Why Blimp Technology Doesn’t Scale Up Well
Blimps face a special scaling problem that arises from the way that hoop stress affects the strength of materials required for the envelopes. “Hoop stress” is a force acting on the wall of a vessel containing a fluid. The origin of the term comes from the wooden stave barrels that were held together by metal hoops. As larger barrels and vats were constructed with larger diameters, the strength of the hoops had to be increased in proportion to the diameter.
In a nonrigid airship, or blimp, the gas wants to form a bubble. Consequently, the most hoop stress forms on the sides of the airship. The formula for hoop stress (approximately), in a “thin-walled” vessel is:σ = (piDi – peDe)/2t
Where:
σ = hoop stress
pi = internal pressure of the vessel
pe = external pressure of the vessel
Di = internal diameter of the vessel
De = external diameter of the vessel
t = thickness of the vessel’s wallIn a blimp, as in a car tire, but not in a rigid airship, there must be a significant difference between pi, the internal pressure on the envelope of the airship, and pe, the external pressure on the airship; again, this pressure difference gives the airship (or the tire) its shape and stiffness. As the diameter of the airship gets larger D, the hoop stress rises proportionally, and the envelope of the airship needs to be made thicker/stronger to resist the forces trying to pull it apart.
Expanding the diameter of the blimp is problematic because the extra thickness of the envelope increases its weight and reduces the benefit of the non-rigid structure. Assuming that strength is a function of the wall thickness, the weight of the envelope scales with roughly the cube of the dimension (the area with the square and the thickness with the dimension; the product of these, which is the weight, scales with the cube). Gross lift also scales with the cube of the dimension, and does not particularly outrun weight. Consequently, blimps lack the tendency to become dramatically more efficient as they become larger. In the words of one engineer, the scaling problem of the blimp becomes like “a dog chasing its tail.”
Heavier envelopes are also harder to fabricate (stiffer materials) and more difficult to transport for assembly of the blimp. In order to reduce the strength of the materials required for a larger diameter blimp, catamaran designs have been created that have two or three lobes. These blimps are usually referred to by the builders (Lockheed-Martin and HAV) as “hybrids” because they are designed to use both aerostatic and aerodynamic lift to fly. The smaller diameters of the two lobes reduces the hoop stress and allows these blimps to lift more than an equivalent single cigar-shaped envelope.
Economies of size also impact the competitive distance of operations. Without being able to gain efficiency by scaling up, blimps will have difficulty becoming a profitable mode of transport for intercontinental shipping.
See also Inflation of a Spherical Rubber Balloon (equations and curves) and