The evolution of helicopters

Earlier work with the constructal law has shown that it is possible to predict and correlate the speed-mass data of all animals (insects, birds, mammals, fish, and crustaceans), including airplanes, athletics, and inanimate flow systems.^1–5 All these designs of movement on the globe evolve. Airplanes do not evolve by themselves—they evolve as a duo, with the humans who design them and use them. Evolving along with the flying animals is the “human and machine species.”

The history of airplanes illustrates in our lifetime the evolutionary design of all fliers, animal, and human made, as they move on earth: farther, faster, more efficiently, and with greater lasting power (sustainability). Recent work has shown that the evolution of airplanes is predictable from the constructal law of design and evolution in nature.^1,6 The main features of aircraft evolutionary design predicted from the constructal law are the speed, engine size, fuel load, range, and aspect ratios (wing span vs fuselage length, wing profile, fuselage profile). The same theory accounts for the alignment of 1950 aircraft data in Gabrielli and von Kármán's chart of specific power vs speed,^7,8 which along with the broader method of evolutionary design continues to be of interest in the aircraft literature.^9–14 The constructal law further predicts the time arrow of the change from propellers to jets, in the same way that for animal design it predicts the change (and the increase in movement complexity) from swimming to running and, finally, flying.

In this new article, we report a new domain where the constructal law manifests itself as the evolution of vehicle technology. We show that the classical alignment of helicopter designs can be anticipated based on the constructal law, and that it can be added to the grand evolutionary design of animal and vehicle movement on the globe.

The current findings can also be applied to foreseeing evolution of the emerging Unmanned Aerial Vehicles (UAVs). Starting from the last decade, the UAVs are gaining rapid popularity, which is attributed to the rapid advance and maturing of information technologies and autonomous capabilities.^15,16 Many military and civil endeavors have served to showcase the potential of UAVs, such as aerial photography and selfie, border surveillance, highway traffic monitoring, wildfire management, agricultural chemical spraying, and other disaster response needs. An UAV, either rotorcraft or fixed-wing vehicle, is operated without pilots and does not carry any passengers. Nevertheless, the navigation is still the controlled body with the power source, which uses the dynamic lift and thrust based on fundamental aerodynamics.¹⁷ 

We start with the dimensions and performance data of helicopter models during their 60-year history (Table I). The data are collected from Ref. 18 and the Type Certificate Data Sheet of FAA and EASA. Figures 1–4 show at first glance that during the evolution of helicopter technology, very sharp correlations have emerged between design features and body size.

Each of Figures 1–4 display the helicopter data of Table I with two symbols. The black circles indicate military helicopters. The empty circles are for the rest of the data compiled in Table I. The purpose of this two-frame display of the body-size effect on evolutionary design is to show that the correlations that emerge are somewhat sharper when the military models are excluded (note the relatively larger R² values). This finding makes sense because the evolution of military models is driven by an objective (mission) that is not exactly the same as the objective of civilian helicopter models.

For conciseness, the analytical formulas that correlate the data (without the military data) are reported directly on each of the graphs of Figs. 1–4. Indicated is also the R² value of each correlation, which shows that the correlation is statistically meaningful. The corresponding correlations obtained by including the military data are indicated in the respective figure captions. In these empirical formulas, the masses (M, M_e, M_f) are expressed in kg, the propeller radius R_p is expressed in m, and the heating value of the fuel (H) is expressed in shp h/lb (or 5.9 × 10⁶J/kg), where shp means shaft horse power. The engine efficiency η is defined in Section III.

Figure 1 shows that the efficiencies (η) of helicopter engines have evolved such that η is proportional to the engine size (M_e) raised to a power that is less than 1. This is in accord with the prediction based on the constructal law,⁴ according to which η should vary as $Meα$⁠, where α < 1.

In Fig. 2, we see the correlation of the engine size (M_e) versus vehicle size. The two frames, together, reveal an approximate proportionality between engine size and body size, and, in addition, a ratio M_e/M that is in the order of 1/10. This finding is the same as in the engine size versus body size scaling exhibited by the evolution of airplanes.⁶ 

Figure 3 shows that the engine size and the fuel load have emerged to be proportional over a size range that spans one full order of magnitude. The engine mass is roughly one third of the fuel load mass over this entire range. This too agrees with the trend uncovered for the evolution of aircraft.⁶ 

Figure 4 reveals the correlation that emerged between the helicopter propeller radius (R_p) and the vehicle size, which is represented by the maximum take-off mass (M). The figure shows that the propeller radius varies monotonically with the vehicle size, where R_p emerged as proportional to M^0.3. Because the vehicle size M is proportional to the vehicle length scale cubed (L³), the proportionality between R_p and M^0.3 means that R_p is essentially proportional to L.

The geometric meaning of the body-size scaling revealed by Fig. 4 is that the propeller radius scales with the length scale of the vehicle, and that all helicopters (large and small, old and new) are geometrically similar. This conclusion is the same as the one reached in the study of the evolution of aircraft, where all aircraft evolve to be geometrically similar, with the wing span almost the same as the fuselage length.⁶ 

The geometric similarity of old and new helicopter models is evident in Fig. 5. Furthermore, the figure shows that during the past five decades the specific fuel consumption (SFC) has decreased to half of its original level. This too is in accord with the evolution of the specific fuel consumption of commercial aircraft (measured in liters of fuel spent for one seat and 100 km flown).⁶ The specific fuel consumption plotted in Fig. 5 is defined in Section III.

