AE 429 - Aircraft Performance and
Flight Mechanics
Rate of Climb
Time to Climb
Rate of Climb R/C
Now let’s analyze a steady climb
–
Forces include a gravity component now
dV
m ∞ = T cos ε − D − W sin θ
dt
m
V∞2
= L cos φ + T sin ε cos φ − W cos θ
r1
T = D + W sin θ
L = W cos θ
V∞
The rate of
climb (R/C) is
the vertical
component of
velocity
1
Rate of Climb R/C
For low climb angles (up to about 20°)
–
We can assume
–
So we work on the drag equation, multiplying by V∞
to calculate CL
cos θ ≈ 1
TV∞ = DV∞ + WV∞ sin θ
Excess Power
TV∞ − DV∞ (T − D )V∞
=
= V∞ sin θ
W
W
–
In climb, the rate of climb is the vertical component of
velocity
R / C = V∞ sin θ
Maximum R/C
–
–
Maximize excess power
Minimize weight
RC=
Maximum angle of climb
–
–
Maximize excess thrust
Minimize weight
Rate of Climb R/C
(
= q∞ SCD ,0 +
R / C = V∞ sin θ =
Preliminary design
2
L
θ = sin −1
≡
(T − D ) (T − D )
W
≈
W
(T − D )V∞
W
)
W cos θ
= q∞ S CD ,0 + K
q∞ S
2
=
KW 2 cos 2 θ
q∞ S
(T − D )V∞
R / C = V∞ sin θ = V∞
W
L
W cos θ
=
q∞ S
q∞ S
CL =
D = q∞ SCD = q∞ S CD ,0 + KC
(T − D )V∞
W
= V∞
T D
−
W W
T 1
W
− ρ ∞V∞2
W 2
S
cos θ ≈ 1
OK for
−1
CD ,0 −
2K W
cos 2 θ
2
ρ∞V∞ S
θ ≤ 50
2
Maximum Climb Angle
θmax
sin θ =
cos θ ≈ 1
T D T
D
T
1
−
=
−
=
−
W W W L / cos θ W L / D
sin θ max =
T
1
−
W (L / D)
=
max
T
− 4CD ,0 K
W
1
L = W cos θ = ρ∞V∞2 SC L
2
L = W cos θ max =
CL =
C D ,0
K
from L/D max
C
1
ρ∞Vθ2max S D ,0
K
2
Speed at max θ for jet propelled airplane
Vθ2max =
2
ρ∞
K W
cos θ max
C D ,0 S
Rate of climb at max θ
( R / C )θ
max
= Vθmax sin θ max
Max Angle to Climb
If thrust available is constant with V
–
–
Maximum climb angle occurs at minimum drag
This speed also gives best acceleration in level flight
Thrust is approximately constant for
a pure turbojet -- not a very good
approximation for other powerplants
θ=
(TA − D )
W
Vθ max
3
Rate of Climb R/C
Typically, climbs are not flown at constant V∞
T = D + W sin θ +
W dV∞
W dV∞ dh
= D + W sin θ +
g dt
g dh dt
Recognizing
dh
= R C = V∞ sin θ
dt
and rearranging
T −D
T −D 1
W
sin θ =
=
V∞ dV∞
W k
1+
g dh
V dV∞
k = 1+ ∞
g dh
T −D
1 dV∞
V∞ sin θ
= sin θ +
W
g dh
Rate of Climb R/C
For our idealized jet airplane, best rate of climb
does not occur at minimum power required
–
–
Maximum rate of climb occurs at the velocity where
excess power is greatest
The velocity for maximum rate of climb is determined for
any aircraft by
Plotting the power required versus true airspeed
Overplotting the power available versus true airspeed
Choosing the velocity where the distance between the two
curves is greatest
4
Rate of Climb R/C
The chart below illustrates this procedure
Best R/C
speed
Best angle
of climb speed
Rate of Climb R/C
Max Rate of Climb
d (R / C)
dV∞
V( R / C )
max
CD ,0 −
T 3
W
=
− ρ∞V∞2
W 2
S
( R / C )max =
3ρ∞ CD ,0
1/ 2
−1
CD ,0 +
1+ 1+
3ρ∞ CD ,0
(W / S ) Z
2K W
ρ∞V∞2 S
2K W
=0
ρ∞V∞2 S
1/ 2
(T / W )(W / S )
=
Z ≡ 1+ 1+
−1
T 1
W
− ρ∞V∞2
W 2
S
R / C = V∞
T
W
3
( L / D )max (T / W )
3/ 2
1−
2
2
Z
3
−
6 2 (T / W )2 ( L / D )2 Z
max
3
( L / D )max (T / W )
2
2
5
Rate of Climb R/C: Effect of the
Altitude
This chart shows the
