Pilot Training - Theory of Flight: Performance

102. Definition.-The term "performance" is broad enough to include all the characteristics of an airplane in flight. Ordinarily, however, the term is used in a restricted sense to include only those flight qualities directly measurable quantitatively such as speed, rate of climb, etc. Those qualities such as stability, controllability, etc. which are not readily measured in figures are not included in performance as thus defined. The performance characteristics are summarized as follows:

a. Maximum speed in level flight at any altitude.

b. Rate of climb at any altitude.

c. Time of climb to various altitudes.

d. Ceiling, the maximum altitude attainable.

e. Endurance, the maximum duration of flight.

f. Range, the maximum flight distance.

g. Landing speed.

h. Length of roll on landing.

i. Length of roll on take-off.

103. Work and power.-Where the action of a force on a body produces motion, the amount of work done is the product of the force times the distance through which the body moves along the line of action of the force.

Work (ft. lbs.) =  Force (lbs.) x distance (feet)

Power is the rate at which work is done.

Power (ft. lbs./sec.) = Work (ft. lbs.) / Time (sec.)

The unit of power commonly employed is the horsepower.

1 HP=550 ft. lbs./sec.

104. Performance calculations.-In making calculations of performance, many simplifying assumptions are customarily made to reduce the tremendous labor involved in making exact calculations. These assumptions are justified on the ground that the errors involved are so small as to warrant neglecting them. The assumptions made are

a. The motion in flight is unaccelerated.

b. The thrust of the propeller acts in a horizontal direction and is equal to the total drag of the airplane.

c. The lift of the wings is constant and equal to the total weight of the airplane.

d. The lift forces on the fuselage and tail assembly are so small that they may be neglected.

In the further discussion of performance, the foregoing assumptions will be considered in effect without further reference to them.

105. Power required at sea level.-By definition,

The power required for level flight, P, is the sum of the power required to pull the wings through the air plus the power required to overcome the parasite resistance.

For any assumed value of V, the corresponding wing drag and parasite drag may be determined and the power required calculated from equation 59. The power required may be calculated for a number of different airspeeds and a curve plotted of power required against airspeed. Since we are accustomed to think in mile per hour units rather than in feet per second units, the curve is usually plotted against airspeed in miles per hour (fig. 83).

In determining the wing drag, wind tunnel test data on the airfoil section concerned must be available, and since these data are plotted against C_L it is necessary to determine C_L, at the speed of flight from the characteristics of the airplane.

Example: Monoplane wing cellule

W = 6,000 lbs.

S=300 sq. ft.

b=45 ft.

AR = b²/S - 6.7 5

A_e= (at high speed) = 5 sq. ft.

Airfoil section-N. A. C. A. 2218-09

Required: C_L at 140 miles per hour at sea level

Solution: 140 m. p. h. = 205 ft./sec.

Required: Power required by the wings at 140 in. p. h. at sea level. Solution: From the aerodynamic data on the profile section N. A. C. A. 2218-09 (fig. 46) when C_L=0.398,

Required: Power required by the parasite resistance at 140 m. p. h. at sea level.

Solution:

Required: HPr, of the complete airplane at 140 m. p. h. at sea level. Solution:

106. Power required at altitude.-For a given angle of attack of the wing, C_L, and C_D are constant regardless of the altitude. For level flight at any altitude at a given angle of attack.

The airplane must fly faster at altitude to support itself at a given angle of attack because the density of the air is less.

For a given angle of attack, , increases in the same ratio thatis reduced, so that the drag at altitude is the same as at sea level.

The power required at altitude is proportional to the product D V and will increase with altitude for a given angle of attack.

Example: Find the power required at 10,000 feet for the airplane of

60 at C_L = 0.398

Solution:

The curves of horsepower required in figure 84 are obtained by the tabulation and calculation of a large number of points of HP required for different airspeeds and altitudes. The general shape of the power required curve is similar for all airplanes.

107. Power available at sea level.-The power output at full throttle of an engine at sea level is nearly proportional to the engine speed. With a fixed pitch propeller, the engine speed is less and its power output correspondingly less at climbing speeds than in level flight at full speed. With controllable pitch propellers the engine speed may be kept constant, and its power output constant throughout the flight range.

The power available is equal to the engine power multiplied by the propeller efficiency. The propeller efficiency is a function of several variables, and may best be determined by the use of graphical curves for the type of metal propellers now in service (fig. 85).

Cp is defined as the power coefficient of the propeller and is expressed by the equation

Required: Propeller efficiency at 140 in. p. h.

