = sin

Here, we are assuming that the a/c may continue to accelerate up to a desired holding speed, at varying (likely reducing) thrust levels. Account for change in V as done earlier. When constant or slowly varying, V constant or slowly varying at holding speed, then use standard climb equations,

sincos

sin

+=+=

L

D

CC

WT

WDT

where

2

20

,

21

cosLDDL KCCC

SV

WC +==

We can solve for as only unknown in implicit equation. Eventually, you'll have to account for W decreasing as fuel consumed

tTTSFCW averageF = for Jet (where TSFC is thrust specific fuel consumption) tHPBSFCWF = for Prop. (where BSFC is brake specific fuel consumption)


Balanced Field Length(BFL) The BFL is defined as the runway length such that the distance to continue a takeoff following recognition of a critical engine failure at speed V1 is equal to the distance required to stop if the t/o should be aborted.

Only one match exists for accelerate-stop and continues t/o distances for a given set of local conditions According to FAR25, field length to be greatest of

accelerate and go distance accelerate and stop distance 115% of all-engine-operating distance to a 35' height


The FARs specify no requirement on BFL, however, it is an important concept as it represents a minimum field size that the aircraft can safely use. Conversely, for any size field, a decision speed based upon satisfaction of the BFL conditions generally represents the safest choice for V1 . Procedure for determining the BFL 1. Guess a value for decision speed V1 . 2. Determine the associated value for engine failure speed VEF. 3. Assume that a failure occurs exactly at this speed, such that VEF* = VEF (and VEF = V1). 4. Calculate the field lengths for the two cases:

a. accelerate-and-go (takeoff continued): b. accelerate-and-stop (takeoff aborted):

5. Compare the two field lengths, and 6. Iterate on V1 until the two field lengths become equal: SAG = SAS 7. The common field length is the balanced field length: SBFL .


Landing Performance

The landing phase of an aircraft's operation consists of three segments

Approach Flare Ground roll

One can refer to FAR25 regulations with regard to the landing phase of a transport aircraft. In this case landing distance is defined as distance necessary to come to a complete stop from a point 50ft above landing surface. Further requirements include:

1. A/C must be in landing configuration (e.g. flaps developed, gear down). 2. Steady approach VA> 1.3 VSL must be maintained down to the 50' height.

Typical glide-slopes are 3o~4o for transport a/c up to 6o~7o for STOL a/c. 3. Landing must be made without excessive vertical accelerations or velocities. 4. Landing distance must be determined with critical engine inoperative, if a

decelerating device like reverse thrust is dependent upon this engine. Airborne Transition (Approach & Flare) One can calculate airborne transition SLA with reasonable accuracy using an analytical approach (vs. numerical method described earlier for t/o transition). We will consider two different landing methods

1. Flown-on landing 2. Stalled-on landing


EXAMPLE #3

Evaluate the following case for takeoff distance using an approximate method(use energy method):

a) Sea level, no wind

ISA conditions/ roll 0.05 (short grass)

Airfield

Constant speed propeller, n = 2700 rpm, dprop = 6 ftTo.S/L = 500 lb; T/To = 1 - 0.3 J ; T(h)/TS/L = , J=V/(n dprop)

Engines

Single-engine light aircraftWing span 32 ft; wing reference area 180 sq. ft.; efficiency factor (t/o) e = 0.75 Aircraft weight 2800 lbMean height of wing over ground in roll, 8 ftCDo 0.04 (takeoff config.); CLmax 1.7 (@ T/O); CLg 0.9 (@ ground roll)VLOF at 1.15 VS,T/O ;V2 (at 50 ft) 1.3 VS,T/O

Aircraft

EXAMPLE #4Evaluate the following cases for stalled-on landing distance using an approximate method:

a) Sea level, no windb) Sea level, 10 kt tail wind (1 kt = 1.688 fps)

ISA conditionsdecel 0.25 (braking)

Airfield

Single-engine light aircraftWing span 32 ft; wing reference area 180 sq. ft.; efficiency factor (t/o) e = 0.75 Aircraft weight 2500 lbMean height of wing over ground in roll, 8 ftCDo 0.05 (landing config.); CLmax 1.85 (@ landing conf.); CL(flare) = 1.15 CL (approach) ; approach angle of 5 oCL (ground roll) 0.3VA at 1.25 VS,LD

Aircraft


Range (R) Total distance (measured with respect to the ground) traversed by an aircraft on one load of fuel. W0 : gross weight of the a/c including everything (fuel, payload, crew, structure) Wf : weight of fuel; this is an instantaneous value, and it changes as fuel is consumed during flight Wempty : weight of everything else the structure, engines(with all accessory equipment), electronic equipment(including radar, computers, communication devices, etc.), landing gear, fixed equipment(seats, galleys, etc.), and anything else that is not crew, payload, or fuel W1 : weight of the a/c when the fuel tanks are empty @any instant during the flight, W = W1 + Wf

Wf is decreasing: ff W

dtdW

dtdW &== (negative values)

Range is intimately connected with engine performance through the specific fuel consumption and decreasing fuel Wf is defined before as

tTTSFCW averagef = for Jet (where TSFC is thrust specific fuel consumption) tHPBSFCWf = for Prop. (where BSFC is brake specific fuel consumption)

Note: Over the years, conventional engineering practice has quoted the specific fuel consumption in the inconsistent units of pounds of fuel consumed per horse power per hour; these are the units you will find in most specifications for internal combustion reciprocating engines. Before making a calculation which involves specific fuel consumption, we always convert the inconsistent units of SFC to the consistent units of c. "c" will designate the specific fuel consumption with consistent units.


