Adrian Bejan is a Professor of Mechanical Engineering and Materials Science at Duke University and as an offshoot from his thermodynamics research he has pondered the question why evolution exists in natural i.e. biological and geophysical, and man-made i.e. technological realms. To account for the progress of design in evolution Prof. Bejan conceived the constructal law, which states that

For a finite-size flow system to persist in time (to live), its configuration must evolve (freely) in such a way that it provides easier access to the currents that flow through it.

In essence a new technology, design or species emerges so that it offers greater access to the resources that flow i.e. greater access to space and time. The unifying driver behind the law is that all systems that output useful work have a tendency to produce and use power in the most efficient manner.

The Lena Delta. Photo Credit Wikipedia [1]

Given Prof. Bejan’s specialty in thermodynamics it is no surprise that the law uses the analogy of a flow system to describe the evolution of design. In nature the branches of rivers carry water, nutrients and sediments to the sea, and if given enough freedom, over time evolve into a river delta that provides a source of life for an entire area. Similarly, our lungs facilitate flow of chemical energy between air and blood and have evolved into a complex multi-branch system that aims to improve the flow of currents within it.

A difficulty in studying natural evolution is that it occurs on a time-scale much greater than our lifetime. However, in a recent study published in the Journal of Applied Physics Prof. Bejan and co-workers show that the shorter technological evolution of airplanes allows us to witness the phenomenon from a bird’s-eye view. Interestingly, as a “flying machine species” the evolution of airplanes follows the same physical principles of evolution that are observed in birds and that can be captured elegantly using the constructal law. For example, the researchers found that

  • Larger airplanes travel faster. In particular the flight velocity of aircraft is proportional to its mass raised to the power 1/6 i.e. V = k M^{1/6}
  • The engine mass is proportional to body mass, much in the same way that muscle mass and body mass are related in animals
  • The range of an aircraft is proportional to its body mass, just as larger rivers and atmospheric currents carry mass further, and bigger animals travel farther and live longer.
  • Wing-span is proportional to fuselage length (body length), and both wing and fuselage profiles fit in slender rectangles of aspect ratio 10:1
  • Fuel load is proportional to body mass and engine mass, and these scale in the same way as food intake and body mass in animals.

This overall trend is depicted nicely in Figure 1 which shows the size of new airplane models against the year they were put into service. It is evident that the biggest planes of one generation are surpassed by even bigger planes in the next. Based on economical arguments it can be assumed that each model introduced was in some way more efficient in terms of passenger capacity, speed, range, i.e. cost-effectiveness than the previous generation of the same size. Thus, in terms of the constructal law the spreading of flow is optimised and this appears to be closely matched with the airplane size and mass. Similarly, Figure 2 shows that both birds and aircraft evolve in the same way in that the bigger fly faster. Thus, the evolution of natural and technological designs seems to have converged on the same scaling rules. This convergent design is also evident in the number of new designs that appear over time. At the start of flight the skies were dominated by swarms of insects of very different design. These were followed by a smaller number of more specialised bird species and today by even fewer “aircraft species”. Combining these two ideas of size and number, it seems that the new are few and large, whereas the old are many and small.

Figure 1. Evolution of airplane mass versus time

Figure 1. Evolution of airplane mass versus time [2]

Figure 2. Evolution of animal flight speed versus body mass

Figure 2. Evolution of animal flight speed versus body mass [2]

The key question is why engines, fuel consumption or wing sizes should have a characteristic size?

Any vehicle that moves and consumes fuel to propel it depends on the function of specific organs, say the engines or fuel ducts, that interact with the the fuel. Because there is a finite size constraint for all these organs the vehicle performance is naturally constrained in two ways:

  1. Resistance i.e. friction and increasing entropy within the organs. This penalty reduces for larger organs as larger diameter fuel ducts have less flow resistance and larger engines encounter less local flow problems. Thus, larger is generally better
  2. On the flip side the larger the organ the more fuel is required to move the whole vehicle. But the more fuel is added the more the overall mass is increased and the more fuel you need, and so on. This suggests that smaller is better.

From this simple conflict we can see that a size compromise needs to be reached, not too small and not too large, but just right for the particular vehicle. In essence what this boils down to is that larger organs are required on proportionally large vehicles and small organs on small vehicles. Thus, as more and more people intend to travel and move mass across the planet the old design based on small organs becomes imperfect and a more efficient, larger design for the new circumstances is required.

Overall, the researchers conclude that the physical principles of evolution define the viable shape of an aircraft. Thus, the fuselage and the wing must be slender, the fuselage cross-section needs to be round and the wing span must be proportional to the fuselage length. A famous outlier that broke with these evolutionary trends of previous successful airplanes was the Concorde with its long fuselage, massive engines and short wingspan. Rather than attempting to achieve an overall superior solution the designers attempted to maximise speed, and thereby compromised passenger capacity and fuel efficiency.  Only 20 units were ever produced and due to lack of demand and safety concerns the Concorde was retired in 2003. Current aircraft evolution manifested in the Boeing 787 Dreamliner, 777X and Airbus A350 XWB are rather based on combining successful architectures of the past and with new concepts, that allow the overall design to remain within the optimal evolutionary constraints. Thus, it is no surprise that in an attempt to make aircraft larger and at the same time more efficient, the current shift from metal to carbon fibre construction is what is needed to elevate designs to a higher level.

 

References

[1] http://en.wikipedia.org/wiki/Lena_Delta_Wildlife_Reserve#mediaviewer/File:Lena_River_Delta_-_Landsat_2000.jpg

[2] Bejan, A., Charles, J.D., Lorente, S. The evolution of airplanes. Journal of Applied Physics, 116. 2014.

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