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### MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

### RANGE AND ENDURANCE

Thrust and Power Requirements

April 28, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

EXAMPLE: BEECHCRAFT QUEEN AIR

- The results we have developed so far for lift and drag for a finite wing may also be applied to a complete airplane. In such relations:
- CD is drag coefficient for complete airplane
- CD,0 is parasitic drag coefficient, which contains not only profile drag of wing (cd) but also friction and pressure drag of tail surfaces, fuselage, engine nacelles, landing gear and any other components of airplane exposed to air flow
- CL is total lift coefficient, including small contributions from horizontal tail and fuselage
- Span efficiency for finite wing replaced with Oswald efficiency factor for entire airplane
- Example: To see how this works, assume the aerodynamicists have provided all the information needed about the complete airplane shown below

Beechcraft Queen Air Aircraft Data

W = 38,220 N

S = 27.3 m2

AR = 7.5

e (complete airplane) = 0.9

CD,0 (complete airplane) = 0.03

What thrust and power levels are required of engines to cruise at 220 MPH at sea-level?

How would these results change at 15,000 ft

OVERALL AIRPLANE DRAG

- No longer concerned with aerodynamic details
- Drag for complete airplane, not just wing

Entire Airplane

Wing or airfoil

Tail Surfaces

Engine Nacelles

Landing Gear

DRAG POLAR

- CD,0 is parasite drag coefficient at zero lift (aL=0)
- CD,i drag coefficient due to lift (induced drag)
- Oswald efficiency factor, e, includes all effects from airplane
- CD,0 and e are known aerodynamics quantities of airplane

Example of Drag Polar for complete airplane

4 FORCES ACTING ON AIRPLANE

- Model airplane as rigid body with four natural forces acting on it
- Lift, L
- Acts perpendicular to flight path (always perpendicular to relative wind)
- Drag, D
- Acts parallel to flight path direction (parallel to incoming relative wind)
- Propulsive Thrust, T
- For most airplanes propulsive thrust acts in flight path direction
- May be inclined with respect to flight path angle, aT, usually small angle
- Weight, W
- Always acts vertically toward center of earth
- Inclined at angle, q, with respect to lift direction
- Apply Newton’s Second Law (F=ma) to curvilinear flight path
- Force balance in direction parallel to flight path
- Force balance in direction perpendicular to flight path

GENERAL EQUATIONS OF MOTION (6.2)

Free Body Diagram

Apply Newton’s 2nd Parallel to flight path:

Apply Newton’s 2nd Perpendicular to flight path:

LEVEL, UNACCELERATED FLIGHT

L

- JSF is flying at constant speed (no accelerations)
- Sum of forces = 0 in two perpendicular directions
- Entire weight of airplane is perfectly balanced by lift (L = W)
- Engines produce just enough thrust to balance total drag at this airspeed (T = D)
- For most conventional airplanes aT is small enough such that cos(aT)~ 1

D

T

W

LEVEL, UNACCELERATED FLIGHT

- TR is thrust required to fly at a given velocity in level, unaccelerated flight
- Notice that minimum TR is when airplane is at maximum L/D
- L/D is an important aero-performance quantity

THRUST REQUIREMENT (6.3)

- TR for airplane at given altitude varies with velocity
- Thrust required curve: TR vs. V∞

PROCEDURE: THRUST REQUIREMENT

- Select a flight speed, V∞
- Calculate CL

Minimum TR when airplane

flying at (L/D)max

- Calculate CD

- Calculate CL/CD
- Calculate TR

This is how much thrust engine

must produce to fly at selected V∞

Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil

THRUST REQUIREMENT (6.3)

- Different points on TR curve correspond to different angles of attack

At b:

Small q∞

Large CL (or CL2) and a to support W

D large

At a:

Large q∞

Small CL and a

D large

THRUST REQUIRED VS. FLIGHT VELOCITY

Zero-Lift TR

(Parasitic Drag)

Lift-Induced TR

(Induced Drag)

Zero-Lift TR ~ V2

(Parasitic Drag)

Lift-Induced TR ~ 1/V2

(Induced Drag)

THRUST REQUIRED VS. FLIGHT VELOCITY

At point of minimum TR, dTR/dV∞=0

(or dTR/dq∞=0)

CD,0 = CD,i at minimum TR and maximum L/D

Zero-Lift Drag = Induced Drag at minimum TR and maximum L/D

HOW FAST CAN YOU FLY?

