Unit 2: Performance of Level Flight, Climb & Glide - Subjective Questions

Equation along the flight path:

For an aircraft of weight , thrust , drag , velocity , and climb angle , Newton's second law along the flight path gives

For a steady climb, the velocity is constant, so . Therefore,

The vertical component of velocity is the rate of climb:

Using ,

Since power available is and power required is ,

Thus, the rate of climb is equal to the excess power divided by aircraft weight.

The graphical method uses curves of power available and power required plotted against flight velocity at a specified altitude.

• Power available is .
• Power required is .
• The vertical separation between the curves represents excess power:

The rate of climb at any velocity is

The velocity at which the vertical separation between the two curves is greatest is the best rate-of-climb speed, denoted by . Therefore,

As altitude increases, power or thrust available generally decreases, reducing the maximum excess power and hence the maximum rate of climb.

The rate of climb is expressed as

Using the parabolic drag polar,

and level-flight lift approximation , the power required may be written as

where

Hence,

For maximum rate of climb, differentiate with respect to and set the result equal to zero:

Thus,

The solution gives the speed for maximum rate of climb. For an ideal propeller aircraft with approximately constant power available, , and the condition reduces to the speed for minimum power required.

The absolute ceiling is the altitude at which the maximum possible rate of climb becomes zero.

At this altitude,

Since

it follows that

at the operating speed corresponding to the limiting condition. On a power-versus-speed graph, the power-available curve is just tangent to the power-required curve. There is no excess power for climbing.

On a plot of maximum rate of climb against altitude, the absolute ceiling is obtained by extrapolating the curve to the altitude where it intersects the line . At the absolute ceiling, the aircraft can theoretically maintain level flight at only one limiting speed but cannot climb further.

Service ceiling is the altitude at which the maximum rate of climb falls to a specified small value rather than zero. A commonly used value for many aircraft is , although the exact criterion depends on regulations and aircraft category.

Difference between the ceilings:

• At the service ceiling, equals the prescribed minimum usable climb rate.
• At the absolute ceiling, .
• The service ceiling is lower than the absolute ceiling.
• The aircraft retains a small amount of excess power at the service ceiling.
• At the absolute ceiling, power available equals the minimum power required, leaving no excess power.

Both ceilings can be found from a graph of maximum rate of climb against altitude.

The instantaneous rate of climb is

Therefore, the differential time required to climb through an altitude increment is

Integrating between altitudes and gives

For minimum time to climb, the maximum rate of climb available at each altitude is used:

If the rate of climb varies approximately linearly with altitude,

then

which gives

The time rises rapidly as the aircraft approaches its absolute ceiling because approaches zero.

The graphical method follows from

The procedure is:

• Determine the maximum rate of climb at a number of altitudes.
• Calculate the reciprocal at each altitude.
• Plot on the horizontal or vertical axis against altitude on the other axis.
• Find the area under the versus curve between and .

That area equals the climb time:

For numerical estimation, the area may be evaluated using the trapezoidal rule:

Consistent units must be used throughout the calculation.

A climb hodograph is a plot representing the horizontal and vertical components of aircraft velocity during climb.

For flight speed and climb angle :

where is horizontal velocity and is vertical velocity. Each operating condition produces a point , and the locus of these points forms the hodograph.

The diagram can be used to identify:

• Maximum rate of climb: the point having the greatest vertical coordinate .
• Maximum climb angle: the point for which the line drawn from the origin to the hodograph has the greatest slope.
• Horizontal speed during climb: read directly from the horizontal coordinate.

The slope of a line from the origin to any point is

so a tangent from the origin identifies the maximum climb-angle condition.

For a steady climb, the force equation along the flight path is

Hence,

For fixed aircraft weight, the climb angle is maximum when the excess thrust is maximum:

Therefore,

For a shallow climb, , giving

The associated speed is called the best angle-of-climb speed, . Maximum climb angle provides the greatest altitude gain per unit horizontal distance and is especially important for obstacle clearance after takeoff. It is different from maximum rate of climb, which depends on maximum excess power.

Maximum climb angle:

• Occurs at maximum excess thrust, .
• Gives the greatest altitude gain per unit horizontal distance.
• Corresponds to the best angle-of-climb speed .
• Is important for clearing obstacles.

Maximum rate of climb:

• Occurs at maximum excess power, .
• Gives the greatest altitude gain per unit time.
• Corresponds to the best rate-of-climb speed .
• Is important for reaching altitude in minimum time.

The governing relations are

and

For many conventional aircraft near sea level, , although their variation with altitude depends on the propulsion system.

