K.Haroon Site Admin
Joined: 11 Dec 2005 Posts: 1311

Posted: Thu Apr 05, 2012 6:01 pm Post subject: Climb Performance 


Forces in a Climb
The balance of forces in a steady climb show thrust is acting upwards and an element of weight is adding to the drag
As the thrust assists the lift, the lift required is less than in level flight. Verify mathematically by the formula Lift = W.cos gamma
For a steady speed to be maintained the thrust and the two retarding effects of aerodynamic drag and the weight element must be equal.
If Thrust = T, Drag = D and Weight = W, then as a formula it can be written as:
T = D + W sin gamma
or
Sin gamma = T  D / W
It means that climb angle (Sin gamma) depends on the excess thrust (i.e. thrust less drag) and the weight.
Tan gamma = Opposite / Adjacent
Opposite = Height gained
Adjacent = Distance covered on ground.
Height gained against distance covered on ground is the climb gradient.
Thus Tan gamma = climb gradient
For small angles Adjacent is nearly the same as Hypotenuse and Opposite/Adjacent = Sin relation.
So we get an approximate formula for climb gradient, which is:
Sin gamma = T – D / Weight
or
Climb gradient = T – D / Weight
Meaning that the greatest climb gradient is obtained when a greatest difference exists between thrust and drag and the weight is least.
Climb Gradient as Percentage:
Climb gradient = T–D / Weight
Climb gradient % = T–D / Weight x 100
Q.1. A four jetengine aeroplane with a mass of 150,000 kg is established on climb with all engines operating.
The L/D ratio is 14.
Each engine has a thrust of 75,000 Newton.
g = 10m/sec square
What is the gradient of climb expressed as a percentage?
Solution:
Climb gradient % = T–D / Weight x 100
Thrust = 4 engines at 75,000 N each = 300,000 N.
Weight = Mass x g = 150,000 x 10 = 1,500,000
L/D = 14 so D = L/14
Assuming the Lift = Weight.
Drag = 1,500,000 / 14 = 107143
Putting the values in the formula:
Climb gradient % = T–D / Weight x 100
Climb gradient = 12.86 %
Q.2. A four engine aeroplane has a thrust of 45,000 N per engine.
Drag is 69,525 N
Minimum acceptable gradient in the second segment of the climb, which is calculated with an engine out, is 3%.
g = 10m/sec square
Determine the maximum takeoff mass limited by the second segment climb gradient?
Solution
Climb gradient % = T–D / Weight x 100
or
Weight = T–D / %gradient x 100
Note: Due to engine out case, thrust will be from 3 engines and not 4 i.e. 45,000 x 3 = 135,000
Weight = (135,000  69,525)/3 x 100
Weight = 2,182,500 N
Mass = Weight / g
Mass = 2,182,500 / 10 = 218,250 N
Best Angle of Climb Speed  VX:
Best angle of climb speed will be when the excess thrust is greatest.
Plotting the thrust and drag curve on a graph, we can see that best angle of climb speed (VX) is at a point where there is a maximum difference between the two curves (i.e. maximum difference between thrust and drag).
Since the thrust lines differ for jets and propeller aircraft, their VX is different.
Vx for jet aircraft is close to Vmd (minimum drag speed).
Where Vx for propeller aircraft is lower than Vmd, somwhere close to the stalling speed (about 1.1VS).
Factors Affecting Climb Gradient:
In reference to the above figure, anything that reduces the distance between the thrust and drag curves, reduces the climb gradient and viceversa.
Altitude and Temperature
The drag for a given EAS (close to IAS) stays the same with variations in temperature and pressure.
However the thrust decreases with low pressure and high temperature.
So High Altitude (i.e. low pressure) or High Temperature decreases the climb gradient.
The reduction in temperature at higher altitudes has the effect of increasing the climb gradient but reduction in pressure is more significant, so overall as the aircraft climbs, the climb gradient is reduced.
Mass
Increase in mass  Increases in lift  Increase in lift induced drag  Reduction in excess thrust  Reduced climb gradient.
Flaps
Increase in flap setting  Increases in drag  Reduced climb gradient.
Change of lift produced produces no benefit to climb gradient. Climb gradients are best when the aircraft is flapless.
Wind
The climb gradient in aerodynamic terms is considered in the air mass and is therefore not affected by wind. It is an air gradient.
If the gradient is related to ground distance a headwind will increase the flight path climb gradient whereas a tailwind will decrease it.
In this case the gradient should be referred to as the Flight Path Angle.
Bank Angle
Increase in bank angle  Increase in induced drag  Reduced climb gradient.
Factors Affecting the Value of VX:
Temperature and Pressure:
EAS of VX is unchanged with altitude.
