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### Airship Buoyancy Equations and Calculator

Airship Buoyancy Equations and Calculator

Archimedes’ principle states that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced. If an object immersed in a fluid is heavier than the fluid displaced, it will sink to the bottom, and if lighter, it will rise.

Figure 1 Free body diagram of an object.

From the free-body diagram of Fig. 1, it is seen that for vertical equilibrium,

Eq. 1

∑ Fz = = FB - Fg - FD

where FB is the buoyant force, Fg the gravity force (weight of body), and FD the force required to prevent the body from rising. The buoyant force being the weight of the displaced liquid, the equilibrium equation may be written as

Eq. 2

FD = FB - Fg = γb V - γ0 V = ( γf - γ0 ) V

where γf is the specific weight of the fluid, γ0 is the specific weight of the object, and V is the volume of the object.

Specific weight is

Eq. 3

γ = p / RT

where:

γ = ft3/lbm
p = pressure, pressure
R = Universal gas constant, ft·lbf/(lbm)(°R)
T = absolute temperature, Rankine

Example calculation:

An airship has a volume of 3,700,000 ft3 and is filled with hydrogen. What is its gross lift in air at 59°F (15°C) and 14.696 psia? Noting that γ = p / RT

Eq. 4

${F}_{D}=\left({\gamma }_{f}-{\gamma }_{0}\right)V=\left(\frac{p}{{R}_{a}T}-\frac{p}{{R}_{{H}_{2}}}\right)V$

Eq. 5

${F}_{D}=\frac{pV}{T}\left(\frac{1}{{R}_{a}}-\frac{1}{{R}_{{H}_{2}}}\right)$

Eq. 6

${F}_{D}=\frac{144×14.696×3,700,000}{59+459.7}\left(\frac{1}{53.34}-\frac{1}{766.8}\right)$

Eq. 7

Source:

Marks Standard Handbook for Mechanical Engineers

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