Unit 3: Range And Endurance - Subjective Questions

Range is the horizontal distance traveled by an airplane using a specified amount of fuel. Endurance is the total time for which the airplane can remain airborne using that fuel.

The fundamental definitions are:

where is the ground speed.

Important factors include:

• Fuel carried: A larger fuel-weight fraction generally increases both range and endurance.
• Aerodynamic efficiency: A high lift-to-drag ratio reduces the thrust required.
• Power or thrust-specific fuel consumption: Lower fuel consumption improves performance.
• Propulsive efficiency: Particularly important for propeller-driven airplanes.
• Aircraft weight: Weight decreases continuously as fuel is consumed.
• Altitude and air density: These affect true airspeed, drag, engine operation, and fuel consumption.
• Wind: Headwind reduces ground range, while tailwind increases it; ideal still-air endurance is not directly changed.

In a propeller-driven airplane, the engine supplies shaft power, which the propeller converts into useful thrust power. Propeller efficiency is

where is thrust, is true airspeed, and is shaft power. In steady, level flight, , so

The fuel-consumption rate is proportional to shaft power:

where is the power-specific fuel consumption.

Therefore:

• Range depends on distance obtained per unit fuel. Since distance rate is , the airspeed cancels from the specific-range expression, making high especially important.
• Endurance depends on time obtained per unit fuel and is maximized by minimizing required power .
• High , low , and a large fuel-weight ratio improve both quantities.
• As fuel is consumed, weight and the speed required to maintain an optimum lift coefficient decrease.

For steady, level flight,

The useful propulsive power is . If is shaft power,

Thus,

If is the weight of fuel consumed per unit shaft power per unit time, the aircraft weight changes according to

The specific air range is

Using ,

The specific endurance is

where is the power required. Hence, maximum range requires low drag, whereas maximum endurance requires low power required.

Assume steady, level flight with approximately constant propeller efficiency , power-specific fuel consumption , and lift-to-drag ratio .

Fuel-weight consumption is

Since ,

In level flight, , and therefore

Substitution gives

Integrating from initial weight to final weight ,

Hence, the Breguet range equation is

It shows that propeller-airplane range improves through:

• Higher propeller efficiency
• Lower power-specific fuel consumption
• Higher aerodynamic efficiency
• Higher fuel-weight ratio

For maximum range, the aircraft should operate at the lift coefficient that maximizes , subject to the assumptions used in the derivation.

For a propeller-driven airplane,

At constant altitude and lift coefficient,

Also,

Therefore, the power required is

The time increment becomes

Integrating from to gives

Thus,

For given weight limits, density, and wing area, endurance is maximized by maximizing

which is equivalent to minimizing the power required.

For a propeller-driven airplane, fuel flow is proportional to shaft power:

Maximum range:

Specific range is

Thus, for constant and , maximum range requires minimum drag. In level flight, this is equivalent to

or maximizing .

Maximum endurance:

Specific endurance is

It is therefore maximized at minimum power required, corresponding to

The maximum-endurance speed is lower than the maximum-range speed because minimum power occurs at a higher lift coefficient than minimum drag.

In practical operation, the optimum indicated airspeed decreases as weight decreases, so the airplane must gradually slow down to remain at the optimum lift coefficient.

For maximum range of a propeller-driven airplane, maximize

Differentiating with respect to and setting the result to zero gives

Therefore,

For maximum endurance, maximize

Differentiation gives

Hence,

At the same weight and altitude,

Thus,

or

Therefore, a propeller-driven airplane achieves maximum endurance at a lower speed than maximum range.

The standard propeller Breguet equations normally assume:

• Steady, level, unaccelerated flight, so and .
• Constant or representative propeller efficiency .
• Constant power-specific fuel consumption .
• Constant optimum aerodynamic condition, such as fixed or fixed .
• No fuel loss other than that represented by the engine fuel-flow model.
• Atmospheric properties remain constant when a constant-altitude expression is used.
• Climb, descent, reserve fuel, maneuvering, and takeoff fuel are neglected.
• Wind is absent when calculating air range.

In actual flight, and vary with power setting, altitude, propeller speed, and airspeed. Compressibility, changing Reynolds number, non-parabolic drag, and operational constraints also alter the optimum condition. Consequently, the Breguet equations are best used for preliminary estimates or individual cruise segments; detailed mission analysis generally requires numerical integration with realistic engine and aerodynamic data.

A jet engine is modeled as a thrust-producing engine. Its fuel-weight flow is approximately proportional to thrust:

where is thrust-specific fuel consumption. In steady, level flight, , so

The important consequences are:

• Endurance depends on the time obtained per unit fuel and is improved by minimizing thrust required, or drag.
• Range depends on distance per unit fuel and therefore includes airspeed:

• High , low , and a large fuel-weight ratio improve performance.
• At a fixed altitude, maximizing jet range requires a compromise between high speed and low drag.
• As fuel is consumed, weight decreases. To remain at a fixed optimum lift coefficient, the airplane must reduce speed, climb to lower density, or use a combination of both.
• Altitude strongly influences true airspeed and actual engine fuel consumption.

