1 Introduction

In the previous report,an estimation of the minimum wing loading of the

Phoenix Jet was performed and the maximum (T/W) value was also ascer-

tained. It was also decided that the powerplant for the Phoenix Jet would

be Rolls Royce BR710C4-11.

The following values were obtained,

W/S=3056.41 N/m

2

(W/S) take-o equivalent=3137.4 N/m

2

Max. T/W=0.387

In this report,a new estimate of the aircrafts take o gross weight is performed

factoring into account the wing loading value obtained. The geometric sizing

of the wing,tail and control surfaces is also performed.

2 Third Weight Estimate

In second weight estimate,take-o weight of the aircraft has been estimated

using basic empirical relations and historical data.

In the current weight estimate a more rigorous set of relations are used

to obtain the take o gross weight in an iterative manner. The new relations

factor into account the wing loading value obtained in the previous report.

2.1 Empty Weight Fraction Estimate

The empty weight to the gross weight ratio (W

E

/W

0

)is estimated using an

empirical but improved statistical relation. The expression is given by

[2]

,

WEW0

= (a+ b W c1o + ARc2 + (T

W)c3 + (

W

S)c4 +M c5max) Kvs (2.1)

From reference [2],for a typical business jet,we have,

a=0.32 T/W

0

=0.387

b=0.66 W

0

/S=3137.4 N/m

2

c1=-0.13 M

max

=0.87

c2=0.30 AR=7.5

c3=0.06 K

vs

=1.00 (for xed sweep)

1

c4=-0.05

c5=0.05

By substituting the above values we get,

WEW0

= (0.32 + 0.7576 W0.1300 ) (2.2)

2.2 Fuel weight fraction estimate

The amount of fuel required during a mission depends on the mission pro-

le. The fuel fraction can be estimated based on approximations of fuel

consumption and aerodynamics.

Figure 2.1: Mission Prole of Phoenix Jet

[8]

*Figure obtained from reference [8]

For analysis,the mission prole shown in gure 2.1 is segmented into var-

ious phases. These phases are (1) Engine start,warm up; (2) Taxi; (3) Take-

o; (4) Climb; (5) Cruise; (6) Loiter; (7) Reserved cruise and (8) Descent and

Land. W

i

denotes weight of the aircraft at the end of phase i e.g. W

1

denotes weight of aircraft at the end of Engine start and warm up phase.

2.2.1 Phase 1: Engine start,warm up

The weight ratios at the end of phases Engine start,warm up,Taxi,Take-o

and Climb are chosen by following the standards given in Reference [5].

2

W1W0

= 0.999 (2.3)

2.2.2 Phase 2: Taxi

W2W1

= 0.995 (2.4)

2.2.3 Phase 3: Take O

W3W2

= 0.995 (2.5)

2.2.4 Phase 4: Climb

An empirical relation for the climbing and acceleration segment weight frac-

tion by relating it to the Mach number is given by

[2]

,

W4W3

= 1.0065 0.0325M (2.6)M is the Mach number for a subsonic jet accelerating to a Mach number

of 0.87 (cruise Mach number).

Using the value of Mach number we get,

W4W3

= 0.978 (2.7)

2.2.5 Phase 5: Cruise

For a cruising aircraft,the fuel weight fraction can be determined quite well

using the Breguet range equation. For a jet,the range can be calculated by

[2]

,

R = (L

D)(V

C) ln(

W5W4

) (2.8)

which gives,

W5W4

= eRCV ( L

D)(2.9)

Where R is the required design range which is 12000km for the specied

mission of the Phoenix Jet,C is the specic fuel consumption (SFC) of the

aircraft. The typical C value for business jet is found to be 0.5hours

-1[5]

.

3

The cruise speed of Phoenix jet is taken to be 0.87 Mach number (approx.

258m/s (923.94 Km/hr) over 12,192 m (40000 Ft.) altitude). The (L/D)

ratio is calculated by using the following equation

[2]

,

(L

D)cruise =

1

qCDoWS

+WS

qpiARe(2.10)

Actual wing loading at cruise is taken i.e. W/S = 3056.41 Nm

-2

.

C

Do

=0.015 and Density(at 40,000 ft)= 0.30229 kg m-2.