For propeller-driven /reciprocating engine Specific fuel consumption "c" (in terms of power): weight of fuel burned per unit power per unit time

=p

Wc f

&(weight of fuel consumed for given time increment)/{(power output)(time

increment)} Unit of c

[ ]))(/( sslbft

lbc = or [ ] sWNc = where W = J/s and J = work = (N m)

However, inconsistent units of lb of fuel consumed per horsepower per hour is also used as

[ ])( hrhp

lbBSFC = and [ ] skWNSFC =

so,

[ ]ft

BSFCs

hrslbft

hphrhp

lbBSFCc 136005503600

1/550

1

=

=

or

[ ] [ ]m

SFCs

smN

NSFCskW

NSFCc1000

1

1000=

=

=

where W=J/s and J = work=(N m) It is always a good idea to convert the inconsistent units of SFC to the consistent units of [c].


For Jet engine, we use thrust rather than power. Thrust specific fuel consumption

=T

Wc ft

&(weight of fuel consumed for given time increment)/{(thrust output)(time

increment)} Unit of ct

[ ]sslb

lbct1

)(== or [ ] ssN

Nct1

)(==

However, TSFC (use hour instead of second)

[ ]hrhrlb

lbTSFC 1)(

== , [ ] sTSFC

shr

hrlblbTSFCct

136003600

1

=

=

Relationship between c and ct

Now we can relate )(p

Wc f

&= and )(T

Wc ft

&=

Tpcct

= where pr

App = and PA: power available from the engine ( = TV ) and pr: propeller efficiency

Then, pr

TVp = .

The specific fuel consumption for a reciprocating engine c in terms of an equivalent thrust specific fuel consumption ct is

prt

Vcc =


We can find range for steady, level flight with no wind

dtdsV = or . dtVds =

Also, T

dtdWT

Wc fft

/== & or Tc

dWdt

t

f=

then, WdW

TW

cVdW

TcVds

tf

t

== (since dWf = dW) In steady, level flight condition (L = W and T = D)

WdW

DL

cVds

t

= Integrate from W = W0 (s = 0) to W = W1 (s = R) where W1 : weight of the a/c when the fuel tanks are empty

== 100

W

W t

R

WdW

DL

cVdsR

Then,

= 01

W

W t WdW

DL

cVR for propeller ct becomes c

for level, steady flight without wind From above range equation, we can find that range is

),,,/( 10 WWVcDLRR t = and ),,,(/ hWVfDL = if we know all these values


Assume constant (for preliminary performance analysis) and DLcV t /,,pr

t

Vcc =

1

0lnWW

DL

cVR

t

= "Breguet Range equation"

To maximize R, We want to fly simultaneously at the highest possible and largest L/D (high aerodynamic efficiency)

V

So, we want to fly at max. DLV but condition is different for propeller and jet aircraft.

Range for Propeller-driven airplanes

Since pr

t

Vcc =

1

0lnWW

DL

cR pr

= , velocity is gone For max. range

Fly at max. L/D Have the highest possible pr Have the lowest possible c Carry a lot of fuel (max. W0-W1)

For max. (L/D),

2

0 LD

L

D

L

KCCC

CC

DL

+== and we can maximize CL/CD by

( ) 0)2()/(

22

2

0

0 =++=

LD

LLLD

L

DL

KCC

KCCKCCdC

CCd, then (zero lift drag = drag

due to lift)

2

0 LDKCC =


then, K

CC DL 0=

and 0

0

00

0

2//

maxmax D

D

DD

D

D

L

CKC

CCKC

CC

DL =+=

=

Then,

KCDL

D04

1max

=

valid for any flight conditions (climb, level, turn since it is from aerodynamics of aircraft via drag polar) The velocity at (L/D)max.is dependent on flight conditions Velocity will be different for climbing, turning flight compared to steady, level flight for steady, level flight (L = W)

LSCVWL2

21

== and KCSVW DDL /21

0max)/(

2 = then

2/1

)/(

0

max

2

=

SW

CKV

DDL


Range for Jet-propelled airplanes

1

0lnWW

DL

cVR

t

= , velocity is included

For max. range for jet is dictated by

D

LV not

DL

For steady, level flight

LSCVWL2

21

== or LSC

WV 2=

thus

D

L

CC

SW

DLV

2/12=

.max

D

LV when the airplane is flying at a .max

2/1

D

L

CC

= 01

2/121W

W D

L

t WdW

CC

SW

cR Assume ct,,S, CL1/2/CD are constant. Then

= 01

2/1

2/121 W

WD

L

t WdW

CC

ScR or


)(22 2/112/1

0

2/1

WWC

CSc

RD

L

t

= simplified range equation for a jet aircraft For max. range

Fly at max. CL1/2/CD Lowest ct Fly at high altitude where is small ( is not shown in propeller range equation. so

Jet a/c should fly at high altitude), max. altitude is absolute ceiling where R/C = 0. Carry a lot of fuel

How to achieve (CL1/2/CD) max.

