Wydział Mechaniczny Energetyki i Lotnictwa Politechniki Warszawskiej - Zakład Samolotów i Śmigłowców
Project 2 – feasibility study
This project consists of two parts:
1) Estimation of the wing and thrust (power) loading
2) Costs analysis
1. Estimation of the wing and thrust (power) loading
It is the most convenient to estimate wing loading (W/S) and thrust loading (W/T) of the
future aeroplane with application of the plot presenting thrust loading as a function of wing
loading for assumed requirements (Fig.1).
B
A
Fig.1 Example of the inverted thrust loading as a function of a wing loading.
Lines presented in this plot represent equations describing assumed requirements. White area
(Fig.1) constrained by these lines represent wing and thrust loading combinations satisfying
assumed requirements.
Preparation of the plot begins from an arbitrary assumed range of wing loadings (W/S). This
range of wing loadings should cover wing loadings observed during trends analysis. One has
to calculate and draw curves T/W=f(W/S) resulting from certain requirements. At the end
decision has to be made which point from the area of satisfactory solutions will be the most
advantageous for designed aeroplane. Taking takeoff weight estimated in the first project and
wing and thrust loading defined here one can calculate wing reference area and powerplant
thrust or power. As a result certain propulsion system can be selected.
Quite frequently requirements appear contradictory. Designer has to decide which of them is
more important in such cases. For example wing loading optimal for cruise usually is greater
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than maximum wing loading allowed by landing distance requirement. The longest landing
distance requirement has to be satisfied first since the aeroplane can land only on existing
(defined) airstrips. In the case of a contradiction with legal airworthiness regulations,
regulations are always more important since any aeroplane has to satisfy these regulations if
legal operations are predicted. One is not forced to quit any requirements completely since the
closest allowed solution can be always chosen. For example maximum wing loading allowed
by maximum landing distance requirement (pt. B in the Fig.1) can be selected in the problem
described above. On the other hand point A should be selected if designer has to apply the
weakest engine available.
In the case of propeller driven aeroplane power can be considered instead of the thrust. It is
possible to replace thrust in any equation by the following formula:
η#
V
where:
T – powerplant thrust
N – powerplant power
η - propeller efficiency (0,8 can be assumed for constant RPM propellers)
T=
(1)
1.1 Takeoff
Maximum takeoff distance requirement can be analyzed with application of the following
empirical formula:
T
≥
W
1 1
W
0,133 ⋅  
 S  TO C L max σ
s TO
1 1
W
− 3,834 ⋅  
 S  TO C L max σ
#
#
 
(2)
where:
W–
S–
T–
CLmax –
σSTO –
weight
wing reference area
thrust
maximum lift coefficient of the aeroplane
air density ratio (air density at the airport altitude / air density at the sea level), σ=1
can be assumed in the current project
takeoff distance
Takeoff distance should be assumed considering the aeroplane type and dimensions of the
airstrips where the aeroplane is supposed to operate.
Maximum lift coefficient of the clean wing can be estimated with application of Fig.2.
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(a)
(b)
Fig.2 (a) maximum lift coefficient as a function of the airfoil thickness and (b) airfoil
thickness as a function of the Mach number. (Corke)
High lift device type
(takeoff)
CLmax
High lift device type
(landing)
CLmax
Fig.3 High lift devices effect on maximum lift coefficient. (Abbot and Denhoff )
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Effect of high lift devices application can be estimated with help of Fig.3. Please note that
smaller deflections are usually applied for takeoff and greater for landing. The reason for this
is a need for lower drag in the case of takeoff. Therefore it can be assumed that configurations
presented on the left side of the Fig.3 are takeoff configurations and these presented on the
right side are landing configurations.
Moreover increase of the maximum lift coefficient exists only on this part of the wing where
high lift devices are installed. Therefore it should be reduced proportionally to the ratio of the
wing area with high lift device to the total wing area.
1.2 Landing
Similarly to the takeoff maximum landing distance requirement can be analyzed with
application of the following empirical formula:
W σC L max (s L − 122 )
≤
S
0.75
 # 
m2 