Solution: 140 m. p. h.=205 ft./sec.

From the chart of propeller characteristics figure 85.

Required

:

for other assumed airspeeds, Pa may be computed and the curve plotted for the entire range of flight speeds (fig. 83).

108. Power available at altitude.-At altitude the indicated power output of the airplane engine at a given r.p.m. decreases in direct proportion to the decrease in density ratio, unless by supercharging the power output is boosted. With fixed pitch propellers, the unsuperchargred engine loses speed with increase in altitude, the net result being further decrease in power output due to decreased r. p.m.

Example: Supercharged engine. critical altitude 10,000 feet.

Required: Pa at 10,000 feet at 140 m. p. h.

Solution: From formula 80, Cp at sea level = 0.052 =Cp0.

For a constant speed propeller at V = 140 m. p. h.

For other assumed airspeeds, Pa may be computed and the curve plotted at altitude in the same manner as at sea level.

109. Maximum speed in level flight.-The curves of Pr and Pa for any particular altitude of flight intersect at two points which define the minimum and maximum speeds respectively in level flight. The maximum speed is the characteristic which is of prime importance in performance. For airplanes with unsupercharged engines, the maximum speed in level flight decreases with altitude. For airplanes with supercharged engines, the maximum speed in

level flight increases with altitude up to the critical altitude of the supercharger, after which the maximum speed decreases with further increase in altitude.

110. Rate of climb.-The excess of horsepower available over horsepower required at any speed determines the rate of climb at that speed. If climb is not desired, the power available is reduced by retarding the throttle until a balance is reached between the power available and the power required. Where excess power is available, the rate of climb is determined from the equation,

The airspeed of best climb at any given altitude is that airspeed at which the greatest amount of excess power is available. This may be determined from inspection of the graph showing the Pa and Pr curves. (See fig. 83.)

Example: At 140 miles per hour at sea level                                                                                          (87)

The rate of climb curve when plotted against altitude is a straight line for unsupercharged engines (fig. 86). It is also a straight line for supercharged engines if the climb is made at constant throttle setting. For climbs made at constant manifold pressure to the critical altitude of the supercharger, and above the critical altitude of the supercharger at constant throttle, the rate of climb will be a straight line above the critical altitude and a curve below the critical altitude (fig. 87).

111. Time of climb.-The time of climb is a performance characteristic that is usually listed as the time of climb to some definite altitude, such as time of climb to 15,000 feet. On the performance chart of the airplane the entire curve showing time of climb to all altitudes below the ceiling may be shown.

112. Ceiling.-The absolute ceiling is the limiting altitude which the airplane may attain. At this altitude, no excess power remains available for climb and the rate of climb becomes zero. The altitude at which the rate of climb is 100 ft./min. is arbitrarily defined as the service ceiling.

113. Endurance.-a. The endurance of an airplane may be defined as the time it can remain in the air without refueling. It depends on the fuel capacity and the fuel consumption. The installed fuel capacity of a given airplane is fixed by the size of the gas tanks, and the load carrying capacity of the airplane. The fuel consumption will vary according to the operating speed and altitude. In Air Corps specifications it is customary to specify endurance at full throttle at a definite altitude or at a definite operating power output of the engine.

b. Where maximum endurance is required, it is necessary to determine the airspeed for minimum fuel consumption for a given gross weight. It may best be determined by actual flight test. A curve may be plotted of fuel consumption against airspeed and the airspeed for minimum fuel consumption picked by inspection from the curve (fig. 88).

c. As fuel is consumed, the gross weight of the airplane decreases and the airspeed for most economical fuel consumption drops very nearly along a straight line. This curve may be obtained from tests for fuel consumption at reduced loads corresponding to weights at half-fuel load and no-fuel load.

d. The maximum endurance will then be obtained by flying the airplane at a gradually decreased airspeed as the weight decreases. In general, the maximum endurance will be obtained by flying at speeds of 15 percent to 20 percent above the stalling speed.

114. Range.-a. The range of an airplane is the distance it can fly without refueling. It depends on

(1) Fuel capacity.

(2) Fuel consumption.

(3) Velocity and direction of the wind.

b. In Air Corps specifications it is customary to prescribe range under definite conditions of altitude and engine operation.

At operating speed, range =  operating speed x endurance at operating speed.

c. Maximum range is a complex function of the airplane, engine, and propeller characteristics. It may be determined by a flight test procedure similar to that in determining endurance. As in the determination of endurance, the fuel consumption test should be repeated at half fuel load and no-fuel load. The most economical speed of flight at any condition of fuel load is determined graphically from the fuel consumption curve for any condition of wind. (See fig. 88.)