2

2/12/1

0 LD

L

D

L

KCCC

CC

+= and max. DL

CC 2/1

can be found by

( )( ) 0

)2(21

)/(22

2/12/122/1

0

0 =+

+

=

LD

LLLLD

L

DL

KCC

KCCCKCC

dCCCd

, then and 230 LD

KCC =

KC

C DL 30=

therefore

4/1

3

max

2/1

03

143

=

DD

L

KCCC

and the velocity at which (D

L

CC 2/1

)max. is achieved is

2/1

)/(0

max2/1

32

=

SW

CKVD

CC DL , for steady, level flight condition


Now, we can compare and max

2/1 )/( DL CCV

max)/( DLV

2/1

)/(

0

max

2

=

SW

CKV

DDL and

2/1

)/(0

max2/1

32

=

SW

CKVD

CC DL We see that = 3

max2/1 )/( DL CC

V 1/4 or = 1.32 max)/( DL

Vmax

2/1 )/( DL CCV

max)/( DLV

When the aircraft is flying at (CL1/2/CD) max, it is flying 32% faster than when it is flying at (L/D)max.

Reflecting on the product (L/D) in V1

0lnWW

DL

cVR

t

= , we see that for max. range for jet, although the aircaft is flying such that L/D is less than its max. value, the higher is a compensating factor.

V

We assumed V, L/D, CL1/2/CD are constant during the flight. But, during the flight, fuel is being burned, and W is decreasing. L = W = (1/2) V2 SCL. RHS should decrease. Because of the assumption that V, L/D, CL1/2/CD are constant, only the can be changed. "Stair stepping" increases range. We can consider range in terms of pounds of fuel consumed per mile. The smaller the number of pounds of fuel consumed per mile, the larger the distance the aircraft can fly, that is, the larger the range


Propeller driven aircraft

=

VHPBSFC

miconsumedfueloflb

pr

R

)( ,

where

hbhpconsumedfueloflbBSFC = and AR HPHP = , Apr HPbhp = and = mph V

So, min. pounds of fuel consumed per mile are obtained with min. (Slope shown below)

VHPR /

Points of maximum range and endurance on the power required curve for propeller driven aircraft Min. value of slope. This point corresponds to the conditions for max. range for a propeller driven aircraft.

Since then ,= VTP RR RR TVHP

.

So, min. V

HPR corresponds to RT

Min. corresponds to flight @max. L/D. RT


WLDW

WDDTR === and

DL

WTR =

Drag versus velocity Velocity instability

Velocity instability : Velocity stability


Velocity instability and stability curve


Jet propelled aircraft

=

VTTSFC

miconsumedfueloflb R)( ,

where

hrlbconsumedfueloflbTSFC =

So, min. pounds of fuel consumed per mile are obtained with min. V

TR (slope shown

below)

Points of maximum range and endurance on the thrust-required curve

Min. value of slope = max. range for jet

2/122

21

21

L

DD

LD

R

CCWSSC

SCWVSC

VT

===

so, V

TR is a min. when 2/1L

D

CC is a min. or

D

L

CC 2/1 is a max.


Endurance Amount of time that airplane can stay in the air on one load of fuel.

We know Tc

dWdtorTc

dtdW

t

ft

f == Since T = D, L = W for steady, level flight

WdW

cDL

DcdW

dt ftt

f 1== Integrate from t = 0 where W = W0 to t = E where W = W1

WdW

DL

cWdW

DL

cE f

W

W t

fW

W t == 01

1

0

11

If the detailed variation of ct, L/D, and W are known, then integrate numerically. For preliminary performance (assume ct,and L/D are constant)

WdW

DL

cE f

W

Wt= 01

1

1

0ln1WW

DL

cE

t

=


Endurance for Propeller-driven airplanes

Since pr

t

Vcc =

WdW

CC

VcE f

W

W D

Lpr

=0

1

For steady, level flight LSC

WV 2= (where T = D, L = W)

Then,

WdW

CC

WSC

cE f

W

W D

LLpr= 01

2 or 2/3

2/30

12 W

dWC

CSc

E fW

W D

Lpr=

Assume constant ,, cprD

L

CC 2/3,

)(2 2/102/1

1

2/3 = WW

CCS

cE

D

Lpr

Conditions for maximum endurance for propeller-driven airplanes

Fly at max. D

L

CC 2/3

Have the highest pr Lowest specific fuel consumption Carry a lot of fuel (max. W0 W1) Fly at sea level (where air density is largest value)


Conditions for max. D

L

CC 2/3 or how to achieve (CL3/2/CD) max.

2

2/32/3

0 LD

L

D

L

KCCC

CC

+= , differentiate with respect to CL

( )( ) 0

)2(23

)/(22

2/32/122/3

0

0

=+

+=

LD

LLLLD

L

DL

KCC

KCCCKCC

dCCCd ,

then 231

0 LDKCC = ( zero lift drag equals one-third of the drag due to lift)

and KC

C DL 03=

4/34/3

max

2

2/3

max

2/30

000

0

0

34

13

)/3(

=+=

+=

KC

CCCKC

KCCC

CC D

DDD

D

LD

L

D

L

or

4/3

3/1max

2/3

0

341

=

DD

L

KCCC

For steady, level flight (L = W), the velocity at which (CL3/2/CD) max. is

LSCVWL2

21

== and KCC DL /3 0=

2/1

.max)/(C0

2/3L 3

2

=

SW

CKV

DCD and we know

2/1

)/(

0

max

2

=

SW

CKV

DDL

or

.max)/(.max)/(

4/1

.max)/(C76.0

31

2/3L

DLDLCVVV

D=

=


Endurance for Jet-propelled airplanes

We know 1

0ln1WW

DL

cE

t

= Conditions for maximum endurance

Fly at max. L/D Lowest specific fuel consumption Carry a lot of fuel (max. W0 W1)

The simplest way to think about endurance is in terms of pounds of fuel consumed per hour. The smaller the number of pounds of fuel consumed per hour, the longer the airplane will be able to stay in the air; that is, longer the endurance. SUMMARY OF PROPELLER-DRIVEN AIRPLANES

hbhpShaftconsumedfueloflbSFC = , )( bhpShaftHP prA =

For steady, level flight (HPA = HPR)

))(( RHPSFChourconsumedfueloflb

Therefore, min. lb of fuel consumed per hour are obtained with min. HPR.