Thrust reduces with increasing temperature or reducing pressure but the drag curve does not change as an EAS with altitude and temperature.
Since the drag curve does not change, the speeds stay the same.
Mass:
Increase in mass  Increase in lift induced drag  Drag curves shifts to the right  VS and Vmd Increase  VX Increases.
Therefore as mass increases, Vx for both the jet and propeller driven aircraft also increase.
Flaps:
Increase in flap setting  Increase in profile drag  Drag curves shifts to the left  VS and Vmd Decrease  VX Decreases.
Increase in flap setting causes an increase in the lifting ability but actual lift remains the same as it only balances the weight. Since the actual lift remains the same, lift induced drag also remains the same.
Drag Curve
Takeoff Safety Speed, V2:
V2 is the takeoff safety speed (takeoff climb speed at 35ft).
For Class B aircraft it is the highest of:
• Vmc and 1.1 Vmc for a twin
• Vs1
• Safe under all reasonable conditions
Class A aircraft use VsR instead of Vs1 and for Class A this speed (V2min) is the highest of:
• 1.13 VsR for twoengine and threeengine turboprop powered aeroplanes and jet aircraft without provisions for obtaining a significant reduction in the one engine inoperative poweron stall speed.
• 1.08 VsR for turboprop aeroplanes with more than three engines and jet aircraft with provisions for obtaining a significant reduction inithoeone engine inoperative poweron stall speed.
• 1.10 Vmc
• VR plus the speed increment attained before reaching a height of 35 ft above the takeoff surface.
• A speed that provides adequate manoeuvring capability.
See also a post about V2/VS Ratio and VSR
Effect of combining these limits is to produce a V2min that is often constant at light weights.
However it increases with increase in weight.
Since V2min has to be higher that the "speed increment on VR", it can produce high values of V2min at light weights.
The value of V2 using these criteria will be a low value and will be on the wrong side of the drag curve (unstable speed).
Since Vx is low on propeller driven aircraft, V2 will be close to Vx.
However in case of jets V2 will be some way below Vx. Thus V2 is not best angle of climb speed on a jet.
So incase of jets, why V2 and not the best climb speed?
The answer is that it would take extra distance to accelerate to the optimum speed.
Economically it is easier to sell jets that can operate to V2 (min standard) from short runways (8,00010,000ft) as compared to those that have a better initial climb speed but require longer runways (11,00012,000ft).
However, V2 is only the initial target speed in a jet climb if an engine fails on the takeoff run. Without engine failure a jet is usually accelerated to
V2+10 (V4), for a better angle of climb, pitch attitude and incase of an engine failure a better climb gradient than calculated for the situation where an engine fails at VEF.
Climb Limit:
Net takeoff flight path starts at the screen height for the class under consideration.
Screen height for class A is 35ft for a dry runway and 15ft for a wet runway.
Screen height for class B is 50ft.
Takeoff flight path (for class A) begins at 35ft above the takeoff surface at the end of the takeoff distance (reference zero).
Takeoff path ends at a point which is higher of the following:
 A point 1500 ft above the takeoff surface.
 A point where transition from takeoff to enroute configuration is completed at VFTO.
Takeoff flight path is divided into several segments.
There are some minimum climb gradients to be met on takeoff (nothing to do with obstacle clearance).
The limiting gradient (usually but not always) is the gradient in segment 2 of the takeoff flight path.
Climb gradient in segment 2 may not be less than:
• 2.4% for twoengined aeroplanes.
• 2.7% for threeengined aeroplanes.
• 3.0% for fourengined aeroplanes
Climb gradient is affected by Weight, Altitude and Temperature hence there will be a MTOM limit to ensures that the most severe climb gradient requirement is achieved.
Climb Segments:
The net takeoff flight is the gross takeoff flight path reduced by the regulatory requirement of:
• 0.8% for a twin engine aircraft.
• 0.9% for a three engine aircraft.
• 1.0% for a four engine aircraft.
Maximum acceleration height is limited to the time the engines are certified at maximum takeoff thrust (5 or 10 mins depending on type).
Thus segment 3 must be completed by that time limit point and therefore restricts the maximum acceleration height.
Turns in the Takeoff flight path are permitted but avoided whenever possible.
No turns may be made below the height of 50ft or half a wingspan, whichever greater.
Bank angle is limited to 15 deg below 400ft and 25 deg thereafter.
Turning reduces the gradient and increases the stalling speed.
Turns at 25 deg bank angle usually demand an allowance of 10kt on top of minimum speed for 15 deg bank.
Regulations for Turning Departures
Best Rate of Climb  Vy
Rate of climb is the height gained per unit of time (feet per minute).
Rate of climb is affected by the TAS and the climb angle (in a steady climb).