As shown in studies of the evolution of commercial aircraft and animals,^6,8,19–21 theory can deepen our understanding of the origin of body-size scaling. We start with the observation that a hovering aircraft such as a helicopter can move in all directions. Chief among these is the vertical direction: the main function of the aircraft is to hover, i.e., to maintain its altitude above ground. Secondary is the sliding movement in the horizontal direction. The simplest model is the one that retains the fewest and most important features of the actual physical system. This is why we begin with the assumption that the hovering body is stationary at its altitude, while consuming fuel to maintain itself in this position for the longest time possible.

Thermodynamics shows that larger flow systems function less irreversibly, because their flows encounter smaller obstacles, such as wider ducts and larger heat transfer areas in heat exchangers. The monotonic effect of size on efficiency was predicted in Ref. 4. The mathematical conclusion is that if the size of the engine is represented by its mass M_e, then the energy conversion efficiency of the engine evolves such that it increases monotonically with size

where c₁ is a constant factor. The α exponent is comparable with 1, and must be less than 1 because the η curve must be concave with respect to M_e: this is because in accord with thermodynamics, the efficiency cannot surpass an ideal level, a ceiling. The more mature the engine technology, the higher the efficiency, the closer to the ideal level, and the smaller the α exponent. The engine efficiency is defined as

where P is the shaft power from engine, t is the time of hovering, H is the heating value of the fuel, and M_f is the mass of consumed fuel. The specific fuel consumption (SFC) is the quantity of fuel consumed in order to produce one unit of power in one unit of time²² 

By comparing Eq. (3) with Eq. (2), we see that

or, $ηH=1/SFC$⁠. By using the helicopter data compiled in Table I, we found the correlation shown in Fig. 1.

The rotor hovering efficiency (η_p) is defined as the ratio of the minimum possible power required to hover (induced power) to the actual power required to hover (shaft power). The total hover power is a value that can be obtained only by measurement. Unfortunately, we did not have access to measurements of performance. In any case, care must be taken when comparing rotors. Only rotors with the same disk loading should be compared. Testing is the only way to figure out the relationship between the η_p and the radius of the propeller. Noteworthy is the study²³ that reported the static testing of micro propellers. A load cell and a torque transducer were used to measure the thrust and torque created by the propeller. The results show that a larger-diameter propeller tends to be more efficient, which is in accord with the body-size effect anticipated with the constructal law.^1,4 At the design loading of the rotor, a value of η_p = 0.55–0.60 is typical. Because of this narrow range, in the following analysis we treat η_p as a constant.

The size of the hovering aircraft is represented by its total mass M, which accounts for everything that hovers, engine (M_e), propeller (M_p), fuel (M_f) and the rest of the body frame (M_b)

Assume that the engine mass M_e varies, while the other masses do not vary. Consequently, the total mass changes with the engine mass. We explore the idea that there is a certain relationship between the engine mass and the total helicopter mass when considering that best performance means maximum hovering time for a given amount of fuel. To start, from Eq. (2) we find that the engine power output is

The engine power is responsible for the force (the thrust, T) that holds the hovering body at constant altitude. The relationship between P and T is

where T and V are the thrust and the induced air velocity, respectively.²² The induced velocity is V = (T/2ρA)^1/2, where ρ is air density, and A is the rotor disk, i.e., the circular area swept by the blades of the rotor. The vertical equilibrium of the hovering body requires

Combining Eqs. (6)–(8), where t is the duration of the hovering flight, we obtain

where $K=0.56 Mfηpg−3/2(2ρ)1/2$ is treated as constant, and M varies linearly with M_e as shown in Eq. (5). The maximum hovering time is obtained by maximizing t with respect to M_e, and the result is

In conclusion, the evolutionary designs should tend toward vehicles with a certain proportion between engine size and total body size. This is in accord with the empirical correlation found in Fig. 2 and is the same as the proportionality between muscle mass and total body mass in animal design^19–21 and the proportionality between engine mass and total mass in airplane design.^1,6

In Fig. 4, we saw that a larger helicopter carries larger blades. A relation between R_p and M is²² 

Under hovering conditions, C_T/σ can be thought of as the lift coefficient per blade. The number of blades is N. Here, it is assumed that the mean angle of attack of the blade is a constant. The thrust coefficient C_T is equal to T/[ρA(R $Ω$⁠)²], where $Ω$ is angular speed of the rotor, σ is the rotor solidity, which is equal to Nc/(πR_p) where c is the chord of the blade. Equation (11) becomes

and, in view of T = Mg, we arrive at the proportionality

This is in agreement with Fig. 4, which shows that if a power function is used for curve-fitting, then R_p emerges as proportional to M^0.3. This means that R_p is roughly proportional to the length scale of the helicopter, which is proportional to M^1/3. This agrees with a correlation of data reported in Ref. 24.

In summary, the evolution of helicopters adds itself to the universal phenomenon of evolution,^1,25 which is exhibited by all flow systems that are free to morph as they flow: animate, inanimate, and engineered. The latter are the technology evolutions responsible for empowered humans—the evolving human and machine species.^1–3 The application of the constructal law to the evolution and spreading of UAVs recommends itself as a subject for future investigation.

The authors thank the Research Grants Council, Hong Kong, for the financial support under Contract No. C5010-14E. Professor Bejan also thanks the U.S. National Science Foundation for supporting his research activity during this period.