== VCCWVTP

DLRR /

where LL

SCVWLSCWV 2

21,2 ===

Then, 32322

/ LD

LDLRR SC

CWSCW

CCWVTP ===

We can relate D

LR C

CandP2/3


Min. power required occurs when the aircraft is flying such that D

L

CC 2/3 is a max.

We already know the conditions for max. D

L

CC 2/3

24/3

2/3.max

2/3

31,3

41

0

0

LDDD

L KCCCKC

C =

=

2/1

.max)/(C0

2/3L 3

2

=

SW

CKV

DCD , .max)/(.max)/(C 76.02/3L DLC VV D =

SUMMARY OF JET PROPELLED AIRPLANES The specific fuel consumption for a jet-propelled aircraft is based on thrust

hThrustconsumedfueloflbTSFC =

For steady, level flight TA=TR

TSFCThconsumedfueloflb

R =


Min. pounds of fuel consumed per hour are obtained with min. TR

Min. TR occurs at TR = D = WWD

)/( DLWTR =

so, (TR)min. occurs at (L/D)max.

We already know the conditions for max. D

L

CC

2.max

0

0

,4

1LD

DD

L KCCKCC

C ==

2/1

.max)/(C0

L

2

=

SW

CKV

DCD


Graphical summary of conditions for maximum range and endurance

From previous analyses

321 76.076.0 VVV == 24 32.1 VV =

where and .max)/(.max)/(C 76.02/3L DLC VV D = .max)/(.max)/(C 32.12/1L DLC VV D =


The effect of wind on range We already know V = Vg - Vw where VV = (air speed) Then V = Vg - Vw for tail wind V = Vg + Vw for head wind Endurance is not influenced by the wind. The airplanes relative velocity V is simply that for max. endurance.

1

0ln1WW

DL

cE

t

= and )(2 2/102/112/3

= WWC

CSc

ED

Lpr Range is affected by wind. Example V = 100 mph, Head wind of 100 mph. Then, Vg = V - Vw = 0. The ground speed is zero. The aircraft just hovers over the same location, and the range is zero. R = f(Vg) Let s denote the horizontal distance covered over the ground.

dtdsVg = and dtVds g=

Range for jet aircraft 1

0lnWW

DL

cV

Rt

g=

Range for propeller aircraft 1

0lnWW

DL

VV

cR gpr

=

We can find the values of V that correspond to max. range for a jet & propeller aircraft including the effect of wind by differentiating above equations with respect to V and setting the derivatives equal to zero.


The best range value of V with tail wind is lower than that for no wind.(V = Vg - Vtw) The best range value of V with head wind is higher than that for no wind.(V = Vg + Vhw) We know that the range is determined by the pounds of fuel consumed per mile covered over the ground.

gpr

R

VHPBSFC

miconsumedfueloflb

= . for propeller aircraft

g

R

VTTSFC

miconsumedfueloflb =.

for jet

Minimum number of pounds of fuel consumed per mile, which corresponds to max. range,

is obtained with min. g

R

VHP or

g

R

VT .

Effect of headwind on best range airspeed for a propeller driven aircraft


Effect of tailwind on best range airspeed for a propeller driven aircraft

Effect of tailwind and headwind on best range airspeed for a jet The best range value of V with tail wind is lower than that for no wind.(V = Vg - Vtw) The best range value of V with head wind is higher than that for no wind.(V = Vg + Vhw) HPR and TR (=drag) are depend on V (depend on aerodynamics of the aircraft)


Example #5 Estimate the maximum range and maximum endurance at 30,000 ft for the Gulfstream IV.Also calculate the flight velocity required to obtain this range and endurance.

Rolls-Royce Tay turbofanTSFC @ 30,000 ft = 0.69 lb/(hr lb)

Engines

Gulfstream IVWing reference area 950 sq. ft. ; AR = 5.92 ; K = 0.08Aircraft weight 73,000 lbCDo 0.015 ; Maximum usable fuel weight = 29,500 lb

Aircraft

Example #6

Given a headwind of 30 kt, determine the maximum-ground-range airspeed at a current aircraft weight of 41000 lb and cruise altitude of 10000 ft. How does this value compare to the nominal no-wind case ?

