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This Powerpoint presentation describes a Class II methodology for drag calculation of transport planes
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PRJ-22 Aerospace Design
Weight Estimation Using Class II Drag Calculation Method
Prof. Bento S. de Mattos
V20 – June 2009
Content
• Fundamental principles
• Zero-lift drag
• Induced drag
• Wave drag
• Class exercise
• Weight estimation
Total drag of the airplaneRef.: TORENBEEK, E. – “Synthesis of Subsonic Airplane Design”,
Kluwer, 1982 (pg. 368).
4
Total drag of the airplane
Prof. Mason, Virginia Tech
Total drag of the airplane
Prof. Mason, Virginia Tech
6
Total drag of the airplane
• Zero-lift, or parasite, drag
• Induced drag, or drag due to lift
• Drag due to compressibility, or wave
drag
• Total drag
CD,0 = CD,0,form + CD,0,friction
CD,i = kCL2 =CL
2/peA
CD,w= CD,w(M)
CD = CD,0 + CD,i + CD,w
7
Drag and Required Power
XDrag
Speed – V Speed - V
Induced Drag
Di ~ W2( b V)-2“Parasite” (viscous) Drag
Dp ~ Swet V2
Total Drag = Dp + Di
Power ( P)required =
D x V
V*prop V*jet V*prop
Power
V* = Optimum Speed to Fly for Maximum Range
Pavailable
Vmin Vmax
Power (P) = Thrust (T) x Speed (V)
8
CD0 CD0
CD0
CD0
CD0
CD0
CD0
9
Drag Breakdown of a Complete Airplane Configuration
10
Wing drag
Vertical tail drag
Flap drag
Nacelle drag Landing gear drag
Fuselage drag
Horizontal
tail drag
Aerodynamic drag components acting on
aircraft
11
Class I Method
Common Cfe Values
ref
wetfeD
S
SCC 0
12
Theoretical Background – Class II Method
A more useful measure of the parasite drag is the equivalent flat-plate drag
area, f . This quantity is exactly what it suggests–a flat plate of area, f , will
have the same drag as the airplane (when the plate is positioned
perpendicular to the wind). Thus, the total parasite drag is just
DP = f q
where q is the dynamic pressure.
You can find f by doing a component drag buildup. Each exterior component
of the airplane is considered separately, and the f of each is found. Then the
total f is determined by summing the component drag areas. In general, the
equivalent flat-plate area of the ith component can be computed from
ref
wet
iifref
iD
S
SQFC
Sf
C i
ii0
13
Theoretical Background – Class II Method
ref
wet
iifDS
SQFCC i
ii0
Friction coefficient
Form factor Interference factor
Area ratio
14
tw
,
2
,0 0.664 0.664
1 Re
2
f lam
x
xc
VxV
t
,
0,
0
1.328
Re
l
f lam
F lam l
l
c dx
C
dx
Local skin friction
coefficient
Average, or
integrated, skin
friction coefficient
Laminar flow over a smooth flat plate
X=0 X=l
Skin friction drag
15
laminar transitional turbulent
Laminar, transitional, & turbulent Flow
104 105 106 107 Rex
10
4
1
1000cfTurbulent
cf=0.74Rex-1/5
Laminar cf =1.328Rex-1/2
Transitional
X0
16
For 106 < Rel < 109 use:
Skin friction drag in turbulent flow
CF
106 107 108 109 Rel
0.0045
0.0035
0.0025
M=0
M=1
CF is small but the
dynamic pressure and
wetted area are large
65.0258.2
10 )144.01(Relog
455.0
MC f
17
DW,0 = DW,f + DW,p = zero-lift drag
DW,0 = CF,turbSwet q + k CF,turbSwetq
DW,0 = (1 + k) CF,turbSwet q
DW,0 = KWCFSwet q
Drag build-up by components
Consider, as an example, the drag build-up for the wing
(subscript w)
The drag estimate is based on a multiple of the friction
drag, KW, the form factor for the wing
18
Skin friction calculation Re, , /FC f M l k
Admissible surface roughness (Table 4.1.5.1-A):
l/k = reference length (in.)/surface roughness height (in.))
