Breguet's range formula

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With the Breguet'sche coverage formula , named after Louis Charles Breguet , which can range from motorized aircraft are calculated. The original Breguet formula refers to aircraft with propeller drive ( piston engines and propeller turbines with low residual thrust), which, however , can also be applied to aircraft with jet engines with appropriate modification . The calculated range value is only exact for constant aircraft mass, constant airspeed (with jet engines), constant power (with piston engines) and constant flight altitude.

definition

Breguet's range formula specifies the range for propeller-driven aircraft, according to

${\displaystyle R=E\cdot {\frac {\eta _{P}}{b_{P}\cdot g}}\cdot \ln \left({\frac {m_{0}}{m_{0}-m_{t}}}\right)}$

Where: R = range; - propeller efficiency; - performance-specific fuel consumption; = Glide ratio ; m ₀ = aircraft mass at the starting point of the calculation; m _t = amount of fuel consumed during the cruise segment. Consistent units must be used, as the specific fuel consumption is often given in different units. This formula can also be used for turboprop engines with low residual thrust . ${\displaystyle \eta _{P}}$ ${\displaystyle b_{P}}$ ${\displaystyle E}$

If you adapt Breguet's formula accordingly (see derivation), you can also use it to calculate the range for aircraft with jet propulsion :

${\displaystyle R=E\cdot {\frac {V}{b_{F}\cdot g}}\cdot \ln \left({\frac {m_{0}}{m_{0}-m_{t}}}\right)}$

Where: R = range; = Airspeed; - Thrust-specific fuel consumption (jet engine); = Glide ratio; m ₀ = aircraft mass at the starting point of the calculation; m _t = amount of fuel consumed during the cruise segment. Consistent units must be used, as the specific fuel consumption is often given in different units. ${\displaystyle V}$ ${\displaystyle b_{F}}$ ${\displaystyle E}$

For flights in the area close to noise ( Ma > 0.7), the flight speed v must be corrected according to v = Ma · a , where Ma is the Mach number and a is the speed of sound at the altitude. If one or more flight parameters change during cruise, Breguet's formula is only a more or less precise approximation.

This can be remedied by dividing the cruise flight into several segments, whereby the accuracy of the calculated range value can be increased with the number of segments. It should also be noted that different glide ratios must be used for propeller-driven aircraft and turbo-jet aircraft for maximum range .

Derivation

Breguet's range formula with propeller drive is based on the performance- specific fuel consumption ( specific fuel consumption ):

${\displaystyle b_{P}={\frac {{\dot {m}}_{B}}{P}}={\frac {{\dot {m}}_{B}}{F\cdot V}}}$

${\displaystyle P}$- Propulsion power [kW]; - thrust [N]; - power-specific fuel consumption [kg / kW · h]; - Airspeed [km / h] ${\displaystyle F}$ ${\displaystyle b_{P}}$ ${\displaystyle V}$

It is

${\displaystyle {\dot {m}}_{B}={\frac {dm_{B}}{dt}}}$

the fuel mass flow [kg / h] and the flight time. ${\displaystyle t}$

The speed can be written as the ratio of the distance covered ( - distance [km]) and the time required for this: ${\displaystyle s}$

${\displaystyle V={\frac {ds}{dt}}}$

The following applies to the mass decrease in level flight:

${\displaystyle dm=-b_{P}\cdot F\cdot V\cdot dt}$

This cuts out and the distance differential is: ${\displaystyle dt}$

${\displaystyle ds={\frac {1}{-b_{P}\cdot F}}dm}$

With the condition of horizontal flight there is an equilibrium of forces between:

Buoyancy and weight in the vertical direction and analogously in the horizontal direction between: ${\displaystyle A}$ ${\displaystyle G=m\cdot g}$

Thrust and resistance . ${\displaystyle F}$ ${\displaystyle W}$

It follows that the ratio of drag to lift is equal to the ratio of thrust to weight:

${\displaystyle {\frac {W}{A}}=\epsilon ={\frac {F}{G}}\to F=\epsilon \cdot m\cdot g}$

with - glide ratio [-] and = gravitational acceleration [m / s²] ${\displaystyle \epsilon }$ ${\displaystyle g}$

This means that the thrust in the distance differential can be replaced by:

${\displaystyle ds={\frac {1}{-b_{P}\cdot \epsilon \cdot m\cdot g}}dm}$

By integrating the starting mass up to the mass at the end of the observation : ${\displaystyle m_{Start}}$ ${\displaystyle m_{Ende}}$

${\displaystyle ds=\int _{m_{Start}}^{m_{Ende}}{\frac {1}{-b_{P}\cdot \epsilon \cdot m\cdot g}}\,\mathrm {d} m,}$

one obtains the Breguet's range formula with (ideal) propeller drive :

${\displaystyle R={\frac {1}{b_{P}\cdot \epsilon \cdot g}}\ln \left({\frac {m_{Start}}{m_{Ende}}}\right)}$

Since the propeller works differently efficiently depending on the design and operating condition, its efficiency can also be taken into account: ${\displaystyle \eta _{P}}$

${\displaystyle P=\eta _{P}\cdot P^{*}}$

with - engine power [kW]; - Propeller efficiency [-] ${\displaystyle P^{*}}$ ${\displaystyle \eta _{P}}$

This gives Breguet's range formula with propeller drive to:

${\displaystyle R={\frac {\eta _{P}}{b_{P}\cdot \epsilon \cdot g}}\ln \left({\frac {m_{Start}}{m_{Ende}}}\right)}$

Breguet's range formula with jet propulsion is based on the thrust-specific fuel consumption:

${\displaystyle b_{F}={\frac {{\dot {m}}_{B}}{F}}}$

${\displaystyle F}$- thrust [N]; - Thrust-specific fuel consumption [kg / N · s]; - Airspeed [m / s] ${\displaystyle b_{F}}$ ${\displaystyle V}$

The further procedure is analogous and you get Breguet's range formula with jet propulsion :

${\displaystyle R={\frac {V}{b_{F}\cdot \epsilon \cdot g}}\ln \left({\frac {m_{Start}}{m_{Ende}}}\right)}$

In determining the range, it is assumed that the constant speed and constant lift coefficient are maintained , which would inevitably lead to a reduction in density through gain in altitude in the course of the flight (by reducing the flight weight).

Individual evidence

1. ↑ ^{a b} Egbert Torenbeek: Synthesis of Subsonic Airplane Design . Kluwer Academic, Dordrecht 1982, ISBN 90-247-2724-3 .
2. ^ ^{A b} Daniel P. Raymer: Aircraft Design: A Conceptual Approach . AIAA Education Series, Washington DC 1992, ISBN 0-930403-51-7 .
3. ^ ^{A b} Jan Roskam: Preliminary Configuration Design and Integration of the Propulsion System (=  Airplane Design . Part II). Roskam Aviation, Ottawa 1985.