AR=7.5 and e=0.8

(L

D)cruise = 15.35 (2.11)

This gives the value of W

5

/W

4

as follows,

W5W4

= 0.6134 (2.12)

2.2.6 Phase 6: Loiter

While loitering the aircraft ies at maximum L/D ratio.The expression for

obtaining maximum L/D is as follows

[2]

:

(L

D)loiter = (

L

D)max =

( LD

)cruise

0.866= 17.725 (2.13)

The weight ratio at the end of loiter phase is calculated using the En-

durance equation,

E = (L

D)(

1

C) ln(

W6W5

) (2.14)

which gives the following expression for the weight ratio,

W6W5

= eECLD(2.15)

E,the time taken for loiter,is 20 minutes for Phoenix jet and the value of

C is found to be 0.4hours

-1

from reference [2]. This gives ,

W6W5

= 0.9925 (2.16)

4

2.2.7 Phase 7: Reserved cruise (Alternate)

It is assumed that the alternate airport is located within the radius of 150 Km

from previously destined airport. Due to the short distance,aircraft cannot

cruise in the normal cruise speed. Assuming the acceptable time taken for

reaching alternate airport is 15 minutes,implies that the required cruising

speed should be 600 Km/hr. Therefore,R = 150 Km,V = 600 Km/hr. Using

Breguet range equation we get,

W7W6

= 0.9941 (2.17)

2.2.8 Phase 8: Descent and Landing

The weight ratio at the end of this phase is chosen by following the standards

given in reference [2].

W8W7

= 0.995 (2.18)

Having calculated the weight fractions for the various phases,the cascade

of the fractions is shown below,

W8W0

=W8W7

W7W6

W6W5

W5W4

W4W3

W3W2

W2W1

W1W0

= 0.5824 (2.19)

Typically a 6% is allowed for reserved and trapped fuel. The fuel fraction

is estimated using equation from reference [2] given by,

WFW0

= 1.06(1 W8W0

) = 0.4426 (2.20)

Wcrew +Wpayload = 1740kgW0 = Wcrew +Wpayload +

WFW0

W0 + WEW0 W0The resultant of the above two equations is used to obtain a recursive

relation which is iteratively solved. The equation is as follows,

W0(new) = 1740 + 0.4426 W0 + (0.32 + 0.7576 W0.1300 ) W0 (2.21)An iterative process was executed in MATLAB

TM

for an initial guess W

0

= 25,000kg. Design take-o weight i.e the above third weight estimate after

5

iterations was found to be W

0

= 38,380kg,which is 3055kg (8.6%) greater

than the value obtained from the second weight estimate(W

0(old)

=35,325kg).

Substituting this W

0

value in equations (2.20) and (2.2) respectively gave fuel

weight W

F

= 16,986kg and Empty weight W

E

=19,654kg. The Figure(2.2)

shows the graph between W

0

and number of iterations.

Figure 2.2: Plot of W

0

vs Iterations

3 Geometry Sizing

3.1 Fuselage

Fuselage is the central body of an aircraft,to which the wings and tail assem-

bly are attached and which accommodates the crew passengers and cargo

[1]

.The

sizing of the fuselage involves determining the geometry of the fuselage using

design parameters,historical data and statistical data.

1. Fuselage Width: It is set by the number of seats abreast,seat width and

6

the aisle width. The depth is set to accommodate the cargo containers

below the oor and the head room above the aisle.

For the Phoenix Jet,2 seats abreast conguration is chosen to provide

maximize comfort. A typical executive seat is 0.7 m wide. Providing a

generous 0.8 m aisle width,the cabin internal width would be 2.2m.

Since a pressurized cabin is used a circular section is preferred for the

fuselage. This is because it resists the internal pressure loads by hoop

stress most eectively in this conguration. Giving a 0.2 m allowance

for the pressure cabin structure the total cabin width becomes 2.4m

CabinWidth = (0.7 2 + 0.8 + 0.2)m = 2.4m (3.1)

Since the fuselage has a circular cross section the total cabin width can

be taken as diameter of the cross section.