k(in.)=
0.02 to 0.08 x 10-3 polished metal
0.40 x 10-3 camouflage paint
6 x 10-3 dip-galvanized metal
106 107 108 Re,cut-off
106
105
104
103
l/k M=0 M=1
For Re<Re,cut-off
19
Skin friction drag in turbulent FlowCF
106 107 108 109 Rel
0.0045
0.0035
0.0025
M=0
M=1
Rel/l~2.4x106 per foot
at M=0.85 and h=35kft
Rel =Vl/ < Re,cut-off
CF
Reynolds Number
lV Re
Sutherland's formula can be used to derive the dynamic viscosity of an ideal gas
as a function of the temperature:
GasC [K] T0 [K]
μ0
[10−6 Pa s]
air 120 291.15 18.27
nitrogen 111 300.55 17.81
oxygen 127 292.25 20.18
carbon dioxide 240 293.15 14.8
carbon
monoxide118 288.15 17.2
hydrogen 72 293.85 8.76
ammonia 370 293.15 9.82
sulfur dioxide 416 293.65 12.54
helium 79.4 273 19
Valid for temperatures between 0 < T < 550 K with an
error due to pressure less than 10% below 3.45 Mpa.
Sutherland’s constant and reference
temperature for some gases.
Prof. Bento S. de Mattos
Cut-off Reynolds Number
053.1)(21.38Re l/kcutoff Subsonic:
Transonic:16.1053.1)(62.44Re Ml/kcutoff
Source: Raymer
t
1
1/25.012 exp, rootWingwet ctSS
22
Wetted area Calculation
Wing
Taper ratioFrootref dchordSS exp
F
FF
d
l
23
Wetted area Calculation
Fuselage
2
3/2
,
11
21
FF
FFFuswet ldS
p
24
Wetted area Calculation
Engine Nacelles
25
Wetted area Calculation
Engine Nacelles
ppplugwet DlS p7.0,
3/5
, 18.0113
11
g
g
g
eg
gggengaswetl
D
D
DDlS p
n
ef
n
l
nn
hll
n
lnncowlingfanwet
D
D
l
l
dl
Dl
l
lDlS 115.18.035.02,,
Wetted Area
Prof. Bento S. de Mattos
Component Approximate Wetted Area Aw
Form Factor
28
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
k
Fineness Ratio λ=Length/diam.
kfactor in fuselage drag
correlation
NAA
Datcom
Torenbeek
Hoerner
Fuselage form factor as determined by
different investigators
(Kf=1+k)
Form factor
Fuselage:
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930090976_1993090976.pdf
New link for NACA Technical Report 824.
4003
601
f
ffus
F
Body with blunt base
Closed body (CD,b=0)
d
base
fus
fusfus D
ref
wet
fusfusfD CS
SQFCC
0
30
Wing form factor as determined by
different investigators
k - factor in wing drag correction
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.05 0.1 0.15 0.2
Thickness Ratio t/c
k
Jenkinson
Torenbeek
NAA
Hoerner
DATCOM 2
DATCOM 1
(Kw=1+k)
cos34.1/100/6.0
128.018.04
m
t
W Mctctx
FF
Form factor
Wing:
Chordwise location of the maximum thickness (should be used a actual length in m)
Nacelle:
Sweep angle of the wing measured at the line that is generated by joining the maximum thickness location of the airfoils (xt)
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930090976_1993090976.pdf
New link for NACA Technical Report 824.
Interference Factor
Interference Factor
Interference Factor
Prof. Bento S. de Mattos
Interference Factor
Prof. Bento S. de Mattos
Roskam suggests that the value for the wing-fuselage interference factor should be
taken from the graph below
Interference Factor
Interference Factor
Estimated values of interference drag originating in the corners of various
tail configurations
Source: Horner, Fluid-dynamic Drag
Miscellaneous Parasite Drag
The drag of some items does not quite fit the form
must be computed in an alternative manner. Table below gives values of f /Afrontal,
the equivalent flat plate drag area normalized by the projected frontal area, of
landing gear components. Deployed flaps have a drag area given roughly by
with bflap representing the span of the flap and δflap the flap angle in degrees.
Recall that takeoff flap setting is approximately 25◦ whereas a δflap of 50◦ is
used on landing. Fuselage-mounted speed brakes have f /Afrontal = 1, while
wing-mounted speed brakes, or spoilers, have f /Afrontal = 1.6.
Estimated equivalent flat-plate drag
areas for landing gear components.
(From Raymer.)
Miscellaneous Parasite Drag
Drag Caused by Flap Deflection
Prof. Bento S. de Mattos
Definition of flap drag @ 0.70 CL max
41
DCD,flaps = 0.9 (cflap /c)1.38 (Sflap /S) sin2 d (slotted flaps)
Alternative Formulation for the Calculation of
Drag Coefficients for Flaps
Prof. Bento S. de Mattos
Prof. Bento S. de Mattos
Miscellaneous Parasite Drag
Windmilling Engine and Propeller Drag
When computing engine-out performance, it is necessary to include additional
drag from stopped or windmilling engines or propellers. While operating, these
propulsion components are considered to have no drag since the reported thrust
includes a decrement for forces generated in the drag direction. Propeller drag
depends on the solidity, σ, given by
where B is the number of blades, cavg is the average blade chord, and R is the
propeller radius. Note that this quantity just equals the ratio of blade area to
disk area. The drag area is given by
A windmilling jet has an equivalent drag area given by 0.3 times the face area.
