NASA/TM-2010-216681

Approximate Pressure Distribution in an 

Accelerating Launch-Vehicle Fuel Tank

Michael P. Nemeth

Langley Research Center, Hampton, Virginia

March 2010
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NASA/TM-2010-216681

Approximate Pressure Distribution in an 

Accelerating Launch-Vehicle Fuel Tank

Michael P. Nemeth

Langley Research Center, Hampton, Virginia

National Aeronautics and 

Space Administration

Langley research Center

Hampton, Virginia 23681-2199

March 2010
Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
1

Summary

A detailed derivation of the equations governing the pressure in a generic liquid-fuel launch 

vehicle tank subjected to uniformly accelerated motion is presented. The equations obtained are 

then for the Space Shuttle Superlightweight Liquid-Oxygen Tank at approximately 70 seconds 

into flight. This generic derivation is applicable to any fuel tank in the form of a surface of 

revolution and should be useful in the design of future launch vehicles.

Introduction

An important consideration in the design of launch vehicles is the pressure exerted on the 

wall of a fuel tank during flight. For example, studies of the Space Shuttle Superlightweight 

Liquid-Oxygen (LO2) Tank shown in figure 1 have been presented that include the effects of the 

LO2 pressure on its nonlinear and buckling behavior under flight loads.1, 2 In these very complex 

studies, the LO2 pressure acting on the tank wall was approximated by using several simplifying 

presumptions that are based on the specific characteristics of the vehicle dynamics, at specific 

times in the flight trajectory. For example, stratification of the cryogenic fluid was neglected and 

the fluid was presumed to be homogeneous and, as a result, to have a uniform density. 

Additionally, fluid sloshing effects were neglected. The fluid was taken to be incompressible and 

inviscid, and flow effects were neglected. Moreover, every particle of fluid in the tank was 

presumed to be subjected to uniformly accelerated motion. 

The objective of this memorandum is to present a detailed derivation of the fomulas used in 

references 2 and 3 to simulate the the LO2 pressure field in the Space Shuttle Superlightweight 

Liquid-Oxygen Tank. Toward this objective, a generic presentation of the fundamental equations 

is given first, followed by an example problem that corresponds to approximately 70 seconds into 

flight, given in reference 1. This generic derivation is applicable to any fuel tank in the form of a 

surface of revolution and should be useful in the design of future launch vehicles.

Derivation

In the analysis presented subsequently, the fluid is presumed to be incompressible and 

inviscid, and to have a uniform density. The tank and the fluid are presumed to be moving together 

as a uniformly accelerated rigid body without sloshing, and the tank has an internal ullage 

pressure pu

. For these presumptions, the pressure field, p(x, y, z), is given by (see reference 3) 

 (1)

In this equation, ρ is the density of the fluid, is the acceleration vector, is the body force 

associated with gravity, and is the gradient operator for the (x, y, z) coordinate system fixed to 

the generic tank shown in figure 2. Associated with the (x, y, z) coordinates are the corresponding 

− ∇p + ρg = ρa

a ρg

∇
2

unit vectors . It is convenient to express equation (1) as

 (2a)

where

 (2b)

is the net acceleration vector. 

To obtain the desired expression for the pressure field acting on the tank wall, it is convenient 

to first determine the magnitude and direction of net lateral acceleration, denoted by η, as shown 

in figure 3. The magnitude of the net lateral acceleration is given by

 (3)

and the corresponding direction is given by the unit vector

 (4a)

where

 (4b)

and

 (4c)

Next, the coordinate system (x, η, ζ) is introduced, as shown in figure 3. The (0, η, ζ) coordinates 

shown in this figure span the same plane as the (0, y, z) coordinates. With the (x, η, ζ) coordinates, 

equation (1) is expressed in a simpler component form as

 (5a)

 (5b)

 (5c)

From equation (5c), it follows that the functional dependence of the pressure field is given by

i, j, k

− ∇p = ρα

α = a − g

αη = αy

2

+ αz

2

η = cosψ j + sinψ k

cosψ = αy

αη

sinψ = αz

αη

− ∂p

∂x = ραx

− ∂p

∂η = ραη

− ∂p

∂ζ = 0
3

 (6)

On a surface of constant pressure, the pressure differential dp must obey

 (7)

Using equation (5c) with equation (7) yields

 (8)