2. Fuselage Length: The empirical relation giving the fuselage length as

a function of gross weight is given by the following relation,

F.L. = aW c0 (3.2)

(F.L.=Fuselage Length)

From the fuselage length data obtained from similar airplanes,a graph

between ln(F.L.) vs ln(W

0

) was plotted and the values of a and c were

obtained using a linear curve t.

Table 3.1:Fuselage length and W

0

of similar aircrafts

[9]

Name F.L.(m) W

0

(kg)

Dassault Falcon 7X 23.19 31,752

Gulfstream G500 29.28 38600

Gulfstream G550 29.38 41,277

Bombardier Global 5000 29.5 42,071

Bombardier Global Express XRS 30.3 44,500

Gulfstream G650 30.4 45,177

Bombardier Global 8000 33.71 47,536

The equation used is given by

ln(F.L.) = ln(a) + c ln(Wo) (3.3)

7

The plot is given in Figure (3.1). From the graph the value of c=0.809

and a=5.374*10

-3

are obtained.

Figure 3.1: Plot of ln(F.L.) vs ln(W

0

)

Using the values of a and c obtained from the plot and the W

0

value

obtained in the third weight estimate we get the fuselage length value

to be 27.47 m.

The length of the cabin is determined by the seat pitch. The seat pitch

is taken as 1.8m which is the eective length of a reclining luxury seat.

The number of passengers in Phoenix Jet is 10(including air-hostess).

Since 2 seats abreast conguration was chosen,the length of the cabin

is 9 m

[3]

. The length of the service module consisting of catering and

restrooms is taken to be 4 m

[4]

. The fuselage length is the sum of cabin

length,service module length and the front & rear prole shaping(non

cabin length). The front accommodates the ight deck and the rear

provides attachment for the tail surfaces . The value of the non cabin

length is chosen as 15 m in comparison to the values obtained from

similar aircraft.

The total fuselage length is hence obtained as (9+4+15)=28 m. This

8

Table 3.2 : Non cabin length of similar aircrafts

[9]

Name Non Cabin Length(m)

Dassault Falcon 7X 11.29

Gulfstream G500 16

Gulfstream G550 16

Bombardier Global 5000 16.56

Bombardier Global Express XRS 15.57

Gulfstream G650 16.13

Bombardier Global 8000 15.54

value obtained is found to comparable to the fuselage length value

obtained using the empirical relation.

3.2 Wing

3.2.1 Aspect Ratio

Aspect ratio is dened as the ratio of the square of wing span to that of the

wing area. For a typical business jet the aspect ratio is expected to be 7.5

[2]

.

3.2.2 Wing Area and Wing Span

The expression for obtaining wing area is as follows,

S =W0

(WS

)takeoff(3.4)

W

0

=38,380kg (W/S)

take o

= 3137.4 N/m

2

(From Previous Report).

We obtain S=119.88 m

2

We use the following formula for obtaining the wing span(b),(From de-

nition of aspect ratio)

b =

(S AR) (3.5)This gives wing span as b=29.985 m.

Table (3.3) gives the wing area and wing span for similar aircraft.

From the Table (3.3) we observe that the wing span and area of the

Phoenix Jet is comparable to the wing span and area of similar aircraft.

9

Table 3.3: Wing span and Wing Area for Similar Aircrafts

[9]

Aircraft Wing Span(m) Wing Area(m

2

)

Dassault Falcon 7x 26.21 70.7

Gulfstream G500 28.49 105.6

Bombardier Global 5000 28.65 94.94

Gulfstream G650 30.41 119.2

3.2.3 Taper Ratio

The ratio of the chord length of the airfoil section at the wing tip to that of

the airfoil section at the wing root is dened as taper ratio. The selection of

the taper ratio involves many considerations.

A constant chord rectangular wing is easier and cheaper to manufacturebut is aerodynamically less ecient.

An elliptical wing is the most aerodynamically ecient but not easy tomanufacture.

Taper ratios of order 0.2-0.5 are found to be slightly less ecient thanelliptical wing congurations.

Moreover,an increased taper leads to reduction in the root bendingmoment as the centre of lift moves inboard as taper is increased.