Prof. Bento S. de Mattos
Landing gear Drag
Prof. Bento S. de Mattos
(SI)
nits)(British U
785.0
785.0
S
KM
qS
D
S
KW
qS
D
Analysis of flight measured drag, for a number of civil transport aircraft, indicates that the
following values of K should be used
In British units with W in lbf
deflection flap fullfor 108.1K
deflection flap zerofor 103.3
3
3
x
xK
In SI units with M in kg
deflection flap fullfor 1031.0K
deflection flap zerofor 1057.0
3
3
x
xK
The reduction from K with flap deflection is
assumed to be linear
nits)(British U 105.13.3 3
x
MaxF
FKd
d
Source: ESDU
For the calculation of landing gear dreag when the geometry is unknown a empyrical formula
can be used
45
CDi CDi
CDi
CDi
CDi
CDi
CDi
46
What is?
•Induced drag is the drag related to the generation of lift.
•It results from the angle of attack induced from the 3-D flowfield.
•For an elliptical wing:
•For a non-elliptical wing, we have introduced Oswald's Efficiency Factor, e
•So is a measure of the reduction in efficiency over the optimum elliptical wing
case.
•In other words, the elliptical lift distribution is optimum in terms of induced
drag.
A
CC L
Dip
2
Ae
CC L
Dip
2
47
Oswald’s Factor e
M = Mach number
= Taper ratio of the reference wing
ne = number of engines placed under the wing
A = Aspect ratio
(t/c) – averaged maximum thickness ratio of the wing
25 = Sweepack angle at ¼ chord
Prof. Bento S. de Mattos
48
Induced drag @ supersonic regime
Prof. Bento S. de Mattos
49
Reducing Induced Drag
Prof. Bento S. de Mattos
Winglets
Wings of higher aspect ratio
50
Reducing Induced Drag
Prof. Bento S. de Mattos
Ae
CC L
Dip
2
d
1
1e
d
Source: Anderson, Aircraft Performance and Design.
Swept wings
Induced Drag
012
2
CCCCC
C LL
Di
p
d
1
11
2Ce
Source: Schlichting. H. und Truckenbrodt , Aeodynamik des Flugzeuges, Zweiter Band.
Induced drag of symmeric twisted trapezoidal wing
a) Planform with linear twist
b) Induced drag of the untwisted wing
c) and d) Twist contribution to induced drag
a
b
c
Aspect ratio
d
1
1
C
2
1
0
C
2C
angle induced1
52
Reducing Induced Drag
Prof. Bento S. de Mattos
Aircraft L/Dmax Wing Span (m) Aspect Ratio
WWII Bomber
Boeing B-29 16.8 42.98 16.8
B-24 J Liberator 12.9 33 11.5
Boeing B-17G 12.7 31.64 7.58
Martin B-26F 12 21.64 7.66
WWII Fighter
Lockheed P-38L 13.5 15.84 8.26
P-51D Mustang 14.6 11.28 5.86
Me 262A 14.09 12.53 7.23
53
CDwave CDwave
CDwave
CDwave
CDwave
CDwave
CDwave
54
Wave drag @ supersonic regime
Example: CDWave Comparison With Two Methods for a SSBJ
• Fuselage length = 198 ft
• Height of Airplane with landing gears down = 33 ft
• Span of the delta wing = 103.17 ft
• Wing aspect ratio
= 2.6 with wingtips Up
= 1.9 with wingtips Down
55
Rallabhandi and Mavris’s Approach
Below are given an analytical expression
for the wave drag assuming the aircraft
body to be Sears-Haack body.
Wave drag @ supersonic regime
wet
DSl
VolC
4
2
wave
128
p
British System
56
Induced drag @ supersonic regime
Raymer’s Approach
• Assumptions:
-Correlates aircraft wave drag to an
equivalent Sears-Haack body at Mach = 1.2.