Rate of climb increases if:
 Speed increases at a given climb angle.
 Climb angle increases for a given speed.
Rate of climb = TAS x Sin gamma
Sin gamma = TD/W (as mentioned above)
So Rate of Climb = TAS x (TD)/W
TAS x thrust = Power Available.
TAS x Drag = Power Required.
So Rate of Climb = (Power Available  Power Required) / Weight
According to the figure, Vy for a jet is quite high and can vary from its optimum without much affecting the rate of climb.
On the otherhand, Vy for a prop is quite low. Speeds above the optimum do not affect rate of climb much but speeds below the optimum reduces the climb rate.
Factors that affect Rate of Climb:
Altitude and Temperature
In a climb  Thrust reduces  TAS increases  Product of Thrust x TAS reduces  Power available reduces.
In a climb  Drag constant  TAS increases  Product of Drag x TAS increases  Power required increases.
Power available reduces  Power required increases  Rate of climb decreases.
Thus increase in altitude and increase in temperature reduces the rate of climb.
Mass
Rate of Climb = (Power Available  Power Required) / Weight
Rate of climb is inversely proprtional to mass.
Flaps and Gear
Increase in drag  Increases in power required  Decrease in rate of climb.
Wind
It has no effect.
Rate of Climb Formula:
Rate of climb = TAS x Gradient
Factors Affecting Value of Vy:
Altitude and Temperature
As altitude increases, best rate of climb TAS increases.
Change in air density with altitude means that for a given TAS the EAS and IAS reduce.
Reduction in EAS and IAS is faster than the increase in TAS
Vy is ideally expressed as an IAS.
So Vy (as an IAS) decreases as height increases.
EAS and TAS diverge less in lower temperatures and more in high temperatures.
Thus Vy decreases less with height in low temperatures and more with height in high temperatures.
Mass
Increase in mass  increase in induced drag  left side of drag curve (power required curve) shifts to the right  VY increases.
Flaps and Gear
Increase in flaps or extending the gear  increase in profile drag  right side of drag curve (power required curve) shifts to the left  VY decreases.
Flying for range in a jet, it is best to climb wih climb thrust at the recommended speed until the cruising altitude.
If the climb is continued, a height will come where there is no excess power available.
Vy will reduce to VMD (which is Vx). This will be the absolute ceiling.
Vx is always a lower value than Vy except at the absolute ceiling where Vx = Vy.
Ceiling:
At high subsonic speeds drag created by shock wave formation starts to change the shape of the power curve at a speed known as McDR (mach critical drag rise).
McDR is above Vy and does not affect the climb.
However as altitude increases, McDR occurs at a reducing EAS.
It eventually drops down to Vy.
If power is still avaialable it is possible to continue climbing at McDR
Ultimately EAS falls to some minimum control speed near the stalling speed.
Holding this EAS  Mmo is reached  Coffin corner is reached.
Service ceiling is lower than the absolute ceiling.
For jets aircraft, it is the height at which the rate of climb falls to 500 ft/min.
A structural limit on the pressure hull may impose a pressurisation ceiling limit.
Piston and Turboprop Climb
As compared to a piston, a turboprop maintains the power output to high speeds due to the intake ram effect.
Lower intake temperatures increases the engine's thermal efficiency and thus a better turboprop power output at height.
Ultimately the propeller efficiency is affected (at TAS between 300400 kts).
Service ceiling for a propeller aircraft is the height where the rate of climb has fallen to 100 ft/min.
Angle of Attack in Climb:
Maximum angle of attack occurs at the stall.
Climb at fixed IAS  Angle of attack is constant.
Climb at fixed Mach number  IAS reduces  Angle of attack Increases.
Noise Abatement:
NADP 1:
• Climb at V2+10 to V2+20 kt to 3000 ft.
• 800 feet  Thrust reduction.
• 3000 feet  Accelerate to enroute climb speed retracting flaps on schedule.
NADP 2:
• Climb at V2+10 to V2+20 kt.
• 800 feet  Accelerate to VZF (zero flap speed) and retract flaps on schedule.
• Flaps up  Thrust reduce
• Climb at VZF+10 to 3000 feet.
• 3000 feet  Accelerate to enroute climb speed.
NADP A:
• Climb at V2+10 to V2+20 kt to 3000 ft.
• 1500 feet  Thrust reduction.
• 3000 feet  Accelerate to enroute climb speed retracting flaps on schedule.
NADP B:
• Climb at V2+10 to V2+20 kt.
• 1000 feet  Accelerate to VZF (zero flap speed) and retract flaps on schedule.
• Flaps up  Thrust reduce
• Climb at VZF+10 to 3000 feet.
• 3000 feet  Accelerate to enroute climb speed.
Noise Abatement Procedures 