This decreases the structural weight of the wing considerably. There-fore we go with a taper ratio() of 0.2

3.2.4 Sweep Angle

Sweepback is mainly used to reduce drag from local ow velocities at or

near to supersonic speeds. Sweeping the wing planform (usually backwards)

allows thicker wing sections to be used and delays the onset of critical Mach

number. For Phoenix Jet,the sweep angle (leading edge LE) is taken tobe 33

0

from the graph taken from reference [2],(see Figure 3.2) and for a

Mach number of 0.87. The quarter chord sweep angle(C/4) is given by theequation

[2]

,

tan LE = tan C/4 +1

AR(1 + )(3.6)

10

Figure 3.2: Leading Edge Sweep vs Maximum Mach No.

[2]

AR= Aspect Ratio and =Taper Ratio.C/4 = 29.27

0

3.2.5 Root and Tip Chord Length

Figure 3.3: Illustration of Root and Tip Chord length

[2]

The root chord length is given by the equation:

11

Croot =2S

(1 + )b(3.7)

where S is the wing area,b is the wing span and is the taper ratio.Substituting S=119.88 m

2

,b=29.985 m and =0.2 ,the Croot

=6.663 m.

The tip chord length is given by the equation :

Ctip = Croot (3.8)Hence the value of C

tip

=1.3326 m.

3.3 Tail

The tail assembly of the aircraft is also called the empennage. The tail is

essentially a device that provides stability and control to the aircraft. The

eectiveness of the tail,which is mostly used for trimming the aircraft,depends

on the lift produced by the tail and the distance of the tail from the centre

of gravity of the airplane.

Figure 3.4: The conventional tail assembly of an aircraft.

[6]

*Figure obtained from Reference [6]

The front of the horizontal portion of the tail is the horizontal stabilizer.

It provides pitch stability to the aircraft. The rear section is called the

12

Figure 3.5: Rear View of the T-Tail.

[6]

*Figure obtained from Reference [6]

elevator and is usually hinged to the horizontal stabilizer. The elevator is a

movable airfoil that controls the up-and-down motion of the aircraft's nose.

The vertical structure consists of the vertical stabilizer,which constitutes the

front portion and provides yaw stability to the aircraft and the rudder,which

attached at the rear of vertical stabilizer and provides directional control.

A T-tail layout has been chosen for the Phoenix Jet aircraft which is shown

in Figure 3.5

[6]

The T-tail has been chosen based on data available from similar airplanes

(all possess a T-Tail) and also certain advantages of the T-tail.

In a T-tail conguration the horizontal stabilizer is mounted on top of the

n(vertical portion of tail). Owing to this design layout,the tail is kept

out of the inuence of engine wake. This provides smoother ow over the

tail,predictable design characteristics and better pitch control. The T-tail

also provides a better glide ratio and experiences lesser fatigue because of the

nullied eect of downwash on the tail. The ns of T-tails should be stronger

to withstand the forces acting on the horizontal section,as a result,the T-tails

are generally heavier than the conventional tails. The T-tail also accounts

for a smaller vertical tail because of the end plate eect.

The typical aspect ratio of the horizontal tail is taken as 5 and that of the

vertical tail is taken as 1.2

[7]

. The taper ratios for the horizontal tail (H)istaken as 0.5 and that of the vertical tail section (V ) is taken as 0.9[7]

.

13

The leading edge sweep of the horizontal tail is set to be 5

0

greater than

that of wing

[2]

. This is done to ensure that the tail stalls after the wing and

also ensures that the critical Mach number of tail is greater than that of the

wing. Hence the sweep of the horizontal tail is taken as 38

0

. The vertical

tail sweep is taken 35

0

(slightly greater than the wing sweep angle).

Additionally the horizontal tail is 10% thinner than that of the wing to

increase the critical Mach number of the tail.

The primary purpose of the wing is to counter the moments due to the wing.

Hence the tail size is expected to be partially dependent of the wing size.

The tail volume coecient is a parameter which gives an estimate of the size

of the tail using the wing size. The expression for the tail volume coecient

is given by the following formula

[2]

,

cvt =LvtSvtSwbw(3.9)

cht =LhtShtCwSw(3.10)

Figure 3.6: Pictorial Representation of moment arm

[2]

L

ht

and L

vt

are the moment arms measured from the tail quarter chord to the

wing quarter chord.

b

w

and S

w

are the wing span and wing area respectively.