Aircraft de Havilland DHC-5D (Buffalo)

Wing span 96 ft ; Wing reference area 945 sq. ft. ; e = 0.82

CDo 0.024 ; CLmax 1.6 ; Max. t/o weight 43000 lb ; Max. fuel weight 14000 lb;

Max. Payload weight 12000 lb ; Operating empty weight 25000 lb

Engines Two General Electric CT64-820-4 turboprops

BSFC at 10000 ft, 0.5 lb/hr-hp

Cruise propulsive efficiency 85 %


Note on Weight In performing the calculation for range, note that the various components that comprise the aircrafts overall weight

MTOWWWWWpayloadfuel LPFemptyE

++=)()( /)(

(max. takeoff weight) Here, WE refers to the empty operating weight (airframe, engines, trapped oil/fuel, fixed equipment, crew), WF the mission fuel weight (fuel to be used plus required reserve), and WP/L being the disposable payload weight. The aircraft will have limited capacity for carrying fuel in tanks, and for storing cargo onboard. In calculating overall range, one must account for fuel consumption as the mission profile can be broken down as follows

)(),(),()()lim,/( reservelanddescentdelayholdcruisebcot FFFFFFWWWWWW ++++=

Subtract the appropriate weight for mean weight estimates in calculating attained distance, using the appropriate specific fuel consumption for the particular flight segment.


Cruise Performance Cruise performance depends on the requirements for a given mission, e.g. max. range, max. endurance, min. trip time, min. cost. This performance may be characterized by a number of parameters. Recall Specific Air Range from flight mechanics

===

kgkm

mV

tmtV

usedfuelcedisairSAR

ff &&tan

where L

Df C

CWTSFCTTSFCm ==& for Jets. (T = D = W(CD/CL) )

In terms of equivalent air speed (LS

EEppVVV

/

0 )(2,/ == )

wherepressuretotalpratiotempLShratiodensityratiopressure :,.:)),/(/)((:),(: 0 ==

Equivalent air speed: airspeed at mean sea-level in the standard atmosphere that has the same dynamic pressure as the true airspeed at local density. The equivalent airspeed is significant for performance since flying at a constant EAS is effectively flying at a constant dynamic pressure, where in certain regimes the aerodynamic forces (L, D, etc.) tend to remain constant

D

LE

CC

WTSFCVSAR 1=

From this expression, it is evident that cruise performance improves with

higher altitudes higher L/D lower TSFC & W


The product of SAR and aircraft weight W is a performance parameter and it is often plotted

versus EV

D

LE

f CC

TSFCV

mVWWSAR =

= & with different weight curves (W/). Thus, at a given weight, altitude and velocity, once can ascertain the SAR.

Other formats illustrating fuel flow versus airspeed may be provided by manufactures of aircraft

Also, there will be a drag divergence limit on max. Mach no.


Cost consideration In addition to fuel costs, one may have to account for other flight operations costs, i.e., that associated with operations & maintenance. In doing so, aircraft performance at cruise may have to be adjusted.

Typical Direct Operating Costs Commercial transport Fuel/ Oil 38% Aircrew (flight & cabin) 24% Ground personnel & maintenance 25% Depreciation 12% Insurance 1% Total 100%

In addition to direct operating costs for flight operations, one may also have indirect operating costs such as administration, sales & customer service, and depreciation of ground facilities. Actual dollar figures for DOC (Direct Operating Cost) & IOC (Indirect Operating Cost) are sometimes hard to pin down, and some costs may have no bearing on aircraft design or performance. Operation and maintenance costs are usually given as costs per flight hour, therefore do have a bearing on performance.

History of operating costs for U.S. air transports


Vmc (Minimum Cost Cruise) For the minimum-cost cruise case, one would fly at a specific speed for a given altitude, weight & wind. Consider the parameters related to cost;

== Bfuelofmassunit

tA ,cos hourly cost[aircraft + crew operation & maintenance (exclude

fuel)] Cost to fly for time t : $ = tBtmA f +& and given t = s/Vg , $ =

ggf V

sBVsmA +&

Given L

LDLDf C

KCCWTSFCCCWTSFCTTSFCm

20)/(

+===&

+=

SV

KWSVCTSFCm Df2

22

212

10

& , since SV

WCL2

21

= , (L = W for cruise)

For simplicity, let V = Vg (no wind), so that

$ = VsBs

SVKWVSCTSFCA D +

+ 3

2

221

0 For minimum,

24

2 )(62

$00 V

BsSV

sKWTSFCASsCTSFCAdVd

D ==

Multiply by V4/s, divide by SCTSFCA D 02 ,

2

224

)(1220

00SC

KWVSCTSFCA

BVDD

=


Solve for V2 via quadratic solution, and take square root

0

00

22

cos.min 121

DDD

t CKWTSFCAB

SCSCTSFCABV +

+= then

+

+= 0

0

22

cos.min 121

DD

t CKWTSFCAB

TSFCAB

SCV Note that for high subsonic speeds, one may be getting into drag divergence, requiring

modification in above. )(0

MCD When M > Mcr,

Drag is increased by factor 1.0

1 crMM + Check case for, no hourly cost, i.e., B = 0 (consider only fuel cost)

.max

0

0

0

/2

cos.min 32.132121 DL

DD

Dt VC

KS

WCKWSC

V ===

where 0

32

DCK

SW

= =1.32 .max2/1 )/( DL CCV max)/( DLV With earlier result for max. range, for jets (Vmax. range). As a result, we can see that Vmc (min. cost) increases above Vmr (max. range) as hourly cost coeff. B increases.

This explains why one operates to the right of the max.

==kgkm

mV

usedfuelcedisairSAR

f&tan

point, at a higher velocity.