Substituting equations (5a) and (5b) into equation (8) yields

 (9)

which is easily integrated to yield the equation of plane

 (10)

where C is a constant. It is useful to point out that the net acceleration vector in the x-η plane is 

given by

 (11)

and its slope in this plane is αx

/αη, which is the negative reciprocal of the slope of the plane of 

constant pressure given by equation (10). Thus, the planes of constant pressure are perpendicular 

to the net acceleration vector in the x-η plane. The angle of inclination, β, of the lines of constant 

pressure is given by 

 (12)

as seen in figure 4. By using equations (3) and (9), equation (12) becomes

 (13)

p = p x, η

dp = ∂p

∂x

dx +

∂p

∂η

dη +

∂p

∂ζ

dζ = 0

dx

dη = −

∂p

∂η

∂p

∂x

dx

dη = − αη

αx

x +

αη

αx

η = C

α = αxi + αηη

tanβ = − dx

dη

tanβ = αη

αx

= αy

2

+ αz

2

αx
4

In addition, figure 4 indicates that

 (14a)

and

 (14b)

To obtain an expression for the pressure gradient that can be easily integrated, it is convenient 

to let denote the coordinates of points in the plane perpendicular to the planes of constant 

pressure, as shown in the figure 4. For this coordinate system, equation (2a) is expressed simply as

 (15)

Integrating this equation gives

 (16)

where is a constant. At the top of the fluid surface, , where

 (17)

where equation (14a) has been used. At , the pressure field is equal to the ullage pressure 

pu, as depicted in figure 4. Therefore, equations (16) and (17) yield

 (18a)

for

 (18b)

The coordinate is expressed in terms of the (x, η, ζ) coordinates by

 (19)

Substituting equation (19) into equations (18) gives

cosβ = αx

α

sinβ = αη

α

x, η

− dp

dx = ρ α

p = − ρ α x + C

C x = xf

xf = xf cosβ = xf

αx

α

x = xf

p = pu + ρ α xf

αx

α − x

x ≤ xf = xf

αx

α

x

x = cosβ x + sinβ η = αx

α

x +

αη

α

η
5

 (20a)

for

 (20b)

Next, the η coordinate is expressed as (see figure 3)

 (21)

where equations (4) have been used. Substituting equation (21) into equations (20) gives

 (22a)

for

 (22b)

For a tank in the form of a surface of revolution, the y and z coordinates of a point (x, y, z)

on the surface are expressed as

 (23a)

 (23b)

where is the horizontal radius of the tank wall and θ is the polar angle, as shown in figure 

5. By using equations (23) with equations (22), the pressure field acting on the tank wall is given 

by

 for (24a)

and

 for (24b)

where 

 (24c)

p = pu + ραx xf − x − αη

αx

η

x +

αη

αx

η ≤ xf

η = cosψ y + sinψ z = αy

αη

y +

αz

αη

z

p = pu + ραx xf − x − αy

αx

y − αz

αx

z

x +

αy

αx

y +

αz

αx

z ≤ xf

y = r x cosθ

z = r x sinθ

r x

p x, θ = pu x > xs x, θ

p x, θ = pu + ραx xs x, θ − x x ≤ xs x, θ

xs x, θ = xf − r x

αy

αx

cosθ +

αz

αx

sinθ
6

Example Problem

Formulas similar to equations (24) were used in the structural analysis of the Space Shuttle 

Superlightweight External Tank (SLWT). In particular, an early booster ascent loading condition 

was examined in reference 1 that corresponds to approximately 70 seconds into flight and a LO2

tank that is approximately seven-eighths full. At this instant of the trajectory, the acceleration field 

was approximated by a uniform field that is given in terms of the coordinates shown in figure 5 

by αx

 = 2.011g, αy = 0.049g, and αz = 0.440g, where g is the magnitude of uniform gravitational 

acceleration. For this acceleration field, equations (3) and (4) give ψ = 83.65 degrees for the plane 

of the resultant lateral acceleration vector. Likewise, equation (13) gives β = 12.42 degrees for 

the inclination of the fluid surface, in the x-η plane, shown in figure 4. These two angles are 

independent of the shape of the tank.

The SLWT LO2 tank, without its foam insulation, is shown in figure 6 and the corresponding 

geometry and dimensions of a corresponding idealized wall reference surface are shown in figure 

7. The major tank-wall components consists of a forward ogive, an aft ogive, a cylindrical barrel 

section, and an aft dome. The aft dome consists of a truncated elliptical dome section with a 

spherical cap. Each of these tank elements has a circular cross section. 

