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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

*∑ F _{z} = = F_{B} - F_{g} - F_{D}*

where *F _{B}* is the buoyant force,

*F*the gravity force (weight of body), and

_{g}*F*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

_{D}Eq. 2

*F _{D} = F_{B} - F_{g} = γ_{b} V - γ_{0} V = ( γ_{f} - γ_{0} ) V*

where *γ _{f}* is the specific weight of the fluid,

*γ*is the specific weight of the object, and

_{0}*V*is the volume of the object.

Specific weight is

Eq. 3

*γ = p / RT*

where:

*γ =* ft^{3}/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 ft^{3} 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 *

Eq. 5

${F}_{D}=\frac{pV}{T}(\frac{1}{{R}_{a}}-\frac{1}{{R}_{{H}_{2}}})$Eq. 6

${F}_{D}=\frac{144\times 14.696\times 3,700,000}{59+459.7}(\frac{1}{53.34}-\frac{1}{766.8})$Eq. 7

${F}_{D}=263,300\text{}lbf\text{}(1.17\times {10}^{6}N)$Source:

Marks Standard Handbook for Mechanical Engineers

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