Download 11.53: PROBLEM DEFINITION Situation: A helium

11.53: PROBLEM DEFINITION
Situation: A helium-filled balloon moves through air (see the problem statement for
all the details).
Find: Terminal velocity of the balloon.
Properties: at T = 15◦ C:
ρair ≈ 1.22 kg/m3 ;
ρHe = 0.169 kg/m3
PLAN
Apply equilibrium with the weight, drag force and buoyancy force.
SOLUTION
Velocity from drag force equation
V0 = (2FD /(CD Aρ))1/2
Equilibrium
Fnet = FD − Wballoon − Whelium + Fbuoy = 0
FD = +0.15 − (1/6)πD3 (γ air − γ He )
= +0.15 − (1/6)π × (0.50)3 9.81(ρair − ρHe )
Solve for drag force
FD = +0.15 − (1/6)π(0.50)3 × 9.81(1.22 − 0.169) = −0.52 N
Assume CD ≈ 0.40 Then
V0 = ((2 × 0.52/(0.40 × (π/4) × 0.52 × 1.22))1/2
= 3.29 m/s
Check Re and CD :
Re = V D/ν = 3.29 × 0.5/(1.46 × 10−5 ) = 1.13 × 105
From Fig. 11-8, CD ≈ 0.45 so one additional iteration is necessary.
V0 = 3.11 m/s upward
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