Determination of the pressure acting on tank wall requires an expression for the horizontal 

radius r(x), appearing in equation (24b), for each tank component. The geometry of the ogive 

section, consisting of the forward and aft ogives, is shown in figure 8. The ogive is formed by 

rotating a segment of an eccentric circle about the x-axis. This tangent line of the segment at 

x = h2 is parallel to the x-axis. Inspection of this figure reveals

 (25a)

and

 (25b)

From equation (25a) it follows that

 (26)

Substituting equation (26) into equation (25b) gives the desired expression

 for (27)

For a given value of r1 shown in figure 7, the height h1 is found from equation (27) as

 (28)

x = h2 + R sinφ

r x = R cosφ − 1 + r 2

cosφ = 1

R R2

− x − h2

2

r x = R2− x − h2

2

+ r 2 − R h2 ≤ x ≤ h1 + h2

h1 = r 2 − r 1 2R − r 2 + r 1
7

For the barrel section, inspection of figure 7 gives

 for (29)

The corresponding expression for the truncated elliptical section of the aft dome is obtained 

directly by using the equation for an ellipse in x-r coordinates along with the geometry given in 

figure 7. The result is

 for (30)

Likewise, the expression for the spherical part of the aft dome is obtained by using the equation 

for a circle whose origin is at r = 0 and x = Rs

 - h3 - h4, where Rs is the radius of the spherical 

cap shown in figure 7. From this equation, it follows that

 for (31a)

where

 (31b)

A "to scale" graphical image based on equations (27)-(31) is shown in figure 9. For this image, a 

value of r1 = 0 was used to obtain the sharp point at the nose of the tank. The corresponding value 

of h1 given by equation (28) is 418.5 in.

The approximate pressure distribution on the interior surface of the LO2 tank is given by 

equations (24), where x is the axial coordinate shown in figure 7 and r(x) is given by equations 

(27)-(31). The fill level for this example is given by x = xf

 = 305.9 in., and the density ρ is given 

by ρg = γ, where γ = 0.04123 lb/in3 is the specific weight of the LO2. The ullage pressure for this 

example is pu = 19.6 psi and the corresponding tank dimensions are given in figure 7. Pressure 

calculations were made for these specific values and for a closed, pointed ogive with r1 = 0, and 

are shown in figures 10 and 11. The results in figure 10 show the variation of the pressure acting 

on the shell wall as a function of the polar angle θ. Five curves are shown in the figure that 

correspond to different axial positions along the tank. The straight line shown for x = 500 in. is 

in the unfilled region of the ogive where the pressure is equal to the 19.6 psi ullage pressure, and 

is independent of θ. The curves shown for x = 250, 100, and 0 in. correspond to a location in the 

ogive below the fluid surface, a location within the barrel section, and the location where the 

barrel and aft dome meet, respectively. The sinusoidal shape of these curves corresponds to the 

inclination of the fluid surface caused by the uniformly accelerated motion. The straight line 

shown for x = -114 in. is almost at bottom of the aft dome, where the variations of the pressure 

with θ are negligible.

r x = r 2 0 ≤ x ≤ h2

r x = r 2 1 − x

h3

2

1 − r 3

r 2

2

− h3 ≤ x ≤ 0

r x = Rs

2

− x − Rs + h3 + h4

2

− h3 + h4 ≤ x ≤ − h3

Rs = r 3

2

+ h4

2

2h4
8

The results in figure 11 show the variation of the pressure acting on the shell wall as a 

function of the axial coordinate x. Two curves are shown in this figure that correspond to 

θ = 83.65 and 263.65 degrees. These two tank meridians span the plane of maximum lateral 

acceleration. The differences in the two curves are also associated with the inclination of the fluid 

surface. The two curve intersect at x = -114.9 in., the center of the aft dome, where a maximum 

pressure equal to 54.5 psi occurs.

Concluding Remarks

A detailed derivation of the equations governing the pressure in a generic liquid-fuel launch 

vehicle tank subjected to uniformly accelerated motion has been presented. The equations 

obtained have been applied to detemine the pressure acting on the shell wall of the Space Shuttle 

Superlightweight Liquid-Oxygen Tank at approximately 70 seconds into flight. This generic 

derivation is applicable to any fuel tank in the form of a surface of revolution and should be useful 

in the design of future launch vehicles.

References

1. Young, R. D.; Nemeth, M. P.; Collins, T. J.; and Starnes, J. H., Jr.: Nonlinear Behavior of 

Space Shuttle Superlightweight Tank under Booster Ascent Loads. J. Spacecraft and 

Rockets, vol. 36, 1999, pp. 820-827.

