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The maximal total range is the maximum distance an aircraft can fly between takeoff and landing, as limited by fuel capacity in powered aircraft, or cross-country speed and environmental conditions in unpowered aircraft. The range can be seen as the cross-country ground speed multiplied by the maximum time in the air. The fuel time limit for powered aircraft is fixed by the fuel load and rate of consumption. When all fuel is consumed, the engines stop and the aircraft will lose its propulsion. Ferry range means the maximum range the aircraft can fly. This usually means maximum fuel load, optionally with extra fuel tanks and minimum equipment. It refers to transport of aircraft without any passengers or cargo. Combat range is the maximum range the aircraft can fly when carrying ordnance. Combat radius is a related measure based on the maximum distance a warplane can travel from its base of operations, accomplish some objective, and return to its original airfield with minimal reserves.

Contents

1 Derivation

1.1 Propeller aircraft 1.2 Jet propulsion 1.3 Cruise/climb

2 See also 3 References

Derivation[edit] For most unpowered aircraft, the maximum flight time is variable, limited by available daylight hours, aircraft design (performance), weather conditions, aircraft potential energy, and pilot endurance. Therefore, the range equation can only exactly calculated and will be derived for propeller and jet aircraft. If the total weight of the aircraft at a particular time

t

displaystyle t

is

W

displaystyle W

=

W

e

+

W

f

displaystyle W_ e +W_ f

, where

W

e

displaystyle W_ e

is the zero-fuel weight and

W

f

displaystyle W_ f

the weight of the fuel (both in kg), the fuel consumption rate per unit time flow

F

displaystyle F

(in kg/s) is equal to

d

W

f

d t

= −

d W

d t

displaystyle - frac dW_ f dt =- frac dW dt

. The rate of change of aircraft weight with distance

R

displaystyle R

(in meters) is

d W

d R

=

d W

d t

d R

d t

= −

F V

displaystyle frac dW dR = frac frac dW dt frac dR dt =- frac F V

, where

V

displaystyle V

is the speed (in m/s), so that

d R

d t

= −

V F

d W

d t

displaystyle frac dR dt =- frac V F frac dW dt

It follows that the range is obtained from the definite integral below, with

t

1

displaystyle t_ 1

and

t

2

displaystyle t_ 2

the start and finish times respectively and

W

1

displaystyle W_ 1

and

W

2

displaystyle W_ 2

the initial and final aircraft weights

R =

t

1

t

2

d R

d t

d t =

W

1

W

2

V F

d W =

W

2

W

1

V F

d W

displaystyle R=int _ t_ 1 ^ t_ 2 frac dR dt dt=int _ W_ 1 ^ W_ 2 - frac V F dW=int _ W_ 2 ^ W_ 1 frac V F dW

. The term

V F

displaystyle frac V F

is called the specific range (= range per unit weight of fuel; S.I. units: m/kg). The specific range can now be determined as though the airplane is in quasi steady state flight. Here, a difference between jet and propeller driven aircraft has to be noticed. Propeller aircraft[edit] With propeller driven propulsion, the level flight speed at a number of airplane weights from the equilibrium condition

P

a

=

P

r

displaystyle P_ a =P_ r

has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency

η

j

displaystyle eta _ j

and specific fuel consumption

c

p

displaystyle c_ p

. The successive engine powers can be found:

P

b r

=

P

a

η

j

displaystyle P_ br = frac P_ a eta _ j

The corresponding fuel weight flow rates can be computed now:

F =

c

p

P

b r

displaystyle F=c_ p P_ br

Thrust power, is the speed multiplied by the drag, is obtained from the lift-to-drag ratio:

P

a

= V

C

D

C

L

W

displaystyle P_ a =V frac C_ D C_ L W

 ; here W is a force in newtons The range integral, assuming flight at constant lift to drag ratio, becomes

R =

η

j

g

c

p

C

L

C

D

W

2

W

1

d W

W

displaystyle R= frac eta _ j gc_ p frac C_ L C_ D int _ W_ 2 ^ W_ 1 frac dW W

 ; here W is the mass in kilograms, therefore standard gravity g is added. Its exact value depends on the distance to the centre of gravity of earth, but it averages 9.81 m/s2. To obtain an analytic expression for range, it has to be noted that specific range and fuel weight flow rate can be related to the characteristics of the airplane and propulsion system; if these are constant:

R =

η

j

g

c

p

C

L

C

D

l n

W

1

W

2

displaystyle R= frac eta _ j gc_ p frac C_ L C_ D ln frac W_ 1 W_ 2

Jet propulsion[edit] The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship

D =

C

D

C

L

W

displaystyle D= frac C_ D C_ L W

is used. The thrust can now be written as:

T = D =

C

D

C

L

W

displaystyle T=D= frac C_ D C_ L W

 ; here W is a force in newtons Jet engines are characterized by a thrust specific fuel consumption, so that rate of fuel flow is proportional to drag, rather than power.

F = −

c

T

T = −

c

T

C

D

C

L

W

displaystyle F=-c_ T T=-c_ T frac C_ D C_ L W

Using the lift equation,

1 2

ρ

V

2

S

C

L

= W

displaystyle frac 1 2 rho V^ 2 SC_ L =W

where

ρ

displaystyle rho

is the air density, and S the wing area. the specific range is found equal to:

V F

=

1

c

T

C

L

C

D

2

2

ρ S W

displaystyle frac V F = frac 1 c_ T sqrt frac C_ L C_ D ^ 2 frac 2 rho SW

Therefore, the range (in meters) becomes:

R =

1

c

T

C

L

C

D

2

2

g ρ S

W

2

W

1

1

W

d W

displaystyle R= frac 1 c_ T sqrt frac C_ L C_ D ^ 2 frac 2 grho S int _ W_ 2 ^ W_ 1 frac 1 sqrt W dW

 ; here

W

displaystyle W

is again mass. When cruising at a fixed height, a fixed angle of attack and a constant specific fuel consumption, the range becomes:

R =

2

c

T

C

L

C

D

2

2

g ρ S

(

W

1

W

2

)

displaystyle R= frac 2 c_ T sqrt frac C_ L C_ D ^ 2 frac 2 grho S left( sqrt W_ 1 - sqrt W_ 2 right)

where the compressibility on the aerodynamic characteristics of the airplane are neglected as the flight speed reduces during the flight. Cruise/climb[edit] For long range jet operating in the stratosphere (altitude approximately between 11–20 km), the speed of sound is constant, hence flying at fixed angle of attack and constant Mach number causes the aircraft to climb, without changing the value of the local speed of sound. In this case:

V = a M

displaystyle V=aM

where

M

displaystyle M

is the cruise Mach number
Mach number
and

a

displaystyle a

the speed of sound. W is the weight in kilograms (kg). The range equation reduces to:

R =

a M

g

c

T

C

L

C

D

W

2

W

1

d W

W

displaystyle R= frac aM gc_ T frac C_ L C_ D int _ W_ 2 ^ W_ 1 frac dW W

where

a =

7 5

R

s

T

displaystyle a= sqrt frac 7 5 R_ s T

 ; here

R

s

displaystyle R_ s

is the specific heat constant of air 287.16

J

k g K

displaystyle frac J kgK

(based on aviation standards) and

γ = 7

/

5 = 1.4

displaystyle gamma =7/5=1.4

(derived from

γ =

c

p

c

v

displaystyle gamma = frac c_ p c_ v

and

c

p

=

c

v

+

R

s

displaystyle c_ p =c_ v +R_ s

).

c

p

displaystyle c_ p

en

c

v

displaystyle c_ v

are the specific heat capacities of air at a constant pressure and constant volume. Or

R =

a M

g

c

T

C

L

C

D

l n

W

1

W

2

displaystyle R= frac aM gc_ T frac C_ L C_ D ln frac W_ 1 W_ 2

, also known as the Breguet range equation after the French aviation pioneer, Breguet. See also[edit]

Flight length Flight distance record Endurance

References[edit]

G. J. J. Ruijgrok. Elements of Airplane Performance. Delft University Press.[page needed] ISBN 9789065622044. Prof. Z. S. Spakovszky. Thermodynamics and Propulsion, Chapter 13.3 Aircraft
Aircraft
Range: the Breguet Range Equation MIT turbines, 2002 Martinez, Isidoro. Aircraft
Aircraft
propulsion. Range and endurance: Breguet's equat

.