57
Wave drag @ supersonic regimeExample:
CDWave calculation for a SSBJ fuselage by the two methods
from previous slides
ft3
ft
58
Exercise
ERJ 145 CD0 Calculation for the Wing and Fuselage
• Wing reference area = 51.12 m2
• Wing taper ratio = 0.2543
• CMA of the wing = 2.9 m
• MMO @ 37000 ft = 0.78
• Fuselage length = 27.93 m
• Fuselage diameter = 2.28 m
• Sweepback angle @ 1/4 chord = 22.72º
• Chord @ root = 4.09 m
• (t/c)root = 14%
• (t/c)tip = 9.5%
• (t/c)averaged = 12%
General Data
Prof. Bento S. de Mattos
Calculation will be performed for the cruise
condition (Number of Mach = 0.78)!
60
ERJ 145 CD0 Calculation for the Wing
Friction coefficient is given by
We consider in this example that the laminar portion of the wing covers
10% of its whole exposed area (usually this figure is around 5%). That
means we must take klam = 0,10.
Calculation of the Reynolds number
7
6
1025.11027.18
9.2230343.0Re x
mCMAV
smkg
sm
cruise
turbflamlamflamf CkCkC ,, )1(
716.1
053.1
16.1053.1 107.678.0003.0
290062.44)(62.44Re x
mm
mmMl/kcutoff
We take 1.25x107 because is the lowest Reynolds number!
Prof. Bento S. de Mattos
61
ERJ 145 CD0 Calculation for the Wing
Friction coefficient is given by
We obtain the Cf,lam and the Cf,turn
The friction coefficient is the given by
turbflamlamflamf CkCkC ,, )1(
333
,, 1051.21075.29.010345.01.0)1( xxxCkCkC turbflamlamflamf
3x
7, 10 0.3756
10 x 1.25
328.1
Re
328.1 lamfC
3
65.0258.27
10
65.0258.2
10
, 1075.278.0144.011025.1log
455.0
144.01Relog
455.0
xM
C turbf
Prof. Bento S. de Mattos
ERJ 145 CD0 Calculation for the Wing
We consider the interference factor QW=1.01 (from the Roskam’s Graph)
The form factor of the wing can be calculated using the following
expression
With Xt = 0,35xCMA = 0,35 x 2,90 m = 1.1015 m
The wetted area is given by
The area ratio can easily be calculated:
33
,0 1096.501.17.1384.110 x 51.2 xC wingD
7.112.51
8.862
2,
m
m
S
S
ref
wwet
384.15.15cos78.034.112.010012.0015.1
6.01
28.018.04
o
WFF
28.018.04cos34.1/100/
6.01 m
t
W Mctctx
FF
2
exp, 8.862543.01
2543.04736.1114.025.018.412
1
1/25.012 mctSS rootWingwet
t
Prof. Bento S. de Mattos
ERJ 145 CD0 Calculation for the Fuselage
The Reynolds number for the fuselage is
Re is lower that the cut-off Reynolds number. Thus, the usual Reynolds number
must be taken into the equations for the calculation of Cf,lam and Cf,turb.
Considering that the fuselage presents a rough surface, the cut-off Reynolds number must also
be calculated to taken this characteristic into account
(Raymer)
8
6
310206.1
1027.18
93.27230343.0Re x
lV
mskg
msm
m
kg
fuscruise
816.1
053.1
16.1053.1 1031.378.000635.0
2793062.44)(62.44Re x
mm
mmMl/kcutoff
3
65.0258.28
65.0258.2,
10962.178.0144.0110206.1log
455.0 ...
144.01Relog
455.0
xx
MC turbf
4x
8, 10 1.209
10 x 1.206
328.1
Re
328.1 lamfC
334
,, 10778.110962.19.010209.11.0)1( xxxCkCkC turbflamlamflamf
ERJ 145 CD0 Calculation for the Fuselage
The form factor of the fuselage can be calculated by
The wetted area can be obtained by using a Torenbeek’s formulation
(DATCOM 78)
We consider the interference factor QF=1.
