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Space Shuttle External Fuel Tank Model*

* This model was provided by Dr. Olivier de Weck from MIT and originally appears in: T. Schuman, O. L. de Weck, and J. Sobieski, 2005, “Integrated System-Level Optimization for Concurrent Engineering with Parametric Subsystem Modeling,”    1st Multidisciplinary Design Optimization Specialist Conference, Austin, TX, AIAA, AIAA-2005-2199.

= an input variable that you can change

= overall objective function (ROI) that you are trying to maximize

= intermediate objective functions that you can maximize or minimize

= constraints that must be satisfied (formulated to be < or = 0)

VALUE

= a nominal (baseline) value that is constant in the model

Design Variable Values

Units (N,cm, m, km, kg) unless otherwise noted

1atm = 10N/sq cm

Design Variables - nondimensional

Ln

Rn

t1n

t2n

t3n

h/Rn

1.5

1.8

0.25

0.25

0.25

3

Nominal Values of Design Variables

L

R

t1

t2

t3

h/R

h=R*(h/R)

spcweight

4150

450

0.7

0.8

0.75

1

kg/cm cu

Actual values of design variables

2430

0.0028

L

R

t1

t2

t3

h/R

stress allow.

cm

cm

cm

cm

N/cm sq

6225

810

0.175

0.2

0.1875

3

40000

cost Al-alloy

cost Seam

payload

orbit height

orbital speed

pressure

dollar/kg

dollar/cm

N

km

km/sec

pi

N/cm sq

6

0.001

30000

250

8

3.1416

70

l=(R^2+h^2)^(1/2)

2561.4449

ANALYSES

Surfaces and Volumes

Length of Weld Seams

H-Sphere: surface=2*pi*R^2;

volume=2/3*pi*R^3

Seam length in Cyl.=4*L

24900

4122407.52

1113050030

Seam leng. in Sph.=2*pi*R

5089.392

Cylinder: surface = 2*pi*R*L; volume=pi*R^2*L

Seam len. cyl&sph=2*pi*R

5089.392

31681465.2

1.2831E+10

Seam. Len. Cyl&Cone=2*pi*R

5089.392

Cone: surface=pi*R*l w/o bottom; vol = (1/3)*pi*R^2*h

Seam len. Cone = 4*l

10245.7796

6518098.6

1669575046

Seam len. total = sum of above

50413.9556

Tank surface = Sphere +cyl+cone

Volume=Sphere Vol.+Cyl.Vol.+cone vol.

Nom Vol=

42321971.3

<---OBJ

15613618482

MUST BE CONSTANT

2926400400

Nom Surf Area

13905909.9

L to be varied accordingly

CON--->

4.33543478

<--g=Vol/VolNom-1=0

Weights and Costs of Material

Tank material weight = ((Cyl.surface*t1)+(Sphere surf.*t2)+ConeSurf*t3)*(spcweight)

OBJ--->

21254.4679

Nominal Tank Weight=

27737.7969

Tank material cost =  (Cyl. Surface*t1*spcweight*(unit cost for t1) + (Sphere surf.*t2*spcweight*(unit cost for t2)+

(ConeSurf*t3*spcweight*(unit cost for t3))

Unit cost per weight cost function.

Cost Cyl.

104896.1808

Cost. Sphere=

15494.74474

Cost Cone=

23044.88809

Cost Tank Mat.=

143435.8137

Cost. Tank. Nomin.

168056.3956

Cost of Seams

Cost seam=(dollar/m)*(seam length)*(seam cost function)

Seam for unequal t's, compute using averaged t

STRESSES

dollar/m

Cylinder

12

Stress 1,  hoop=pressure*R/t1

324000

Stress  2, long. = pressure*R/(2*t1)

162000

Cost Seam Cyl=

349567.988

Equivalent stress = (str.1^2+str.2^2-str.1*str.2)^1/2

Cost seam Sphere=

70874.873

CON--->

g=

6.014805771

280592.231

Cost seam cone=

143256.17

Sphere

t av cyl&sph

0.1875

Stress1 = stress2 = pressure*R/(2*t)

283500

tav cyl&cone

0.18125

Equivalent stress

283500

Cost seam cyl&sphere=

71159.7199

CON--->

g=

6.0875

Cost seam cyl&cone=

71303.9326

Cone

Total seam cost=

706162.683

Stress 1= pressure*R/t3

302400

Nomin Total Seam cost=

343367.803

Stress2=pressure*R/(2*t3)*(l/h)

159378.794

Equivalent stress =

262013.764

CON--->

g=

5.550344107

VIBRATION Constraint on 1st bend. mod.

Total Costs

VibFactor=v1*(R^3*t1/(Tank wght*(L+R+h)^3)^(1/2)

Tank cost TOTAL= total seam+total mat.

v1=

10000

VibFact=

0.7183573

849598.4971

<---OBJ

NominVibFac=

1.33627161

Nominal Tnk Cost

AllowedMinimum. of VibFac=

0.8

511424.1984

CON--->

g=

0.102053381

<--1-VibFac/MinVibFac

Aerodynamic drag penalty on payload for tank cross-section, for wetted surface

and for the cone bluntness

Tank Cross-section A = pi*R^2

A

2061203.76

Delta Payload = -p1*(A-Ao)/Ao*(cone drag)-p2*(WetSurf-NominalWetSurf)/NominalWetSurf

Nominal Payload

p1

Ao

p2

cone drag

30000

-38630.5288

<---OBJ

14300

636174

5000

0.281319081

Cone Drg Nomin

1.018336291

Return on investment data

Fixed cost to launch 1kg to orbit , dollars, k=

20000

Cost to launch Nom. payload=k*Nom. pld

600000000

Fixed cost other than tank=Cost to launch Nom.pld-Cost of Nom. tank=

599488575.8

599488575.8

True launch cost=(Fixed cost other than tank)+(cost of tank)

600338174.3

Tank design influences profit in two ways: delta of payload & delta of cost/launch.

Payload actually launched=Nominal payload-(Tank weight-Nominal Tank weight)+Delta due to drag

-2147.199859

<---OBJ

Charge per kg launched=(fixed cost to launch)*(1+profit)

profit

21000

0.05

Customer pays= (payload actually launched)*(Charge to launch 1 kg, dollar)

-45091197.03

ROI per launch=(Customer pays-True launch cost/(true launch cost)

-1.075109661

<---OBJ

Auxilliary Functions - No need to change these

Cost function for material, WEIGHT COST FUNCTION

a

b

c

x offset

1.15

-0.33

0.165

0.1

f=

a+

b*x+

c*x^2

x=

0

0.9

1.9

f result

1.15

0.98665

1.11865

actual x

0.1

1

2

use this function as follows:

Cost = Volume*(cost per unit volume)*(a+bx+cx^2)

substitute in the above x=(t-x offset)

Cost function for seam, SEAM COST FUNCTION

a

b

c

x offset

1.2

-0.42

0.25

0.1

f=

a+

b*x+

c*x^2

x=

0

0.9

1.9

f result

1.2

1.0245

1.3045

actual x

0.1

1

2

use this function as follows:

Cost = length*(cost per unit length )*(a+bx+cx^2)

substitute in the above x=(t averaged-x offset)

Cone drag function d=b+a*exp(1-c*x)

a

b

c

x

1.4

0.25

1.6

4

0.256323213

substitute x= 1+(h/l-(h/l)o)/(h/l)o