For propeller driven aircraft,

)/( LDprpr

Rf CCW

VWBSFCVTBSFCPBSFCm ===&

where prpr

apra

VTPPPP === , , then

+=

VS

KWSVCBSFCm Dpr

f

212

1 230

& , let V = Vg and given B,

For simplicity, let V = Vg (no wind), so that cost to fly for time

$ = VsBs

SVKWSVCBSFCA D

pr

+

+ 2

22 2

21

0 where A = cost/unit mass of fuel and B = hourly cost. For minimum in order to find Vmc (min. cost),

23

2

4$00 V

BsSV

sKWBSFCASsCBSFCAdVd

prD

pr

==

Multiply by V3/s and divide by pr

D SCBSFCA

0

,

2

2

cos.min4

cos.min )(40

00SC

KWVSCBSFCA

BV

Dt

D

prt

=

Solve iteratively. When B = 0, no hourly cost

0

2cos.minD

t CK

SWV = ; agrees with earlier result for max. range case for props.


We already know that

2/1

)/(

0

max

2

=

SW

CKV

DDL and this is max. range for props and

max. endurance for jet. Here again, as B goes up, so does Vmc(min. cost) increases above Vmr(max. range). Having looked at costs, later well look at revenue generation and see where we might optimize in that respect, remembering: net profit = gross revenue operating costs.


Minimum Fuel Cruise, Variable Properties Recall for min. fuel consumption in cruise, fly at max. specific air range with no wind. We defined SAR = (air distance)/(fuel used)

Max. SAR = Max.

TTSFC

V or Max.

PBSFCV

Max. SAR = Max.

DTSFC

V or Max.

DBSFCpr where Pa = P = TV =DV

Max. SAR = Max.

D

L

CWC

TSFCV

or Max.

D

Lpr

CWC

BSFC

where T = D = W(CD/CL), W = L TSFC = lb/(lb hr) = 1/hr, BSFC = lb/(hp hr) Up to now, we have assumed in analysis that properties like TSFC, pr and BSFC are relatively constant, but in fact we know they vary as function of V or M. To illustrate importance of allowing for property variation in optimization, consider an example for a jet, with linear variation:

VaTSFCTSFC 20 += , so that

SAR = [ ] )()/(2 )/(2 22020 0 LDL LLDL KCCWSCWaTSFCCSCW

CWC

VaTSFCV

++=+

Invert and find min. SAR-1 and 0)(1

=

LdCSARd

By doing this, we can show that K

CC DL 3

0= or when a230 LD

KCC = 2 = 0. This is for max. CL1/2/CD: best range for jet


Now consider prop. case, where one might not expect BSFC to vary much with V, but might expect pr to vary substantially at lower V's. Consider an example of propeller driven aircraft which has linear variation of

Vapr 10 += so that

SAR = [ ][ ] )(

)/(22

10

0 LD

LL

D

Lpr

KCCWBSFCCSCWa

CWC

BSFC ++=

We want to maximize SAR.

Then, 0)( =LdC

SARd

By doing this, we can show that K

CC DL 0= or when a20 LD KCC = 1 = 0. (Max. range

for prop: max. CL/CD) As a third case to consider, allow for zero-lift drag to increase above the drag-divergence Mach no. M

0DC

DD, via parabolic rise [ ] DDDDDD MMMMCCC i >+= ,)(1 200

DDiDDMMCC = ,

00

So, for min. fuel (max. range) cruise by jet in drag divergence, (whereSpM

WCL 27.0 = ). Since

22222 7.0)2/1()2/1()2/1(,, pMMpMaVRTpaMaV ====== Then,


SAR = )/()/(

1

0 LLDLDKCCCWTSFC

MaCCWTSFC

V+

= We can substitute [ ]2)(1

00 DDDDMMCCC

i+= and then find min of SAR-1, in terms of

M rather than CL.

Then, 0)(1

=

dMSARd

.

The nominal incompressible value, at C = 0

DD

D

range MKCpSWM >= 3/7.0

0

.max


Minimum Fuel Mission (Climb/Cruise/Descent) In considering the overall mission for aircraft with respect to fuel consumption concerns, one must also consider the climb and descent portion of the flight along with the cruise segment (for now, we'll assume that fuel consumed in takeoff, loiter and landing is fixed). For longer trips, one can likely choose in isolation the best cruise max. range Mach no.(Mmr) and cruise altitude for min. fuel and max. range. In these cases, it is typically best to get to the best cruise altitude as quickly as possible, thus one may climb at or near R/Cmax from climbout to altitude, a common assumption in preliminary design. One may want to consider a min. fuel climb approach in getting to that desired altitude, for further fuel saving:

=

=

CR

WMin

dtdhdtdW

Mindh

dWMin fff

/// &

Thus, best interim climb speed V at a given h would be ascertained from 0/

=

CR

WdVd f& .

Recall for jet, W

VDTCRTTSFCWf)(/, ==&

(where R/C = V sin = (Pa Pr)/W = (TV-DV)/W ) Thus, want to minimize such that

0)(/

=

=

VDTWTTSFC

dVd

CRW

dVd f&

for best climb V, or alternatively maximize such that,

0)(/ =

=

WTTSFC

VDTdVd

WCR

dVd

f& , for jet


Similarly then for props, where (or alternatively, Af PBSFCW =&pr

f

TVBSFCW =& ), then for maximizing,

0)(/ =

=

WPBSFC

VDTdVd

WCR

dVd

f& , for props

Descent portion of the flight When Pr > Pa, aircraft will descend rather than climb. If there is no power at all, aircraft will be in gliding, or un-powered flight. [Engine quits (failure out of fuel), un-powered glider &sailplanes]. Difference between sailplane and glider

Sailplane: expensive, high-performance un-powered aircraft Glider: crude, low-performance un-powered aircraft

For the descent portion of the flight (from say hcruise to ~ 1500 ft above ground level, ready for fuel approach in landing), one would also in general like to minimize fuel consumption. Assuming the aircraft's flight descent angle D doesn't fall below specified maximums (say ~4o), an "unpowered" max. ground range approach would be typical, i.e., for no wind, Force diagram for an un-powered aircraft in descending flight

For steady, unaccelerated descent Glide angle is a function of the lift-to-drag ratio, the higher the L/D, the shallow the glider angle


.maxmin )/(

1tan,sin,cosDL

WDWL === ; max. horizontal distance covered over the ground (max. range). At a given h, this is the case for max. horizontal distance covered over the ground (max. range) We already know that (L/D)max. occurs when induced drag = zero lift drag.

