Constraints on aircraft climb performance are also specified in the federal air regulations.
These include a minimum landing climb gradient with all engines running, and minimum
climb gradients with one engine inoperative during three take-off segments, an approach
segment, and an enroute case.
These regulations are discussed in the section of FAR Part 25 included in these notes.
They are summarized in the table below:
Required Climb Gradient
|Number of Engines:
|First Take-Off Segment
|Second Take-Off Segment
|Final Take-Off Segment
The flight conditions are as follows:
First Take-Off Segment is with the critical engine inoperative, take-off thrust,
landing gear extended, flaps in take-off position, V = Vlo, and weight
that exists at the time gear retraction is started (essentially the take-off weight).
Second Take-Off Segment is similar to first segment climb except that gear
is up, V = 1.2 Vs, and the altitude is 400 feet above the ground.
Final Take-Off Segment also has one engine inoperative, but the others are
operating at maximum continuous thrust rather than at take-off thrust. The altitude
is that achieved when transition to enroute configuration is accomplished (flaps,
slats, gear up) or 1500 feet (whichever is higher). Speed is 1.25 Vs at
the weight at the end of the take-off segment.
Enroute Climb also requires one engine out, although there are requirements
for two engine-out performance of 3 and 4 engine aircraft. One may choose a favorable
speed, and an altitude that is sufficiently high to clear obstacles.
Approach Segment is again with one engine out and take-off thrust. Gear is
up. Flaps are retracted a bit to increase stall speed by 10% above the stall speed
with landing flap deflection. With this flap setting the airplane is flown at V =
1.5 Vs at the landing weight.
Landing Segment is the only case with all engines operating. Gear is extended,
flaps in landing position, V = 1.3 Vs and thrust that is available 8 secs.
after the throttle is moved from idle to take-off thrust position.
The second segment climb and, for two engine aircraft, the enroute climb are often
the critical design requirements affecting the required engine thrust and wing aspect
Detailed requirements for climb are described in FAR 25.115.
Estimating the Climb Gradient
Climb performance is specified in terms of the climb gradient, the ratio of climb
rate to forward speed. For small angles of climb, the climb gradient and the flight
path angle are essentially the same:
If the speed V is constant, the rate of change of potential energy must be equal
to the product of V and the net force in the direction of motion:
W V sin g = (T - D) V or: g ª
(T - D) / W
When the aircraft is flown at a fixed Mach number or equivalent airspeed, the true
airspeed changes. In this case, the total energy change is:
W/g V dV/dt + W V sin g = (T - D) V
so, g ª (T - D) / W - 1/g dV/dt
and, dV/dt = dV/dh dh/dt = dV/dh V sin g
After some algebra:
g ª [ (T - D) / W ] / (1 + V/g dV/dh )
The value of dV/dh depends on the type of operation as shown below:
||V/g dV/dh (approx.)
||Above 36,089 ft
||Below 36,089 ft
||Above 36,089 ft
||Below 36,089 ft
Engine-out Climb Performance
In computing FAR 25 climb performance, the effects of one engine inoperative must
include not only a decrease in thrust, but an increase in drag due to:
1) windmilling drag of inoperative engine or windmilling or feathered drag of propeller.
Modern propellers on larger aircraft would always be equipped with automatic feathering
2) rudder and aileron drag associated with counteracting asymmetric thrust.
At low speeds, the windmilling drag of a high bypass ratio turbofan may be estimated
empirically by the expression:
Dwindmill = .0044 p Ac
where: p is the ambient static pressure, and Ac is the inlet area.
The second component of the drag increment may be estimated by computing the induced
drag of the vertical tail when it is carrying the lift needed to trim the asymmetric
yawing moment due to the failed engine:
Lv is the trimming load on the vertical tail
hv is the vertical tail height
yengine is the distance from fuselage centerline to critical engine
T is the take-off thrust for the critical engine
lv is the vertical tail length (distance from c.g. to vertical tail a.c.)
The total drag increment is the sum of the windmilling term and the trim drag.
These climb gradients are determined for all applicable weights, altitudes, and temperatures.
From this data, the maximum permissible weight for a given condition are established.
Normal climb to cruise altitude is carried out at the speed for best overall economy
(high speed climb) which is considerably faster than the speed for maximum rate of
climb, which, in turn, is much faster than the speed for maximum climb gradient.
If fuel quantity is limiting, climb may be performed at the speed for best fuel economy
(long range climb speed), a speed between the best overall economy climb speed and
the best gradient climb speed. Speed schedules are selected to be easily followed
by the pilot with available instrumentation. Recent introduction of automatic flight
directors, makes this task easier. The computed climb rates are integrated to produce
time, fuel, and distance to climb to any altitude.
For approximate calculations, the additional fuel to climb to altitude (as compared
with cruising the same distance at the cruise altitude) can be approximated by adding
an increment to the total cruise fuel. This increment has been determined for a wide
range of weights for the DC-9-30, the DC-8-62, and the DC-10-10. The results, expressed
as a percentage of take-off weight are summarized in the following figure.
For different aircraft such as SST's we might think more fundamentally about the
cause of this fuel increment. With a rough estimate of the overall propulsion efficiency,
we can express the extra fuel used in terms of the change in kinetic and potential
energy. The net result, expressed as a percentage of take-off weight, is:
Wclimb_fuel_inc / Wto (%) = h(kft) / 31.6 + [V(kts) / 844]2
This agrees with the plot above, indicating a 1.3% increment for flight at M =
.8 and 30,000 ft, while for an SST that climbs to 60,000 ft and Mach 2.4, the increment
is over 4.5%.