*Power Units*. Power is the rate of doing work. If a force of 10 pounds
is applied to a body moving at the rate of 300 feet per minute, the
power will be expressed by 10 x 300 = 3000 foot-pounds per minute. As
the figures obtained by the foot and pound units are usually
inconveniently large, the "Horsepower" unit has been adopted. A
horsepower is a unit that represents work done at the rate of 33,000
foot-pounds per minute, or 550 foot-pounds per second. Thus if a certain
aeroplane offers a resistance of 200 pounds, and flies at the rate of
6000 feet per minute, then the work done per minute will be equal to 200
x 6000 = 1,200,000 foot-pounds. Since there are 33,000 foot-pounds of
work per minute for each horsepower, the horsepower will be:
1200000/33000 = 36.3.
As aeroplane speeds are usually given in terms of miles per hour, it
will be convenient to convert the foot-minute unit into the mile per
hour unit. If H = horsepower, R = resistance of aeroplane, and V = miles
per hour, then H = RV/375, the theoretical horsepower, without loss. If
an aeroplane flies at 100 miles per hour and requires a propeller thrust
of 300 pounds, then the horsepower becomes:
H = RV/375 = 300 x 100/375 = 80 horsepower. This is the actual power
required to drive the machine, but is not the engine power, as the
engine must also supply the losses due to the propeller. The propeller
losses are generally expressed as a percentage of the total power
supplied. The percentage of useful power is known as the "Efficiency."
The efficiency of the average aeroplane propeller will vary from 0.70 to
0.80. If e = propeller efficiency expressed as a decimal, the motor
horsepower becomes: H = RV/375e. To obtain the motor horsepower, divide
the theoretical horsepower by the efficiency. Using the complete formula
for the solution of an example in which the flight speed is 100 M. P.
H., the resistance 225 pounds and the efficiency 0.75, we have:
H = RV/375e = 225 x 100/375 x 0.75 = 80 horsepower.
*Power Distribution*. Since power depends upon the total resistance to
be overcome, part of the power will be used for driving the lifting
surfaces and a part for overcoming the parasitic resistance. The power
required for driving the wings depends upon the angle of incidence,
since the drag varies with every angle. The wing power varies with every
flight speed, owing to the changes in angle made necessary to support
the constant load. The power for the wings will be least at the speed
and angle that corresponds to the greatest lift-drag ratio. Owing to the
low value of the L/D at very small and very large angles, the power
requirements will be excessive at extremely low and high speeds.
As the parasitic resistance increases as the square of the speed, the
power for overcoming this resistance will vary as the cube of the speed.
It is the parasitic resistance that really limits the higher speeds of
the aeroplane, since it increases very rapidly at velocities of over 60
miles per hour.
The total power at any speed is the sum of the wing power and power
required for the parasitic resistance. Owing to variations in the wing
drag and resistance at every point within the flight range, it is
exceedingly difficult to directly calculate the total power at any
particular speed. The wing drag and the resistance should be calculated
for every speed, and then laid out by a graph or curve. The minimum
propeller thrust, or the minimum total resistance, occurs approximately
at the speed where the body resistance and wing drag are equal. The
minimum horsepower occurs at a low speed, but not the lowest speed, and
this will differ with every machine.
[Illustration: Fig. 1. Power Chart of Bleriot Monoplane, With Outline of
Wing Section. The Results Were Taken From Full Size Tests Made by the
English Government.]
Fig. 1 is a set of performance curves drawn from the results of tests on
a full size Bleriot monoplane. At the bottom the horizontal row of
figures gives the horizontal speed in feet per second. The first column
to the left is the horsepower, and the second column is the resistance
or drag in pounds. The four curves represent respectively the body
resistance, wing or "plane" resistance, horsepower, and total
resistance. The horizontal line "AV" shows the available horsepower. It
will be noted that the body resistance increases steadily from 9 pounds
at 50 feet per second, to 180 pounds at 100 feet per second. The wing
resistance, on the other hand, decreases from 350 pounds at 56 feet per
second to a minimum of 130 pounds at 83 feet per second. It will be
noted that the angles of incidence are marked along the wing-drag curve
by small circles. The incidence is 6° at 75 feet per second, and 4° at a
little less than 85 feet per second.