KCSWV

D

range /2

0

.max = Note that even though there may be zero nominal thrust (indeed spoilers may be deployed at same point to slow a/c), idling engine are still burning fuel, at a min. allowed rpm to keep accessories, etc, functioning, at some ) idlewf /&


As noted earlier, for longer trips, one can analytically isolate the cruise segment choices for Vcr, Mcr or hcr, from the climb and descent portions. However, for shorter trips, one must start to look at the whole mission together, in order to arrive at a good choice for hcr and Mcr for overall fuel consumption concerns. One typical rule of thumb is to apply a one-third rule below a minimum distance set for a given a/c, whereby the ground segments for climb, cruise and descent would each be roughly one-third of the overall trip distance. Simply to apply, it turns out to be a reasonably economical approach for a number of aircraft. In a full analysis for a given aircraft for min. fuel consumption, one would apply a computer program to iterate for the best hcr, etc. for the trip distance A to B, for a given payload delivery.

A 1/3 1/3 1/3

1/3 1/3 1/3

B6 B5 B4 B3 B2 B1

hcr


PROBLEM #7

Allowing for the variation of specific fuel consumption at an altitude of 40000 ft as shownbelow, find the maximum-range Mach number (compare to the nominal, constant TSFCcase). You may assume no wind and neglect drag divergence, and the current weight of theaircraft is 500000 lb.

Four Pratt & Whitney JT9D-7A turbofansTSFC at 40000 ft, = 0.3 + 0.00036 V (ft/s), lb/hr- lb

Engines

Boeing 747-200Wing reference area 5500 sq. ft. ; K = 0.05CDo (M < MDD) 0.02 ; MDD = 0.83 ; CLmax 1.7

Aircraft

PROBLEM #8

Determine the maximum-range airspeed at a current aircraft weight of 41000 lb and cruisealtitude of 10000 ft, if one allows for the variation of propulsive efficiency as noted below.How does this value compare to the nominal constant pr case ?

Two General Electric CT64-820-4 turbopropsBSFC at 10000 ft, 0.5 lb/hr-hpCruise propulsive efficiency pr = 0.5 + 0.00095 V (ft/s)

Engines

de Havilland DHC-5D (Buffalo)Wing span 96 ft ; Wing reference area 945 sq. ft. ; e = 0.82CDo 0.024 ; CLmax 1.6 ; Max. t/o weight 43000 lb ; Max. fuel weight14000 lb; Max. Payload weight 12000 lb ; Operating empty weight 25000 lb

Aircraft


Accelerated Flight Static performance (no acceleration) how fast it can fly, how far it can go? Now, we want to know how fast can it turn? Level turn The curved flight path is in a horizontal plane parallel to the plane of the ground: altitude remains constant.

Flight path & forces for an aircraft in a level turn

The relationship between forces required for a level turn. The aircraft is banked through the roll angle . The necessary condition for a level turns Lcos = W : under this condition, the altitude of the a/c will remain constant.


Another way of stating this necessary condition is

rFr

(resultant force) = Vector sum of Lr

&Wv

Governing equation of motion is

sin2

LR

Vm = : Centrifugal force R

Vm2

is balanced by the radial force sinL Two performance characteristics of greatest importance in turning flights are

The turn radius, R The turn rate dtd / = ; local angular velocity of a/c along the curved flight path

Above are important for combat a/c. For supersonic dog-fighting capability the a/c should have smallest possible R and fastest possible turn rate, . The aircraft is turning due to the radial force, rF

r.

The larger the magnitude of this force, rFr

, the tighter & faster will be the turn.

rFr

is the horizontal component of the lift, sinL . As L increases, rF

rincreases for two reasons

1. The length of the lift vector increases 2. increases because, for a level turn, cosL must remain constant (=W)

Hence Lr controls the turn. When a pilot goes to turn the aircraft, pilot rolls the a/c in order to point the lift vector in the general direction of the turn.

)/(1cosWLL

W == where L/W is the load factor (= n) So, )/1(cos 1 n= The roll angle depends only on the load factor (if you know n, then you know , and vice versa)


The turn performance of an aircraft strongly depends on the n from equation of motion for a level turn

sin2

LR

Vm =

Solve for R, sinsinsin222

===

ngV

gV

LW

LmVR

We already know that n1cos = and from the trigonometric identity of . 1sincos 22 =+

We have 1sin1 22

=+

n

or 1111sin 22 == nnn . Then, the turn radius is expressed as

12

2

=

ngVR

To obtain the smallest possible, R

highest possible n (L/W) lowest possible . V

For the turn rate . (angular velocity ,,is related with radius R and ; =R) V V

RV

dtd ==

Then,

=Vng 12

To obtain the largest possible turn rate

1. highest possible n 2. lowest possible V

(same criteria for smallest possible R) n (=L/W) and are only two explicit factors to determine R and . VBut, n and depend on aircraft design characteristic wing loading (W/S), (T/W), drag polar, air density.

