scaling analysis of propeller-driven aircraft for mars

Scaling Analysis of Propeller-Driven Aircraft

for Mars Exploration

Tianshu Liu

ISAS, JAXA, Sagamihara, Japan

&

Department of Mechanical & Aeronautical Engineering

Western Michigan University, Michigan, USA

Objective

To give a criterion for feasible cruising

flight of propeller-driven aircraft on Mars

and provide a guideline for the design of

such a Martian aircraft

Martian Atmosphere

Density

0142.01087.5 4 HM

Pressure

2.7743.42 HpM

Martian Atmosphere

Temperature Speed of Sound

8.28622.6 HTM 3.2681.3 HaM

Martian Atmosphere

Dynamical Viscosity

Adaption of PIW-VIW Relation in Flight Testing:

Effects of Density and Gravity for Mars Flight

where the standard velocity and the generalized power:

2/1

2/1

)/(

)/(

ref

ref

WW

VVIW

2/3

2/1

)/(

)/(

ref

ref

WW

PPIW

)(

2)(

2

12

3

0VIWS

KWVIWSCPIW

ref

refD

Power-Velocity Relation on Earth

Related to Martian Parameters

where the velocity and the power are related to those on Mars

EE

EEEDE

VS

KWVSCP

23

0

2

2

1

2/1

2/1

)/(

)/(

EM

EMME

WW

VV

2/3

2/1

)/(

)/(

EM

EMME

WW

PP

Scaling Relations: Power and Velocity

where

2/1

min, / SWaV EPE 2/3

min, / SWbSP EPE

4/1

0

2

4/1

3

4

DEC

Ka

a

KaCb

E

ED

2

2

1 3

0

Pmin,E

2/1

M

E

2/1

E

M

2/1

E

2/1

M

E

2/1

E

MPmin,M V

g

g

S

W

g

gaV

S/Pg

g

S

W

g

gbS/P Pmin,E

2/1

M

E

2/3

E

M

2/3

E

2/1

M

E

2/3

E

MPmin,M

Scaling Relations: Power and Velocity

Scaling Relations: Lift Coefficient

In cruising flight on Earth and Mars,

the lift coefficient remains the same, i.e.,

Pmin,MRmax,M V35.1V

Pmin,MRmax,M P146.1P

E,L

2

M

E

M

E

E

MM,L C

V

V

g

gC

E,LM,L CC

Scaling Relations: Propulsive Power

where the power coefficient is defined as

E

M

E,P

M,P

E,prop

M,prop

c

c

P

P

Propulsive Power Scaling:

3

P

3

AS3

Ppropprop VScVSr

J

C4P

)J/C4(r)VS/(Pc 3

PASprop

3

propP

Power Criterion for Cruising Flight on Mars

Propulsive Power Available:

Propulsive Power Required:

where the following scaling relations are used

6/7

EPE,prop WP 3/2

ES WS

6/7

E

E

M

E,P

M,P

PM,prop Wc

cP

6/7

E

2/1

M

E

2/3

E

M

2/1

S

Rmax,M Wg

gb146.1P

Power Criterion for Cruising Flight on Mars

Power Ratio Criterion:

1P/P Rmax,MM,prop

1g

g

c

c

b873.0

P

P2/3

E

M

2/3

M

E

E,P

M,PP

2/1

S

Rmax,M

M,prop

M

1c

c

b106.4

E,P

M,PP

2/1

S3

M

Further Formulation of Power Criterion

for Cruising Flight on Mars

Weight Constraint:

others,Emotor,Eprop,Ewing,EE WWWWW

322113E CxCxCxS/W

where A/WC prop,E1

E,propmotor,E2 P/WC

S/)WW(C others,Ewing,E3

Further Formulation of Power Criterion

for Cruising Flight on Mars

Power Ratio Criterion:

1

CxCxC

xxC

2/3

32211

214M

where M1 S/Ax S/Px E,prop2 S/Wx E3

Disk-to-Wing

Area Ratio

Propulsive Power

Per Unit Wing Area Wing Loading

E3

Pprop

M

3

Pprop

E

3

4JC

JC

)S/A(b

106.4C

Further Formulation of Power Criterion

1)x,x(frC

eARC 21

E,prop

M,prop2/3

E,propmotor4/1

0D

4/3

5M

where

2/3

232121

2121

C/Cxx)C/C(

xx)x,x(f

Power Ratio Criterion:

E3

P

M

3

P

E

2/1

E4

5JC

JC

)S/A(1045.1C

Further Formulation of Power Criterion

where

2/3

232121

2121

C/Cxx)C/C(

xx)x,x(f

Power Ratio Criterion:

1)x,x(frC

CC 21

E,prop

M,prop2/3

E,propmotor

0D

2/3

E,L

6M

Requirements for Cruising Flight on Mars:

0D

2/3

E,L C/C motorr M,prop1C 3C

E3

P

M

3

P

E

2/1

E5

6JC

JC

)S/A(1063.6C

Power Criterion for Cruising Flight on Earth:

A Reduced Case Power Ratio Criterion at Altitude H:

1)H(

c

c

b873.0

2/3

SLSL,P

H,PP

2/1

SH

Condition for the Max Altitude of Typical Aircraft:

1H

263.0b095.1/)H( 3/2

P

3/1

S

3/2

SLmax

km9.11Hmax

Upper Bound of Total Weight

Upper bound by setting 1M

where the weight function is

3/2

21

3/1

2

3/2

121 )yy1(yy)y,y(g

mpw,Emotor,E1 W/Wy mpw,Ewing,E2 W/Wy

3/1

wing,S

3/2

prop,A

3/2

E,propmotorE

3/2

41 )rg(CB

3/13/2

E,prop

3/23/2

4E SPAC)W(UB

)y,y(gWB)W(UB 21mpw,E1E

Upper Bound of Total Weight: Weight Function

Table 1. Scaling Laws and Derived Results for Propeller Aircraft on Earth

Quantity Scaling Law

Upper Limit or Max TO Weight W282.1WMTO

Wingspan 3/1

wing W462.0b

Wing Area 3/2

wing W0262.0S

Mean Chord 3/1W0567.0c

Aspect Ratio 8.15

Body Length 3/1

body W41.0l

Max Body Diameter 3/1

0 W0481.0d

Wet and Wing Area Ratio 4.35

Body Fineness Ratio 8.52

Wing Loading 3/1W53WL

Reynolds Number 2/14

c W102.6Re

Cruise Speed 6/1

Rmax W52.15V

Cruise Power 6/7

aircraft,Rmax W67.1P

Power Available 13.1

prop W25.5P

Engine Weight 8944.0

engine W327.0W

Max L/D 11.9

Oswald Efficiency 9.06.0e

Parasite Drag 044.002.0CwingPara S,D

Induced Drag e/0032.0CwingIn S,D

Propulsive Efficiency wingS,Dprop C32.29

Note:

(1) Units: Newtons for weight, m for length, m2 for area, m/s for velocity, Watts for power,

kg/m3 for density, and Newtons/m

2 for loading.

(2) The mean relative errors associated with the scaling laws are indicated in Liu (2006).

Scaling Laws for Typical

Propeller Aircraft on Earth

Disk-to-Wing Area Ratio for

Typical Propeller Aircraft on Earth

Service Ceiling for Typical Propeller Aircraft

on Earth

Parametric Domains for Cruising Flight of

1- and 2-Propeller Aircraft on Earth

Parametric Domain for Cruising Flight of

The Helios Prototype on Mars

Table 2. Parameters of Several Propeller-Driven Aircraft

Parameters S P S/A b Cruise maxH

M

Typical Aircraft 0.0262 5.25 0.18 0.1 11.8 km 0.039

Helios 0.564 0.67 0.24 0.025 16.1 km 0.12

Sample Martial

Aircraft

0.096 31 0.55 0.12 21.5 km 1.07

Design Parameters of the JAXA Martian Aircraft

Wing Parameters

wb 2.42 m Wing span

wc 0.48 m Mean wing chord

AR 5.11 Wing aspect ratio

S 1.15 m2 Wing area

wing,S 0.343 kg/m2 Wing surface density

Design Parameters of the JAXA Martian Aircraft

Propulsion Parameters

propD 0.636 m Propeller diameter

proph 5 mm Mean blade thickness

A/S prop 0.3 Propeller solidity

prop 1797 kg/m3 Propeller material density (carbon fiber)

E,prop 0.8 Propeller efficiency on Earth

M,prop 0.8 Propeller efficiency on Mars

n 2 Number of propellers

E)S/A( 0.18 Reference disk-to-wing area ratio

motorr 2720 W/kg Motor power-to-mass ratio

batteriesE 420 kJ/kg Energy density of batteries

Initial Input Aerodynamic Parameters

0DC 0.02 Zero-lift drag coefficient

e 0.8 Oswald efficiency

Upper Bound of Total Weight

of the JAXA Martian Aircraft

Parametric Domain for Cruising Flight of

the JAXA Martian Aircraft

Relation between Motor Power and Mass:

Specific Power of Motor

JAXA

Noth et al. (2009)

Table 4. Mass Distribution of Sample Propeller-Driven Martian Aircraft

Em 4.24 kg Total mass

wing,Em 0.394 kg Wing mass

prop,Em 0.175 kg Propeller mass

motor,Em 1.16 kg Motor mass

others,Em 2.51 kg Other mass

Output Parameters of the JAXA Martian Aircraft

Table 5. Performance Parameters of Sample Propeller-Driven Martian Aircraft

M 1.07 Power ratio on Mars

Rmax,MV 65 m/s Cruising velocity on Mars

MR 260 km Cruising range on Mars

ME 1.1 h Cruising endurance on Mars

Rmax,MP 84 W Cruising power required on Mars

M,propP 90 W Propulsive power available on Mars

M,LC 0.46 Lift coefficient on Mars

S/WM 14.5 N/m2 Wing loading on Mars

Conclusions

The power ratio criterion is given for cruising

flight of propeller-driven aircraft on Mars

The power ratio criterion is validated by

examining typical propeller-driven aircraft

and the Helios on Earth in several

interesting cases

The power ratio criterion indicates that

the preliminary design of the JAXA Martian

aircraft could be feasible for cruising flight

on Mars

Conclusions

The power-to-weight ratio of a DC motor:

= 3000 W/kg

The propeller efficiency:

= 0.8

Specific Requirements for JAXA Martian Aircraft:

The lift-to-zero-lift-drag ratio:

= 26 0DE,L C/C

Minimizing the weights of wing

and propellers

Further Topics

Aerodynamics optimization:

Wing/airfoil design and testing

Propeller optimization/design and testing

Optimization under suitable constraints

on solar power source, batteries, structures

etc.