The available horsepower "AV" is 42. This is shown as a straight line,
although in the majority of cases it is slightly curved owing to
variations in power at the higher speeds, and to variations in the
propeller efficiency. At 90 feet per second the actual horsepower curve
crosses the line of available horsepower "AV." Beyond this point
horizontal flight is no longer possible, as the power requirements would
exceed the available horsepower. It will be noted that the lowest total
resistance occurs near the point where the body and wing resistance
curves intersect, or in other words, where the body and wing resistance
are equal. The minimum horsepower takes place at 63 feet per second, or
at a point nearly 1/3 between the lowest flight speed and the highest
speed attained by the available horsepower in horizontal flight (90
ft/sec).
The actual range of flight speeds is limited to points between the
intersection of the "Horsepower required" curve, and the "Available
horsepower" curve. By increasing the propeller efficiency, or by
increasing the power of the motor, the available horsepower line is
raised and the flight range increased.
*Horsepower For Climbing*. Up to the present we have only considered
horizontal flight. The power available for climbing is the difference
between the power required to maintain horizontal flight at any speed,
and the actual horsepower that can be delivered by the propeller. Thus,
if the actual power delivered by a motor through the propeller is 85
horsepower, and the power required for horizontal flight at that speed
is 45, then we have: 85–45= 40 horsepower available for climbing. Since
the difference between the driving power and the power required for
horizontal flight is less at extremely low and high speeds, it is
evident that we will have a minimum climbing reserve at the high and low
speeds. Consulting the power curve for the Bleriot monoplane, we see
that the power required at 56 feet per second is 40 horsepower, and at
85 feet per second it is 38 horsepower. At the low speed we have a
climbing reserve of 44–40 = 4 H. P., and at the higher speed 44–38 = 6
H. P. The maximum available horsepower "AV" is 44 horsepower. The
minimum horizontal power required is found at 63 feet per second, the
climbing reserve at this point being 44–28 = 16 horsepower. At 55 feet
per second, and at 90, we would not be able to climb, as we would only
have sufficient power to maintain horizontal flight.
If W = total weight of aeroplane, c = climbing speed, and H = horsepower
reserve for climbing, then the climbing speed with a constant air
density will be expressed by: c = 33000H/W. Assuming that the weight of
the Bleriot monoplane is 800 pounds, and that we are to climb at the
speed of the greatest power reserve (16 horsepower), our rate of climb
is:
c = 33000H/w = 33000 x 16/800 = 660 feet per minute. It should be
understood that this is the velocity at the beginning of the climb.
After prolonged climbing the rate falls off because of diminishing power
and increasing speed. Much depends upon the engine performance at the
higher altitudes, so that the reserve power for climb usually diminishes
as the machine rises, and hence the rate of climb diminishes in
proportion.
The following table taken from actual flying tests will show how the
rate of climb decreases with the altitude. These machines were equipped
with 150 H. P. Hispano-Suisa motors. It will be noted that the S. P. A.
D. and the Bleriot hold their rate of climb constant up to 7800 feet
altitude, which is a feat that is undoubtedly performed by varying the
compression of the engine.
Besides increasing the power, the rate of climb can also be increased by
decreasing the weight of the aeroplane.
*Maximum Altitude*. The maximum altitude to which a machine can ascend
is known as its "Ceiling." This again depends on both the aeroplane and
the motor, but principally on the latter. It has been noted that
machines having the greatest rate of climb also have the greatest
ceiling. Thus the ceiling of a fast climbing scout is higher than that
of a larger and slower machine. Based on this principle, a writer in
"Flight" has developed the following equation for ceiling, which, of
course, assumes a uniform decrease in density. Let H = maximum altitude,
h = the altitude at any time t after the start of the climb, and a = the
altitude after a time equal to twice the time t, then:
H = h / (2 - a/h)
Approximate values of h and a may be had
from the following table, which are the results of a test on a certain
aeroplane:
*Time* (Minutes)
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5
*Altitude* (Feet)
0.03300 6150 8730 10760 12610 14190 15530 16650 17600
If we assume that the height is 10760 feet after the first 10 minutes,
and that the altitude after twice this time (20 minutes) is 16650 feet,
then the maximum ceiling attained will be:
H = h / (2 - a/h) = 10760 / 2 - 16650/10760 = 23,770 feet. The use of
this formula requires that the climb be known for certain time intervals
before the ceiling.