2. Nemeth, M. P.; Young, R. D.; Collins, T. J.; and Starnes, J. H., Jr.: Nonlinear Behavior of 

Space Shuttle Superlightweight Tank under End-of-Flight Loads. J. Spacecraft and 

Rockets, vol. 36, 1999, pp. 828-835.

3. Fox, R. W. and McDonald, A. T.: Introduction to Fluid Mechanics. Third ed., John Wiley 

& Sons, 1985, pp. 74-77.
9

USA

United States

External tank

Solid rocket booster

Orbiter

Aft orbiter

attachment

fitting

Aft SRB

attachment

 point

Liquid-oxygen tank

Intertank

Liquid-hydrogen tank

attachment fitting

Forward orbiter

Figure 1. Space Shuttle external tank components.

Foreward SRB

attachment

 point
10

LO2

x

y

z

Figure 2. Baseline coordinate frame of generic fuel tank.

 Ullage pressure, pu

Liquid fuel

Free surface

of liquid fuel

Generic fuel tank

y

z

η

ζ

ψ

Figure 3. Coordinate frame of generic fuel tank associated with the resultant

αzk

αηη

αy j

 lateral acceleration.

Resultant lateral

acceleration

vector
11

x

η

LO2

α

β

x

η

xf

Figure 4. Coordinate frame of generic fuel tank associated with lines of constant

 pressure in the plane of resultant lateral acceleration.

Line of constant

pressure

 Ullage pressure, pu

xf

LO2

x

y

z

r(x)

xf

x

θ

Figure 5. Cylindrical coordinate frame for a generic fuel tank in the form of a

 surface of revolution.
12

Figure 6. Space Shuttle Superlightweight Liquid-Oxygen Tank.

 Forward ogive

 Cylindrical barrel section

 Aft ogive

 Aft dome
13

x

r

r2 = 165.5 in.

r1 = 25.7 in.

h1 = 389.3 in.

h2 = 103.6 in.

h4 = 11.9 in.

r3 = 70.0 in.

+

R = 612.0 in.

h3 = 103.0 in.

Figure 7. Geometry and dimensions of Space Shuttle Superlightweight Liquid-Oxygen Tank.

+Rs = 211.8 in.

r(x)

x

Truncated

elliptical

dome

Spherical cap

Forward and

Barrel section

aft ogives
14

r

x

φ

R 

R - r2

x

h2

R sinφ

r(x)

r2

r1

h1

Figure 8. Geometry of the ogive. 
15

Figure 9. Graphical output of equations (27)-(31) for the Space Shuttle Superlightweight

Liquid-Oxygen Tank with the dimensions given in figure 7, except r1 = 0. 

x

LO2

r

Angle, θ, degrees

Pressure,

p, psi

x = -114.9 in.

x = 522.1 in.

x = 103.6 in.

x = 305.9 in.

x = 500 in.

x = -114 in.

x = 0 in.

x = 100 in.

x = 250 in.

 Ullage pressure, pu = 19.6 psi

Figure 10. Pressure acting on the shell wall of the Space Shuttle Superlightweight

Liquid-Oxygen Tank as a function of the polar angle θ (see figure 5). 
16

Pressure,

p, psi

Axial coordinate, x, in.

θ = 83.65 degrees

θ = 263.65 degrees

 Ullage pressure, pu = 19.6 psi

Figure 11. Pressure acting on the shell wall of the Space Shuttle Superlightweight

Liquid-Oxygen Tank as a function of the axial coordinate x (see figure 5). 

x

LO2

r

x = -114.9 in.

x = 522.1 in.

x = 103.6 in.

x = 305.9 in.

y

z

θ = 83.65 degrees

θ = 263.65 degrees

Cross-section

x = constant

r

Cross-section

θ = constant
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Approximate Pressure Distribution in an Accelerating Launch-Vehicle Fuel 

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Nemeth, Michael P. 

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NASA Langley Research Center

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14. ABSTRACT

A detailed derivation of the equations governing the pressure in a generic liquid-fuel launch vehicle tank subjected to 

uniformly accelerated motion is presented. The equations obtained are then for the Space Shuttle Superlightweight 

Liquid-Oxygen Tank at approximately 70 seconds into flight. This generic derivation is applicable to any fuel tank in the form 

of a surface of revolution and should be useful in the design of future launch vehicles.

15. SUBJECT TERMS

Shell structures; Fuel tank; Structural design; Launch vehicle; Tank pressure

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