(3)
Thrust loading does not appear in this formula because landing is performed with powerplant
working in idle mode, therefore available thrust is not relevant. As a result requirement of the
maximum landing distance appears as a vertical line on Fig.1, i.e. for any T/W, W/S is the
same.
Values of sL, CZmax and σ can be estimated as previously.
ATTENTION: Wing loading calculated from equation (3) is performed at the end of the flight
when fuel tanks are almost empty, therefore it is not comparable with wing loading during
takeoff. One has to calculate the wing loading during takeoff that allows obtaining wing
loading from formula (3) during landing. It can be done with application of the following
procedure:
1) Calculate landing (W/S)L from formula (3)
2) Calculate wing area with application of landing (W/S)L and landing weight from the
formula:
S=
WL
W
 
 S L
(4)
3) Calculate takeoff (W/S)TO with application of formula
W
W
  = TO
SL
 S  TO
(5)
Wing loading described by equation (5) can be presented at the Fig.1. The maximum landing
distance requirement will be satisfied if current wing loading is smaller than calculated above.
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1.3 Climb rate
An aeroplane usually climbs after takeoff. Quite frequently the climb rate is an important
requirement because of air traffic management and safety. Therefore airworthiness regulations
specify climb rates for some aeroplane types and configurations. Climb rate may be
determined also by its application.
Climbing can be described by the following equation:
2
1
W
 T
W
qC D 0 +  
=0
−
−G
 S
 S  qπAe  W
(6)
where :
G = sinγ
dH/dt – climb rate
A – wing aspect ratio (from project 1)
e ~ 0,8 – Oswald coefficient
q = ρV2/2 – dynamic pressure
It can be assumed that the largest
climb rate is achieved for airspeeds equal approximately Vmin+0,25(Vmax-Vmin), Vmax can be
taken from trends analysis (project 1), Vmin and dH/dt from trends analysis or from
airworthiness regulations (e.g. CS 22.49, CS 22.65, CS 23.49, CS 23.65, CS 25.117, CS VLA
49, CS VLA 65)
ATTENTION: Some regulations contain requirements not only for maximum climb rates (e.g.
CS 25.121).
Solving the equation (6) for W/S gives:
2
C


 T
 T
− G  − 4 D0
− G ± 

πAe
W W

W

=
2
S
qπAe
(7)
That means equation (6) makes sense only for T/W greater than:
C D0
T
≥G+2
W
πAe
(8)
This condition can be put on the Fig.1 as a horizontal line to make sure that the final plot is
reasonable. However it is not a line representing the climb rate requirement yet. One can
obtain necessary formula by solving the equation (6) for T/W:
T 
 S  W 1 
 + G
≥  qC D 0   +  
W 
 W   S  qπAe 
(9)
This line should be also put on the Fig.1.
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1.4 Minimum airspeed
Minimum airspeed is very important requirement since flying below this airspeed leads to the
stall. Equation for minimum airspeed requirement can be derived directly from the equation
for the lift force:
W
≤ q ⋅ CL
S
(10)
As previously q is a dynamic pressure, however airspeed necessary to calculate it is equal to
the minimum airspeed that can be taken from trends analysis or airworthiness regulations (e.g.
CS 22.49, CS 23.49, CS VLA 49). Because of safety reasons it is recommended to apply
slightly reduced lift coefficient 1,1*CL = CLmax
1.5 Cruise and other conditions
Some requirements result not from regulations or available resources but from the aeroplane
mission. Cruise efficiency can be given as an example as well as maximum airspeed or
maximum turn rate etc. Fig.1 should contain all relevant requirements of this kind. Some
useful equations can be found in the file „useful equations”.
Let’s analyze cruise efficiency here. In the case of propeller driven aeroplanes maximum
range will be achieved for:
W
= q π ⋅ A ⋅ e ⋅ CD 0
S
(11)
In the case of jet aeroplanes:
C D0 πAe
W
=q
S
3
(12)
In both cases airspeed for dynamic pressure calculation can be taken from trends analysis
since certain cruise airspeed is also frequently required.
ATTENTION: In both cases cruise wing loading is not comparable with takeoff wing loading,
therefore similar procedure has to be applied as in the case of landing. The only difference is
that wing loading (W/S)Cruise calculated from equations (11) or (12) refers to the weight
achieved in the middle of the flight, hence WCruise = WTO – 0,5Wfuel
2. Cost analysis
Cost analysis should be prepared with application of the method presented in T. Corke
„Design of Aircraft” Chapter 12.
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