Problem *3.99
[Difficulty: 4]
Given:
Spherical balloon filled with helium lifted a payload of mass M=230 kg.
At altitude, helium and air were in thermal equilibrium. Balloon diameter is
120 m and specific gravity of the skin material is 1.28.
Find:
The altitude to which the balloon rose.
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)
t
z
D
M
Assumptions:
(1) Static, incompressible fluid
(2) Static equilibrium at 49 km altitude
(3) Ideal gas behavior
Taking a free body diagram of the balloon and payload:
Σ
F
z
F
buoy
M
He
g
⋅
−
M
s
g
⋅
−
Mg
⋅
−
=
0
=
Substituting for the buoyant force and knowing that mass is density times volume:
ρ
air
g
⋅
V
b
⋅
ρ
He
g
⋅
V
b
⋅
−
ρ
s
g
⋅
V
s
⋅
−
⋅
−
0
=
ρ
air
V
b
⋅
ρ
He
V
b
⋅
−
ρ
s
V
s
⋅
−
M
−
0
=
The volume of the balloon:
V
b
π
6
D
3
⋅
=
The volume of the skin:
V
s
π
D
2
⋅
t
⋅
=
Substituting these into the force equation:
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 Fall '07
 Lear
 Fluid Mechanics

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