The area ratio can easily be calculated: 523.312.51
8.862
2,
m
m
S
S
ref
fwet
33
,0 1076.6 523.308.1110778.1 xxC fuselageD
Prof. Bento S. de Mattos
65
Preliminary Weight Estimation
Prof. Bento S. de Mattos
66
2. Preliminary Weight Estimate
WTO = WE + WTFO + WPLC + WF,USED + WF,RES
=Take-off Weight
WE =Empty Weight
WF = WF,USED + WF,RES
= Weight of Fuel Used+ Weight of Fuel Reserve
= Total Fuel Weight
WPLC =WPL+WCREW = Weight of Payload +Weight of Crew
MTFO = WTFO / WTO=(Trapped Fuel and Oil Weight)/WTO
MFUEL = WF/WTO= Fuel Fraction
Commercial Airplane Design 67
Solve for the empty weight knowing WPLC
WE = (1 – MTFO – MFUEL)WTO – WPLC = aWTO + b
WTO
WE
0
-WPLC
(1-MTFO-MFUEL)
increasing
Empty weight vs take-off weight relation
Fuel fraction
needed for
mission,
including
reserves
68
WF = WTO – WFINAL=WTO – (Weight at End of Mission)
WF/WTO = MFUEL= 1 – WFINAL/WTO = 1 – MFINAL
Fuel Needed for Mission
1 2 3
4
5 6
78
9
10
11
Mission Profile
Normal
Diversion
Prof. Bento S. de Mattos
69
1 2 3
4
5 6
78
9
10
11
Mission Profile
0.99 0.99 0.995
exp[-RCj/V(L/D)] exp[-Cj/(L/D)] exp[-Ralt.Cj/V(L/D)]
0.98 0.99 0.98 0.99
0.992
Segment weight fractions Wi / Wi -1
Cruise
Loiter
Prof. Bento S. de Mattos
70
11
10 1
nFINAL i
FINAL
iTO i
W WWM
W W W
MFINAL =(W11/W10)(W10/W9)(W9/W8)….(W2/W1)(W1/W0)
,
, ,1F USED
F USED FINAL F RES
TO
WM M M
W
,
,
LAND NOM
FINAL F RES
TO
WM M
W
5 9,
,
1 61 1
1F RES i i
F RES
i iTO i i
W W WM
W W W
Final Weight Fraction
Fuel Weight
Fraction Used
Nominal Landing Weight
Reserve
Fuel
Fraction
71
L/D Calculation
Cruise!
Prof. Bento S. de Mattos
72
L/D Calculation
For a jet airplane the L/D for maximum endurance (to applied for the loiter phase) can be simply obtained by
Prof. Bento S. de Mattos
73
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2000 4000 6000 8000 10000
Range (mi)
1-M
fin
al
Normal Mission Fuel Fraction vs Range
This is the nominal value of the ratio WF,USED/WTO
1-MFINAL = 0.00316(R-800)1/2
74
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 2000 4000 6000 8000 10000
Range (miles)
1 -
Mfi
na
l+M
res
Total Fuel Fraction vs Range
1-MFINAL+MRES=0.0048R1/2
Nominal ratio of total fuel carried to take-off weight, MFUEL
75
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 200,00
0
400,00
0
600,00
0
800,00
0
1,000,0
00
1,200,0
00
1,400,0
00
Take-Off Weight (lbs)
Mtf
o
Weight Fraction of Trapped Fuel & Oil
MTFO=0.227(MFUEL)2/3(WTO)-1/3
Correlation for the weight fraction of trapped fuel and oil
76
Correlation of empty weight vs take-off
weight for 45 airliners
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0
Take-off weight, Wto (klbs)
Em
pty
we
igh
t, W
e (
klb
s)
Actual weights
logWe=(logWto - A)/B
We=0.5Wto
Prof. Bento S. de Mattos
77
Correlation of empty weight vs. take-off
weight for 45 airliners
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0
Take-off weight, Wto (klbs)
Em
pty
we
igh
t fr
ac
tio
n, W
e/W
to
We/Wto = 1.59/(Wto/1000) .̂0906
Prof. Bento S. de Mattos
78
WE=aWTO-WPLC
Historical
correlation
WE=0.504WTO
WE
0
-WPLCWTO
Market survey aircraft
Estimating Aircraft Empty Weight
Estimating Cruise Fuel Consumption
Performance
Max operating Mach number 0.83
Max operating altitude 41,000 ft (cabin altitude: 8,000 ft)
Take-off field lenght 6,500 ft (SL / ISA + 15°C / MTOW)
Landing field 5,000 ft (SL / MLW = 90% of MTOW)
Range with max payload 2,200 nm (overall fuel volume for 3,200 nm version)
External noise FAR 36 Stage IV minus 15 db
IPET7 Airliner
80
Estimating Cruise Fuel Consumption
41000 ft
0,150
0,170
0,190
0,210
0,230
0,250
0,270
0,290
0,40 0,50 0,60 0,70 0,80 0,90
Mach
SR
[n
m/k
g]
MTOW 90% MTOW 80% MTOW
Long Range MMO
SR vs. Mach number 41000 ft
0,00
2,00
4,00
6,00
8,00
10,00
12,00
14,00
0,40 0,50 0,60 0,70 0,80 0,90
Mach
M*L
/D
MTOW 90% MTOW 80% MTOW
Mach*L/D vs. Mach number
The number of Mach for maximum specific range (SR) is not the same as that for
maximum M*L/D because sfc increases with speed
IPET7
IPET7