V


Constraints on Load Factor (n) As the aircrafts is increased, the magnitude of the lift must increase. As L increases, the drag due to lift increases. Hence, to maintain a sustained level turn at a given velocity and , the thrust must be increased from its straight & level flight value to compensate for the increase in drag. If this increase in thrust pushes the required thrust beyond the maximum thrust available from the power plant, then the level turn cannot be sustained at the given & . In this case, to maintain a turn at the given , will have to be decreased in order to decrease the drag.

V V

Since the load factor n is a function of , cos1=n at any given velocity, the maximum

possible load factor for a sustained level turn is constrained by the maximum thrust available. Maximum load factor nmax. can be calculated as follows.

)(21 22

0 LDKCCSVD += for a level flight T = D.

Also, LSCVnWL2

21

== or SVnWC L 2

2

= .

Substitute

+=

2

22 )2(

21

0 SVnWKCSVT D and solve for n.

2/1

2

2

/21

)/(21

0

=

SWC

VWT

SWK

Vn D

above equation gives the load factor (hence ) for a given & T/W. V


The maximum value of n is obtained by T = Tmax. or (T/W)max.

2/1

2

.max

2

.max /21

)/(21

0

=

SWC

VWT

SWK

Vn D

for a given , n can only range between 1


At point B, the aircraft is flying at (L/D)max.

WT

DL

WD

DL

WLn === . When Tmax. is inserted then, .max.max )(W

TDLn =

Relationship between nmax. and CL.

As decreases, CV L increases. (As decreases, the magnitude of L is maintained by increasing C

VL, by increasing )

CL cannot increase indefinitely it is limited by stall CLmax. The velocity at CLmax.is reached is denoted by point A At lower velocities, less than point A, the maximum load factor is constrained by

CLmax., not by available thrust

When nmax. is constrained by CLmax., the nmax.is SWC

Vn L/2

1 .max2.max = (the solid curve

to the left of point A is obtained by this)

Also, .max

max1cos

n= .


Constraints on V We know that for high performance for turn, should be as small as possible. However,

cannot be reduced indefinitely without encountering stall. Hence, the stall limit is a constraint on .

V

V

V The stalling velocity is a function of (n = L/W)

LSCVnWL2

21

== and LSC

nWV 2= .

When CL = CLmax., then = V stallV

.max

2

Lstall SC

nWV = more general equation for stall velocity When the aircraft is at a bank angle , the stalling velocity is increased above that for straight & level flight (n = 1). Hence, when the aircraft is in a level turn with a load factor n > 1, the stalling velocity increases proportionally to n1/2.


Minimum Turn Radius The conditions for minimum R(radius) are found by setting dR/d = 0 V

From 12

2

=

ngVR , we can rewrite R as

2

2 21,

12

== VqngqR (dynamic pressure)

Load factor n is a function of velocity (n = f( )), so n = f(q). VWe can find min. radius by

0)1(

)1(212

222

2/122

=

=

ngdqdnnqngng

dqdR

or

0)1(212 2/122 = dqdnnqngng

or

012 =dqdnqnn . Now we need to find

dqdn

.

From previous derivations, we know n as a function of . V

=

SWC

qWT

SWKqn D

/)/(02

Differentiate with respect to q gives


2)/()/(2/

0

SWKqC

SWKWT

dqdnn D=

After substitution,

0)/()/(2

)/(1)/()/( 2

2

2

2

00 =+SWK

CqSWK

WTqSWK

CqWT

SWKq DD

Combining and cancelling terms, we get

1)/(2

)/( =SWK

WTq

or WT

SWKq/

)/(2= . Then,

)/()/(4)(

.min WTSWKV R = : min. turning radius

Above equation gives the value of which corresponds to the min. turning radius. V The load factor corresponding to above velocity is found by substituting above into n = f(V)

222

222

)/(4

2)/()/(

)/(4)/()/()/)(/(2

/)/(000

WTKC

SWKWTCSWK

SWKWTWTSWK

SWC

qWT

SWKqn DDD ==

=

or

2)/(4

2 0.min WT

KCn DR = Above equation gives the load factor corresponding to the min. turning radius.

Finally, the expression for min. turning radius is obtained by using 12

2

=

ngVR


1)//(421

)/()/(4

1 222

min

0.min

min

==

WTKCgWTSWK

ngVR

DR

R

Then,

2min )//(41)/()/(4

0WTKCgWT

SWKRD

== So, best Rmin. occurs at

at sea level (max. ) small (W/S) large (T/W) good aerodynamic characteristics good streamline( low ), low K

0DC


Maximum Turn Rate

=Vng 12

max. is obtained by differentiating above equation and setting the derivative equals to zero. Same method as for min. turn radius. We will get

4/12/1

0

.max

)/(2)(

=DC

KSWV

1/

0

.max=

DKCWTn

=

2/1

max0

2/

/ KC

KWT

SWq D

Wing loading (W/S) and thrust-to-weight (T/W) ratio dominate the values of R min & max. For good turn performance (low R min & high max.), W/S should be low (increase S) and T/W should be high. For low K high AR. High turning performance Min. turn radius and max. turn rate are important performance characteristics for a fighter airplane

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