Climb Performance

Introduction

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:	4	3	2
Flight Condition:
First Take-Off Segment	0.5%	0.3%	0.0%
Second Take-Off Segment	3.0%	2.7%	2.4%
Final Take-Off Segment	1.7%	1.5%	1.2%
Enroute Climb	1.6%	1.4%	1.1%
Approach Segment	2.7%	2.4%	2.1%
Landing Segment	3.2%	3.2%	3.2%

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 = V_lo, 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 V_s, 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 V_s 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 V_s at the landing weight.

Landing Segment is the only case with all engines operating. Gear is extended, flaps in landing position, V = 1.3 V_s 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 ratio.


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:

Climb Operation	Altitude	V/g dV/dh (approx.)
Constant Vtrue	All	0
Constant Vequiv	Above 36,089 ft	0.7 M2
Constant Vequiv	Below 36,089 ft	0.567 M2
Constant M	Above 36,089 ft	0
Constant M	Below 36,089 ft	-0.133 M2

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 provisions.

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:
D_windmill = .0044 p A_c
where: p is the ambient static pressure, and A_c 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:

where:
L_v is the trimming load on the vertical tail
h_v is the vertical tail height
y_engine is the distance from fuselage centerline to critical engine
T is the take-off thrust for the critical engine
l_v 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.

Operational Climb

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:
W_{climb_fuel_inc} / W_to (%) = 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%.