aa283 ch08 multistage rockets

7/18/2019 AA283 Ch08 Multistage Rockets

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AA283

Aircraft and Rocket Propulsion

Chapter 8 - Multistage Rockets



7.1 Notation



7.2 Analysis

(8.2)

(8.3)

(8.4)



(8.5)

(8.6)

(8.7)



8.3 The variational problem



8.4 Example - Exhaust velocity andstructural coefficient the same for all stages.



There is very littleadvantage to using

more than about

three stages.



C 1 = 2500 C

2 = 4250 C

3 = 4250

V 3 =C

1 Ln

1+ ! 1

" 1+ !

1

# $ %

& ' ( +C

2 Ln

1+ ! 2

" 2 + !

2

# $ %

& ' ( +C

3 Ln

1+ ! 3

" 3 + !

3

# $ %

& ' (

V 3 = 2500 Ln

1+ 0.321

0.05 + 0.321

! " #

$ % & + 4250 Ln

1+ 0.466

0.071+ 0.466

! " #

$ % & + 4250 Ln

1+ 0.603

0.191+ 0.603

! " #

$ % & =10429 M / sec



The final velocity of a multistage system can be expressed as

V n = C

i ln

M 0i / M

Pi

M 0i / M

Pi !1

" # $

% & ' i=1

n

(Consider a two stage design

V 2 = C 1 Ln

M 01 / M

P1

M 01 / M

P1 !1

" # $

% & ' +C 2 Ln

M 02 / M

P2

M 02 / M

P2 !1

" # $

% & '

M s1 =! 1

1" ! 1

# $ % & ' ( M P1 M s2 =

! 2

1" ! 2

# $ % & ' ( M P2

M 01 = M

S 1 + M

P1 + M

S 2 + M

P2 + M

L

M 02 = M

S 2 + M

P2 + M

L

Alternative approach.

Express payload mass in terms of propellant masses and payload fraction

M L =

!1" !

# $ %

& ' (

1

1" ) 1

# $ %

& ' ( M

P1+

!1" !

# $ %

& ' (

1

1" ) 2

# $ %

& ' ( M

P2



M 01

M P1

=1

1! "# $ %

& ' (

1

1! ) 1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' ( M

P2

M P1

#

$ %&

' (

M 02

M P2

=1

1! "# $ %

& ' (

"1! )

1

# $ %

& ' (

1

M P2 / M

P1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' (

#

$ %&

' (

Express stage mass ratios in terms of propellant mass ratios

V 2 = C

1 Ln

M 01 / M

P1

M 01 / M

P1 !1

" # $

% & ' +C

2 Ln

M 02 / M

P2

M 02 / M

P2 !1

" # $

% & '



V 2 = C

1 Ln

1

1! "# $ %

& ' (

1

1! ) 1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' ( M

P2

M P1

#

$ %&

' (

1

1! "# $ %

& ' (

1

1! ) 1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' ( M

P2

M P1

#

$ %&

' ( !1

#

$

%%%%

&

'

( ( ( (

+C 2 Ln

1

1! "# $ %

& ' (

"1! )

1

# $ %

& ' (

1

M P2

/ M P1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' (

#

$ %&

' (

1

1! "# $ %

& ' (

"1! )

1

# $ %

& ' (

1

M P2

/ M P1

# $ %

& ' ( +

1

1! ) 2

# $ %

& ' (

#

$ %&

' ( !1

#

$

%%%%

&

'

( ( ( (

For given values of

! ,C 1,C

2,"

1,"

2

The final velocity is a function of the propellant ratio.

V 2 = F

M P2

M P1

! " #

$ % &

It is now just a matter of differentiating with respect to the propellantratio to identify a maximum.



Application - Mars Sample Return Campaign

2020-2026(ish)

Lander to Surface

MAV

Mars Orbit Rendezvous

Return to Earth

•

Next major step in Mars Science• Requires international collaboration

• Multiple new developments

– Mars Ascent Vehicle (MAV)

– Sample acquisition and handling

– Precision entry descent and landing

Ashley Chandler ’s

PhD project



Critical Challenge: Mars Environmental Conditions

0 5 10 15 20 25!120

!100

!80

!60

!40

!20

0

20

40

Local Time of Day, hours

S u r f a c e T e m p e r a t u r e ,

C

Min: !111C

Max: 24C

Average:!

60C

•

Diurnal/seasonal minima and maxima (-111C to 24C) – Nitrous oxide freezing point is -90.8 C.

– Paraffin-based fuel is primarily crystalline with a low glass transitiontemperature at -108 C.

Data from the NASA AmesResearch Center Mars Global

Climate Model for HoldenCrater.



Use a two stage design for the Mars ascent vehicle

! 1 = 0.13

! 2 = 0.155

C 1 = 2883

C 2 = 3026

! = 0.147

M P2

/ M P1

V 2

0.354

Note that the finalvelocity is actually

quite insensitive to

the propellant mass

ratio giving the

designer quite a bit

of flexibility.

Notional values.