The rate of climb is

Effect of weight:

• Greater weight directly increases the denominator.
• More lift is required, increasing induced drag and power required.
• Maximum rate of climb decreases.
• Best climb speeds generally increase.
• Time to climb and service-ceiling limitations become more severe.

Effect of altitude:

• Air density decreases with altitude.
• Engine thrust or power available generally falls, particularly for unsupercharged engines.
• Maximum excess power decreases.
• Maximum rate of climb progressively reduces.
• At the service ceiling it reaches a prescribed low value, and at the absolute ceiling it becomes zero.

The precise variation depends on engine type, propeller efficiency, compressibility, and atmospheric conditions.

During a steady power-off glide, thrust is zero and the aircraft descends at a constant speed along a path inclined by the glide angle below the horizontal.

Resolving weight parallel and perpendicular to the flight path gives

Dividing these equations,

or

The horizontal and vertical velocity components are

where is the rate of descent or sink. For a shallow glide, and . Thus, a high lift-to-drag ratio produces a shallow glide angle and a large horizontal distance.

For a steady glide through a vertical height and horizontal distance , the glide-path geometry gives

The force relations for a power-off glide give

Therefore,

and the glide range is

For maximum range,

This ideal expression assumes:

• Still air
• Constant aerodynamic efficiency
• A steady glide
• Negligible altitude required for maneuvering or landing

A headwind reduces ground range, while a tailwind increases it, even though the aerodynamic ratio is unchanged.

Maximum glide range occurs at the shallowest glide angle. Since

this requires maximum , or equivalently maximum .

For the parabolic drag polar,

The lift-to-drag ratio is

Differentiating with respect to and setting the derivative equal to zero gives

Thus, at maximum range,

At this condition, parasite drag equals induced drag. The maximum lift-to-drag ratio is

The minimum rate of sink is the lowest possible vertical descent speed in a steady power-off glide. It allows the aircraft to remain airborne for the maximum time.

For a shallow glide,

Since ,

Therefore, minimum sink occurs when the power required is minimum:

In coefficient form, minimum sink is obtained by maximizing

For the parabolic drag polar , differentiation gives

Hence, the minimum-sink condition occurs at a higher lift coefficient and lower airspeed than the maximum-range glide condition.

Speed for maximum glide range:

• Maximizes .
• Produces the shallowest glide angle.
• Maximizes horizontal distance per unit altitude lost.
• For a parabolic polar, .

Speed for minimum sink:

• Maximizes .
• Minimizes vertical descent speed.
• Maximizes time airborne from a given altitude.
• For a parabolic polar, .

Since flight speed varies inversely with ,

Thus, minimum-sink speed is approximately of the maximum-range glide speed under the ideal parabolic-polar assumptions.

The shallowest glide angle is the smallest angle between the descending flight path and the horizontal. It produces the maximum horizontal distance for a given loss of altitude.

For a steady power-off glide,

Therefore, is minimum when is maximum:

For a parabolic drag polar,

maximum occurs when

Hence,

The shallowest glide angle and maximum-range glide consequently occur at the same aerodynamic condition in still air.

Wind changes the aircraft's velocity relative to the ground but does not directly change its aerodynamic lift-to-drag ratio.

• In still air, maximum range is obtained by flying at the speed for .
• A headwind reduces ground speed and therefore reduces ground distance covered before a given altitude is lost. The pilot generally flies faster than the still-air best-glide speed.
• A tailwind increases ground speed and ground range. A somewhat lower speed than the still-air best-glide speed may improve ground range, subject to stall margin and operating limitations.

If is positive for a tailwind, the ground-range ratio is proportional to

Thus, maximum air-distance performance and maximum ground-distance performance need not occur at the same indicated airspeed when wind is present.

The drag polar is

Drag is

For a shallow climb or glide, , so

Substitution gives

Power required is , hence

The first term is parasite power and increases with ; the second is induced power and decreases with .

This relation is important because:

• Maximum rate of climb depends on maximum excess power .
• Minimum sink in a glide occurs at minimum .
• Time to climb depends on the rate of climb produced by excess power at each altitude.

The key climb-performance relations are:

and

Their implications are:

• Climb angle is governed by excess thrust, .
• Rate of climb is governed by excess power, .
• Maximum climb angle occurs where excess thrust is greatest.
• Maximum rate of climb occurs where excess power is greatest.
• As altitude increases, available thrust or power usually decreases, reducing climb performance.
• At the absolute ceiling, maximum excess power is zero and therefore .
• The climb time between two altitudes is

Consequently, reduced excess power increases climb time, and the theoretical time required to reach the absolute ceiling becomes unbounded as approaches zero.