*Gliding Angle*. The gliding angle of the wings alone is equal to the
lift-drag ratio at the given angle. The best or "Flattest gliding angle"
is, of course, the best lift-drag ratio of the wing—say on the average
about 1 in 16. The gliding angle of the complete machine is considerably
less than this, owing to the resistance of the body and structural
parts. This generally reduces the actual angle to less than 12, and in
most cases between 6 and 8. Expressed in terms of degrees, tan ø = R/W
where R = head resistance and W = weight in pounds.
Fig. 2 is a diagram giving the gliding force diagram. The plane descends
along the gliding path AC, making the angle of incidence (ø). When in
horizontal flight, the lift is along OL and the weight is OW. When
descending on the gliding path the lift maintains the same relation with
the wing, but the relative angle of the weight is altered. The weight
now acts along OG. The drag is represented by OD, with the propeller
thrust OP equal and opposite to it. With the weight constant, the lift
OL is decreased by the angle so that the total life = L = W cos ø. The
action of the weight W produces the propelling component OP that gives
forward velocity. The line AB is the horizontal ground line. If the
total lift-drag ratio is 8, then the gliding angle will be 1 in 8, or
measured in degrees, tan ø = R/W = 1/8 = 0.125. From a trigonometric
table it will be found that this tangent corresponds to an angle of 7° –
10’. It should be noted that R is the total resistance and not the
wing-drag.
[Illustration: Fig. 2. Gliding Angle Diagram Showing Component of
Gravity That Causes Forward Motion. The Gliding Angle Depends Upon the
Ratio of the Resistance to the Weight.]
*Complete Power Calculations*. Knowing the total weight and the desired
speed, we must determine the wing section and area before we start on
the actual power calculations. This can either be determined by
empirical rules in the case of a preliminary investigation, or by actual
calculation by means of the lift coefficients after the approximate
values are known. Sustaining a given weight, we can vary the angle,
area, wing section, or the speed, the choice of these items being
regulated principally by the power. Given a small area and a great angle
of incidence, we can support the load, but the power consumption will be
excessive because of the low value of the L/D ratio at high angles. If
small area is desired, a large value of Ky due to a high lift-wing
section is preferable to a low lift wing at high angles. In general, the
area should be so arranged that the wing is at the angle of the maximum
lift-drag ratio at the rated speed. A low angle means a smaller motor,
less fuel, and hence a lighter machine. This selection involves
considerable difficulty, and a number of wing sections and areas must be
tried by the trial and error method until the most economical
combination is discovered.
[Illustration: Graphical Gliding Diagrams of Several Aeroplanes Recorded
in British Army Contest of 1912.]
The first consideration being the total weight, we must first estimate
this from the required live load. This can be estimated from previous
examples of nearly the same type. Say that our required live load is 660
pounds, and that a live load factor of 0.30 is used. The total weight
now becomes 660/0.30=2200 pounds. To make a preliminary estimate of the
area we must find the load per square foot. An empirical formula for
biplane loading reads: w = 0.065V - 0.25 where V = maximum speed in M.
P. H., and w = load per square foot. If we assume a maximum speed of 90
M. P. H. for our machine, the unit loading is w = (0.065 x 90) - 0.25 =
5.6 pounds per square foot. The approximate area can now be found from
2200/5.6 = 393 square feet. (Call 390.) The minimum speed is about 48
per cent of the maximum, or 43 M. P. H. We can now choose one or more
wing sections that will come approximately to our requirements by the
use of the basic formula, Ky = w/V².