MAV System

Two StageSolid

40% MarginHybrid

20% MarginHybrid

S t a g e

1

Mass (kg) 215.6 145.89 128.86

Structural Coefficient 0.170 0.189 0.167

Max. Pchamber (MPa) 4.83 1.72 1.72

Delivered Isp (m/sec) 2813 2952 2951

!V (m/sec) 2530 1675 1675

S t a g e

2

Mass (kg) 86.2 91.37 83.57

Structural Coefficient 0.175 0.169 0.147

Max. Pchamber (MPa) 6.45 1.38 1.38

Delivered Isp (m/sec) 2813 2977 2976

!V (m/sec) 1845 2700 2700

Total Ideal !V (m/sec) 4375 4375 4375

Maximum Diameter (m) 0.442 0.541 0.524

Vehicle Length (m) 2.56 3.829 3.747

Payload Mass (kg) 36 36 36

Gross Lift Off Mass (kg) 301.8 273.3 248.4

Igloo Mass (kg) 50 0 0

System Total Mass (kg) 351.8 273.3 248.4

System Parameters - proposed design

! = 0.132



Volumetric Constraint

•

Hybrid MAV can be repackaged to fit within the volumetric and

center of gravity constraints

Volume Constraint

In order to confirm the design it is necessary to fly it to orbit.



Kepler ’s equations govern the motion of objects near gravitating

bodies. This is called the two body problem.

!! x(t )+ M Planet G

x(t )

r(t )3 = 0

!! y(t )+ M Planet G y(t )

r(t )3 = 0

!! z(t )+ M Planet G z(t )

r(t )3 = 0

Kepler ’s Equations

G = 6.67 !10"11

m3/kg-sec2

r t ( ) = x t ( )2

+ y t ( )2

+ z t ( )2

F = !G Mmr2

r

r" # $ % & '

Universal gravitational

constant



Orbital Period

Kepler ’s theory gives



Orbital Periods of the Planets about the Sun



Mars Ascent Vehicle - launch to orbit

Equations of motion

!! x(t )+ m MARS

G x(t )

r(t )3 +

F x DRAG

(t )

m(t )!

F xTHRUST

(t )

m(t )= 0

!! y(t )+m MARS

G y(t )

r(t )3 +

F y DRAG (t )

m(t )!

F yTHRUST (t )

m(t )= 0

!! z(t )+m MARS

G z(t )

r(t )3 +F z DRAG (t )

m(t )!

F zTHRUST (t )

m(t )= 0

Mars radius=3.376 x 106 m

Mars mass=6.418 x 1023 kg

g MARS =

m MARS

G / r2= 3.756 m/sec2

! MARS

=

r3

mG= 948.03Mars time scale: sec

Mars velocity scale: U MARS

=

mG

r= 3561 m/sec

x

y z

G = 6.67 !10"11 m3/kg-sec2

Universal gravitational constant

Vehicle mass

m(t )



Aerodynamic drag

C D

C D0

M

Drag

t /!

CD0=0.2



Angle between velocity vector and planet radius

V

r

x

y z

Launch point

!



Surface speed at the

equator=241.17 m/sec

Vertical launch12.5 sec

First stage

gravity turn

36.6 sec

Coast620 sec

Second stagegravity turn to orbit

39.4 sec

d !

dt ROCKET

= 0.038 degrees/sec

d !

dt

= 0.132 degrees/sec

d !

dt = 0.0007 degrees/sec

Gravity turn

Launch trajectory

F THRUST

!V = 0

V = ( ! x, ! y, ! z)

See the paper Universal

Gravity Turn Trajectories onmy website.



Launch and orbit trajectory

Maximum altitude

557.9 km

Minimum altitude 527.5 km



The mass ratios can be written in terms of the payload fraction as follows.

M 01 / M

P1 =

1

1! "# $ %

& ' (

1

1! ) 1

+1

1! ) 2

M P2 / M

P1( )+1

1! ) 3

M P3 / M

P1( )# $ %

& ' (

M 03 / M

P3 =

1

1! "# $ %

& ' (

1

1! ) 3

+"

1! ) 1

1

M P3 / M

P1

# $ %

& ' ( +

"1! )

2

M P2 / M

P1

M P3 / M

P1

# $ %

& ' (

#

$ %&

' (

M 02 / M

P2 =

1

1! "

#

$ %

&

' (

1

1! ) 2

+"

1! ) 1

1

M P2 / M

P1

#

$ %

&

' ( +

1

1! ) 3

M P3 / M

P1

M P2 / M

P1

#

$ %

&

' (

#

$ %

&

' (

V 3 = C

1ln

M 01 / M

P1

M 01 / M

P1 !1

" # $

% & ' +C

2 ln

M 02 / M

P2

M 02 / M

P2 !1

" # $

% & ' +C

3ln

M 03 / M

P3

M 03 / M

P3 !1

" # $

% & '

Three stage design



For given values of

! ,C 1,C

2,C

3,"

1,"

2,"

3

The final velocity is a function of of the propellant ratios.

Differentiate with respect to the two variables toidentify a maximum.

V 3 = F

M P2

M P1

, M

P3

M P1

! " #

$ % &



! 1 = 0.116

! 2 = 0.117

! 3 = 0.148

C 1 = 3040 m /sec

C 2 = 3210 m / sec

C 3 = 3350 m /sec

" = 0.01125

#V = 10,800 m /sec

Payload mass = 85 kg

M 01

= 7557 kg

Three stage launch vehicle for small satellites

aa283 ch08 multistage rockets

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