(1) Lay off the wind to scale on the X axis, positive if a head wind, and negative if a tail wind.

(2) From the end of the wind vector draw a line tangent to the fuel consumption curve for the immediate weight of the airplane.

(3) The point of tangency is the airspeed for maximum range.

d. The airspeed of flight for maximum range should correspond to the most economical speed for the immediate weight of the airplane. In general, the operating speeds for maximum range occur in the vicinity of 40 percent above the stalling speed. Head winds increase and tail winds decrease the operating speed for maximum range.

115. Landing characteristics.-a. Landing speed is customarily considered to be the stalling speed at sea level, and may be obtained from solution of the formula

The term W/S is the wing loading and is measured in pounds per square foot. In order to obtain low landing speeds, low wing loading and high maximum lift coefficients are required. These requirements

lead to large wings and otherwise undesirable wing profiles, so that the tendency is always to design for as high a landing speed as the skill of the pilot and considerations of wear and tear on materiel will permit.

b. The length of roll upon landing is a function of the landing speed and the rate of deceleration. Braking systems have been developed to a high degree of effectiveness and the development of the tricycle landing gear promises to permit the use of large decelerating forces from the braking system immediately upon contact with the ground. The limiting factor on the size of modern airports at the present time is not the landing speed and length of roll of the modern airplane, but rather the take-off characteristics.

116. Take-off characteristics.-The standard Air Corps performance test for take-off is based on the assumption that take-off speed occurs at 90 percent C_Lmax and that this speed is maintained to clear obstacles at the start of the climb. The distance required to take off is a function of

a. Wind.

b. Take-off speed (90% C_Lmax).

c. Ground friction.

d. Power loading (W/P).

e. Accelerating force (propeller thrust).

Power loading is defined as the gross weight divided by the rated engine horsepower and is measured in pounds per horsepower. The higher the power loading the more sluggish the take-off.

The propeller thrust is the accelerating force at take-off. The fixed pitch propeller is a poor device for securing adequate thrust during the initial stages of the take-off, and the modern heavily loaded aircraft of large size is dependent on the controllable pitch propeller for satisfactory take-off characteristics.

117. Factors affecting performance.-a. The chief factors affecting performance may be listed as follows:

(1) The aerodynamic characteristics of the, wings, the wing profile, aspect ratio, and wing arrangement.

(2) The wing loading (lbs. per sq. ft.).

(3) The power loading (lbs. per horsepower).

(4) The fineness, or the ratio S/A_e.

(5) The variation of engine power with altitude.

(6) Propeller efficiency.

(7) Fuel consumption.

b. The reduction of wing drag to a minimum is accomplished by selecting a suitable section profile. The choice as between monoplane and biplane has been definitely in favor of the cantilever monoplane for all modern high performance aircraft. The reduction of induced drag through increase in aspect ratio is of greatest importance in heavily loaded airplanes or airplanes designed to operate near their ceiling. It is less important in high speed airplanes of the racer type. The effects of variation in wing loading are

(1) Low wing loading leads to superiority in climb and ceiling and permits low landing speed.

(2) High wing loading leads to superiority in maximum speed.

c. The effect of high power loading is disadvantageous to the performance characteristics of climb, speed, ceiling, and take-off. High power loading means poor performance.

d. The effects of parasite resistance are most disadvantageous at high speed. It is of greatest importance to keep as much of the parasite resistance as possible out of the slipstream. Airplanes designed for maximum performance in climb or at low cruising speeds can afford to compromise with parasite resistance to achieve other ends, but parasite resistance is a major limiting factor to high speed performance.

e. The variation of engine power with altitude leads to reduced performance at altitude. The use of the supercharger leads to increased weight and consequent slight sacrifice in performance at sealevel, but to greatly increased climb, speed, and ceiling at altitudes above sea level.

J. Increased propeller efficiency leads to more power available for a given power plant and a corresponding all around increase in performance. Since a propeller can only be designed to give its maximum efficiency for a particular condition of flight, it is important that the propeller design be such as to make available the maximum efficiency at the place where it will be needed most. By permitting the engine to operate under more favorable conditions, controllable pitch propellers have greatly increased airplane performance at altitude and in climb at sea level.

g. Decrease in specific fuel consumption increases both endurance and range. Fuel economy is of paramount importance in airplanes designed for maximum possible range and endurance, and relatively less important in aircraft designed for short range operation only.