Vacuum airship

Francesco Lana de Terzi's flying boat concept c.1670

A vacuum airship, also known as a vacuum balloon, is a hypothetical airship that is evacuated rather than filled with a lighter than air gas such as hydrogen or helium. First proposed by Italian monk Francesco Lana de Terzi in 1670, the vacuum balloon would be the ultimate expression of displacement lift power.

Principle

An airship operates on the principle of buoyancy where air is the fluid in contrast to a ship where water is the fluid. The density of air at standard temperature and pressure is 111KPa and 1 L of displaced air has sufficient buoyant force to lift 1.28 g. Airships use an airbag to displace a large volume of air; the bag is usually filled with a lightweight gas such as helium or hydrogen. The total lift generated by an airship is equal to the weight of the air it displaces, regardless of the materials used in its construction or the gas used to fill the airbag; However for flight it is necessary for the total lift capacity to exceed the ship's weight, which includes the weight of the gas used to fill the airbag.

Using the molar volume, the mass of 1 L of helium (at 1 atmospheres of pressure) is found to be 0.18 g, since every displaced liter provides 1.28 g of lift the effective lift is reduced by 14%.

Vacuum airships would theoretically replace the helium gas with a near-vacuum environment and would theoretically be able to provide the full lift potential of displaced air. The main problem with the concept of vacuum airships however is that with a near-vacuum inside the airbag, the outside pressure would exert enormous forces on the airbag, causing it to collapse if not supported. Though it is possible to reinforce the airbag with an internal structure, it is theorized that any structure strong enough to withstand the forces would invariably weigh the vacuum airship down and exceed the total lift capacity of the airship, preventing flight.

Material constraints

Following the analysis by Akhmeteli and Gavrilin:

The total force on a spherical shell of radius $R$ by an external pressure $P$ is $\pi R^{2}P$. Since the force on each hemisphere has to balance along the equator the compressive stress will be

$\sigma =\pi R^{2}P/2\pi Rh=RP/2h$

where $h$ is the shell thickness.

Neutral buoyancy occurs when the shell has the same mass as the displaced air, which occurs when $h/R=\rho _{a}/(3\rho _{s})$, where $\rho _{a}$ is the air density and $\rho _{s}$ is the shell density, assumed to be homogeneous. Combining with the stress equation gives

$\sigma =(3/2)(\rho _{s}/\rho _{a})P$.

For aluminum and terrestrial conditions Akhmeteli and Gavrilin estimate the stress as $3.2\cdot 10^{8}$ Pa, of the same order of magnitude as the compressive strength of aluminum alloys.

Unfortunately this disregards buckling. Using the formula for the critical buckling pressure of a sphere

$P_{cr}={\frac {2Eh^{2}}{\sqrt {3(1-\mu ^{2})}}}{\frac {1}{R^{2}}}$

where $E$ is the modulus of elasticity and $\mu$ is the Poisson ratio of the shell. Substituting the earlier expression gives a necessary condition for a feasible vacuum balloon shell:

$E/\rho _{s}^{2}={\frac {9P_{cr}{\sqrt {3(1-\mu ^{2})}}}{2\rho _{a}}}$

The requirement is about $4.5\cdot 10^{5}kg^{-1}m^{5}s^{-2}$.

This cannot even be achieved using diamond ($E/\rho _{s}^{2}\approx 1\cdot 10^{5}$). Dropping the assumption that the shell is a homogeneous material may allow lighter and stiffer structures (e.g. a honeycomb structure).

Arthur De Bausset

From 1886 to 1900 Arthur De Bausset attempted in vain to raise funds to construct his "vacuum-tube" airship design, but despite early support in the United States Congress, the general public was skeptical. Illinois historian Howard Scamehorn reported that Octave Chanute and Albert Francis Zahm "publicly denounced and mathematically proved the fallacy of the vacuum principle", however the author does not give his source. De Bausset published a book on his design and offered $150,000 stock in the Transcontinental Aerial Navigation Company of Chicago. His patent application was eventually denied on the basis that it was "wholly theoretical, everything being based upon calculation and nothing upon trial or demonstration."