Problem *3.100
[Difficulty: 3]
Given:
A pressurized balloon is to be designed to lift a payload of mass M to an altitude of 40 km, where p = 3.0 mbar
and T = 25 deg C. The balloon skin has a specific gravity of 1.28 and thickness 0.015 mm. The gage pressure of
the helium is 0.45 mbar. The allowable tensile stress in the balloon is 62 MN/m
2
t
M
D
Find:
(a) The maximum balloon diameter
(b) The maximum payload mass
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
F
buoy
ρ
g
⋅
V
d
⋅
=
(Buoyant force is equal to mass
of displaced fluid)
p
ρ
R
⋅
T
⋅
=
(Ideal gas equation of state)
π
D
2
∆
p
/4
π
Dt
σ
Assumptions:
(1) Static, incompressible fluid
(2) Static equilibrium at 40 km altitude
(3) Ideal gas behavior
The diameter of the balloon is limited by the allowable tensile stress in the skin:
Σ
F
π
4
D
2
⋅
∆
p
⋅
π
D
⋅
t
⋅
σ
⋅
−
=
0
=
Solving this expression for the diameter:
D
max
4t
⋅
σ
⋅
∆
p
=
F
buoyant
M
b
g
Mg
z
D
max
4
0.015
×
10
3
−
×
m
⋅
62
×
10
6
×
N
m
2
⋅
1
0.45 10
3
−
⋅
bar
⋅
×
bar m
2
⋅
10
5
N
⋅
×
=
D
max
82.7m
=
To find the maximum allowable payload we perform a force balance on the system:
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 Fall '07
 Lear
 Fluid Mechanics

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