At high speed, Ky = 5.6/(90 x 90) = 0.000691. At low speed, Ky = 5.6/(40
x 40) = 0.003030. We must choose the most economical wing between these
limits of lift, and on reference to our wing section tables we find:
It would seem from the above that the chosen area is a little too large,
as the majority of the L/D ratios at high speed are poor, the best being
11.00 of the U.S.A.-1. The angles are small, being negative in most
cases at high speed. While the lift-drag of the R.A.F-3 is very good at
low speed, it is very poor at high, hence the area for this section
should be reduced to increase the loading. The R.A.F.-6 and the U.S.A.-1
show up the best, for they are both near the maximum lift at low speed
and have fair L/D ratios at high speed. It will be seen that for the
best results there should be a series of power curves drawn for the
various wings and areas. This method is too complicated and tedious to
take up here, and so we will use U.S.A.-1, which does not really show up
so bad at this stage. Both the R.A.F.-6 and the U.S.A.-1 have been used
extensively on machines of the size and type under consideration. While
we require Ky = 0.003030, and U.S.A-1 gives 0.003165, we will not
attempt to utilize this excess, as it will be remembered that we should
not assume the maximum lift for reasons of stability.
The wing-drag at high speed will be 2200/11.0 = 200 pounds, and at low
speed it will be: 2200/10.4 = 211 pounds. Since the maximum L/D is 17.8
at 3°, where Ky is 0.00133, the least drag will be: 2200/17.8 = 124
pounds. This least drag will occur at V = V5.6/000133 = 65 M. P. H.
The wing drag for each speed must now be divided by the correction
factor 0.85, which converts the monoplane values of drag into biplane
values. Since this is practically constant it does not affect the
relative values of Kx in comparing wings, but it should be used in final
results. For this type of machine we will take the total parasitic
resistance as r = 0.036V². At 90 M.P.H., r = 0.036 x 90 x 90 = 291.6
pounds. At 65 M. P. H., the resistance is: 0.036 x 65 x 65 = 152.1. At
the extreme low speed of 43 M. P.H. we have r - 0.036x43 x 43 = 66.56
pounds. The total resistance (R) is equal to the sum of the wing-drag
and the parasitic resistance. At 90 M. P. H. the total resistance
becomes 200 + 291.6 = 491.6 pounds. At 65 M.P.H. the total is 124 +
152.1 = 176.1, and at 43 M.P.H. it is 211 + 66.56 = 277.56 pounds. The
horsepower is computed from H = RV/375e, and at 90 M. P. H. this is : H
= 491.6 x 90/375 x 0.80 = 147.5 H. P. where 0.80 is the assumed
propeller efficiency. At 65 M. P. H. the horsepower drops to H = 176.1 x
65/375 x 0.8 = 38.1 H. P., assuming the same efficiency. In the same way
the H. P. at 43 M. P. H. r is 39.8.
A table and power chart should be worked out for a number of sections
and areas according to the following table. The calculations should be
computed at intervals of 5 M. P. H., at least the lower speeds. Wing
drag is not corrected for biplane interference:
*Weight and Power*. The weight lifted per horsepower varies in the
different types of aeroplanes, this difference lying principally in the
reserve allowed for climbing and horizontal speed. A speed scout may
carry as little as 8 pounds per horsepower, while a slow two-seater may
exceed 20 pounds per horsepower. A rough estimate of the horsepower
required may be had by dividing the total weight by the weight per
horsepower ratio for that particular type. Thus if the unit H. P.
loading is 16 pounds and the total weight is 3200, then the horsepower
will equal 3200/16 = 200 horsepower. Assuming that the live load w’ is
0.32 of the total weight W, then W = w’/0.32. If m = lbs. per H.P., then
H = W/m or H = w’/0.32m. Taking the case of a training machine where m =
20, and the live load is 640 pounds, the approximate horsepower will be:
H = w’/0.32m = 640/0.32 x 20 = 100 horsepower. A speed scout carrying
320 pounds useful load, with m = 10, will require H = 320/0.32 x 10 =
100 horsepower.