The new and large fly the farthest

British Airways and Air France mutually retired the Concorde supersonic jet in 2003. Both companies cited rising maintenance costs as being the reason, which in turn were compounded by falling demand after the Paris crash in 2000 and a general downturn in civil aviation after 9/11. Now, American and French scientists have found that Concorde was in fact an allometric outlier that stood out design-wise at the cost of its feasibility and, presumably, its maintenance. Perhaps it grounded itself.

One thing Adrian Bejan (Duke University), J.D. Charles (Boeing) and Sylvie Lorente (Toulouse University) seem to be in awe of throughout their analysis is that the evolution of commercial airplane allometry seems deterministic (allometry is the study of the relationship between a body’s physical dimensions and its properties and functions). This is awesome because it implies that the laws of physics used to design airplanes are passively guiding the designers toward very specific solutions in spite of creative drift, and that successive models are converging toward a sort of ‘unified model’. This paradigm sounds familiar because it could be said of any engineering design enterprise, but what sets it apart is that the evolution of airplane designs appears to be mimicking the evolution of flying animals despite significant anatomical and physiological differences.

One way to look at their analysis is in terms of the parameters the scientists claim have been guiding airplane design over the years:

1. Wingspan
2. Fuselage length
3. Fuel load
4. Body size

Among them, fuel load and body size are correlated along the lines of Tsiolkovsky’s rocket equation. It says that, for rockets, if two of the following three parameters are set, the third becomes immovably fixed in a proportional way: energy expenditure against gravity, potential energy in the propellant, and the fraction of the rocket’s mass made up by the propellant. According to Bejan et al, there is a corresponding ‘airplane equation’ that shows a similar correlation between engine size, amount of fuel, and mass of the whole vehicle. The NASA explainer finds this association tyrannical because, as Paul Gilster writes,

A … rocket has to carry more and more propellant to carry the propellant it needs to carry more propellant, and so on, up the dizzying sequence of the equation

Next, there is also a correlation between wingspan and fuselage length corresponding to an economy of scale such as what exists in nature. Bejan et al find that despite dissimilarities, airplanes and birds have evolved similar allometric rules on the road to greater efficiency, and that like bigger birds, bigger airplanes are “more efficient vehicles of mass”. Based on how different airplane components have evolved over the years, the scientists were able to distill a scaling relation.

S/L ~ M^1/6 g^1/2 ρ^1/3 σ^1/4(ρ_aV²C_l)^-3/4 2^1/4 C_f^7/6

Be not afraid. S/L is the ratio of the wingspan to the fuselage length. It is most strongly influenced by ρ_a, the density; σ, the allowable stress level in the wing; g, the acceleration due to gravity; and C_f, the fixed skin-friction coefficient. More interestingly, the mass of the entire vehicle has a negligible effect on S/L, which pans out as a fixed S/L value across a range of airplane sizes.

Similarly, the size of a plane’s engine has also increased proportional to a plane’s mass. This would be common sense if not for there being a fixed, empirically determined correlation here as well: M_e = 0.13M^0.83, where M_e and M are the masses of the engine and airplane, respectively, in tons.

In terms of these findings, the Concorde’s revolutionary design appears to have been a blip on the broader stream of traditional yet successful ones. In the words of the authors,

In chasing an “off the charts” speed rating the Concorde deviated from the evolutionary path traced by successful airplanes that preceded it. It was small, had limited passenger capacity, long fuselage, short wingspan, massive engines, and poor fuel economy relative to the airplanes that preceded it.

That the Concorde failed and that the creative drift it embodied couldn’t achieve what the uninspired rules that preceded it did isn’t to relegate the design of commercial airplanes to algorithms. It only stresses that whatever engineers have toyed with, some parameters have remained constant because they’ve had a big influence on performance. In fact, it is essentially creativity that will disrupt Bejan et al‘s meta-analysis by inventing less dense, stronger, smoother materials to build airplanes and their components with. By the analysts’ own admission, this is a materials era.

Bigger airplanes fly farther and are more efficient, and to maximize fuel efficiency, are becoming the vehicles of choice for airborne travel. And that there is a framework of allometric rules to passively maximize their inherent agency is a tribute to design’s unifying potential. In this regard, the similarity to birds persists (see chart below) as if to say there is only a fixed number of ways in which to fly better.

From the paper:

Equally important is the observation that over time the cloud of fliers has been expanding to the right . In the beginning were the insects, later the birds and the insects, and even later the airplanes, the birds, and the insects. The animal mass that sweeps the globe today is a weave of few large and many small. The new are the few and large. The old are the many and small.

References

The evolution of airplanes, J. Appl. Phys. 116, 044901 (2014); DOI: 10.1063/1.4886855