effect of altitude
– At higher true
airspeeds, PR
decreases
with altitude
– However, PA falls off
faster than PR
– The best climb speed
usually decreases
slightly with altitude
Rate of Climb R/C
For a propeller aircraft
–
Maximum rate of climb occurs at the V∞ where maximum excess
power occurs
Maximum R/C does not occur at VminPR
However, if PR is assumed constant with V, R/Cmax does occur at VminPR
DV∞ is approximately the power required in the climb
6
Rate of Climb R/C
For propeller aircraft
–
Maximum angle of climb occurs at the V∞ for which maximum
excess thrust occurs
Maximum climb angle (which is used to clear obstacles on takeoff)
occurs at a velocity < VminTR
Rate of Climb R/C
For a given
altitude
–
–
For any type
of airplane,
excess power
determines R/C
Induced drag
changes PR
R/C=
PA − PR
W
R/C changes with
velocity
Plots like the one on left
allow determination of
best R/C speed
7
Rate of Climb R/C
Climb performance hodograph
–
Vertical velocity versus horizontal velocity
–
Notice the difference in velocity for θmax and the velocity for (R/C)max
Rate of Climb R/C: High performance
climb
–
Lift forces Eq.
–
Drag forces Eq.
–
Solving the two
equations for θ
L = W cos θ
T − D − W sin θ = 0
solve each for qS
set qS = qS
solve the resulting
quadratic for θ
sin θ =
−
CL2T
W CL2 + CD2
(
CL2T
W CL2 + CD2
(
)
2
)
−
CL2T 2 − CD2W 2
W CL2 + CD2
(
)
8
Gliding flight
Forces in a power-off glide
D = W sin θ
–
Dividing drag by lift
tan θ =
–
L = W cos θ
1
L
D
tan θ min =
1
L
D max
( L / D ) max = 1/
4CD ,0 K
Once again, an important
performance parameter
is set by L/D
1
the smallest θ gives
L = ρ∞V∞2 SCL = W cos θ
2
maximum gliding range
this maximum range occurs when L/D is maximum
for V∞ constant
airplanes with good aerodynamic efficiency (high
L/D) can glide 20-50 times as far as their altitude
Does not depend on wing loading or altitude.
VV = V∞ sin θ
V∞ =
2 cos θ W
ρ∞ CL S
( L / D ) max = 1/
Equilibrium
glide
velocity
VH = V∞ cos θ
V∞ =
2 cos θ W
ρ∞ CL S
Equilibrium
glide
velocity
D = W sin θ
DV∞ = W sin θ V∞ = WVV
VV =
DV∞ PR
=
W
W
4CD ,0 K
cos θ ≈ 1
9
Ceilings
How high can
the airplane
climb?
Absolute Ceiling: 0 fpm
Service Ceiling: 100 fpm
Absolute Ceiling
10
Ceilings
The ceiling is the altitude at which R/C has
reached some minimum value
–
Absolute ceiling
Is defined as the altitude at which the R/C = 0
Is dictated when PA is just tangent to the PR curve
–
Service ceiling
is defined as that altitude where R/Cmax = 100 ft/min
is the practical upper limit for steady, level flight
–
Procedure
calculate values of R/Cmax for different altitudes
plot R/Cmax versus altitude
extrapolate this latter curve to 100 fpm and 0 fpm to get
the service and absolute ceilings
Time to Climb
Time to climb
Needs to be short
Calculating R/C
–
–
R/C
t2
t1
–
dh
dt
dh
R/C
dt
–
V sin
Integrating
n
h2 dh
h
dt
h1 R / C
i 1 R/C
Calculating time-to-climb
graphically
i
plot (R/C)-1 versus h
Approximate the area under
the curve
Subtract time to climb to the
starting altitude
R/C
tmin
max
tmin
0
a
dt
Altitude x 10-3
bh
h2
0
a
dh
bh
1
ln a
b
bh2
ln a
11