14

Cw

is the wing mean chord length.

c

vt

and c

ht

are the vertical and horizontal tail volume coecients respectively.

S

vt

and S

ht

are the vertical and horizontal tail areas respectively .

The typical values of c

vt

and c

ht

for a jet aircraft is 0.0855 and 0.95 respectively

[2]

.

The tail arm is dened as the distance between the wing quarter chord point

and the tail.

The Phoenix Jet has aft-mounted engines (taken by comparing with similar

aircrafts) and hence the tail arm(for both horizontal and vertical tail) is taken

as 45% of fuselage length

[2]

. The moment arm then is calculated as 12.6 m.

b=29.985 m and Wing area=119.88 m

2

By rearranging Equations (3.9) and (3.10) we obtain tail areas as :

S

vt

= 24.249 m

2

S

ht

=36.13 m

2

AR

vt

=1.2 and AR

ht

=5.

Using the aspect ratio values and wing areas we obtain the span of the

tails as,

b

vt

= 5.39 m b

ht

=13.44 m. (b

vt

and b

ht

are spans of vertical tail

and horizontal tail respectively).

By using Equation (3.7) and (3.8) an estimation of the tail root and

tip chord length is done and the values are as follows,(the corresponding

area,span and taper ratios are used to obtain the chord length).

C

root(vt)

=4.735 m C

tip(vt)

=4.26 m

C

root(ht)

=3.584 m C

tip(ht)

=1.79 m

3.4 Control Surfaces

An aircraft possesses 3 main control surfaces- rudder( for directional (nose)

control) ,elevator (for pitch control) and the ailerons (for roll control).

The aileron chord is assumed to be 20% of wing chord

[2]

. The rudder and

tail are assumed to have 35% of the corresponding tail chord

[2]

. The chord

length for the rudder and elevator are chosen as above to partially nullify the

eect of control ineectiveness of T-tail at very high angle of attack.

The elevator and rudder have 90% of the corresponding tail span beginning

from the fuselage

[2]

. The aileron span is taken as 0.4 using the chord ratio of

0.2 from Figure 3.7

15

Figure 3.7: Graph of Aileron-wing span ratio vs chord ratio

[2]

The following dimensions were obtained for dierent control surfaces,

Aileron:

b=11.994 m C=0.7995 m

Rudder:

b=4.851 m C=1.52 m

Elevator:

b=12.096 m C=0.932 m

* b and C represent the span and chord length respectively. C is calculated

using the mean chord length of the corresponding surface.

4 Conclusion

The third weight estimate of the Phoenix Jet was performed and the gross

take o weight was obtained using an iterative method. The values obtained

are as follows,

W

0

=38,380 kg. (Take o Gross Weight )

W

F

=16,986 kg. (Fuel Weight at take o)

16

WE

=19,654 kg. (Empty weight of Phoenix Jet)

The fuselage length was calculated and is found to be 28m. The geometry

sizing of the wing,tail and control surfaces was performed.Some important

geometric parameters are listed below,

Wing:

S=119.88 m

2

(wing area) and b=29.985 m (wing span) AR=7.5

Tail:

S

vt

= 24.249 m

2

S

ht

=36.13 m

2

AR

vt

=1.2 and AR

ht

=5.

b

vt

= 5.39 m b

ht

=13.44 m. (where ht and vt represent horizontal

and vertical stabilizers respectively.)

17

References

[1] www.thefreedictionary.com

[2] D.P Raymer(1995),Aircraft Design: A Conceptual Approach,Publisher

- AIAA Education Series.

[3] www.google.com

[4] Lloyd.R.Jenkinson-(2003),Aircraft Design Projects for Engineering Stu-

dents,Butterworth Heinemann Publications

[5] Dr. Roskam-(1985),Aircraft Design,Publisher-Roskam Aviation and En-

gineering Corporation.

[6] www.wikipedia.org

[7] http://adg.stanford.edu/aa241/stability/taildesign.html

[8] Santosh Ballal Amarnath, Ngugen Thanh Tue, Rui Tang-21st century

Business Jet-Aircraft Design Project-Departmental of Mechanical Engi-

neering, The University of Adelaide

[9] www.planes.ndthebest.com

18