Power and Thrust for Cruising Flight

[Pages:23]Power and Thrust for Cruising Flight

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

Copyright 2018 by Robert Stengel. All rights reserved. For educational use only.



1

U.S. Standard Atmosphere, 1976



2

1

Dynamic Pressure and Mach Number

= air density, function of height = e-h

sea level

a = speed of sound = linear function of height

Dynamic pressure = q V 2 2 Mach number = V a

3

Definitions of Airspeed

Airspeed is speed of aircraft measured with respect to air mass

Airspeed = Inertial speed if wind speed = 0

? Indicated Airspeed (IAS)

IAS =

( ) 2 p - p stagnation

ambient

SL =

( ) 2 ptotal - pstatic

SL

!

2qc SL

,

with

qc

!

impact

pressure

? Calibrated Airspeed (CAS)*

CAS = IAS corrected for instrument and position errors

=

( ) 2 qc corr #1

SL

* Kayton & Fried, 1969; NASA TN-D-822, 1961

4

2

Definitions of Airspeed

Airspeed is speed of aircraft measured with respect to air mass

Airspeed = Inertial speed if wind speed = 0

Equivalent Airspeed (EAS)*

EAS = CAS corrected for compressibility effects =

( ) 2 qc corr # 2

SL

True Airspeed (TAS)*

Mach number

V TAS = EAS

SL (z)

=

IAScorrected

SL (z)

TAS M=

a

* Kayton & Fried, 1969; NASA TN-D-822, 1961 5

Flight in the Vertical Plane

6

3

Longitudinal Variables

7

Longitudinal Point-Mass Equations of Motion

? Assume thrust is aligned with the velocity vector (small-angle approximation for )

? Mass = constant

V!

=

(CT

cos

-

CD

)

1 2

V

2S

-

mg

sin

(CT

-

CD

)1

2

V 2S

-

mg

sin

m

m

!

=

(CT

sin

+

CL

)

1 2

V

2S

-

mg

cos

CL

1 2

V

2S

-

mg

cos

mV

mV

h! = -z! = -vz = V sin

V = velocity = Earth-relative airspeed

r! = x! = vx = V cos

= True airspeed with zero wind = flight path angle

h = height (altitude)

r = range

8

4

Conditions for Steady, Level Flight

? Flight path angle = 0 ? Altitude = constant ? Airspeed = constant ? Dynamic pressure = constant

0

=

(CT

-

CD

)

1 2

V 2S

? Thrust = Drag

m

0

=

CL

1 2

V

2S

-

mg

mV

h = 0

? Lift = Weight

r = V

9

Power and Thrust

Propeller

Power

=

P

=

T

?V

=

CT

1 2

V 3S

independent

of

airspeed

Turbojet

Thrust

=

T

=

CT

1 2

V 2S

independent

of

airspeed

Throttle Effect

T

=

T T max

=

CTmax qS T ,

0 T 1

10

5

Typical Effects of Altitude and Velocity on Power and Thrust

? Propeller

[Air-breathing engine]

? Turbofan

[In between]

? Turbojet

? Battery

[Independent of altitude

and airspeed]

11

Models for Altitude Effect on Turbofan Thrust

From Flight Dynamics, pp.117-118

Thrust

=

CT

(V

,

T

)

1 2

(h)V

2S

( ) =

ko

+

k1V n

1 2

(h)V 2ST ,

N

ko = Static thrust coefficient at sea level k1 = Velocity sensitivity of thrust coefficient n = Exponent of velocity sensitivity [ = -2 for turbojet]

(h) = SLe-h, SL = 1.225 kg / m3, = (1 / 9, 042) m-1

12

6

Thrust of a Propeller-Driven

Aircraft

With constant rpm, variable-pitch propeller

T

= PI

Pengine V

= net

Pengine V

P = propeller efficiency I = ideal propulsive efficiency

( ) ( ) = TV T V + Vinflow = V V + Vfreestream 2

netmax 0.85 - 0.9

Efficiencies decrease with airspeed Engine power decreases with altitude Proportional to air density, w/o supercharger

13

Reciprocating-Engine Power and

Specific Fuel Consumption (SFC)

P(h)

PSL

=

1.132

(h)

SL

-

0.132

SFC Independent of Altitude

? Engine power decreases with altitude ? Proportional to air density, w/o supercharger ? Supercharger increases inlet manifold pressure, increasing power and extending maximum altitude

Anderson (Torenbeek) 14

7

Propeller Efficiency, P, and Advance Ratio, J

Effect of propeller-blade pitch angle

Advance Ratio

J= V nD

where V = airspeed, m / s n = rotation rate, revolutions / s D = propeller diameter, m

from McCormick

15

Thrust of a Turbojet Engine

T

=

m V

02*# 1,% 32+$

o o-

1

&# (%

t

'$t -

1

& ( '

(

c

-1) + t

o c

-1/2 / .

42 -15

62

m! = m! air + m! fuel

( ) o = pstag

( -1)/

pambient

;

= ratio of specific heats 1.4

t = (turbine inlet temp. freestream ambient temp.)

c = (compressor outlet temp. compressor inlet temp.)

from Kerrebrock

Little change in thrust with airspeed below Mcrit

Decrease with increasing altitude

16

8

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