- Maximum flight speed occurs when thrust available, TA=TR
- Reduced throttle settings, TR < TA
- Cannot physically achieve more thrust than TA which engine can provide

Intersection of TR

curve and maximum

TA defined maximum

flight speed of airplane

FURTHER IMPLICATIONS FOR DESIGN: VMAX

- Maximum velocity at a given altitude is important specification for new airplane
- To design airplane for given Vmax, what are most important design parameters?

Steady, level flight: T = D

Steady, level flight: L = W

Substitute into drag equation

Turn this equation into a quadratic

equation (by multiplying by q∞)

and rearranging

Solve quadratic equation and set

thrust, T, to maximum available

thrust, TA,max

FURTHER IMPLICATIONS FOR DESIGN: VMAX

- TA,max does not appear alone, but only in ratio (TA/W)max
- S does not appear alone, but only in ratio (W/S)
- Vmax does not depend on thrust alone or weight alone, but rather on ratios
- (TA/W)max: maximum thrust-to-weight ratio
- W/S: wing loading
- Vmax also depends on density (altitude), CD,0, peAR
- Increase Vmax by
- Increase maximum thrust-to-weight ratio, (TA/W)max
- Increasing wing loading, (W/S)
- Decreasing zero-lift drag coefficient, CD,0

AIRPLANE POWER PLANTS

Two types of engines common in aviation today

- Reciprocating piston engine with propeller
- Average light-weight, general aviation aircraft
- Rated in terms of POWER
- Jet (Turbojet, turbofan) engine
- Large commercial transports and military aircraft
- Rated in terms of THRUST

THRUST VS. POWER

- Jets Engines (turbojets, turbofans for military and commercial applications) are usually rate in Thrust
- Thrust is a Force with units (N = kg m/s2)
- For example, the PW4000-112 is rated at 98,000 lb of thrust
- Piston-Driven Engines are usually rated in terms of Power
- Power is a precise term and can be expressed as:
- Energy / time with units (kg m2/s2) / s = kg m2/s3 = Watts
- Note that Energy is expressed in Joules = kg m2/s2
- Force * Velocity with units (kg m/s2) * (m/s) = kg m2/s3 = Watts
- Usually rated in terms of horsepower (1 hp = 550 ft lb/s = 746 W)
- Example:
- Airplane is level, unaccelerated flight at a given altitude with speed V∞
- Power Required, PR=TR*V∞
- [W] = [N] * [m/s]

POWER REQUIRED

At point of minimum PR, dPR/dV∞=0

POWER REQUIRED

- V∞ for minimum PR is less than V∞ for minimum TR

WHY DO WE CARE ABOUT THIS?

- We will show that for a piston-engine propeller combination
- To fly longest distance (maximum range) we fly airplane at speed corresponding to maximum L/D
- To stay aloft longest (maximum endurance) we fly the airplane at minimum PR or fly at a velocity where CL3/2/CD is a maximum
- Power will also provide information on maximum rate of climb and altitude

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)

Recall PR = f(r∞)

Subscript ‘0’ denotes seal-level conditions

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)Propeller-Driven Airplane

Vmax,ALT < Vmax,sea-level

RATE OF CLIMB (6.8)

- Boeing 777: Lift-Off Speed ~ 180 MPH
- How fast can it climb to a cruising altitude of 30,000 ft?

RATE OF CLIMB (6.8)

Governing Equations:

RATE OF CLIMB (6.8)

Vertical velocity

Rate of Climb:

TV∞ is power available

DV∞ is level-flight power required (for small q neglect W)

TV∞- DV∞ is excess power

EXAMPLE: F-15 K

- Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact at high speed (kinetic energy weapon)
- F-15 can bring ALMV under the ground track of its target, as opposed to a ground-based system, which must wait for a target satellite to overfly its launch site.