For steady, level flight,

If is thrust-specific fuel consumption, then

The specific endurance is

Using

this becomes

The specific air range is

or

Therefore:

• Jet endurance is improved by maximizing .
• Jet range is improved by maximizing for the applicable flight constraint.
• At fixed altitude, depends on , and the aerodynamic range factor becomes .

For a jet airplane, fuel-weight flow is

During steady, level flight, , and

Therefore,

Rearranging,

Assuming and remain constant, integration from to gives

Hence,

The equation shows that jet endurance increases with:

• A larger lift-to-drag ratio
• Lower thrust-specific fuel consumption
• A larger initial-to-final weight ratio

For constant , maximum jet endurance occurs at , which is the minimum-drag condition.

For a jet airplane in steady, level flight,

Since ,

Using

we obtain

If , , and are constant during the cruise segment, integration gives

Thus,

This equation is often associated with an ideal cruise-climb strategy because maintaining constant speed and lift coefficient while weight decreases generally requires a reduction in air density. Range improves through high speed, high , low , and a high fuel-weight ratio, although speed cannot be increased independently without changing drag and engine performance.

For a jet airplane,

At constant altitude and lift coefficient,

and

Therefore,

Substitution gives

Integrating from to ,

Hence,

For specified weights, altitude, and wing area, maximum constant-altitude jet range requires maximizing

The airplane must reduce speed as weight decreases to maintain the selected constant .

Maximum endurance:

Jet fuel flow is proportional to thrust, and in level flight . Therefore, endurance is greatest at minimum drag:

Maximum range at fixed altitude:

Specific range is

Since and ,

Thus, maximum fixed-altitude range requires

This occurs at a lower and higher speed than the maximum-endurance condition.

For the classical constant-speed Breguet equation, range is proportional to . If is prescribed, maximizing maximizes range. In realistic compressible cruise, the relevant criterion is often expressed as maximizing while accounting for variation in thrust-specific fuel consumption.

For maximum jet endurance, maximize

Differentiation gives

Thus,

For maximum fixed-altitude jet range, maximize

Taking the derivative and equating it to zero gives

Therefore,

and

At the same weight and altitude, , so

Hence,

A jet airplane therefore achieves maximum fixed-altitude range at a higher speed than maximum endurance.

The difference arises from the fuel-flow model:

• For a propeller airplane, fuel flow is proportional to power, .
• For a jet airplane, fuel flow is proportional to thrust, or drag .

The principal optimum conditions are:

For a parabolic drag polar:

• Propeller maximum range:
• Propeller maximum endurance:
• Jet maximum range:
• Jet maximum endurance:

Thus, a propeller airplane flies slower for maximum endurance than for maximum range. A jet also flies slower for maximum endurance than for maximum fixed-altitude range, but the aerodynamic criteria are interchanged because of the different dependence of fuel flow on power and thrust.

At an optimum aerodynamic condition, is fixed. The required true airspeed is

Therefore:

• Increasing weight increases optimum speed as .
• As fuel is consumed, the optimum speed decreases as at constant altitude.
• At higher altitude, density decreases, so the same optimum requires a higher true airspeed.

Under ideal Breguet assumptions:

• Propeller range depends primarily on , , , and .
• Propeller endurance at constant altitude contains a factor and terms involving .
• Jet endurance depends on , , and .
• Constant-altitude jet range contains and .

Higher altitude can increase jet true airspeed and ideal range, but actual benefits depend on engine fuel consumption, available thrust or power, compressibility effects, wind, and operational limits.

Let be true airspeed and let be the wind component along the flight path, taken as positive for a tailwind. Ground speed is

Thus:

• With a tailwind, , so ground speed and ground distance covered per unit time increase.
• With a headwind, , so ground speed and ground distance covered per unit time decrease.
• If the aircraft maintains the same airspeed, altitude, and engine condition, its fuel flow and ideal time endurance are essentially unchanged by steady wind.
• Ground range increases with tailwind and decreases with headwind.
• For a fixed destination, a headwind increases flight time and fuel required, while a tailwind decreases them.
• If the airplane is allowed to change speed for best ground range, the optimum speed is generally increased in a headwind and reduced in a tailwind.

The aircraft must have to make forward progress over the ground.

Let the wind component be positive for a tailwind and negative for a headwind. Then

For a propeller-driven airplane,

Hence, its instantaneous specific ground range is

For a jet airplane,

Therefore,

Consequences include:

• When , the expressions reduce to the still-air specific-range relations.
• A tailwind increases specific ground range because .
• A headwind decreases specific ground range because .
• The wind correction differs between propeller and jet airplanes because propeller fuel flow contains the additional airspeed factor through power required.

Let be positive for a tailwind and negative for a headwind, with ground speed .

For a propeller-driven airplane, maximize

assuming and are constant. Taking a logarithmic derivative,

Thus, the optimum satisfies

For a jet airplane, maximize

which gives

or

Interpretation:

• In a headwind, , the airplane should generally fly faster than its still-air maximum-range speed to reduce time exposed to the adverse wind.
• In a tailwind, , the optimum speed is generally lower because the favorable wind supplies part of the ground speed.
• The actual speed must remain within stall, structural, engine, compressibility, and operational limits.
• These relations assume a steady wind and neglect changes in fuel-consumption parameters with speed.