November 28, 2010

Fig. 1: The four forces of flight: lift,
weight, thrust, and drag. For stable level flight, all forces
must balance out. |

There are four forces in play during an object's flight: lift, weight, thrust, and drag. [1] To initiate a climb, the aircraft's upward force due to lift must exceed its downward force due to its weight in order cause upward acceleration. However, during stable level flight, lift and weight will be equal and cancel. Thus the only force the aircraft must overcome to maintain its flight is drag. The aircraft must spend energy in the form of thrust equal to the drag to maintain its trajectory.

There are two primary types of aircraft: aerostatics (lighter-than-air - blimps, zepplins, hot air balloons) and aerodynes (heavier-than-air - airplanes, helicopters, gliders). Aerodynes generate the necessary lift through means of an airfoil. A fluid (in this case air) flowing past an airflow will generate a surface force. [1] The perpendicular component of this force is called lift while the parallel component is drag. As a result, the amount of drag an aerodyne produces is roughly proportional to the amount of lift it must generates which is related to its weight. This relationship is described by the Lift-to-Drag (L/D) ratio, a property of the aircraft design. Typical L/D ratios range from 20:1 for a passenger jet to 65:1 for an advanced glider. [1,2] In contrast, aerostatics generate lift through a buoyancy force because they are lighter than air. Therefore, the amount of drag generated by an aerostatic craft is not proportional to its lift but mainly due to only the body's shape. For the aircraft to maintain its level flight, it must exert a thrust equal to its drag.

The required power to produce this thrust is calculated by

For conventionally powered aircraft, this energy is supplied from its liquid fuel.

Powering an aircraft solely using solar power limits
its maximum thrust. The total amount of power available to an aircraft
is the product of the solar constant (roughly 1.366 kW/m^{2} [3]
) and the incident area of the aircraft. A Boeing 737 has a total
wingspan area of 102m^{2} [4] with a L/D ratio of roughly 20:1
[1] and travels at an average cruise speed of 740 km/hr. [4] The total
solar power available to the aircraft assuming 100% conversion
efficiency is therefore 140 kW. (Current solar cell technologies have a
theoretical upper limit of 33% efficiency although the carnot efficiency
on the conversion of sunlight to electricity is 95%.)

Therefore the maximum amount of drag in level cruise flight the aircraft can overcome is 645 N:

With a L/D ratio of 20:1, the maximum lift the aircraft can produce is 20 x 645 N = 12900 N. This translates into a maximum loaded aircraft mass of 1315 kg:

Considering an average human male weighs approximately 90kg, this is the weight of 15 people, not including any weight of the actual aircraft structure itself. It is clear that large scale air transportation at jet speeds is not possible using solar energy alone in this configuration. Notice that maximum thrust is inversely proportional to the cruise velocity. If the aircraft cruised at even 1/4th the speed (185km/hr), this maximum aircraft mass will only increase by a factor of 4 to 5260 kg or the weight of 60 people (not including the aircraft weight). Maximum thrust is directly proportional to the L/D ratio. Similarly, increasing this to 60:1 from 20:1 will only increase the maximum aircraft mass by a factor of 3. It is clear that large scale air transportation at jet speeds is not possible using solar energy alone in this configuration.

In contrast to heavier-than-air craft, aerostatics produce lift through a buoyant force by being lighter than air. The drag of a blimp is not proportional to the lift it generates and is instead governed by the drag equation [5]:

where the density of air at altitude rho=0.89
kg/m^{3} (at 3k m ASL) [6], u is the velocity of the craft, A is
the drag area and C_{d} is the coefficient of drag as determined
by the craft's exterior design. For the Goodyear "Puritan" blimp, A =
8.8 m^{2}, C_{d} = 0.05, u = 21.5 m/s. [7] Plugging
these values in gives a cruise drag force of 90.5 N:

This is equal to the amount of thrust needed to overcome this drag. The power required for this thrust is again calculated as in the previous example to be 787 W:

This particular airship is 127.5ft long and 36.4ft
wide resulting in a crude rectangular incident area of 431
m^{2}. Multiplying this area by the solar constant results in
589 kW of solar power available to the blimp. If the solar energy
conversion efficiency was even 0.4%, this would be more than enough to
power the blimp in level flight.

Another related discussion is the topic of human powered flight. The resulting energy balance is identical to the previous airplane discussion with the thrust energy input coming from human power instead of the sun. An amateur cyclist can produce about 200 W sustained. [8] Assuming a person's mass to be 90 kg and the aircraft mass to be (unrealistically) negligible, the total amount of lift needed to sustain level flight is 883 N:

With a superb glider-like L/D ratio of 65:1, this translates into 883N / 65 = 13.5N of drag. With a maximum energy input of 200 W, this results in a maximum aircraft speed of 15 m/s or 33.6 mph:

© Michael Liu. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] H. H. Hurt, *Aerodynamics for Naval
Aviators* (Washington, 1965).

[2] M. D. Maughmer, T. S. Swan and S. M. Willits,
"Design and Testing of a Winglet Airfoil for Low-Speed Aircraft,"
J. Aircraft **39**, 4 (2002).

[3] "Solar Constant," Encyclopædia Britannica (Chicago, 2010).

[4] C. Brady, *The Boeing 737 Technical Guide*,
(Frodsham, 2009).

[5] G. Batchelor, *An Introduction to Fluid
Dynamics*, (Cambridge, 2000).

[6] F. E. Fowle, "Smithsonian Physical Tables,"
Smithsonian Miscellaneous Collections **71**, 1 (1921).

[7] F. L. Thompson and H.W. Kirschbaum, "Report 397: The Drag Characteristics of Several Airships Determined by Deceleration Tests," NACA-TR-397, National Advisory Committee for Aeronautics Annual Report 17 (1932), p. 665.

[8] S. Jacobsen and O. Johansen, "An Ergometer
Bicycle Controlled by Heart Rate," Medical and Biological Engineering
and Computing, **12**, 5 (1974).