EXAMPLE: HIGH ASPECT RATIO GLIDER

q

To maximize range, smallest q occurs at (L/D)max

A modern sailplane may have a glide ratio as high as 60:1

So q = tan-1(1/60) ~ 1°

How far can we fly?

How long can we stay aloft?

How do answers vary for propeller-driven vs. jet-engine?

RANGE AND ENDURANCE

- Range: Total distance (measured with respect to the ground) traversed by airplane on a single tank of fuel
- Endurance: Total time that airplane stays in air on a single tank of fuel
- Parameters to maximize range are different from those that maximize endurance
- Parameters are different for propeller-powered and jet-powered aircraft
- Fuel Consumption Definitions
- Propeller-Powered:
- Specific Fuel Consumption (SFC)
- Definition: Weight of fuel consumed per unit power per unit time
- Jet-Powered:
- Thrust Specific Fuel Consumption (TSFC)
- Definition: Weight of fuel consumed per unit thrust per unit time

PROPELLER-DRIVEN: RANGE AND ENDURANCE

- SFC: Weight of fuel consumed per unit power per unit time

- ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per hour

- Minimum lb of fuel per hour obtained with minimum HP
- Maximum endurance for a propeller-driven airplane occurs when airplane is flying at minimum power required
- Maximum endurance for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL3/2/CD is a maximized

PROPELLER-DRIVEN: RANGE AND ENDURANCE

- SFC: Weight of fuel consumed per unit power per unit time

- RANGE: To cover longest distance use minimum pounds of fuel per mile

- Minimum lb of fuel per hour obtained with minimum HP/V∞
- Maximum range for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum

PROPELLER-DRIVEN: RANGE BREGUET FORMULA

- To maximize range:
- Largest propeller efficiency, h
- Lowest possible SFC
- Highest ratio of Winitial to Wfinal, which is obtained with the largest fuel weight
- Fly at maximum L/D

PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA

- To maximize endurance:
- Largest propeller efficiency, h
- Lowest possible SFC
- Largest fuel weight
- Fly at maximum CL3/2/CD
- Flight at sea level

JET-POWERED: RANGE AND ENDURANCE

- TSFC: Weight of fuel consumed per thrust per unit time

- ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per hour

- Minimum lb of fuel per hour obtained with minimum thrust
- Maximum endurance for a jet-powered airplane occurs when airplane is flying at minimum thrust required
- Maximum endurance for a jet-powered airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum

JET-POWERED: RANGE AND ENDURANCE

- TSFC: Weight of fuel consumed per unit power per unit time

- RANGE: To cover longest distance use minimum pounds of fuel per mile

- Minimum lb of fuel per hour obtained with minimum Thrust/V∞
- Maximum range for a jet-powered airplane occurs when airplane is flying at a velocity such that CL1/2/CD is a maximum

JET-POWERED: RANGE BREGUET FORMULA

- To maximize range:
- Minimum TSFC
- Maximum fuel weight
- Flight at maximum CL1/2/CD
- Fly at high altitudes

JET-POWERED: ENDURACE BREGUET FORMULA

- To maximize endurance:
- Minimum TSFC
- Maximum fuel weight
- Flight at maximum L/D

SUMMARY: ENDURANCE AND RANGE

- Maximum Endurance
- Propeller-Driven
- Maximum endurance for a propeller-driven airplane occurs when airplane is flying at minimum power required
- Maximum endurance for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL3/2/CD is a maximized
- Jet Engine-Driven
- Maximum endurance for a jet-powered airplane occurs when airplane is flying at minimum thrust required
- Maximum endurance for a jet-powered airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum
- Maximum Range
- Propeller-Driven
- Maximum range for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum
- Jet Engine-Driven
- Maximum range for a jet-powered airplane occurs when airplane is flying at a velocity such that CL1/2/CD is a maximum

USEFUL APPROXIMATION (T >> D, R)

- Lift-off distance very sensitive to weight, varies as W2
- Depends on ambient density
- Lift-off distance may be decreased:
- Increasing wing area, S
- Increasing CL,max
- Increasing thrust, T

sL.O.: lift-off distance

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